LAZAR DRAGON
MATHEMATICAL
METHODS
IN
AERODYNAMICS
ll
KLUWER ACADEMIC
PUBLISHERS
EDITURA ACADEMIEI
ROMANE
MATHEMATICAL METHODS IN AERODYNAMICS
Mathematical Methods in
Aerodynamics
by
LAZAR DRAGO$,
Roqsonime Academy.
H charm Romania
L7
KLUWER ACADEMIC PUBLISHERS
EDITURA ACADEMIEI ROMANE
DORDRECHT/BOSTON FLONDON
BUCURE$11
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ISBN 1-4020-1663-8
ISBN 973-27-0986-3
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Printed in Romania
Table of Contents
xiii
Preface
I The Equations of Ideal Fluids
..
. ......................
.... ................. .
The Equations of Motion
1.1.1 Elements of Kinematics
1.1.2 The Equations of Motion ...
1.2 The
1.2.1 Helmholtz's equation. Bernoulli's integral
1.1
.
.
... .... .........
Potential Flow .............................
1.2.2
1.2.3
The Linear Theory .........................
I
1
3
5
5
6
8
... .......... .. .....
The Shock Waves Theory ............ .............
1.3.1
The Jump Equations ........................
11
11
ilugoniot's Equation
13
15
16
17
19
1.2.4
1.3
............
The Equation of the Potential ...................
1
1.3.2
1.3.3
1.3.4
1.3.5
1.3.6
The Acceleration Potential
. ..
. . .
. .
. ..
.
. . . . . .
.
. .
.
.
.
The Solution of the .lump Equations ..............
.
Prandtl's Formula ....................... ..
The Shock Polar ..........................
The Compression Shock past a Concave Bend
..........
9
2 The Equations of Linear Aerodynamics and its Fundamental Solutions
21
2.1
2.2
21
21
2.1.2
2.1.3
2.1.4
2.1.5
2.1.6
22
24
26
28
29
30
The Steady Solutions ....................... 30
.... ...
.................
34
............ ...
The Fundamental Solutions for the Fluid at Rest ........
On the Interpretation of the Fundamental Solution .......
The Fundamental Solutions of the Steady System ............
2.2.2
2.2.3
2.2.4
2.2.5
2.2.6
2.2.7
2.4
. .........
The Equation of the Potential ..... ... ...........
.. ............ .
The Linear System ......
The Uniform Motion in the Fluid at Rest ..... ... ... .
The Equations of Motion ........ ...
The Equations of Linear Aerodynamics .............
The Fundamental Solutions of the Equation of the Potential ......
2.2.1
2.3
.................
. .........
The Equations of Linear Aerodynamics
2.1.1 The Fundamental Problem of Aerodynamics
Oscillatory Solutions ...... ...........
Oscillatory Solutions for Al = 1
The Unsteady Solutions . . . . .
The Unsteady Solutions for At = I
.
.
. .
. .
. .
.
32
. .
.
. . .
.
.
41
42
43
2.3.1
The Significance of the Fundamental Solution
2.3.2
2.3.3
2.3.4
2.3.5
2.3.6
The General Form of the Fundeunental Solution .........
2.4.2
2.4.3
.
. .
.
. . .
. .
.
. ...........
...........
44
44
..... ....
45
46
47
48
48
50
The Determination of the Velocity Field .............
51
Other Fortes of the Components V and W
53
The Subsonic Plane Solution .... ....
The Three-Dimensional Subsonic Solution
The TwoDimensional Supersonic Solution
.
.
. . .
. .
. .
.
.
The Three-Dimensional Supersonic Solution ...........
The Fundamental Solutions of the Oscillatory System
2.4.1
36
The Determination of Pressure ... ... ............ 50
..
.
... ....
vi
2.4.4
2.4.5
2.5
The Fundamental Solutions in the Case Af = I .........
Fundamental Solutions of the Unsteady System I .......... ..
55
57
57
58
..
Cauchy's Problem ........ ............ . .. ..
The Perturbation Produced by a Mobile Source ... . .....
Fundamental Solutions of the Unsteady System If .. ... .....
2.5.1
2.5.2
2.3.3
2.3.4
2.6
The Incompressible Fluid ..................... 55
Fundamental Solutions .
Fundamental Matrices .
. .
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61
62
64
64
66
.
2.6.1
2.6.2
The Fundamental Matrices .. ............. .....
The Method of the Minimal Polynomial .... .... ... ..
3 The Infinite Span Airfoil In Subsonic Flow
3.1
3.1.2
3.1.3
3.1.4
3.1.5
3.1.6
3.1.7
3.1.8
3.1.9
3.2
3.3
69
. ...... . .... ... ...
.. ... .... .... ..
A Classical Method ........................
The Fundamental Solutions Method . ....... ... ..
..
The Airfoil in the Unlimited Fluid
3.1.1 The Statement of the Problem
.
.
69
69
70
.
.
The Function f (z). The Complex Velocity in the Fluid
The Calculation of the Aerodynamic Action . . . . . .
.
.
.
.
72
75
76
.
. .
.. ....................
.
77
Examples ........ .
The General Case ......................... 80
Numerical Integrations ... ...... .............. 81
.
The Integration of the Thin Airfoil Equation with the Aid of
Gauss-type Quadrature Formulas . ...
. ....... .....
81
The Airfoil in Ground Effects ....................... 82
3.2.1 The Integral Equation .. ..... ....... .... ..... 82
3.2.2 A Numerical Method .................. . ... .. 85
3.2.3 The Flat Plate ........................... 85
3.2.4 The Symmetric Airfoil .. ..................... 86
The Airfoil in Tunnel Effects .. ... . .... ... . ......... 88
.
. .. .
88
3.3.1 The Integral Equation
. ..
90
3.3.2 The Integration of the Equation (3.3.9)
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91
. .
.. .... ... . ......
The Numerical Integration .............. ......
92
92
95
The Integral Equation ................. ......
97
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.
. . .
.
.
.
. . .
Numerical Results . . . . . . . . . .
3.4 Airfoils Parallel to the Undisturbed Stream
3.4.1 The Integral Equations
. .
3.3.3
.. ..
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. ... .
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.
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.............................. 97
3.4.2
3.5 Grids of Profiles
3.5.1
3.5.2 The Numerical Integration
3.6 Airfoils in Tandem
. . . . .
. .
... .. ..................
100
.
. .... . ................
.
3.6.1
3.6.2
3.6.3
3.6.4
.
.
.
.
.
.
.
.
The Integral Equations
The Determination of the Functions f, and ff
.
.
.
.
.
.
.
.
101
.
101
.. .... .. 103
. ....... ..... 104
.................
.........
The Lift and Moment Coefficients
Numerical Values
.
.
.
105
4 The Application of the Boundary Element Method to the Theory
of the Infinite Span Airfoil In Subsonic Flow
4.1
109
... . .... ... .............. 109
.. ... ... . .......... ....... 109
.. . ............ .. 110
The Equations of Motion
4.1.1 Introduction
4.1.2 The Statement of the Problem
.
4.1.3
.
.
The Fundamental Solutions .. ..
. ............... 112
vii
4.2
Indirect Methods for the t idimited Fluid Case ............. 113
4.2.1
The integral equation for the Distribution of Sources ...... 113
4.2.2
The Integral Equation for the Distribution of
Vortices
4.2.3
4.2.4
4.2.5
4.2.6
4.3
4.4
The Determination of the Unknowns .............. .
The Circular Obstacle
. .
.
.
.
.
. . .
.
. . .
. . .
.
. . .
.
.
.
115
115
117
120
The Elliptical Obstacle ...................... 121
4.3.1
4.3.2
4.3.3
4.3.4
4.3.5
The representation of the solution ............... .
122
. .................... ....
125
4.3.6
4.3.7
4.3.8
Appendix ...... ................... ... .. 129
The Integral Equation ............. ... .. .... 123
.
The Circulation ..
The Discretization of the Equations ............... 126
The Lifting Profile ......................... 126
The Local Pressure Coefficient .................. 128
Numerical Determinations ................. ..
. .................... ..
The Representation of the Solution . ............. .. 131
131
131
.
.
The Airfoil in Ground Effects
4.4.2
4.4.3
4.4.4
1.4.5
The Integral Equation .................. .. .. 134
The Computer Implementation ... .......... ..... 135
.
136
..
The Treatment of the Method . ..
.
.
.
.
. . .
.
.
. .
.
.
.
The Circular Obstacle in a Compressible Fluid ......... 137
.. .... . ... ................ . .. 138
.1.4.6
Appendix
4.5.1
The Representation of the Solution ............. ..
.
The Airfoil in Tunnel Effects .... ................ ... 140
4.5.2
4.5.3
4.5.4
4.5.5
4.6
.. ....................... .... .
The Direct Method for the Unlimited Fluid Case ............ 122
4.4.1
4.5
.
The Boundary Elements Method ................ .
The Integral Equation ................ ....... 144
The Verification of the Method ...... ... ......... 146
Appendix ...... ................... ..... 149
Other Methods. The Intrinsic Integral Equation .......... .
4.6.1
140
.
Green Functions ...... ......... ........... 142
150
.
The Method of Regularization .................. 150
5 The Theory of Finite Span Airfoil in Subsonic Flow. The Lifting
Surface Theory
5.1
The Lifting Surface Equation
5.1.1
5.1.2
5.1.3
5.1.4
155
.
... ... ............. .. 155
.
The Statement of the Problem .................. 155
Bibliographical Comments ..................... 158
The General Solution ................ ... .. .. 159
The Boundary Values of the Pressure ... ........... 161
..
163
.The
U.S Boundary Values of the Component w .
. .
.
.
. ... .. .. .
. . .
.
.
.
x.1.6
The Integral Equation
=..1.7
.5.1.8
The Plane Problem ....... ..... .. ... ... ...
. .
.
.
.
.
.
.
. .
. .
164
Other Forms of the Integral Equation .............. 166
5.1.9 The Aerodynamic Action in the First Approximation
. .
.
.
. .
.
168
169
5.1.10 A More Accurate Calculation ................... 171
5.2
5.1.11 Another Deduction of the Representation of the General Solution 173
Methods for the Numerical Integration of the Lifting Surface Equation 175
5.2.1
5.2.2
The General Theory ........................ 175
Multhopp's Method ............. ........... 178
Viii
5.3
5.4
..
179
180
The Third Method ......................... 181
5.2.3
The Quadratum Formulas Method
5.2.4
5.2.5
The Aerodynamic Action ... .. .
. .
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.............. 184
. ...................... 184
Ground Effects in the Lifting Surface Theory
5.3.1
The General Solution
5.3.2
The Integral Equation
5.3.3
The Two-Dimensional Problem .... .... ....... ... 188
.
.
The Wing of Low Aspect Ratio
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189
. .
193
.
5.4.2
The Integral Equation ......... ........ ... .. 189
The Case h = h(x) .......... . .. ... .... ... 192
5.4.3
The General Case
5.4.1
.
.
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6 The Lifting Line Theory
6.1
186
19T
Prandtl's Theory .............................. 197
6.1.1 The Lifting Line Hypotheses. The Velocity Field
. ...... 197
6.1.2 Prandtl's Equation .
.. ..... .............. 200
6.1.3 The Aerodynamic Action ...... ........ . ...... 202
.
.
.
.
...................... 203
6.1.4 The Elliptical Flat Plate
6.2 The Theory of Integration of Prandtl's Equation. The Reduction to
Fredholm-Type Integral Equations ....... ..... ........ 205
6.2.2
The Equation of Trefftz and Schmidt ............... 205
Existence and Uniqueness Theorems ............... 209
6.2.3
Foundation of Glauert's Method
6.2.4
6.2.5
The Minimal Drag Airfoil
6.2.1
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210
Glauert's Approximation ..................... 212
..... .. .............. 212
6.3 The Symmetrical Wing. Vekuas Equation. A Larger Class of Exact
Solutions
... ....... .... . ... .. . ............. 214
The Integral Equation ....... .. . ... .......... 215
6.3.1
Symmetry Properties ........................ 214
6.3.2
6.3.3
Vekua's Equation
......................... .
217
The Elliptical Wing ........................ 220
6.3.5 The Rectangular Wing .
.
.
221
6.3.6 Extensions ..................
. ...... ... 222
Numerical Methods . ..
.
223
6.4.1 Multhopp's Method ........................ 223
6.4.2 The Quadrature Formulas Method .. ... ..
.. ...... 228
6.3.4
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6.4
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.. ... ................. 231
Various Extensions of the Lifting Line Theory .............. 234
6.5.1 The Equation of Weissinger and Reissner .. .......... 234
6.5.2 Weissinger's Equation. The Rectangular Wing ......... 236
The Lifting Line Theory in Ground Effects .. ...
.. 2.18
.
6.5
6.6
6.4.3
The Collocation Method
6.6.1
The Integral Equation .......... ...... ....... 238
The Elliptical Flat Plate ...................... 240
6.6.2
6.6.3
6.7
Numerical Solutions in the General Case
............. 241
The Curved Lifting Line ................
6.7.1
6.7.2
6.7.3
The Pressure and Velocity Fields
The Integral Equation . . .
.
.... ...... 242
.................
242
. ..
.. . ..
246
The Numerical Method .... . ................. 247
. . . .
. . . .
.
. .
ix
7 The Application of the Boundary Integral Equations Method to the
251
Theory of the Three-Dimensional Airfoil In Subsonic Flow
7.1 The First Indirect Method (Sources Distributions) ........... 251
7.1.1
7.1.2
7.1.3
7.1.4
7.1.5
7.1.6
7.1.7
7.1.8
7.2
The Integral Equation
The Integral Equation
. .................
. ... .
. .... .............. 253
253
.
.
.
.
The Discretixation of the Integral Equation ........... 255
The Singular Integrals
. ... ................. .. 258
The Velocity Field. The Validation of the Method ....... 258
The Incompressible Fluid. An Exact Solution .......... 259
..
. .
. .
. . . . .
263
The Expression of the Potential . .
The Second Indirect Method (Doublet Distributions). The Incompress.
.
.
.
.
ible Fluid .................................. 265
7.2.1
7.2.2
7.2.3
7.2.4
7.3
The General Equations ... . .................. 251
The Integral Equation ..... ... ............. .. 265
The Flow past the Sphere. The Exact Solution ......... 267
The Velocity Field ......................... 268
The Velocity Field on the Body. N. Marcov's Formula ..... 268
The Direct Method. The Incompressible Fluid ............. 271
7.3.1 The Integral Representation Formula ........ .. ..... 271
7.3.2
7.3.3
7.3.4
7.3.5
..... . ................
.. .....................
... 275
The Integral Equation
Kutta's Condition
274
The Lifting Flow .......................... 276
The Discretization of the Integral Equation ........... 279
8 The Supersonic Steady Flow
8.1
8.1.1
8.1.2
8.1.3
8.1.4
8.1.5
8.1.6
8.2
283
The Thin Airfoil of Infinite Span .. ................ ... 283
The Analytical Solution ...................... 283
The Fundamental Solutions Method ....... .... .... 286
The Aerodynamic. Action
287
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The Graphical Method ....................... 289
The Theory of Polygonal Profiles ................. 290
..
294
Validity Conditions .
Ground and Tunnel Effects .. .. .................. .. 295
8.2.1 The General Solution ............... .... .... 295
298
8.2.2 The Aerodynamic Coefficients ................. .
The Three-Dimensional Wing .............. ... ... .. 300
.
.
.
.
.
8.3
.
8.3.1
Subsonic and Supersonic Edges ..... ............ .
8.3.2
8.3.3
8.3.4
8.3.5
8.3.6
8.3.7
8.3.8
8.3.9
The Representation of the General Solution ........... 302
The Influence Zones. The Domain Di ....... .. ..... 304
The Boundary Values of the Pressure ........... . .. 305
The First Form of the Integral Equation .
. .
.
. .
.
.
. .
. .
.
300
306
The Equation D in Coordinates on Characteristics .... .. 308
The Plane Problem ......................... 310
The Equation of Heaslet and Lomax (the 11L Equation)
.
.
.
311
The Deduction of HL Equation from D Equation ...... 313
8.3.10 The Equation of Homentcovschi (II Equation) . .... .... 318
8.4
... .... .... 320
Abel's Equation .......................... 320
The Theory of Integration of the H Equation ..
8.4.1
8.4.2
.
The Solution of the H Equation in the Domain of Influence
of the Supersonic Trailing Edge ...... ........ ... .
321
X
8.4.3
The Solution in the Domains of Influence of the Subsonic Lead-
ing Edge .......................... .. ... 323
The Wing with Dependent Subsonic Leading Edges
and Independent Subsonic Trailing Edges . . .
. . . . . .
324
8.4.5 The Wing with Dependent Subsonic Trailing Edges . .. ... 326
8.4.6 The Solution in the Zone of Influence of the Subsonic Edges
under the Hypothesis that the Subsonic Leading Edges are letdependent . . . . . . . . . . . . . . . . . . . . . . . . . . 327
8.4.7 The Wing with Dependent Subsonic Trailing Edges . . . . . . 337
8.4.4
.
.
8.5
.
.
.
.
.
The Theory of Conical Nloticaas ...... ...... ..........
Introduction ............... .. ... .. ... .
8.5.1
339
8.5.2
8.5.3
340
.
8.5.4
8.6
.
The Wing with Supersonic Leading Edges ..... .......
The Wing With a Supersonic Leading FAlge and with Another
Subsonic Leading or Trailing Edge .. .... . ...... ...
The Wing with Subsonic Leading Edges .... .........
Flat Wings ... .
... .. .. .... .... ... ...... .. ...
The Trapezoidal Wing with Subsonic Lateral Edges ...... 352
The Trapezoidal Wing with Lateral Supersonic Edges ..... 355
The Triangular Wing. The Calculation of the Aerodynamic
Action . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
.
.
.
.
The Equations of the Transonic Flow .......... .... ... ..
359
:3.59
.. . .... .. .. 359
9.1.1
The Presence of the Transonic Flow ....
9.1.2
The Equation of the Potential .. ... .... .... . ..... 361
The System of 'transonic Flow . ... . ....... . ... .. 364
The Shock Equations
.
.
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..
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.
.. ......... ... .. 368
The Plane Flow ..... ............... . .......... 369
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9.2.2
The Fundamental Solution
The General Solution
. .
9.2.3
9.2.4
9.2.5
The Lift Coefficient
The Symmetric Wing
The Solution in Real .
9.2.6
The Symmetric Wing
9.:3.1
The Fundamental Solution .......... . .. .... ... .M1
The Study of the Singular Integrals ........ ....... 386
9.2.1
.
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.
.
. ..
371
376
376
377
. .. ..
......... ......... .. .. 380
The Thre"Dimensional Flow ... ..... . ......... ..... 383
9.3.2
9.3.3
9.3.4
9.4
347
8.6.3
8.6.4
9.1.3
9.1.4
9.3
:343
The Angular Wing with Supersonic Leading Edges .... ... 347
9 The Steady Transonic Flow
9.2
342
8.6.1
8.6.2
.
9.1
:339
.
.
.
... ... ... . ....... ..... 387
.......... :389
The Lifting Line Theory .. ... ........ .. .......... 392
The General Solution
.
Flows with Shock Waves .. ...... .. ..
.
9.4.1
9.4.2
The Velocity Field ......................... :392
The Integral Equations ......... .
.. .. 394
. .
.
.
. .
.
.
10 The Unsteady Flow
397
... ... . .... . .. 397
10.1.1 The Statement of the Problem ..... . .. .... .... .. 397
10.1.2 The Fundamental Solution ..... . ............ .. 397
10.1 The Oscillatory Profile in a Subsonic Stream
10.1.3 The Integral Equation
.
.
.
.
.
.
. .
.
.
. .
. .
. .
.
.
.
.
.
.
.
.ie.19
10.1.4 Considerations on the Kernel ..... ....... .. ..... 402
xi
...... 404
10.2 The Oscillatory Surface in a Subsonic Strewn .....
10.2.1 The General Solution ............ ......... . 404
10.2.2 The Integral Equation ....................... 405
.
10.2.3 Other Expressions of the Kernel Function ......... .
.
10.2.4 The Structure of the Kernel .
.
10.2.5 The Sonic Flow ........
. .
. .
.
. . .
.
.
409
412
. .................. 413
.
.
.
.
.
.
.
.
.
10.2.6 The Plane Flow ........ . .................. 414
10.3 The Theory of the Oscillatory Profile in a Supersonic Stream ..... 415
10.3.1 The General Solution
10.3.2
.
. .
.
.
.
.
. ..... .
.
.
.
. . .
.
. .
The Integral Equation and its Solution ....... .......
10.3,3 Formulas for the Lift and Moment Coefficients
10.3.4 The Flat Plate .............. .
. . . .
. ..
. ..........
.
.
415
418
421
423
424
.. ..........
10.4 The Theory of the Oscillatory Wing in a Supersonic Stream ...... 426
.
10.3.5 The Oscillatory Profile in the Sonic Flow
10.4.1 The General Solution ............ .... ... . ... 426
10.4.2 The Boundary Values of the Pressure .............. 428
10.4.3 The Boundary Values of the Velocity, The Integral Equation
.
430
10.4.4 Other Expressions of the Kernel ................. 433
. ....... ............ . ....... 435
10.4.5 A New Form
10.4.6 The Plane Problem . . . . . .
10.5 The Oscillatory Profile in a Sonic Stream
.. .. .... .
.
.
. .
.
.
. .
436
.. .......... .... 438
.
.
10.5.1 The General Solution. The Integral Equation .. ........ 438
10.5.2 Some Formulas for the Lift and Moment Coefficients
.
.
.
.
. .
441
10.6 The Three-Dimensional Sonic Flow .................... 442
..
..
.. 442
.. .................. 443
10.6.3 The Plane Problem ......................... 446
10.6.4 Other Forms of the Kernel .
.. . ... .
447
10.6.1 The General Solution
. .
.
. .
.
.
.
. .
.
.
.
.
.
11 The Theory of Slender Bodies
11.1 The Linear Equations and Their Fundamental Solutions ..... ..
.
.
.
.
.
.
.
.
.
.
10.6.2 The Integral Equation ...
.
. . .
. .
11.1.1 The Boundary Condition. The Linear Equations ...
11.1.2 Fundamental Solutions .
.
449
449
. .... 449
... .... ........ . .... .
11.2 The Slender Body in a Subsonic Stream ... ............. .
452
454
. ... . ...... ........ 454
11.2.1 The Solution of the Problem
11.2.2 The Calculus of Lift and Moment Coefficients .....
..... 456
11.3 The Thin Body in a Supersonic Stream ................. 458
11.3.1 The General Solution ..... .................. 458
11.3.2 The Pressure on the Body. The Lift and Moment
. ............... 461
. ... .. ........ . ........... .. 463
Coefficients .............
11.3.3 The wing at zero angle of attack .... ............. 463
11.3.4 Applications
A Fourier Transform and Notions of the Theory of Distributions
465
.. ............... .. 465
A,1 The Fourier Transform of Functions
A.2 The Spaces V and S .. ................... .... .. 466
A.3 Distributions
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
467
A.4 The Convolution. Fundamental Solutions ................ 470
A.5 The Fourier Transform of the Functions from S ............ 472
A.6 The Fourier Transform of the Temperate Distributions ......... 473
xii
............ 475
.............. 478
A.7 The Calculus of Some Inverse Fourier Transforms
A.8 The Fourier Transform in Bounded Domains
B Cauchy-type Integrals. Dirichlet's Problem for the Half-Plane. The
Calculus of Some Integrals
481
Cauchy-type Integrals ........................... 481
.................. 482
B.3 Plemelj's Formulas ............................. 483
B.l
B.2 The Principal Value in Cauchy's Sense
................
483
........ 485
B.4 The Dirichlet's Problem for the Half-Plane
B.5 The Calculus of Certain Integrals in the Complex Plane
B.6 Glauert's Integral. Its Generalization and Some
Applications ................................. 489
B.7 Other Integrals
............................... 491
C Singular Integral Equations
C. i
493
The Thin Profile Equation ......................... 493
C.2 The Generalized Equation of Thin Profiles
................ 496
C.3 The Third Equation ............................ 498
C.4 The Forth Equation ............................ 502
C.5 The Fifth Equation ............................. 504
D The Finite Part
509
D.1 Introductory Notions ............................ 509
.............................
509
................ 510
D.2 The First Integral
D.3 Integrals with Singularities in an interval
D.4 Hadamard-Type Integrals ......................... 513
D.5 Generalization ............................... 515
E Singular Multiple Integrals
517
F Gauss-Type Quadrature Formulas
521
.............................
521
F.2 Formulas of Interest in Aerodynamics .................. 525
F.3 The Modified Monegato's Formula .................... 529
FA A Useful Formula .............................. 530
F.1 General Theorems
Bibliography
533
Index
571
Preface
The researchers in Aerodynamics know that there is not a unitary method
of investigation in this field. The first mathematical model of the airplane wing, the model meaning the integral equation governing the phenomenon, was proposed by L. Prandtl in 1918. The integral equation
deduced by Prandtl, on the basis of some assumptions which will be
specified in the sequel, furnishes the circulation C(y) (see Chapter 6).
Using the circulation, one calculates the lift and moment coefficients,
which are very important in Aerodynamics. The first hypothesis made
by Prandtl consists in replacing the wing by a distribution of vortices
on the plan-form D of the wing (i.e. the projection of the wing on the
plane determined by the direction of the uniform stream at infinity and
the direction of the span of the wing). Since such a distribution leads
to a potential flow in the exterior of D and the experiences show that
downstream the flow has not this character, Prandtl introduces as a supplementary hypothesis another vortices distribution on the trace of the
domain D in the uniform stream. The first kind of vortices are called
tied vortices and the second kind of vortices are called free vortices. On
the basis of this model one developed later the main theories of Aerodynamics namely the lifting surface theory (after 1936, more precisely
in 1950, when Multhopp gave the equation of this theory), the lifting
surface theory for the supersonic flow (after 1946) and the lifting theory
for oscillatory wings and surfaces for the subsonic, sonic and supersonic
flow (after 1950). In the framework of the last theory the wing is replaced by doublets distributions. From a physical point of view, there is
no reason for replacing the wing with vortices or doublets distributions.
It is true that the vortices are detaching from the wing, but these are
effects , not causes of the presence of the wing. The fact that these
replacements lead to correct results shows how subtle was Prandtl's in-
tuition. We specify that the distributions on D and its trace do not
result from the equations of motion (they have been introduced outside
the mathematical model). Taking into account this inconvenient, we
have shown in (5.7) how it can be removed. We have to consider that
the wing and the fluid constitute an interacting material system. If we
want to study the fluid flow, then according to Cauchy's stress principle
xiv
(the principle of the internal forces; see for example [1.11), p.35), we
have to assume that there exists it forces distribution on the boundary,
which has against the fluid the same action like the wing itself. We shall
replace therefore the wing with it forces distribution instead of a vortices, sources or doublets distribution and we shall find out the density
of this distribution such that it should have the same action against the
fluid like the wing itself. We shall proceed by imposing to the fluid flow
determined by the forces distribution to satisfy the slipping condition
on the wing, condition which is also satisfied by the flow determined by
the wing. In this way it follows an integral equation for determining
the forces density. This equation constitutes the mathematical model
for the wing we have in view. This method is an unitary one and it
is based only on the classical principles of mechanics (in fact. Cauchy's
stress principle). It may be applied to all configurations: see [5.7) for
the wing in a subsonic stream, [8.4] for the wing in a supersonic stream,
[10.15], [10.16], [10.17) for the oscillatory wings in subsonic, sonic or
supersonic stream etc. All these results are given in this book (see chapters 5, 8, 10, 11). We called this method (in[5.7)): the fundamental
solutions method. It may be utilized to all cases in which one can calculate the fundamental solutions of the equations of motion. We have to
notice that in the framework of this method, the existence of the vortices
downstream the wing follows from the model (i.e. from the equations
of motion) and it must not be introduced artificially. In the sequel we
shall present some of the models of aerodynamics. For two-dimensional
configurations, in a subsonic stream, the models are one-dimensional
singular integral equations considered in the sense of Cauchy's principal
value. One may integrate analytically only the equation of thin profiles in a free stream. For other geometries one determines numerical
solutions with the aid of Gauss-type quadrature formulas (see Chapter
3). For three-dimensional wings in a subsonic stream, the models are
two-dimensional integral equations with strong singularities, which are
defined in the sense of Finite Part (see Chapter 5). For other geometry (for example the wing in ground effects) the models are generalized
equations. All these models are solved only numerically. For the wing
in it free stream, Multhopp's method is available. In this book we introdace a more general method - the quadrature formulas method. In the
last part of Chapter 5 one presents the theory of low aspect wings which
was extended by the author to the general case of asymmetrical wings.
The lifting line theory may be deduced from the lifting surface theory
with the aid of Prandtl's assumptions (6). This theory is developed
xv
by presenting analytical and numerical methods for solving Prandtl's
equation; one considers also extensions of this theory, all the methods
representing one-dimensional integral-differential equations. The author
shows how these equations may be reduced to integral equations with
strong singularities and for this type of singularities he gives a Gausstype quadrature formula, which allows the equation to be reduced to a
linear algebraic system which is solved numerically. This method, which
is very general. allows to obtain numerical solutions both in the case of
the lifting line (Chapter 6) and the case of the lifting surface (Chapter
5). In the case of supersonic flow, the integral equations are solved analytically. For the three-dimensional wing (the lifting surface) we present
in Chapter 8 a nice solution given by D. Homentcovschi in 18.16]. The
integral equations describing the flow past oscillatory wings and profiles
(chapter 10) have the same nature like the equations utilized in the case
of steady flow but the kernels are more complicated. However for the
sonic and supersonic flows these equations may be solved exactly by
means of the Laplace transform, as it is shown in [10.17]. Chapter 9,
devoted to the transonic motions. begins with a new asymptotic deduction of the equations of motion. The two and three-dimensional integral
equations are obtained following the papers of the author and D. Homentcovschi. The theory of subsonic and supersonic flow past slender
bodies (in Chapter 11) relies also on the fundamental solutions theory.
In Chapter 2 one deduces the equations of the linear aerodynamics, on
the basis of an asymptotic analysis. assuming that the small parameter
depends on the thickness of the profile. In the classical aerodynamics
this deduction is performed under the assumption that the unknowns
and their derivatives have the same order of magnitude, but this fact
cannot he a priori assumed. Then one calculates the fundamental solutions for the equation of the potential (paper [2.11]) and the fundamental
solutions for the systems of equations of aerodynamics : the steady system[2.8], the oscillatory system [10.17], the unsteady system [2.6], (2.7].
On these solutions will rely the theories from the forthcoming chapters.
The models we have already presented are the so called classical or linear
models. They are suitable for the thin wings and thin profiles because
they rely on the following assumptions: 1) one uses a linear boundary condition, 2) the boundary condition is imposed on the support of
the wing (the segment (-1.1] for the profile, the plan-form D for the
three-dimensional wing), :3) the equations of motion are linearized. The
development of the scientific computing allows us to develop more exact methods. Indeed we can give up to the first two assumptions using
xvi
the boundary integral equations method (BIEM). also called the boundary element method (BEN), which was employed for the first time by
Hess and Smith [7.9], [7.10]. The integral equations on the boundary
are obtained imposing the exact boundary condition on the boundary
of the wing. The integral equation is discretize d using, for example, the
collocation method. One obtains an algebraic system which is solved numerically. The linearization of the equations of motion is necessary only
in the case of compressible fluids. The theory that we have developed is
thus valid for every body in an incompressible fluid and for at thin body
in a compressible fluid. Two chapters from this book, Chapter 4 for the
2d airfoil and Chapter 7 for the 3d airfoil are baLsed on our papers (L.
Drago.1 and A. Dinu). The comparison between the known analytical
results and the numerical results shows a very good agreement. In the
Appendices we give some results concerning The Distributions Theory,
The Singular Integral Equations Theory, The Principal value and The
Finite Part, Gauss-type Quadrature Formulas, etc.
In every work one finds, in a certain measure, both the achievements
of the predecessors and of the researchers contemporaneous with the
author. Among the people which have directly collaborated with tae,
I have to mention at first my professors Victor Va lcovici and Caius lacob, who introduced me in the field of aerodynamics. I also mention
my younger colleagues Nicolae Marcov, Liviu Dint, Dorel Homentcovschi, Adrian Carabineanu, Victor Tigoiu, Vladimir Cardoi, Gabriela
Marinoschi, Stelian Ion and Adrian Dinu. They were my students
at the University of Bucharest, but I learned a lot from their papers.
Some of them were my fellow - workers in the aerodynamics research,
many of them stimulated me with their youth and their way of thinking
in our seminars from the Faculty of Mathematics of the University of
Bucharest. I am very grateful to all of them.
My special gratitude goes to Adrian Carabineanu for his work in
performing the English translation of the book, to Adrian Carabineanu
and Stelian Ion for typesetting the monograph in Latex and to Victor
'igoiu for his activity in finalizing the 195D Grant with the World Bank.
I acknowledge that the hook was sponsored by MEC-CNCSIS Contract 49113/2000, Grant 195D with World Bank.
LAZAR DRACOS
Chapter 1
The Equations of Ideal Fluids
1.1
1.1.1
The Equations of Motion
Elements of Kinematics
In this Chapter we present the equations governing the flow of ideal
fluids. On the basis of these equations we shall develop the theory in the
forthcoming chapters of the book. It is well known (see, for exannple,
[1.11J), that the fluid flow is defined by the di eomorphism
x = X(t, X),
(1.1.1)
where X is the vector of position of a particle P in the reference
configuration (for fluids this is the initial configuration), and x is the
vector of position of the same particle at the moment t. For a fixed
X and a variable t , the equation (1.1.1) furnishes the motion law for
the particle having the vector of position X. Hence the velocity and
the acceleration of the particle will be given by the formulas
v (t, X =
It X (t, X ),
a (t, X =
cit.
v (t, X).
(1.1.2)
The fluid is a continuum medium. It means that the support of the
initial configuration is it domain Do. The image of this domain by the
diffeomorphism (1.1.1) will be denoted by D and one demonstrates
[1.11J that it is a domain. The functions X;(t, Xi, X2i X3) appearing in
(1.1.1) belong to the class C2(D0) and the Jacobian is
0(X1, X2, Xa)
V(XI, X2, X3)
# (}
(1.1.3)
The velocity field defined in (1.1.2) may be discontinuous in isolated
points, on abstract. curves or across abstract surfaces. This kind of
surfaces will be named shock waves.
THE EQUATIONS OF IDEAL FLUIDS
2
The fluid flow may be described by functions defined on Do, i.e.
functions having the form 46(t, X) , or by functions defined on D,
i.e. functions having the form F(t, x). The first presentation is called
the material description, because it utilizes quantities attached to the
material particles, the second is called the spatial description, because
it furnishes information about the particles which are located at the
moment t in the points of a domain D.
The material derivative of the quantity 0 attached to the fixed
and it is given by the formula
particle X, is denoted by
46(t, X).
(1.1.4)
In order to obtain the derivative of F when X is fixed, we have to
take into account that F depends on the coordinates X, through the
functions Y{. Using the derivation rule for the composite functions, we
obtain for the material derivative
OF OF d X;
OF
OF OF
+(v
V)F. (1.1.5)
F(t' x) dt + as d t
+V'r7x;
t
For studying the motion of fluids we employ the spatial description.
The main quantities are: the density (or specific mass) p(t, x ), the pres-
,
T.
sure p(t. x ), the velocity field v (t, x ), the temperature T(t, x ), the
entropy s(t, x) etc. The acceleration which is the material derivative
of the velocity is obtained by means of the formula:
a = at + (v 0)v .
(1.1.6)
Utilizing the derivation rule for the determinants, from (1.1.3) one
obtains Eider's theorem
J = J div v
(1.1.7)
and then (see for example [1.11]) Reynolds's formulas for continuous
inotions (i. e. motions characterized by fields belonging to the class)
C1((to,tII X DWff
dtJDF(t,x)dv=
r
-
1
ID
(F'+Fdivv)dv=
r
(
1.1.8)
BtF dv+ / DFv nda
fD [OF + div(Fv)J dv =
JD
and Reynolds's formula for motions with shock waves
dtI
F(t,x)dv= J_a_dv.f-J
E++E_
F v neda-J fFlld da,
s
( 1.1.9)
3
THE EQUATIONS OF MOTION
- F_, and d is the displacement velocity of the
surface of discontinuity S (see formula (2.5.1) and figure 2.5.1 from
where OFJ = F.,.
(1.110.
The Equations of Motion
1.1.2
The principle of conservation of mass is
d
dt,Dpdv=O (V)DCD.
(1.1.10)
For continuous motions one utilizes the formula (1.1.8) and one obtains the equation of continuity
p+pdivv =0.
(1.1.11)
The general expression for the principle of variation of the momenturn is
t L"= dPwdv
-jDpnda+ JDpf dv (V)DCD,
(1.1.12)
f representing the force per unity of mass. For continuous motions,
from (1.1.11) and (1.1.6), one obtains Euler's equation
p[
+ (v V) v] =pf -grade-
(1.1.13)
The balance equation of the energy (the first principle of the thermodynamics) is
j
(
aD
JOD
+ /Dpf - vda- /D q nda (V)DCD,
(1.1.14)
e representing the specific internal energy, and q, the flux of heat vector. For continuous motions, taking into account (1.1.11) and (1.1.13),
we deduce
Pe = -pdiv v - div q .
(1.1.15)
In ideal fluids the processes are reversible. Eliminating div v by
the aid of equation (2.1.11) and employing the second principle of thermodynamics (1.11], p.54, one obtains the fundamental equation of thermodynamics
de=Tds - pdv,
(1.1.16)
THE EQUATIONS OF IDEAL FLUIDS
4
where s is the specific entropy and v is a notation for 1/p.
Eliminating div v from (1.1.15) and taking into account (1.1.16),
we deduce the following remarkable form of the equation of energy
pTs= - divq.
(1.1.17)
This shows that if it is possible to neglect the change of heat (it is the
case of aerodynamics where the velocities are great), then
s = 0.
(1.1.18)
The equation (1.1.18) indicates that s is constant on trajectories,
the constant varying from one trajectory to the other. One calls such
a motion isentropic motion.. If there exists a configuration where the
entropy constant is the same everywhere, then in every configuration
arising from the first one, the constant will be the same everywhere.
Such a motion is called homentropic or isentinpic everywhere.
The perfect gas is characterized by the following equations of state
p=pRT,
(1.1.19)
The first one (which is obtained from the laws of Boyle-Mariotte and
Gay-Lussac) is the thcrmic equation (or Clapeyron's equation) and the
second is the caloric equation. It is easy to prove (see for example [1.11)
p. 57-58) that for this gas we have
p }+Co, e=
P7 //J
1
-+C1.
-Y-1p
(1.1.20)
c, being the so called specific heat at constant volume and y = cp jc
where cP is the specific heat at constant pressure. For the air 7 = 1.405.
From the expression of a it follows for the homentropic motion:
p = kp'',
(1.1.21)
k representing a constant. The quantity c, defined by the formula
=
(dp/
'
(1.1.22)
has the dimension of a velocity. One shows (see for example [1.10)) that
it gives just the speed of propagation of the surfaces of discontinuity of
the pressure (sound waves). For the ideal gas in homentropic motion it
follows
c2
= 7plp.
(1.1.23)
5
THE POTENTIAL FLOW
The Potential Flow
1.2
1.2.1
Helmholtz's equation. Bernoulli's Integral
If the tnassic forces possess a potential f = grad II (in aerodynamics
these ford (representing the weight of the air) are neglected) and if the
fluid is characterized by a thermodynamic law having the form p = p(p),
where p is a derivable function, defined for p > 0 such that ;7(p) > 0,
then we deduce
f- Igrad p= grad
II -1 pp)
Utilizing the identity (v V) v = curl v x v + grad (v 2/2), Euler's
equation (1.1.13) becomes
/ f
+curlvxv=grad(n-1
dp-2v2).
jt
P
\\
(1.2.1)
This is Hetnahoftss equation.
The flow of a fluid is irrotational in a domain D if everywhere in
D. we have curl v = 0. This equation constitutes the necessary and
sufficient condition for the existence of a differentiable function jp(x, x),
such that
v=grad 44.
(1.2.2)
Such a flow is called potential. Applying the operator curl in (1.2.1), we
eliminate the term in the right hand side of the equation. The resulting
equation is integrated [1.11] as follows
0
P
(.VX)
=
X(t,X),
(1.2.3)
WO and m) representing the vortex (2w =- curly) and respectively
the density in a reference configuration. The formula (1.2.3) shows that
if the motion is potential in a configuration, it remains potential in every
theorem).
configuration arising from the first one
We can therefore put curl v = 0 in (1.2.1). It results Bernoulli's
integral
Ot+2vz+ J LP -n=C(t) 4t=8W Ot.
(1.2.4)
One may give to the spatial constant C(t) the value zero ([1.11],
p.90). Neglecting the messic forces, Bernoulli's integral is
tt+2vz+
J
Pp=0.
(1.2.5)
6
THE EQUATIONS OF IDEAL FLUIDS
The motion of a fluid is stationary or steady if the velocity field does
not depend explicitly on t. Nowr (1.2.4) becomes
2v2+ J ap-n=C.
(1.2.6)
P
In this case the constant cannot be zero.
For the perfect gas in homentropic motion we have the formula
(1.1.21). With a convenient notation of the constant, from (1.2.6) we
deduce
v- + 'Y P = 7 PO = _ ,
(1.2.7)
2
T::_1 -P
7-lpo 7-1
pa, pn and co representing the pressure, the density and the square of
the sound velocity for the fluid at rest. From (1.2.7) one deduces that
in the compressible fluid there is a superior limit v
of the velocity
which is obtained for p = 0 and a critical value v, which is obtained
for v = c. These values are
VCr = Co
Vmax = Co
V7
+2
_
(1.2.8)
For v < v ., the flow is subsonic and for vrr < v < vmm, the flow is
supersonic.
1.2.2
The Equation of the Potential
The equation of the potential is obtained from the equation of continuity (1.1.11), Bernoulli's integral (1.2.5) and the equation (1.1.21),
assuming that v has the form (1.2.2). Calculating the material derivative of (1.2.5), we obtain:
dt(5t+2tr2)+P=0.
(1.2.9)
Taking into account that p depends on t through the agency of
p and utilizing the notation (1.1.22), we deduce
_± d
dpP
P
(Oe
+ 2v2)
(1.2.10)
Replacing in the equation of continuity, we obtain the equation of the
potential
c2A -(v .V)(v -V)o -2(v V).6e-#u=0,
(1.2.11)
*r11F: POTENTIAL FLOW
where c' depends on 40.
We find this dependence in the case of the perfect gas in homentropic
flow. Taking into account (1.1.21), Bernoulli's integral (1.2.5) becomes
v2
4+
+
)-i
{n7-t -1710-t}
= o,
(1.2.12)
representing the density in a reference state. From (1.1.21) and
(1.1.22) it follows cc = kyp0-1, and from (1.2.12) we deduce:
po
1
(+v2)]7_1
=p
11
P=poll-7 1 (fit+2v2)Jy-1
(1.2.13)
,
(1.2.14)
CS
+2v2),
(1.2.15)
representing the pressure and the sound velocity in the
CO
reference state of density po.
For the steady flow one obtains the equation
Po and
c2A4'-(v .V)(v cI)4'=0
(1.2.16)
and the formulas
//
1
P=po1--y21
}
Jy-1
y
P=poll-y 21V1\-V-1
,
(1.2.17)
c2 = c9
-
y-1 v2
2
(1.2.18)
where po, po and co are quantities corresponding to the fluid at rest
(v = 0).
One demonstrates [1.10], p. 207, that the equation (1.2.11) is hyperbolic and the equation (1.2.16) is elliptic for the subsonic flow (v2 < c'-)
and hyperbolic for the supersonic flow (c2 < v2). The equality v2 = c2
occurs only on curves in the two-dimensional flow or surfaces in the case
of three-dimensional flow. These varieties are separating the domains
where the flow is subsonic from the domains where the flow is supersonic. This kind of motion is called transonic. We denoted tit = I v 12.
8
THE EQUATIONS OF IDEAL FLUIDS
1.2.3
The Linear Theory
When the equations (1.2.11) and (1.2.16) are utilized in order to
determine the perturbation produced by a thin body in a fluid having a
known flow, they may be linearized Let us consider for example, that
the uniform flow of a fluid having the velocity U,,, i, the pressure p,,
and the density p,,,, is slightly perturbed by the presence of a body. We
shall denote by
v = UU i + v'
(1.2.19)
the velocity field for the perturbed flow ( v' is the perturbation of the
velocity field). We assume that all the coordinates of v' have the
same order of magnitude a (e representing a small parameter which
characterizes the body). Hence we assume that.
(1.2.20)
where I- VI is bounded and e << 1. According to Lagrant o-Cauchy's
theorem, the perturbed flow is potential, since it arises from a potential
flow. Hence, (3)¢, such that v = grad O and from (1.2.19), it follows
v' = grad p and
-0 =U,,,r+V.
(1.2.21)
The condition (1.2.20) is equivalent to
Sit) =
(Yr .
;Py, s5z) .
(1.2.22)
Replacing in the formulas above and neglecting the terms O(E2), we
deduce
v = (U,o + E ;5;r, E v, 63z), 2J2 = U200 + 21J. pa + 0022)
(1.2.23)
and then
(1.2.24)
U p7= - 2U.v*1 - pc1 = 0,
(1.2.25)
(1 - Af`)Vls +Wv + Vz_ = 0,
where Af(= U,,,/c,,,) is Mach's number in the perturbed flow. One
easily find that the equation (1.2.24) is hyperbolic, and the equation
(1.2.25) is elliptic if Al < 1 and hyperbolic if Al > 1. The motions
characterized by Al < I are named subsonic, and the motions with
Af > 1 arc named supersonic.
The pressure is deduced from Bernoulli's integral, (1.2.5), which may
be written as follows
1
2
actual
v2) 0o
+
p
f
clp
-;p
= 0.
(1.2.26)
THE POTENTIAL FLOW
9
if we neglect the massic forces. It obviously results
de+2 (v2-U2)+f dr=0,
(1.2.27)
Let us assume that the relation
p=Poo+0(e).
(1.2.28)
is also valid. Then we deduce
p(p) = poo +
J
r dp
rx 1
=
I
loo (p
`poo)
dr
+ 1 --
J
pW JJ j 0
I
P
0(e2) = p + P c2px
pay
+ O( r2)
0(6
c?C p
2) .
poll
(1.2.29)
Replacing (1.2.21), (1.2.23) and (1.2.29) in (1.2.27), we find in the linear
approximation the formula for the determination of the pressure
P - P,x = -f).G%e + U,otp,.),
(1.2.30)
Since the previous formulas will be deduced in a different manner
in Chapter 2, in dimensionless variables, we are going to introduce here
these variable (x', y'. z', t'), by means of the formulas
(x, y, z) = Lo(r`, y', z'),
t = (Lo/UU)t',
y: = Uaip'p - P = PoUp',
(1.2.31)
Lo representing a reference length which is not specified yet. So, the
equation (1.2.22) becomes
j112Vt.e. = 0.
(1.2.32)
and the formula (1.2.30).
l), = - (vi. + co=.) .
1.2.4
(1.2.33)
The Acceleration Potential
Another function, often utilized in aerodynamics, is the acceleration
potential ;', introduced by Prandtl. Its existence is ensured for the
unlimited fluid characterized by the lawn p = p(p), by the equation
a = -brad /
.11
dam'
P
(1.2.34)
TILE EQUATIONS OF IDEAI. FLUIDS
10
resulting from (1.1.6). (1.1.13) and (1.2.1). Denoting
a= grad !, ,
(1.2.35)
we get
v" = -
J
clp'
+ C(t) .
(1.2.36)
where C(t) is an arbitrary function depending on t, which can be
determined Lw imposing the value of 0 for a certain state. For example,
if the fluid is incompressible and one consider, that t/' vanishes at
infinity (where p = p,), it follows C = p,/p, whence
(1.2.37)
V, = (P. - P)//).
Since t' represents the perturbation of the pressure, it is usually called
the prrssum function.
The relation between 4) and 0 may be deduced from the equations
(1.2.5) and (1.2.26). We have
r
= grad
grad (wt + (1/2)v2) _ -grad J
(1.2.38)
If the flow is uniform at infinity, with the velocity U., it results
tl =Vet + (1/2){tom - U )
(1.2.39)
In the framework of the linearized theory this relation becomes
V'= P +U.oP.-
(1.2.40)
Taking into account the permutahility of the operator a/Ot + U,,(9/0x
with the operators appearing in (1.2.24), it follows that '0 satisfies the
sane equations (1.2.24) or (1.2.25).
From (1.2.29) and (1.2.39), it results that ,, has the same expression
(1.2.37) in the case of the linear approximation for compressible fluids.
In order to write the boundary conditions, we have to indicate the
relation between 0 and the coordinates of the velocity. For steady
motions, we obtain from (1.2.40), (taking into account that rv must
vanish at infinity) :
Uc
= jW4)dx,
Ucw
as-dX.
(1.2.41)
For unsteady flow, deriving (1.2.40), we get
.2.42)
(1.2.42)
THE SHOCK WAVES THEORY
11
Performing the change of variables t, x --+ r, £ :
r=t,
t;=x-U".t
(1.2.43)
and denoting
w(t, x, y, -) = w (T,
0(t, x, y) z) _ V"'(r,
U,
,
y, z) ,
(1.2.44)
we deduce ay//Or = N//&r, whence, integrating
Ta
(1.2.45)
00
Returning to the variables t and x, we get
w(t, x, y, z)
=
f
+U.r',y,z)dr' _
(1.2.46)
fe
1.3 The Shock Waves Theory
1.3.1
The Jump Equations
The principles of motion for the continuous media are expressed in
terms of balance equations. The general form of a balance equation is
dt,npQdv -
Jen
R nda+JnpSdv,
(1.3.1)
where D is a material domain, Q and S are tensors having the
same order and R is a tensor having the order greater with a unity.
For the ideal fluid, the principle of conservation of mass (1.1.10) has the
form (1.3.1), with
Q=I, R=0,S=0.
(1.3.2)
The principle of variation of the momentum (1.1.12) is obtained for
Q=v, R=-pI, S= f,
(1.3.3)
I representing the unity tensor, and the balance equation for the energy
(1.1.14) for
Q=a+(1/2)v2,R=-pv-q,S= f v.
(1.3.4)
THE EQUATIONS OF IDEAL. FLUIDS
12
Let us see now the balance equation (1.3.1) in case that D is crossed
by a surface of discontinuity with the displacement velocity d ([1.11),
p.28). Applying the derivation formula (1.1.9), the equation (1.3.1) gives
.I [
UL 51
pQ)-AS1 l dv + i;:++E.. pQ(v
n)-Rnda-
llpQQdda=O.
We pass to the limit superposing F,+ and F_ on S (fig. 1.3.1). In
this case, n+ becomes n, and n_ becomes -n.
Fig. 1.3.1.
If we assume that the integrand from the first integral is bounded,
the first term tends to zero, because vol D± ---. 0. Hence one obtains:
jDPQ(v.n_d)_Rnoda=07
(1.3.5)
where we denoted
(1.3.6)
The limit values encountered here are continuous on S (they are not
continuous across S), and S is arbitrary, because the balance equations
are formulated for every D. By virtue of the fundamental lemma ([1.11)
p.31), from (1.3.5), we deduce
OpQ(v n - d) - Rn = 0.
(1.3.7)
13
THE SHOCK WAVES THEORY
This is the equation which leads to the balance equation. It establishes
the connection between the limit values from the two parts of the shock
waves. If across a surface the fields have discontinuities, this surface is
a shock wave.
Utilizing (1.3.7) for (1.3.2), one obtains;
=0,
(1.3.8)
for (1.3.3)
Opv(v n - d) +pn, = 0,
(1.3.9)
and for (1.3.4),
(1.3.10)
The conduction laws utilized in practice (particularly Fourier's law)
make impossible the discontinuity for q. Hence we shall utilize the
equation (1.3.10) as follows
1
p(e+2v2
=0.
(1.3.11)
To the previous equations we shall add the inequation:
Ops(v . n
- d)Q > 0,
(1.3.12)
coming from the Second Principle of Thermodynamics ([1.111, p.50). For
the ideal fluid we have an equality.
1.3.2
Hugoniot's Equation
The jump Equation (1.3.11) is quite complicated. It can be replaced
by a simple equation (Hugoniot's equation), which establishes a connection only between the thermodynamic quantities from the two sides
of the shock wave. In order to deduce these equations, for the sake of
simplicity we introduce the propagation speed P = d - v n. In this
way, the jump equations become
r
OPPI =0,
(1.3.13)
OpPv-png =0,
(1.3.14)
=0,
(1.3.15)
THE F.QUATIOSS OF IDEAL FLUIDS
14
[pPsO <0.
(1.3.16)
From these equations we have to find the unknowns behind the shod;
wave and its propagation speed if we know the state of the fluid in front
of the shock wave. For the salve of simplicity too, we mark by the index
1 the limit values (on S) in front of the shock wave and by the index 2
the limit values behind the shock wave. The normal to S is positively
orientated from the region 2 toward region 1. We exclude the situation
Pi = P-2 = 0, because in this case the surface of discontinuity does not
cross the fluid (it is a material surface moving together with the fluid).
For deducing Hogoniot's equation, we notice that from (1.3.13) it
follows that the quantity p P is continuous across the shock wave. N e
denote by
rn=pjP1=P2P2
(1.3.17)
this quantity. Since in the jump equations we employ only the velocity
on the shock wave in a current point of the wave, we write
v=v,,n+vet.
(1.3.18)
v'2 =
vn representing the normal component and vi the tangential component
(situated in the plane determined by n and v). We denoted by t the
versos of the tangent.
Projecting (1.3.14) on the tangent and on the normal to the shock
wave and taking into account that by hypothesis, in T 0, we deduce
UVt =0,
rnOv,, 11
- H =0.
(1.3.19)
The first equation shows that the tangential component of the velocity
remains constant across the shock wave. From the definition of the
propagation speed P and from the conservation relation (1.3.17), we
deduce
Vn =d-P=d-nzT, r- 1/p,
(1.3.20)
the second equation from (1.3.19) becoming
rn2OTO +
0.
(1.3.21)
From (1.3.15) taking into account the equations (1.3.19), (1.3.20) and
(1.3.21), we notice that:
pPvt 9 =III
vie) = 0,
OpP'el1 = -21n2dOrl + rn3[r20 = 2dOpE
- Innllpll (rl + r'2)
THE SHOCK WAVES THEORY
15
From these equations and from (1.3.15), we get:
2Ohl = OpO(T1 +r2),
(1.3.22)
where h = e + pr is the enthalpy ([1.11], p.55). This is the first form
of Hugoniot's equation. Performing the calculations in (1.3.22), we get
also the form
(1.3.23)
2Oej + rlJ(PI + P2) = 0,
and. for the perfect gas, when e is given by (1.1.20), one finds the third
expression:
P2
P1
=
b+ 1)p2 - (7 - 1)PI
(1.3.24)
(7 + 1)pt - Of - 1)P2
which is useful in applications.
The inequality (1.3.2) shows that if m > 0 (the shock wave surpasses the fluid), then 81 < 82; if in < 0 (the fluid surpasses the shock
wave), then 82 < s1. In all cases behind the shock wave the entropy is
not decoeasing. It increases or remains constant.
1.3.3
The Solution of the Jump Equations
In the sequel we are going to demonstrate, for the perfect gas, that
giving the state of the medium in front of the shock wave and the propagation speed Pt, one may determine completely the state behind the
shock wave. To this aim, we introduce the numbers:
M1 = Pt/cl ,
M2 = P2/c2,
(1.3.25)
where c1 and c2 represent the sound velocity in front of the shock
wave respectively behind the shock wave (CI = 7Pi/Pl, 4 = 7P2/p2).
By hypothesis, M1 is known and M2 is unknown. From (1.3.21), we
deduce:
P2 - P1 = in2(rl _,r2) = p1P1 (1 - r2/rl) = 7P1Mi (1 - 72/r1) .
The function (pa/pl) -1 may also be obtained from (1.3.24). Comparing the expressions we get:\
T2-1=
71
2
(M1-11,E-1=
27
7+1
/ Pt
7+
(M2 -1).
(1.3.26)
From the relation P1r2 = P2r1i it follows:
f
-1)I(1-Jt't2)=11th-1
(1.3.27)
THE EQUATIONS OF IDEAL FLUIDS
16
From the perfect gas law p = pRT and from (1.1.20), we obtain:
- 81
82
C11
fi-C )(p1)'
= in
J
(i)
r
111
r2/rl and p2/pt being replaced from (1.3.26).
Prandtl's Formula
1.3.4
For the stationary wave (d = 0), we have v = -rnr, such that the
equation (1.3.11), for the perfect gas, becomes
Q1v2+
ry
PO
y-1P
2
=0.
(1.3.29)
The expression written here intervenes in Bernoulli's integral (1.2.7).
Taking into account the meaning given there to
v,,,,,x
, we have the
relation
P1=1 2
2+
iv max=l 2vt.7-1pt
1
P1=12v2msx
2
11
(1.3.30)
which shows that vmax is invariant across the shock wave. Taking into
account that
0, from (1.3.30) we deduce:
1
y
P1
PI
2 t[ 2 0 _ 'r' - l pj
P11}2
1l
(1.3.31)
/
We eliminate p2/pt by the aid of equation (1.3.24) and pre/pl, by the
aid of relation PIV1.. = P2v2n which results from (1.3.17). Employing
once again (1.3.30), we deduce:
('y + 1)v1nv2n - (/ - 1)v? = 2yP1/Pl
1)(v 1
- vl ).
Writing vu = v2i = vi, we obtain Prandtl's fonnula
v1nt'2n=
-llvmax-Vi
y+1
ry+1
)=t2.r(1.3.32)
the significance of vt. being that of (1.2.8). In the expressions vma
and va. utilized here, cp is the sound velocity for the fluid in front of
the shock wave (at rest). The relation (1.3.32) gives c2,, when the state
of the fluid in front of the shock wave is known. For the normal shock
According to this relation, if
(:;( = 0), the relation becomes vttt2 =
vt is supersonic, then v2 is subsonic.
THE SHOCK WAVES THEORY
17
The Shock Polar
1.3.5
Buselnann (Vortrage arcs dem Gebiete des Aerod amik, Aachen,
1929), gave a graphic method for constructing v2 if the angle with
v1 is known. To this aim, we consider in the current point Al from
the shock wave (fig. 1.3.2) the AIX axis, pointing in the direction and
sense of v1 and the AIY axis in the plane determined by v1 and the
normal n to the shock wave.
Fig. 1.3.2.
We use the normal pointing towards the state (2); the change of
the sense of the normal does not affect the jump relations (excepting
the relation (1.3.16), whose signification is known), because in these
relations the normal intervenes linearly. Because of the continuity of
the tangential component of the velocity, v, lies also in the plane
determined by v1 and n and it will have on the tangent versor t
the same projection AIQ like vl. We denoted by PI and P2 the
extremities of the vectors v 1 and v2 and by 0, the angle between
v2 and v1. Our aim is to find the geometric locus of the point P2
when 0 is varying. We shall use, obviously, Prandtl's formula (1.3.32).
Denoting by (X, Y) the coordinates of the vector v2, from the figure,
it results
X = v2 cos 0, Y = 1v2 sin 0, vl = v1 sin o, vt = v1 cos B ,
v2., = v-± sin(v - 0) = X sin Or - Y cos a .
(1.3.33)
So, Prandtl's formula becomes:
Xsin2a-YsinorCosa= vi
- 7+1 VICOS2a
(1.3.34)
THE EQUATIONS OF IDEAL FLUIDS
18
From the triangle P1P2R, where the angle P1P2R = a, it follows
tana = (vi - X)JY, such that (1.3.34) becomes
(1.3.35)
Y2(b- X) = (vl X)2(X - a),
-
%%'here we denoted
2
a=
Vt
,
b=v`,+
vl
2
y+1
vi.
(1.3.36)
From the assumption that vt is supersonic, it follows that a < vi,
we deduce vj < b < v..
and from vi <
The curve representing eq. (1.3.35) is symmetric with respect to
MX, it intersects the MX axis in the points A(a) and Pt (v1) (which
are inverse with respect to the sonic circle x2 + y'' = v,2,,), the last point
being double and has a vertical asymptote in B(b). The curve is real
for a < X < b and it is represented in figure 1.3.3. It is called foliurn
of Descartes. It is the shock polar.
Fig. 1.3.3_
Let us show now how we construct v2 when we know the angle
0 made with vl. First of all one constructs the shock polar. This
is determined only by the state (1). The same state determines the
angle ©p made by the tangent MT with the MX axis. If 0 < Bo,
there exist three intersection points 1,1_,13, of the radius vector with
the polar, but only one is P2. The points on AT are eliminated
because they correspond to instable shocks. One eliminates the points
THE SItOCI' WAVES THEORY
19
1:1
because. generally, the shocks are compressive. (v2 < vt ). P2 will
coincide therefore with 12. If C is the intersection of the polar with
the sonic circle and 9° is the corresponding polar angle, then v2 will
be supersonic if 0 < B, and subsonic, if 0 > O.
Once 11 determined, it results the direction of the versor t (the orthogwua from M on P1P2) and then the angle a. The density, the
pressure. the temperature and the entropy behind the shock wave is obtaincNi front (1.3.26) and (1.3.28), setting M1 = -vln/et = -MI sin or,
where .111 = v1/c1 is Mach's number in front of the shock wave.
1.3.6
The Compression Shock past a Concave Bend
We consider a supersonic flow having the velocity v1 i, the density
p) and the pressure P1 in the presence of a concave bend having the
opening b (fig. 1.3.4a)). The wall ME produces a compressive shock,
the discontinuity line MM' being characterized by the unknown angle
a. Behind the shock, the velocity which has to be tangent to ME, will
make the angle 6 with MX . If 6 < Bo, the position of P2 will be
given by 11 from the polar corresponding to this motion. It follows like
above, a and the flow behind the shock.
If 6 > B(), one cannot satisfy for v2 the condition to be parallel to
AI E. As the experience confirms, the assumption of a rectilinear shock
wave Al Al' cannot be taken into consideration. In this case one admits
the existence of a detached curvilinear shock wave which is formed in
front. of Al (fig. 1.3.4b)). The experience confirms this assumption.
Flence the detached shock waves are formed in front of the dihedron
(S > A() (fig. 1.3.5a)), or in front of the bodies with rounded leading
edge (fig. 1.3.5b)). As an information we give the following values :
(10 = 10° for M1 = 1.42 and Bo = 22.55°, for M1 = 2. Depending on
Fig. 1.3.4.
the shape of the body, i.e. on the values of 0, behind the shock wave,
THE EQUATIONS OF IDEAL FLUIDS
20
C
0<00
Pig. t.3.5.
we meet regions where V2 is subsonic (11M2 < 1), if 9 > 9, and regions
where u2 is supersonic, if 9 < O. When 9 is big, the compression is
big. When 9 is passing to small values, it appears a detente and the
velocity becomes again supersonic. On the direction MV the shock
is normal (vi v2 = v,2 and v2 is subsonic. The subsonic regions are
separated from the supersonic ones through sonic lines CD and C'D'.
Behind the shock wave, the flow is transonic.
The shock waves theory will be present in the transonic flow and the
hypersonic flow.
Chapter 2
The Equations of Linear Aerodynamics and its
Fundamental Solutions
2.1
2.1.1
The Equations of Linear Aerodynamics
The Fundamental Problem of Aerodynamics
The fundamental problem of aeroclynamica consists in determining
the perturbation produced in a given state of a fluid by a certain motion
of a body. The given state of the fluid is called basic state or unperturbed
state. The unperturbed state of the fluid may be the rest state, the
uniform flow state, or more generally, the state given by the flow with
an imposed non-uniform velocity field. In this book the unperturbed
state will be either the rest or the uniform flow.
In his turn, the body may he fixed, moving uniformly or may have
a general imposed motion. Obviously, a fixed body in at rest state of
a fluid does not produce any perturbation. The most common are the
case when the unperturbed fluid moves uniformly and the body is fixed
and the case when the fluid is at rest and the body has a given uniform
motion. As we shall see in 2.1.6, these cases are equivalent, from the
mathematical point of view. In both cases, the resulting perturbation
will he stationary.
If the perturbing ixxly has a non-uniform motion, the perturbation
will be non-stationary, whatever should be the state of the fluid. In the
sequel we shall consider only the cases when the unperturbed fluid is at
rest or has an uniform flow.
The problem of determining the perturbation may be practically
solved only in the case of small perturbations, when we can neglect their
products (we keel) only the principal parts of the equations). In these
cases, the linear systems of equations obtained for the perturbations may
be investigated either with the methods of the classical analysis or with
the methods of the theory of distributions. The systems will be linear
22
LASEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS
with constant coefficients in case that the basic state is uniform and
they will be linear with variable coefficients in case that the basic flow
is not uniform. Since, as we have already mentioned, the last case will
not be treated in this book we give here some references, [2.4], [2.14],
[2.15], [2.20], (2.22), for the reader interested in this subject.
The order of magnitude of the perturbation is determined by the
basic flow and by the shape of the body. For some basic flows, a slender
body with it small incidence changes slightly the flow, i.e. produces small
perturbations (governed by linear systems of equations). We cannot
establish in advance the conditions of validity of the linear theory. This
will be done after determining the solution of the linearized equations,
imposing not to obtain results which cannot be accepted from a physical
point of view. In this way, from Chapters 3 and 8, it will follow that in
the case of steady flow, the linear theory is not valid when the basic flow
has approximately the sonic velocity (M = 1), or has an hypersonic
velocity (M > 3) even if the body is slender with a small incidence. In
these cases one has to employ the non-linear equations for determining
the perturbation.
A special feature has the unsteady flow because the system of equations of motion is hyperbolic and it is well known (see, for example,
[1.6]), that for this sort of problems, Cauchy's problem is correct, i.e.
the solution depends continuously on the initial data. If these data are
small, the perturbations will remain small at every instant.
2.1.2 The Equations of Motion
First of all we assume that the basic motion (the unperturbed motion)
(the Ox axis is
of a fluid is an uniform motion with the velocity
taken to be on the direction and in the sense of the unperturbed stream
and we denote by i the versor of this axis), the pressure po,, and the
density p,,. We denote by xi, yi, zi the generic spatial coordinates, by
t i the time and we introduce the dimensionless coordinates x, y, z, t
by mean., of the relations
(2.1.1)
(xi, yi, zi) = Lo(x, y, z), Uwtl = Lot.
where LO is a characteristic length which has to be specified in every
problem we have in view.
We assume that the uniform motion defined above is perturbed by
the presence of a body which has a prescribed motion. Let
F(ti, xi, yi, zt) = 0.
(2.1.2)
23
TILE EQUATIONS OF LINEAR AERODYNAMICS
be the equation of the surface of the moving body.
Denoting by V 1, p1 and pt the velocity, the pressure and the
density for the perturbed flow, we shall write:
V 1 = UU(i + v), Pt = P.- + P"Uoop, pi = Poa(1 + P),
(2.1.3)
the first terms defining the basic motion, and the last ones, the perturbation. Obviously we have:
lien (v, p, p) = 0.
_-.-OD
(2.1.4)
We neglect the heat changes (in aerodynamics this assumption is
plausible, because the variation of the phenomena are very rapid and
there is not enough time for the heat change) and we assume that the
fluid obeys to the perfect gas law. In these conditions, the perturbed
flow will be determined by the following equations:
01+p1divlV1 =0, sl =0, P1V1+grad, pi =0,
(2.1.5)
s1 representing the specific entropy and "the point", a notation for the
material derivative:
(2.1.6)
s 1 = c,.1n(Pl /Pi) + C, f = of /0t1 + (V 1 V 1)f .
The index 1 attached to the differential operators indicates that the
derivatives are calculated with respect to x1, yl, zl
Taking into account the expression of the entropy, from the equation
(2.1.6), we get:
P1p1 = 'YPt11,
(2.1.7)
so that we can eliminate pl from the first equation (2.1.5). Hence, we
have to take into consideration the system
(2.1.8)
01 + ypldiv3 V1 = 0, p1Vl +grad, p1 = 0
which, taking into consideration (2.1.1) and (2.1.3) becomes:
A!2[pt+(1+u)p=+VVlpy+wp;]+(1+-tM2p)(u?+vy+w=)
+p)[ut+(1+u)U=,+ ruy+wu,]+ps=0,
+p)[Vt++u)vs+vuy+wvZ]+Py=0,
(1+p)[wt+(1+u)w2+trwy+wWz]+p: = 0,
= 0, (2.1.9)
(2.1.10)
(2.1.11)
(2.1.12)
24
LINEAR AERODYNAMICS. F'UNDA'MENTAL SOLUTIONS
where (u, v, w) = v, pt = dp/at, Ps = Op/tax,... and
M = U00/c., coo =
'YPoo/Poo ,
(2.1.13)
Al representing Mach's number in the basic motion and 'y, the ratio of
the specific heats
The condition for the perturbation surface (2.1.2) to be a material
surface is F = 0 (the Euler-Lagrange criterion)and it may be written
as follows
Ft +(l+u)F,+vFy+wF =0.
(2.1.14)
This condition must be satisfied for F(t, x, y, z) = 0.
The system of equations (2.1.9) - (2.1.12), together with the boundary condition (2.1.14), will determine the perturbation (p,v).
A. The Linearization around the Uniform Motion
2.1.3
The Equations of Linear Aerodynamics
We assume now that the equation of the perturbing surface is
z = h(t, x, y) = ei (t, x, y),
(2.1.15)
where e is a small parameter and h(t, x, y) is a known function
with continuous first order derivatives (fig. 2.1.1a)). If the perturbation
surface is cylindrical with generators parallel to Oz (fig. 2.1.1b)), then
the equation of the profile determined in the xOy plane is assumed to
have the form
y = h(t, x) = J (t, x) .
(2.1.16)
In this case, the perturbation will be plane.
For the surface (2.1.15), we write F = eh(t, x, y) - z in (2.1.14).
One obtains
Eht + E(1 + u)h= + evhy = w
(2.1.17)
which has to be satisfied for z = Eh(t, x, y). The principal part from
the left. hand side of the equality (2.1.17) has the order of e. We deduce
that the right hand part must have the same order of magnitude. Hence
it follows
w(t,z) y,eh) = e'w(t)x,y,eh).
We consider that this relation is valid all over the fluid, whence
w(t, x, y, z) = e1 F(t, x, y, Z).
(2.1.19)
25
TIIE EQUATIONS OF LINEAR AERODYNAMICS
--s
0
a)
Fig. 2.1.1.
Taking into account (2.1.18). the principal part of the boundary condition (2.1.17) is
h, (t, x, y) + h..(t, x, y) = w(t, x, y, 0) .
(2.1.20)
Taking into account (2.1.19), we deduce that the principal part of
the product from (2.1.12) has the order e. It follows therefore that p
has the same order. Hence,
p(t, x, y, -) = Ep(t. X, r, =)
,
(2.1.21)
the residual equation from (2.1.12) being
Dw+p: =0,
(2.1.22)
where D is the material derivative operator for the unperturbed motion:
D = a/at + 9/ax .
(2.1.23)
Taking into account (2.1.21), from (2.1.10) and (2.1.11), we deduce
u(t, x, y, z) = F-u(t, x, y, z) , v(t, x, y, :) = e'u(t, x, y, z)
(2.1.24)
and the residual equations
Du+p,,=0, Dv+py=0.
(2.1.25)
Now, the equation (2.1.9) becomes
?J2Dp+ dive = 0.
(2.1.26)
This equation together with the equations (2.1.22) and (2.1.25) which
have the vectorial form
Dv + grad p = 0,
(2.1.27)
LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS
26
constitute the fundamental system of the linear aerodynamics. Obviously, this system has to be integrated with the boundary condition
(2.1.20) and the condition at infinity upstream
lim (p, v) = 0.
(2.1.28)
If the perturbation is plane, the condition (2.1.20) will be replaced
by
ht(t, x) + h=(t, x) = v(t, x, 0)
.
(2.1.29)
If the perturbation surface is fixed (i.e. h does not depend explicitly
on t), the perturbation will be stationary, determined by the system
1v12p= + div v = 0 ,
vx + grad p = 0
limo(p, v) = 0
2 30)
.
(.1
and by the boundary conditions
w(x, y, 0) = hr (x, y) ,
(2.1.31)
v(x, 0) = hr (x) .
(2.1.32)
Obviously, the equations (2.1.26), (2.1.27) and (2.1.29) could be easily
obtained from (2.1.9)-(2.1.12) and (2.1.14), supposing that all the perturbations have the same order of magnitude. But this thing has to be
demonstrated. In the case of transonic flow, for example, the perturbations have different orders of magnitude (see Chapter 9).
2.1.4
The Equation of the Potential
Applying the operator curl in (2.1.27), we obtain:
Dcurlv = 0.
(2.1.33)
Taking into account the significance of the operator D, we deduce that
curl v is constant on the parallels to the Ox axis, the constant generally
varying from one parallel to the other. Since at infinity upstream (x =
= -oo) the constant vanishes on every parallel to Ox, it follows that
curl v = 0 on every trajectory coming from --cc. This property is not
true for trajectories detaching downstream from the body. This fact will
be better put into evidence by the fundamental solutions.
In the irrotational zone we have
v = grad cp(t, x) .
(2.1.34)
THE EQUATIONS OF LINEAR AERODYNAMICS
27
Taking into account this representation of the field v, from (2.1.27) we
deduce
p = -D<p = -(vt + ip:) .
(2.1.35)
The function f (t) which should be added in the right hand side
of the equation may be considered equal to zero, since W is determined
with the approximation of an arbitrary additive function of t. From
(2.1.26) and (2.1.35) it follows Dip = M2D2co or, explicitly
(1 - M2)fpxs + Wy, +'pz:. - 2M2WfZ - M2Vu = 0.
(2.1.36)
This is the equation of the potential.
In case of the stationary flow, from (2.1.33) it results
curly = f(y,z),
f representing an arbitrary field which must be considered zero because
for x -- -oo we have curl v = 0. Again it is true the observation that
curl v is not zero on the parallels at the Ox axis detaching downstream
from the body. In the irrotational zone we have therefore
v = grad W(x) .
(2.1.37)
From the first equation (2.1.25), taking into account (2.1.28), it follows
p(x) = -u(x) .
(2.1.38)
Replacing (2.1.37) and (2.1.38) in the first equation (2.1.30), it follows
(1 - M2)9u + ,pyy + ,P:: = 0.
(2.1.39)
This equation may be obviously deduced from (2.1.36). For the incompressible fluid (M = 0) these equations become
Al P = 0.
(2.1.40)
Applying the operator grad in (2.1.39), we get:
(1 - M2)v22 + vyy + USX = 0.
(2.1.41)
Hence, the coordinates of the velocity (and the pressure) satisfy the same
equation (2.1.39).
The equations (2.1.36) and (2.1.39) have been obtained in another
way in Chapter 1. There we utilized the Lagrange-Cauchy theorem
in order to prove that the perturbed flow is potential. Here, without
28
LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS
utilizing this theorem, we demonstrated that in the first approximation
the perturbation is potential. In addition, we see here that downstream
the perturbing body, the perturbation does not possess this property
any longer.
B. The Linearization around the Rest State
2.1.5
The Linear System
Let us assume now that the basic state of the fluid is the rest state,
which is perturbed, as above, by a prescribed arbitrary motion of a
body. We denote by po and po the pressure and the density of the
fluid at rest. We denote also by ca = -ypo/po the square of the sound
velocity for the same state. Since, in this case, there is no characteristic
velocity, it is not recommended to use the variables (2.1.1) and (2.1.3).
It is natural to use dimensional coordinates and to put
V1 = V, P1 = PO + POP, Pt = Po(1 + p),
(2.1.42)
litn (p, v, p) = 0.
(x}-00
(2.1.43)
The system of equations of motion has the shape
p+(1+p)divv=0, (1+p)P=(c2 +7p)p,
(1+p)v+grad p=0.
(2.1.44)
If the equations of the perturbing surface have the form (2.1.15), we
shall have the boundary condition
Eli, + EU) X + Evii, = w
(2.1.45)
which has to be satisfied for z = Eh(t, x, y). We deduce
w(t, x, y, d) = eii (t, x, y, ch)
(2.1.46)
and the boundary condition
w(t, x, y, 0) = ht (t, x, y) .
(2.1.47)
Assuming like in (2.1.3), that (2.1.46) is valid all over the fluid,
w(t, x, y, z) = ew(t, x, y, z),
29
THE EQUATIONS OF LINEAR AERODYNAMICS
from the projection of the second equation from (2.1.44) onto the Oz
axis, we deduce
P(t,x,v,z) = C PO, z, v, z)
and the residual equation
wt+pz=0.
Acting in the sequel like in subsection (2.1.3), we obtain at last
p t +divv = 0 , p t = c2pt , vt + grad p = 0
(2.1.48)
We deduce the system
pt+c2divv=0, vt+gradp=0
lim (p, v) = 0
III--=
(2.1.49)
and the solution p = c2p. The system (2.1.49) has to be integrated with
the boundary condition (2.1.47).
From (2.1.49) one obtains the fundamental equation of acoustics
pa - cRAp = 0.
2.1.6
(2.1.50)
The Uniform Motion in the Fluid at Rest
Let us consider the particular case of the uniform motion of the per-
turbing body in the fluid at rest. We assume that the motion is performed with the velocity U0 in the negative sense of the Ox axis.
Putting
V, = Uov, pi = poll + A, Pt = po + poop
(2.1.51)
and utilizing the variables (2.1.1), we get the system:
Mop+(1+yAf,p)divv=0, (1+p)Mop=(1+yM p)P,
(1+p)v+gradp=0,
(2.1.52)
where MQ is Mach's number for the fluid at rest. As above, one deduces
the boundary condition (2.1.47) and the system (2.1.52) reduces to the
residual form
Af2pt+divv=0, vt+gradp=0
and M p = p.
(2.1.53)
30
LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS
In a frame of reference R' = Cx'y'z' (fig. 2.1.2) solidary with the
body, having the axes parallel and having the same sense with the axes
of the frame R = Oxyz (the frame R' is inertial), the system (2.1.53)
has the form of the steady system (2.1.30). Indeed, we pass from the
frame R to the frame R' by means of the Galilean transformation
t'=t,
=x+t, Y ,=Y, z'=z.
(2.1.54)
With this transformation, the system (2.1.53) gains the form of the
system (2.1.26). (2.1.27) and the condition (2.1.47), the form (2.1.20).
But, since we deal with a translation of the body, h in (2.1.47) has
the form h(x + t, y) which is transformed in h(x', y'). The boundary
condition in the variables x', y', z' will have therefore the form (2.1.31),
so that it will determine a steady motion. The system of equations of
motion in z', y', z' will have the form (2.1.30).
Hence we have demonstrated that the problem of determining the
perturbation produced by a body moving uniformly with the velocity
-Uoi, in a fluid at rest, is equivalent to the problem of determining the
perturbation produced by the same fixed body in an uniform stream
with the velocity Uoi.
2.2
The Fundamental Solutions of the Equation of the
Potential
2.2.1
The Steady Solutions
This subsection is written on the basis of the paper [2.111. The fundamental solutions of the equation (2.1.39) are the solutions of the equation
(1 -
e ,+
(2.2.1)
a representing Dirac's distribution. These solutions are, obviously, dis-;
tributions. Utilizing the Fourier transform method, we shall obtain ternperate solutions. Taking into account (A.5.1), valid also for temperate
solutions, we obtain using the notation a2 = IaI2,
(a2
- M202).? = -1,
(2.2.2)
whence
E_
1
a2-M2ni
--V-'
1
[(1
-A12)a +a2+a
(2-2.3)
FUNDAMENTAL SOLUTIONS OF THE EQUATION OF THE POTENTIAL 31
In the subsonic case M < 1, we shall denote
fl= 1-M2,
(2.2.4)
and in the supersonic case (M > 1),
k= M2-1.
(2.2.5)
In the subsonic case one utilizes the formulas (A.7.10) and (A.7.11).
One obtains the following fundamental solutions:
x+
4w
E
1
2Aft
In
1(y
+z)
,
n = 3,
x f2 +y2, n = 2,
(2.2.6)
(2.2.7)
For the two-dimensional case we have not written the additive constamt C - in j3 appearing in (A.7.11), because the fundamental solution
is determined with the approximation of a solution of the homogeneous
equation.
In the supersonic case, one utilizes the formulas (A.7.14) and (A.7.15).
One obtains the following fundamental solutions:
1 H(x - ky2+z2)}
27r
.6 = -
x-k(y+z)
n=3,
H(x - klyl), n = 2.
(2.2.8)
(2.2.9)
H repn-senting, as we have considered in Appendix A, Heaviside's func.
tion. From the definition of this function (A.3.13), it follows that the
three-dimensional solution is different from zero only for
x > kk/,/2 +z2.
(2.2.10)
This inequality implies
x > 0, xs > k2(y2 + zs).
The set of points from the space verifying these inequalities constitutes
the interior of the cone with the vertex in the origin of the system of
coordinates (the perturbation point) and the symmetry axis along the
Ox axis (fig. 2.2.1a)). This cone is called Mach's cone. In fact, it is the
characteristic cone associated to the partial differential equation (2.1.39)
LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS
32
Y
b)
a)
Fig. 2.2.1.
in the hyperbolic case. The radius of the cone is x/k, and the angle p
(the semi-opening) is determined by the formula
tan y =
xxk
=k
= Ml -11
(sin µ = M J
.
(2.2.11)
In the two-dimensional case, the solution is different from zero only
inside the dihedrou made by the plane; x = ±ky where x > 0 (fig.
2.2.1b)). This is Mach's dihedrun.
2.2.2
Oscillatory Solutions
The fundamental oscillatory solutions are defined by the equation
(1 - M2)En + Cm + e - 2M26 - M2Ea = 6(x)exp(iwt) , (2.2.12)
associated to the equation (2.1.38). They will have, obviously, the shape
E = E(x)exp(iwt),
(2.2.13)
where
(1- M2)E;= + E,,,, + E.: - 2iwAf2E= + w2M2E = 6(x).
(2.2.14)
Performing the change of functions E - e:
E = exp(Ax)e
(2.2.15)
and nullifying the coefficient of the derivative e=, one obtains the equar
tion:
(1 - M2)e", + eyy + e1t + ae = exp(-Ax)6(x),
(2.2.16)
FUNDAMENTAL SOLUTIONS OF THE EQUATION OF THE POTENTIAL 33
where
A
ikM,
w2M2
a= 1-M2
(2.2.17)
In the subsonic case (M < 1) one performs the change of variable
(x, y, 0 (X, Y, Z):
X =x, Y = fly, Z=Az
(2.2.18)
and one takes into account (A.3.11). One obtains the equations:
cxx + cyy + e z + k2e = exp(-AX)8(X ), n = 3,
e x x + eyy + k2e = ftuexp(-)LX)b(X), n = 2,
(2.2.19)
where k = WA1/02 = wM.
In (2.2.19) we have Helmholtz's non-homogeneous equations. It is
well known (see, for example, [A.12], §9), that Helmholtz's equation
has two fundamental solutions in the three-dimensional case and the
same number of solutions in the two-dimensional case, the choice of the
solution depending on the type of oscillation defined by the equation.
Taking into account that we obtained Helmholtz's equation looking for
oscillations having the form (2.2.13), it follows (see, for example, [1.41)
Chapter 7, §2) that the following solutions have a physical meaning:
ea
=
exp(-ikIXI)
4i(XI
1 fr(2)
e2 = 4
(kIXI) ,
63 representing the solution for the three-dimensional case and 'e2 the
solution for the two-dimensional case. Hoe) is Hankel's function.
Performing the convolutions of these solutions with the right hand
member from (2.2.19) (A.4.6 formula), we get the following fundamental
solutions
e2 = 4'-Ho2)(kjXI),
e3 =
so that, taking into account the changes already made, we find:
E3
ik(Mx - R) ,
(2.2.20)
E2 = -'blIt HQ(kR)exp (kMx) ,
with the notations
R= a +f2(y2+z2), 17 = Vx2 ++Q2y2.
34
LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS
In the supersonic case ; (Al > 1) one performs the change of variable
X =x, Y=ky, Z=kz.
(2.2.21)
The equation (2.2.16) becomes
exx - eyy - ezz + v2e = -exp(-AX) 6(X),
(2.2.22)
where v = wM/k2. In (2.2.22) we have the non-homogeneous KleinCordon-Fock equation. The fundamental solutions of this equation are
also known (JA.8J [A.111). Performing the convolutions of these equations with the right hand member of (2.2.22), we get:
Y Y
X27tH (
e3
C2
y- -2+-Z Z2) cos X
X.
Z.
V
Z2
--y2)
2kH(X - IYI)Jo (vvX2
such that finally it follows
-
E
E3
1
H `x - k Vy2 +zs)
cosv
x -k y +z)
x_k (
kf.M2
.i.i)exp,
1-M=x)
(2.2.23)
(v
2L,
H(x - IyI)Jo
x2 - k2y2) exp (12X)
Due to the presence of Heaviside's function in (2.2.23), it is obvious
that these solutions will be different from zero only in the interior of
Mach's cone with the vertex in the origin for x > 0, respectively in the
interior of Mach's dihedron (with the edge on Oz) for x > 0.
2.2.3
Oscillatory Solutions for M = 1
If M = 1, the equation (2.2.14) becomes
Ev + E..s - 2iwE= + w2E = 6(--).
(2.2.24)
Introducing the Fourier transform 2 with respect to the variables y
and z one obtains
A
2iwE=-(w2-az-a3)E_-6(x).
FUNDAMENTAL SOLUTIONS OF THE EQUATION OF THE POTENTIAL 35
The solution of this equation (see (A.21)) has the shape E = H(x)E,
where
2iwE. - (w2 - az It follows
0,3 ) E(o)
E = O , = -1/2iw.
E - - H(x)
w2
2iwP
- a3 - a3.
[
(2.2.25)
2iw
For obtaining E we take into account that
L
0
exp (-au2)d u =
(2.2.26)
we notice that
+00
exp (-iAa - Ba2)da =
FOG
exp -B (a + 2B
l)a
-
da
///
42
I.
One obtains:
E
4irx) p
(- i'd r2) ,
(2.2.27)
with the notation r2 = z2 + y2 + z2.
In the two-dimensional case we have:
[__(x
x 2+y2)],
11
(2.2.28)
where 2o- 2Riw = 1.
The solutions (2.2.27) and (2.2.28) may also be obtained as limits of
the subsonic solutions (2.2.20) for M - 1. For obtaining (2.2.27), we
notice that
M2
2
jj(Mx - R) _ - lim M+ R = -U ,
limy
(2.2.29)
x being positive, as it follows from the supersonic solution (2.2.23). For
obtaining (2.2.28) one takes into account the asymptotic behaviour of
Ho2) for great values of the argument ((1.42)).
HO(2(k1)
t/2
(-h)
exp (-ik R +
i)
(2.2.30)
and one performs a calculcalculus similar to (2.2.29). In the two-dimensional
case, the limit of the supersonic solution for M --+ 1 is obtained in
(10.171.
36
LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS
The Unsteady Solutions
2.2.4
In the sequel we are going to determine the fundamental solutions of
the equation (2.2.14), i.e. the solutions of the equation
(I - M2)e11 + £yy + E, - 2Af2£u - Af 2£tt = 6(t, x)
.
(2.2.31)
This equation determines the perturbation produced in the uniform
stream, defined in 2.1, by a source of potential acting at the moment
t = 0 in the origin. The problem is plane (two-dimensional) if the source
is uniformly distributed on the Oz axis.
Applying the Fourier transform, we deduce
M2d2£/d t2
- 2M2ial d E/d t + (a2 - A12c )£
(2.2.32)
We know from Appendix A that the solution of this equation has the
form £ = H(t)E, where H(t) is Heairiside's function and k, is a
solution of the problem
MV P/dt2 - 2M2ia1 dE/d t + (a2 - Al2a2 )E = 0,
k(0) = 0 (dE/d t)(0) = -AI--2.
(2.2.33)
We get:
A'IE =
_sin J a jf- ti t
e t°'t
.
(2 . 2 .34)
Utilizing the formulas (A.7.14) and (A.7.15), we deduce
£(t , x) = -- Irt 6(t-MR) ,
1
2r
where
R=
H(t - MR)
n=3 ,
n = 2,
(2.2.35)
(2 . 2 . 36)
t2 - M2R`
(x-t)2+1,2+z2, R= (x-t)2+12.
We shall not write any longer the factor H(t) from the right hand
member, because obviously this member is different from zero only for
t>0.
Further we shall perform a detailed investigation of the solutions
(2.2.35), (2.2.36). We denote
h(t)=t-AIR.
;'.2.37)
FUNDAMENTAL SOLUTIONS OF THE EQUATION OF THE POTENTIAL 37
We are interested to find the zeros t; of this function, as we are going
to utilize the formula:
(2.2.38)
from [A.10], page 20. We have also to know the sign of the function h,
because the two-dimensional solution differs from zero only for h > 0.
One notices in (2.2.37) that the zeros t; are positive. Obviously.
h(0) < 0 and h(oo) = (1 - M2)oo. For the graphic representation of
the function h : (0, oo) - R, we have to separate the cases M < I and
M > 1. The zeros of the function h(t) are
tt-
M2x+M x +(1(2.2.39)
1
112
In the subsonic case (Al < 1), h(t) has a single positive root namely
h(t)
t,
t
Fig. 2.2.2.
t+. The graphic of the function h is represented in figure 2.2.2. Utilizing the formula (2.2.38) and taking into account that
t+=Ai (x-t+)2+y2+z2,
we deduce
--
1 t+
4r t
x
b(t - t+)
+ (y` + x )
(2.2.40)
,
in the three-dimensional case and
H(t - t°.)
1
27r
t -A
(2
,
.
2 41)
.
38
LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS
in the two-dimensional case. Here we denoted to = t f(z = 0).
The solution (2.2.40) was given for the first tine' in [2.11], and the
solution (2.2.41) is given in [1.2]. A given point P (fig. 2.2.3) perceives
in different manners the perturbation produced at the moment t = 0
in the origin of the system of coordinates in the three-dimensional case
and the perturbation produced at the moment t = 0 uniformly on the
O: axis, in the two-dimensional case. In the three-dimensional case the
perturbation is perceived at the moment t+ and only at this moment
(see the solution (2.2.40)). In the two-dimensional case, the perturbation
is perceived by P continuously, beginning from the moment t° (see
the solution (2.2.41)). The explanation of this difference is that in the
case of the plane problem, one admits that, at the moment t = 0, the
entire Oz axis emits perturbations, to representing the moment when
the perturbation emitted by the origin reaches the point B(x, y, 0) and
t > to , the period when the perturbations emitted at the moment t = 0
by the other points of the Oz axis reach P. In figure 2.2.3, Q is the
position of the source (moving with the stream) at the moment to, the
distance QP being given by the formula J(x - t.)2 + y2 = M'lt+
Fig. 2.2.3.
In the supersonic case, we have to determine the zeros of the function
h'(t). They are given by the equation
(x-te)2+y2+z2 = M(to-x).
(2.2.42)
Noticing that x < to, it follows that there exists a single zero, namely
to = x +
y2 -+z2/k.
(2.2.43)
A simple calculation gives
h(to) = x - k
y2 -+z 2 .
(2.2.44)
FUNDAMENTAL SOLUTIONS OF TILE EQUATION OF THE POTENTIAL 39
We shall distinguish three cases (fig. 2.2.4):
h(to) = 0,
h(to) > 0,
h(to) < 0.
(2.2.45)
Ni)
t,
2
x>k(y+x)
b)
a)
c)
Fig. 2.2.4.
In the first case the fundamental solution is
£3
1 t+b(t - t+) + t_8(t - t_)
t x- ky+
41r
0,
E2
_
_
1
2;r
1
t -M (-x
t)
t<t+,t_<t
t+ <t<t_,
Tf P '
(2.2.46)
,
(2.2.47)
Ea was given for the first time in (2.11]. One interprets this solution as
follows: given a point P(x, y, z) in the interior of Mach's cone, there
exist two and only two moments t+ and t_ when the perturbation produced in the origin at the moment t = 0 affects this point (fig. 2.2.5).
The moments t+ and t_ are calculated as functions of the coordinates
of P by means of the formula (2.2.39). In the two-dimensional case,
P(x, y, 0) is affected during the interval It, to_) the explanation being
similar to that given in the subsonic case. In this interval, the expression
under the square root is positive.
40
LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS
In the three-dimensional case, figure 2.2.5 presents the intersection
of the z = 0 plane with Mach's cone and with the spheres having the
centers Q1 and Q2. In the two-dimensional case the figure indicates
the intersection of the same plane with Mach's dihedron and with the
cylinders having the radii M- t9+ and Mtoi . Since from the relation
sing = Q,P;/OQ; we deduce sinp = M-1 it follows that p is Mach's
angle, i.e. the two spheres (cylinders) are inscribed into Mach's cone
(dihedron).
Fig. 2.2.5.
If P belongs to Mach's cone, there exists only one moment t+ =
= t_ = to = xM2/k-2, when the point P is affected by the perturbation.
At that moment, the source is located in the intersection of the Ox axis
with the normal at the cone in the point R.
If P is in the exterior of the cone, it is not affected by the perturbation.
Hence, for the unsteady flow, the perturbed zone is also in the interior
of Mach's cone (respectively Mach's dihedron), but unlike the case of
steady or oscillatory flow (when an arbitrary point P from the zone
was affected all the time) in the unsteady motion it is affected only at
two moments t+ and t_ (in the interval between these moments in
the two-dimensional flow) or in a single moment if P is on the cone
(dihedron). This happens because in the first two cases the fundamental
solutions correspond to sources acting all the time in the origin (on the
Oz axis), while in the last case the solutions correspond to sources acting
only in the moment t = 0.
FUNDAMENTAL SOLUTIONS OF THE EQUATION OF THE POTENTIAL 41
2.2.5
The Unsteady Solutions for M = 1
If M = 1, the solutions of the equation (2.2.31) will be (2.2.35) and
(2.2.36) with M = 1. The function h(t) = t - R has an unique
zero (it' > 0) namely the moment tj = r2/2x. It follows that the
fundamental solutions
E3
-_
5 ( t8;rt x2
)
2x
)
4rrx a (t
- 2x
(2.2.48)
1 H(2xt-x2-y2)
2n
2zt. _ x2 _ y2
vanish only for x > 0.
Every point P from the half-space (half'-plane) x > 0 is affected
by the perturbation. In the three-dimensional case, the perturbation
emitted by the origin at the moment t = 0 is received in the point
P(x > 0, y, z) at the moment ti, and only at this moment. If the
perturbation is emitted by the Oz axis (the two-dimensional flow), an
arbitrary point P(x > 0. y, 0) receives this perturbation continuously,
beginning with the moment tj = (x2 + y2)/2x.
Fig. 2.2.6.
The moment of the first reception tj is given by the equation
(x - ti), + y2 + z2 = ti.
(2.2.49)
The graphic of this equation is a sphere (a cylinder in the two-dimensional
case) with the center on the Ox axis in the point tj and the radius
42
LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS
t1 (figure 2.2.6). The point t1 is obtained graphically, taking the intersection of the Ox axis with the mediator plane of the segment OP
(the mediator line of OP in the two-dimensional case).
2.2.6
The Fundamental Solutions for the Fluid at Rest
As we have seen in subsection (2.1.5) , the propagation of a small
perturbation in the fluid at rest, is described by the equation (2.1.50).
The fundamental solutions of this equation will be defined by
Eu-c2(E +Eyy+t )=6(t,z).
(2.2.50)
They will give the pressure produced by an unitary perturbation in the
origin of the system of coordinates at the moment t = 0. Applying the
Fourier transform in (2.2.50), we get:
d2 ?/d t2 + c2a2E = d(t) .
(2.2.51)
Utilizing (A.3.17), we deduce that the solution of the equation (2.2.51)
is E = H(t)E, where
d2E/d t2 + c2a2E = 0, E(0) = 0, k(0) = 1.
(2.2.52)
Solving this problem, we find:
sinclcxIt
cla
By virtue of formulas (A.7.14) and (A.7.15), we obtain:
6(ct - r)
C3
=
£2 =
- 6(ct - r)
4zrc2t
1
2rc
H(ct - r)
41rrr
(2.2.53)
c t- -r
E1 representing the solution in the three-dimensional care and t2 the
solution in the two-dimensional case. The solution E shows that the
perturbation produced in the origin at the moment t = 0 is concentrated, for t > 0, on a spherical surface having the radius ct and the
center in origin. The perturbation propagates as a spherical wave having
the velocity c. On the sphere of radius r the perturbation is only at
the moment t = r/c. We notice the validity of Huygens' principle. The
FUNDAMENTAL SOLUTIONS OF THE EQUATION OF THE POTENTIAL 43
solution E2 shows that the perturbation emitted on the Oz axis at the
moment t = 0 lies at a moment t > 0 in the interior of the cylinder of
radius ct. There is a foregoing front of the wave propagating with the
velocity c, but there is no posterior front. Unlike the three-dimensional
case, behind the foregoing front, the perturbation differs from zero at
every moment t. In this case Huygens' principle is not valid any longer.
We know from 2.2.4 the explanation of this fact.
2.2.7 On the Interpretation of the Fundamental Solution
The equation
p+pdivv = pgb(x)
(2.2.54)
may be interpreted as the equation of continuity when there is a source
with the intensity pq in the origin. Indeed, integrating on every domain
D containing the origin and taking into account (A.7.3), we get:
(p+pdivv)dv=pq.
(2.2.55)
ID
The integral gives the variation of the mass from D in the unity of
time. This is given by the intensity of the source. In every domain D
which does not contain the origin, the mass is preserved.
The presence of the term b(x) in (2.2.54) represents the cause of
the motion. Since the term has a spherical symmetry, it follows that the
flow will have this property too. Hence v = F(r)x/r and v = grad V
with YP = f F(r)dr.
An uniform flow, with the velocity U,,i is also potential, hence
the flow resulting by overlapping the uniform flow over the flow due to a
source is also potential. According to the calculus from subsection 1.2.2,
it follows
c20O -
(V - V)(V V)¢ = 95(x)
Setting q = eq, the equation may be linearized and for q = 1 one
obtains (2.2.1). The solution of the equation (2.2.1) could therefore
represent the perturbation produced into the uniform stream defined by
M, by a mass source of intensity p, placed in the origin of the system
of coordinates.
In the same way, the equation (2.2.31) could be obtained from the
equation
A+ pdiv v = pgb(t, x),
(2.2.56)
which would represent the equation of continuity in case that a source
of intensity pq is acting in the origin at the moment t = 0.
44
LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS
2.3 The Fundamental Solutions of the Steady System
2.3.1
The Significance of the Fundamental Solution
In order to put into evidence the physical significance of the fundamental solutions of the steady linear system, we consider the equations:
Af 2px + div v = 0,
vx + grad p = F,
(2.3.1)
lim (p, v) = 0.
As it is already known, these equations determine, in the first approximation. the perturbation produced into the uniform stream defined in
2.1 by a force density F. For being able to utilize these equations in
the case of a force of intensity f = (fl, f2, f3), applied in a point C, we
have to define a density F whose action against the fluid must have
the same torsor (resultant and resultant moment) like the force f. We
state that this density is
F = f b(x - t),
(2.3.2)
where 6 is Dirac's distribution. Indeed, taking into account (A.7.3),
we deduce that the torsor of this density is
f fb(x - lr)dx = f, f x x fb(x - F)dx = E x f ,
(2.3.3)
i.e. just the torsor of the force f.
Hence the system
M2p,, + div v = 0, v= + grad p = f b(x)
lim (p, v) = 0,
(2.3.4)
x--.-oo
determines the perturbation produced in the uniform stream, defined in
2.1, by the force of intensity f = E 7f, applied in the origin of the system
of coordinates. By definition, this system determines the fundamental
solutions of the steady linearized system of aerodynamics.
One obtains the plane solutions if one considers that the force having
the constant intensity f is uniformly distributed along the Oz axis,
being parallel to the xOy plane, i.e. f = (fl, f2i 0). In this case we
have the same conditions for every z; and the perturbation is plane.
THE FUNDAMENTAL SOLUTIONS OF THE STEADY SYSTEM
45
The General Form of the Fundamental Solution
2.3.2
We are interested in those solutions of the system (2.3.4) which can
be obtained by means of Fourier transform. They are, obviously, distributions. Utilizing the formulas (A.6.4), from (2.3.4) we deduce:
A12alp+et v=0, ialy+iap=-f.
(2.3.5)
From (2.3.5) we deduce:
P" =
is f
(2.3.6)
a2 - M2a'I
Then, from the second equation (2.3.5) it follows
f
ink
(-ia)(-ia f)
(2.3.7)
iai(02-M2a1)
Utilizing (A.6.9) and (2.2.3) from (2.3.6) we deduce:
P(x, y, z) = -(f . V)F_i [a2 - A12ai = (f V)E.
I
(2.3.8)
Taking into account (A.7.7), and (2.3.7) we obtain:
v = fH(x)5(y,z) - V(f 0)S-' [ini(cr2 1 Af2c 2)J
(2.3.9)
But
X
[iaia
a2 - A1201] =
_ AI2a2)J
E,
(2.3.10)
whence, integrating with respect to x, by virtue of Lebesgue's theorem
[A.9}, it follows
J.--`1
ial(a"-
' .1(2 "
1fI
=
f Ed x.
(2.3.11)
The integration limits have been appropriately imposed in order to
satisfy the last condition from (2.3.4). It results therefore
v(x, y, z) = f H(x)6(y, z) + VV,
where
r
f o0
Ed x.
(2.3.12)
(2.3.13)
46
LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS
The formula (2.3.12), which is valid both in the subsonic and supersonic
cases, shows that the perturbation is potential, excepting the Ox axis
for x > 0 (in the two-dimensional case, one excepts the xOz plane for
x > 0), where the first term does not vanish. Hence the perturbation is
not potential downstream the point (or the fine) where the perturbing
force is acting.
From (2.3.8) and (2.3.12) it also results:
u(x, y, z) = ft H(x)ft, z) - p(x, y, z) .
(2.3.14)
With the exception mentioned above one obtains (2.1.38).
In the sequel we shall utilize also some expressions of the components
of the velocity which do not result from (2.3.12). In the two-dimensional
case, performing the change az = a2 - M2a1 - (1 - M2)ai in the
component v resulting from (2.3.7), we deduce:
V=-
a
ia2f
+ (1 - M2)
iaif2
a2-1b12a
(2.3.15)
whence
(2.3.16)
v(x, y) _ - ftEy + (1 - M2)f2El .
Analogously, in the three-dimensional case, replacing a3 =
Maa1 - (1 - M2)a1 a2, in the component w resulting from (2.3.7),
-
we obtain:
w
_
ioaf1
-a2-M2
1
ia1f3
+(1-M2 )a2-M2a1
(2.3.17)
1a2f3
ia2a3f2
crl(a2 M2a1) a1(a2 - M2at) '
-
whence
,,2
w(x, y, z) = -f1Es + (1- M2)f3e2 - f2 ys
Ed x + f382yy f Ed x ,
'f-C00
00
(2.3.18)
with the notations
= 82/8y8z, .. .
2.3.3 The Subsonic Plane Solution
If the perturbation is plane and the free flow is subsonic, E has the
expression (2.2.7). From (2.3.8) one obtains
P(x, y) -
1 xf1 + 132yf2
27x/3 x2 + /32y2
(2.3.19)
47
THE FUNDAMENTAL SOLUTIONS OF THE STEADY SYSTEM
and from (2.3.16),
0x
(2.3.20)
- yf
+g2y
2
If the force f is distributed on a straight line parallel to the Oz
q), then the
axis intersecting the xOy plane in the point
2nx2
v(x, y) =
.
perturbation will be determined by the system
12px+divv=0, v=+gradp= f6(x-t,y-q),
(2.3.21)
litn (p, v) = 0.
After performing the change of variables (x, y) -, (xo, yo):
xo=x-e,yo=y-17 ,
(2.3.22)
the system (2.3.21) is transformed into the system (2.3.4). Hence, for
(2.3.21) one obtains
I
P(x, y) = 2T'8
xofl +
X02
02yof2
+ p2ya
,
v(x, J) =
Q x0f2 - yofl
27r XT o
2y02
-
(2.3.23)
These solutions have been obtained in [2.10].
2.3.4
The Three-Dimensional Subsonic Solution
In this case, E is determined by (2.2.6). From (2.3.8) one obtains
4ap(x, y, z) = --(fia: + flay + f30z)(1/R) ,
(2.3.24)
where
R=
x2+f2(y2+z2).
(2.3.25)
Taking into account that
_
dx
J (x2 + a2)3/2
x
a2(x2 + a2)1/2
(2.3.26)
we deduce
j'a()dX=_22
(2.3.27)
1
R dx
-y+z2
48
LINEAR AERODYNAMICS. EUNDAMEN"TAL SOLUTIONS
Employing these results, we deduce from (2.3.13) the expression of the
potential
cP(x'y,z)
1
fi _
= 4zr R
(1+j)12-22 (1+)f3]
y
y2+z
(2.3.28)
which gives the possibility to calculate the components of the velocity.
From (2.3.18) we also obtain:
/
4-1r8,(I?)-4 fsa=[
w(x,y,z)=
(2.3.29)
- -,(f2a: - f3ay)y2 + z2
Cl
+R
.
Considering A = 1, we find the solutions for the incompressible fluid.
These results have been obtained in [2.8).
2.3.5
The Two-Dimensional Supersonic Solution
In this case, E is given by (2.2.9). Taking into account (A.7.17),
from (2.3.8) we deduce
2kp(x, y) _ (-f, + k f2sign y)a(x - klyl)
(2.3.30)
,
and from (2.3.16),
2v(x, y) = (-f1sign y + kf2)6(x - klyl).
(2.3.31)
Obviously, the perturbation differs from zero only in Mach's dihedron.
The solution was given and utilized in [2.10].
2.3.6
The Three-Dimensional Supersonic Solution
In this case, E is given by (2.2.8) and
(2.3.32)
P(x, y, z) = (flame + fear + fsa.)E .
Denoting s = k y2 + z2 and taking into account (A.3.9) and (A.3.14),
we deduce
ay T Ed x = a H(x - s)
o
H(x - 8)(9y
J
rl
dx
Js I X7 9
2 z2E
- = 11+
x
7X=F==
R
(2.3.33)
f
49
THE FUNDAMENTAL SOLUTIONS OF THE STEADY SYSTEM
such that,
By means of formula (2.3.12) we determine the velocity field. The
similarity between this potential and the subsonic potential is striking
(2.3.28).
Let us show now that both p given by (2.3.32) and the velocity field
resulting from the potential (2.3.33), are different from zero only in the
interior of Mach's cone. The assertion follows immediately if we use the
formula
d [H(x)
dx xa )j
= x(x .A) = -aH(+i
,
A # 0,1,2,...
(2.3.35)
demonstrated in the theory of distributions (see, for example, [A.5],
§2.2). Indeed, we can write:
1 H(x - s)
1
(2.3.36)
27.- (x - s)1/2 (x + s)1/2
such that the derivatives of £ will have the factor H(x - s).
From (2.3.18), it follows for w :
(2.3.37)
w(x, y, z) _ -(fl(9t + k2f3a=)£ + (f2as - f3ay)./2 +/z'£
In the sequel, we shall utilize also another expression of the component w. This follows writing
W
ia3f1
a2 - M2c,
_
ia2a3f2
al (a2 - J%f 2a1)
k2iai - ia2
oil (a2 - Af 2ai)
f3
instead of (2.3.17). Utilizing (2.3.11), one obtains:
Fx
w(x, y, z) =
f2t.1
£dx - f3(k28L - 01y) J
z
£d x.
(2.3.38)
50
LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS
The Fundamental Solutions of the Oscillatory System
2.4
2.4.1
The Determination of Pressure
The fundamental oscillatory solutions are defused by the system:
M2(pt + p=) + div v = 0
vt + yr + grad p = f d(x)exp (i wt)
(2.4.1)
lim (p, v) = 0.
z-+-oo
They will be complex. Since the system is linear, the real part of the
solutions will correspond to the case when exp (i wt) will be replaced
by coswt, and the imaginary part will correspond to the case when
exp (i wt) will be replaced by sin wt. The solutions of the system (2.4.1)
determine the perturbations produced in the uniform stream defined in
2.1, by a force having the periodic intensity f exp (i wt), applied into
origin in the three-dimensional case and uniformly on the Oz axis in
the two-dimensional case.
Obviously, the solution of the system (2.4.1) has the form
p = P(x)exp (iwt), v = V(x)exp(iwt) ,
(2.4.2)
P and V satisfying the system
AI2{iwP-}-Ps)+divV =0
iwV + V,, + grad P = f 8(x)
(2.4.3)
Urn (P, V) = 0.
s-+-oo
Applying the Fourier transform in (2.4.3), we get
A12(w-as)P=a'V, (w--al)V-aP=-if,
whence
P
_
ia- f
M2(as - w)2 - a2'
V
_
if
a
ia- f
as - w + as - w M2(as - w)2 -- a2
(2.4.4)
Applying the Fourier transform, from equation (2.2.14) we deduce:
E = -F-s [a2 - M2(as - w)2j
,
P = (f V)E.
(2.4.5)
THE FUNDAMENTAL SOLUTIONS OF THE OSCILLATORY SYSTEM
51
The pressure P will be expressed by means of the solutions (2.2.9) and
(2.2.23). In the two-dimensional subsonic case, we shall have:
P= -0(f V)[Ho2)(k1)exp(ikdlx)] ,
(2.4.6)
and in the throe-dimensional subsonic case,
P
Q)
{expJux - R)]
(2.4.7)
4r(f
k being given in (2.2.19) and R, R in (2.2.20).
In the two-diinensional supersonic case, we have:
P = -(f' V) [H(x - klyl)Jo (v x2 - k2y2)exp (-ivMx)] ,
(2.4.8)
and in the three-dimensional supersonic case,
1
P
(f 0)[H(x-Z:
21r
y2 +z2)
1
cosy x--k2( +z7)
exp(-ivMx)]
\/X- 2 - k (y +:)
(2.4.9)
v being given in (2.2.22).
Taking into account the formula (2.3), we deduce that the solution
(2.4.9) differs from zero only in the interior of Mach's cone.
The Determination of the Velocity Field
2.4.2
Since the velocity field V vanishes at -oo, we deduce like in
(2.3.12):
f
_ i
_
iai(a2-M'-a? +2alw+w2) - f
1
"
Gdx,
where
G = -f- 1a2
1
- M2a1 + 2alw + w2
Using the change ai - ca -+ aI, we get:
(
]
r Gdx.
-i a 1 w NIz a lW)2 - a2] = -ie'""d 00
(2.4.10)
(2.4.11)
If we replace al by al - w in (A.7.9) we find
ai
- w ] = e-"`H(x)b(y, z)
i
(2.4.12)
52
LINEAR AERODYNAMICS. FUNDAMENTAL. SOLUTIONS
Utilizing (A.6.9), on the basis of formulas (2.4.11) and (2.4.12), from
(2.4.4) we deduce:
V = fe-k"H(x)b(y,z) + Vcp,
(2.4.13)
where
_ -(f
0)e-k'1X
/
G dx.
(2.4.14)
f oc
We notice that the perturbation is potential, excepting the Ox axis for
x > 0, i.e. excepting the trace of the source in the uniform stream.
Let us determine now the distribution G. We notice, applying the
Fourier transform, that it is the solution of the equation
(1 - M2)Gu +
G - 2iwGx - w2G = 6(x).
(2.4.15)
The solution of this equation may be obtained like in 2.2. So, in the
two-dimensional subsonic case we find
G = _ ZI-0 Hoe) (k1) exp (iox) ,
(2.4.16)
and in the three-dimensional case
G
41rR
exp
[ik(_R)]
(2.4.17)
.
In the two-dimensional supersonic case one obtains-
G = -- 2k H(x - klyl)Jo (ui/2 - k2y2) exp (lJx)
= H(x - kEyl)g,
not
(2.4.18)
and in the three-dimensional case
G = - 2 H (x - k
y2 + --2)
x= - k (y' + z-)
cos v
X-
r
- k-(y- + -)
exp (izux)
-
not
H(x-k. Vy2 -+z2) g,
(2.4.19)
w being defined in (2.2.19).
Taking into account the behaviour of Ho2) for great values of the
argument (2.2.30), we deduce that in all cases we have:
lim (G, G=) = 0.
s----oo
(2.4.20)
THE FUNDAMENTAL SOLUTIONS OF THE OSCILLATORY SYSTEM
53
2.4.3 Other Forms of the Components V and 14
In the two-dimensional case from the component V given by (2.4.4)
one eliminates a2 by means of the identity
a2=a2-M112(al -w)2+M2(a1-w)1-ai.
Thus one obtains
if,
ai -W
ft, a2
h12(a1-W)2-a2
+ if2 (M2 -1)a, - 2M2wa1- M2w2
a, -w
M2(a1 - w)2 - a2
whence, utilizing the inversion formulas (A.6.9) and (2.4.11),
V (X, y)
{f,
,
+ f2 [(M2 -1)O + 2M2iwt7, - M2 W2] }
(e-iWX
X
J
Gdx = -f,e';v,x Gy - iw
00
J
m
-fee-i4" llI2iwG - (I - M2)Gx + w2 J X G d xl
.
J
oo
(2.4.21)
We also obtain this form using the expression of V given in (2.4.13)
and (2.4.14),
V=
`H(x)6(y, z) - f14
f/
X
x
a-wx
J
\
G d x)
-
00
Gyy, dx,
;aid eliminating Gy, with the aid of the equation (2.4.15). Indeed, in
the two-dimensional case, utilizing (2.4.20), we deduce
G., d x = H(x)d(y) - (1- it f2)Gx + 2iwG + w2 J
Gdx .
From the last taro formulas we find again (2.4.21).
For the supersonic solution, taking into account (2.4.33), with the
definition of g given in (2.4.18) we have
s
r
-
8y
Gdx=H(x-kIyI)
r
gdx,
f Ivl
r
r
(x Gdx=H(x-kjyj)OOJ x gdx,
J oo
lyi
(2.4.22)
54
LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS
such that V given by (2.4.21) is zero outside Mach's dihedron.
In a similar manner, in the three-dimensional case we deduce for the
component W
W (x, y, z) = - f ie " (G c - iw rx
G$ d x)
J oo
- f2e'11x foo G,, d xx
-he-i- 2"-(1-M2)Gx+(w2-822y)
x
Gdx,
00
(2.4.23)
We notice that using the notation:
s=k y 2 ++z2
(2.4.24)
we may write in the supersonic case
W (x, y, z) = H(x - s)w(x, y, z) ,
(jX)
(2.4.25)
where taking into account the definition of g from (2.4.19),
w = -fie
ir'"
k.ix
(.
- f3e-u.,x [2iwg+k2g+(,2_ay)jXgdxj
whence we deduce that W is different from zero only in the interior of
Mach's cone with the vertex in the origin. Indeed, utilizing the formulas
(A.3.9), (2.3.35) and noticing that s8/8y = k2y8/8s, we may write :
82yy /x Gdx=822
[H(x-s)
JT gdxJ =
00
= [H(z
- s)8(k2y/s)8,j gd= H(x - s)8J g d x
and from
GdxJ =H(x-S) {g=-iw f g2dx)
8s\\`G-iwJ
00
it follows the formula (2.4.25).
.J.
55
THE FUNDAMENTAL SOLUTIONS OF THE OSCILLATORY SYSTEM
2.4.4
The Incompressible Fluid
The (oscillatory) solutions for the incompressible fluid may be obtained considering Al = 0 in (2.4.4), or directly in the subsonic solution.
So, in the two-dimensional case we deduce from (2.4.5):
P(x, Y) =
I xf! + 02
(2 . 4 . 26 )
2
The general form of the equation (2.4.15) for M = 0 is
G = g exp (iwx),
(2.4.27)
where Og = 0. One obtains therefore:
G(x, y) =
exr (i x)
ln(x2 + y2) .
(2.4.28)
For the three-dimensional problem we have:
(f V) T ,
P(x, y, z)
with the notation r =
2.4.5
G(x, y, z)
ex (iwx)
(2.4.29)
x2 + y2 +-2.
The Fundamental Solutions In the Case M = 1
If Al = 1, E is the solution of the equation (2.4.23), and G, the
solution of the equation
Gyy + C_. - 2iwG,. - w2G = 8(z) .
(2.4.30)
C is determined like E. They have the form:
E = H(x)e, G = H(a)g.
(2.4.31)
In the two-dimensional case. e and g have the expressions:
e=-
a
.exp
(x+ T
\ g=-
2
exp
)J ,
l2 (x- (2.4.32)
with the notation utilized in (2.4.26) for a. The solution given by (2.4.5)
and (2.4.21) is
P(x, Y) = H(x)(fies + f2eg)
(2.4.33)
V (x, y) = H(x)v(x, y),
56
LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS
where
v(x, y) = - f le-iwx [gy - iw J xgy (r, y) d r]
-
Q
(2.4.34)
[2ig + w2
f2C-
g(r, y) d TJ
Ix
J0
In the three-dimensional case, utilizing (2.2.6) we deduce:
p-(
1
C = - ----- ex
4irx
iw
2x
t2
_
1
J
,
-1
4'rr
2
2
x- y+Z
iw
exp
x
2
(2.4.35)
Taking into account (2.4.5) and (2.4.23), it follows
P(x, y, z) = H(x)(fie. + f2e,, + f3e=)
(2.4.36)
W(x, y, z) = H(x)w(x, y, z),
where
-fie-Lox
w(x, y, z) =
rgz - iw
jx1
''j
x { - fee
gy d x-
//
- f3e-twx [2iwg + (w2 - ayy)
x
g dx
.
J
Q
The integrals from (2.4.37) have a strong singularity in the origin, but
they are convergent, as it results from the following calculus indicated
by V.Iftiinic:
!
/Oz
gd r
fJf
47r
bin ,
4T
hm
e\O
r -exp
!02
r
I i (br
`
fT r exp { i (6T
F
lL
r= exp (ibr) - 1
exp
2l) l d r -2\1Jff
T!
f
J
dT
1
(-i- I d r + EimQ fJ T exp f\-- i
idt.
The first limit exists because exp (-ice /r) is a bounded function. In
the second integral we perform the change of variable r = 1/t and we
integrate by parts. We obtain:
inn J T
ce/z
(_!)dr=j
exp (-iC'-t)
jt =
57
FUNDAMENTAL SOLUTIONS OF THE UNSTEADY SYSTEM 1
x
iC2
= 2 exp (x )
f
0c
2
dt
exp (-ic t) t2 .
~ ir2
The last integral is obviously convergent.
The solution (2.4.33) was given in {10.20) and the solution (2.4.36),
in [10.17]. They can be also obtained as limits of the subsonic solutions.
Indeed, for P given by (2.4.6) and (2.4.7) one utilizes (2.4.27) and
(2.4.28). A similar calculus may be performed for G given by (2.4.16)
and (2.4.17). Passing to the limit for M \ 1, we get g given by
(2.4.32) respectively g given by (2.4.35). This section was written
entirely on the basis of the papers [10.17]-[10.20].
Fundamental Solutions of the Unsteady System I
2.5
2.5.1
Fundamental Solutions
In this section we determine the fundamental solution of the system
(2.1.51), i.e. the solution of the system
pt+c2divv=0, vt+gradp= f6(t.x)
(2.5.1)
liim(p,v)=0.
00
We already know that this system determines the perturbation produced in the fluid at rest, having the density po and the pressure po, by
a force having the intensity f, applied instantaneously (at the moment
t = 0) in the origin of the system of coordinates (on the Oz axis in the
case of the two-dimensional problem).
Utilizing Duhamel's principle, we deduce that the solution of the
system (2.5.1) has the form:
(p, v) = H(t)(P, V) ,
(2.5.2)
where H(t) is Heaviside's function and (P, V) is a solution of the
system
Pt+c2divV =0,
P(0, x) = 0,
lim(P, V)
00
Vt+gradP=0
V (0, a) = f 6(x)
(2.5.3)
0.
Applying the Fourier transform, we deduce the system
(2.5.4)
58
LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS
which has to be integrated with the conditions
P(0) = 0, V (O) = f .
(2.5.5)
The solution of this system is
f)(1-coscla(tl
P=cia. f5inlcall It, V = f+ia(ia
J
2
,
such that, by inversion, we have:
P(t,x) _ -c(f . V).F-i
Csinclaltl
.
)
1C O
(2.5.6)
(1- Cos cla It 1
V (t' x) = fo(x) + V (f V ).F`'
C,,2
.
Utilizing the formulas (A.7.14) and (A.7.22), with the notation r = lxl,
we deduce the following fundamental solution for the three-dimensional
problem:
P(t,x)=-(f.V)d(ct-r)
41rt
(2.5.7)
V(t,x) = f6(x) + V(f V)
H(ct - r)
47rr
and for the two-dimensional problem,
P(t, x) = -
c
H(ct - r)
(f V) ct2-r'
21r
(2.5.8)
r) Inct+ VCr t -r
V(t,x)=f6(x)+V(f
These are the solutions of the system (2.5.3).
2.5.2
Fundamental Matrices
The solution of the system (2.5.1) has been determined for the first
time by means of matrices in (2.6]. As we shall see, in some applications
it is preferable to use fundamental matrices. We introduce therefore the
matrices:
0c200
00c20
000c2
0000
1000 ] 0000
Q= 0000 ,R- 1000 ,S= 0000
1000
0000
0000
,
(2.5.9)
FUNDAMENTAL SOLUTIONS OF THE UNSTEADY SYSTEM 1
VT
= fir,
U, V, W1, FT
59
= (0. f1, f2, f3J,
u, v, w representing the components of the vector v and fl, f2, f3, the
components of the vector f. The system (2.5.1) is written as follows:
Vt+QV=+RV,+SVL =F6(t,x)
(2.5.10)
V(t, oo) = 0.
Introducing the matrix L. such that
V .- LF,
(2.5.11)
we deduce:
Le + QL,, + RLy + SL,. = E6(t, x)
(2.5.12)
L(t, oo) = 0,
E representing the unit matrix with 4 x 4 elements. Utilizing
Duhamel's principle, we deduce that the solution of the equation (2.5.12)
has the form
L = H(t) K(t, x),
where
(2.5.13)
Kt+QKx+RKy+SKs=0
(2.5.14)
K(0, x) = EJ(x)
,
K(t, oo) = 0 .
Applying the Fourier transform to the problem (2.5.14), we deduce:
k,=.U, k(0,a) = E,
(2.5.15)
where
0
A=icr1Q+i02R+ia3S=i
C2a1 C202 C2a3
01
0
0
0
a2
0
0
0
0
a3
0
0
The solution of the problem (2.5.15) has the form:
K = Eexp(At).
(2.5.16)
It is well known that for determining the function exp (At) one may
employ two classical methods: the method of matrix functions and the
LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS
60
method of the minimal polynomial. We shall utilize herein the method
of the minimal polynomial described in 2.6.2. The eigenvalues of the
matrix A are Al = 0, \2,3 = ±i laic, the first one being multiple of
order two. One deduces that the minimal polynomial has the form
a(A - a2)(-\ - a3) = a(a2 + a2c'2) .
Comparing this form with (2.6.13), we obtain
(?p=a2=0,
ai =-a2c2
Hence, the equations (2.6.18), (2.6.1.9) and (2.6.17) become
-
9oft
2c2921
go = 0,
92(0) = 9'2(0) = 0, !f2(0)
go (0) = 1,
9i = -c12c292 + 90,
91(0) = 0 .
K has the form Ego + Agl + A-2g2. We get:
k = E+ Asmclalt + A21 - COSCktIt
C202
Clal
Using the inverse Fourier transform we obtain:
K(t,x)=E6(x)+AY-1 I
sinclalt+A2.._t
1 -c:asclaltclal
c2a2
J
where, for the three-dimensional case,
0
A
C2ar Clay c2as
ox
ay
0
0
0
0
0
0
a_
0
0
0
Utilizing the formulas (A.7.14) and (A.7.22), we deduce that the
matrix K has the form:
H(ct - r)
(2.5.17)
K(t,x) = Eb(x) + 4tAb(ct - r) + 4ac2A2
71
In the two-dimensional case we have:
K(t, x) = Eb(x) +
+
1
27rc2
1
21rc
A
H(ct - r) +
c t - r'
VC-It-
A2 H(ct-r)Inct+ r
-r
(2.5.18)
FUNDAMENTAL SOLUTIONS OF THE UNSTEADY SYSTEM 1
A=-
61
0 CZC7z c28y
0
0
ax
a, 0
0
The formulas (2.5.13), (2.5.17) and (2.5.18) give the fundamental
matrices.
2.5.3
Cauchy's Problem
We shall prove in the sequel that the solution of the problem
Vt + QVz + RVt, + SV. = F(t, x)
V(0, x) = 0,
(2.5.19)
V(t, oo) = 0
(2.5.20)
is
c
V(t,x)=1 K*Fdt,
(2.5.21)
0
where
K*F= f 3K(t-r,x-4)F(r,4)d4=f K(r,4)F(t-r,x-4)d4
is a convolution.
Indeed, taking into account (2.5.14), we have:
Vt=K*FLt +
t
f K,*Fdr,
0
t
QVz+RVv+SVs=J (QKx+RKv+SK=)*Fdt,
0
V +QVs+RVy+SVz
F(t,x).
fR3
The conditions (2.5.20) are obviously verified, because K(t, oo) = 0.
Taking into account (2.5.17) and (2.5.18), with the notation p =
_ fit', it follows for the solution of the problem (2.5.19), in the three-
62
LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS
dimensional case,
V(t,x)
.fu F(-r,x)d-r +
4c A
r
c
Jr3 F(t-
-r, x - t)6(cr - p) d t +
, A2 I d r fa F (t-
irC"
-r,x - ls)
3
H(cr-P)d
P
(2.5.22)
and in the two-dimensional case,
r1
V (t, x)
= foF(r,x)dr+2xc AI
u
-T,x~4)
H(crr _
I dTI2F(te
p) d4+2
VR-T P2
A2
p2dt.
-r,x-E)H(cr-P)III Cr+
P
(2.5.23)
2.5.4
The Perturbation Produced by a Mobile Source
We shall apply the above formulas for determining the perturbation
produced in the fluid at rest (having the density po and the pressure po)
by a force of intensity f . whose application point moves uniformly, with
the velocity v, in the direction of the Ox axis and in the negative sense.
This problem, also considered in [2.6), is important in aerodynamics
because it models the leading edge of an uniformly moving airplane in
the air at rest.
The solution is obtained from (2.5.22) replacing
F(t, x) = Fo6(x + vt)6(y, x),
(2.5.24)
63
FUNDAMENTAL SOLUTIONS OF THE UNSTEADY SYSTEM 1
where Fo' = (0, fl, f2, f3). One finds
V(t,x)=FDH(x,+vt)o(y,a)+4I
R,)
AFD.I
D
t
l A2Fo f
+zirc-
H(
dr+
b(cr
_
RT) d
(2.5.25)
where
R, _
x + vt - vr)2 + y2 + z2 .
(2.5.26)
For calculating these integrals we have to study the behaviour of the
function
h(r) = c-r - R,r ,
(2.5.27)
on the interval (0, oo). We shall proceed like in 2.2.4. Because h(O) < 0
and h(oo) = (c2 - v2)oo. we have to separate the subsonic flow case
(v < c) from the supersonic flow case (v > c). The zeros of the function
h(r) are given by the formula
(c2
where
- v2)r+ = -v(x + vt) ± R,
R = [(x + vt)2c2 + (c2 - v2)(y2 + z2), 1/2
(2.5.28)
.
(2.5.29)
In the subsonic case, only r+ is positive. If the velocity of the source
is supersonic (v > c), the function h(r) has two zeros (r+ < 'r-) if
/to > 0 and no zero if !to < 0. We denoted ho = h(ro), ro representing
the zero of the derivative of the function h,
vro=x+vt+
cr0
Vv-
(2.5.30)
It results,
vho = c(x + vt) -
(v2 - c2)(y2 + z2).
(2.5.31)
Utilizing the relation crfh'(r ) = ±R which may be verified directly,
and the formula (2.2.38), we deduce
f
' 6(cr - R')
T
0
dr =
cR-1 H(t -,r+),
if v < c,
cR-1JH(t-r+)+H(t-r_)J, if v>candho>0,
10,
if v > c and ho < 0.
(2.5.32)
64
LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS
In the same time, taking into account the definition of Heaviside's, function we have:
JtH(CRr) dRr
v
- v In c(x +
vt)
c(x + vt) + R
R,
if(v<c)U(v>c,ho>0,t<r_),
if v > c, ho > 0, t >T-,
-;-
if v>c,ho<0.
0,
(2.5.33)
Now the solution (2.5.25) is explicit. The solution of this problem
may be also obtained directly (without using the formula (2.5.22)) [1.10].
The interpretation of the solution is similar to the interpretation from
(2.2.4).
Fundamental Solutions of the Unsteady System II
2.6
2.6.1
The Fundamental Matrices
In this section we determine, using the papers [2.71 and [2.9], the
fundamental solutions of the system (2.1.26), (2.5.27), i.e. the solutions
of the system
M2(pt+p=)+divv=0
vi + v;r + grad p = f a(t, x)
(2.6.1)
limp, v) = 0.
00
Introducing the matrices
Q=
1
1
0
0
0
0
00
10 '
01
000M-2
00x1'20
1 M'200
R=
00
0
10
0
00 0
0
0
0
_
' S
000 0
000 0
100 0
and the matrices V and F defined in (2.5.9), the system (2.6.1) may
be written like in (2.5.10). We shall proceed like in the previous section.
Considering
V = H(t)K(t, x)F,
(2.6.2)
65
FUNDAMENTAL SOLUTIONS OF THE UNSTEADY SYSTEM II
one obtains the following equation for k :
Kt = AK,
where
K(0, a) = E,
(2.6.3)
al a1M-2 02M -2 a3M-2
al
A=i
a2
CM
1
0
0
0
0
1
0
0
1
The roots of the characteristic polynomial are:
Al = ial,
A2,3 = i (al ± lalM-1),
(2.6.4)
the first root being double. One verifies that the minimal polynomial
has the form
(A-A1)(A-A2)(A-,13) = (A-ial)3+a21tf-2(a-ia1).
Comparing with (2.6.13) it results:
a2 = 3i a1, al = 3a2 - 02M-2, ao = i a1(a2M-2 -
a2).
(2.6.5)
k has the form
K = Ego(t) + Ag1(t) + A2g2(t) ,
(2.6.6)
the functions 90, 91, 92 being determined by the equations
gg2 - a292 - a19o2 - aog2 = 0, 92(0) = g2(0) = 0, 92(0) = 1,
(2.6.7)
goo = ao92, 90(0) = 1,
901 = a1.92 + go, 91(0) = 0.
Determining these functions, .%v find:
go = I 1 -
ioM
lal t
la) sin M
M2
- a?a2
/
I
1 - o0c1
lalt`1
M JJ
/
m sin lal t - 2ia?M2
a2
1 -cos lal t J
91
=
9'2
(al 1
exp(ialt).
= a/ - aft
_
M
M
exp(i alt),
exP(i a1 t),
1112
1
(2.6.8)
LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS
66
In the three-dimensional case one utilizes the formulas (A.7.15) and
(A.7.23). From (2.6.6) it follows the fundamental matrix:
K(t, x) = Eb(2- t)b(y, z) + 4
+
(E
t
R)+
(EOY +
+2A8,.+A')11(tA1R
-R}
(2.6.9)
where R = V(2--- t)2 + YT-A- z2 and
A = -
AI'28y M 2(9:
8r A1'2O
0.
a2
0
8=
0
0
0
0
Sr
0
ay
In the two-dimensional case one utilizes the formulas (A.7.15) and
(A.7.23). From (2.6.6) one obtains the following fundamental matrix:
A12
K(t, x) = Eb(x - t, y) + -(EO + A)
2"
+
where R=
(
A122
2-r
H(t -- AM)
jt72
+
A12R2
(O_ + 2A87 + A2}H(t - AIR) In t i
+_
t2 - A12
nip
(2.6.10)
- t)- y2 and
8?
A
AI'28,. M-28y
8=
0.1
0
8y
0
8t
lit [1.10] and [2.7] one gives some applications which are important
for aerodynamics. We mention the mobile source on the direction of the
unperturbed stream and the mobile source on the direction perpendicular to the same stream.
2.6.2
The Method of the Minimal Polynomial
In the sequel we present the method utilized for determining the
function exp (At). Let A be a matrix with n x n elements and let
At, ... , Ak(k < n) be its eigenvalues, i.e. the roots of the characteristic
polynomial
A(A) = det (AE - A)
.
FUNDAMENTAL SOLUTIONS OF TIIE UNSTEADY SYSTEM II
67
Obviously,
0(a) = (A - A1)m1
... ()1 - A )mk ,
(2.6.11)
where ml + ... + ink = n.
The minimal polynomial associated to the matrix A is by definition
the polynomial with the smallest degree which has the property
P(A) = 0.
(2.6.12)
Let
-... - a1,\ - ao
P(A) = am be this polynomial. Obviously, m < n, because .(A) = 0.
am-la,n-I
(2.6.13)
From (2.6.12) and (2.6.13) it results
m-1
Am=
EajA''
j=0
and step by step
m-1
Am+p
= E aln) A3 ,
p = 0, 1, 2, .. .
jO
Taking into account that the exponential is a power series, we deduce
that there exist the functions
go(t), ... , g.-1(t) .
such that
exp(At) = goE+g1A+...+gm_1A"-1
(2.6.14)
We have:
A exp (At) = goA +... +
+9,n-1(ao E + al A + ... + a,,,-1 Am-1 ) ,
gm_2A'n-1+
( 2. 6 .15)
On the other hand,
A exp(At) = (d/(It) exp(At) = g(E+g'A+...+g',,,-lAm-1. (2.6.16)
Since P(A) is unique, from (2.6.15) and (2.6.16) we obtain:
90=ao9m-1791 =a19+n-1+go,...,9in_1 =a+n-19m-1+9m-2 (2.6.17)
68
LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS
We deduce that 9m_1 is the solution of the differential equation
1rn?1
= am-1g(n1 1' -#- ... + olgVn- 1 + (109,n-1 ,
(2.6.18)
with the following initial conditions:
g,,
1(0) = girl -1(0) = ... = girl1(0) _ 0, g
(in-1)
1' (0) = 1 .
(2.6.19)
The functions go, ... , gin-2 are determined from (2.6.17) with the
cu conditions
90(0) = 1,91(0) = 0,---,9,,.-2(0) = 0-
(2.6.20)
imposed by (2.6.14).
In fact, for determining the minimal polynomial, one takes into ac-
count that it is a divisor of the characteristic polynomial, hence it has
the form:
P(A) = (A - A0," (A - A2)u'2 ... (A - Ak)' =
= A"-on,-1Arn-1 -...-ao,
(2.6.21)
nj < mj and n1 + n2 + ... + nk = in. To conclude, we
try with divisors having the form (2.6.21), beginning with the simplest
where
(ill = n2 = ... = nk = 1) in order to satisfy the equation P(A) = 0.
from the University of Bucharest
We mention that dr. *t.
has communicated us this method.
Chapter 3
The Infinite Span Airfoil in Subsonic Flow
The Airfoil in the Unlimited Fluid
3.1
3.1.1
The Statement of the Problem
In this subsection we determine the perturbation produced by a thin
infinite cylindrical body in a subsonic uniform stream having the velocity UQ,i, the pressure pQ, and the density p,,. We assume that
the generators of the cylinder are perpendicular on the velocity of the
unperturbed stream which is parallel to the xOy reference plane. The
cylindrical bodies are bodies with constant cross-section and the infinite
cylinders are cylinders which are long enough, such that the effects of
the end conditions over the flow in the rOy plane may be neglected.
Under these assumptions ( the conditions which determine the perturbation are not varying in time and they are the same in every plane
parallel to xOy), the perturbed flows of the fluid will be stationary and
plane.
Ua
Y
Fig. 3.1.1.
% Te utilize the variables r , y, z introduced in (2.1.1) and the per-
turbations p and v defined by (2.1.3). In figure 3.1.1 one presents
the profile determined by the intersection of the cylindric body with the
xOy plane. We denote by
y = la(x) ± hl(x)
(3.1.1)
the equations of the curves C+ (the upper surface) and C_ (the
70
THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW
lower surface). The curve having the equation y = h(z) gives the so
called skeleton of the profile. For imposing the boundary conditions we
use the projection of the profile onto the Ox axis (the direction of the
unperturbed stream). Defining Lo from (2.1.1) as the half length of this
projection and taking the origin of the axes of coordinates in the middle
of the projection, it follows that the functions h(x) and h1(x) have
to be defined on the segment [-1,+1]. We assume that these functions
have continuous first order derivatives and that the derivative h(x)
satisfies Holder's condition (see Appendix B). The cylinder is thin if
h(x) = Fh(x),h1(x) = Eh1(x)
e representing a small parameter.
We notice that whatever would be the equations of the curves C+
and C_, they may be written like in (3.1.1). Indeed, if the equations
would be y = h±(x) = ehf(x), they might take the form (3.1.1) putting
h+(x) + h_ (x) = 2h(x), h+(x) - h- (x) = 2h1(x).
Taking into account the equations (3.1.1) the following conditions
have to be satisfied
v(x, ±0) = h(x) ± 14 (x), Ix I < 1
(3.1.2)
as it follows from (2.1.32).
3.1.2 A Classical Method
The classical methods for solving this problem rely on the assumption
that the perturbation is potential. We have therefore
v = grad Sp(x, y).
(3.1.3)
The potential p which is defined in the exterior of the segment [-1, +11
from the xOy plane, is the solution of the equation
(3.1.4)
P"P:: +'p1,s, = 0
with the boundary conditions
jpi,(x, f0) = h'(x) ± h4 (x) ,
JxJ< 1
(3.1.5)
and the condition at infinity
z
lim00(P_, spy) = 0
(3.1.6)
71
THE AIRFOIL IN THE UNLIMITED FLUID
which follows from (2.1.28).
Using Clauert's transformation x, y -, X, Y:
X=x,
Y=fiy,
(3.1.7)
one obtains from (3.1.3) and (3.1.4)
u=six -- U,
v =A(PY=QV
sOxx +'PYY = 0
(3.1.8)
(3.1.9)
in the exterior of the segment [-1,+11 from the XOY plane. The
boundary conditions (3.1.5) and the conditions at infinity (3.1.6) become
V (X, fO) = h'(X) f hi (X),
1X 1 < 1
lira (U,V)=0.
X -»- cc
(3.1.10)
The problem was solved by many authors (see for example Birnbaum
[3.51, [3.61, Sohngen [A.34), Sedov [1.38], p.47, Iacob [1.21), p.661), by
means of one of the following two methods: reduction of the problem to
an ntegral equation which is then solved or the reduction of the problem
to a boundary value problem in the complex plane. We also mention
the methods relying on the expansion of the solution in a Fburier series.
(Glauert [3.211 and Atei. inger (3.491).
In the sequel we give one of the most natural methods for reducing
the problem to an integral equation. We represent the holomorphic
function W = U - iV in the Z = X + iY plane with the cut [-1, +1]
on the real xis, by means of the Cauchy integral
W(Z)=-L
r -f(t)+I t(t)dt,
(3.1.11)
where f(t) and fl(t) are unknown real Holder functions. It is easy
to understand the significance of this representation. W(Z) Is the sum
of the complex velocity determined by a continuous superposition of
squrm having the intensity fm on the segment [-1, +1) and the
complex velocity determined by a continuous superposition of vortices
having the circulation f, on the same segment. In fact, the integral
containing f may also be regarded as a continuous superposition of
doublets on the segment [-1, +1] from the (Z) plane.
Obviously; W(Z) is a holomorphic function in the exterior of the
segment [-1, +11 and it vanishes for Z -+ oo. Hence, U and V
THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW
72
satisfy the equation (3.1.9) and the condition (3.1.10). In order to prove
this we pass to the limit considering Z - X ;k i 0, where X E (-1, + 1).
Using Pleme(j's formulas (B.3.1) and separating the real part from the
imaginary one we obtain:
2f(X)+2- /
U(X,±0)=
V(X,±0) =
2ft(X) -
t
2zr,
it()dt
(3.1.12)
tfit) dt.
(3.1.13)
The mark"' " indicates the principal value in Cauchy's sense (Appendix
B). From (2.1.38), (3.1.8) and (3.1.12) it follows
p(x, +0) - p(x, -0) = U(X, -0) - U(X, +0) = f (x).
(3.1.14)
This relation puts into evidence the significance of the function f (x).
It will be fundamental for the determination of the lift and moment
coefficients.
Imposing the conditions (3.1.10) in (3.1.13), adding and subtracting
the relations just obtained, we find
Q
T
/ tdt
_
= -2h' ( X)
JX i < 1 ,
Oft(X) = -2h4(X) JXi < 1.
(3 . 1 . 15)
(3.1.16)
The relation (3.1.16) determines the function Of, and (3.1.15) represents an integral singular equation for the determination of the function
Of. It is the well known equation of thin profiles which was put into evidence by Birnbaum in 1923 and solved by Sohngen in 1939. Its solution,
given in Appendix C, will be written in (3.1.25).
We notice that the thickness of the profile is ensured by the distribution of sources (ht = 0
ft = 0) and the discontinuity of
the pressure (hence the lift and moment coefficients) is ensured by the
distribution of vortices.
3.1.3
The Fundamental Solutions Method
There is no physical justification for replacing the wing by a continuous distribution of sources and vortices. It is more natural to replace
the wing by a continuous distribution of forces and to determine the
73
THE AIRFOIL IN THE UNLIMITED FLUID
intensity of these forces, in order to obtain the same action against the
fluid like in the case of the wing. Indeed, the fluid and the wing are
two interacting systems. If we want to determine the motion of the first
subsystem, we have to replace the action of the second subsystem by a
distribution of forces. This is the basic idea of the fundamental solutions
method.
We shall replace the wing by a continuous distribution of forces f =
(fl, f) on the segment [+1, -1], having the intensity f a priori
unknown (the intensity will depend on the point). In fact, for obtaining
a plane perturbation, the forces have to be uniformly applied on parallels
to the Oz axis, intersecting the xOy plane in the points of the segment
[-1,+11. Taking into account the formulas (2.3.23), it will result the
following perturbation in the fluid:
p(x,y) =
(x, v) _
1
27r#I
f
+1 xof1()
X02
+ #2y2
(3.1.17)
/+t xof(t) -yfi(E)d
21r ./
1
xo + 82y2
In view of imposing the boundary conditions (3.1.2) we have to calculate
ylimo v(x, y) for jxj < 1. Denoting by v(x, ±0) these limits, we have
u(x, f0) =
Q
lim
2ir y--±o
f+1 xof W 1
(3.1.18)
xo + 02y2
The first integral represents the tangential derivative of a simple layer
potential and the second the normal derivative of the same potential.
Although the formulas which give the limits of these derivatives are
known, we prefer to calculate them directly.
If we pass to the limit in the first integral making y = 0, we cannot
simplify with x0 because for
= x it is zero. We shall divide this
integral in three parts. Since for a e small enough, we may approximate
f
on the segment (x - e, x + E) with f (x), we deduce performing
the substitution - x = u,
:+C xoM)
_
#2y2
+`
ud u
I-C U2 +Qty 2
74
THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW
the integrand being an odd function. It remains
+1
lim
xof W
d+
r:- E f
d- -lim
E-.0 l J
(
V-±0 J1 xo2+
1
zo
1
r1 f(_)
Ji x-4
f
d
J
=
d
excepting
zero,
The second limit from (3.1.18) is
f1
Jx.c xo
the
(3.1.19)
interval
(x - s, x + e) where the integrand is infinite for y = 0 and f = x.
Approximating again f j (C) by f, (x), or applying the mean formula,
since f i is integrable and multiplied by an integrable factor having a
constant sign, we get:
Yli
m
J
lm Y
114
+a
fi(x)vuy f,
+
uZ
v(x,
0)
°J
y
-Z0'
f
±f1(x) -
d
(_)
Imposing the boundary conditions (3.1.2), adding and subtracting, we
get:
Q
f
(4)
d _ -2h'(x),
(3.1.21)
(x).
(3.1.22)
fi(x) _
We also have
P(x, fU) =
fi(e
f2f(x) + 2WQ
(3.1.23)
such that f (x) has the same significance like in (3.1.14). The sntegral
equations (3.1.15) and (3.1.21) coincide.
In fact, the representation (3.1.11) may also be obtained from the
fundamental solution. This representation is valid for the points which
do not belong to the semi-axis Ox (x > 1). Indeed in these points we
have u = -p, such that
u- v=
1
1
2
i -f (S) + i P
J_i
-Z
.11(x)
df
'
(3.1.24)
75
THE AIRFOIL IN THE UNLIMITED FLUID
where Z = x + i fly. We see from (3.1.16) and (3.1.22) that instead
of Of, from (3.1.11) we put here fl. The equivalence just established
shows that the formula (3.1.11) is not valid on Ox (x > 1). On this
half-line the flow is not irrotational.
We mention finally that the proof of Plemelj's formulas (which are
necessary for obtaining the integral equation (3.1.15)) is anyway more
difficult than the proof of the formulas (3.1.19) and (3.1.20).
3.1.4 The Function f (x). The Complex Velocity in the Fluid
The solution (bounded in the trailing edge) of the integral equations
(3.1.15) and (3.1.21) are obtained from the formula (C.1.9). We have
therefore:
(x)=
Of
2 V -x' 44 /iTih'(t)d t.
tt x
l+xf
(3.1.25)
1
Obviously, for the symmetric profile (h = 0) it results f = 0.
One obtains the complex velocity substituting (3.1.16) and (3.1.25)
in (3.1.11). Since, taking into account the formulas (B.5.5) and (B.5.7),
we deduce
_
dt
11-t
1+t (t-Z)(s-t)
+1
1
+1
1
t
1
a
1
Z+t(tZ+st)dt=
Z
1
s-Z Z+1'
it follows
W(Z)
h'
I
-- A J-11 t'(Zdt+
i
+zr(3
Z
1
Z+1
f,
+1
(3.1.26)
1+ t h` (t)
1-tt-Z d t
For the incompressible fluid (f3 = 1) one obtains the formula
(t)
u-iv=-1AJf 1 hit-z
dt+
i
+
-
z_ i
+1 J
+l /rTih (t)
1
(3.1.27)
1-tt-zdt'
where z = x+iy. The formula (3.1.27) was given for the first time by Sedov [1.38], p.51 and deduced in a different manner by lacob [1.21), p.664,
76
THE INFINL'I'E SPAIN AIRFOIL IN SUBSONIC FLOW
which solved the problem (3.1.9), (3.1.10) reducing it to a boundary
value problem in the complex plane. It is not simpler to solve the bound-
ary value problem than solving the singular integral equation (C.1.1).
Moreover, the complex velocity field is not of much interest in aerodynamics. It is utilized only for determining the jump of the pressure on
the profile (which is calculated directly in the framework of the method
of the integral equations).
3.1.5
The Calculation of the Aerodynamic Action
In some papers the aerodynamic action is calculated by means of a
curvilinear integral on the contour of the profile. This calculation is
wrong for contours with angular points. A correct calculation is performed using a control contour (surface in the three-dimensional case)
surrounding the profile (wing). We shall perform this calculation where
it will be absolutely necessary. Here we shall give a simple calculation,
observing that in the first approximation, the action of the fluid comes
from the jump of the pressure pl(xl, -0) - pi(xi, +0) = Dpi 1, which
gives the lifting force, parallel to the Oy axis. Taking into account the
formulas (2.1.1) and (2.1.3) on the unity of length of the cylinder, it
follows the lifting force
L= I
(IPi I dxI = P.UULo
J
I i [!pO d x
(3.1.28)
and the following momentcalculated with respect to the point x°:
MI
=fJ to (xi - x°) x pt ?d xi ,
where j represents the versor of the Oy axis. We obtain MI = Mk,
where
rfi
M = poU020Lo J
(r - x°)OpOdx.
(3.1.29)
We denoted
Bp0 = P(x, -0) - p(x, +0)
(3.1.30)
Instead of the dimensional quantities L and M, it is preferable to
use the dimensionless quantities CL and chl, named the lift coefficient
respectively the moment coefficient. These aerodynamic coefficients are
defined by the formulas:
L
M
CL,
= (l/2)PooU2 (2Lo) '
cA'
= (1/2)p.U,
(3.1.31)
77
ME AIRFOIL IN THE UNLIMITED FLUID
It is preferable to use the dimensionless aerodynamic coefficients because
the numerical calculations are performed for dimensionless quantities.
From (3.1.14), (3.1.28)-(3.1.30), and (3.1.31) it results
CL
=-
f
+1
f (x)d x, cAt =
i
-12
J
'
(x - x°)f (x)d x .
(3.1.32)
i
Finally, utilizing the solution (3.1.25), (B.5.4) and (B.6.9) we get
cL-
h(3.1.33)
r+i
JY
CAf=-i13 J-1
f V
±tth'(t)dt- 1
2x°cL.
1
(3.1.34)
In the case of the profiles which are symmetric with respect to the Ox
axis, (h = 0), cL and cM vanish.
Obviously we cannot use the method for calculating the drag because
it has the order e2
3.1.6
Examples
The flat plate. For the flat plate having the angle of attack
(fig. 3.1.2a)), the equation (3.1.1) is
E
y = -xtgE = -Ex. It follows
h(x) = -.ex whence
CL=
27re
ire
CM= -(x° + 2)A
1
(3.1.35)
These formulas were given for the first time by Glauert (1928) and
Prandtl (1930). For the incompressible fluid (M = 0) one obtains
CL = 27rE,
1l
cAf = - z° + 2) Ire
(3.1.36)
We notice that CL is increasing (c,%f is decreasing) for M / 1 (fig.
3.1.3) (the lift is increasing because of the compressibility). In the vicin-
ity of M = 1 (starting approximately with Al = 0.8) the lift has very
great values, in contradiction with the reality. We deduce that in the
vicinity of M = 1 the linear theory is not valid any longer. Therefore,
for the transonic flow (Chapter-9) we shall utilize other equations. From
(3.1.25) one obtains the jump of the pressure
THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW
78
U.
b)
r
c)
Fig. 3.1.2.
CI
2ttE
M
0.8
1
Fig. 3.1.3.
/if = -2e
1
V +
x
(3.1.37)
The parabolic profile (fig. 3.1.2b)). This profile is obtained for
h(x) = e (I - x2), h1 = 0. Utilizing (B.6.9) we find
CL =
Ze
73
°f=-
7r
Cm
1 - xa
(3.1.38)
with the same interpretation for cG.
The S-profile (fig. 3.1.2c)). For h(x) = e(x - x3) and hl (x) = 0,
we obtain
CL
- x°)
A , CH = 4,6 (
j
,
f = V-x + Tx
1+x
--X 2 (3.1.39)
1
The profile with thickness, having the shape of an elliptic
sector For a profile having the shape of a sector bounded by two arcs
THE AIRFOIL IN THE UNLIMITED FLUID
79
of ellipse having the small semi-axes e2 < ei (fig. 3.1.4), we have
h+(x) _ -62 VI -X
h_(x)
,
x7,
(3.1.40)
2h(x) = -(i, + s2) 1 - x
2h1(x) = (El -- 62W1 -X*
,
.
uw
Fig. 3.1.4.
For f, and f ,
taking into account the representation (3.1.16) and
(3.1.25) we deduce
Aft = (F1 - E2)
lx x
v-
3f =
,
-
dt
t
1-t 1-t t -x
X
e2
1
?r
+xI
f'
1-x
t-x
where
1+t
£l
x
dt +
1
1-x
(3.1.41)
+1 dt
1
1-t
Per forming the change of variable 1 -- t = u and taking into account
(D.2.3), we get
7* +l 1ddu
1
,/
1- t = `r2
u = in 2.
(3.1.42)
Utilizing (B.5.8) we get
1(3f =
El i Ex
/
I
\
x .
In 2 + x In l + x)
(3.1.43)
Ale also have
£1 +F2
2-
dt
_ 61+12(2--1n2),
(3.1.44)
CMS _ -
£12x°(2
- In 2).
2/3
The action of the fluid is equivalent to a lifting force passing through
the origin.
80
THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW
3.1.7 The General Case
It is well known that the Chebyshev polynomialsof first kind
T,,(t) = cos(n arccost)
(3.1.45)
constitute a basis on the interval (-1, +1). We shall consider the series
expansion of h(t) in this basis:
h'(t) = E an cos(n arccos t).
(3.1.46)
t=ooso
(3.1.47)
Putting
it follows
co
h'(cos a) _ E an cos no
n-U
whence
j" h'(cos8)d6,
ao
an = 2
0
7r
7T
jh1(cos0)cosn0d0,
(3.1.48)
n= 1,2,...
Substituting (3.1.46) in (3.1.33) and (3.1.34) and performing the
change of variable (3.1.47), one obtains
2n
cf,
ao +
al
2
a2\
7r
V cdt = - a ao + al + 2 J.
(3.1.49)
Theses formulas were given by Homentcovschi in (A.201.
It is important to calculate the distribution of the jump of the pressure on the profile. Using the change of variable (3.1.47) and Glauert's
formula (B.6.6) we deduce
+l 1 + t h'(t)
°Q
(1 + cos a) cos no-
J
1-tt-xdt=EanJ
n-o
coso-cos9
U
=
(3.1.50)
= 7r cot
8 °O
an sin n9,
x = coo 8.
-o
Replacing in (3.1.25) it follows:
pf(x)=2Eansinn8
00
n=0
It is Glauert who proposed this type of solution.
(3.1.51)
THE AIRFOIL IN THE UNLIMITED FLUID
3.1.8
81
Numerical Integrations
In the case of an arbitrary profile, for calculating the integrals (3.1.33)
and (3.1.34) one may employ quadrature formulas (F.2.24). Using the
notations
_
tQ-cos
2a-1 a=1,...,n
2n + 1'
(3.1.52)
,
one obtains
n
> (1 +
41r
13(2n + 1)
cL
(3.1.53)
CM
-
n
2n
-T3
1
a
(1 + ta)tali (tQ) - 2 CL.
(2n -+I) a-
1
For calculating the integral from (3.1.25) one utilizes the formula
(F.3.1). In the collocation points
2i7r
xj =
t = 1,...,n,
Cos Fn + 1 ,
(3.1.54)
it results
f(T,)
4
p(2n
+'
1)
X. E
1
+
t
Q=1 Q
- x1 h (tQ)
(3.1.55)
The results obtained with these formulas are much more accurate than
the results obtained with other methods.
3.1.9
The Integration of the Thin Airfoil Equation with the
Aid of Gauss-type Quadrature Formulas
We may use the quadrature formulas from Appendix F to create and
extremely efficient method for determining the solution of the integral
equation (3.1.15). There were made many such attempts in various
papers but nowhere one can find the good solution because there is not
prescribed the behaviour of the solution in the points ±1. We know
from Appendix C that the solution of the above mentioned equation
depends on the behaviour imposed in ±1. The solution satisfying the
Kutta-Joukovsky condition in the trailing edge is
/3f (t) _ -E
+ t F(t)
1
(3.1.56)
82
THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW
Using (F.2.19) and posing h = -Eh, we reduce the equation (3.1.15) to
the algebraic system
n
Ai,F, =hl, j = 1,...,n,
(3.1.57)
a=1
where Fa = F(ta),
h1
= h (xj), A,a =
t-
2n + I '
2n + 1 Qa-
z1 =
x,'
(3.1.58)
+ 17r'
In (3.1.57) we have a linear algebraic system with n equations
for n unknowns Fl, ... , Fn. We can create a computer code for
Cos
solving the system. The coefficients AJa, the weight points to and the
collocation points xj are the same for every profile. For a given profiles
we have to change only the column with the elements h1, ... , K. After
determining the unknowns Fa, the lift and moment coefficients result
from the formulas (3.1.32) and (F.2.18).
CL = - +1 f(t)d t = F
t F(t)d t =
j_1I
l+t
Q
(3.1.59)
1
cl=-Z J
+1
1
WE
tf(t)dt=-2ENA1,
where
n
ta)F0,
NL
a=1
(3.1.60)
a=I
For verifying the method we utilized the analytic solution for the
flat plate (3.1.37). From (3.1.56), it follows that for hJ = 1 we must
have F = 2. The results obtained with the numerical method described
above, with n = 20 gave for F. values situated between 0.999 and
2.001 and for NL and NM the value 0.999. Hence the method is
extremely efficient [3.12].
3.2
3.2.1
The Airfoil in Ground Effects
The Integral Equation
When an airplane is landing or taking off we have to take into account
the ground effects. Some of the first papers devoted to these effects belong to Tomotika and his fellow-workers (3.44], [3.45]. We have also to
83
THE AIRFOIL. IN GROUND EFFECTS
mention the papers of Pancenkov [1.33], [1.34], [3.36], and especially the
papers of Plotkin and his fellow-workers [3.37]-(3.39}, where one gives
the integral equation of thin profiles in ground effects and one proposes
approximate solutions. The is considered a small parameter. Widnall
and Barows (5.37J and Tuck [3.46] used asymptotic methods for investigating the problem. In fact one encountered two small parameters: the
arrow of the airfoil and the distance from the airfoil to the ground. The
fluid was considered incompressible.
In the sequel, following 13.131, we shall utilize the method of funda-
mental solutions, in order to obtain the integral equation for the compressible fluid and for the airfoil with thickness. The small parameter is
the arrow of the airfoil. In case that the distance from the airfoil to the
ground is also small, we have to elaborate a new theory.
We use the notations from the previous subsection. We denote by
a/2 the distance from the airfoil to the ground (fig. 3.2.1). The perturbation has to satisfy the following boundary conditions:
v(x, f0) = h(x) ± hI (x),
v(x, -a/2) = 0
U.
Ix1 < I.
(3.2.1)
- oo < x < oo .
(3.2.2)
-I
+1
x
T a/2
an
-I
---
- -
+1
Fig. 3.2.L.
According to the method of fundamental solutions, we have to replace the airfoil by a continuous distribution of forces (ft,
on the
segment [-1, +1] from the y = 0 axis. For satisfying the boundary
condition (3.2.2) we shall consider a symmetric distribution of forces
(fl, -f
on the symmetric segment (-1, +11 of the y = -a axis
and we shall determine the intensity of the distributions from the condition (3.2.1). Taking into account that the perturbation produced within
the uniform stream having the velocity U,i by the force (fl, f2) is
(2.3.23), it follows that the two distributions will determine in the fluid
84
THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW
the perturbation
1 J-tf
+t
P(-T' y) =
+
v(x, y) =
1
xoft W + J32yf (t)
x02 +,82y2
f
+t
xofi(Q - 132(y + a)f (C)
+ M(y + ;)-2
A f+t z0f(0 - bf1(0 d
xa +U
1
-
(3.2.3)
1 r+t xofW + (y + a)ft (_)
21r
x0#2(N+a)2
1
f and f j
having to be determined. One easily verifies that this
representation satisfies the condition (3.2.2). We denoted xo = x - .
Only the first integrals become singular. when passing to limit for
y
±0, Using the formulas (3.1.19) and (3.1.20), we deduce
p(x, +0) - p(r., -0) = f (x) ,
f(o d t-
t
v(x, f0)
_
2ft(x)
+ 27r
(3.2.4)
,
(3.2.5)
+1
io+m
11-r
where in = fla.
adding, we get:
Imposing the conditions (3.2.1), subtracting and
ft(x) _ -2hi(x)
ar
FI fro)
2l
jxi < 1
t1 oo+ in
dt = H(x),
(3.2.6)
(3.2.7)
where
H(x) = h'(x) -
A ,l
+t,,,2d
1
xo
(3.2.8)
The equation (3.2.7) is the integral equation In Appendix C it was
called the generalized equation of the thin airfoil. For the incompressible
fluid (/3 = 1) and for the airfoil without thickness (ht = 0) it coincides
with the equation given by Pancenkov and Plotkin.
85
THE AIRFOIL IN GROUND EFFECTS
3.2.2
A Numerical Method
The equation (3.2.7) is obviously a singular integral equation. As we
have shown in Appendix C, it may be reduced to a Fredholm equation,
but the problem remains still unsolved because there are not available
general methods for solving this type of equations (excepting the method
of successive approximations). As we have already shown in 13.131, the
equation (3.2.7) may be solved numerically utilizing the quadrature formulas from Appendix F. Looking for the solutions of (3.2.7) having the
form (3.1.56), putting H = -e and using (F.2.18) and (F.2.19), one
obtains the linear algebraic system
n
7 1j
.
(3.2.9)
a=1
where FQ = F(t0),
T-1 j = H(xj),
B ra
1
(t° - 1)(xj - t0)
2n + I (xj - tu)2 + m2
(3.2.10)
Aj,,, t,, and xj being given in (3.1.58).
From the system (3.2.9) we determine the unknowns F1,. .. , F,,.
Now the lift and moment coefficients (3.1.32) (for x° = 0) will be
obtained from the formulas
CL
J_
!+1
CAI
2{3
I.
(3.2.11)
_
+
t tF(t)d t
2 Nit ,
where we have utilized the notations (3.1.60).
One may write computer programs for solving the system (3.2.9).
The coefficients Aja and Bj0 do not depend on the shape of the
airfoil, such that the program may be utilized for every airfoil. One
changes only the matrix with one column Wj .
3.2.3
The Flat Plate
In the case of the flat plate with the angle of attack
we have
77j = 1. We solve the system (3.2.9) and then we determine NL and
THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW
86
NM with the formulas (3.1.60). We write the aerodynamic coefficients
(3.2.11) as follows
CL = cL NL,
(3.2.12)
CAf =
where c' and cM are the coefficients for the free stream (a = oo).
5.
NL1
4
3
2.
1.0.6
Mp0
}
1.
0.0
4.0
2.0
6.0
8.0
0.0
4.0
2.0
6.0
8.0
a
a
a)
b)
Fig. 3.2.2.
The variation of NL and NAf versus a is shown in figures
3.2.2a) and 3.2.2b). The compressibility effects on the lift and moment
are greater than it is shown in the above mentioned figures, because A
also intervenes in c -L4
cy. Obviously, the ground effect is great
when a is small and it quickly decreases when a increases.
3.2.4
The Symmetric Airfoil
As we have seen in (3.1.5), for the symmetric airfoil in a free stream
with zero angle of attack, the lift and the moment vanish. The situation
is different for the wing in ground effects. As we may observe from
(3.2.7) and (3.2.8), i i the case of symmetric airfoil (h = 0, hl 0 0), H
does not vanish, hence the solution of the equation. (3.2.7) is different
from zero. For the Joukovsky symmetric profile considered in 11.28),
[3.37], [3.13)
y = +E(1 - X) VI - x2
(3.2.13)
THE AIRFOIL IN GROUND EFFECTS
87
we obtain using the notation hl = -(1 - x) 1- x' and the formulas
(F.2.12) and (F.2.18):
H =-m
;r J-1 (xj - e)2 + m2
in
P
d_
1 - (-2,
'1
2m
to (1 - trt )
+1F-(,)2+fn2 - 2n+I
E(xj-tn)2)+m2'
(3.2.14)
In this case one presents NL and N11 versus
a in figures 3.2.3a) and 3.2.3b) for n = p = 20. The lift coefficient is
where
cos y7r
p+
negative i.e. the resultant is a force pointing towards the ground. Hence
when the airplane is landing or taking off it becomes heavier. The pilot
has to take into account this fact.
- al.bo
- - - Msean6
I
: _
F-
a'l
0.001 l
000
1.60
/
-=-
l
4.00
5.20
a30
410
OAO.
I
000
400
I
1
a
1.60
'20
4.90
640
900
b
a)
Fig. 3.2.3.
The fact that both in the case of the flat plate and the case of the
symmetric profile, the lift and the moment become very great when a is
very small, is not true in reality. Hence, for small values of the parameter
a we have to elaborate a new theory based on two small parameters
(see [5.37]).
88
3.3
3.3.1
THE INFINI'T'E SPAN AIRFOIL IN SUBSONIC FLOW
The Airfoil in Tunnel Effects
The Integral Equation
The experiments for determining the aerodynamic parameters are
performed in wind tunnels. We have therefore to take into account the
influence of the walls of the tunnel an the aerodynamic characteristics,
of the airfoil. Since many papers dedicated to this subject are secret, we
cannot give the history of the research in this field. We have cited in the
bibliography some authors, without consulting their papers. We shall
present therefore, only the model that we gave in [3.14]. This model can
be easily obtained with the method of fundamental solutions and it is
in the spirit of the theory previously presented in this book.
We formulate the problem as follows: an uniform stream, having the
velocity U ,i, the pressure pao and the density p,,, flowing between
two infinite flat plates parallel to the O.T. axis, encounters a thin airfoil
of infinite span with the generatrices parallel to the Oz axis. The fluid
is compressible, and the velocity of the uniform stream is subsonic. One
requires to determine the perturbation and the influence of the stream
on the airfoil. We utilize the variables (2.1.1) and (2.1.3). Let (3.1.1)
represent the equations and a the distance between the plates (walls)
(fig. 3.2.1). For determining the perturbation, we have to impose the
following boundary condition:
v(x, ±0) = h'(x) ± hl(x),JxJ < 1
(3.3.1)
v(x, ±a/2) = 0, -oo < x < oo
(3.3.2)
Y
+11 w2
xx
I /2
Fig. 3.3.1.
In order to utilize the method of fundamental solutions, we shall
f)(t)
replace the airfoil by a continuous distribution of forces
defined on the segment [-1, +1). For satisfying the boundary conditions
(3.3.2), we have also to take into account symmetric distributions on the
images of the strip [-1,+1) in the planes y = ±a/2 and symmetric
distributions on the images of the images in the planes y = ±3a/2 etc.
89
THE AIRFOIL IN TUNNEL EFFECTS
(the method of images). In this way one obtains the following general
representation of the perturbation:
+oo
P(x, Y) - 27rp
na)f(e)
d
xo + 02(y - na)2
+1
1
J-
,
(3.3.3)
(
v x, y) =
p }OO f
27r
1
(-1)"Xof(t) - (y ro + (y - na)2
with xo = x - . We can easily verify that v given above satisfies
(3.3.2). For imposing the conditions (3.3.1) we have to pass to the
limit considering y -+ 10, -1 < x < 1. The only singular integrals
correspond to n = 0 and they are calculated using the formulas (3.1.19)
and (3.1.20). Taking into account the equalities (1.161:
00
_
1
k2 + n2
n=1
zr cosh kn
2k sinh kir
1
2k2 '
A
2k sink ka
"-1 k2 + n2
2k2 '
we deduce:
1
p(x, f0) = f 2 f (x) +
v(x,±0) _
f xo
)d
+
(3.3.4)
+1
1
7rp
-L
J_1
2f1(x) + 2r ,
rLd+
(3.3.5)
+p
where m = ap
Kl (xo) _
I fi
K(xo)f(t)dt`,
-o
K(xo) = m sink-' (m xo' -
,
coth \\(m xo//
,
\
XO
(3.3.6)
From (3.3.4) we deduce again the significance of the function f :
f (x) = p(x, +0) - p(x, -0),
(3.3.7)
90
THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW
and from (3.3.5) and (3.3.1),
fl(x) = -2hi(x),
(3.3.8)
+I
x
2, K(xo)f()d= h'(x)
.
(3.3.9)
1
The formula (3.3.8) determines fl and the equation (3.3.9), the
function f. The integral equation of the problem is in fact the generalized equation of the thin profiles. The kernel K has no singularity for
t = x. We notice that for a - oo (m -+ oo) the equations (3.3.4)
and (3.3.9) are reduced to the equations corresponding to the airfoil in
a free (unlimited) airfoil.
3.3.2
The Integration of the Equation (3.3.9)
We utilize the quadrature formulas from Appendix F for integrating
the equation (3.3.9). Putting h = -eh, we shall look for solutions f
having the form (3.1.56). Utilizing (F.2.18) and (F.2.19), we reduce the
equation (3.3.9) to the system
n
j=1,...,n,
>2CfaFo =hj.
(3.3.10)
0=1
where F. = F(ta), hi = h(xj),
C'° = m 2n
1
sinh-1 I -(xj
- to,)}
,
(3.3.11)
§i xj being given in (3.1.58). After determining the unknowns
F1,.. . , F, from (3.3.10), the lift and moment coefficients may be obtained by means of the formulas:
tQ
(3.3.12)
cA, _
CL =
NL and NA, having the expressions given in (3.1.60). The distribution
of the pressure on the airfoil may be obtained from (3.3.4), taking into
account (3.3.8). We get
p(to,T0)=ff&
2J3
l+ta
1
rr/3
1 hl(x)dx1
tp - x
THE AIRFOIL rN TUNNEL EFFECT'S
91
The values of the pressure in the points to, may be determined in
the wind tunnel by means of pressure plugs. We have thus the possibility
to verify the theory presented herein.
We may use a computer for solving the system (3.3.10). Since the
coefficients C1,, do not depend on the shape of the airfoil, we have to
change in the program only the column containing the elements 1 .
Numerical Results
3.3.3
= 1. In
For the flat plate having the angle of attack e we have
this case, the formulas (3.3.12) become cL = cL NL, cm = cMNM, cOLO
and cc representing the coefficients corresponding to the unlimited
fluid.
- M. b 0
»
1
MWI"06
za
-I
I
1.4
1.3
a
00
02
0.4
0.6
01
1.0
'10.1
00
01
0.4
06
02
1.0
to"
h)
a)
Fig. 3.3.2.
The figures 3.3.2 present the variations of NL and Niv versus the
width of the tunnel for M = 0 (incompressible fluid) and M = 0.6.
We notice that the lift in the wind tunnel is greater then the lift in the
unlimited fluid and it decreases when the width of the tunnel increases.
For the same width, the lift is an increasing function of M. The theory
presented herein allows us to determine these variations.
92
3.4
3.4.1
THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW
Airfoils Parallel to the Undisturbed Stream
The Integral Equations
In the literature, the problem of the uniform flow past a configuration
of airfoils has been solved for the incompressible fluid and for airfoils
whose exterior may be mapped conforma.lly on the exterior of a circle.
There were given especially solutions for the biplane [1.36], [3.38]. We
shall consider in the three forthcoming subsections, following the paper
[3.15], the general problem of the compressible fluid in subsonic flow and
airfoils with thickness of arbitrary shape.
YA
UM
a/2
x
Fig. 3.4.1.
We shall consider in this subsection, two airfoils parallel to the undisturbed stream (fig. 3.4.1), not necessarily identical. The free stream has
the velocity U,,,i, the pressure p,, and the density pp,;,. One utilizes
the variables (2.1.1) and (2.1.3) and one considers that the equations of
the airfoils are:
y = h(x) . h1(x),
Y = 1(x) ± 11(x),
krl < 1
(3.4.1)
1 XI < 1
(3.4.2)
the first corresponding to the upper airfoil and the second to the lower
airfoil. The dimensionless distance between the projections of the airfoils
is denoted by a. Utilizing the method of fundamental solutions, we
shall replace the upper airfoil by a distribution of forces of intensity
and the lower airfoil by a distribution (gl, g)(t), both of them
(fl,
defined on the segment [-1, +1]. Starting from the above mentioned
perturbation produced by a force in the free stream (the fundamental
solution (2.3.23)), we deduce that the perturbation produced by the two
AIRFOILS PARALLEL TO THE UNDISTURBED STREAM
93
distributions is:
+t
1
Ax, U) - 2yrQ
xof 1(e) + QZ(y - a12)f (t)
1- 1
xa +,Bs(y - a/2) s
d t+
+t
X091 (t) + O2(y + a/2)9(t)
1
+ 27rp J-1
d
(y + a/2)2
xo +
(3.4.3)
x, y) =
(
p
T,,-- 1
+1 xof (e) - (y - a/2)ft(e)
d t+
xp +,02(y - a/2)2
1
+ Q r+t xo9(t) - (y + a/2)gt () d t,
2tr f 1
xo + /32(y + a/2)2
where x0 = x - .. The distributions of forces will be determined from
the boundary conditions
When y
v(x, (a/2) f 0) = h'(x) ± hi (x),
fix( < 1
(3.4.4)
v(x, -(a/2) f 0) = !'(x) ± li (x),
jxj< 1
(3.4.5)
(a/2) ± 0), the first integrals from the representation
(3.4.3) become singular and they have to be calculated with the formulas
(3.1.19) and (3.1.20). One obtains
P(x, (a/2) f 0)
2 f (x) +
+1
1
+ 2aQ
' fl
,
270 T-1
-;To
(3.4.6)
xo.91 + 7n,89
xo + jn2
_1
1
v(x, (a/2) 10) _
1 ft (x) + 2a
T-1
d+
d
f (')d t+
xo
(3.4.7)
1
1 Qxog -
f
+ 2rr ,l
1
mgt
xo + m2 d
,
where rn = ap. From (3.4.6) it follows
f(x) = P(x,(a/2) + 0) - p(x,(a/2) - 0),
(3.4.8)
and from (3.4.4) and (3.4.7)
fi(x) = -2h'1(x)
(3.4.9)
94
THE INFINITE SPAN AIRFOIL ICY SUBSONIC FLOW
J
t f x()) d
t
+ 2 , i t xo +(r roe d = H(x),
where
(3.4.10)
+I
H(x) = h'(x) +
2
d
.
(3.4.11)
In the same way, passing to the limit in (3.4.3) when y -i -(a/2)±O
(in this case the second integrals become singular), from the boundary
condition (3.4.5), we obtain
g(x) = p(x, -(a/2) + 0) - p(x, -(a/2) - 0),
'(mot 9(e)
1
gi(x) = -21c(x),
+t
f( )2dC
+
I
0
I
X2 +,fn
(3.4.12)
(3.4.13)
= L(x),
(3.4.14)
0
where
L(x) = l'(x)
-
,1
t
I x('2d.
(3.4.15)
The distributions of forces on the two chords are determined by solving
the system of integral singular equations (3.4.10) and (3.4.14). From
(3.4.9) and (3.4.13) the functions H(z) and L(x) are known. The
field of pressure on the profiles are given by the formula (3.4.6) and the
corresponding formula for p(z, -(a/2) ± 0).
Symmetric Airfoils. If the two airfoils are symmetric with respect
to the Ox axis, then
l(z) = -h(x), 1I (x) = hI (x) .
(3.4.16)
It follows L(x) = -H(x) whence
xz(+tn
r,
I
If
gdt;+
o
J
(3.4.17)
I
Since the solution of the generalized equation of thin profiles is
unique, it results f = -g, and the 'system of equations (3.4.10) and
(3.4.14) reduces to a single equation
2
Ifro) d 1
2
f
+I
I
+(n)
'TO f
s
d4= H(x)
(3.4.18)
o
which is just the equation (3.4.7) of the airfoil in ground effects. The
result is natural. The lift and moment coefficients for the upper airfoil
95
AIRFOILS PARALLEL TO THE UNDISTURBED STREAM
are given by the formulas (3.2.11). For the lower airfoil they have the
opposite sign. For the entire configuration we have cL = cAt = 0.
Identical Airfoils. If the airfoils are identical, then
1(x) = h(x) - 2a,
it(x) = hj(x).
(3.4.19)
Moreover, if the airfoils have no thickness, then L = H. Subtracting
the equations (3.4.10) and (3.4.14), it results g = f whence
A
f
I_ z2 +(m)2d = h'(x).
1
1
27r
(3.4.20)
This equation has the form of the generalized equation of thin profiles
and it can be integrated numerically like in (3.2.2) and (3.3.2).
3.4.2
The Numerical Integration
In order to solve numerically the system (3.4.10) and (3.4.14), we
shall use the quadrature formulas from Appendix F. For thin airfoils,
the functions H(x) and L(x) have the form H(x) = -eH(x), L(x) _
= -iL(x), hence we shall look for the following type of solutions
Qf (t) = -E V 1
+t
F(t),
3g(t) = -E
1 + tG(t)
(3.4.21)
which satisfy the Kutta-Joukowsky condition on the trailing edge. Utilizing (F.2.18) and (F.2.19), the system (3.4.10) and (3.4.14) is reduced
to
n
E(Aj.F.-Bj.Go)=Hl, ,7=1,...,n,
a=1
(3.4.22)
n
>(A1oGo-BjoFa)=1i,
a=1
A,,,, tQ and x, are given in (3.1.58) and Bj, in (3.2.10). Like always,
F. = F(t0), G. = G(t0), H, = 77(x,), LJ = L(x,). We have to find
out the unknowns Fl,. .., Fn, Gl, ... , Gn from the system (3.4.22). The
coefficients Aj0 and B 0 are the same for all the airfoils; only H
and L, are varying.
The lift and moment coefficients for the entire configuration are given
by the formulas
THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW
96
CL
=-1
1+1
+1
(f+9)dt=
/(F+G)dt--Ni
1+t
(3.4.23)
+1
CAI
--2J-,
+1
t(f+9)dt=2Q11
where
t
l+t(F+ c)dt =
-ASE2Q1V2,
n
N, =
1
2n + 1
:(1 - ta)(F. +G(),
Q=1
(3.4.24)
n
N2
2n 1 1
1)(F.+G.).
0=1
We notice that N, and N2 contain only the unknowns Fa + Ga
which may be determined from the system
Aja(Fa + Ga) - Bja(Fa + Ga) = Nj + Lj
(3.4.25)
a=l
of n equations (j = 1, ... , n) with n unknowns.
The unknowns Fa and Ca are separating only when we calculate
the pressure on the two airfoils. For example, the field of the pressure
on the upper airfoil is obtained from (3.4.6) with the formula
p(t,., (a/2) f 0) = f
29E
f
1 -to F.
0 - 4)G(ti)
f3(2n + 1)
(ta - t{)2 + m2
- x)li(x)
h', (x) dctQx(ta-x)2+m2d
x,
iro f!"
1
t
cos
2iIr
2n+1'
(3.4.26)
to being given in (3.1.58). In the same way we may determine the field
of the pressure on the lower airfoil.
The numerical determinations have been performed for the biplane
having the angle of attack E(h' = 1' = -E) (fig. 3.4.2a)) and for the
symmetric biplane (h' = -1' = -e) (fig. 3.4.2b)) taking n = 10. In
GRIDS OF PROFILES
97
a)
b)
Fig. 3.4.2.
the first case HJ = L, = 1, and in the second 77J = -Tj = 1. The
coefficients A, , Bin depend only on the parameter m = a#.
In the second case we obtain N1 = N2 = 0 for all the values given
to m (it is natural). In the first case, the values of N1Q'1 and N2Q-1
depend on a and M. They are given in tables 1 and 2. In the first
line we may find the values of 13-1N1 and 0-1N2 for the monoplane.
We notice that the lift coefficient for the biplane is much greater then
the lift coefficient for the monoplane and it increases with a. The lift
increases also with M. The same conclusions are true for the moment
coefficient.
Table 2
Table I
The values of N,$-1
a=0.5
a=1
a=5
3.5
3.5.1
M=0
M=0.6
1.00
1.461886
1.25
1.741425
1.70842
1.981013
2.034937
2.46087
The values of N2
a=0.5
a=1
a=5
1
M=O
M=0.6
1.00
1.624288
1.25
1.941250
2.209987
2.477736
1.827670
1.991025
Grids of Profiles
The Integral Equation
The classical problem may be also solved in a simple manner by
means of the method of fundamental solutions. One obtains again the
generalized equation of thin profiles. Let us consider a grid of identical
airfoils having the equations
y = na + h(x) ± hl (x) ,
n=0,±1,±2,...,
(3.5.1)
98
THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW
which perturb the uniform subsonic stream defined in section 2.1.1.
Since every airfoil produces the same perturbation, we shall replace
every profile by the same distribution of ford (fl, f) (c). Since the
perturbation produced by the force (fl, f2) in the uniform stream is
(2.3.23), we deduce that the perturbation produced by the grid is
1 C`
P(x, U) =
+t xoft(s) + Q2(y 1-1
(3.5.2)
Q
21r
u(x, A =
+°°
-00
f
+' xof W - (U - na)fl W d
x0 + (32(y - na)2
L
where xo=x-t;.
The profile corresponding to n = 0 will be called the reference
Passing to limit, when y -, ±0 the integrals corresponding
to n = 0 become singular and they are calculated with the formulas
profile.
(3.1.19) and (3.1.20). Employing the notation k = xo/m. we get
+00
n*o
-00
x°
A0 + m2n2
- 2x°
-00
nx
n
-
1
2x°
1r
k2 - n2 - m2 \2k
m2
=
+00
00
n
coth -xo m
7T
coth k7r -
1
2
1
K(xo),
r
TO
2n=0'
(3.5.3)
whence:
'6
P(x, ±0) = f 2 f (x) + j,-
f x°) d t+
f1
(3.5.4)
+z
1
+2- j h (xo)fi (4)d t,
P(x, +0) - P(x, -0) = f W,
(3.5.5)
d+
1
u(x, f0) _
2
Q
ft (x) + 2 J
t
o
(3 .5.6)
+i
+2 fl
99
GRIDS OF PROFILES
Imposing the boundary condition
v(x, f0) = h'(x) ± M, (x)
(3.5.7)
fi(x) = -2hi(x)
(3.5.8)
we deduce
+1
2 j K(xo)f(t)dt
If (o) d t +
= h'(x)
(3.5.9)
1
This is the integral equation of the problem. It has the same form like
the equations (3.2.7), (3.3.9) and (3.4.10), (3.4.14), excepting the non
- singular kernel. Obviously, from (3.5.3) we deduce that the kernel
K(xo) is not singular. After determining f by means of (3.5.9), the
field of pressure on the reference profile will be obtained from (3.5.4), and
(3.5.5) will be utilized for the calculation of lift and moment coefficients.
In order to obtain the pressure on the profile corresponding to n = I
and to impose the boundary condition on this profile, we must pass to
the limit in (3.5.2) y - a :E 0. In this case, the integrals corresponding
to n = 1 become singular. With the change of variable y - a = Y we
have for example
rx,
+t
Ax, a t 0) =
lim
to J- xxo + #2Yf d +
27r,3 Y
+
+00
1
n#1
27 r3
2
-00
+1 xof1 +,82(1 - n)af
d
,l-1 x0 +1112(1 -n)2
Putting I - n = nl in the last integral we deduce
+00
n'At
-00
_ +ao
1
To' + m2(1 -
-a0
00
1-n
+00
"#1
n)2
n,00
X.2
+ m2(1 - n)2
1
m2n'
-0
such that, taking into account (3.5.3), we obtain for p(x, a t 0) the
same expression like for p(x, ±0). In the same way, for v(a, a f 0) one
obtains the same formula like for v(x, ±0). In this way, imposing the
boundary condition
v(x,a±0) = h'(x) ±14(x),
(3.5.10)
100
THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW
one obtains the equations (3.5.8) and (3.5.9). We draw the conclusion
that if f1 and f are determined by (3.5.8) and (3.5.9), the boundary
conditions arc satisfied on each profile and the field of pressure coincides
with the field on the reference airfoil.
The lift and moment coefficients are calculated on every profile by
means of the formulas
cj =-J
1
1
2
1
3.5.2
r
f(t)dt, cal=---
+
tf(t)dt.
(3.5.11)
t
The Numerical Integration
If f haa, the form (3.1.56) and It has the form -sh, using the
quadrature formulas (F.2.18) and (F.2.19), we obtain from the integral
equation (3.5.9) the system
n
Di,,Fa = Itj, j = 1, ....:a ,
(3.5.12)
a=1
where
T 1 - t Goth
in 2n + 1
71
in
(xj - ta) ,
(3.5.13)
to and xj being given in (3.1.58). Considering for CL and cAf the
form (3.1.59), we obtain for NL and N,%j the expressions (3.1.60). For
example, for a grid of flat plates having the angle of attack E(hj = 1),
the variations of the coefficients Nf and N,tt versus a, are given in
figures 3.5.1.
We notice that the coefficients cL and cAt are increasing functions
of a and Al. They tend asymptotically to I when a - cc, hence the
lift and moment coefficients tend asymptotically to the
taken in
the case of a single profile.
The pressure in the control points (nod(s) is obtained from the formula
P(tu>f0) =
213
1
7T,9
x-
tQda.-
(3.5.1 l)
/'
,ro
-1
AIRFOILS IN TANDEM
101
.o
N
12
1.02
a%
IN
09$
014
091
066
011
US
0n
00
a
000
020
010
0.60
030
040
1.00 X10-,
a)
000 am 040
060
0.*0
1.00 X10'
b)
Fig. 3.5.1.
3.6
3.6.1
Airfoils in Tandem
The Integral Equations
In the sequel we shall determine the perturbation produced in the
uniform stream defined in 2.1 (under the assumption that it is subsonic
(M < 1)), by a configuration of airfoils in tandem (fig. 3.6.1), having
the equations
y=11(x)±h1(z), al <x<bl,
(3.6.1)
a2 < z < b2 .
(3.6.2)
y =12(x) f h2(z),
For the incompressible fluid this problem was studied by Chaplygin and
Sedov (1.38) who determined the complex velocity. If we reduce the
problem to integral equations and we solve them like in 13.151, we can
determine directly the quantities of interest in aerodynamics (the field
and the jump of the pressure on the airfoils, the lift and moment coefficients). Moreover, the fluid may be compressible and we don't need to
use elliptic integrals. In the sequel, we shall present the results from the
last paper cited above.
Replacing the airfoils by distributions of forces having the intensities
(gl . fl) respectively (g2, f2) and taking into account the fundamental
THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW
102
Y
K
a,
b,
a3
b,
Fig. 3.6.1.
solution (2.3.23), we get the following representation of the perturbation:
jbX0g1(t)+#2yf1(C)
1
P( x, y) = 2aQ
1
x0 + /y2
r"' xo92(
+2Q 1
S
) + 02Yf2(t) d
xo + Q2 y2
,
(3.6.3)
v(x,y)=
xofiW --y91( )d
xo + 132y2
2,r J a
Q rb'
+ 2a az
x0 +
dtt
y2
We kept the same notations like above. Passing to limit for al < x < b1
and taking into account (3.1.19) and (3.1.20), one obtains:
P(x,f0) = 2fi(x) + 27r#
xo
J a2
(3.6.4)
p(x, +0) - p(x, -0) = f 1(x)
(3.6.5)
f,a,
0
Q, f bi fi(t)
v(x, t0) =
bz
o1( )dC+
91(x) + 27r J
n,
xo
d+
fb
aZ
xo d '
(3.6.6}
Imposing the boundary condition
v(x, ±0) =1 (x) ± hi (x)
it follows
91(x) = -2hi(x)
Q
fl(O)
27r1 axo
1
d
2a Jxo
2xo
L d=11(x) (3.6.8)
Q
f
+
+
(3.6.7)
AIRFOILS IN TANDEM
103
In the same way, for a2 <x < b2 we deduce:
P(x,±0)=f2f2(x)+2a
V(x, f0) = -T 192W +
2
27r
-
C
g 0{dt+2
)
., f )dt+ 2
Mo
22(i)dt(3.6.9)
Y J., x0
f oQd(,
z
whence, utilizing the boundary condition
v(x, f0) = !'(x) f h2(x)
we obtain
p(x, +0) - p(x, -0) = MX),
(3.6.10)
92(x) _ -214(x),
(3.6.11)
2f
a,
i
t
'f
: _ (x).
f
( 3. 6 . 12)
Hence the functions fl(x) and f2(x) will be defined by the system
(3.6.8) and (3.6.12). This is an interesting mathematical problem, since
the first equation is defined for al < x < bl and the second is defined
for a2 < z < .b2.
3.6.2
The Determination of the Functions f, and f2
For f, the equation (3.6.8) has the structure (C.1.5) which has the
solution (C.1.9). Utilizing (B.6.11) and the last formula from (B.5.4) we
deduce:
Of, (x) = Lt (x) + x
xt-
t
2(f)d
f
V
- bt
(3.6.13)
where
L1(z) = a
b-a
t
n,
(3.6.14)
i
Substituting fl given by (3.6.13) in (3.6.12) and taking (B.6.11)
and the last formula from (B.5.4) into account, one obtains the following
integral equation for f2:
2 r.2 V/F - a'
f2(t)dt=L2(x), a2<x<b2,
(3.6.15)
THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW
104
La(x)
r
x-
,(x)
=
a1
b1
-!
- a,
h'
7r Ja,
Y
-- x
4b1 --
clt
.
(3.6.16)
The type of the equation (3.6.15) is (C.1.1) and its solution has the form
(C.1.9). One obtains:
2 ((x - b1)(b2 - x)11/'r,ll (x - a t)( - aa)
/3f2(x) =
/t- a
l.
L2(t)dt
t-x
t
l
or, taking (3.6.16) and the first formula from (B.6.11) into account,
13f2(2) =
2
(x - bi)(b2 _ x)
7r
x - '1)(x - n2)
+
(
( -a l}(aa
(b1 - .)(b2 -
t-n 1)(t-a- I.'
)
(t-b1)(b2-t) t - .r
a
)
- x`l +
a2 < x < b2 .
dt
(3.6.17)
Once 13f2 determined, from (3.6.13) it results f3f1:
(b1 - x)(ax -
fllx) ! 2
a) r
(x - al)(x -
J
a2
2
V(()
- ) li(S}
xd +
--a1}(a2
(t - a1)(t - a2) 12(E)
(t - b1)(ba - t) t - x
a1 < c < b1.
dt
(:3.6.18)
In this way we determined the solution of the system (3.6.8) and (3.6.12).
3.6.3 The Lift and Moment Coefficients
From the formulas (3.6.5) and (3.6.10), we deduce for the lift and
moment. coefficients:
b;
1 f;(x)dx, cAr
cL = -
1
b,
xfi(a)dx
(3.6.19)
Utilizing the solutions (3.6.17) and (3.6.18), changing the order of integration and taking (13.6.5) into account, we find
c1
bi
-2j
+
a,
b=
102
E)1t(ti)dti
(b1 --
/(t - al)(t - a2)12(t)cit
(t - b1)(b2
- t)
.
(3.6.20)
AIRFOILS IN TANDEM
105
Utilizing the same solutions, changing the order of integration and
taking (B.G.6) into account, we get for the moment coefficient:
C.%/
( - al)(a2 - )
L
-13
(bi
b2 FLI- al)(t-a2)
bi)(t - t)
+,._
n
alt (od +
,
tt2(t)d t
+
al+as-b1 -b2
4
CL C. (3.6.21)
Obviously, the formulas (3.6.20) and (3.6.21) are generalizations of the
formulas (3.1.33) and (3.1.34).
Utilizing (B.5.14) we may prove by induction that in the case of n
profiles in tandem we have:
r
cL
b
kt
/i_ak1(t)d
k=1
1
(3.6.22)
"
n
cet=-1R fj11
k=1
3.6.4
k
t
t
Q
I n
bktlJ(t.)dt+(ak-bk)
k=1
Numerical Values
In the case of two flat plates having the angle of attack fs
(fig.
3.6.2a) or 3.6.2b)), the integrals (3.6.20) may be expressed with the aid
of elliptic functions. The first one has the form of the integral 252.21
from (1.4), page 105, and the second has the form of the integral 256.21
from the same book, page 122. For more complex configurations, these
kinds of expressions are not available. However, in all cases, the integrals
intervening in the expressions of c z, and ctit (for every n), may be
calculated numerically by means of the quadrature formulas (F.2.24).
For example, if n = 2, performing the change of variable C -+ t:
=
b1-al
al+bl
2
+ 2
-1<t<+1
(3.6.23)
THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW
106
a)
b)
Fig. 3.6.2.
and utilizing (F.2.24) we deduce:
_
(b - ) (b'i - ) d - 2n+ 1 or=l (1
Jo 1
I1
Jl =
n
bl - at
al)(a2 -
1
1
(a2f)) {d f
t)(62
al + bl
2
(3.6.24)
-
-
it
a to
b - to
+ ta)
(bi - al)2 n
1 + 2(2n
E to(1 + to)
+ 1)
:to
0=1
a = 2a2 - (al + bl)
b
,
=
bl - al
252 - (al + bi)
bl - al
where to are given by (3.1.52).
Similarly we get:
'2
1
2
1
J2
,
(t - al)(t - a2)
(t - bl)(b2 - t)
=a2 2
C=
-' 1 + t
2n+1 am1
(t - bl)(b2 - t)
7r
J2
(t-al)(t-a,) = b2-a2 n
to
(3.6.25)
c-tom
2(2n+1)
2al - (a2 + b2)
b2-a2
td t
c
d=
Vd-to'
0=1
251 - (a2 + b2)
b2-a2
Hence for two flat plates in tandem having the angle of attack -- we
107
AIRFOILS IN TANDEM
have
0ir2
CL =
£(11 +12),
cM = re`(Jt +J2)
(3.6.26)
13
where 11, 12, J1, J2 may be calculated numerically using a computer.
For example, for at = 1, b1 = 2, a2 = 3 and b2 = 4 one obtains:
4
11 = 0.370671 and 12 = 0.629328. For a1 = 1, b1 = 2,
and b2 = 5 one obtains: I1 = 0.415458 and 12 = 0.584541. For
a1 = 1, b1 = 2, a2 = 6 and b2 = 7 one obtains: I1 = 0.449746
and 12 = 0.550253. In all the cases we have 11 + 12 = 0.999999, hence
the lift has practically the value of the lift for a single plate and this is
true even if the distance between plates increases. Since --11 + 12 takes
the values 0.258657, 0.169083, 0.100507 it results that if the first plate
has the angle of attack -c and the second e, the lift decreases when
the distance between plates increases. If the first plate has the angle
of attack -s and the first has the angle of attack e, the lift becomes
negative.
The previous formulas enable us to obtain numerical values for cL
and cAl in the case of n airfoils with different shapes.
Chapter 4
The Application of the Boundary Element
Method to the Theory of the Infinite Span
Airfoil in Subsonic Flow
4.1
4.1.1
The Equations of Motion
Introduction
The theory exposed in the previous chapter relies on the following
three assumptions:
10 The linearization of the boundary condition.
2° The linearized condition is imposed on the chord of the profile, not
on the boundary (as it is natural to do).
3° The linearization of the equations of motion.
The assumptions are plausible for thin airfoils, otherwise they may
be the cause of great errors.
The application of the boundary integral equations method (BIEM),
which is also called the boundary elements method (BEM), enable its to
give up the first two assumptions. Hence we shall utilize the non-linear
boundary condition which will be imposed on the contour (boundary)
of the airfoil.
For the incompressible fluid, we shall employ the exact equations
(4.1.4) i.e. the equation of continuity and the equation of irrotationality,
such that, in this case, the mathematical model (the equations of motion
and the boundary conditions) will be valid for every airfoil (not only for
the thin ones). In the case of the compressible fluid, we shall utilize the
linearized equations, hence the results will be valid only for thin profiles,
but even in this case the results are better then the results obtained by
adopting the first two assumptions.
110
THE BOUNDARY PLEXENTSNIFTHOD
The first step in solving a problem with the boundary integral equations method (or the boundary elements method) consists in reducing
the boundary value problem to a boundary integral equation (whence it
follows the first name of the method); the second step consists in solving approximately the integral equation, replacing the boundary with a
polygonal line (whence it follows the second name). In addition we have
to mention that there exists a direct method and an indirect method.
In, the direct method we utilize the equations of motion for deducing a
representation of the solution in the domain occupied by the fluid with
the aid of the solution on the boundary. Then we pass to the limit on
the boundary. In the indirect method, we replace the wing with a distribution of fundamental solutions (sources, vortices, doublets) on the
boundary and and we determine their intensity imposing the boundary
conditions to be verified; in this way one obtains an integral equation
for the intensity of the fundamental solutions; the method is therefore
indirect. As we shall see in 4.2, it is easier to use the indirect method,
but the results are less accurate. It is more difficult to utilize the direct method (4.3) but the results are more accurate. Therefore, in the
forthcoming subsections we shall use mainly this method.
4.1.2
The Statement of the Problem
The problem was already formulated in the previous chapter. A
subsonic stream having the velocity U,,.i (the Ox, axis has the
direction of the unperturbed stream), the pressure p. and the density
p is perturbed by the presence of a cylindric body with the generatrices
perpendicular on the x1Oyu plane, having a given cross section (see fig.
4.1.1). One requires to determine the perturbation and the action of the
fluid against the body. Denoting by X, Y the dimensionless coordinates
introduced by means of the relation
(xi,yi) = L0(X,Y),
(4.1.1)
Lo being an unspecified reference length, and by U,,,,V and pc,.U"'2'P,
the perturbation of the velocity and the perturbation of the pressure,
we have:
V1 =UU(i+V), P1 =p +pWU,P.
(4.1.2)
The system which determines the perturbation is (2.1.30), written
with capital letters. Projecting the second equation on OX and taking
the damping condition at infinity into account, we deduce
P = -U.
(4.1.3)
111
THE F . UNrIONS OF MOTION
y
U.,
x
Fig. 4.1.1.
In this way we deduce the equations which determine the perturbation:
/28U/ax + aV/aY = 0, 8V/ax - au/ay = 0,
(4.1.4)
U and V representing the coordinates of the vector V. and ,0 having
the usual significance ([f = l - Af ).
On the boundary C, of the airfoil we shall impose the condition
(l +U)N,y +VNy = 0,
(4.1.5)
N,V and Ny representing the coordinates of the inward normal to C1.
Performing the change of variables
x=X. y=0Y
(4.1.6)
u= f3U, v=V,
the system (4.1.4) becomes
au/ax + aV/ay = 0,
av/ax - au/8y.
(4.1.7)
In order to transform the condition (4.1.5), Ave notice that if the
boundary C1 has the parametric equations X = X (S), Y = Y(S)
with S increasing when the curve C1 is traversed like in figure 4.1.1,
then
N = (dY/dS, -dX/dS).
(4.1.8)
Performing the change of variable (4.1.6), we have x = x(X (S)),
y = y(Y(S)) and
THE BOUNDARY ELEMENTS METHOD
112
Cl and the transformed curve C being traversed in the same sense (a
increases with S). It follows
NX
- # dS'
Ny
=n
a-S
(4.1.9)
,
such that (4.1.5) becomes
(,3 + u)nx + l2vny = 0 on C.
(4.1.10)
One also imposes the damping condition at infinity
lim(u, v) = 0.
(4.1.11)
00
4.1.3 The Fundamental Solutions
The first step in applying the BIEM consists in determining the fundamental solution of the system of equations (in our case, the system
(4.1.7) ). We call source-type solution the solution of the system
au'/ax + 8v /ay
a(x - t, y - 11),
(4.1.12)
&v'/Ox -au.la!l = 0.
The fact that the perturbation represented by 5 intervenes in the
equation of continuity justifies the name of the solution. We apply the
Fourier transform and like in Chapter 2 we obtain:
U0
1
x-t
27r(x-e)2+(y- n)2'
v*
_
I
y
q
21r(s-t)2+(y-17)2
(4 1.13)
We call vortex-type solution the solution of the system
au /Ox + W /ay = 0,
(4.1.14)
Ov'/Ox Obviously one obtains:
W = -v*, v = u' .
(4.1.15)
113
INDIRECT METHODS FOR THE UNLIMITED FLUID CASE
Indirect Methods for the Unlimited Fluid Case
4.2
The integral equation for the Distribution of Sources
4.2.1
Taking (4.2.13) into account, we deduce that if we replace the airfoil
having the contour C (fig 4.1.1) with a distribution of sources of intensity f (x) apriori unknown, the perturbation of the velocity in the
point Al(t) from the fluid will be given by the formulas:
v( ) _
2-4
-1
(4.2.1)
,Cf(x)Ix-4I2ds.
In order to determine the intensity f (x) we impose the boundary
condition (4.1.10). To this aim, we have to calculate the boundary
values in (4.2.1) when C -+ zo a current point Qo of the boundary C.
We are going to prove that if f (x) satisfies the Holder condition on
C, i.e. there exist two positive constants A and p (p < 1) such that,
for every two points Q(=j) and Q(x2) belonging to C, we have:
If (z1) - f (w2)1 < Alaet - x21',
then
- 2 . PC f(x)
v(xo ) = lim
_ - 2 f (xo)n °
-2
- 2a Pf (m)
(4.2.2)
d8=
(4 .2 .3)
X
Isa
X0
d
=012
it,
in every regular point Geo. We denoted
PC
= li , Pi-C
(4.2.4)
where c is the arc cut out from C by the circle of radius E and center
Qo (the arc QI Q2 from fig. 4.2.1).
We denoted by no the inward normal at C in Qo.
Indeed, taking the definition (4.2.4) into account, we notice that it
suffices to evaluate the integral on c. We have
limo
Eli
( _ 2 - f f (x)
\
=
Ii
n
{
L= liX0 PC
12-4
TV JC
I-T-C
ds=
f
[f (x) - f (xo)) I
d aJ - f Z
o)
44.2.5)
THE BOUNDARY ELEMENTS METHOD
114
Fig. 4.2.1.
Taking into account that f satisfies the condition (4.2.2), we deduce
that the first integral from the right hand side of the equality (4.2.5)
vanishes when E -- 0. For calculating L we shall replace the arc
Q1Q2 with the segment on the tangent Q'Q'2 (fig. 4.2.1) and we shall
put on this segment:
x = xO + STO,
s E [-E, +EJ .
(4.2.6)
We assume, for the sake of simplicity, that M(C) tends to Qo(xo) on
the direction of the normal n°. We have therefore
4 =xo--9n° cu q>0.
(4.2.7)
It follows
L= Um
17-0
r
+C
E
o+ r1no
s2 .
,2
q.e.d.
Imposing the condition (4.1.10), we obtain from (4.2.1) and (4.2.3)
the following integral equation:
(flo2
+ N2n02) f (xo)+
+
(x - xo)n° +f2(y - bo)n°
Uds=2pn2.
cf(x)
Ix-xo12
(4.2.8)
INDIRECT' METHODS FOR THE UNLIMITED FLUID CASE
115
From this integral equation we are going to determine the intensity of
the sources. For the incompressible fluid (M = 0) it may be solved
exactly for the circular obstacle (4.12).
4.2.2
The Integral Equation for the Distribution of
Vortices
If we replace the airfoil with a distribution of vortices having the
intensity (circulation) g(x) , defined on C, then, according to the formulas (4.1.15), we obtain the following expressions for the components
of the perturbation u and v in a point M(t) from the fluid:
da,
u(0 =
2
where l; = (1:, rl), x = (x, Y) Taking the formula (4.2.6) into account, we obtain:
Y-YO d s
W012
U(X0) = 2g(xo)ny +
9(x) Ix
-
,
(4.2.10)
f
2 g(xo)n:
v(xo)
- -1 P9(m) 1 -Z
X0
12
d
and from the boundary condition (4.1.10):
-M2g(xe)niny+
+1
ir
'(x)p2(x-xo)ny-(V -bb)nrd8=2f3n=,
c
(4.2.11)
Ix - xol
where M is Mach's number in the unperturbed flow. For the incompressible fluid (A! = 0) we have a first kind integral equation.
4.2.3
The Boundary Elements Method
One utilizes the following collocation method in order to solve the
equations (4.2.8) and (4.2.11): one approximates the boundary C by
it polygonal line {!,,}(j = 1 T), with the end points on C (fig.
'TIME BOUNDARY ELEMENTS METHOD
116
Fig. 4.2.2.
4.2.2) and one approximates on every segment Li, the unknown f
(respectively g) by the value fj (resp. gi) from the midpoint of the
segment. Denoting by :xf the vector of position of the midpoint, we
have fi = f (x°), (j = L N). We may also consider a linear variation of
f on L. In this case we have to determine the two constants from the
values of f in the end points of the segment L,. i.e. from the values
of f on C.
Now the equation (4.2.8) may be written as follows:
( 02
+/32802) f(xo) +
1
+_7
r
(x - X0 710 + )`12(y - ?ko)tty
'Y
fi
L,
ds = 28nr .
Ix - xo1Imposing
the equation to be satisfied in every midpoint, i.e. putting
successively xo = x9 . i = 1N, we get the system:
N
ai fi +
Ai i f) = Ai,
i
(4.2.12)
where
tai = nr(xP) + '3ly1y(x))
(4.2.13)
AiJ
n'r(x:))Uii + /3lnv(x°)V i
+
Ai = fln (xn) ,
117
INDIRECT METHODS FOR THE UNLIMITED FLUID CASE
with the notations
Ucj = / u'(x,x°)da
L
(4.2.14)
Y1 = jv(zz?)ds1
f
u' and v' being given in (4.1.13). The linear algebraic system (4.2.12)
consisting in N equations will determine the N unknowns f{.
Now the perturbation of the velocity in the points Po(x°) follows
from (4.2.3)
N
1
vi
=-2f.n(x°)-EV1jfj
J-1
(4.2.15)
V>, = (U+f,V )
In the same way, from the equation (4.2.11) we get the system
N
hi9, + 1: Bij9j = A, , i = 1,(4.2.16)
j:1
where
2b; _ -1tnz(x°)n,,(x°) ,
(4.2.17)
Bij = #'nr(x°)U,j - n(x°)Vij.
From (4.2.10) we obtain the following velocity field
N
1
w = 29jnr(x0)+Evj9j
j..1
(4.2.18)
1
Vf
4.2.4
N
t'4s(x'i')-Uijgj.
-
The Determination of the Unknowns
The integrals (4.2.14) may be computed numerically using quadrature formulas. They may be also calculated exactly for every shape of
THE BOUNDARY ELEMENTS METHOD
118
the contour C. Since the integrals UUJ, V, are singular we shall calculate their Finite Parts. For avoiding the singular integrals, we may
utilize the regularization method 4.8.
For obtaining a parametrization of the segment L,, we shall denote
by (x1, yl) and (x2,112) the coordinates of the end points PI, respectively P2, considered in the sense of traversing C (fig. 4.2.2). Then,
the coordinates of the generic point on L., will be
y°+-2ylt, -1<-t<1,
x°-i-x22xlt,
(4.2.19)
where, obviously,
x°
X2 2 x1
yj=112
2 111
(4 . 2 .20)
.
Utilizing the formula
ds = 1(&-)2 -+ (dy)
,
we deduce
gds=l,dt, lJ =
(x2-x1)2+(112-111)2.
(4.2.21)
Denoting
4a = I,b= (x°-x°)(x2-xl)+(i4 -Y?)(112-Yi),
c = (xjI - x? )2 + (yp - y1)2,
-
+1
k
1
tkdt
ate+bt+c'
k = 0,
(4.2.22)
1,
we get
(x - x°)2 + (y - y°)2 = ate + bt + c
(4.2.23)
whence,
U,,
=
Io
47f [ (xj_ x°)
+x2
2
x1J
l]
,
(4.2.24)
ii4i[(y°- y° ) Io+2 yi
V
119
INDIRECT METHODS FOR THE UNLIMITED FLUID CASE
Taking the running sense on C into account, we shall have in
(4.2.13), (4.2.15), (4.2.17) and (4.2.18):
n,,(x°) _
_-V1
r+y(x 0) = x1 !J X2
1
ii V2
(4.2.25)
1 11z`)
n,,(xo) _
n (x?) = x I
j.
where (xl'), yi')) and (X( 2" Y2()) are the coordinates of the end points
of the segment Li and 1; is the length of this segment, i.e.
xz')
- xi'1)2 + (yz;l - y1'))2]1/2
(4.2.26)
.
We have to notice that in (4.2.24) the integrals Ik may be calculated
exactly. Indeed, since
<0,
A =- b=-4ac =
we have
to =
V'--5
Il - I
2a
In
arctan.
(4.2.27)
dc-a'
a+b+c -
(4.2.28)
b
a-b+c a/
arc t an vr---s
c-a
Taking (4.2.23) into account, we deduce
a + b + c = (x2 - x°) + (y2 (4.2.29)
a-b+c= (xl -x°)2 +(yi -y°)2.
For i = j we have b = c = 0, such that the integrals Ik become
singular. Utilizing the formulas (D.2.3) we obtain
Io = -2/a,
Il = 0.
(4.2.30)
So, the coefficients ai, At,, A{, b1 and B1j depend on the coordinates
of the end points of the segments L, on C. Hence, we may solve the
systems (4.2.12) and (4.2.16) using a computer.
120
TllE BOUNDARY ELEMENTS METFIOD
4.2.5
The Circular Obstacle
For testing the two methods presented above and the direct method
from the following subsection, we shall use the exact solution for the
incompressible flow past a circular obstacle. This solution is already
known. If 0 is the origin (center) of the obstacle and R its radius,
then the complex velocity in dimensionless variables is (see (5.2.4) from
11.111)
R
UI -iVI =U01- Z2)
\\
.
(4.2.31)
t
Taking R as reference length and putting z = exp(iO) it results
u = - cos 2O ,
y _ -sin 2O
(4.2.32)
In figures 4.2.3 and 4.2.4 we compare the numerical values obtained
for u by mens of (4.2.15) and (4.2.18) with the values obtained by
means of (4.2.32). We observe that the solution obtained by means of
the distribution of vortices is better. It is also useful to compare the numerical values obtained on the three ways for a quantity of aerodynamic
interest. This is the local pressure coefficient defined by the formula
A P.
Cy _
= (1/2)P..oUx ,
(4.2.33)
with the notations from (4.1.2). Taking into account that for the incompressible fluid, the system (4.1.7), for which we have determined the
approximate solutions, is the exact system of the equations of motion, we
shall calculate also Cn exactly. This may be obtained from Bernoulli's
integral
1 V2
Pi
1 r r2 . Poo _
(4.2.34)
Poa
2
Poo
It follows
Cp=1-V2=1-(1+ u)2-t'2.
(4.2.35)
Using this formula, from (4.2.15) we get C,, in the case of the distribution of sources, from (4.2.18) we get. C,, in the case of the distribution
of vortices and from (4.2.32) we get C,, exactly. We also calculate Cn
with the direct method from the following subsection. The graphic rep-
resentations show that the direct method gives better results than the
indirect methods and among these ones, the method based on the distributions of vortices is better.
INDIRECT METHODS FOR THE UNLIMITED FLUID CASE
121
t
1.
I
es
m
As
.03
.1s
.1s
a
.1
a
n 6e m IJI If Im )m >r )10 1L ))a us
a
W b b Ip Ip 110 7q 30 M RD 30 No
Fig. 4.2.3.
4.2.6
Fig. 4.2.4.
The Elliptical Obstacle
The profiles with smooth boundaries (for example the circle or the
ellipse) are non-lifting profiles. The profiles with angular points determine a circulatory motion of the fluid and therefore it appears the lift.
The circulation (and the lift) are determining from the Kutta-Joukovsky
condition (see for example, (1.101 p. 179), hence these profiles are lifting
profiles. In the first case we encounter d'Alembert's paradox. As we
shall see in the following subsection, the indirect methods are utilized
mainly for non-lifting bodies, while the direct formulation is suited for
writing the k - J condition.
It is well known the exact solution for the elliptic obstacle in the
incompressible fluid. The complex potential from the (Z1) plane is
given by the formula (see for example, (1.101 p.178).
F(zi) =
2 {( zi + za'°+
zl
+ (zi '
at + bt e1°l
(4.2.36)
t
at - bj
where at and bl are the semiaxes, 4 = at - b1, and a is the angle of
attack. Deriving (4.2.36) we deduce the complex velocity which has the
same form in dimensionless variables z = Z, a, b, c . Hence it follows
1+u-iv=
]e`1°+
2[(1+
(4.2.37)
+(l `
z
a+bei°l
a - b
122
THE BOUNDARY ELEMENTS METHOD
We determine the components u and v separating the real part
from the imaginary one. The graphic representations for the exact solution and for the solution obtained with the first indirect method, are
given in [4.7J. We notice that the results are almost identical.
4.3
The Direct Method for the Unlimited Fluid Case
4.3.1
The representation of the solution
As we have specified in 4.1, in the framework of the direct method
we represent the velocity field in the fluid with the aid of its values on
the boundary of the body and then we pass to the limit. In order to
obtain this representation, we shall consider a domain D exterior to
the boundary C and limited by a circle CR having the radius big
enough, such that C should be situated in the interior of CR. For f
and g continuously differentiable function C', we have the identity
[f (aulax + avlay) + 9(8v/ax - 8u/8y)Jda = 0
(4.3.1)
which follows from (4.1.7). Noticing that
fau/ax = a(fu)/ax - ua f /ax, .. .
and applying Gauss's formula, we obtain the identity
J[u(Of/ax - 89/8y) + v(8f/ey + 09/8x)Jda =
(4.3.2)
=
j[u(fn
- 9ny) + v(9nx + f ny)Jds .
the integral on 47R vanishing when R - oo, because of the behaviour
at infinity of f and g which will be identified with u' and v' and
because of the condition (4.1.11).
For f = u' and g = -v' (4.1.13), taking (4.1.12) into account, we
deduce:
u(f) =JC u(x) [u'(x, )n (x)
(4.3.3)
+v(x) [u'(x, Ony(x) - v'(x, )n=(x)] }ds
,
THE DIRECT METHOD FOR THE UNLIMITED FLUID CASE
123
and for f = v' and g = u'
2L(x) [v`(x, S)ns(x) C
(4.3.4)
+v(x)[u'(z,4)n=(x) + v'(x,C)ny(x)]}d8.
This is the integral representation of the solution, valid for every
point M(4) from the exterior of C.
4.3.2
The Integral Equation
For obtaining the boundary integral equation, we have to pass to the
limit in (4.3.3) and (4.3.4), letting the point M(r;) to tend to the generic
point Qo(zo) belonging to C. To the limit, the integrals containing
the fundamental solution (u', v') become singular. Hence we have to
adopt the definition (4.2.4). We calculate the integrals on c, as follows:
lim
Ju(z) [u*(x,F)n (x) + v*(x,C)nyr(x)]ds
lim
J [u(x) - u(xo)] [u'(x,4)n=(x)+
(4.3.5)
+v (x, 4)nr(x)]ds + u(xo)Li
L 1 = lim
t-M. 1c
[u' (x,
,
where
v`(x, 4)ny(x)]ds .
If c =
is replaced by the segment Q'1Q'2 (fig. 4.2.1), then we
Z
have the parametrizations (4.2.6) and (4.2.7) and n = no. Hence,
L=
27r re
s2
+
ds = 2
.
Passing to the limit, the first integral from the right hand side of the
equality (4.3.5) vanishes when e -+ 0.
124
THE BOUNDARY ELEMENTS METHOD
Similarly we have:
Eli
m
J
v(x)[u*(x,Lr)nv(x)
- v`(x,t)nt(x)]ds
_ slim J [v(x) - v(xo)] [u`(x, )ny(x)-v*(x, t)nz(x)]ds + v(xo)L2 , where
L2 = lim
=
(4.3.6)
J[u*(x)n(x) +v(x,)n(x)]ds =
ny - -r,0, no.)
r
s2
it
+
ds = O.
172
Therefore, from (4.3.3) we deduce:
2u(xo) _
{u(x) [u*(x, xo)ns(x) + v*(x, xo)ny(x)]+
(4.3.7)
+v(x) [u* (x, xo)nv (x) - v* (x, xo)nz(x)] }ds
.
In the same way, from (4.3.4) we obtain:
2V(xo)
{u(x)[u*(z,xo)nz(x) - u*(x,xo)nv(x)]+
(4.3.8)
+v(x) [u* (x, xo)n=(x) + v*(x, xo)ny(x)] Ids.
The formulas (4.3.7) and (4.3.8) constitute a system of two singular
integral equations on C for the unknowns u(xo) and v(xo).
Now we introduce the function
G = (13 + u)ni, - vnx .
(4.3.9)
We may utilize on C the condition (4.1.10). It follows
Q+u= n2n2G, v= -nz
v
2
2C.
(4.3.10)
nv
Replacing these relations in (4.3.7) and (4.3.8) and taking into account
the relations
PC. (u* n,,
+ v*nv)ds = -1/2;
P(u*nv - v*nr)ds = 0,
(4.3.11)
THE DIRECT METHOD FOR THE UNLIMITED FLUID CASE
125
which will be demonstrated in (4.3.7), we get:
u ( xo )
= ,3 + 2
P
C
(v '
n,n
V
u ')Gd s
- M2 n2 + #2n2
x
y
4.3.12)
u (xo
)=2TS
and then
G(xo) - 2
\
- u '-A12 n2+
2n2v ' Gds
x
y
)
I
PC
+ hf2
[u'nox + v'rty+
n22
nzny
+ g2nyh (v n o
'
- u ' ny° )JGd s = 2An y ,
(4.3.13)
with the notation no = n(xo). The equation (4.3.13) is the singular
integral equation of the problem. Putting M = 0, one obtains the
equation for the incompressible fluid.
4.3.3
The Circulation
In the case of the profiles with angular trailing edge, we have to
utilize the circulation. Taking the sense defined on C1 into account
and denoting by stl) and n(') the (dimensional) versors of the tangent
and inward normal, we have
41),
sod = -11y1), 41) =
(4.3.14)
such that
rl =
V1 si'idwi'
(U1nyll
- Vinzll)ds(') =
= -U0LoP[(1 + U)Ny - VN,;Jds.
With he changes of variables (4.1.6) and (4.1.9), we deduce
[' 1 = -
U
0 ° j[(0 + u)ny - vn=Jds.
(4.3.16)
Denoting r1 = U,,,Lor, r representing the dimensionless circulation
and utilizing (4.3.9), it results the simple formula:
r = - PCds.
(4.3.17)
For the incompressible fluid one obt ins:
r=-
G ds.
(4.3.18)
These formulas also emphasize the significance of the function G.
THE BOUNDARY ELEMENTS METHOD
126
4.3.4
The Discretization of the Equations
Acting like in 4.2.3, we reduce the equation (4.3.13) to the algebraic
system
N
i = 1, 2, ... , N ,
(1/2)G{ + F A;jG,i = J3;,
(4.3.19)
j=1
where
Ci = C(x°), A = Onv(x°),
A,j = -nh(x°)Utj - n,(x°)V J--
(4.3.20)
nz(x°)n1r(x°)
n2(x') nr(x°)V, -- nv(x°)U,j]
_M2 n2(xj),
Uy and
being defined (4.2.14).
After determining the unknowns G1 from (4.3.19), we may determine the circulation from the formula:
N
-/3I' = E 1jGj ,
(4.3.21)
j=1
representing the length of the segment Lj. We deduce the components of the velocity
1j
+ u; =
A2ny(x°)G=
n2(x°) + A2n2(x°)
ns (x°)Gi
n2(xe) +
n2(x°) '
(4.3.22)
from (4.3.10). Obviously, for n(x°) and n(x°) we shall utilize the
expressions (4.2.25) and for U;j and [;j, (4.2.24).
4.3.5
The Lifting Profile
If the profile has an angular point (fig. 4.3.1), or a cusp, like the
Joukovsky-type profiles, then we have to determine the circulation using
the Kutta-Joukovsky rule.
The circulation determines the lift. For determining the circulation,
we impose the equality of the pressure in the points P, (on the upper side of the angle) and P; (on the lower side of the angle) when
THE DIRECT METHOD FOR THE UNLIMITED FLUID CASE
127
Fig. 4.3.1.
these points (Pp and Pi) tend to the angular trailing edge P1. From
Bernoulli's integral it follows that the equality of the pressures implies
the equality of the tangential components of the velocity, (according to
the slipping condition the normal components are null). Taking into
account the orientation of the versor of the tangent s(1) on the upper
and lower sides of the angle (fig. 4.3.1), we deduce the condition:
Vi(P,).stt}+V1(Pi),alt}
=0.
(4.3.23)
With the notation (4.3.9), we have:
V1 slit =
U)sal + VsY)J =
UM
+u J"'J
13
+vn=1ds
pJdS
d5
such that (4.3.23) implies
G(Pi) = 0.
(4.3.24)
Consequently, for such a profile, we have to add the condition (4.3.24)
to the system (4.3.19). In order to solve this problem we introduce [4.18J
the regularization variable A and we obtain the system:
A+(1/2)Gi+AijGj =;3i, i = 1,...,N,
(4.3.25)
G(P,) + G(Pi) = 0,
for the unknowns C1,. .. , Gtr , A. in this way the number of equations
equals the number of unknowns. For big values of N we obtain a small
A i.e. (A - 0).
THE BOUNDARY ELEMENTS METHOD
128
4.3.6 The Local Pressure Coefficient
As we have seen in 4.2.5, the local pressure coefficient for the incompressible fluid (4.2.33) is (4.2.35). Let us deduce the expression of
the local pressure coefficient for the compressible fluid. For the potential flow we have the formulas 11.101 page 206 which become, using the
notations from this chapter:
Vi ;fir
Pi = Pu
2
Pl-potl-ry-1
l
C?
:
c2o
V12\i
(4.3.26)
J
2
_co2t1-72
For the uniform flow that we have in view, these formulas give
7-1UW 7-l
2
cam)
7-1 U2
Pco - Po
2
T
(4.3.27)
co
=cfltl-7-1 U"
cclo
2
czo
From the last relation we deduce
k
,y-1
1
Uz
7-tlli2+
2
(4.3.28)
such that, denoting
r=72142,
(4.3.29)
it follows
1
k(7 - 1)
=1+1.
z
(4.3.30)
Introducing the parameter k in the formulas (4.3.26) and (4.3.27), we
THE DIRECT METHOD FOR THE UNLIMITED FLUID CASE
129
get:
P1=PO I-
V"k
00
\
r
(4.3.31)
Poo=P0(1-721k}T
,
7-1
Poo = Po 1-2
k)
T
These formulas will be replaced in (4.2.27). Taking out the factor
(7 - 1)k/2 and utilizing (4.3.30), it results
_
2
2 (
)]'-T-11.
(4.3.32)
Cp =
j -- 1+ ry 2 1 M2 (
U2
.yM2
For the incompressible fluid (M = 0) one obtains the formula
(4.2.29). For the compressible fluid, V12/U.2 from (4.3.32) will be
replaced by (1 + u/#)2 + v2.
We shall employ the local pressure coefficient in order to define the
lift coefficient which gives a global measure of the action of the fluid
against the profile
CL =
'f
Li
Cl,nylidat ,
(4.3.33)
where Ll is the length of the contour C1.
It is also important to get the expression of the critical velocity
v, because the subsonic character of the flow is maintained only if
Vl < vi,, . This velocity is obtained from (4.3.26) putting vl = cl and
utilizing the definition of k. We obtain:
U02
4.3.7
11+721 (M2 -1)]
.
(4.3.34)
Appendix
In this subsection we shall prove the formulas (4.3.11). To this aim,
we denote by C, the arc (semicircle) of the circle of radius e with the
center in Q0, interior to the profile (fig. 4.3.2) and we notice that we
have the parametrization
x-xo=Ecos9, y-yo=EsinO
(4.3.35)
00<0<00+ir
THE BOUNDARY ELEMENTS METHOD
130
Fig. d.3.2.
which, for u', z}' (4.1.13) implies:
U* (x, xa)
1 cos 0
2;r
v' (x, x)
E
1 sin y
27,,
(4.3.36)
E
We also consider the circle CR with the center in Qo, and the
radius R big enough, such that the profile C is in the interior of the
circle. On this circle we have the parametrization
x-x0=Rcos4
,
,-
such that
9l'
lcoscp
= in R
y-yp=RsincF,
V*
(4.3.37)
1Sill
27r R
In the domain D. exterior to the contour C - c + Cs and interior
'
to the circle CR, we may integrate the equation:
au'/ax + av'/(9y = a(.r - x0, y - yo)
(4.3.38)
Applying Green's formula we get:
(u'n., + v'ny)ds + Prr (0n, + v'ny)ds+
(4.3.39)
+
(u'n2 + Ti'ny)ds = 1.
THE AIRFOIL IN GROUND EFFECTS
131
or,
PC
lu*(x, xo)ns(x) + v'(x, xo)ny(x)jds +
1
+ 2n
f"'
o
2a
1
dO
+-
j dip = 1.
,
0
One obtains the first of the relations (4.3.11). Integrating on D the
equation
8u'/8y - 8v-/8x = 0,
one obtains the second of the relations (4.3.11).
4.3.8 Numerical Determinations
In figure 4.3.3 we compare u calculated by means of the direct
method with u calculated exactly. We observe that the values are
almost the same. In figure 4.3.4 we give 4 graphic representations: u
exact, u direct, u sources, u vortices. We notice that u obtained
by means of distributions of sources (4.2.15) does not give very good
results and u obtained by means of distributions of vortices (4.2.18)
gives better results. The values of u obtained with the direct method
are constantly in the vicinity of exact values of u .
Is
tS
as
a3
.1
Ti rr-FT lTF
43
TG
as
a
a o IN laI10us 110
210 210 010 110
a
to
.a
10 Ib ISO to ]b M 7s Im in IM
Fig. 4.3.3.
4.4
4.4.1
Fig. 4.3.4.
The Airfoil in Ground Effects
The Representation of the Solution
When the airplane is landing or taking off we have to take into account the influence of the ground. We shall consider this problem herein
'1'm; BOUNDARY ELEMENTS METHOD
132
Y;
x>
Fig. 4.4.1.
replacing the ground with the x10:1 plane (fig. 4.4.1). We have the
same problem like in the previous sections. Art uniform subsonic stream,
having the velocity U,,,, the pressure p and the density p", flowing
in the yt > 0 half-space, is perturbed by the presence of an infinite
cylindrical body, having the generatrices parallel to :r, IOzt, the Oxt
axis having the direction of the stream. The x10y1 plane determines a
cross section whose boundary C1 is assumed to be a thin profile. Using
the smite notations like in 4.1 we have to integrate the system (4.1.7) in
the ha lf-plane y > 0, with the boundary condition (4.1.10) on C and
the supplementary condition
zr(x, 0) = 0,
(4.4.1)
(V)X.
In this case, instead of the fundamental solution (u', v') from 4.1.3 we
shall utilize Green's distribution (U', V') determined by the system
au; 119X l aVV/ay = 6(x -
y -11)
(4.4.2)
(9DU/ay = 0,
I) representing a point front the y > 0 half-plane, and the
boundary condition
U'(a,0) = 0.
V+(r.0) = 0.
(V)x.
(4.4.3)
This distribution is obtained by means on the fundamental solution
(4.1.13) using the method of images. Denoting
2r,
(x - )2 + (y - rj)2
(4.4.4)
vt (T, y, , tl) _
1
27r (x
y-rI
- )2 + (y - 1)2
133
THE AIRFOIL IN GROUND EFFECTS
we have
Uf = u*(x,11,Cq) f u*(x,y,C-q)
(4.4.5)
Vf = v'(x,y, ,q) ±v'(x,y,C -q)
Indeed, u' (x, y, t, q) and v'(x, y, t, q) verify the system (4.1.12) for ev-
ery t in the xOy plane. Obviously, u'(x, y, t, -q) and v' (x, y, t, -q)
verify the system
8u'/8x+ 8v'/811= b(x - C y +q)
(4.4.6)
8v'/8x-8u'/Oy=0.
When t is in the superior half-plane (y > 0) this system becomes
8u'/8x + 8v'/ay = 0
(4.4.7)
8v'/8x + 8t,'/8y = 0.
With these specifications one easily finds that (4.4.5) verifies (4.4.2). It
is also very easy to verify the conditions (4.4.3).
Let us return now to the representation of the solution. Denoting by
D the domain from the superior half-plane y > 0 which is exterior to
C and delimited by a half-circle CR with the center in the origin, the
diameter on Ox and the radius big enough in order to contain C in
the interior (fig. 4.4.1), we deduce like in 4.3, for R - oo, the identity
[u(8f 1,9x
- 8g/8y) + v(8f/8y + 8g/8x)] ds =
(4.4.8)
IL)
=:
j [(u f + vg)n= + (v f - ug)ny] ds +
J
u(x, 0)9(x, O)dx.
0000
We took into account the fact that on the diameter of the half-circle we
have n = (0, -1) and that the integral on CR vanishes for R - oo
when f and g are distributions having the form U., V.
Putting now f = U+ and g = -V;, from (4.4.2) and (4.4.3), we
deduce
u( ) = J (u(x) [U+(x, 4)n1(x) + V+(x, )ny(x)] +
c
+v(x) [U+(x, C)n,(x) - V+(x, 4)ns(x)] Ids.
(4.4.9)
THE BOUNDARY ELEMENTS METHOD
134
In the same way, for f = V. and g = U , from (4.4.9) we obtain
vW =
PC {u(x) [V -*(x, F)nr(x)
- V*
(4.4.10)
+v(x)[U-w (x, )ns(x) + V`(x,Ony(x)]}ds.
Formally, the solution is identical to (4.3.3) and (4.3.4). It differs only
by Ut, V}.
4.4.2
The Integral Equation
Acting like in 4.3, we deduce
f
(1/2)u(xo) = PC {u(x)[U+(x,xo)n:(x) + V+ (m, xo)ny (x)] +
+v(x)[U..(z,xo)ny(x)
(1/2)v(xo)
_'
- V+(x,xo)ny(x)]}ds,
(4.4.11)
{u(x) [V=` (x, xo)n: (x) - U* (x, xo)ny(x)] +
+v(x) [U' (x, xo)n=(x) + V! (m, xo)nv(x)} }ds.
Utilizing the identities
PC (U}nt + V.ny)ds =
-2, P(U_ny - V'n=)ds = 0
(4.4.12)
which will be demonstrated at the end of this section, we obtain for G
defined by (4.3.9), the following integral equation:
(1/2)G(xo) - PC {U (x, xo)n.(xo) + VV (x, xo)ny(xo)+
+1bllni(z)+
(ny,x) [V_(x,xo)n=(x)-
(4.4.13)
-U, .(x, xo)n.,(x)] }Gds = Qnv(xo),
where the mark "prime" has the same significance like in (4.2.4).
The formulas (4.3.10) which determine u and v in function of C,
remain valid.
135
THE AIRFOIL IN GROUND EFFECTS
4.4.3
The Computer Implementation
The equation (4.4.13) is discretized like in 4.3.4. One obtains the
system
N
(1/2)G; +
i1
A;}Gj= B;,
I = 1, ... , N ,
(4.4.14)
where
AU = -n,,(x°)U,j o
o
-M2ny(x')+Q2n2(xq)( (x)V
(4.4.15)
+
with the notations
U, =
J Lf
U; (Z' x0)ds , V j =
j
V(x, x9)ds,
(4.4.16)
s
B; remaining unchanged. The system (4.4.14) will determine the unknowns C1,.. . , GN. From (4.3.22) it follows u;, v;, (i = 1, ... , N). The
circulation is obtained with the formula (4.3.21).
For lifting airfoils, we also consider the Kutta-Joukovsky condition
(4.3.24). We solve the problem introducing the regularization variable
A and acting like in 4.3.5.
In order to obtain the expressions of the unknowns U, j and Vtj*,
we use the notations (4.2.22) and the relations
Y= (xs -
x°)(x2 - x1) + (y° + y°)(y2 - y1),
c1= (xi -x°)2+(y;+y°)2
Ik =!-i1
(4.4.17)
ate+bit+
'
In this way one obtains
(z-x°)2+(y+y?)2=ate+b't+c',
4[(xjo -x0)(Io
U,3
Vi'*
=
-Lj
41r
kla)+x22x1(Ilfli)f
12 yl (Il f li)J
[Y-v?Iodv?+Y?I6+
(o
(4.4.18)
THE BOUNDARY ELEMENTS METHOD
136
n(x0) and n(xti) being defined in (4.2.25).
For It) and It one obtains the expressions (4.2.28). Since
A = b2 - 4ac'
(?1j + Y9)(x2
- XI) - (xj - xe))(?M -
x!1)]2
< 0,
it follows
4, =
a
2Qr arctan
,
C' -a
(4.4.19)
1t=2a III +b+c'-afwhere
,arctar.
-©
c' - a
,
a+b'+c' = (>'2-x°)2+(J2+J9)2
(4.4.20)
a -b<+cl =
For i = j the integrals Io and It become singular and they have
the values (4.2.30).
4.4.4
The Treatment of the Method
The problem presented in this subsection was studied in (4.10] and
we present with small modifications the solution given in that paper.
For testing the method we use the exact solution which is available for
the circular obstacle in ground effects (fig. 4.4.2), (4.12].
If the center of the circle is in the point (0, b) and the radius is a (a <
b), then, employing the usual notations, we have the exact solution:
u - iv= -2 E Z2 C"
+
n>1
where the sequences
+4
E
n>t
cn b»
Z2
+ b2
(4.4.21)
and {cn} are defined by the following
{bn}
formulas:
2
2
b"+1=b-b+a
c,++1=(b+bC11
b,,,
n)2n--0,
>
(4.4.22)
b1=b,
c1=a2.
Decomposing (4.4.21) in simple fractions, we get
at - iv = - E Gn [(z
- bn)2 +
n>1
i1
(1
(4.4.23)
+ ibn )2}
THE AIRFOIL IN GROUND EFFECTS
137
YA
U-
Fig. 4.4.2.
whence, separating the real part from the imaginary one, we deduce
u
= - L'`
n>1
x22-(y-b,,)2 + x2-(y+bn)2
f x2 + (y
- bn )212
[x2
+ (y + b.)212
(4.4.24)
v= -2xEG& rx y - b
+ (y - bn)212 +
n>1
y +bn
Jx2
+ (1/ + bn)?12
Putting
r=acos8, y=b+asinO,
we find uIC and vJ,.
Using the formula (4.2.35), we represent in figure 4.4.3 the exact
solution by a continuous curve and the approximate solution (in the case
of the incompressible fluid, rt = 0) by small squares. We considered
r = 0, a = 1 and b = 1.1. Since the two solutions practically coincide,
we conclude that the method presented herein is very good. It may be
applied for every shape of the airfoil.
4.4.5
The Circular Obstacle in a Compressible Fluid
In this case there is not available any exact solution, such that we have
only the information provided by the BIEh1. We have to take care to use
THE BOUNDARY ELEMENTS METHOD
138
4
C,
0
-4
.8
1
-12J
ace METHOD
EXACT
-16.J
-20 J
-24.
r
0
90
180
I
270
360
Fig. 4.4.3.
for the radius of the circle values that are small with respect to the other
characteristic lengths (for example, the distance to the ground) because
only in these circumstances we can employ the linearized equations. In
figure 4.4.4, we represent the coefficient CL defined by (4.3.33) for
the circular obstacle with the radius a = 1, considering 60 nodes on the
boundary, . Obviously, the compressibility determines the increase of the
lift (as we have already seen in the case of the free fluid). We considered
7 = 1.405. Giving the distance to the ground, we may calculate (in the
absence of the circulation) the values of M for which the flow remains
subsonic (one utilizes vQ given by the formula (4.3.34).
4.4.6
Appendix
In order to prove the formulas (4.4.12) we take into account that in
the entire plane, the solutions (U7, VV) satisfy the systems
aUU/ax+av}/ay=a(x-t,y-17)± 5(x-C y+ ii)
(4.4.25)
aVV/ax - aUl/ay = 0,
where 7> 0.
Like in 4.3.7, we denote by Cf the arc of circle interior to the profile
C, having the center in Qo(zo) and the radius a and by CR the
THE AIRFOIL IN GROUND EFFECTS
139
000C,
-0.02
-0.01
-0.06
-0.08
-0.10
-012
-0.14
-016.
-02000
10
0.5
1.5
2.0
30
2.5
3.5
H
Fig. 4.4.4.
circle having the center in Qo and the radius R big enough, such that
the profile C belong to the interior of the circle. The domain exterior to
the curve C - c + CF and interior to the circle CR will be denoted by
D. We shall integrate on this domain the first equation from (4.4.25)
and and we shall apply Green's formula. One obtains:
2 = P[U+(x, xo)nr(x) + V. (x, xo)nv(x)Ids+
+ find P (U+r1r + V. n.,,)ds +
-00
f)(U*nr +V. nv)ds .
1z
(4.4.26)
With the parametrizatiou (4.3.37) we deduce on CR:
:c - :r.0
COs
(r-xo)2+(y- ))2 - R
x-x'0
_
(x - xo)2 + (y + yo)2
Cos
/
'
_
T. - xU
R2 + 411o(y - yo) + 4Jo
I 1 +4sin
R
z
R
+42)
1
=
Cos
+O(R `)
whence,
u+ =
RV
-
+ O(Rr2), V+
si
+ O(R--)
140
THE BOUNDARY ELEMENTS METHOD
V. = O(R-2) .
Utilizing the parametrization (4.3.35) on C1 we have for example
U! = O(R-2),
y-yo+2yo
y+yo
(x-xo)2(y+yo)2 (x-xo)2+(y-vo)2+4yo(y-1m)+4y
2
(,+,,sin
2yo j (1 + Esyoe +
2yo
2
bI
+ 0(c).
Hence,
Uf
_
1
=tar
C0619+0(6),
s
_
V1
sine
1
e + 4Ayo
+0(6).
Taking into account that on C. we have d s = Ed 9 and on CR,
d s = Rdw, we get:
6o+x
lim
(U+n= + V nv)d s =
PC.
limo PC
tar ,
(U..n + V+ny)d s=
1
d9
2t
I f0 de = 2.
Replacing all these in (4.4.26) it follows the first relation (4.4.12).
4.5
4.5.1
The Airfoil in Tunnel Effects
The Representation of the Solution
Assuming that the airfoil is situated between two planes parallel to
the direction of the unperturbed stream (fig. 4.5.1), we have to integrate the system (4.1.7) in the domain exterior to the contour C and
bounded by the two planes, with the boundary conditions (4.1.10) and
the supplementary condition
(4.5.1)
v(x, 0) = 0, v(x, a) = 0 (V)x,
Denoting by D the domain of motion delimited by the sides x = fd
(fig. 4.5.2), we deduce for d - oo the identity
ID [u(Of I8x - 89/8y) + v(8f/8y + 89/&)I da =
= JR/u + gv)n, + (f v - 9u)nv)ds+
+J
+00
00
q(x, 0)u(x, 0)dx 00
g(x, a)u(x, a)d x .
+00
(4.5.2)
141
THE AIRFOIL IN TUNNEL EFFECTS
Fig. 4.5.2.
We took into account (4.5.1) and the fact that for x = td the integrals
vanish when d --+ oo if f and g behave like the solution of (4.1.13)
and it and v have the property (4.1.11).
For f =1.t and g = -V+, where (17+, V+) verifies in the domain
occupied by the fluid, the system (4.1.12) and the relations
V+(x, 0) = 0,
V+(x, a) = 0,
(4.5.3)
we get from (4.5.2):
u(4) =
lc 1u(x)[U+(x,01=(x)
(4.5.4)
+v(x) [U+(x, 4)ny(x) -V+(=. F)n7(x)] )d s.
Analogously, for f = V_ and g = U_, where (U_,V_) verify the
system (4.1.12) and have the supplementary property
U_(x,0) = 0,
U_(x,a) = 0,
(4.5.5)
we get from (4.5.2):
v() _
(4.5.6)
+v(x) [U-(x, t)rns(x) - V-(x, Ony(x)] }d s.
142
THE BOUNDARY ELEMENTS METHOD
Hence we obtain for u(x) and v(4) the same representation like in
(4.4.1), with the difference that (UU, V V ) become (U_+, V ).
4.5.2
Green Functions
The system (4.1.12) determines the fundamental solutions of the system (4.1.7). These are distributions. The fundamental solutions which
are defined only in a portion of the plane and satisfy the boundary conditions are named Green functions, or, more properly, Green distributions.
The solutions (U+,V+) and (U_,V_) are therefore Green functions.
For obtaining these solutions we employ the method of images. If M(t)
is a point of the strip -oo < z < +oo 0 < y < a (fig. 4.5.1), then
the solution (without restrictions) of the system (4.1.12) is (4.1.13). As
we already know, it represents the perturbation produced in the entire
plane xOy by a source located in M. For satisfying the first condition
of (4.5.3), we have to place a source having the same intensity (taken to
be equal to the unity) in the symmetric point M(t, -ri). For satisfying the second condition of (4.5.3), we have to place sources having the
same intensity in the points which are the symmetric points of M and
M with respect to the y = a axis, i.e. in the points Mi (t, 2a - rl),
Mi (t, 2a + q). These sources disturb the condition on y = 0, such that
we have to place sources in the symmetrical points of Mi and Mi
with respect to the y = 0 axis, i.e. in the points M+1 (t, -2a + i) and
-2a - ,) etc. The perturbations produced by these sources are
x-t
1
+
V(")+
27r (x-t)2+[(y-(2na±q)12
1
y-(2nat9)
(4.5.7)
27r (x - t)2 + [y - (2na ±, )12
and they satisfy the system
OUP }/Ox + OV n)/Oy = a(x - t, y _ (2na ± ii))
(4.5.8)
OVA" i /Ox + OD'") /ay = 0.
In the strip -oo < x < +oo, 0 < y < a, all the perturbations
satisfy the homogeneous system, excepting the perturbation (Uo , V+)
THE AIRFOIL IN TUNNEL EFFECTS
143
which satisfies the corresponding non-honwgeneous system.
lI+(x, ) = 2;r
(x'
+ [y - (2nn + 11)]'
+
(4.5.9)
r:
I
'V-4'
{ (r
=-00
y - (2nn + rl)
}
(x - )2 + [y - (2nn + n)]2
2,r n=-00
y - (2na - r1)
[y- (2na-17)]2
+{:r.
satisfies the system (4.1.12) and the boundary conditions (4.5.3). It is
just the solution utilized in (4.5.4).
In the same way we deduce that the solution
U-(x, ) = 2;r
{ (x - C)2 + [y - (2na + 11)12
-
x-t
(4.5.10)
)-2,
L
I
n=-1
(2na
+ 11)
(a-ti)2+ [y-(2na+1))]2+
y-(2na-11)
+ (x - )2 + [y - (2na -11)]1
satisfies the system (4.1.12) and the boundary condition (4.5.5). This
is just the solution utilized in (4.5.6). The series (4.5.9) and (4.5.10) are
the Green functions of the problem.
The sums intervening in (4.5.9) and (4.5.10) may be calculated using
the formula
= ircot az,
(4.5.11)
where z = x + iy. Separating the real part, from the imaginary one, we
144
THE BOUNDARY ELEMENTS METHOD
get from (4.5.11):
+oo
y
y2 + (x - n)2
= 7r
sink 27ry
cosh 2iry - cos 21rx
(4.5.12)
+00
0
x-n
sin 2irx
n=-ao y2 + (x - n)2 = A cosh 2iry - cos 27rx
Using these relations we obtain the expressions of the Green functions:
sinhQ(x-
(x - ) - cos Q (y - q)
cosh
Q
71
f=4n cosh
1
sinh -(x
- .)
a (x - ) - cos a (y + +))
(4.5.13)
7r
_
V f (x, ) =
sin -(y - q)
n
1
sin
7;(y
cosh Q(x - e) -
4.5.3
+
A
`a
+,q)
7r(y + n)
The Integral Equation
In the sequel we shall proceed like in 4.3.2, i.e. we pass to the limit
-i xo E C in (4.5.4) and (4.5.6), we introduce the function G by
means of the formula (4.3.9) and we utilize the relations
+(x, xo)n.(x) + V+ (x, xo)ny(x)]d s = - 2
(4.5.14)
lu-(x,xo)ny (x) - V-(x,xo)n.(x)1ds = 0,
145
THE AIRFOIL IN TUNNEL EFFECTS
which will be demonstrated in Appendix 4.5.5. One obtains the following
integral equation:
(1/2)G(xo) - c {U_(x,xo)n(x)
V+(x, xo)nt,(x)+
+M2ni(X) + f(X) [V-(x, xo)nx(xo)-
(4.5.15)
-U+(x, xo)n,(xo)]}Gds = Onv(xo) ,
and discretizing, we deduce the system
N
(1/2)Gi + E AijGj = Bi, j = 1, ... , N,
(4.3.16)
i.t
where Aij is A!. from (4.5.15) with the notatipns
U= f Dx, x°)ds, l= J V f(x, x°)d s
.
(4.5.17)
;
..
In order to determine these coefficients we consider on Lj the
parametrization (4.2.19). Further we have two possibilities: the exact calculus like in 4.2.4, or the approximate calculus, with Gauss-type
quadrature formulas (A.55], for the resulting integrals
+l Ut(t)d
1-'
Uii =
2
t, Vj = 1_j t1
f V±(t)d t .
2
t
(4.5.18)
t
In case that i = j, these integrals become singular for t = 0, because
of the terms
sinh a [x(t) - x?d t
+1
t
cosh
a
sin
t
ui
[x(t) - x°] - cos'-` [y(t) - A
a
a
(y(t) - y°) d t
[x(t) - x°] - cos [y(t) - y°j
cosh
a
a
146
THE BOUNDARY ELEMENTS METHOD
but, taking into account that the integrands are odd functions, we deduce that they vanish. We have therefore
1.
+1
8a
i
Uf=± `
}-
`" -
sink - [x(t) - x9 'd t
a
cosh
[x(t) -;J - cos a [y(t) - y°]
(4.5.19)
a
sin
+e
Co.),
a
[y(t) - y,)jd t
n
7r
[x(t) - x,)[ - cos [y(t) - yp[
a
a
These integrals may be also calculated numerically, using the Gauss-type
quadrature formulas.
4.5.4 The Verification of the Method
In order to perform this verification the shall use the solution given
by the complex potential
f (z) = z + Ctanh
(4.5.20)
z
where C and a are real constants. Let us study the flow determined
by this potential. Employing Euler's formulas we deduce the identities
sinh iy = i sin y,
cosh iy = cos y
cosh 2 Ax cos2 Ay + sinh2 Ax sins Ay = 2 (cosh 2Ax + oos 2Ay),
cosh2 Ax cost Ay
- sinh2Ax sin Ay = 2 (1 + cosh 2Ax cos 2Ay),
(4.5.21)
such that, separating the imaginary parts, we get from (4.5.20):
C sin 2Ay
- y + cosh 2Ax + cos 2Ay '
(4.5.22)
We denoted A = 7r/a. Obviously the straight lines y = 0 and y = a
are streamlines ('P = 0 and 'I' = a). Another streamline is the line
y = a/2(41 = a/2) but not entirely because for x = 0 the fraction from
(4.5.22) is not determined.
Further we shall study the velocity field. From (4.5.20) it results
w(z) = u
iv = I +
AC
cosh2 Az
,
(4.5.23)
THE AIRFOIL IN TUNNEL EFFECTS
147
whence, utilizing (4.5.21),
u.
= 1 + 2AC
1 + ch2 Ax cos 2Ay
(cosh 2Ax + cos 2Ay)2
(4.5.24)
V = 2AC
sinh2 Ax sin 2Ay
(cosh 2Ax + cos 2Ay)2
Now we return to the streamline y = a/2. This line may be continued ('I' = a/2) on the set of points (x, y) where we have:
a
z
=
-y+
C sin 2Ay
cosh 2Ax + cos 2Ay
(4.5.25)
This curve is symmetric with respect to the Oy axis, because the equation is even in the x variable. It is also symmetric with respect to the
y = a/2 axis, because the points having the ordinates y = a/2 + yo and
y = a/2 - yo, where
_
yo
C sin 2Ayo
cosh 2Ax - cost Ayo
(4.5.26)
simultaneously belong to the curve. Hence the curve having the equations (4.5.25) is an oval which orthogonally intersects the axes y = a/2
and x = 0. In the point P(-a, a/2) we have therefore u = 0. From
(4.5.24) it follows the equation
sinh2An = AC,
(4.5.27)
for the half-diameter a as a function of C. The semi-diameter 6 on
the Oy axis may be determined from (4.5.26) as follows
_
C sin 2A,0
1-cos2Af3'
whence it results a simpler equation
$tan AQ = C.
(4.5.28)
We draw the conclusion that the complex potential (4.5.20) characterizes
the uniform flow with the velocity (1, 0), in the channel having the sides
y = 0 and y = a, in the presence of the oval having the diameters 2a
and 2/3 with the center on the Oy axis (fig. 4.5.3).
It is not difficult to see how this potential was obtained. One knows
that the uniform flow at infinity, in the presence of a doublet, determines
THE BOUNDARY LLPMENTSMEITHOD
148
y=0
L. H=8110
x=0
Fig
the flow past a circular obstacle. Let us study now the circular flow in
the presence of a doublet in a channel. For the sake of simplicity, we
consider the doublet situated in the origin and y = a/2, y = -a/2, the
equations of the lines which are the walls of the channel.
These right lines become streamlines for the flow determined by the
doublet, if we add, according to the method of images. symmetric doublets. The potential f0(z) describing the flow produced by these doublets is
A(z)_...+
I
z+ia
+1+
+...
1
=
_ro
_ let
-cot Trz
Ia
IT
1
Ir
irz
u
a
-coth
Ave take
fo(z) = CcothA>. A = 7r/a,
C representing it real o constant. If the Ox axis coincides with the
lower wall. as we have previously considered. then the potential becomes
fo(z) = Ceoth A
(:_ i a2)
= Ct.unh Az .
Adding the potential of the uniform flow having the velocity (1,0) we
obtain (4.5.20).
In the paper [4.9] that we utilized for writing this subsection, we
made tests for the local pressure coefficient C,,, defined by (4.2.35). In
figure 4.5.4 we present the values of Ci, calculated exactly (by means of
(4.5.24) and the values calculated with the numerical method presented
above with H = a/10. The maximum relative error is 0.73% in
the case of a discretiration with 80 nodes. For the circular obstacle,
symmetrically placed in a tunnel, we present the compressibility effect
in figure 4.5.5 where one indicates the variation of the maximum velocity
THE AIRFOIL IN TUNNEL EFFECTS
149
i-
I -o-
-
na i
\
0 J
Fig. 4.5.4.
_
Fig. 4.5.3.
against the distance to the walls for 'r = 1.405 in the cases M = 0 and
M = 0.3 and in the absence of the circulation (F = 0). We considered
60 equidistant nodes on the circumference.
In [4.9) one presents the graphic representations for C, in the case
of the NACA-4412 profile at 0 angle of attack and Mach=0.5 in a free
stream and in a tunnel. The example is instructive, because, in this
case the trailing edge is angular, and we have to impose the equality
of the pressures on the two sides of the edge in order to determine the
circulation.
4.5.5 Appendix
In order to prove the first formula (4.5.14) we shall notice that (U+, V+)
(x, y, xo, yo) satisfy in the domain D, exterior to the contour C-c+C1,
and bounded by the walls y = 0 and y = a and by the segments
x = ±d, the equation:
aU+/49x+aV+/Oy=6(x-xo,y-yo).
Integrating this equation over D and applying Gauss's formula one
obtains:
J (U+nx + V+n1)d s + ii;t a 1c,
+ d im
ra
J0
(U+n" +
s+
U+(d, y, xo, yo)d p + slim J t7+(-d, y, xo, yo)d p = 1,
-oo J.a
(4.5.29)
THE BOUNDARY ELEMENTS METHOD
150
the integrals on y = 0 and y = a vanishing by virtue of the relations
(4.5.1). The integrals on C6 are calculated utilizing the parauiietrizntions (4.3.35) which imply
sin0+C,
cos0+0(e),
Uf=
I
I
where
C_
1
2ryo
yn
1
1 + a2
n=0
(yo/a)2 - n2
To the limit, when a -i 0, this is 1/2. One calculate the last two
integrals taking into account that
d t
Id
_
2
(a. - b)tan (t/2)
a -b
a+bcost
arctan
a -b'
To the limit, when d -- 0, every integral is 1/2. One obtains the
first formula (4.5.14). Analogously, we obtain the second formula if we
integrate the equation
aTJ_/ay
- 8TH-/ax = 0
on the same domain D.
4.6
4.6.1
Other Methods. The Intrinsic Integral Equation
The Method of Regularization
The methods that we have already utilized have been given by L.
Drago§ [4.6] and by L. Drago§ and A. Dinu [4.7], [4.8], [4.9], [4.10].
Specific to these methods is the fact that one utilizes physical variables
the velocity and pressure fields), such that from the solution one obtains
directly the elements of interest in aerodynamics (the velocity and the
pressure on the profile). Other methods (Biudolino a.o. [4.2], Morino
and Luo [4.20]) utilize the real potential and others (Griello a.o. [4.15],
Carabineanu [4.4]), the stream function. All these methods may be
utilized only for incompressible fluid. The theory that we are going to
present in the sequel will be also applied to the incompressible fluid, but
it has the advantage to utilize the physical fields. Moreover, the singular
integrals are avoided and one utilizes the intrinsic elements of the flow
(the tangential component of the velocity). Therefore it is suitable to
call this method the method of regularization [4.11].
OTHER METHODS. THE INTRINSIC INTEGRAL EQUATION
151
Denoting
V = Un(i+v),
(4.6.1)
we shall determine the perturbation from the equations
dive=0, rotv=0
(4.6.2)
with the boundary condition
(4.6.3)
and the condition at infinity
limn v(x) = 0.
(4.6.4)
We may write the equations (4.6.2) as follows
div (v - c) = 0, rot (v - c) = 0,
(4.6.5)
c representing a constant vector. We put into evidence the normal and
tangential components of the velocity:
v = (v n)n + (v s)s =
v,8.
(4.6.6)
By virtue of the condition (4.6.3) we may write
v = -n?n + v,s.
(4.6.7)
Rewriting the scalar formulas from 4.3 as a vectorial formula we
deduce that for every two continuously differentiable functions or distri-
butions f and g , by virtue of the equations (4.6.5) (k - the versor of
the Oz axis), we have:
ID if div (v - c) + (gk) rot(v - c)]da = 0,
(4.6.8)
D representing, like, in 4.3.1, the domain exterior to C and interior to
the circle Cn having the radius R big enough. Utilizing the formulas
div f (v - c) = (v - c) grad f + f div (v - c)
(4.6.9)
div [gk x (v - c)] = (v - c) rot (gk) - gk rot (v - c)
and applying Gauss's theorem, we obtain the identity:
J(v -c) gradf-rot(gk))da=
f(v - c) (fn-nxgk)ds,
}- CR
(4.6.10)
152
THE BOUNDARY ELEMENTS METHOD
n being the normal (pointing outwards the domain D (i.e. the inward
pointing normal with respect to C and the outward pointing normal
with respect to CR).
We shall write the distribution (4.1.13) as follows
v,-
1
x - xn
2r,
Ix - xo12
(4.6.11)
and the system (4.1.12) that it satisfie :
divv'=b(x-x()), rote'=0,
(4.6.12)
the equations being valid for every point xn from D + C. For (f, g)
(u', -v'), one obtains from (4.6.10) the projection of the identity
(v - c)divv'da = J
{[n (v - c)]v + [n x (v -- c)] x v}ds,
+CR
JJJD
(4.6.13)
on the Or axis. For (f,g)
(v'.u') we obtain the projection of the
same identity on the Oy axis. Hence the relation (4.6.13) is valid. By
virtue of (4.6.12) we obtain from (4.6.13)
v(xa) - c = f
{[n (v - c)]v' 4. [ri x (t+ - c)] x v}tl s. (4.6.14)
TCu
llere we shall put c = v(xo) - vo and we shall evaluate the integral
on CJt. Utiliziug the parametrization (4.3.37), we shall deduce for the
projection of this integral on the Ox axis
xli
{n - (v - vo)u' + (u - ua)(u'n7 + v'n,)2m
-riI[(ei - 110W + (v - vo)v']}d3 =
1
2=r j (u - t!o)dV = -110,
because for R , oo, u vanishes according to the condition (4.6.4). One
deduces a similar formula for the projection on the Oy axis.
Utilizing these results one obtains from (4.6.14) the following representation
V(x0) = PC {[n (v - vo)]v' + In x (v - vo)] x v' }d s.
(4.6.15)
The integral is not singular because the factor v - vo tends to zero
when x xo E C. This is a regularized integral. The formula (4.6.15)
is valid for both xo in the fluid and on the boundary C.
OTHER METHODS. THE INTRINSIC INTEGRAL EQUATION
153
We notice utilizing (4.6.7) that on C we have
nxv=v,k
(4.6.16)
In - (v-vo)]v' = (-n=+
such that, after elementary calculations, from (4.6.15) one obtains
V(xo) _
P(v,k x v' + v;((v' . s°)n-(n v')s° - (n s°)v'] - nxv'-
(4.6.17)
-79x((n° v`)n - (n . v')n° - (n n°)v']}d s.
For obtaining the unknown v, outside the integral, we shall perform
the operation no x (4.6.17) and we shall take into account that we have
no_ soy, n o_
y - -soy,
(4 6 18)
no x v' = (so v')k
(4.6.19)
:-
.
.
We deduce
no x s°
whence
11°=
4.6.20)
((s° . v')n' - (so v')ns]d s.
This is the integral equation of the problem. It is a regularized (nonsingular) integral equation because when x - x°, the denominators and
the numerators
(n° v')v°) of the integrand simultaneously
vanish (see (4.6.11)). The equation (4.6.20) was obtained in a different
manner by V.Cardng (4.5]. It is, obviously an intrinsic equation. This
equation may be also solved by means of the boundary elements method.
One obtains, like above, a linear algebraic system of N equations with
N unknowns.
In the case of lifting airfoils the following boundary condition is
added (see also (4.3.23)).
-+Pf.
(4.6.21)
154
THE BOUNDARY ELEMENTS METHOD
Taking (4.6.1) and (4.6.18) into account, we apply this condition as
follows:
v. (P.) + v.(A) = ny(P.) + ny(Pi) ,
(4.6.22)
P, and Pi being sufficiently close of Pf. From the equation (4.6.20)
and the condition (4.6.22) we shall determine v, on C. 'Since we have
more equations than unknowns, we shall treat the problem like in 4.3.5,
introducing the auxiliary variable A.
The reader may find examples in [4.11].
Chapter 5
The Theory of Finite Span Airfoil in Subsonic
Flow. The Lifting Surface Theory
5.1
5.1.1
The Lifting Surface Equation
The Statement of the Problem
We assume that an uniform flow, having the velocity
(the Ox axis has the direction and the sense of the uniform flow), the pressure
p,, , the density p,, and Af(= U,./c,.) < 1, is perturbed by the
presence of a finite span wing, perpendicular on the flow direction. Any
body which has a characteristic dimension much larger then the two
other dimensions is considered to be a wing. We call the span of the wing,
the length of the wing taken along the direction of the large characteristic
dimension (figure 5.1.1).
Fig. 5.1.1.
One requires to determine the action of the fluid against the wing.
In fact. as we have already seen when we studied the two-dimensional
case, for determining the action, one has to calculate the perturbation
THE LIFTING SURFACE THEORY
156
fields p(x) and v(x).
We employ the Cartesian variables x,y, z introduced in (2.1.1) and
the fields p and v introduced by (2.1.3). The Oy-axis has the span
direction and the origin 0 is situated in the middle of the wing. The
Oz-axis is taken perpendicular to the Ox and Oy-axes in order to
detennine a Cartesian, positive oriented frame.
We assume that the projection of the wing onto the xOy-plane
is a simple connected domain D, with a piecewise smooth boundary
&D such that every straight line parallel to the Ox-axis (which has
the direction of the unperturbed flow) should intersect the boundary in
almost two points Q. and Q f. As we may see in figures 5.1.2 and 5.1.3,
the lateral edges of the wing may be exceptions from this assumption if
they are parallel to the Ox -axis . The intersection of 8D with a
lateral edge parallel to the Ox -axis consists of a point, two confounded
points or a straight segment. Assuming that the two lateral edges consist
of a point (denoted respectively by B and B') we notice that they
divide the boundary into two arcs
The front arc BB' which is attacked by the stream is called the
leading edge and the rear arc is called the trailing edge. The equation
of the leading edge (consisting of the points Q,) is x = x_(y), and
the equation of the trailing edge is x = x+(y). Hence Qa has the
Cartesian coordinates (x- (y), y) and Qf (x+ (y), y).
For wings with lateral edges consisting of simple or confounded points
we obviously have:
x+(±b) = x-(±b),
(5.1.1)
where 2b is the span (in dimensionless variables). We shall call these
ones I wings, and the wings whose lateral edges consist of straight segments will be called II wings. For the II wings the condition (5.1.1) will
be replaced by the condition of the continuity of the pressure along the
edge (the Kutta-Joukowski condition (see (5.1.30)). In figures 5.1.2 we
indicated the domain D for some I wings having lateral edges consisting
of simple points (the delta wing, figure a), the gothic wing, figure b), the
trapezoidal wing, figure c), the rhombic wing, figure d), the swallow tail
wing, figure e) or lateral edges consisting of double confounded points
(the elliptical wing, figure f)).
In figure 5.1.3 we present the domain D for the arrow shaped wing
(with lateral edges consisting of straight segments).
The domain D is named the plane form of the wing or the planar
form of the wing.
157
THE LIF"TINC SURFACE EQUATION
b)
a)
d)
f)
e)
Fig. 5.1.2.
We notice that the wing has two surfaces: the upper surface, denoted
by S+ and the lower surface denoted by S_ . The equations of these
THE LIFTi G SURFACE THEORY
158
surfaces may be written (for the sake of simplicity) in the following form
z = h(x, y) f hj (x, y) ,
(x, y) E D .
(5.1.2)
Indeed, considering z = f (x, y) the equation for S+ and z = g(x, y)
the equation for S_, then, setting
2h(x, y) = f (x, y) + g(x, y)
(5.1.3)
2h1(x,y) = f(x,y) - g(x,y),
we obtain (5.1.2).
B.
Fig. 5.1.3.
The wing is thin if and only if h and h1 have the form
h(x, y) = A-(x, y),
h1(x, y) = Fhl (x, y)
,
(5.1.4)
E(<<1) being a real parameter (max h, h1 in D), and !t $i hl
bounded functions. We also assume that h(x,y) and hl (x, y) are
defined on D and have the first order derivatives with respect to x,
denoted by h,,, hi=.
The two surfaces S+ and S_ do not intersect each other if and
only if
h(x, y) + hl (x, y) > h(x, y) - hi (x, y).
5.1.2
(5.1.5)
Bibliographical Comments
The bibliography concerning this subject is extremely rids and it
is not possible tp mention it entirely. In fact, this is not necessary
'ME LIFTING SURFACE EQUATION
159
because the fundamental solutions method, utilized herein is different
from the previously utilized methods. As we have already mentioned
in the introduction, the methods utilized so far rely on the equation
of the potential (we assume that the perturbation is irrotational, the
wing being assimilated to a distributions of bound vortices or doublets.
Downstream, behind the wing, where the experience shows that the flow
is not irrotational, one introduces in an artificial manner a free vortices
distribution. The perturbation of the velocity results from the velocity
field induced by the two distributions. This is Prandtl's model [6.21) for
the lifting line theory, which assumes that the wing may be replaced by
the segment BB'. The lifting surface theory, which does not replace
the domain D with the segment BY, was developed after 1936 by
Prandtl [5.26], (5.27), Weissinger [5.35], Reissner [5.29], Multhopp [5.24],
Flax, Lawrence [5.12), Truckenbrodt (5.32), Mangler and Spencer (5.22),
etc. We also mention the unpublished paper [5.6] of P.Cociirlan. Some
synthesis of the research in this domain may be found for example in
[1.1), [1.2], (1.38) and [1.41]. D. Homentcovschi [5.16] presents, for the
compressible fluid, a theory relying on the equations of motion written
in distributions, without assuming that the flow is potential. Another
theory is given in (5.7]. Here one introduces the fundamental solutions
method. It is not necessary to assume that the flow is potential and a
vortices sheet is formed behind the the wing. These properties result
from equations.
5.1.3
The General Solution
As we have already mentioned in 3.1.3, there is no physical reason to
replace the wing by a distribution of vortices. It is reasonable to replace
the wing by a distribution of forces which should have against the fluid
the same action as the wing. Indeed, the fluid and the wing must be
considered as an interacting material system. According to Cauchy's
tension principle (see [1.11), p.35), there exists a ford distribution f
on the wing (or on D) which has the same action against the fluid like
the wing itself. The perturbation of the fluid flow will be determined
by this distribution and by the slipping condition on the surface of the
wing. We try to employ a forces distribution on D, having the form
f = (f1,0,f)(,t1)
If we manage to satisfy the boundary conditions, on the basis of the
uniqueness theorem, we deduce that we found the solution of the prob-
160
THE LIFTING SURFACE THEORY
Obviously, there exist various distributions which enable us to
reach the solution. In chapter 7 we shall employ distributions on the
boundary of the body.
From (2.3.24) we deduce that the perturbation of the pressure due
1em.
to the force f applied in the point (t, q) is
ft(c rt)8 + f(c n)8 u-.) ,
(5.1.6)
xo=x-t,yo=y-v1, Ri= xo+(yo+z2),
(5.1.7)
P(x,IV,x)
J
where
A continuous superposition of such forces on D, will give
[fi(ii)- + f
P(x, y, z) = -1
n) ez
]
JJD
since the problem is linear.
For the potential jp, from (2.3.28) we deduce
( X y, z) = T 7r-
JD
[f
(R1
) -- uo+
z z2
(_) d t d n ,
+ RI) f (t,
n)1 d
(5.1.8)
dn
(5.1.9)
and for the velocity field from (2.3.12)
v(x, y, z) = 6(z) Jf f(, t7)H(xo)b(yo)d d r< + Oip
(5.1.10)
which can be written explicitly:
u(x, y, z) = 6(z) Jj h (, r1)H(xo)a(yo)d d r- p(x, y, z)
v(x,y,z) =
AD I fl
,
(f,n)' (i;)
-f(t,rl)Z [y
(i+9.)] }ddt
2
(5.1.11)
w(x, y, z) = 6(z)
IfD f (t, tt)H(xo)5(yo)df d n+
+4- fD f fi(t,n)g ()dd_
r1
-4Rffnf(F,n)az
[1l0
z2
THE LIFTING SURFACE EQUATION
161
For w, from (2.3.29) we get:
w(x,y,z)
4rr
lID
[fi(i); (RI}
-Q2f(E,rl)
CRI)
JdC dq+
+4 AD f(C17)Zy
(5.1.12)
(l+ Xg
Jo2+YO
Ri)J
z2
dtdn
From formula (5.1.10) we deduce that the perturbation is potential,
excepting the strip from the xOy - plane, -b < y < b, x > x_ (y),
where the first term does not vanish. We notice that the existence of the
vortices sheet behind the wing, which was a hypothesis in the classical
theories (see Prandtl [6.21]), represents herein a consequence following
from the equations of motion. As the experience confirms the existence
of such a sheet, it means that the experience confirms the mathematical
model based on the fundamental solutions method.
5.1.4
The Boundary Values of the Pressure
In view of calculating the boundary values of the pressure when z -'
±0 and (x, y) E D, we notice that the first term from (5.1.8) represents
the tangential derivative of a simple layer potential and the second,
the normal derivative of the same potential. From the potential theory
one knows (see for example, [1.6), [1.39]) how to obtain the boundary
values of these derivatives but we prefer to calculate them directly. After
deriving we have to calculate
It
limo
f
i
(s, y) E D
(5.1.13)
=-
lim
z-'.±O JD
(x, y) ED
dd
f(t,n)
.
1
We notice that setting z = ±0 and (x, y) E D, Rl vanishes when
the generic integration point Q(t, n) coincides with M(x, y) E D. The
integral Il becomes singular. Denoting by DE the disk with the radius
THE LIFTING SURFACE THEORY
162
E and the center h1, we shall write and we shall consider by definition
,
hm
(5.1.14)
U IJD
110 = JJ-D, +AD,
if the limit exists. For calculating the integral on Dt, one makes the
change of variable
O7D,
q -+ r, 0:
- x = Or c os 0, tj - y = r sin 0
(5.1.15)
0<r<e,
0<0<21r
and one notices that if f is a continuous function, for a small enough,
we may approximate f on the disk with f (x, y). We deduce:
AD,
1
and then, the formula
dtj=
fJ
r2 cos
f(xJ 1!) 12,,
0c (
f1
r2
+z 2)3/2
d9dr=0
=o1 f(e,r1)0dedt,
R
(5.1.16)
D
where
R= Vxg+
(5.1.17)
As it is shown in (E.0.12), the last integral exists.
As regarding 12, it vanishes outside a vicinity DD of M. We have
therefore
I2
Zo
:
(x. y) E D
.f(C,t]) z dCdti
lID
74
2s
=f
li
n)
fo
z
Jo
(5.1.18)
rd 0d r
= f2tr f (x, y)
Jo (r2 + z2)3/2
(;, v) E D
Taking into account these results, from (5.1.8) we deduce:
p(x, y, f0) =
-
0
ID f i (4, r]) ? d d of 2f (x, y), (x, y) E D. (5.1.19)
This is the formula for the pressure on the upper and lower surfaces of
the wing. We deduce the formula
(5.1.20)
p(x, y, +0) - p(2, y, -0) = f (x, y) ,
which gives us the significance of the function f (x, y). Like in the twodimensional case, f is the jump of the pressure on the wing. This is a
fundamental result for the calculus of the aerodynamic action.
THE LIFTING SURFACE EQUATION
5.1.5
163
The Boundary Values of the Component w
If one utilizes (5.1.11), then taking the limit, when z -+ ±0, the first
term disappears because 5(±0) = 0. It is more difficult to calculate the
limit of the last term because it contains the derivative with respect to
z. The calculation is performed in (5.221 and in 11.111, p.208, 209. We
utilize here the expression (5.1.12) for w. In this expression, the limits
of the first two terms are given by the formulas (5.1.17) and (5.1.18) and
the derivative with respect to y from the last term commutes with the
limit. We have therefore,
lira
13
a8y yo + z
I1
a
z
(1 + La
Rl )] dC drl =
(s, y) E D
(5.1.21)
8
_
j22K(xty,z7ii)dt,
y) ED
+Cr,
with the notation
x+(7)
K(x,y,z,r1)=1_
(5.1.22)
00
After setting z = 0 in the last expression of 13, we notice that
we may not simplify with yo, because it vanishes for 77 = y. We shall
divide the integral into three parts:
J-bb
_ 1b £+Jy+E+
JyyEa
The last integral vanishes because for F small enough we may consider
K(x, y, z, v7) = K(x, y, z, y) whence
+E
TY-C
yoz
K(x,y,z,rl)dn=-K(x,v,z,y) f +E
+z =0.
E u 2du2
For the integrals on (-b, y - s) and (y + e, +b) we may pass to the
limit, setting z ± 0 and we may then simplify by yo, because it does
THE LIFTING suitFACE THEORY
164
not vanish. It follows
13 =
a
Hen I I
`J b
3y
= line
+ r rb\ ! (r., rl, 0, 0
d rJ
Jy+il
1C(x,'y, 0, y - E)
l
f
+b\l
,,...s
+
t3
=
Yo
1i
(x, y.
L'
+ \J-b + Jv+c/ ay \ yo /
0,
y + `)
(5.1.23)
d tJ
Expanding into a Taylor series the first two terms and taking into
account the definition of the "Principal Value" (D.3.1) and the definition
of the "Finite Part" (D.3.6), we deduce
3'/`' b
1
+1
tb
h'(x,
II
_JJDf
-7( o- `1+
Rddy+2
ayh(x,y,0,J1)di, _
JJ,
f(e,iJ)x
(5.1.24)
Taking into account the expressions of 11,11 and 13 from (5.1.12) we
obtain:
u,(x, y, ±0)
5.1.6
= f fl (?', y) - 41, 1fn f (y, i) (l +
)
d d r1 . (5.1.25)
The Integral Equation
F om the boundary conditions (2.1.33) and the equatiolLS (5.1.2) we
deduce:
ua(x, y, f0) = Jr=(3:. y) t h' (x, y)
,
(:r, y) E D .
(5.1.26)
Replacing the last relation in (5.1.25), subtracting and adding the two
relations, the first corresponding to the upper sign (the condition on the
upper surface) and the second corresponding to the lower sign, we get
(5.1.27)
4
_ ly JJ n
C1
yo
\
+
0)
d diJ = hr(x,y),
R
(x, y) E D.
(5.1.28)
THE LIFTING SURFACE EQUATION
165
The relation (5.1.27) determines directly f, (w, y) and the relation
(5.1.28) constitutes an integral equation for determining the function f .
This is the well known equation of the lifting surface. It is obviously a
singular integral equation with a strong singularity. The mark * is
for the "Finite Part". For the incompressible fluid, The equation has
been written for the first time by Multhopp in 1950 [5.24]. A rigorous demonstration is given by Mangler in the Appendix of the above
mentioned article. The demonstration is quite complicated because the
author utilizes the representation
w(x,y,z)=-47r
(5.1.29)
which follows from (5.1.11). Short deductions are presented in [1.1],
[1.38].
For wings with lateral edges consisting from straight segments (figure
5.1.3) the equation (5.1.28) is integrated with the conditions
f (x, ±b) = 0
(5.1.30)
which, taking into account (5.1.20), means the continuity of the pressure
(the Kutta-Joukowski condition). On the trailing edge
x = x+(y), -b < y < +b
(5.1.31)
one imposes the boundedness condition for the function f (x, y). An
equation equivalent to (5.1.28) was given by Lawrence and Flax in 1951
[6.11]. Formally, it can be obtained from (5.1.28) using the identity
)=
R
(I+0
yo
xo+R _ 8 rxo+Ro
xpyo)
8y
1 \ xoyo
8
(5.1.32)
It follows:
4v
1
A
f(
AC 17)
(1+)ddtih'(xY)1
(x, y) E D.
(5.1.33)
Sometimes it is preferable to use this equation instead of (5.1.28), because it has weaker singularities (Cauchy-type singularities). A rigorous
deduction of equation (5.1.33) from (5.1.28) must start from the definitions of the Finite Part and Principal Value . Using the function
C(y) = -
/
+(n)
s . (n)
f (t, t1)d e ,
(5.1.34)
TILE LIMNG SURFACE THEORY
166
whose significance will be given in 6.1, one demonstrates the identities
r f(art) d z; d q - -
= Iv
yJD
dy
ff
X
f(C rl) d
(5.1.35)
d d(5.1.36)
ff
aUo
d
e
which lead to the equivalence in view (see also 6.1).
5.1.7 Other Forms of the Integral Equation
1. We are going to give another form for the expression (5.1.12) of the
component w . In this expression the integrals are not singular, so we
may employ Green-type formulas. We have
(1+.)]dd1l=
fin
_aly 1 70, +_ - i
yo
z2 {1
+
a)}
drl
d
(1+)]ddY!
1111 Lf YD_2
R1
+
y02
+
JJ D
a yoZa
YO
(1 + RI ) d d l =
r 1 Jo
J
= PODf(4,r!)yo
a
.a
YO
1/o+z2
(5.1.37)
Using the identity:
(
ax `Ri)
1
l/ + 3 R
z
8 (_
i xo
Of
oR
(5.1.38)
and Green's formula, we deduce
I
fDf(c,rl)8x
(_)ddT!= -POD M,17)
+132z2ffD x(oR1)dedn-JJ1
l
Lf
(5.1.39)
YO
T9Ridr
dry.
167
THE LIFTING SURFACE EQUATION
Replacing (5.1.37) and (5.1.39) in (5.1.12) and passing to the limit, we
obtain
1
1
w(x, y, ±O) = t 2 f 1(x, y) +47rJaL
-
f (c n) (
yo
\l +
R
moo) d e+
(5.1.40)
+4n JJD 5; yo \1 +
o} dt dn,
whence it follows the integral equation:
R
f (C n)
1
47rJoD
yo
(5.1.41)
of
+4R JJD
yo
+
J de do ° h=(x,y)
This is the equation given by Homentcovschi equation herein, utilizing
Green's formulas before passing to the limit is preferable to that of
Homentcovschi which utilizes Green's formulas for singular integrals.
Formally, the equation (5.1.41) is deduced from (5.1.28) by means of
the identity (5.1.32) as follows:
ACT))
AD
(i+ xo
_
1r
8
(1+
R
1
R
(1+
l
o)d
/
`
dn(5.1.42)
2. In other papers (see for example, 11.21) we find the equation:
47r
AD 1 yo
I+ o) d
do - h=(x, y) .
(5.1.43)
It can be obtained from Homentcovschi's equation, imposing the condition
f (x, y) = 0 on 8D.
(5.1.44)
Usually this condition may be imposed either on the trailing edge or
on the leading edge and on the lateral edges. Indeed, imposing the
168
THE LIFTING SURFACE THEORY
condition (5.1.44) both on the leading and trailing edges means that for
the plane profile obtained through it section of the wing with a plane
parallel to the xOz-plane, one has to determine the jump of the pressure
which vanishes at the both edges. But it is well known that the jump
of the pressure for the plane profile satisfies the equation (C.1.1) which
has a bounded solution at both edges only if the free term Satisfies
the condition (C.1.13). Hence the condition (5.1.44) is valid only after
imposing some restrictions for hx(x, y).
However, for wings whose trailing (leading) edge is a straight line
perpendicular on the Ox-axis, like, for example, the delta wing with the
base representing the trailing (leading) edge perpendicular on Ox, or
the trapezoidal wing with the big base representing the trailing (leading)
edge perpendicular to Ox, the entire integral equation (5.1.41) reduces
to the integral equation (5.1.43), because on the perpendicular trailing
(leading) edge we have d = 0, and on the lateral edges we may impose
the condition (5.1.41). For the rectangular wing the equation (5.1.43) is
exact.
5.1.8
The Plane Problem
In order to obtain the solution of the plane problem from the threediniensional solution we assume that the intersection of the wing with
each plan parallel to the xO.-; -plane is a profile having the same shape
(figure 5.1.1). This means that the equations (5.1.2) have the form
z = h(x) ± hl (x) ,
(V)y,
(5.1.45)
and the domain D is the rectangle (- I < x < 1, -b < y < b) with
no (half of the chord of the wing wsas taken as the reference length
La). From these hypotheses it follows the general representations of the
solutions (5.1.8) and (5.1.12),
b
fi = fi(b),
f = fW
Taking into account that
J
dy
y
(a2 + ff2y2)3/2
(12 f a'- + / y2
we deduce
lim r
b-..oo_b
_ (!)(17)_$(229)
R
2:co
169
THE LIFTING SURFACE EQUATION
lin)
b- -oo ./ b
Oy
1
y02
V+
(1 +
Z2
2
=- lim r+b0 (
1,
R
y°
)J cl vi =
(5.1.47)
(i+)Jdi=o.
Jl[yo
So, the above mentioned representation becomes
I
+I
nQ
xofi(o)+42
#2Zf
dE
Q r+1 xofg) - zpIWd,
w(C z) = 2a
JI
xO+/32z2
(5.1.48)
and v(x, z) = 0. Taking into account that here Oz has the position
of the Oy -axis from the problem considered in 3.1,we find again the
formula- (3.1.17).
The integral equation is obtained from (5.1.28). For calculating the
integral
)r1
b
b yo
(1 +
Lo dtl,
R
we use the definition of the Finite Part and the identity (5.1.32) valid
on each of the intervals (-b, y - s), (y + E, b). Thus we obtain
I+b 1
b
yo
(1 +°J d
lint f x0 + ,
e-'0 l xoyn
R
+x yRl+b c -
2
(1+
Uf
Ix011
/-bby ( l+
R)drl=-o.
whence
urn
(5.1.49)
The integral equation r educes to (3.1.21).
5.1.9
The Aerodynamic Action in the First Approximation
We indicated in [1.11], p. 79 the way to calculate the action of the
fluids against the bodies by applying the transport equations of the
momentum and the moment of momentum to the fluid filling the domain
THE LIFTING SURFACE THEORY
170
bounded by the body surface E and the control surface E0 surrounding
E. This calculus is absolutely necessary when the body has corners,
like in the case of thin bodies. However, in a first approximation the
calculations may be done taking into account that the action of the fluid
is given by the jump of the total pressure
OPi 0 = Pt (x1, yl, -0) - PI (x, yi, +0)
which determines the lift L. On the unit of area the force is pljk
(fig. 5.1.4), k representing the versor of the Ozl-axis, and on the
entire area of the domain D1, L = Lk, where
L=
f 1 pl I
1dy1 =
f fD POdxdy .
(5.1.50)
Fig. 5.1.4.
For the resulting moment one obtains the formula
M
=
zl x Opt I kdxtdyl = PooUZLo fJ z x QpO kdxdy, (5.1.51)
where according to (5.1.20),
64 = P(x, y, -0) - p(x, y, +0) _ -f (x, b) .
(5.1.52)
171
THE LIFTING SURFACE EQUATION
Denoting by At the area of the domain DI (which is the domain D
in dimensional coordinates) we have
Al =
dx1dy1 = LoJ1 dxrdy = LOA.
(5.1.53)
U
IL,
For the lifting coefficient et, one obtains the formula
cl
def
L
(1/2)paoU.A1
= - A AD f(x,y)dxdy,
(5.1.54)
and for the moment coefficients
(5.1.55)
2
Aao
dot
der
_11f
=
Z
Aao
l,layf(x, J)dxdy,
xf (x, y)dxdy,,
(5.1.56)
where al = a0Lo is the medium chord of the domain D1 on the
direction of the unperturbed stream. As we shall see in the sequel, the
drag (the projection of the resultant on the direction of the unperturbed
stream ) and the gyration moment (the projection of the moment Ozlaxis) have the order of magnitude e2, for this reason they are not present
in this calculation. M? is the rolling moment, and Afy, the pitching
moment.
5.1.10 A More Accurate Calculation
We obtain a more accurate calculation if we have in view that the
resultant and the resultant moment are determined by the jump of the
tension vector i.e. by the formulas
R = - Jf pt nOdxtdyl , M = - fJ x1i x OplnO&rtdyl ,
(5.1.57)
I
w here n is the outward normal on E. The presence of the normal
enables us to take into account the shape of the wing. If the equation
of the wing is h(x, y) - z = 0(h = FIT), the normal n is
+h
k
n=kri1+hr+hy
=hri+hyj-k+
ta'' + h2
2
172
THE UFTINC SURFACE THEORY
Neglecting the terms of order O(E3) and taking into account that h
is h(x, y) f h1(x, y), it follows
- p'nD = 2pco [h1=i + hlyj + (hxh1 + hyhlu)k] -POOU,2, IpI (h=i + by j - k) + pooU.2, < p > (hi1i + hlyj)
,
where we denoted
< p >= p(x, y, +0) + p(x, y, -0) .
(5.1.58)
Hence the lift is:
R. = -PpoUULo ff .f (y)dxdy + 2p.Lo ff
(h,,hl +
hyhiy)dxdy,
(5.1.59)
and the drag
Rr = 2PooL2
IfD h1:dxdy + p UULo Jf f (x, y)h,,dxdy+
D
(5.1.60)
f < p > hudxdy.
+p,oUO2OL0 ifD
For the rolling moment one obtains the formula:
h!= = -P.UULo LID yf (x, y)ddy +2pLy(hh+
(5. 1.61)
for the pitching moment one obtains
= pULx f (x, y)dxdy - 2pLx(hhl,, + hhly)dxdy,
jj
(5.1.62)
and for the gyration moment,
AI. = 2pooL03
Jj ( xhl- yhl)dxdy+
+PoU;,Lo JJ f(xhy - yhx)dxdy+
.l D
(5.1.63)
+PooUo,LO f f < p > (xhty - yh1m)dxdy,
D
where
<p>=--
xo
h1xR3dEdrl,
D
(5.1.64)
THE LIFTING, SURFACE EQUATION
173
as it results from (5.1.19) and (5.1.27).
If we keep only the terms of order O(E), it follows:
R.- _ -Poo U.2 Lo2
Rr = 2pxL
JJ
JJ D f
(x, y)dxdy,
1i1=(x, y)dxdy,
(5.1.65)
(A fr, Afy) = p
A-Y'X)f (x, y)dxdy,
D
M, = 2pmLo Jf (xh- yhtr)dxdy.
For the lifting surface (hl = 0) the formulas (5.1.55)-(5.1.59) become
R.
fD f (x, y)dxdy,
R: = -PU Lo f f f(x,y)h.(x,y)dxdy,
D
(5.1.66)
(M1. A.fy)
=p
LG
ff(-y, x)f (x, y)dxdy,
A9, = 2pxLo ff (xhy - yhr)dxdy..
D
Rz, Air §i My being O(s), and Rr and Al;, O(E2).
5.1.11
Another Deduction of the Representation of the Gen-.
eral Solution
In the sequel we shall deduce again the representation of the solution
(5.1.8), (5.1.12). We start from D. Homentcovschi's idea (exposed in
(A.81) to utilize the Fourier transform for bounded domains (A.6]. The
method synthesizes the problem of determination of the fundamental
solution and the problem of replacing the wing with a forces distribution.
In addition it justifies the assimilation of the wing with a distribution
of forces having the form (fl, 0. f ). Indeed, employing the formulas
THE LIFTING SURFACE THEORY
174
(A.8.2) to the system of linearized aerodynamics (2.1.30) and taking
into account that D is a surface of discontinuity, we get
0=-ia1A
(5.1.67)
0=-iali) - iap- Pk,
with the notations
[jpj)e,(QI x+a2Y)d x d
y,
(W' P) = JJ D (
where, taking into account (5.1.20), (5.1.25) and (5.1.27),
awj = w(x, y, +0) - w(x, y, -0) = 2hi=(x, y)
(5.1.68)
W = p(x, y, +0) - p(x, y, -0) = f (X., Y)
From (5.1.67) one determines first p and then w. One obtains
ia3P - ia1W
a2-M201
(5.1.69)
ia3W
W =ate
+,Q2 a
ia1P
ia2P
+ a ( 2 - M22)
Utilizing the expressions of P and W, we find:
1Q3
p-
J fp(t, r?) a2-A?2aj e(-t4+0tin)9 dv7is l
dd d11,
-2 [1D hit (e, t?) a2 - M2a2
whence, taking into account (A.6.9),
P(x, y, z)
fJD'02
,-I
i(o,+ozq)
a2 -
2ai d dy+
(5.1.70)
rrrr
+2JJD
hlt
,rl)
az
a2 - M2a2
Employing (A.7.2) we obtain
P(x,y,z) =-4a
+ 27r
f fD
rr
JJD
(_)d+
(5.1.71)
hlt (, q) 8x (R1 d dr7
NUMERICAL I`TECRATION OF THE LIFI1NC SURFACE EQUATION
175
where R1 is that from (5.1.7). We obtained in this way the representation (5.1.8).
Similarly, utilizing (2.3.11) and (2.3.27), we have:
F`1
fly
_
ial(a2 - M1ai)
(5.1.72)
r
0
aJ-110
1)dr
47r
ya+z2
yo
(i+)
jai
such that R, having the same expression, it follows:
w(x,y,z)
=-T7r
-
1,! n
Yj)
dFdij-
/32
+ 4T 11
(5.1.73)
(R1) di;d;+
0 [ ?100+Z2
yo
(1+)].d rl,
which is just the representation (5.1.12). We have to notice that from
(5.1.70) one deduces the inversion formulas
1
[Y2
at(G2 - h12a21) 1
4z y2
y
1+
X
r.' +
(a" +z-)
(5.1.74)
which could be utilized in 2.3.
5.2
5.2.1
Methods for the Numerical Integration of the Lifting
Surface Equation
The General Theory
There are not yet known exact solutions of the equation (5.1.28). The
first numerical solution was given by Multhopp in 1950 15.24]. Previously
the same author had given in 1937 the approximate solution (to which
one assigned his name), for the lifting line equation (Prandtl's equation
(6.1.16)). Multhopp's method relies on the Gauss-type quadrature formulas for non-singular integrals. At that time there were not available
quadrature formulas for singular equations. For the singularity appearing in the lifting surface equation, Multhopp utilizes a series expansion
176
THE I..IFTINC SURFACE THEORY
with Chebyshev polynomials of first kind, which is truncated in order
to obtain an algebraic system. The method is analogous to Glauert's
method for Prandtl's equations, except that the sin functions are replaced by Chebyshev polynomials. In 1958, at a Meeting in Fort Worth,
Hsu gave a quadrature formula for integrals with a strong singularity
and employed this formula for the singularity from the equation (5.1.28).
However, in Hsu's formula the unknowns are present even in the collocation points and this is a drawback. Starting from a formula given by
Monegato [A.52), Drago§ gives [6.5] the formula (F.3.5) where there are
present supplementary unknowns in the collocation points. One utilizes
successfully this formula for solving Prandtl's equation in 6.5 and for
solving the lifting surface equation in [5.10] and [5.11]. These solution
will be presented in 5.2 and 5.3. In 5.2 we shall sketch the solution of
the equation S via the collocation method.
We have to solve the equation:
I
4rr
N(xo, lyo)dt dr1= -hr(x, y),
i f(
Q
(5.2.1)
o
where
N(xo,yo)=1+
To
V2'or+
Y4
(5.2.2)
with the following conditions:
f (t, ±b) = 0,
x-(fb) < < x+(±b)
(5.2.3)
-b < rl < +b.
(5.2.4)
The first two conditions mean the continuity of the pressure on the
f (x+(rl), q) = 0,
lateral edges in case that they are straight segments (if the lateral edges
are represented by points the conditions disappear by virtue of (5.1.1)).
The condition (5.2.4) ensures the boundedness of the pressure along the
trailing edge. One performs the following reasoning: each intersection
of the wing with a plane parallel to xOz determines a thin profile;
as it is known from 3.1 for such a profile one imposes the boundedness
condition on the trailing edge.
In order to utilize the quadrature formulas we shall perform the
change of variables
(x, y) - (u, v)
TI) --4 (a.
defined by the equations
x = a(y)u + c(y) Zr = a(ri)a + c(q)
(5.2.5)
y
by
rl = b13
177
NUMERICAL INTEGRATION OF THE LIFTING SURFACE EQUATION
with
a(y) =
x+(y)
- z-(y)
2
,
c(y) =
x+ (y) + x-(y)
2
(5.2.6)
Writing the equation (5.2.1) as follows:
r
1
T
UFO
+(n)
f
q)N(xo, yo)d
do = -4xhz(x, y)
and taking into account that
e(t,g)
= a(F)b,
.9(a, /3)
we deduce
f
a(A)
1
(v - A)2
f (a, /3)N(u, v, a, Q)da] dfl = 9(u, v),
(5.2.7)
where denoting by a(v), c(v), f (a, /3) etc., the functions a(y), c(y),
f((, q) in the new variables, we have
a(v)u + c(v) - a(/J)a - c(/3)
N(u, v, a, /3) = 1 +
1/2
{[a(v)u + c(v) - a(Q)a - c(f)J' + k2b2(v - /3)21
(5.2.8)
(5.2.9)
g(u, v) = -4rrbhi(u, v),
with the notation k
The Kutta-Joukowski conditions
(5.2.3) become
-1 < a < +1) ,
f(1,/3)=0, -1</3<+1).
(5.2.10)
(5.2.11)
The solution of the equation (5.2.7) depends on the behaviour that we
impose at the extremities of the intervals (±1). For a fixed 0 , one
obtains within the wing a thin profile. For such a profile we have imposed
in 3.1 the behavior given by
++ a . Along the span one imposes in
the behaviour
1 --02. Such a behaviour ensures that the conditions
(5.2.10) are satisfied. We shall seek for solutions f (a/3) having the
shape
f(a,A) = f -l
++aF(a,A),
(5.2.12)
178
THE LIMING SURFACE THEORY
with F bounded. For F one obtain the equation
JT (p1- v a(13) I- t '1 + a
F(Q, 3)N (ti: vj rx $)da1 d8 _
-1<u,v<+1.
=y(u.v),
(5.2.13)
Utilizing the quadrature formula (F.2. [8), this equation becomes
ni
1 ^F
t
2. (1 - a)
2
r)
,
a(;T):"(u.. v,(xi,O)F(ai, 1)d 8 =
(5.2.14)
= (tin + 1)g(u.v),
where
`=cos2m=
(5.2.15)
i= lm.
a
+1'
From now on Nttlithopp's method and the quadrature: formulas method
are separating.
Multhopp's Method
5.2.2
Using the formula (F.4.2) for (f3 m
v)-2
,
equation (5.2.14) becomes
p
--4irEE(1+k)(1i=1 k-i
J1
-or)Uk(u)
JJJ
- 3 `u(f3)N(a. v,
!
= (2m + 1)g(u,v).
(5.2.16)
One calculates the integral from this equation utilizing (F.2.12). One
Obtains
r
-.1-r
it
E(l+I)(I
r=1 k-l,t=1
(5.2.17)
=(2m+ 1)(n.+ 1)9(u, V),
-1 < u,v <+1,
NUMERICAL INTEGRATION O THE LIFTING SURFACE EQUATION
with
cosn+l'
7
=r
.
179
(5.2.18)
Introducing the notation
H(ai,f3j) _ (1 - ai)(1 -QJ)a(#j)F(ai,f3j),
(5.2.19)
the system (5.2.17) becomes
m rn
p
-4irrG.rL.r
(1
k)Uk(v)Uk(Yj) (u, v,ai,Qj)H(ai,QJ) _
i=1 j=1 k-i
(5.2.20)
=(2m+1)(n+1)g(u,v),
-1<u,v<+1.
There are m x n unknowns, H(ai, /3j ). In order to obtain the same
number of equations in (5.2.20) we shall assign m values to u and n
values to v, obviously, all of them in the interval (-1, +1).
As we shall see later, the aerodynamic coefficients are functions of
H(ai, /3j ). It is sufficient to find out these unknowns. We may write
computer programs for solving the system (5.2.20).
5.2.3
The Quadrature Formulas Method
Utilizing in equation (5.2.14) the formula (F.3.5), we obtain the equation
m n
tar E F` [1i=1 j=1
(1 - ai)(1
()3j - Ak)2
m
-7r(2m + 1)(n + 1)2 E(1 - ai)a(.3k)N(u, Qk, ai, Qk)F(ai, Qk) _
iml
= (2m + 1) (n + 1)g(u,13k) ,
(5.2.21)
where
Aj=ten+1,
for k = T.
(5.2.22)
180
THE LIFTING SURFACE THEORY
In (5.2.21) we have a system of n linear algebraic equations with
m x n unknowns F(a1, /3j ). Imposing the system to be verified in m
points
ul
-
oOS
Vir
2m+1
=al,
t=T,
(5.2.23)
the number of equations equals the number of unknowns. The system
will be written as follows
m n
Bik, a=Tm k=I-n(5.2.24)
Aekij Ha 13
i=1
where we denoted
Alki;
(2 [1- (-1)j+k]
=t
()3j -)3k)2
-(2m + 1)(n + 1)21
j
-A
}N(atflkoi/3i)
(5.2.25)
Btk = (2m + 1) (n + 1)9(a1,Pk)
5.2.4
The Aerodynamic Action
The lift and moment coefficients are calculated by the aid of formulas
(5.1.54)-(5.1.56). Using (5.2.12), (F.2.18) and (F.2.12) we obtain:
CL
A
ff
D
Af
+1
=Af
f
+L +1
f(t,
1- fl2a(Q)
1
41r2b
m
[f
1
l f(a,
d/3 =
+1
1
1+
aF(a, l)da] dQ =
(5.2.26)
"
A(2m+ 1)(n+ 1) FF H(a"Pj)'
w1 j=1
where H(a;, (3j) are (5.2.19). Formula (5.2.26) is valid for both
Multhopp's method and the quadrature formulas method.
NUMERICAL INTEGRATION OF THE LIFTING SURFACE EQUATION
181
In a similar way we obtain:
2
=
IL of
4R
Aao
+I
J_ I
ja(,3)f (a,13)da df3 =
n
in
2b2
f
+I
aoA(2m + 1)(n + 1) E
E Qj H(ai,13j)
i=1 j=1
,
(5.2.27)
cy
Aao
2b
- Aao
11
f
r')dCdr' =
+1
+I
J-1
4-Ir
2
ta(,B)a + c(R)I a(A)f (a, Q)da d/.3 =
b
aoA(2m+1)(n+1)
ao
m
r( (
it
(5.2.28)
q
ti_1 j=1
representing the length of the medium chord and
di =
CD = - A !ID
41r2 b
A(2n+1)(n+1)
n
(5.2.29)
n
i=1 j=1
For the flat plate of incidence E, we have h(x, y) = -Ex then g(u, v) _
47rb. For the rectangular flat plate we have x_ = -1, x+ = 1, a = 1,
c=0,
u-a
N(uvapl=1+
, , ,
5.2.5
(5.2.30)
The Third Method
Some numerical experiments show that it is not always indicated to
impose the behaviour along the direction a under the form
We shall use in (5.2.7) the following quadrature formula
+1
m
f (a, Q)N(u, v, a, /3)da = > f(ai, i9)Ki (u, v, f3) ,
1
i=1
(5.2.31)
182
THE LIFTING SURFACE THEORY
2i
where a; = - m , i = 0, 1'.. . , m are equidistant nodes on the interm
val (-1,+1), and
K1 (u, v, /3) =
joi-I
N(u, v, a, fl)da =
2
m
1
a(Q) /[a(v)u + c(v) - a(A)a; -- c(/3)1+ k2b2(v
+
- p)2+
(a(v)u + c(v) - a(/3)a{-1 - c(,6)J2 + k2b2(v - /3)2.
(5.2.32)
The behaviour in the span direction remains the same (like in the previous methods), hence:
1--#2F(&,,6),
f (a, A) =
(5.2.33)
where F(a, 3) is finite for rQ = ±1. We shall use the same formula
(F.3.5) with respect to p. So, the equation (5.2.7) furnishes the following discretized form of the equation (5.2.7)
,n
n
HtkiiF(a (3:) = bh=(ailk) ,
(5.2.34)
i=I j=1
where
Htk;j _
=
1
4(n + 1)
{i_(_i+k]
2
Ak)2a(13i)Hj(ac,,6k, Qj)
(Qj -
,
k
(5.2.35)
and
n +l 1
H10919 =
a(/3k)Ha(at, Qk, /3k) .
(5.2.36)
The algebraic system (5.2.34) has m x n linear equations with m x n
where #j =coo 17r
unknowns
n+l*
NUMERICAL INTEGRATION OF THE LIFTING SURFACE EQUATION
183
The lift and moment coefficients are calculated by means of formulas
CL
_
-
2b
+1
A
11
+1
1-,2a(fl)F(a,A)da dQ =
J_1
1
2bir
(n + 1)A
j
(1-1,)a(13j) J
F(a, $$)da
1
(5.2.37)
1
n
4bnr
(1 in
n+ 1 )rr1A Jul i=1
C,
22 2
1
1+1
#a(13)
J1
=
1- J32F(a, /3)da dfl _
(5.2.38)
nm
4b2ir
E I6ja(0j)(1 - AJ)F(ai, Qi)
= (n + 1)mAao,
Jul i=1
Aao
f
1111
a(#)[a(A)a + c(Q)]
1- 02fla,,O)da d/3 =
n cm
-4b,r
In + 1)mAan Jul iml
EEa(AJ)(1-
_Q1)(2i-'m-1a(pi)+c(fii),F(ai,QP)(5.2.39)
In the paper 15.101, from which we presented this method, there are
presented computer programs for the elliptical flat plate
h(x, y)
= -cx, x2 + e < 1, b = 2
and for the wing whose projection on the Oxy - plane is an rectangle
-b<y<b, 0<x<1, x(y)=0, x+(y)=1,
and whose normal section is an arc of parabola having the equation
h(x,y) = e(1 - x2),
e <<l,
(5.2.40)
THE LIFTING SURFACE THEORY
184
The fluid in considered incompressible. For the rectangular wing one
obtains an analytic solutions in the framework of the theory dealing
with the wings of low aspect ratio. In this way we have a test solution
for the third method. One deduces that this method furnishes very good
results.
5.3
5.3.1
Ground Effects in the Lifting Surface Theory
The General Solution
The ground effects in the lifting surface theory were taken into consideration in (5.311 and [5.371 where one utilizes asymptotic methods.
An approach of this subject in the framework of curvilinear lifting line
theory may be found in [1.32], [1.331. The present subsection is elaborated following (5.8], where one gives the general theory. One utilizes
the fundamental solutions method.
The geometry of the problem is presented in figure 5.3.1. The origin
of the reference frame is located in the middle of the span, the Ox-axis
has the direction of the unperturbed stream and the Oy-axis has the
span direction. The ground is considered to be the plane 1I having
the equation z = -d/2. The unperturbed flow is characterized by the
velocity Ui , the pressure p(,,, and the density po and it is considered
to be subsonic like in 5.1. The field of velocity v1, the pressure p1 and
Fig. 5.3.1.
the density p1 for the perturbed flow have the form (2.1.3). One utilizes
185
GROUND EFFECTS IN THE LIFTING SURFACE THEORY
dimensionless variables. The equation of the upper and lower surfaces
are given by (5.1.2). It results the following boundary conditions
w(x, y, ±0) = hr(x, y, ±hiz(x, y)
(5.3.1)
(x, y) E D
(5.3.2)
w(x, y, -d/2) = 0, (V )x, y.
For satisfying the last condition we shall utilize the images method.
This means to replace the wing with a forces distribution f + = (fl, 0, f )
defined on D and with a symmetric distribution f - = (fl, 0, -f) defined on the domain D', which is the symmetric of D with respect
to the plane II. In order to write the general form of the perturbation
fields p and w we have at first to write the form determined by the
concentrated forces f+ in P+(4+) and f- in P_(4-). We have
therefore to determine the solution of the equations
M28p/8x + divv = 0
(5.3.3)
8v/8x + grad p = f+b(x - 4+) + f -b(x - 4-)
corresponding to the system (2.3.4). The system (5.3.3) is linear and its
solution is the sum of the corresponding solutions from (2.3.4). Taking
into account (2.3.24) and (2.3.29), and the form of the fields f + and
f -, we deduce:
p(x, y, z)
=-4
w(x* y, z)
=4
0T
(f,
+fa ) `R+l
8z ( R+)
4 f8 x
47r
(fie -fe)
(i)'
R+) +
+47r 5; [(y_+)2+(z_z+)2 (i+xR+ )J+
2
+4 Oz
f
R_)+4 f8x (R_)y - 17
4ir 8y (y - 17- )2
where
R+=
(x - t:1)2+
(z - C-)2
+
-- (1 +
x R-
2n*)2 +(z - (±)2).
l
)J
'
(5.3.4)
(5.3.5)
186
THE LIFTING SURFACE THEORY
The points P+ and P- are symmetric if £+ _ {- = f, tl+ = n- = n,
t+ _ C, C- _ C - d. When P+ are in D((= 0), the symmetric points
P-((- _ -d) are in V.
Assimilating the wing with a continuous superposition of forces defined on on D, it results the following general solution:
p(x, y, z) _
- 47r JJv [fi,-+f,q)-}
()d_
d-9(5.3.6)
rr
f (e, n) 8 l ((Rid) dkc di
J
w(xy, z)
4s
IL
+47r'ID
[f(t,
n)ex
JJDf(t,rl)
4r AD
A2f(t, n)
]
(R,'
) de cI +
(L)d+
(i+)}d_
[yo-0
If,V,17)z +
1
+47r
-
02f (f, n)
8J
!0-
Lyo+
dC
(1+ -10-d/J dt
dn,
(5.3.7)
where xo=x-t, yo=y-Hand
Rt =
xo+(32(yo+z2), Rld= 90 +p2[yo+(z+d)2J.
(5.3.8)
This is the general representation of the perturbation, the functions
f, (t, r)) and f (t, t) being determined from the boundary conditions
(5.3.1). The condition (5.3.2) is obviously satisfied.
5.3.2
.
The Integral Equation
Acting like in 5.1, we shall pass to limit considering z -- ±0, (x, y) C-
E D. To the limit, only the integrals related to R are singular. Using
the formulas (5.1.16), (5.1.18) and (5.1.24), from (5.3.6) and (5.3.7) we
CROU\l) EFFECTS IN THE LIFTING SURFACE THEORY
187
get
P(x, y, ±0) = 4rID fi(f,t,)Rgdt dry ± 2f (x, y) +
+ 4a JJ f1(, q)
ddi 4j r
R3 dti,
,
(5.3.9)
P(x, y, +0) - P(-T, V, -0) = f (x, y),
and
w(x, y, f0) =
2 f1 (x, y)
47
if
-
47r ,IJD
R
(i+)d_
f
ff f (t, t)N(xo,
(Z' Y)
-
d'l
41r
3Nn)dt dr1,
(5.3.10)
where
N
yo) _
(d2-#
) + (d +
;'F
d
2
2
(1 + xd° )
I
(5.3.11)
with the notations
R=
xo + $2y$ , Rd =
xo + 02 (y02 + d2) .
(5.3.12)
Imposing the boundary conditions (5.3.1) we deduce
fi (x, y) = -2hi:(x, y)
(5.3.13)
and then
4Rv
f
(YOT
L1+'0 R)
dk drl + 4,r
AD
f (f, rl)N(xo, yo)dC dy =
(5.3.14)
The equation (5.3.14) may be called the generalized lifting surface equation. It was given in [5.8]. The sign "*" is for the Finite Part. Using
188
THE LIFTING SURFACE THEORY
the identity (5.1.32) we may write the equation as follows:
4n 8y
jf f (,'1)
1+
o)
= h' (x, y)
drl
-
4a
If f (t, r!)N(xo Uo)df do =
.JJD hlfR3
1) dt
dtl .
(5.3.15)
The Two-Dimensional Problem
5.3.3
Like in 5.1.7, in order to obtain the representation of the solution and
the integral equation for the wing of infinite span we assume that the
normal sections in the wing determine profiles having the same shape.
It means that the equations (5.1.2) have the form
z = h(x) ± hl(x)
(V)y
and the domain D is a rectangle (-1 < x < 1, -b < y < b) with
oo. We notice that in the representation of the general solution
b
(5.3.6) and (5.3.7) ff and f have the form f,(£) and f(t).
Relying on formulas like (5.1.46), from (5.3.6) we deduce
P(x, z) -
1
21rp
1
)324f (t) dt+
f,2 xoft Wxo ++ $2z2
r+1 xofl (t) +,32(z + d)1 (t)
0 +$2(x2+d)2
which are just (3.2.3)1. Using formulas like (5.1.47), from (5.3.7) it
results:
w(x,z) __
r11 x01(0+
z2(C)dc _2 J_li xOf +fl
(
+d)2( )41
2a
i.e. just (3.2.3)2.
For obtaining the integral equation we shall use (5.1.45). We also
have
oo do
J
ydq
2
,
l
00
The integral
1=
+00
L
-0o (y[+
d2 _ yo
xo dn
)2 R1
= 0.
(5.3.18)
189
THE WING OF LOW ASPECT RATIO
reduces to integrals having the shape
Jo
(u2 +
r
du
d2) VU-T-+31
Je
du
(u2 + d&)Z u +
which may be calculated using the change of variable u -+ v where
u
V
(5.3.17)
VU-27 71
One obtains
I=
2,0
d
1-
xo
xo
arctan
xo
dQ
(5.3.18)
Using the same change of variable (5.3.17) one obtains also
+O0
xo d
1 (plc
01?
11
9
1
+ ;01
o
xo
x
arctan d
(5.3.19)
such that we finally obtain
oo
-l
(5.3.20)
xo + d2 02
Taking into account the previous results, the integral equation (5.4.12)
becomes
,r+i 1
Jl
1
xo
_
dt
-
r+i xof ()
27r JJ
1
xo + d2#2
L" /+1
clk
=
(5.3.21)
h'(1)
which is just (3.2.7).
In (5.111 one extends the method from 5.2.3 also to the equation
(5.3.14).
5.4 The Wing of Low Aspect Ratio
5.4.1
The Integral Equation
In the sequel we shall pay attention to the ratio
A = (2b)2/A,
(5.4.1)
190
THE LIFTING SURFACE THEORY
called the aspect ratio. We denoted by 2b the span and by A the area
of the domain D. For wings characterized by a small A (wings of low
aspect ratio) one develops herein a theory which leads to the integration
of the lifting surface equation (5.1.33). For wings characterized by a big
A we shall develop in the following chapter the lifting line theory. These
are the two asymptotic theories of the lifting surface equation. If the
square of the span is small with respect to the area A, we deduce that
on the greatest part of the wing we have yo << xo (see fig. 5.4.1) and
we may write
R
Ixoi
xo
xo l
L1 + O (
)J ,
(5.4.2)
o
where R is (5.1.17). On the contrary, when the square of the span is
big with respect to A (see figure 6.1.1), on the greatest part of the
wing we have xo << yo, so that we can make the hypothesis (see 6.1).
Fig. 5.4.1.
R = kayo)
(5.4.3)
Returning to the wings of low aspect ratio, we notice that under the
hypothesis (5.4.2) the equation (5.1.33) may be approodmated by
4rr 8y J JD
f (n) (j+LX-01 df drl =1. (z, y)
(5.4.4)
Noticing that the integrand of this equation is different from zero
only on t < x, we shall consider only wings whose trailing edge is
a straight line perpendicular on Ox like in 5.4.1. Introducing the
unknown function
F(x, q)
JS-(n)
f(s, q)d,
(5.4.5)
THE WING OF LOW ASPECT RATIO
191
utilized for the first time by Jones 15.171, 15.18), the equation (5.4.4)
becomes
8
1
v+(=)
{
dry = h=(x, y) ,
2w 8y Jy_ (x)
(5.4.6)
yo
where y = y+(x) represents the equation of the leading edge OB, and
y = y- (r), the equation of the leading edge OA.
From (5.4.5) and from the figure one notices that
F(x, y- (x)) = F(x, y+ (x)) = 0.
(5.4.7)
In (5.4.6), utilizing the definition of the principal value, we derive
taking into account (5.4.7) and then we integrate by parts. It follows
a
F(x, n)
ay
_f
dq -
e
F'
1
yo! d
_ 'v+ SF do
- Jy-
an
'
(5.4.8)
where, from now on,
y- = y-(x), y+ = y+(x)
In this way equation (5.4.6) becomes
+ OF dr)
1
Jv-
8q
- h. (x, y)
(5.4.9)
This equation can be also obtained starting from Homenteovsch's
equation (5.1.41) which, in case that one imposes the condition (5.1.44)
on the leading edge, becomes equation (5.1.43). Then we can see that
from (5.4.5) it follows
8F_
8f
(5.4.10)
The equation (5.4.8) is a thin profiles-type equation (C.1.1) and it
has one of the solutions (C.1.9), (C.1.10), (C.1.11) or (C.1.14). The
solution of the equation (5.4.8) must satisfy the condition
J
+OFdq=0
y_ a!
(5.4.11)
arising from (5.4.7). The only solution satisfying this solution is (C.1.11)
with C = 0. Hence,
8F(x,TI)
-2
1
(t-y-)(y+-t)h=(x,t)dt.
t-77
(5.4.12)
192
THE LIFTING SURFACE THEORY
Integrating this solution on the interval (y_, y) and taking into account
(5.4.7) and (B.6.11), we get
F(x, y)
_
'
dq
2ly
(n-y-)(y+-q)
-
+
(t - V -)(y+ -' t)
t
y-
(x, t)dt .
-q
(5.4.13)
As from (5.4.5) it results f (x, y) = F, (x, y), we deduce
f (x, y) =
=2
8
8x
ydq
ru+
(t -
(qy-)(y+-q) y-
ly-
t)
t-q
(x, t)dt .
(5.4.14)
Employing (B.6.11) ones easily check that this solution satisfies the vanishing condition on the leading edge
f(x,y-)=f(x,y+)=0.
5.4.2
(5.4.15)
The Case h = h(x)
In case that
h does not depend on the variable
y, taking into
account (B.5.6) and performing the change of variable
q = y_ cos20 + y+ sin20,
we get
:
Jv-
0 < 0 < 7r/2,
(5.4.16)
' V+
dt7
At - Y-) (y+ - t)dt =
t-q
n - y-)(y+ - q) Iv_7r
Y+-Y- r
2
-a J
y+
v-
dq
On - y-)(y+ - r!)
ydq=sV(y-y-)(y+- y)
Hence,
f(x, y) = 2
[h'/i_
y)]
.
(5.4.17)
193
THE WINO OF LOW ASPECT RATdO
Utilizing (5.1.54)-(5.1.56) and Green's formula we deduce:
CL
- A If
19
[W(x)Vv - v-}(v+ - Y1
f (y-a (b-y)dy=-2Ah'(1)(b-a)2,
A
-S
= - A ,,D ybx
= - Ah'(1)1
cv
A
v=
Y
[h!(x)-,/(y -
v-)(v+ - v)] dxdy =
(v - a)(b - y)dy = -
bWrh'(l)(b
- a)z,
I/I; [zWv'?j- v-)(v+ - v)1 dx dy_
A
If h'(x) vir(v
-y-)(Yu)dxdy =
-
_ -CL -
1
0
[v+(z) - y_(x)12 h'(x)dx,
(5.4.18)
the reference chord ao bung 1.
For example, fnr the rectangular flat plate of incidence -e with the
span 2b one obtains
CL = wk,
c, = -
c= = 0,
!
,
(5.4.19)
and for the triangular flat plate of incidence -£, we have,
CL=xe(b - a),
because y+ = bx,
c, = (b2 - a')
r
,
cy = - 3 xc(b - a),
(5.4.20)
y- sax.
5.4.3 The General Case
We shall write (5.4.14) as follows
f(x, y) =
2b
x8s
r_
(t - -y-)(y+ - t)h.(x, t)I(y, t)dt,
(5.4.21)
THE LIFTING SURFACE THEORY
194
where
Y
d-q
I(y,t) =
(5.4.22)
(r1-y-)(y+-rl) t - n
with y- < y < y+, y- < t < y+. This integral may be calculated
l
fv-
explicitly.
For y < t the integral is not singular.
For
t < y the
integral is singular but noticing that we have 1(y+, t) = 0 (it results
from (B.5.8)) we get
IY+
1
I(y,t)
d'l
y+-rl) t-17
(5.4.23)
which also is not a singular integral.
Since
A
(n-a)(b-n) t-11
(5.4.24)
(t-a)(b-rt)+ (b-t)(q -a)
1
in
(t - a(b-it)- -v(((r1-a)
(t - a)(b - t)
we deduce the fundamental formula
f(x,y) =
=
2 8 f+h
(t-y-)(y+-y)+ (y+-t)(y-y-) dt.
(x,t)
a 8x v-
(5.4.25)
For determining the lift and moment coefficients we have to calculate
the integral
(t
V+
G(a, t) =
- y-)(y+ - y) +
(y+ - t)(y - y-) dy.
J(t-y-)(y+-y)- V(-y+---4(y -y-)
(5.4.26)
Performing the changes of variables
t=y++y-+y+-y-moo, y=y++y-+y+-y-cos9,
2
2
2
2
(5.4.27)
we deduce
Gxt
- y+ 2 y- fo X in
sm2
sm
B+a
2
sin 8 d9,
(5.4.28)
195
THE WINC OF LOW ASPECT RATIO
the integral having an integrable singularity for 8 = a. The integrand
from (5.4.28) is the kernel from the equation (6.2.15). Using for 0 91 a
the expansion (6.2.17), we get:
G(x, t) = w y+
y- sin o =
2
(y+
- t)(t - y-) .
(5.4.29)
It follows
(t - a(b-y)+ (b-t)(y-a)
fb
G(I, t)
=J
In
Ja
IWa)(b-y)-'(b-t)(y-a)
(5.4.30)
=x (b - t(t-a).
Similarly one deduces
b
(t - a) (b - y) +
JQ y
(b - t) (y - a)
dy=
IV'(t-a)(b-y)-i(b-t)(y-a)
(5.4.31)
=att+b2a) (b-t)(t-a).
Utilizing Green's formula and the expression (5.4.25) for f (z, y)
taking into account (5.4.30), (5.4.31), we get:
A
CL
-
-4
,
lb
41
h`(1, t)G(1, t)d t =
rb
a
_
b
cs = - AJa t (b - t)(t - a)h=(1, t)d t + !-+-b ct,
Cy = -CL -
r1
4
rA
Ju
4
r2
- -CL - A
dx
y+
y_
h.(x, t)G(x, t)d t =
ry+
dT
o
f
(5.4.32)
hs(z, t)
(Y+ - Wt - y-)d t .
When h(x, t) does not depend on t we find again the formulas
(5.4.18).
196
THE LIFTING SURFACE THEORY
This problem, for wings symmetric with respect to the Ox-axis, was
studied in [5.171. It was also included in [1.1], [1.2]. The general problem
for the arbitrary wing as it was presented herein, was solved in 15.91.
Chapter 6
The Lifting Line Theory
6.1
6.1.1
Prandtl's Theory
The Lifting Line Hypotheses. The Velocity Field
Prandtl's theory is the first mathematical model for the three - dimensional wing (the finite span airfoil). It was elaborated in 1918 16.211 and
it remained until the years '40 the only theory for this wing. The german
scientist, gifted with an extraordinary engineering intuition, guessed
very well the simplifications which may be performed. Prandtl's method
consists in replacing the wing with a distribution of vortices on its plane
- form (the domain D from 5.1). Since the experience indicates that
downstream the wing the flow is not potential, Prandtl introduced a
vortical distribution defined on S, the trace of the plane-form D in
the uniform stream (figure 6.1.1), the velocity field in the fluid being
determined by the two distributions. The vortices on D are called tied
vortices and the vortices on S are called fee vortices.
This idea continued to dominate the aerodynamics, the models concerning the subsonic, supersonic and transonic, steady or oscillatory
flows, elaborated in the years '50, '60, being conceived on the basis of
this method. In 1975 D. Homentcovschi, utilizing the theory of distributions proved that the hypothesis of the existence of free vortices is not
necessary because it follows from equations. L. Dragon [5.71 obtained
the same result utilizing the method of the fundamental solutions. In
this subsection, we deduce the lifting line theory from the lifting surface
theory, as we proceeded in [5.7], utilizing the following three hypotheses
(the hypotheses of Prandtl's theory):
10 One neglects the thickness of the wing, therefore in (5.1.2) one
considers hl = 0. From (5.1.27) it results fl = 0, such that the
representations (5.1.8), (5.1.11) and (5.1.12) give
P(x,y,z) = -4 7r-
IL f(CTT)1 (-) dcdi
19Z
(6.1.1)
THE LIFTING LINE THEORY
198
u(x,y,z) _ -p(x,y,z)
v(x,y,z)
_-
+
jo
z2(1+Rl)]dfdi7
(6.1.3)
)Io
qJ
w(x, y, z)
(6.1.2)
fD f (C, 17)
8x 1)dt d t7+
JJD0Y
(6.1.4)
-+ z2
(1 + 11 )]d t d n;
2° One considers that the unknown is the circulation C. along the
contour c, resulting from the intersection of the wing with an arbitrary
plane n parallel to the xOz plane at the distance y(-b < y < b)
(fig. 6.1.1). This will be, obviously, a function of y
Fig. 6.1.1.
Pudx,
C(y)=
because on this contour d y = 0 and wdz = O(E2) (it results from
(2.1.17) and (2.1.21) ). Taking into account (6.1.2) and (5.1.20), we
obtain
r
C(y) =
Jx
u(x, y, -0)d x + / + u(x, y, +0)d x
x
We have also
f
X
f(x,
y)d x .
(u)
(6. 1 .5)
x+ (y)
C(n) = -
x_ (y)
f (f, n)d t.
(6.1.6)
PRANDTL'S THEORY
199
From (5.1.1) or (5.1.30) it results
C(±b) = 0;
(6.1.7)
30 The domain D, i.e. the projection of the wing on the xOy plane,
is replaced by the segment [-b, +bJ (fig. 6.7.1); for this reason we call
this theory the lifting line theory.
when
For studying the behaviour of the integrals (6.1.1)-(6.1.4)
x_(n) -+ 0 .- x+(il), we notice that in the vicinity of this segment, for
a given n , the function f (e, n) keeps a constant sign, such that we
may apply the mean formula. For a function h(z, y, z, n), continuous
in f, when x- (q) -+ 04-- x+(n), we have therefore
f ( n)k(x, y, z,
limJ1
r+b
hm
ti)d d r! _
r+(n)
J-b [ Js_ty)
-lim
,
f (t, n)k(x, y, z, t, r1)d
d r!
+b
-b
k(x, y, z, (', n)C(n)d n = - fb k(-T, y, z, 0, n)C(n)d n ,
(6.1.8)
where £ E (x_(rl), x+(n)). Applying this formula in (6.1.1)-(6.1.4) and
taking into account (6.1.7), we obtain
2
P (x,
y, z) = -
4 rb C(n)
d n = -u(x, y, z) ,
f+b
v(x, y, z) = 4
.! b
_- a
4n
w(x,y,z) =-Q2
41r
ly2+z2
J b C('l)
f
1
)]dii
C(17)ay [ o + z2 (1 +
=
rT
o
(6.1.9)
C'(n)yo+z2 ( 1 +
C(n)
rb
-4 J b
Ro
}n,
d
drl-
b
0,07) y02
YO
Z2
(1 + X
d
200
THE LIFTING LINE THEORY
where we denoted
RD =
x2 +#(y2 + z2)
(6.1.10);
Fbr Q = 1 we obtain Prandtl's representation ((1.21], p. 708).
6.1.2 Prandtl's Equation
In the sequel we shall deduce the equation for C(y). This will obviously result from the lifting surface equation by virtue of the above
hypotheses. We shall employ the equation (5.1.33). Taking (6.1.6) and
(D.3.6) into account it results
8
f( , n) d
8y JD
y-n
1
fd n= dy
-d- f' +b
C(q)
-b n-y
-
d
rJ
C(q) d n.
O
-b (n - y)2
Utilizing (D.3.7) and (6.1.7), we also deduce:
.
&
c(_)
= l1+b C'(n)
J
b
'n-yd
(6.1.12)
In the second term from (5.1.33) we take into account that when
x_(n) - 0 4- z+(n), on the greatest part of the domain D, we have
so << 14, before E which intervenes in the definition of the principal
value of the integral with respect to n becomes zero, i.e. before yo
becomes zero. In the principal value we shall perform therefore the
approximation R = fl I. An exact evaluation of this approximation is,
made by D. Homentoovschi in [6.9) as follows
jjf(r)
ryxTo +
dd
XDYO
Taking into account that
sign yo = 2H(yo) -1,
H'(yo) = 6(y - q),
H representing Heaviside's function and 6 Dirac's distribution we
201
PRANDTL'S THEORY
have:
Z_
JJ
20
f (f, n) xovo
+b
I
J-b
JJ
d {d ~ Q
f (txon)
t
LJ
d C,
f (on) s yod fd n =
' f=+ (v) f (t'17)
_
d
5(y - n)d n= 2Q
xo
(y)
(6.1.13)
Hence the equation (5.1.33) becomes:
1
+(v)
,rte C'(n) d n +
sr f b n - y
f (c Y) d
'r s_(v)
xo
= 2h' (a, y)
(6.1.14)
x]-1/2
(6.1.15)
Multiplying this equation by
[x - x-(y)]11.2[x+(y)
-
and integrating it with respect to x on the interval
obtain from the first formula (B.5.4):
AC(y) =
(x-
a(2) 'f+b C'(n)dn+1(y),
J bn-y
(y), x+(y)), we
(6.1.16)
where we denoted
a(y) =
J(y) _ -2
W
x+ (y) - x- (y)
(6.1.17)
2
IT
x(6.1.18)
a(y) representing a half of the chord of the profile at the distance y.
The equation (6.1.16) is the well known Prandtt's equation (the lifting
line equation). This equation, together with the conditions (6.1.7), has
to determine the unknown C(y). It is an integro-diferential singular
equation, with a Cauchy-type singularity. Utilizing (6.1.12), the equation
for C(y) may be written
QC(y) =
a 2y)
t +b
r
b
(n
+
(ny)2 d n
9(y) .
(6.1.19)
In this form, the equation is not integro-differential any longer, it is only
integral, but with a stronger singularity. The mark "s" is for the Finite
Part. As we have already shown in [6.5] [6.6), this form is more adequate
for the numerical integrations (see also 6.4.2, 6.4.3).
202
THE LIFTING LINE THEORY
z
A
Fig. 6.1.2.
The significance of the function j (6.1.18) is given in [1.2J. In the
case of the flat plates having the angle of attack c (fig. 6.1.2) we have
hx = -e whence
(6.1.20)
j(y) = 2irea(y).
6.1.3
The Aerodynamic Action
Taking into account (6.1.6), from (5.1.54) and (5.1.56) we deduce
2
CL
2
+b
C(y)d y, c
+b
Aao 1-b
yC(b)d y, cv = 0.
(6.1.21)
The expression of c, is natural because for the wing reduced to the
segment [-b, +b] the fluid cannot create a moment which should rotate
this axis. By virtue of the first hypothesis (h1 = 0), in the framework
of the lifting surface theory, the quantities Rs and M, which have the
order of magnitude 0(e2) are reduced to those given in (5.1.56) and
(5.1.59). Utilizing (6.1.6) and (6.1.7), we obtain:
R. = -p.U.Lo
C(y)h=(0, y)d y,
b
(6.1.22)
+b
M: = P,,. U,2,. L. f
yC(y)hi(0, y)d y
b
But, from the boundary condition
w(x,y,0) = h,(x,y)
6.1.23)
and from (6.1.9) it results
h=(0, y) = w(0, y, 0) _ -
1
lim (
41r c-.o \
f
V-s
b
+
b
y+c
)
C'M d
Yo
rl ,
203
PRANDTL'S THEORY
because, with the change of variable n - y = u, we have
y+E
Eiti
y-E
C(rl)'
11o
+-Z 2 d r! _ -C'(y) lim
E-0
f
+6
z = 0.
1 y+bCl(I1)d
b
tl
y
rJ ,
( 6.1.24)
we deduce
2
CU
f
A J-b
C(y)w(y)d y,
(6.1.25)
2
MS
cZ
A
(1/2)Poo
`k
2 Alai
Aa_
Jb yC(y)w(y)d y,
representing the area of the domain D and ao, the length of
the dimensionless medium chord along the direction of the unperturbed
stream. In fact, for w we may also utilize the expression
w(y) = QC(y) - ?(y)
(6.1.26)
2wa(y)
which results from (6.1.18) and (6.1.26).
6.1.4
The Elliptical Flat Plate
Until the appearance of the papers of Magnaradze [6.16] and Vekua
[6.28] in the years 1942, 1945 it was available only one exact solution of
Prandtl's equation corresponding to the elliptical flat plate wing. Usu-
ally this solution is obtained as an answer to the following minimum
problem: "To determine among the wings having the same lift, the wing
corresponding to the minimum drag" (see, for example, (1.201). Sometimes one utilizes Glauert's approximate method (see 6.2.4). In the
sequel we shall present a simple method which does not need any special considerations on Prandtl's equation. Let us consider a flat plate
with the angle of attack E, whose projection on the x10y1 plane is an
ellipse having the semi-axes Lo and bLo (figure 6.1.3).
In dimensionless coordinates, the equation of the ellipse is x2+
+y2/b2 = 1. For the edges and the chord it results
y2
xf = f
1 -
2
52
,
a(y) =
1 -
2'
(6.1.27)
204
THE LIFTING LINE THEORY
YI
bLo
L=am
Fig. 6.1.3.
For a flat plate j(y) has the form (6.1.20) hence it is proportional to
a(y). From Prandtl's equation it results that C(y) is also proportional
to a(y). We shall look therefore for solutions having the form:
C(y) = k 1-
a
Eb2
,
(6.1.28)
k being a constant which will be determined by imposing (6.1.23) to
verify (6.1.27). Using the change of variables q = boos 0, y = b cos o,
and Glauert's formulas (B.6.6), it results:
di7
11-y
such that
k
(2TE
A + 7r/2b)
_ir,
(6.1.29)
(6.1.30
Since the area of the ellipse is w b, from the formulas (6.1.23) and (6.1.27)
we obtain
s
cL = k, CD =
, cj = cs = 0.
(6.1.31)
46
Obviously, cD = 0(0). For b - oo one obtains the infinite span flat
plate. From (6.1.23) and (6.1.27) it results:
CL =
21r8
(6.1.32)
205
THE INTEGRATION OF PRANDTL'S EQUATION
6.2 The Theory of Integration of Prandtl's Equation. The
Reduction to Fredholm-Type Integral Equations
6.2.1
The Equation of Trefftz and Schmidt
The general method of solving the integro-differential equations con-
sists in reducing them to Fredholm-type integral equations. As it is
well known, for the last ones a general theory is available (existence and
uniqueness theorems, exact and approximate methods for solving the
equation). The first investigation of Prandtl's equation was performed
by Trefftz in 1921. He reduced the problem of solving Prandtl's equation
to the problem of determination of a harmonic function in the superior
half-plane with mixed conditions on the boundary. We shall prove in the
sequel that such a problem may be reduced to a Fredholm-type equation. To this aim we shall consider the harmonic function U(y, z) in the
superior half-plane z > 0 from the yOz plane, with mixed boundary
conditions in the Z = y + iz complex plane (fig. 6.2.1). We shall also
consider the complex function
F(Z) = U(y, z) + iV(y, z) =
1
2m
: ZdR.
Jb+1 COO
(6.2.1)
z
(Z)
-b
+b
y
Fig. 6.2.1.
Obviously this is an holomorphic function in the (Z) plane with the
cut (-b, +b) and it vanishes to infinity. U(y, z) is therefore a harmonic
function in the half-plane z > 0. It results obviously
U(y, +0) = 0, y E (-oo, -b) U (b, oo).
(6.2.2)
The limit values on the segment (-b, +b) are obtained by means of
Plemelj's formulas. We deduce
2U(y, +0) = C(y), y E (-b, +b).
(6.2.3).
206
THE LIFTINC LINE THEORY
Taking the conditions (6.1.7) into account, we get
d
-dn= f
C(t1)
ZJ
bbtj
b
=-f C(rl)'3;
rl
b
C(rl)aZ
Z)dr)_
Cn
1 Z) d+l -
f
b 71- Zdrl
Deriving we deduce from (6.2.1):
F'(Z) =
I-i
8 = 2m J-b W(Zd q
(6.2.4)
and with Plemelj's formulas
' +b
az (y, +0) = d (y, +0) = 2a f
d,
(6.2.5)
n representing the inward pointing normal of the superior half-plane on
the segment (-b, +b) of the real axis.
Taking (6.2.3) and (6.2.5) into account, Prandtl's equation is transformed into the following condition on the segment (-b, +b):
dU
21U(y, +o)
A(y)
- J(y),
(6.2.6)
A(y) = ra(y), A(y)J(y) =1(y) ,
(6.2.7)
in
(y, +0) =
where, from now on, we denote
So, Prandtl's equation was replaced by the following mixed boundary
value problem: "to determine the harmonic function U(y, z) in the z >
0 half-plane, vanishing at infinity and satisfying the boundary conditions
(6.2.2) and (6.2.6)". Then, the function C(y) will result from (6.2.3).
The mixed problem is reduced to a Fredholm-type integral equation.
Proceeding like in [1.211, p. 713, we notice that because of the condition
(6.2.2), the function F(Z) may be extended by symmetry in the lower
half-plane z < 0. In this way, on the lower margin of the cut (-b, +b1
we shall have the condition
dU
d n (y, -0) =
2QU(y, -0)
+ Ay),
A(y)
(6.2.8)
So, we reduced the above mixed problem to the problem of determination of the harmonic function U(y, z), in the yOz plane, with the cut
THE INTEGRATION OF PRANDTL'S EQUATION
207
[-b, +b) and the conditions (6.2.6) and (6.2.8) on the two margins of
the cut.
It is known (see for example, 11.111 [1.201 [1.31J) that the Joukovsky-
type conformal mapping Z - W:
Z = I
(W + W
(6.2.9)
maps the exterior of the cut [-b, +b] from the (Z) plane onto the
exterior of the circle of radius b and the center in origin from the (W)
plane (fig. 6.2.2), the superior margin of the cut being mapped on the
superior half-circle from the superior half-plane. The points fb are
double and singular. One obtains the correspondence of the boundaries
putting W = be''. We have
y=boost, a=0.
(6.2.10)
Fig. 6.2.2.
For 0 < a < r, (6.2.10) gives the correspondence between the halfcircle r+ and the superior margin of the cut, and for -W < a < 0, the
correspondence between r- and the inferior margin (fig. 6.2.2). Since
from the extension by symmetry it results U(y, +0) = -U(y, -0), we
deduce that U(a) is an odd function on F. With the same application
(6.2.10), the functions A(y) and j(y) become even functions. We
denote them by A(a) , respectively j(o). Now we shall see how the
boundary conditions (6.2.6) and (6.2.8) are transformed. To this aim
we remind to the reader that, after performing a conformal mapping,
208
THE LIFTING LINE THEORY
the ratio of the lengths is given by the modulus of the derivative. More
precisely, let W = f (Z) be a conformal mapping and let M(Z) and
N(Z+AZ) be two neighboring points and M1(W) and N1(W+AW)
their images.
Obviously we have
AMN'
lim I
1
= lim I1Z 1 = lim
I
AZ = If'(Z)1
Hence, returning to our problem and denoting by N the outward
normal to 1', we shall have
dU_dU dN_dUIdW
do
dN (a) do
TN- d
Taking (6.2.10) into account, the boundary conditions (6.2.6) and
(6.2.8) on the two half - circles r+ and r_ give
dU (a) = 2/3I sin a)
dN
A(a)
U(a) - J(a) sin a
(6.2.12)
Denoting W = it + i v, the harmonic function U(y, z) in the yOz
plane becomes the harmonic function U(u, v) in the exterior of the
circle r. This one vanishes at infinity and has the normal derivative
(6.2.12) known on 1'. It is a Neumann problem. Its solution is given by
Dini's formula (see for example (1.20] p. 31). We obtain
U(u,v)=n J_" U(a)Inlbei°-Wlda+ko
dN
(6.2.13)
being an unknown constant. Considering that W tends to a point
be' ° from C, one obtains
U(s) =
b
r+ir
J
JJJ
aN (o )1n I2 sin s
2 a I da + kO .
(6.2.14)
:
Since U(a) is an odd function it results
dN (a)
dN
(-a) ,
and integrating only on the interval (0, 7r), we deduce
U(s)
b f0" ddN
(a)S(s, a)d a,
7r
(6.2.15)
THE IN'I'ECRATION OF PRANDTL'S EQUATION
209
where we denoted
S(s, o) = In
(6.2.16)
am
2
We have (see, for example, 11.16]):
00
S(s, o) _ -2 F sin ks sin ko
km1
(6.2.17)
k
the series being absolutely and uniformly convergent (s 0 o).
In (6.2.15) we did not encounter ke, because we have U(s) _
_ -U(-s) whence U(0) = 0.
Taking into account the relation (6.2.12) in which 2U(o) is replaced
by C(a) according to the condition (6.2.3), one obtains the following
Fredholin-type integral equation
C(S) = A J
C(°)
si) S(s, a)d or + Jo(a)
,
(6.2.18)
where
A=
2bQ
z
Jo(s)
,
2b
x
J(o)S(s, o') sin o d o .
(6.2.19)
To
Hence we reduced Prandtl's integro-differential equation to the Fredhoimtype equation (6.2.18).
The kernel of this equation has an integrable singularity. Equivalent
integral equations have been given by Betz and Gebelein in 1936 and
Trefftz in 1938.
6.2.2
Existence and Uniqueness Theorems
For proving the existence and uniqueness of the solution of Prandtl's
equation , we shall use the first theorem of Fredholm. This may be
enunciated as follows: the equation
b
i
P(s) = A
K(s,)p(o)ds + f(s),
(6.2.20)
fa
has an unique solution for a given value of A and for every fee term
f if and only if the corresponding homogeneous equation admits only
the trivial solution V(s) = 0. Hence, we must show that the equation
210
THE LIFTING LINE THEORY
(6.2.18) which is homogeneous (Jo = 0) has only the trivial solution.
But the homogeneous equation corresponds to the boundary problem
(6.2.12) which is homogeneous (J = 0). Applying Green's formula
/ JD
U)2du dv = -JOD Ud S,
(6.2.21)
where D is an annulus, exterior to the circle r, bounded by an
concentric circle of radius R > b and observing that for R -- oo the
last term vanishes (see, for example, §6.1 from
1 fx
A
we deduce
UdNda < 0.
Utilizing the homogeneous condition (6.2.12), we obtain:
W IAsin
0.
(6.2.22)
This inequality implies U = 0, because, as it results from the definition
(6.1.17), we have A > 0.
Hence, Prandtl's equation has an unique solution. This result is very
important, because, as we have already seen in 6.1.4 and as we shall see
in the sequel, we manage, on various ways, to determine a solution of
this equations. The above result ensures that if we find a solution, this
is the unique solution of the equation.
6.2.3
Foundation of Glauert's Method
The integral equation (6.2.20) is a Fredholm-type equation of the
second kind. This equation has a symmetric kernel if
K(s, a) = K(a, a).
(6.2.23)
The equation (6.2.18) has not a symmetric kernel, but it can be symmetrized. Indeed, multiplying the equation with
(on the inte-
A)
gration interval the quantity under the radical is positive) and, taking
the function as an unknown
c(8) =
sins C(s)
A(s)
(6.2.24)
211
THE INTEGRATION OF PRANDTL'S EQUATION
one obtains the following equation:
c(s) _ -2a
Ja
c(a)
VsTn,;
i
"in a
A
00
sink sin ka
a
()k_i
d a + F;i(S) A(s)
(6.2.25)
00
K=-2E
k i1
sin s smo sin ressin rca
k
A(s) A(a)
is obviously symmetric. The kernel is even degenerate, but not of finite
rank. As it is known (see, for example, [1.22], vol.3, p.193), the integral
equations with degenerate kernel may be reduced to infinite algebraic
systems and the equations with degenerate kernel of finite rank may be
reduced to linear algebraic systems with a finite number of equations.
We are not in this situation, and according to (6.2.25)we shall take only
the property of symmetry of the kernel into account. According to the
theory of Hilbert and Schmidt (see, for example, (1.22] v. 3, p. 243) the
solution of the integral equation may be expanded, with respect to the
eigenfunctions of the kernel, into absolutely and uniformly convergent
series. Hence the solution of the equation (6.2.25) has the form:
00
C(a) =
A(a E At, sin ka.
)k-1
Taking (6.2.24) into account, it results:
00
C(a) _ E Ak sin ka.
(6.2.26)
k=1
The (constant) coefficients Ak will be determined replacing (6.2.26) in
the equation (6.2.18), or easier, performing this replacement in Prandtl'
equation (6.1.18), which, with the change of variables
y = b cos s,
(6.2.27)
1 ) = b cos a
and with the notation C(s) for the function C(y) composed with
(6.2.27)1 etc., becomes
2b/3C(s + a(s)
C'(a)d a=
casa-cuss
2itbaOff(
s s
)
(
6.2.28
)
Before discussing about how to determine the coefficients A, from
(6.2.26) and (6.2.27), we must notice that the form (6.2.26) of the so-
lution C(a) may result directly from (6.2.18), without utilizing the
212
THE LIFTING LINE THEORY
theory of Hilbert and Schmidt. Indeed, taking (6.2.17) into account, the
equation (6.2.18) becomes:
_'inks
C(s) = -2a
fo"
k=1
A(( )) in k sin a d a+
V
+4rb
sin ks
Air)
sin ko sin od a .
I0
The integrals are constants and the solution will have the form (6.2.26).
6.2.4
Glauert'e Approximation
We shall return now to the problem of determination of the coefficients Ai from (6.2.26). Replacing C from (6.2.26) in (6.2.28) and
using Glauert's formula (B.6.6) (herein is the origin of this formula), we
deduce:
E Ak[2bfl sins + k7ra(s)) sin ks = 21rba(s) j (s) sins .
(6.2.29)
k=1
Glauert's approximation consists in keeping the first n terms from
the expansion (6.2.26) and then imposing (6.2.29) to be satisfied for
n distinct values of the variable s. The coefficients Ak are the
solution of an linear algebraic system, but we cannot evaluate the error
of the approximation. Many other approximations have been given in
the literature (see Lotz in [1.24), Carafoli in [1.5)).
6.2.5
The Minimal Drag Airfoil
The foundation of Glauert's method,which consists in establishing
the formula (6.2.26) gives the possibility to give an answer to the following problem of practical interest: to determine among the wings with
the same lift, that one which has the minimum drag. In view of this determination we shall calculate, utilizing the formulas (6.1.21), (6.1.25)
and (6.2.26), the lift and drag coefficients cL and CD. For determining
the lift and the drag, we multiply these coefficients by the same factor
1
2 p00UUA1. Since
2! sin ka sin lad v = irdkl ,
k,1=1,2,...
213
THE INTEGRATION OF PRANDTL'S EQUATION
we deduce
+b
f
Jb
ct, = A
C(y)d y = A f C(a) sin ado, =
JO
'At v.
(6.2.30)
Utilizing Glauert's formula (B.6.6), we obtain:
()
C'(s)s a
1
1
4ab To Cos a - oos or
such that:
CD
/
2
C(V)w(y)d, y = -
b
A
°°
1
2A
1 (ksmka)
ksI
f
4b
sin ka
kAi sin a
w(a)C(a) sin ad a =
o0
Alsinio do =
(6.2.31)
!:1
oc
kAj.
k=t
The formulas (6.2.30) and (6.2.31) indicate that among all the wings
with the same lift (with the same A1), the minimum drag corresponds
to the wings for which A2 = A3 = ... = 0.
The solution of Prandtl's equation for these wings is
C(o) =A1sin or =C(y)=At V,
where Al = C(O) may be determined obviously from the equation
(6.1.26). We have
_C(_)
A(N)
5
= 2w(y) + J(y) _ - 1 At + J(y).
(6.2.33)
In the case of the flat plate, j has the form (6.1.20). It results
J = 2e whence we deduce that the member from the left hand side of
(6.2.33) is constant. Hence,
a(y) = ao 1 -
,
(6.2.34)
the constant ao being determined by the relation
0
+
2b) Al = 2E,
(6.2.35)
214
THE LIFTING LINE THEORY
if one gives Al. The same relation determines Al if one gives ao.
For example, when Al = k like in (6.1.28), it results ao = 1, like in
(6.1.27) and vice versa.
The expression (6.2.33) shows that the wings which have the above
property are the elliptical flat plates.
6.3
The Symmetrical Wing. Vekua's Equation. A Larger
Class of Exact Solutions
6.3.1
Symmetry Properties
Very often in aerodynamics we encounter the case when the wing is
symmetric with respect to the xOz plane. In this situation we have
x*(y) = xt (-y) ,
-b < y S +b
h(x, y) = h(x, -y) ,
(6.3.1)
From (6.1.19) and (6.1.20) it results
a(y) = a(-y),
j(y) = j(-y).
(6.3.2)
Let us prove that we also have
C(y) = C(-y) .
(6.3.3)
Indeed, changing in Prandtl's equation (6.1.18) y by -y and taking
(6.3.2) into account, it results
AC(-y) =
a2 y)
1c
(q) d q +.7(y) .
Cc(q)
n
(6.3.4)
y
Putting in the integral n = -u and observing that
C(q)d17 = dC = C'(u)du,
we deduce
r -16 C'(rl)d
f b 1+31
_
'/'_b
C(u)du =,+b Cl(,i)d+l
J.fb -u+y
b
7I-y
(6.3.5)
Introducing this relation in (6.3.4) and comparing with (6.1.18), we get
(6.3.3).
THE SYMMETIUC'AL WING. VEKt'A'S EQUATION
215
The Integral Equation
6.3.2
We shall present in the sequel the simplest method for obtaining the
equation (6.2.18). The demonstration is inspired from [A.27), where, on
his turn, it was taken from Magnaradze (6.16] and Vekua (6.28). With
the notations (6.2.7) Prandtl's equation is
I 'r+b
C"(n) d
27r ,!-b rl - y
n = 0C(y) - J(y) .
(6.3.6)
A(y)
For the existence of the principal value we have to assume that C'(y)
satisfies Holder's condition on the segment (-b, +b). We shall invert
this equation assuming that the right hand member is known. As it
is known from (C.1.1) the solution C'(y) depends on the behaviour
imposed in the points ±b. We know that we cannot obtain a bounded
solution in the two points without imposing a restriction to the right
hand member. In the same time, because of the symmetry of the wing,
we cannot consider C bounded only in an extremity. Hence C'
is unbounded in the two extremities, i.e. the solution has the form
(C'.1.11). Further, for inverting the equation (6.3.6), A(y) and J(y)
have to satisfy Holder's condition on [-b, +b]. If A(y) and hs(x, y)
(with respect to the y variable) have this property we deduce the same
thing for J(y) . Moreover, a(y) must not vanish on (-b,+b). If
all these conditions are satisfied, then, using the formula (C.1.11), we
obtain
C'([/) _
-2
'r+b
1
b2y2
b
bz -
I' [#'(") - J(n)] d n+
n-y
A(q)
(6.3.7)
B representing a constant which has to be determined. It is zero
because from (6.3.3) we have C'(y) = -C'(-y), whence C'(0) = 0.
Imposing this in (6.3.7) and observing that the integrand is an odd
function, it results the assertion.
Utilizing now the identity
P) --r12 d
dy
In
-y2- 62-n2
i(y-n)+
i(y-n)+ b2-y2+ P-
(6.3.8)
216
THE LIFTING LINE THEORY
and integrating (6.3.7) on the interval (-b, y), from (6.3.3) one obtains:
C(y) _
+b
2
W
1
110A(+l)
b
(6.3.9)
i(y-tl)+ &2y2i(y-q)+ b -y + /bbl-
-J(17)] In
because the modulus is equal to the unity for y = -b.
Performing the change of variable
y = b cos s ,
tI = b cos a
(6.3.10)
and taking into account that
in
-y2-
(y-tl)+
(y-17)+ b2-y2+ b2-q2
is - e is
= In l e-i s _ eio I
e
- S($ a)
we obtain obviously the equation (6.2.18).
Using the notation A(tr-a) we have A(bcoe(7r-a)) = A(-boos a) _
= A(-y). Hence, taking (6.3.2) and (6.3.3) into account, it results
A(tr - a) = A(a),
J(tr - a) _ i(a),
C(tr - a) = C(a)
(6.3.11)
whence:
sin ad a =
I"/2 [#A(a) - J(a)J In
s+a
--/z
pc(o') - J(a)J In
o
cos
cos
2
s - a sin ad a .
2
But,
.
sin
In
sin
$-a
2
s+a
2
cos
s+a
2
s-a
cos
2
= In
sins - sin or
sin s+sin or
THE SYMMETRICAL WING. VEKUA'S EQUATION
217
In this way, the equation (6.2.18) for the symmetric wing becomes
26
C(s) _
J"/2 [(;)
(6.3.12)
- sing
-J(a) In sins
sinada,
sins + sin a
J
with s in the interval (0, n/2).
Vekua's Equation
6.3.3
In 1945, I.N.Vekua [6.28] gave for the symmetric profile whose chord
has the form
bz
a(y) =
2
-y
with p(y) = p(-y) > 0,
,
P(y)
(6.3.13)
(where p(y) is an analytic function on [-b, +b]), a Fredholm-type integral equation which has the great advantage that it may be integrated
exactly for a large class of profiles.
Vekua's method was extended immediately by Magnaradze [6.16] to
wings for which the function p(y) is not necessarily analytic on the
interval f-b,+b]. Since we had not the occasion to read this papers, we
present herein a a synthesis due to Muschelisvili [A.27]. To this aim, we
write the equation (6.3.7), where we considered 13 = 0, as follows
20
A(y) C'(y) +
+b
C(11
f-b
'
'
1
(6.3.14)
= A(y)J1(y) -
tb
a(y)
b2 - y2
R(y,n)C(rl)d>j
b
where
2(3
R(y, rl) =
Ji (y) =
1
(i-r
a(n)
7r 17 - y
2
-
_
a(R-7-ill
y)
(6.3.15)
Ifb +b
62 - y2
J(q)d i1.
(6.3.16)
Ji (y) _ -J1(-y) .
(6.3.17)
rl - y
Obviously we have:
R(y, ) _ -R(-y, -y)
THE LIFTING LINE THEORY
218
Further we shall assume the continuity of the first order derivative
of the function
P(y) =
(6.3.18)
a(y)
In this case, R(y, i) will be a continuous function.
Since according to (6.1.7) we have:
C(q
ay i-bb
r1
11
=
dq =
o d y \Jb ` + Jy+a,
v
11
tab
do _
J-b
17
-
dn,
from (6.3.14) it results
dy [A(y)C'(y)] +
2Q ' +b
J-b rl (y d q
B(y),
(6.3.19)
where
B(y) =
dy [A()Ji() -
lr+b
a(y)
J
R(y, n)C(r1)d n I
.
(6.3.20)
Obviously,
B(y) = B(-y)
(6.3.21)
Eliminating the integral from (6.3.19) by means of Prandtl's equation,
we obtain the following differential equation:
A(y)
b [A(y)C'(y)] + 4/32C(y) = A(y) [B(y) + 4i3J(y)I .
(6.3.22)
Assuming that the right hand member is known, we have in (6.3.22) a
differential linear equation for C(y).
The homogeneous equation has the linear independent solutions
cos s(y), sin s(y), where
20
8(y) =
(6.3.23)
Jo a(rl)
Utilizing Lagrange's method of variation of constants, we deduce
that the equation (6.3.22) has the following solution:
C(y) = Co cos s(y) + Cl sin s(y)+
y
sin [s(y)
+ 2Q f [B('l) + 4QJ(n)]
- s(n)]d n,
(6.3.24)
THE SYMMETRICAL WING. VEKUA'S EQUATION
219
Co and C, being constants. Obviously, Co = C(0).
Calculating C(-y), taking into account that s(y) is an odd function
(its derivative is an even function) and B(q) and J(n) are odd
functions, and imposing (6.3.3), it results C1 = 0.
Introducing B given by (6.3.20) in (6.3.24), performing an integration by parts and observing that the integrated term is zero because
JI(0) = 0,
1-b
it results the following integral equation:
+1
C(y)
K(y, ii) =
a
f
+b
(6.3.25)
K(y, q)C(t7)d n= g(y) ,
R(qj, q)
cos [s(y) - s(rh)]d nl ,
(6.3.26)
To
+2 10yJ
g(y)=Cocoss
sins
s
d+
(6.3.27)
+ [iivi)ccs[a(v) - s(n)1 d q.
The equation (6.3.25) for J9 = 1 is the equation given by Vekua and
Magnaradze. Unlike the equation of Trefftz (6.2.18), this is regular (the
kernel has no singularity). Moreover, in case that the function p(y)
given by (6.3.18) is a rational function, more precisely in case that a(y)
has the form:
a(y)=a
0
-y21+ply2+...+pny2n
9
1+qly +...+gny2"
(6.3.28)
as we shall see in an example, the equation of Vekua and Magnaradze
reduces to an algebraic finite system. This form for a(y) is suitable
for approximating every wing of practical interest. We have to mention
that, for the wings having the form (6.3.28), the case when qt = 92 =
... = q,, = 0, has been solved by H.Schmidt in 1937, 16.241, and the case
when pi = pl = ... = pn = 0 belongs to a larger class, considered by
the author of the present book in 1958, (6.41. For this class one obtains
the exact solution.
Before passing to applications we notice that if
constant, then, taking (B.5.6) into account, we deduce
g(y) = Co cos s(y) + 2k1,
J(tl) = k is a
(6.3.29)
220
THE LIFTING LINE THEORY
where
I (y) _
{sin[s() - s(rl)) -
cos[s(y) - s(n)J
q
dq.
(6.3.30)
6.3.4 The Elliptical Wing
Denoting by a and b the semi-axes of the ellipse from the xOy
plane, we deduce
s
1-b
V
a(y)=aa b2-y2, as=a/b.
Obviously, R = 0 whence K(y, r)) = 0. The equation (6.3.25) gives
directly the solution C(y) = g(y), where g is calculated with the
formula (6.3.29). In I one performs an integration by parts. Since
from (6.3.23) it results
2fi
s'(tl) = ira(y)
we deduce
+wao
I=
b2-y2-bcoos(y).
Since from (6.1.20) and (6.2.7) it results J = 2e, using the notation
4wea o
aao+2/9'
we deduce
C(y) = Co cos s(y) + k
- kb cos s(y) .
(6.3.31)
For determining the constant Co we shall employ the condition C(b) = 0.
Since from (6.3.23) it results
s(y) =
20
wao
arcsin b ,
we deduce Co = kb whence
C(y)=k b2 -y2.
For ao = 1/b one obtains exactly the solution (6.1.30).
(6.3.32)
THE SYMMETRICAL WING. VEKUA'S EQUATION
221
The Rectangular Wing
6.3.5
We shall consider now that a(y) has the form
a( E/)=ao
b2 -
y
21+
(6 . 3 . 33)
,
the real numbers p and q being chosen in order to ensure only positive
values of the fraction one [-b, +b]. In the sequel we shall see that one
imposes pb2 > -1 whence qb2 > -1. From (6.3.15) and (6.3.33) we
deduce
R(nt, n) =
c(q + Th)
(1+pip)(1+pni)'
20(g -
C
p)
(6.3.34)
irao
and from (6.3.26)
K(y, r1) =
' wo(y) +'P1(y)
(6.3.35)
1+pr7
where
acs{s(by} -_aq(m)) 1 +
MY) = c
1
1
d'ri
(6.3.36)
Taking into account that for pb2 > -1 we have
d ,q
q 1 + pb2
1
I (1+p) b2-
1+
arct&n
from (6.3.23), for p # 0, we deduce
s(y)
2Q f q arcein EI +
_ 7rao
Lp
b
1 +'2 J
p- q
p
arctan
y
1+ pb2 l
(6.3.37)
and for p = 0,
(I +
s(y) _
L
c) aresin b - Zy
vfb2 --y2 l
.
(6.3.38)
Replacing K(y, q) given by (6.3.35) in the integral equation (6.3.25)
and observing that the first term vanishes because the integrand is an
odd function, we deduce:
C(y) + tPiny)
r-b
1
C
+(p
dt = Cocoss(y) + 2k1(y).
(6.3.39)
THE LIFTING LINE THEORY
222
The integral is a constant Cl which may be determined by multiplying
(6.3.39) with (1 + py2)-1 and integrating with respect to y on the
interval (-b,+b). We obtain
J
III[
b
1 + P?1
1
(6.3.40)
bbl+py2dy.
-Col bb +a( dy=2A;
Imposing (6.3.39) and the condition C(b) = 0, we deduce the relation
Co cos s(b) - 91(b) CI = -2k1(b).
(6.3.41)
Determining the constants Co and Cl from the system (6.3.40) and
(6.3.41), we find the exact solution of Prandtl's equation
C(y) = Cocoss(y) - !C1SOj(y) +2kI(y).
(6.3.42)
Using the inverse method, i.e. considering various values for the con-
stants p and q and calculating the form of the chord, we may find
important wings for which the exact solution (6.3.42) is valid. So, in
(A.271, considering q = 0 and pb2 = 0, 9, one obtains an almost rectangular wing (the variation of the chord versus the span is very small).
Indeed, we have
y/b
a/boo
0. 1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.00
1.02
1.03
1.05
1.06
1.06
1,03
0.95
0.75
Importantresults ofthis meth
method are givenin [1.331.
6.3.6
Extensions
Modifying Vekua's method, we managed in (6.4) to give the exact
solution of Prandtl's equation for wings whose chord satisfies the relation
ys
a(y) =
P(y)
,
(6.3.43)
where p(Z) is a holomorphic function in the Z = y + i z complex
plane, excepting the vicinity of the point at infinity where one admits
223
NUMERICAL METHODS
the following series expansion:
k>O
p(Z)
=n=-oo
E PAZ"
(6.3.44)
.
The determination of the solution of Prandtl's equation is reduced to
solving a Hilbert-type problem, whose exact solution is given.
The polynomials belong to the class (6.3.44), whence the great importance of this solution. According to Weierstrass's theorem (see for example, [6.13], p.61), every continuous function on the interval [-b, +b]
(i.e., every possible form of the wing) may be approximated by polynomials. Practically, for this one may employ an interpolation method (for
example, Newton or Lagrange's method). Even if p(y) is a polynomial,
in this method it is not necessary to be symmetric, like in the theory of
Vekua and Niagnaradze.
Numerical Methods
6.4
6.4.1
Multhopp's Method
The idea of biulthopp's method consists in approximating the func-
tion C(a) by the trigonometric polynomial P"(c) obtained by the
Lagrange interpolation in the basis (sinkc)kl,...,n. For determining
P", we notice that after introducing the matrices
ST = (sin c,...,sin nc),
aT
= (al,...,an),
it may be written as follows
=
P.
n
(c)
ak sin ka = sTa.
(6.4.1)
kal
The points where one imposes for PP(o) to coincide with C(o)
(the nodes) are given by the uniform grid
aI
n+1 -o,
a2
2a
n+1
-2a,...,c"=
na
n
1 =nor,
(6.4.2)
whidi is usual in the theory of interpolation [6.13] p.20. These are
equidistant on the half-circle with the diameter on the span (fig. 6.4.1).
The points
xJ = cos a1
(6.4.3)
THE LIFTING LINE THEORY
224
+z
x
-1
1
b
y
Fig. 6.4.1.
are the zeros of the Chebyshev polynomial of order two (F.2.8) on the
interval (0,7r).
Denoting c; = C(a1) we have to determine the matrix a from the
system
Sa = c,
(6.4.4)
where
sin ai sin 2a1 ... sin nal
S = sin a2 sin 202... sin nag
c=
(6.4.5)
sin a, sin 20,E ... sin na
One obtains [6.131, p.21
S-1 =
IS.
n2
(6.4.6)
NUMERICAL METHODS
We shall present at the end of the section this calculus. Hence,
n+l
sin v1... sin na1
Cl
... sin na
c
(sin a...sin no) I
sin on
1Ecinko l
2
n+
(6.4.7)
k=1
1(sin a... sin na)
ctsinka
2
+
sin koj s i n
j=1
k=1
For C(a) we have the Wowing expansion:
C(a)
= n +2 1
n
n
ck
sin ka j sin jo.
(6.4.8)
We must notice that this expression could be also obtained from
(6.2.26) approximating the Fourier coefficients according to the definition of the integral for an equidistant division a. Indeed, f om (6.2.26)
we have:
C(Q) sin yoda
Aj
TO
+1
k,i
C(ck) sin jai, .
Since jak = jka = kof, eraplaying Glauert's approximation
Al sin jo,
C(Q) _
j-1
one obtains (6.4.8).
Utilizing Multhopp'e expansion (6.4.8), from Prandtl's equation (6.2.28)
226
THE LIFTING LINE THEORY
and from Glauert's formula, we deduce:
sin s
4b,3
n+1
n
n
ck > sin jka sin js+
k=1
j=1
(6.4.9)
27r
s sins.
sin jka sin js = 2irba s
j=1
k=1
Giving to s the successive values La(£ = 1, ... , n) and taking into
account the formulas (6.4.17) and (6.4.19) from below, we deduce the
system (£ = I,-, n),
BtkCk = Bt,
Oct +
(6.4.10)
k=1
where
2Bt = (n + 1)ira(£k)j(£a).
(6.4.11)
Fbr k $ t,
Btk
-
lra(£a)
2bsin£a
I - (-1)n-t
8
_
I
sine
(k + t)a
2
1
sine (k - £)o`
' (6.4.12)
2
and for k = e,
Att =
7ra(£a)
n(n + 1)
2b sin la
4
(6.4.13)
Determining the unknowns c1, ... , c from the system (6.4.10), one
obtains the solution of Prandtl's equation from (6.4.8).
Since Atk vanishes when k - £ is an even number, the system
(6.4.10) may be separated; more precisely, the unknowns with odd indices may be expressed by means of the unknowns with odd indices and
vice versa. This fact was proved in [1.2]. As it is already mentioned in
(1.2] and in (6.17] an iterative procedure for determining the unknowns is
also established. The author had not at his disposition this paper. The
procedure simplifies in the case of the symmetric wing (C(y) = C(-y)),
i.e. when ck = Cn+l _k (k = 1, 2, ... , [(n+ 1)/2]) or in the case of the antisymmetric wing (ck = -c,+1_k). In the first case the system reduces
to [(n + 1)/2] equations, and in the second to ((n - 1)/2]. The square
brackets indicate the integer part of the number from the interior.
227
NUMERICAL METHODS
In the sequel we shall calculate the sums that intervened in the above
formulas. Let it be for the beginning
n
r
j=1
i=1
11.Ecosrja, 12 - Esinrja.
(6.4.14)
Denoting z = e1r`r, for r 96 0, we obtain
_
z - zn+i
-
=
l- z
1
whence, separating the real part from the imaginary one,
I1 = -1 +
1 - (-I}r
,
2
12 =
1 - (-I)r
2
cot
ra .
2
(6.4.15)
One obtains the sum
n
r
Ejcosrja-
1
j=1
4
I
era
sin 2
(6.4.16)
12 with respect to a. We deduce therefore, noticing that
k - f and k + f are odd or even simultaneously,
deriving
n
n
2 E sin kja sin eja = E cos(k - e) jaj=1
j=1
(6.4.17)
n
-Ecos(k+f)ja= (n+l)6kt.
j=1
With these formulas, the relation (6.4.6) written as follows
S-1S
n2
S2=1
may be immediately proved, since we have:
n
Sz
n
_ E sine ja F sins ja sin 2ja ... sine ja sin nja
j=1
ial
j=1
(6.4.18)
THE LIFTING LINE THEORY
228
Utilizing now (6.4.16), for k q& a we deduce
n
2 E j sin kja sin eja =
j=1
(6.4.19)
4
sin2
and for k = e:
(k+2 e)a
sin2
e
(k
2
n
4 E j sine kja = n(n -}- 1).
(6.4.20)
j=1
Analogously it results:
n
2 E sin jka sin ja = > cos j(ka - a) - > cos j(ka + a) _
j-1
j,1
j
-Re z-Zn+1
Z-Zn+1
_Re 1- f
I-Z
s,,01(ka+.)
(-1)k+1 sin(n + 1)a sin ak
COs a - COS ak
(6.4.21)
If we utilize this identity, for the formula (6.4.8) which gives the
solution of Prandtl's equation , we obtain the final form
C(a) __
1
n+l
n
k=1
k+1
`
sin(n + 1)a sinker
.
coo a - cos ka
(6.4.22)
In [A.23], p. 98-111 one gives a mathematical justification of this
method. More precisely, one demonstrates that under certain circumstances, the iterative procedure that one utilizes for solving the system
(6.4.10) is convergent and the solution (6.4.22) converges uniformly to
the solution of Prandtl's equation.
6.4.2
The Quadrature Formulas Method
In [6.5) we gave a numerical method for solving Prandtl's equation by
means of Gauss-type quadrature formulas. It is well known that these
229
NUM ERICAG METHODS
formulas give the best approximation. The key of this method consists
in writing Prandtl's equation in the form (6.1.21). With the change of
variables
y = bs , q7 = bz
(6.4.23)
and, keeping the notations C(s), a(s) and j(s) for C(bs), a(bs) and
respectively j(bs), the equation (6.1.21) becomes
.
/3C(s)
2b)
1
J-1 (z
(s)s d x + j (s) ,
(6.4.24)
and the conditions (6.1.7),
C(±1) = 0.
(6.4.25)
The solution of the equation (6.4.24) has the form
C(s) =
1
-_3 2 c(s) .
(6.4.26)
Employing the quadrature formulas (F.3.5), the equation (6.4.24)
reduces to the algebraic system
n
EAkici =7k, k = 1nn,
(6.4.27)
i=1
where we denoted
ak = a(zk), Ck = C(xk),
era k
A
2b(n + 1)
(-1):+kl
1-
Akk=fV1 -xk+6
jk = j(zk)
(x1x2
-1
i#k,
zk)2
(6.4.28)
n41; zi=Cosn+1,t
I'n'
the unknowns being c1,... , c,t. The system has to be studied theoretically and solved numerically using a computer. After determining the
unknowns, the lift, drag and moment coefficients, defined in (6.1.21) and
(6.1.25),
CL
C=
2b
win the form
f
+1
1 - s2 c(s)d s, cD = - A
1
= mA J
1
`s
1 - s2 c(s)d s, c;
r1
ZA-
j
1 - s2 w(s)c(s) ds,
11
s
1 - s2 w(s)c(s) ds,
(6.4.29)
230
THE LIFTING LINE THEORY
give with the formula (F.2.12)
CL =
(n + 1)A E(1
i=1
- x?)c" CD = - (n + 1)A isl
"
_ {n + 1)mA E(1- x; 2)xc:,
27rb2
i_I
x?)w,c{
>,
2nb2
2
c= (n + 1)mA E(1 - xd )x:wig
i_1
(6.4.30)
where
w; =
13 VI - x,
.
2aai
(6.4.31)
In (6.4.30) we used, like in (6.1.21), the notations A for the area of
the domain D and m for dimensionless length of the mean chord (in
the direction of the unperturbed stream).
For verifying the method, we applied it in [6.5) to the elliptical Sat
wing, for which the exact solution is known (6.1.28), (6.1.30). With the
notation (6.4.26) it results c(s) = k. Putting in (6.4.27)
Aki =
1- x4Ak;,
c; = k4,
it results that the system
Ak.;ci
= Q+ it/(2b)
(6.4.32)
must have the solution dl = d2 = ... = c;, = I. For b = 10 one obtains
numerically dl = 4 = ...
1000. We deduce therefore that the
proposed method is very good.
In the sequel we shall give the numeric solution for the rectangular
wing (in this case there exists no exact solution). Taking the reference
length Lo in the definition of the dimensionless variables (2.1.1), to
coincide with the half of the chord, we deduce xt = :L1. It results
a(y) = 1 and j(y) = 27rc. Putting c; = 27red', the system (6.4.27)
becomes
Akic!
= 1, k = I- n,
(6.4.33)
where Ak; are given by (6.4.28) where we put ak = 1. The quantities
NUMERICAL 1METHODS
231
of interest in (6.4.30) are
1
kl =_ 71+1
(1-xi)c',
k2=-.1 (1-x?)3,2(c,')2,
(6.4.34)
V(1-x?)x,c;, k4=
n+1 i_i
A
n+1
E(1-x?)3/2x,(cI')2.
i=1
One obtains
CL = 7r2ek1 ,
CD = 7r2e2(kl - k2),
(6.4.35)
2c= = 7r2bek3 , 2c; = 7r2bC2(k4 - k3)
.
For b = 10 we obtain the following values:
fi
1
0.8
kl
k2
k3
k4
0.2347
0.2544
0.0907
0.0856
0
0
0
0
The result c= = c. = 0 is natural because of the symmetry of the
wing. The lift and the drag increase because of the compressibility. This
result is also natural. The value 2.307c obtained here for cL in the case
of the incompressible fluid is smaller than the values 7.29 e, 5.28 £, ... ,
obtained with Glauert's method [1.12], but the values obtained with
Glauert's method come closer to the values given here if A(= 2b/m)
increases, i.e. the span is great with respect to the chord. Just in this
situation the lifting line theory is valid.
We may think therefore that the method we have just exposed is at
the same time very simple and very efficient.
6.4.3
The Collocation Method
The simplest numerical integration method is certainly the collocation
method [6.6]. In case that Prandtl's equation has the form (6.1.19) and
satisfies the conditions (6.1.7), the solution has the form
C(y) =
b2
- y2 c(y) .
(6.4.36)
According to the collocation method, the segment [-b, +b] is divided
into iV elements L; and the function c(y) is approximated on each
232
THE LIFTING LINE THEORY
element with its value c; from the mid - point y° of the segment. So,
the equation (6.1.19) gives
,
N
[
4
c(y) = a(y)
2,8
fd
ix1
(b - y)2 d rI + 2j(y),
(6.4.37)
where, as we have already stated, Cj = c(y?). Imposing this equality to
be satisfied in every point yk, k =I,-, N, one obtains
N
2
Vr2b2
- yk2ck = ak E Aloe: + 2jk k = I
,
,
(6.4.38)
i=1
where
ek = c(yk), ak = a(yko), ?k = ?(yk),
Ak, -
w+1
f
J(y
(6.4.39)
b-q
-
y y;+1 representing the extremities of the segment L;(y1 = -b, yN+1 =
= b). So, (6.4.35) represents an algebraic linear system consisting of N
equations with N unknowns cj.
For calculating Ak{, we notice that for e V (a, ?'J we have
I
a
( 7 7-e )2
d
,
y-e
a -- e
+
nee In [ (ry-e)2 + ( b -e +
+aresin
a
b2_.72s)2 (a-e)
- ry b --a- 7
(6.4.40)
For e E (a, 7) we shall write
II =
o
-
(71e)2 d q
b2-772
s
=J
66
(n - e)2
v°
'I
d1
(6.4.41)
T
(77 -
e)2 d
b -V2
7
(rl - e)2
d ij.
We use the formula (6.4.37) for calculating the last two integrals. Since
(D.3.9)
r-b (q - e)l
d n= -7r ,
(6.4.42)
233
NUMERICAL METHODS
it results that I1 is also determined. Utilizing these formulas, for k 96 i
we obtain:
Aki
V - y?
b2
y
- 77i+I
yk0
yi+1 - yk
yl
(yi+1
02is
b2
72 +
(Vb2_YA.
b2-t/i+l
In
y`
(Vi-Yk)2+
yi
+arcsin
(\
b2-yk2
- yko
yi+1 - yk
y,2
+
+
R--i/i+l
- -Eli F1
b2
y,2
P
(6.4.43)
and for k = is
y2
b2
Aii =- -7r -
Ii
yt I
-
Vbi
b2 - y<+1
y2
+
+ (y +1 I Vb2 - y2
+arcsin
b2
1,2+
- yI
`2
Vb2-y°2+ b2-y, )
/
2]n
t? +
(VFb2
2b
(6.4.44)
Vb2
y°2
+
tJ+i
where we denoted 2ti = yi+1 - yiFor testing the method one utilizes also the elliptic flat wing. For
this wing the exact solution is (6.1.28) with k determined in(6.1.30).
Comparing (6.4.33) with (6.1.28), it results c = k/b. For E = 0.1,,0 = 1
(the incompressible fluid) b = 10 and N = 50, the numerical values
obtained for ci are situated between 0.050 and 0.053, and the value of
k/b is 0.052. For E = 0.0872, A = 1, b = 20 and N = 50 the values
obtained for ci are situated between 0.0243 and 0.0252 and the value
of k/b is 0.0252. The results given by this method are more accurate
when the angle of attack is small. The accuracy also increases when N
increases.
THE LIFTING LINE THEORY
234
Since the method is simple and can be easily applied one can accept
the error of order 10-3 that it contains.
6.5 Various Extensions of the Lifting Line Theory
The Equation of Weissinger and Reinner
6.5.1
In the sequel we shall study new integral equations in order to improve
Prandtl's model. In the past the researchers did not pay too much
attention to these equations because the integrands contain both the
unknown C and the derivative C'. Now we may use successfully
these equations because some fundamental results concerning the Finite
Part are known and, as we have mentioned in 16.5], after replacing the
Principal Value of an integral which contains C' and has a Cauchy type singular kernel by the Finite Part of an integral which contains C
and has an hypersingular kernel, we may employ the numerical methods.
First we shall study the equation of Weissinger [6.30] and Reissner
[5.29]. For obtaining it, we return to the lifting surface equation (6.1.28).
Taking (6.1.6), (6.1.11) and (6.1.12) into account, we deduce
+b
rlydn.
l
Cdn-7dt
J-e
9
ff bod JJ dq+2
(6.5.1)
Utilizing (5.1.35), from (5.1.28) it results the first form of the integral
equation
r+b fi(n) d n +
1
ri - y
1'rj
(1
4w JJD lJo \
_ z o d td r1= rt, (x, y).
R
(6.5.2)
Multiplying this equation by (6.1.15) and integrating it on the interval [x_(y), x+(y)] one obtains
j+b
a(y)C(rl)d71+
brl - Y
+
1
+b
2a J-b
+(v) x+(n)
Tx-)
x-x-(y) Mm) (i_i) dx d d
E rl
f(v),
x+(v)-x
(6.5.3)
j(y) being (6.1.18).
235
VARIOUS EXTENSIONS OF THE LIFTING LINE THEORY
Further we shall perform an approximation which consists in replac-
ing f(f,9) by
C(n)
A(n)
f (f, rl)
x+(n) - f
(6.5.4)
This is just the solution of the two-dimensional problem when h' is
constant. Indeed, according to the formula (C.1.9), for a given y , the
equation of the two-dimensional problem
1+(v) f(f,y)
T
( 6 .5.5 )
= hx (x,y),
n ,-(,) f - x
becomes, in case that h' does not depend on x:
f (x , y)
x+(y) - x
= -h(y) x - x_(y)
(6 5 6)
.
.
Multiplying (6.5.5) by (6.1.15), integrating the result over (x_ (y), X+ (y)]
and utilizing the formulas (B.5.4) one obtains:
-+(v)
C(y) __
- J:-(v)
(6.5.7)
f (f, y)d f = ira(y)h'(y)
Eliminating h' from (6.5.6) and (6.5.7) one obtains (6.5.4). With this
approximation, the equation (6.5.3) becomes:
a(y) J bbC n)di? +27r
YO
T-b
(6.5.8)
TO
where
N (y, n)
:+(v)
_+(»)
xo
rIl R)
x+(ti) - F x- x-(y)
- x-(+))
x+(y) -- x
dxd
f
.
(6.5.9)
Using the notation
r))
No ( y, n) = N(y,
A(y)A(n) = 1
x(y)
f
A(y)A(q) J=_(vl
1
(6.5.10)
_+(,)
J=-(»)
x+(r1) - f
x -x_ (y)
f- x-(rI) V x+(y) - x
d xd
f,
THE LIFTING LINE THEORY
236
we may write the equation (6.5.8) as follows:
CYO
- f eb
)dn+
2A
fC
o
N°(b,n)dn = J(y).
(6.5.11)
and J(y) being defined in (6.2.7). This equation has been
deduced by Reissner [5.29). Other simplifications are performed in the
papers [5.24] [6.30] in order to obtain an approximate solution. The
A(y)
difficulty consists in the presence of both C(n) and C'(n) in the
integrand (the equation is integro-differential). Employing, as we have
already shown, the Finite Part, i.e. taking (6.1.12) into account, we may
write the equation (6.5.11) as follows
C(q)
2 _1
y! `Ni(y, n)d, = J(y) ,
(6.5.12)
where we denoted NI = No - 2. This is the equation we are going to
use in applications. This is an integral equation.
6.5.2 Welssinger's Equation. The Rectangular Wing
In the case of the wings for which the lifting surface equation cannot
have the form (5.1.41) (this class is established in 5.1.7, and it contains
the rectangular wing) one obtains an equation which may be easily utilized in applications. This equation was given by Weissinger [6.31), and
received his name. Starting from the definition (6.1.6), we have:
-i',}(n)f (x+(n) n) + x'-(n)f (x-(n), vi)-
f
- f=-(")
x+(+r) a f
8dt
rx+(n) 8 f
_ J=-0)
(6.5.13)
dt.
We are going to show why the sum
-x+(n)f (x+(n), n) + x' (rl)f (x- (n), n)
can be neglected. The first term vanishes by virtue of Kutta-Joukowski
condition. The last term vanishes for a straight leading edge, perpendicular on the Ox axis (for example this is the case of the rectangular
wing). It also can be neglected when the leading edge is slightly curved
(x'_ (q) = 0). In general we can neglect this term considering that we
have prolonged the theoretic leading edge in front of the real leading
237
VARIOUS EXTENSIONS OF THE LIFTING LINE THEORY
edge and in this zone f = 0. With this approximation, the equation
(5.1.41) may be written as follows:
'+b
1
C'(17)
Ud,1
-a
2w
AD 09 xo
-d
fd
=-
"(x,
N)
(6.5.14)
We make a second hypothesis, having in view to be satisfied in the
case of the rectangular wing of width, let's say, 2a, namely we substitute
a' instead of zo from R. This approximation is justified by the fact
that (x - jj varies from 0 to 2a, hence on the greatest part of the
domain D we have IxoJ = a. So, the equation (6.5.14) becomes:
'+b
1
27r
1-6
dn -
we
-
f
1
27r
f+A
"+
_. aq
(6.5.15)
x0yo
Multiplying by
(a +- x a - x) and integrating with respect to z
on the interval (-a, +a) one obtains
a
-d n+ L jr a
C 'm
2Ja
2
30(y) _ -a
-a
C (rl)d rl
rah(y) ,
a±xh"(x,y)dx.
(6.5.16)
(6.5.17)
Introducing the non-singular kernel
K(pa) =
f,
fd
a+
_ a
yo
go
(6.5.18)
the equation (6.5.4) may be written as follows
C' r7)
Ir
YO
n+
Zax
f C'(rj)K(ya)d tj_ jo(y) .
(6.5.19)
This is the definitive form of Weiseinger's equation. It was created
mainly for the rectangular wing. Before presenting the method of integration of this equation, we have to notice that it is not necessary to
determine C(q). It suffices to find C'(t) because from (6.1.9), integrating by parts, it results
CL
b yC'(y)d y, es = -aQ
= -- A J b
y2C`(y)d y.
(6.5.20)
238
THE LIFTING LINE THEORY
The solution of the equation (6.5.19) has the form
Cr(y)
tiP(J)
(6.5.21)
with sp satisfying the equation
'''+6
1
d i7
b2
b
2 Yo
1
+ lira
f
(t )
h (Ju)d q = lo(U) . (6.5.22)
b2 - rf2
6
Performing the change of variables 'i = brl', y = by' one obtains
1
rrb
'r+t
J 777=7 it
y
sp(rl)
1
d>>
;p(aT)
27ra J
i
i - >>
h0(bJo)d n = .?o(bJ)
(6.5.23)
This is Weissinger's equation. It looks like the generalized equation of
thin profiles (C.2.1). The numeric solution is determined by means of
the formulas (F.2.5) and (F.2.6).
6.6
6.6.1
The Lifting Line Theory in Ground Effects
The Integral Equation
The lifting line theory in ground effects is obtained from 5.3, as well
as Prandtl's theory is obtained from the lifting surface theory. We are
not interested in finding the velocity and pressure fields. The integral
equation is obtained from (5.3.13) utilizing the formulas (6.1.11)-(6.1.13)
and the relation (6.1.8). One finds
1
T7r
+b E
+ f.?+ (y) f y, y)
!
d
d
U
f $ b 7-y y n
(y)
+
(6.6.1)
1
+b
+2- J C(t)i(x, yo)dTl = 2/is(x, y) ,
where
d2fl2x
d2
- yo
yo) _ (d2 +y()2 )R3 + (dl + y02)2 (
\1 +
R2 =
x + e2 ,
e2
= 02(y2 + d2).
x
RI) '
(6.6.2)
239
THE LIFTING LINE THEORY IN GROUND EFFECTS
Multiplying (6.6.1) by (6.1.15) and integrating the relation just obtained
with respect to x on the interval (x_(, ), x+(rj)), we obtain, taking
(B.5.4) into account:
C'(11) d
2#C(y) + a(y) Lb
Y-17
q
r
+
,l
C(n)No(y,
yo)d n = 2j(y) , (6.6.3)
b
where we denoted
s+(v)
x - x_ (y) N(x,
J:^(v)
x+(y)
1
No(y, yo)
(6.6.4)
yo)d x,
j being (6.1.18).
The equation (6.6.3) is the lifting line equation in ground effects. It
was given in (5.8J. It is obviously an integro-differential equation which
generalizes Prandtl's equation (6.1.16) (Um N = 0).
Further we shall make some considerations concerning the kernel
No(y, yo)-Taking (5.3.11) into account and introducing the functions
Iv(y, yo) _ -
+(v)
r
J
z+(y) (yx (xz + e2)"/2
(v)
d x, v = 1
,
(6.6.5)
we obtain:
No(y, yo) = (a +
y)2 [I, (y, yo) - a(y)] +
#2 13(y, yo).
(6.6.6)
Performing the usual substitution
x = s(y) + a(y)t,
(6.6.7)
where
s(y) = x+(y)
x-(y) , a(y) = x+(y)
X-(Y)
(6.6.8)
2
2
the integrals I become
+i
1
IN = -na(y)s(y)
J
l+ t
1
t
dt
P(t)v
aZ(y)
7r
r+i
.!-1
-1+-t
td t
1-t
P(t). ,
(6.6.9)
where
P(t) = a2t2 + last + e2 + s2 .
(6.6.10)
It is well known that the best approximation of these integrals is given
by the Gauss-type quadrature formulas (F.2.24). With this form (6.6.9)
one may study the asymptotic behaviour of the kernel No(y, yo) given
by (6.6.6) depending on the parameter A2 = d2/b2.
240
THE LIFTING LINE THEORY
6.6.2
The Elliptical Flat Plate
Without considering the ground effects, the solution of this problem
is (6.1.28), (6.1.30). In the sequel we shall determine the influence of the
ground. We assume that the wing has the equation x2/e2 + y2/b2 = 1.
We deduce
xf(y)=fb b2-y2,
a(y) =b Vb2
- y2s(y) = 0.
(6.6.11)
Assuming that the span is much larger than the chord (e << b), we
shall neglect the terms of order (e/b)2. From (6.6.9) we deduce I = 0,
whence
`
Integrating by parts, we deduce:
f
+b
+b
C(n)No(y,yo)dn=-a(y)f C'(n)&?yodn,
b
b
and the integral equation (6.6.3) becomes:
2fC(y) + a(y) J b C,n) d q - 2a(y)
J
bb
C'(1) d2 + y02 d n =
4_-Tra(y).
(6.6.12)
For d - oo one obtains Prandtl's equation. (6.1.16).
Taking into account the shape of a(y) from (6.6.11), we shall look
for solutions of (6.6.12) having the form
C(y) = k b2 - y2.
(6.6.13)
The first integral was calculated in (6.1.29). The second may be calcu-
lated with the substitution y = by, n = biz. If b < < d we neglect the
terms O(b/d)4 and we obtain
f
b 2ir
\a)2
b
(n)d2+yody=k
2
We assumed that the span is much smaller than the distance to the
ground. Replacing (6.6.13) in (6.6.12), we obtain the relation which
determines k:
(b `2
Tr
= 1Ezr6 .
(6.6.14)
k 2p+ -n b
-
-I
241
THE LIFTING LINE THEORY IN GROUND EFFECTS
Denoting by k the value of the constant when there are no ground
effects, i.e. (6.1.30), we deduce
k=kak,
where
(6.6.15)
2f3 + (t/b)ir
k0 = 20 + (t/b)ir - (e/b) (b/d)27r
(6 .6 . 16)
Obviously, k > 1.
The lift, drag and moment coefficients are given by the formulas
(6.1.21) and (6.1.25), where for w(y) = w(0, y, 0) we have 15.81:
W(y)
C ,(n)
J
47r
bb
d
rl - 4n
C'(Tl)
b
& yo d n.
(6.6.17)
These formulas (6.1.21), (6.1.25) and (6.6.17) are valid for every shape
of the wing. For the elliptical flat plate wing we obtain:
w(y) = - 4
1
2
+
C
!
(6 .6 . 18)
such that
cL=kocL, cD=klcp, c==c.=c,
with the notation
(6.6.19)
2
k1=ko li(1+2 )>1,
(6.6.20)
lift
Cr and cD representing the
respectively drag coefficients in the
absence of ground. Obviously, both the lift and the drag are increasing
in the presence of the ground.
The coefficients ko and k1 depend on M, t/b, b/d. The numerical
calculations from 15.81 show that for the lift the increase is not significant
but for the drag it is considerable. The ground effect is a decreasing
function of d. For the same values of the ratios t/b and b/d the
influence coefficient ko is an increasing function of Mach's number M.
6.6.3
Numerical Solutions in the General Case
Utilizing the formula (6.1.12), one may write the equation (6.6.3) as
follows:
2#C(y) - a(y)
f
+b
b
c(n) d
n+
(n - y)2
f
+b
C(n)N(y, yo)d n = 2?(y)
b
(6.6.21)
242
THE LIFTING LINE THEORY
As it is known, this is an integral equation, not an integro-differential
one, but the singularity is stronger than in (6.6.3). In (6.6.3) we have a
Cauchy - type singularity and in (6.6.21) we have to consider the Finite
Part of a hypersingularity. But for this kind of equations there are
available quadrature formulas. In order to apply this method, we have
to perform the change of variables y = by, q = bq' for calculating the
integrals on the interval (-1, +1). We obtain
2bIC(y) - a(y)
f
+t
C(n)
Y)2 d n + b21 C(i7)N(y, yo)d =
+1
1
(n -
j(y)
1
(6.6.22)
Since the solution of this equation has the form
C(y) = V171- y2 c(y),
(6.6.23)
we obtain
1 - yj cj - aj
2b/3
1 --17C(17) d n+
Ti
(n - yj)
1
1 - i2c(yl)N(yj, yj - n)d n = 2bjj , yj = cos n
+b21+
+
1,
t
j=T
.
(6.6.24)
Using (F.3.5) and (F.2.12) one obtains the system:
n
Ajcj + E Ajkck = 2bjj , j = 3n-,
(6.6.25)
k=1
where
A.;= MO 1-y;+aj7r
Aak
-a
L
n+1
2
b2k.
7r
n + 1
- yj)2 +
1(yk
- (-1)'+k
n + 1
N(yj, yj - yk) (1 - yk2)
J
(6.6.26)
In the first term from Ajk one excepts k = j. The system (6.6.25) is
solved numerically.
6.7
6.7.1
The Curved Lifting Line
The Pressure and Velocity Fields
In this subsection, we shall pay a special attention to the aspect ratio
A = (2b)2/A introduced in (5.4.1). Usually, if A is small, one applies
243
TILE CURVED LIFTING LINE
the theory from 5.4 concerning the wings of low aspect ratio. If A is
large one applies the lifting line theory. These are the two asymptotic
theories of the lifting surface theory.
As it is known, one of Prandtl's hypotheses consists in replacing the
domain D by the segment [-b, +b] taken along the span (the Oy
axis). This hypothesis is plausible for the wings having the shape of
an ellipse, triangle, trapezium or rhombus (see fig. 6.7.1) but it can be
the source of great errors in the cage of the wings having the shape of a
swallow tail or the shape of an arrow. In the first case it is natural to
replace the wing by the curvilinear median (see fig. 6.7.2), and in the
second case one approximates the wing by the median broken line (fig.
6.7.3). For birds, the nature preferred the curvilinear median. These are
enough reasons for studying in this subsection the curved lifting line. In
YA
b
j'I i
Y
Y.
b
i
b
rt
1
1
0
x
-4 i
t
aX
0
1
1
b)
1
Fig. 6.7.1.
Fig. 6.7.2.
Fig. 6.7.3.
244
THE LIFTING LINE THEORY
this case, one starts too from the general representation (5.1.8)-(5.1.12)
and Prandtl's hypotheses. We assume therefore that the wing is without
fl = 0) and that the unknown is C(y) defined
thickness (hl = 0
by (6.1.5), with the conditions (6.1.7). When the domain D reduces
to the curved line r (fig. 6.7.2) having the equation x = x.(y), the
formula (6.1.8) is replaced by
r`
/ JD f
lim /
il)k(x, y, z, t, q)d d y = -
f
+b
k(x, y, z, x. (q), n)C(i )d +1,
b
(6.7.1)
for x-(n) -. x. (n) - x+ (n)
So, with the notation
R. =
ix - x.(n)]2 + A2(y02 + z2),
(6.7.2)
the formulas (6.1.9) become
z
P(x,y,z) _ -4
C(ri) z dn.
(6.7.3)
rt
v(X' Y, z) = -
I
:Fir =fl
lim.
o-)]
J 1D f (C, n) gEl l yob z
.1 JD f (C W 1+ RI
(1 + R-
)
z
y2 + z2
d
d
)dtdn+
+XOZ
C(n)11+x(n)lal\
l Jb
47r
L
o+z2)dn-
JJ
p2yo
+6
[x - x.(n)]z
- T7r 14 C(rl)
yof-' R3 d n
1
(6.7.4)
Utilizing now the identity
r
x - x`
R. R
_ 8(
8
) &1
z
z
z(x-x') /2yo
y
x-x.`1
R* jl +
z2 R3 _
(6.7.5)
32z ,
THE CURVED LIFTING LINE
245
we obtain, after performing an integration by parts,
j
f
v(x, y, z) =
+b CI(rl)
f1+
x R'(q)] d,,R.
C
(6.7.6)
Q2
47T
/ +b
J
b
zZl=
C(i) R. d q .
Analogously,
w(x,y,z) =wi +w2,
(6.7.7)
where
=4a
fffri)_
j+bx
R,
x}m.(_)ddtl=
(n) d n
4nx*-x.JJDf(C v?) 2[y02
W-2
47r xlim- . If,,, f (C rT) l 1 + Rt 1
lr
,ffDf( , l) o+Z2
YO X02
lice
=
art
l 6r+.2) d d'r
(6.7.8)
(_)dd=
R
1
r+b
4 J b C(") ['x_x,(,i)l 09l ( T2 + Z2 l d
YO
Y7+
J
+Q rb
c(n) yo +'Z2)
dn.
Introducing the identity
$2(x _
R.
yg
+Z2)
110+z2
1
T3
(6.7.9)
Jyoyo
ly +z2
[1+x-x.(n)1I+Q2yO2X1
Rl
J
Rs
)
246
THE LIFTING LINE THEORY
and, integrating by parts, we obtain u2. In fact, it results:
x
w(x' y, z)
('l) d
4a fb C(r1)
R3
+b
-4
f
b+6
q+
4n
bb C(17)
-
--d n-
r
(+I)] d>1.
y0+2 I1 + x
r
b
l`
(6.7.10)
6.7.2
The Integral Equation
We start from the lifting surface equation having the form (5.1.28).
Utilizing (6.1.6) and (6.1.12), we deduce:
Y=dtdn
Eb
(q)
0
(6.7.11)
dr=
C
1
=;F
+b C'(rl)dq.
1
4'rJ-b 11-y
0
b
Taking (6.2.1) into account, we obtain:
12 = -
lim
1
47r
11D
f(
x0 d l;d
R
To
=
(6.7.12)
C(n) x-x.(rl)drl,
1
R°
where
R; =
[x-x.(n)12+(.32y .
(6.7.13)
Utilizing the formula (D.3.7) and taking into account that C(±b)
= 0, we deduce
Iz
1
'r+6
41r ,l -b n
8r
1
u ON {C(rl)
x- x. (n)1
Ro.
Jd_
(6.7.14)
1
'' +6 C11) -x-
4ir.!_b n-y
x.
Ro
d'l+la
247
THE CURVED LIFTING LINE
whore
1
13 = -
Co 8
-
n v 8n
_-
d *!
R°
(6.7.15)
C(n)xx
x(n
4(n)dn
4sr
From (5.1.28), (6.7.11), (6.7.12) and (6.7.14) we obtain the equation:
t bb
K(x, y, n)d n +
4A ,!_b
n (ny
air
C('i)L(x,
y,,7)d n = hs(x, v)
(6.7.16)
where
K(z,y,n)=1+
(6.7.17)
x - x. (n) - ray'. (n)
L(x, y,
(R.)
are non-singular kernels. The equation (6.7.10) was obtained in another
way by Prosadorf and Tordella (6.22]. It is a singular Integro-differential
equation.
For the straight line (x.(q) : 0) one deduces
Ci(r!)
Ib n
+b
47r
j' C'(n) x
- yd n + Oar J
iJ
n - y Rd n - 4
x
C(q)
n = h:
(6.7.18)
x+
. This equation is a first approximation of
the lifting line equation. Fbr deducing the equation (6.1.16) we had
A < < yp on the greatest part of the domain D, such that we might
where R
consider Re = i4jyo(. Here we cannot perform this approximation.
6.7.3 The Numerical Method
Using (D.3.7), one denaozutrates the identity
J-bb (n -
y2K(x, y,
n)dn = Jn vK(x, wn)dn+
(6.7.19)
C(tl) 8
+ 14 n-y b K(x,y,y)dn,
248
THE LIFTING LINE THEORY
such that (6.7.16) becomes:
C(rl) K(x,
y, ri)d rl - 1
7
T-b (11-y)'"
} C(q) 8 K(x, b, y)d q+
J b n-y q
(6.7.20)
This is an integral equation (not an integro-differential one) but with a
strong singularity, for which the Finite Part is considered.
We denote
AI (x, y, y) =
8
5; K(x, y, y)
(6.7.21)
and we perform the substitution y = by', v = br1'. The equation (6.2.14)
becomes:
'(
A
l
C(eta
2 K (x, y, r1) d 1l -
I
b
iJ
1 C(+1)
+
n-y
M(x, y, y) d r1+
(6.7.22)
+ b2 r+1 C(i)
7r
_
L(2:, y, +1)d 17 = 4bh'. (x, y) ,
I
where
K(x , y , rl) =
L(x, y, n)
xx.(rl)
1+
1(x - x,)2 + b2R2yo11/2
x-x,('7)-byox:
_ -F 2 1(x - x.)2 + 62Q2yo13/2
(6 . 7. 23)
M(x,y,y) = 6 W (T, y, Y) Utilizing the quadrature formulas method, we shall take into account
that the solution of the equation (6.7.22) has the form:
C(n) =
--q2 c(r1)
(6.7.24)
and we shall utilize the formulas (F.2.12), (F.3.4) and (F.3.5). Denoting
IM
=cosnk+l, k=in-,
(6.7.25)
one obtains from (6.2.16) the algebraic system
'AJ,ec* = 4bh2(x,g3), j = ln,
A,cj +
L= I
(6.7.26)
249
THE CURVED LIFTING LINE
where
AJ =
+1
2
K(x, n;, nj),
Ask= (1 - rlk)
1[1-(-1) +k
:i+1
(nK(k x, nj n,,)nk)
-
I
2+
(6.7.27)
+b [1- (-i)+k1 M(x,17;, nk) + b2ajkL(x, n;, nk)
J
rlk - ni
For writing explicitly this system we have to know the shape of the
wing. For example, for the flat plate having the shape of an arrow with
the angle of attack e we adopt the broken line model (fig. 6.7.3). We
have f = -E and utilizing the substitution y = by,
0<y<1
-y, -1 <y< 0,
Y,
K(x,y,n) =
L(x,y,n) =
K1=1+xRn, 0<n<l
K2
=1+
x+r7,
R
-1 < n < 0,
(6.7.28)
L,=-#2x-"-yo 0<n<l
L2=-#2X + + y0, -1<n<0,
R3
where
1/2
R= [(z_y)2+b22(y_q)2]
The system (6.7.26) may be solved numerically using a computer.
Chapter 7
The Application of the Boundary Integral
Equations Method to the Theory of the
Three-Dimensional Airfoil in Subsonic Flow
7.1
7.1.1
The First Indirect Method (Sources Distributions)
The General Equations
The superiority of this method in comparison with the classical meth-
ods has been exposed in Chapter 4. Using this method we succeed to
impose the non-linear boundary condition just on the boundary of the
wing. Moreover, it allows to solve numerically the integral equation of
the problem, approximating the boundary by a polygonal line in the
two-dimensional case, or by a polyhedral surface (consisting of panels)
in the three-dimensional case.
We deal with the problem considered everywhere in this book. A
subsonic stream, having the velocity U,,.i, the pressure poo and the
density per. is perturbed by the presence of a fixed body, having a
known surface E. One requires to determine the perturbed flow and
the action of the fluid against the body. Introducing the dimensionless
variables X, Y, Z related by the dimensional variables xl, yl, zl as
follows
(XI, y1, zt) = Lo(X, Y, Z)
and putting
V1=Uoc(i+V), P1=p,, +p,oU,2,.P
(7.1.1)
we obtain the system for the perturbed fields:
R12OP18X + Div V = 0, OV/OX + Grad P = 0.
(7.1.2)
Projecting the last equation on the OX axis, we deduce:
P=-U
(7.1.3)
252
HIEM. THREE-DIMENSIONAL AIRFOIL
Taking into account this result, the first equation from (7.1.2) and the
projections of the second on OY and OZ, give
fl29U/aX + OV/aY + 8W/0Z = 0
(7.1.4)
av/ax - are/ay = o, aw/ax - av/az = 0,
U, V, W representing the coordinates of the vector V. Denoting by
F(X, Y, Z) = 0 the equation of the boundary E, we have to impose the
condition
(1+U)Nx+VNy+WNZ=O, F=0,
where
N_
(7.1.5)
Grad F
(Grad F1
(7.1.6)
With the change of variables
x=X, y=AY,z=QZ
u= j3U,v=V, w=W.
(7.1.7)
the system (7.1.4) becomes
au/ax +8v/ay + aw/az = 0
(7.1.8)
au/ax + 8u/ay = 0, raw/ax - su/az = 0.
(7.1.9)
Performing also a change of variables in F we have
aF/8X = OF/0x, (9F/8Y = (30F/8y 8F/8Z =130F/az ,
such that the boundary equation (7.1.5) becomes
un=+ f32(vny+wns) = -13b?,F = 0,
(7.1.10)
where
grad F
(7.1.11)
Igrad F1
We agree to utilize the inward pointing normal to the body. We also
impose the damping conditions at infinity
lim(u, v, w) = 0.
00
(7.1.12)
The first equation from (7.1.9) represents a necessary and sufficient
condition for the existence of a function io(x, y, z) such that
u=
v = acp/ay .
253
THE FIRST INDIRECT METHOD (SOURCES DISTRIBUTIONS)
From the second equation and from the damping condition it results
w=8o/8z.
With this representation, the equation (7.1.8) gives
App= 0.
It is well known that the fundamental solution of this equation,
tp(x)_-47r 1,
r r=Ix - tI,
(7.1.13)
represents the potential of the flow determined by a source of intensity
f, having the position vector . The velocity field is
v = grad p =
7.1.2
4 xr
(7.1.14)
The Integral Equation
Replacing the body with a continuous distribution of sources on E,
having the unknown intensity f (x), the velocity field in fluid will be
U( ) _
-4; Iff(x) Ix -
3da,
(7.1.15)
t representing the position vector of the generic point M in the fluid.
7.1.3
The Integral Equation
In order to impose the boundary condition (7.1.10) we have to pass to
the limit in (7.1.15) considering that M(4) tends to the generic point
Qo(xo) E E. To the limit, the integral from (7.1.15) becomes singular.
Following the procedure from the two - dimensional case (see 4.2.1), we
shall prove that if f (x) satisfies Holder's condition on E, then
v(xc)
4o
- \ 4r
jf f
(x) I x - 41 3 d a1
=-2f(xo)r+o-,- 11 (X)Ix-xolda,
(7.1.16)
254
BIEM. THREE-DIMENSIONAL AIRFOIL
where
(7.1.17)
6-+O1 JE-O
J JE
a representing the surface cut from E by a sphere E, having the
center in Qo and the radius E.
Indeed, writing
IL =
J
(7.1.18)
f,0+0
we have to calculate
L= lim
JJf(z)x x3 da,
i.e. the last term from (7.1.18). Writing this term as follows
L= lim
t-.z0 J
where
Lo
d a + f (zo)Lo
J (f (z) - f (xo)l x
li n f f f
IX
,
(z)1013da,
(7.1.19)
we notice that the first integral from the expression of L tends to zero
when a - 0 because f satisfies Holder's condition. For calculating
Lo we shall replace o by A, the projection of the surface a on the
plane 11 which is tangent to E in Qo (fig. 7.1.1).
Fig. 7.1.1.
255
THE FIRST INDIRECT METHOD (SOURCES DISTRIBUTIONS)
On this projection we shall use the parametrization
x = xo + r(cos 9io + sin 9 jo), 0 < r < e, 0 < 0 < 27r,
(7.1.20)
io and jo being versors orthogonal to the plane n.
Also, taking into account that the limit is the same on every path
on which 4 --+ xo, we consider the limit on the direction of the inward
normal no = n(xo) to E in Qo. We have therefore
f =xo-qno, q>0.
(7.1.21)
Hence
r 12" r(cos 9io + sin 9jo) + rlno
rd rd 8 = 2zrn
(r2 + 712)3/2
'i-.0 0
L0 = lim
.
(7.1.22)
Now the formula (7.1.16) is demonstrated.
Imposing the condition (7.1.10), we deduce the following integral
equation
{n2(xo) + li2En2(xo) + n2(xo)]}f (xo)+
+T7r
A Zf
ff
(x) (
x-xo)no+fit[(y-yo)noo+(z-zo)noj
da=
2
Ix - xo13
.
(7.1.23}
For the incompressible fluid it becomes
Lf = f (-To) +
27r
A
f (x) (
Ix
no
xxo13 d a = 2n0, .
(7.1.24)
The integrals (7.1.23) and (7.1.24) are singular.
7.1.4
The Discretization of the Integral Equation
We shall approximate the surface E of the body by a set of triangular
panels TJ (j = 1, ..., N) and we shall approximate on every panel Ti ,
the function f by the value f; of the function in the center of mass
G2 (x°) of the triangle. The equation (7.1.23) reduces to
{(no)2 + f 2[('nV)2
1
//r
+ (n°)2]}f(xo)+
(x - xo)n? +,02[(y - yo)ny + (z - zo)n?}
Ix-xo13
0
da-21nx.
256
BIEM. THREE-DIMENSIONAL AIRFOIL
Imposing this equation to be satisfied in the centers of mass GG, i.e.
putting xa = x°(i = 1, ... , N), we obtain the linear algebraic system
N
a,f,+EA,jfj=b;, i=1,...,N,
(7.1.25)
i-1
where we have no summation with respect to i. We denoted
a = n2 (mg) + p2n?(x°) + #2n2 tx°), bi = 2 n (x°)
A=i
= X,in.(x9) + O2Yjnv(x°) + Q2Z,jn:(x1
(7.1.26)
(7.1.27)
0
X`, = 2.-r, 1 M
W1
13da.
(7.1.28)
For determining the quantities n(x°) and X,j we shall denote by mil,
xt2 and x,3 the vectors of position of the vertices of the triangle T{ ;
we choose the sense on the sides of the triangle such that the normal
n(x9) is positively oriented towards the interior of the body. Taking
into account the definition of the vector product, we obviously have
n(xo) =
(x12 - Wit) X (X,3 - Oil)
2S;
'
(7.1.29)
Si being the area of the triangle Ti expressed by means of the coordinates of the vectors x,l, xi2, x13.
The integrals X;j are singular when i = j. We consider at first the
case i 0 j. Denoting by xjl, xj2i xj3 the vectors of position of the
vertices of the triangle T1 j, we shall consider the parametrization of the
triangle
x=xjl+(xj2-xj1)Ai+(xj3-xjl)A2.
(7.1.30)
F\xrther we shall proceed like in 5.3. Introducing the polar coordinates
by means of the formulas
Al = rcos0,
A2 = rsin8 0 < 0 < r/2, 0 < r < p,
(7.1.31)
where p is defined (fig. 7.1.2) by the relation
(cos 0 + sin 0)p = 1
(7.1.32)
and denoting
e(O) = (xj2 - xj1) cos 0 + (xj3 -
0,
(7.1.33)
257
THE FIRST INDIRECT METHOD (SOURCES DISTRIBUTIONS)
Fig. 7.1.2.
we deduce
x-x9=xj1-x°+re(O)
(7.1.34)
I0 -x,012=ar2+br+c,
where
a = lel2, b = 2(xjl - x°) e, c = jxj1
- x912
(7.1.35)
6 = b1- 4ac = -4(xj; - x9)e
- (vjl
<0.
For the element of area of the triangle Tj one obtains
(7.1.36)
d a = 2Sjd Al d A2 = 2rSjd rd 8,
S; being the area of the triangle. Hence,
X;; = S. f"21(xii - x)I1(e) + e(e)I2(8)]de
where
_
rdr
P
Jo (ar2 + br + c)3/2
=
2bp + 4c
6(ap2 + by + c)1 2
_
(7.1.37)
4f
6
(7.1.38)
P
alt (O) - f (art + br + C)3/2
with the notations
J - IP
dr
are+br+c
-
2
%r-7
J1 - b11 - C/C ,
( arctan Zap + b
vr-7
17)
258
13IEM. THREE-DIMENSIONAL AIRFOIL
K
_
dr
)r°
J0 (art + (n + c)3/2
4ap + 2b
2b
6(ap2 + by + c)1/2 + W c
Ones employ the formulas (7.1.29) and (7.1.37) for calculating the
coefficients A1.
7.1.5
The Singular Integrals
The integrals (7.1.28) are singular when i = j. Following the model
from 5.3 we shall write:
T,j = T(12)
+ T(23) + 7 31)
(7.1.39)
where T!kl) is the triangle GjPkP, and we shall consider the parametriza-
tion of the triangle T,(12) putting
x - x° = (x,1 - x° )J11 + (xj2 - x°)a2 .
(7.1.40)
Passing to polar coordinates and denoting
E12=(xj1-x°)cos6+(xj2-x°)sin8,
(7.1.41)
we deduce
x - x,, = rE12,
Ix - x°I = rIE121
(7.1.42)
Utilizing also (D.2.3) it results
X(12)
»
=
0
1
2n
T.02) I x
X0 13
da=
(7.1.43)
=
1
Ir ' J JO
3S
X?j =
s/2 E12 (0)1np(8)
UU
(E121'
(7.1.44)
+
These expressions are utilized for determining the coefficients A.,,..
X123)
7.1.6 The Velocity Field. The Validation of the Method
The numerical values of the velocity field are obtained from (7.1.16)
with the formula
N
2v(x°) _ - f
X y f3
j=1
.
(7.1.45)
THE FIRST INDIRECT METHOD (SOURCES DISTRIBUTIONS)
259
For testing the method we shall use the exact solution in the case of
the sphere placed in an uniform incompressible stream. We know (see,
for exaunple, [I.11 J, p.163) that if the sphere has the radius a and the
center in the origin of the coordinate axes, and the uniform stream has
the velocity Uk , then the potential of the perturbed flow is
a3)
(7.1.46)
and\\\V1
Calculating V1 = grado
= U,,(i+ v) which results from
(7.1.1) under the hypothesis that the fluid is incompressible (A = 1)
and using for tv the values (7.1.45), it results the comparison from
figure 7.1.3 for IVI I /U,,. We notice that the approximate method and
1.
V
1.0
Exact
...._. I talho d
+
a
ao
Fig. 7.1.3.
the exact one give very closed results.
In the paper [7.2], that we have utilized for writing this subsection,
we may find the approximate results for the ellipsoid for various angles
of attack and for various Mach numbers. We may also find approximate
results for the wings whose cross section is a NACA - 64 - A - 008 profile.
7.1.7
The Incompressible Fluid. An Exact Solution
We have seen that, for the incompressible fluid the integral equation
is (7.1.24). In this subsection we shall determine the exact solution of
DIEM. THREE-DIMENSIONAL AIRFOIL
260
the equation in case that the perturbing body is a sphere (7.4).
INg. 7.1.4.
Considering that the points on the sphere (x and Qo(xo) have
the coordinates (fig. 7.1.4):
X = R(sin q, cos 92, sin ql sin 42, cos ql )
0<gl<zr, 0<qq<27r,
(7.1.47)
xo = R(sin qi cos 92, sin qi sin gZ, cos qi)
we deduce for no (the inward pointing normal to the sphere in Qo)
no = -(sin g° cos q2, sin q, sin qz, coe g°) .
(7.1.48)
We have therefore:
f (x) - f (ql, q2),
f (xo) - f (q°, q2)
(7.1.49)
da = R2 sin gldgidq2,
no -- no = - cos qo
and
Ix - xo12 = 1X12 + Ixo12 -- 2x0 xo = 21122(1 - cos9),
(7.1.50)
cos 6 = sin q1 sin q, - cos(Q2 - q2) + cos ql cos go.
(7.1.51)
where
Hence it results
(x - xo) (no) - (x-xo)
(x - xo) no
Ix-x013
da=
= R(1-cosB),
(1 - cos 8) sin gldgl dq2
-sing1dgldq2
2(1-cos0) 2(1-cos9)
2 1-(oO)
(7.1.52)
THE FIRST INDIRECT METHOD (SOURCES DISTRIBUTIONS)
261
With these formulas the equation (7.1.24) becomes
2w
I
f(gi,gz)+47
rJo Jo
f(gi,g2)sin2(igld)q2
=-2cpsq0j.
(7.1.53)
the sign "' indicating that one eliminates the vicinity of the point Qo,
i.e. 0 = 0 according to the definition (7.1.17).
Denoting
1
K(qi, qq, q1, 42) _
4ir
2(1 - cos 9) '
(7.1.54)
D = (0, 7r) x (0, 2r),
we deduce that the integral operator of the equation which determines
the density f (q,, 92), is a bounded operator with respect to the uniform
convergence norm for real functions, continuous on the closed rectangle
b. This boundedness follows from the property
KsingldQ1dg2 = 1.
(7.1.55)
We sludl also prove that the integral operator has the invariance property
J
j(cosqi)Ksintdqidq2 = cosq°.
(7.1.56)
3 the change of variables
For proving these properties we shall perform
(qj, q2) -- (0, A) defined by (7.1.51) and by the following relation, which
is a direct consequence of the sine rule from the spheric trigonometry
sin A sin 8 = sin q1 sin(g2 - q2),
(e, A) E D .
(7.1.57)
Hence we choose the spherical coordinates relative to the point x for
which, instead of the Oz axis we take the direction -no, and instead
of the xOz plane we take the plane of the versors k and nO (fig.
7.1.5). In this way, for the element of area in the generic point Q one
obtains
d a = R2 sin Q1dQ1dQ2 = R2 sin 6dOdA.
(7.1.58)
Using the cosine theorem from the spherical trigonometry, we obtain
cos q1 = cos q° cos 0 + sin q0 sin 9 cos A.
Hence,
r
1 ID
KsingtdQldg2 = lID Ksin8d9dA
(7.1.59)
BIEti1. THREE-DLMMENSIONAL AIRFOIL
262
Fig. 7.1.5.
(2" d A /"`
Jo
o
=I
sin 8d 9
2(1 - cos9)
J f(cosqi)Ksinqidqidq2 = 1
(cos
Jo
n
cos) d A) / (sin 8)K sin dd 9 =
0
/R
2-(
J(cosqi)KsinodldA =
J ID cos 6K sin Bd9d.1+
r2x
+(sin q°) (
-_ 1 '
sin8cbs9
de_
1
qo
qo) 0
3
(1 - CM e)
Utilizing the identity (7.1.56), we may solve the equation (7.1.53) by
means of the successive approximations method. Indeed, putting
f1 _-2cosg0,,
we deduce
f2 = -
f
fnf1(41,92)K(4i,42,91,92)sing1dg1d42
=
_ -3(-2cosgi)
fs =
-If
U
(,)2
(2(41,42)h'sin41d41d92 =
2cosgi)
263
THE FIRST INDIRECT METHOD (SOURCES DISTRIBUTIONS)
I
fk+I = `J
f
Jofk(q ,gs)Ksingtdqidqs =
(3
\k
J
(-2cos401)
whence
00
f(gl,g2) =Ffk+1(gi,92) _
k=o
00
= (-2oosgt)
(7.1.60)
(1)k
=
-i cogt.
k=O
This is the exact solution for the spherical obstacle.
7.1.8
The Expression of the Potential
For testing the integral equation, we must prove that the potential
calculated with the density (7.1.60) coincides with the exact potential
given by (7.1.46). The potential in the generic point M(F) determined
by a source having the intensity f placed in Q(x) is according to
(7.1.13),
WW _ 4r Ix
i
EI '
whence it results that the potential determined by a continuous distribution, of intensity f (c), on the surface E, will have the expression
w(f) = --I Jf IE IM-41 da.
(x)
(7.1.61)
In the case of the sphere E = S(O, R), with the density (7.1.60), the
potential is
Arr) =
3
87r
cos
fJ Ix - qlEI d a .
(7.1.62)
Considering the point M(L) exterior to the sphere, having the coordinates
f = r(sin gicosgs,sinq°singscoo q),r > R,
we get
Ix - Cl' = R2 - 2rReoe6 + r2
(7.1.63)
BIEM. THREE-DIMENSIONAL AIRFOIL
264
where cos O is defined by (7.1.51). Taking (7.1.58) into account, it
results
3
8n
3
R2
cosglsing1dqldQ2
- '?r R cos 8+ r
ID
f
c
2
8?
R-
l Jj,
qi sin Od Od A
R - 2rR cos 0 + r-
On the basis of the formula (7.1.59) and the periodicity of cos A, we
obtain
3
V
3
2m
R2
_ $R
cosrqo1
ff
d A fo
8n(coc; gi)
cosOsinOd0
J
r
=
[(R2 + r2)(R2 - 2rRcvsO + 1.2)1/2_
o
-(R2 - 2rRcosO + r2)1/2 I sinOd0 =
3
16
lr- RI)- -(r+R)3- Ir-R131
and finally, since r > R,
2 T6
This is also the perturbation potential given by (7.1.46).
(7.1.64)
THE SECOND INDIRECT' METHOD (DOUBLET DISTRIBUTIONS)
7.2
265
The Second Indirect Method (Doublet Distributions).
The Incompressible Fluid
7.2.1
The Integral Equation
The theory in this subsection follow the paper [1.11]. We denote by
po(x) the potential of a known flow in the entire space. We assume
that this flow is perturbed by the presence of a body whose support
is the simple connected and bounded domain D. We assume that the
boundary of this domain, E is smooth, such that we may apply Poisson's
formula. We denote by n the outward pointing normal to E. The
potential po is a harmonic function, such that we may write (1.11)
(D.3.2)
J JE [fi(x) n !x 1 F{
1doi
2(pe(4),
IX-41 tin
4ED
to
(7.2.1)
EE
EE,
E representing the exterior of D. When we write F E D we understand
that the point Al whose vector of position is t belongs to D.
Now we assume that the perturbation is produced by the body having the support D and let So(x) be the potential of the perturbation in
E. According to the equation of continuity (7.1.8), cp will be a harmonic
function which has to satisfy to infinity the damping condition
line
S'(F} = 0.
141-Under these conditions, V will have the representation (D.3.10) from
[1.11)
2T
J JE
- (x
x7-
x'
tEE
1
1
t)
0
T+p(x)1
d a=
So(t),
0
the normal being the same like in (7.2.1).
F= xo E
LED,
(7.2.2)
266
BIEM. THREE - DIMENSIONAL AIRFOIL
Adding (7.2.1)3 to (7.2.2)1 we obtain
W(E) =
f
4a J
{ [,Po(x) + v(x)]
1 41
Ix
(7.2.3)
(s o+9)(x)}da, t E E.
1
I
Determining V in order to have
A
A
(7.2.4)
W E E,
co(x) =
the representation (7.2.3) becomes
00 =
f
[PO(x) + W(x)]
Ix
1f
du.
(7.2.5)
Subtracting (7.2.1)2 from (7.2.2)2 one obtains
,P(X0) - Po(xo) =
.1
g (Ix 1
f [,Po(x) + P(x)J n
Introducing the function
j u:
x01
)
E E.
d a, xo
(7.2.6)
E -+ R by means of the formula
p(x) = fi(x) +
(V)x E E,
(7.2.7)
the equation (7.2.6) reduces to
.U(X0) _
2zr J
Jl U(x)
,
\ Ix 1 xol d a =
JPo(xo), (V)--o E E,
(7.2.8)
and (7.2.5) to
da.
(7.2.9)
WW = 4r ,/
/
The equation (7.2.8) is the integral equation which determines the
function µ, and the equation (7.2.9) shows that this function is just
h (); (Ix 1
the doublets density E which replaces the body. We may write the
equation (7.2.8) as follows
lt(xo) +
2a J Js
µ(x)n(x) IxX
x013 d
a = 2cp(xo), (d)xo E E,
(7.2.10)
n representing the normal pointing towards the fluid (the outward
normal to E). This equation which determines the doublets density, is
similar to the equation (7.2.21) which determines the sources density.
Only the free terms and the normals from the kernels are different. In
(7.1.24) it appears n(xo) while in (7.2.10) this is replaced by n(x).
THE SECOND INDIRECT METHOD (DOUBLET DISTRIBUTIONS)
267
The Flow past the Sphere. The Exact Solution
7.2.2
Considering the problem from (7.1.7), i.e. the integral equation in
the case of the uniform flow with the velocity U,,,k past the sphere
S(O, R) and utilizing the formulas (7.1.47) we have:
oo= -- =Rcosq°
(7.2.11)
such that the equation (7.2.10) becomes
ir
i
II ( g , q0 )
±
47r
2w
j1
0
II (gl, 92, ) s1
-
-la- gc
V 2(l
d. B )
= 2R cos q a .
(7 . 2 . 12)
This equation differs from (7.1.53) only by the right hand side term.
The equation (7.2.12) has therefore the exact solution
it = 3 R cos q1
(7.2.13)
.
For testing the integral equation (7.2.10), we shall prove that the
potential ip, obtained from (7.2.9) with the density (7.2.13), coincides
with the exact perturbation potential (7.1.64). Indeed, (7.2.9) becomes
- 4-r1 I Jz
u(x) '
4 d a.
ix 413
Using the notations (7.1.47) and (7.1.63) we have
cp(l;) =
(7.2.14)
(x - la) n = R - r cos 9
x F = Rr cos 6,
da = R2 sin glclgldq2,
such that, utilizing (7.2.13), we deduce
1 3k 1P
4ir 2
0 2n'
(R-rcos0)cosgjsingtd
(R2 - 2Rr cos 9 + r2)3/2
J0
91 d q2.
Passing to the variables 0 and A and taking the formulas (7.1.58),
(7.1.59) and the periodicity of the function cos A into account, it results
3
'P = - 4 R 3cos ql0
fW sin O cos 0(R - r cos 9)
(R2
J
- 2Rr cos 0 + r2)3/2
Performing the change of variable 0 -+ u :
it = (R2 - 2Rr cos 0 + r2)1/2
we deduce
f
x sin0cos0(R -
Jo (R2 - 2Rr cos 0 + r2)3/2 d
and finally, (7.1.64).
6
2
- - 3r2
d0,
268
I3IEM. THREE - DIMENSIONAL AIRFOIL
7.2.3
The Velocity Field
Since the potential has the expression (7.2.14) we deduce:
v(t) = grade = - 4 Jj;(x)gracI£
47r
n(x)] d a =
L lx - 41
l I µ(x) (n grad) 1 Ix -
X12
Ida
and finally
vW =
4zr !
j1`(x) [n(x) - 3(x - ,)(IX -
I2nl
Ix - 413
(7.2.15)
This formula gives the velocity field in the fluid. We deduce that far away
the kernel has the order I4I-3, as it is natural to be (see for example
(6.1.16) from 11.11]). On the boundary of the body, the integrals from
(7.2.15) have strong singularities. We have therefore to transform these
integrals.
7.2.4
The Velocity Field on the Body. N. Marcov's Formula
In the beginning we have to notice that the double layer potential of
constant density equal to 1 defines a piecewise constant function whose
gradient is obviously zero in the continuity points. Indeed, from (7.2.1)
it results:
1
1,
fr
x 1 t)
TT(1)da=
1/2,
0
4ED
E
(7.2.16)
EE,
and for the gradient we have
1
4n
ff n(x) - 3(x -)
da
Ix - tp2 ] Ix - CI3
(7.2.17)
in E (and D). Multiplying this identity by the constant jt(xo) and
subtracting it from (7.2.15), it results
J
P( X)
!t(xo)
Ix-EI
- n
n-3(:c-F) (xIx-412
da
Ix-EI2
(7.2.18)
TILE SECOND INDIRECT METHOD (DOUBLET DISTRIBUTIONS)
269
for AI(D) in E.
For calculating the velocity when AI(F)
Qo(xo), a point on the
boundary E, we shall denote again by or the portion cut from E by
the sphere with the center in Qo and the radius s and we shall use
the definition of Cauchy's principal value
lim
JJa-
JJE
(7.2.19)
if it exists. With no the outward normal in Qo, we put t = xo +qno
(q > 0). For the limit value of the velocity in this point, we have:
=bin
tje(xo) =
1-0 v(xo + qno) = It + 12,
(7.2.20)
for every F > 0. We denoted
11 = lira
1 if
ti--o 47r
-3(x I2
no4r
12(x) -11(xa)
_o
x - 41
da
j
Ix - 412
Ix _ X12 '
(7.2.21)
11 !l( Ix -
4Ixa) [n-
-3(x-4) (x - t) - n
da
Ix-XI2,
Ix-£12
where
= xo + q no. The first limit and the integral may be interchanged because the integrand has no singularities on E - a. Hence
one obtains
1t
=
/e(x) _ 14(x0)
1
47r
E-o
Ix - xol
[n
(7.2.22)
(x - xo) n
Ix-xo12
da
ix-x012
For every e > 0 this integral exists if µ(x) satisfies Holder's
condition. So, we deduce
v(xo) = Jim (1) + 12) = w(xo) + link 12 ,
(7.2.23)
270
RIEM. THREE - DIMENSIONAL AIRFOIL
where
w(xo) =
1
Jf_
/L(x) - JL(xo)
Ix-xol
-3(x - xo)
L
(7.2.24)
J
Ix - xo12
da
IX - X012
For calculating the last limit from (7.2.23), when s is small enough,
we may replace a with its projection A on the tangent plane in Qo
and we shall utilize (7.2.17). It results that we have
x - t = r(cos Oio + sin Ojo) - i'no,
Ix
- t12 = r2
+.q2
,
(7.2.25)
µ(x) - /&(xo) = (V /)(x0) ' (x - xo) + ... _
= rvp(xo) (CDs NO +Sill 8jo,
where V1z(xo) = (V/c)(xo), according to the usual notation in Analysis.
Considering the scalar product
Vy - (cos Bin + sin Ojo) ,
in the basis io,)o we obtain the identity
[V -(cos Oio + sin 8jo)](cos Oio + sin 8jo) =
= [(Viz io)cos8+ (V/z jo)sinO](cos8io + sinOjo) =
(7.2.26)
= (Vp - io)(cos2 Oio + cos8sin 6jo)+
+(Vp . jo)(sin Ocos 8io + sine` 0jo) .
Noticing that some terns vanish after integrating with respect to 0,
from 12 it remains
3
12
t
2-x
r- 0 47 r Jo JO
r3)?
(r2 + i,2 )5/2 X
(Qµ io) x (cost 8io + cos O sin 8jo)+
drdO,
x
+(QJJt jo) x (sill 8 cos 8io + sine 8jo)
I2 = lim 4
J'
(r2 r3'1
[(VJt - io)io + (ViA - ?o)7o)Id r .
't'ilt: l)IRECT ME IIOI). THE INCOMPRESSIBLE FLUID
271
One obtains (Vi)(xo) in the square bracket and we take it off from
the integral. So,
t
I., =
r3
17
(r2+ 2)b/2dr=
Op(io)
Hence. taking (7.2.23) into account, we obtain the velocity on E by
ineans of the formula
v(xo) = 2(V1)(xo) + w(xo),
(7.2.27)
w(xp) being given in (7.2.24). This formula was obtained by N.Markov
in a unpublished paper.
Since P(x) and Qo(zo) from (7.2.25) belong to the plane which is
tangent to r in Qo, we deduce that Vp(zo) is in the tangent plane.
Hence, the boundary condition
(i+v).n=O pe E
determined the following integral equation:
1
-1s(xo)
f Fi(x)
Ix - xol
4A E
da
=-n=, VxoEE
-3(x-xc) no Ix-moll
Ix-xo12
which is an alternative to (7.2.8).
7.3
7.3.1
The Direct Method. The Incompressible Fluid
The Integral Representation Formula
For writing this subsection we used the paper (7.3]. Since we study the
same problem like in the previous subsections, we shall utilize the repre-
sentation (4.6.1). the equations (4.6.2), the boundary condition (4.6.3)
and the condition to infinity (4.6.4). The difference is now that the prob-
lem is three - dimensional. In (4.6.3), C will be replaced by E, the
surface of the perturbing body B. For avoiding the singular integrals,
we shall replace the equations (4.6.2) by the equations
div(v-c)=O, rot(v-c)=O,
(7.3.1)
272
BIEM. THREE - DIMENSIONAL AIRFOIL
c being a vectorial constant.
The system
(7.3.2)
div v' = 6(x - xo) , rot v' = 0
for every point xo E D + E, D representing the domain occupied by
the fluid (the exterior of the body B), defines the fundamental solution
V* =
x - xO
41rIx-x013
1
(7.3.3)
From the equations (7.3.1) we deduce the identity
4,
[ f div (v - c) + g rot (v - c))d v = 0
(7.3.4)
for every two functions, or regular distributions, f and g. We denoted
by Do the exterior of B, bounded by a sphere S(O, R), R being great
enough, such that the body B is included into the interior of the sphere.
Utilizing the identity (4.6.9)1 and the identity
rot [g x (v-c)]
(7.3.5)
and applying Gauss's formula, from (7.3.4) we deduce
( v - c) (grad f - rot g)d v =
(7.3.6)
(v - c) (fn - (n x g)id a,
E
n being the outward pointing normal on Do.
Substituting successively
(f, g) -' (7 - v*, -9 x VI)
(7.3.7)
we find the projections on the axes of coordinates of the following identity:
J (v - c)div v'd v = LER {n (v - c)v' + [n x (v - c)] x v' }d a ,
Do
xoEDo+E
(7.3.8)
273
THE DIRECT METHOD. THE INCOMPRESSIBLE FLUID
proving in this way that it is correct.
Taking (7.3.2) into account, it results
V(x0) - c = f
{n (v - c)v* + [n x (v - c)j x v' }d a .
(7.3.9)
+ER
The integral on ER may be written as follows
v)v+ (n x v) x vjd a - hR
c)v' + (n x c) x v*] d a .
IS,,
T he first term vanishes when R
oo, because of the condition
(4.6.4). For calculating the second term we use the spherical coordinates:
R, 9, cp:
x = R sin O cos V
y = R sin O sin p
0<9<7r.
z=Rcos9,
We obtain
lim. JR [(n c)v' + (n x c) x v')d a =
C
47C
f
0
2A
f
A
sin 9d 9d W = c.
0
Now, the representation (7.3.9) becomes
v(xo) =
j{n
(v - c)v+ [n x (v - c)] x v}d a ,
xoED+E.
Setting c = v(xo) = vo we obtain the representation formula
V(x0) =
j{n. (v - vo)v* + [n x (v - vo)) x v' }d a
.
(7.3.10)
This is a regularized representation, valid both for xo in D and for
xo on E. Obviously when xo E D, the integrand has no singularity.
This property is also valid when xo E E because of the factor v - vo
which vanishes for x -- xo.
This representation formula is fundamental [7.1j.
Utilizing the boundary condition (4.6.3), the formula (7.3.10) becomes
vo=1
E
(v-vo)) x v*}da.
(7.3.11)
274
7.3.2
RIEM. THREE - DIMENSIONAL AIRFOIL
The Integral Equation
The vector
F- nxv
(7.3.12)
which intervenes in the surface circulation
C= J nxvda.
(7.3.13)
E
introduced by Pascal and utilized again by R.von Mises 11.271, [1.20],
has a great importance herein.
We shall deduce the equation for P. To this aim we write the
formula (7.3.11) as follows:
(7.3.14)
vo =
After the vectorial multiplication by no one obtains
Fo = J[_(nz + n - vo)(n° x v') + (n° v')F(7.3.15)
-(n° F)v - (n v')Fo + (vo . v')(n(' x n)ld a.
But, from the boundary condition n v = -it, and from (7.3.12) we
deduce v = -nxn - n x F such that
n - vo = -non ' no - (n, n°, Fo)
(v' vo)(n° x n) = -n°(v` n°)(n° x n) - (n°, Fo, v*)(n° x n).
(7.3.16)
One obtains therefore
Fo = J(._(ni - nn n°)(n° x v°) + (n, n°, Fo)(n° x
+(n° v')F - (n° - F)v' - (n v')Fo-n°(v' n°)(n° x n) - (n°, Fo, v)(n° x n)Jd a .
Utilizing the double vectorial product formula
(UxV)xW
(7.3.17)
275
THE DIRECT METHOD. THE INCOMPRESSIBLE FLUID
it results
(n, n°,
x v°) - (n°, Fo, v*)(n° x n) _
= no x ((n°, Fo, n)v' - (n°, Fo, v')nJ =
(7.3.18)
= no x [(n x v') x (n° x Fo)J = -(no ,n,v')(no x Fo).
Hence, the equation (7.3.10) becomes
F°+
J,
[(n° F)v' - (n° v')F + (n v')Fo] d a+
+ no x Fo) f (n° x n) v'd a = j {n(v" n°)(n x n°)+
+
(7.3.19)
this representing the integral equation of the problem. This is a regularized equation. The integrals are not singular. Although the denominators of the distribution v' vanish for x -+ x°i we have to take into
account that in these points the numerators vanish too. Indeed,
no -.0,
7.3.3
Kutta's Condition
In the sequel we shall study the flow around lifting bodies (with
smooth surfaces). In order to solve this problem, the classical theory,
beginning with the lifting line theory (Prandtl 1918), introduces a vortical surface behind the body. In this way the flow in no longer irrotational
in the exterior of the body, hence the above solution cannot be utilized.
For defining the fluid How with the aid of the solution given herein, we
shall consider a discrete distribution of vortices located in certain points
from the interior of the body [7.3]. In this way, the external flow remains
everywhere irrotational and it will be characterized by the solution given
herein. Nloreover, the circulation C defined by (7.3.13) does not vanish
on the trailing edge. It remains to discuss the location of the vortices.
The Kutta-Joukowskicondition concerning the continuity of the pres-
sure on the trailing edge will be imposed, like in the two-dimensional
BIEM. THREE - DIMENSIONAL AIRFOIL
276
case, writing that the velocity in plane cross-sections, perpendicular to
the trailing edge, is continuous when P, and P, - Pf (4.6.21). Considering in figure 7.3.1 the point P, and the local frame n, a and
Fig. 7.3.1.
(3 = n x s and knowing that V = V,s, we deduce
n x V = V,/3.
(7.3.20)
Hence, the vectorial form of (4.6.21) is:
(n x V)(P,)+(nx V)(P,)-o 0
(7.3.21)
P P; - PI.
Taking into account that V has the form (4.6.1) it results the condition
<F>+<n,> j- <ny>k=0,
(7.3.22)
<¢>=q(P,)+ b(P.).
(7.3.23)
with the notation
7.3.4
The Lifting Flow
One knows from the theory of the two - dimensional potential in
compressible flow that the lift is generated by a vortex of intensity rk
placed in a point from the interior of the body. For example, in the case
of the flow past the circular obstacle with the center in ;.o (see [1.11) p.
117), the potential
Ua, f(z - zo)e-io +
l
R2
_
Z-20
1l
et°
I
characterizes the non-lifting flow and the potential
r
27ri
In (z - zo)
277
THE DIRECT METHOD. THE INCOMPRESSIBLE FLUID
defines the lifting flow, the lift being -pU,:1'. According to this model,
we shall try to generate the lift by means of some vortices from the
interior of the body, the number of vortices being determined by the
number of pairs of panels adjacent to the trailing edge for which we
have to write Kutta's condition (7.3.22). To this aim we need at first
the velocity field w' generated by a vortex line of constant intensity
r. This will be determined by the system
div w' = 0,
rot w' = r6(x) ,
(7.3.24)
w' being a distribution (for this reason it was marked by *).
For
determining the solution we apply the Fourier transform, like in the
case of the system (7.3.2). Using the formulas (A.6.5) we deduce
axiv'=ir,
whence, utilizing (A.6.9),
w'=rx(-
w' =rxVF-'
2
and then (A.7.10) and (7.3.3)
w` = v' x r.
(7.3.25)
If the vortex is located in the interior of the body, in a point having
the vector of position xi, then
w(x) = v'(x - xi) x r,
(7.3.26)
and if there are L lines, since the equations (7.3.24) are linear, it results
L
w'(x) = E v'(x - xk) x rk .
(7.3.27)
k=i
Returning to our problem and considering, for the sake of simplicity,
a single line, we shall write the formula (4.6.1) as follows:
V=U_ (i+w'+v).
(7.3.28)
From div v = 0 and rot r = 0, taking (7.3.24) into account, it results
div v = 0, rot v = -ri(x - xh),
(7.3.29)
278
BIEMM. 'THREE . DIMENSIONAL AIRFOIL
and the equations (4.6.5) will be replaced by
div(v-c) =0 rot(v - c) = -ra(x-xl).
(7.3.30)
The identity (7.3.4) will be written as follows:
Joo
[fdiv(v-c)+g.rot(v-c)]ctv=
,xl E D°
,x1 D°.
0
(7.3.31)
Since the point having the vector of position xl is in the interior of the
body, we shall obtain a second equality identical to (7.3.4), such that all
the consequences follow like above. We have therefore (7.3.10).
From the boundary condition v - n = 0 we deduce
(7.3.32)
such that the formula (7.3.11) becomes
vo=
R
xv'}da
(7.3 .33)
xoED+E.
The relation between v and F is now
v=-(n,, +wn)n-nxF,
(7.3.34)
and the integral equation (7.3.19) becomes
Fo+
JN
[(n° - F)v* - (n° - v')F + (n v')Fo]da+
+ (n° x Fo)
x n) v'd a + wn(xo)
I [(n° v')(n° x n)-
-(n°.n)(n° x v`)]da+J wn(x)(n° x v')da =
E
= f, { n2(n° v*) (n x n°) + [n0. (n - n°) - n.,] (n° x
t
v')}da,
(7.3.35)
where, taking (7.3.27) into account, we have:
wn(xo) _ I no x v'(xo - xi)] r
(7.3.36)
w,,(x) = [n° x v(xo - xi) ] . r.
L
THE DIRECT METHOD. THE INCOMPRESSIBLE FLUID
279
In (7.3.35) the dependence of F and r is linear. In the case of L
vortices, the equation (7.3.35) remains unchanged and the expressions
(7.3.36) become
L
Wn(x0) _ F [no X v'(x0 - xk)] r
11
k=1
(7.3.37)
L
W4(z) _ E In X v'(xo
k=1
- xk)1
r.
In equation (7.3.34), F, r1i rL are unknown.
For specifying the conditions (7.3.21) we shall write the product n x
V as a function of the principal unknown F. From (7.3.28) and (7.3.34)
it results
nxV=Uoo(F+nzj-nyk).
(7.3.38)
The condition (7.3.21) will be written as follows
0
F + (nx
-ny
0
(P,) +
[F+ (n,
(P;) = 0,
(7.3.39)
-ny
for every pair of panels from the trailing edge. It will result therefore L
equations. These ones, together with the equation (7.3.35) discretized
below, will determine the solution F, F1,... , r,. The numerical results
are more accurate when the points P, and P; are closer from the
corresponding point Pf from the trailing edge (see [7.31, p. 364).
7.3.5
The Discretization of the Integral Equation
Like above, one approximates the surface of the body by N panels
(triangles and quadrilaterals) and we approximate on every panel III
the function F by the constant value F,,, that it has in the center
G, (x9) (the collocation method). If we impose to the equation just
obtained from (7.3.35) to be verified in all the centers G,(x°) and we
denote
X;;=J v'(x-x°)da
(7.3.40)
280
BIEM. THREE - DIMENSIONAL AIRFOIL
we deduce, for i = 1
N
1
Fi+ L.. [(ni Fj)Xji - (iii X ji)Fj + (nj Xji)Fi] +
j=1
N
L
+ (ni x Fi) J(ni x nj) Xji +
j
[(ni x x;k) . rk]'
k=1
1
[(ni x nj)(ni Xjj) - (ni x X ji)(ni ' nj)] +
E
j=1
[(nj x xjk) rk] (ni x xji) =
+
j x1 k=1
{
(7.3.41)
- ni(ni.
,j=1
X ji)(ni x nj) + {n(rsi.nj)_n}(ni x Xji) },
In this system, the singular terms Xii disappear, because the integrals from (7.3.35) are regularized. Taking into account the relations
(nj Fj)X3i - (ni ' Xji)Fj = ni x (Xji x Fj)
(ni nj)X ji - (ni Xji)nj = nj x (Xji x N),
we deduce that the system (7.3.41) may be written as follows
N
L
LA,,F,f+EBk{lrk=Ci, i= 1,...,N
j=1
(7.3.42)
k=1
where
AiiFi = Fi+FnEnj Xji+(ni x Fi)E(ni x nj) Xjie
j#i
j#i
AijFj = ni x E(Xji x Fj),
j#i
Bki?rk = [rk(na x Xik)j
E(ni Xji)(ni x nj)+
;oi
+ > {r.
j#i
[ - (ni x Xik)(ni nj) + nj X Xjk] }(ni x Xji),
1
THE DIRECT METHOD. THE INCOMPRESSIBLE FLUID
281
Ci= nixE[n (njxXji)xn1-n Xji!
i0i
the unknowns being F1,... , FN,
r1, .... r1. To this system of N
equations, we add the equations (7.3.39) written for every pair of panels
adjacent to the trailing edge. The number L of vortices equals the
number of pairs; in this way, the number of (N+L) equations obtained
from (7.3.42) and (7.3.39) equals the number of unknowns F1,... , FN,
rl,... , ri.
For calculating the coefficients Xji (7.3.40) one utilizes either the
quadrature formulas (for example, the Gauss-type formulas), or the analytical formulas given by Hess and Smith in [7.10]. Denoting by x,
=21, x23, (24) the vectors of position of the vertices of the panel lIl
which is a triangle (or a quadrilateral), with the definitions
Vill
- y)2 + ( k+1 - 4)2
(xik)2+(yiyk)2+ (Z, -k)2,
Tk=
mk,k+1 =
'.)I + (yll +
(k+1
dk,k+1 =
+1 -
, ek = (xi-2'k)2+(zi-k)2 , hk = (xi-
)2(Yi-yk)2 ,
k
the formulas of Hess and Smith are
Xji = (Xji,Yji,Zji),
+1 - Ylk
Xji =
dk,k+l
k=1
Yji = -
3(4)
-4+1
4s k=1
4 = -
-
` k In rk + rk+i - dk,k+1
dk'k+l
rk + rk+1 + dk,k+1
3(4)
1
41r
rk + rk+1 + dk,k+1
arctan
k=2
'mk,k+lek - hk
(zi
-arctan mk,k+1 + ek+1 '- hk+1
(zi - k)rk+1
k)rk
In fact, Hess and Smith deduced the formulas assuming that the
panels IIj are situated in planes parallel to xOy (4 = 0).
An example is presented in [7.3].
Chapter 8
The Supersonic Steady Flow
8.1
8.1.1
The Thin Airfoil of Infinite Span
The Analytical Solution
In this subsection we study the same problem like in 3.1. The only
difference is that the unperturbed flow is assumed to be supersonic
(M > 1). Hence the uniform flow, having the velocity Ui, the pressure p,. and the density p,, is perturbed by the presence of an
infinite cylindrical body. The xOy reference plane represents a cross
section and the leading edge is the origin of the axes of coordinates. The
length of the projection of the airfoil on the direction of the unperturbed
stream (which does not always coincide to the length of the chord) will
be taken as reference length (fig 8.1.1). Utilizing the coordinates defined
Fig. 8.1.1.
by (2.1.1), we shall denote the equations of the profile determined by
the xOy plane
y = hf(x),
(8.1.1)
284
THE SUPERSONIC STEADY FLOW
the functions h..(x) being defined on the interval
(0, 1J
and and
possessing first order derivatives.
Taking into account the formulas (2.1.3), we deduce that the perturbation (which is obviously steady), will be defined by the system (2.1.32)
and by the boundary conditions (2.1.33). From this system, taking into
account that the perturbation is plane, it results:
u = -p, v= + py = 0, M2pz + ut + vy = 0 ,
(8.1.2)
whence one obtains
k2vtt - vyy = 0,
(8.1.3)
with the usual notation k = M - 1. The boundary condition (2.1.32)
becomes
v(x,±0)=ht(x), 0<x<1,
(8.1.4)
and the damping conditions at infinity
lim (p, u, v) = 0.
(8.1.5)
The equation (8.1.3) is hyperbolic and the families of characteristics C+
and C_ are defined by the equations
x-k-y=c+, x+ky=c_.
(8.1.6)
Obviously, the first family consists of parallel half-lines which make the
angle µ with the Ox axis, the second consists of parallel half-lines
which make the angle -µ with the Ox axis, where
tan p
=1 (sins =
,
oos p =
(8.1.7)
For determining the characteristics passing by a certain point, we write
that the coordinates of the point verify the equations (8.1.15) and and
then we determine the constants c+ and c_. For example, the characteristics detaching from the leading edge have the equations x - Icy = 0
and x+ky = 0. We assume that the profile is in the interior of the angle
having the opening 2µ which is determined by these characteristics.
It is well known that the equation (8.1.3) has the general solution
v(x, y) = F(x - ky) + G(x + ky),
(8.1.8)
such that from (8.1.2) and (8.1.5) it results
kp(x,y) = F - G
(8.1.9)
ku(x, y) _ -F + G.
'I'III: TIIIN AIRFOIL OF INFINITE SPAN
285
'I'll(- function F is defined on the characteristics from the superior half-plane, and the function G on the characteristics from the
inferior half-plane. We have in view the characteristics detaching from
the segment [0, 1) which replaces the profile that represents the source
of perturbations. The solution from the strip I will be determined by
F(x - ky) which will be denoted by F+(x - ky) and the solution from
the strip II will be determined by C(x + ky) which will be denoted by
F_. (x + ky). Hence the solution has the shape
vf(x,y) = FF(xT ky),
kpf(x, y) = ±F±(x T ky),
(8.1.10)
ku±(x,y) = T Ff(xT ky),
the upper sign corresponding to the solution from the y > 0 half-plane
and the lower sign to the solution from the y < 0 half-plane.
In fact one may demonstrate that if upstream of AOA' the flow is
not perturbed, then the solution of the equation (8.1.3) has necessarily
the forum (8.1.10). Indeed, if the assumption is true, it means that OA
and OA' are lines of discontinuity where we have to impose the jump
relations (1.3.8), (1.3.9) and (1.3.10)
Op1V1nO = 0,
Opl'IVln +P1nO = 0,
(8.1.11)
with the notations (2.1.1) and (2.1.3). Taking into account the order of
magnit.ucle of the perturbations, by linearization it results
=0, Ovn,, +pnO =0.
(8.1.12)
On the line OA we have n = (sin p, - cos µ). Taking into account that
in front of OA and OA' the perturbation vanishes, from (8.1.12) we
obtain:
0=using-vcosa+psina
(8.1.13)
U=vsina - pcosa,
the relationships being imposed on the line x = ky. Replacing here u, v
and p given by (8.1.8) and (8.1.9) and taking (8.1.7) into account, from
the second condition we find C = 0. Since in the linear steady theory we
have p = M2p, it follows that the first condition is identically verified.
In the sauce way one demonstrates that in the zone II v = G(x + ky).
Hence the solution of the equation (8.1.3) has the form (8.1.10).
286
THE SUPERSONIC STEADY FLOW
Imposing the boundary conditions (8.1.4), it results
F±(x) = 14(x).
0 < x < 1.
(8.1.14)
These relations determine the functions F±(x) on the segment 10, 11.
In the exterior of this segment we take F±(x) = 0 because v and
p are continuous functions on the Ox axis in the exterior of the
segment [0,11 (it results from (8.1.12), setting n = (0, 1)), such that
F+ = F_, F+ = -F_. Hence the perturbation zones are I and II. For
c E [0, 1], on the half-line having the equation x - ky = c we deduce:
v(x, y) = h+(c),
and on the half-line x + ky = c
kp(r, y) = h+(c),
(8.1.15)
(8.1.16)
u(x, y) = h' (c), kp(x, y) = h' (c) .
Along a characteristic, the velocity and the pressure take the values that
they have in the point where the characteristic intersects the segment
[0,11.
In the sequel we shall prove that all these results may be obtained
directly with the fundamental solutions method, as we could expect,
taking into account that this is a global method.
8.1.2
The Fundamental Solutions Method
We know that. the perturbation produced in the uniform stream hav-
ing the velocity Ui, the pressure p,., and the density po, by a force
density (fl, f2) uniformly distributed along an axis which is parallel to
Oz and intersects the xOy plane in the point of coordinates (F, 0), is
given by the formulas (2.3.30) and (2.3.31). Replacing the profile from
figure 8.1.1 with such a density applied on the segment [0, 1) from the
Ox axis (i.e. on the strip whose cross sect ion is the segment [0, 1] from
the Ox axis), the perturbation will be given by the formulas
i
2kp(x, y) = in [ - fl(E) + kf ()sign y] b(xo - klyl)d
(8.1.17)
2v(x, y) =
J0
[
- fl ()sign y + k f
6(xo - klyl)d
Taking into account the property of the distribution
f
f (t)6(x
- )d
=
f (x),
0,
if:
ifx E (a, b)
ifx E C(a, b),
(8 . 1 . 18)
287
THE THIN AIRFOIL OF INFINITE SPAN
we deduce that if x - klyl = c E (0,1), then we have:
2kp(x, y) = -ft (c) + k f (c)sign y
(8.1.19)
2v(x, y) = - f, (c)sign y + k f (c) .
Conversely, if c E C(0,1), then p = 0, v = 0. The set of points for
which x - kly) = c determines the characteristic half-lines
x f ky = c
(8.1.20)
which detach from the point x = c belonging to the segment (0,1).
On these half-lines, the pressure and the velocity have constant values,
equal to their values in c.
8.1.3
The Aerodynamic Action
From the representation (8.1.10) we deduce that the jump of the
perturbation pressure on the profile
p(x, -0) - p(x, +0)
is given by the formula
kOP1 = -[F+(x) + F-(x)] = -[h+(x) + h'-(x)]
(8.1.21)
which results from (8.1.14). In this way, the lift and moment coefficients
CL
=
L
CM =
(1/2)po,U2Lo'
M
(l/2)poU2Lo'
(8.1.22)
are given by the formulas
CL = - 2 1'[W+ (x) + h'_ (x)Jd x
CA! = -
2
k
(8.1.23)
rx[h+(x)
t
+ h'(x)]d x
the moment M being calculated with respect to the origin of the axes
of coordinates.
We notice from the formulas (8.1.10) and (8.1.23) that the linear
theory cannot be applied in the vicinity of the value M = 1(k = 0).
288
THE SUPERSONIC STEADY FLOW
As a first example, we shall consider again the case of the flat plate
having the angle of attack e with E < y (fig. 8.1.2). Since the
equation of this profile is y = -Ex, we deduce h.4. = h' = -E whence
v = -E, ku = ±e,
kp = T- e,
(8.1.24)
the total velocity and pressure being
A
Pig. 8.1.2.
vl
U [CI± ) i - Ej]
P1 = poc
,
_T P00U2 k
.
(8.1.25)
For CL and cM we obtain
cL =
4e
,
T
2s
CM=T.
(8.1.26)
These formulas have been obtained for the first time by Ackeret in 1925.
The torsor of the aerodynamic forces reduces to a lifting force L, applied
in the middle of the chord of the profile, force which tends to rotate the
airfoil in the trigonometric sense.
The graphic representation of the function cL(M) (fig. 8.1.3) shows
that the linear theory is not valid in the vicinity of the value M =
1, because the lift cannot be infinitely great. One estimates that the
validity of this theory begins from approximately M = 1, 2 and finishes
to approximately M = 2, 5, because the lift cannot be extremely small.
For the zone M = 1(0,8 < M < 1,2) one elaborates the theory
of transonic flow (see Chapter IX), and for M > 2, 5, the theory of
hypersonic flow.
289
THE THIN AIRFOIL OF INFINITE SPAN
CO
O
Fig. 8.1.3.
8.1.4
The Graphical Method
This method relies on the following fundamental theorem: in every
point the perturbation velocity is perpendicular to Mach's line (the characteristic line) which is passing through that point. The demonstration
results at once if we take into account that the characteristic lines have
the versors i of coordinates (cos p, ± sin µ) and we utilize the representation (8.1.10) and the formulas (8.1.7). Indeed, we have
v-if=uoosµfvsinµ=
kF*
±Ftj=0.
(8.1.27)
Let us show for example how we can utilize this theorem for obtaining the graphical representation of the perturbation in the case of
the flat plate. We consider at first the region I from the 8.1.2. In an
arbitrary point M from OA we draw the vector MP = Ui and a
vector V1 parallel to the plate and having an arbitrary magnitude; the
perpendicular on OA which passes through the vertex P of the vector
Ui determines the magnitude of the total velocity V 1 = MQ; the
perturbation velocity Uv is given by the vector PQ. For determining
the perturbation velocity from the region II, in a certain point M'
from OA' ones draw the vectors M'P' = Ui and V'1, the last being
parallel to the plate and having the magnitude unprecised yet; drawing from P (the vertex of the vector Ui) the perpendicular line on
OA' we determine the vector V' = M'Q' and then the perturbation
Ua,v=P'Q'.
Beyond FB and FB' the flow becomes again parallel to Ox.
One knows the non-linear solution of the supersonic flow past a convex dihedron(the Prandtl-Meyer fan). See for example [1.21], [1.34). In
the framework of the non-linear theory, the velocity which is parallel
290
THE SUPERSONIC STEADY FLOW
to Ox before OA , becomes at last parallel to the plate, crossing the
Prandtl-Meyer fan. In the framework of the linear theory this fan is
reduced to the half-line OA.
8.1.5
The Theory of Polygonal Profiles
In the case of polygonal profiles, the solution may be easily determined starting from the solution for the flat plate. The solution, which
obviously is piecewise constant, may be analytical, graphical or mixed.
Ui
Fig. 8.1.4.
We shall consider at first the profile from figure 8.1.4 (E < p) with
and e1CO8E1 + e2COSE2 = 1. According to the
linear theory, the last relation will be replaced by el + t2 = 1. From the
formulas (8.1.14) it results
OP = el, PQ = e2
F+=-El, 0<T<t cosEl =el,
F+ = -E2,
f1COSE1^'e1 <x<1,
(8.1.28)
FI=E, 0<x<1.
We deduce,
k Up0 =
E+E1,
C + E2,
(8.1.29)
291
THE THIN AIRFOIL OF INFINITE SPAN
whence
CL =
I(e+E1e1 +E2e2), CM =
a
kE2(1- e,).
(8.1.30)
Further we shall determine the pressure in different zones starting
from the solution for the flat plate. Taking (2.1.3) and (8.1.24) into
account, we deduce in zone I that
P1 = Poo - POQU2
k
.
(8.1.31)
In zone II we take into account that the deviation with respect to the
direction of the uniform stream from zone I is 62 - E1. Using again the
formula (8.1.24) it results
P2=PI-PI
V
(8.1.32)
Using the formulas (2.1.3), we deduce P1V2 = p ,,,,U2 +... Taking into
account that the factor (E2 - E1) has the order of magnitude of the
perturbations, in the framework of the linear theory we have to retain
E
E
E
P2=P1-Po,U2 2k 1 =Poo_PooUT
In zone III we have:
p3 = poo +
po0U2
.
(8.1.33)
(8.1.34)
For the jump of the pressure we get the formulas (8.1.29). In fact the
formulas (8.1.33) and (8.1.34) coincide with the formulas given by the
analytic method.
We shall calculate now, in a different way, the resultant of the pressure on the wing. The perpendicular to the plate OP oriented towards
the body has the director cosines
nl =
coos (El +
2) , - sin (El + 2 )) _ (-E1i -1).
(8.1.35)
In the linear approximation, the perpendicular to the plate PQ has
the director cosines (-E2, -1), and the perpendicular to OQ pointing
towards the body (e, 1). Hence the resultant of the pressure is
R = (pit, n 1 + p2e2n2 + p3n3)Lo
(8.1.36)
We took into account that the real lengths of the plates are I, LO, 12L0, L0.
For the lift we obtain the formula
2
L = Ry - paoUk'0(e+f1e1 +E2e2).
(8.1.37)
292
THE SUPERSONIC STEADY FLOW
In the framework of the linear theory the drag
(8.1.38)
R. = (E - 1191 - e2E2)LoPoo
vanishes because, from the projection of the contour OPQ on the Oy
axis we deduce
E = 11E1 + 12E2
In fact this is a known result. The drag has the order of magnitude two
CD =
2
(92 + E1t + 6212)
k
(8.1.39)
In the sequel we shall consider the polygonal profile from figure 8.1.5,
symmetric with respect to the OR axis. We assume that all the E-s
are small. Obviously, the profile is considered to be in the interior of
Mach's angle with the vertex in the origin. We deduce like above that
in zone I the pressure is given by the formula
P1 = Poo +
p00U2
(8.1.40)
E,
in zone II by the formula
86
Fig. 8.1.5.
k E2
p - p00U2
,
(8.1.41)
P3=Pa-PzV2E3 kE2 =Poo-PooU2k ,
(8.1.42)
P2 = P1 - P1VzE1
k
in zone III by
293
THE THIN AIRFOIL OF INFINITE SPAN
in zone I',
P1
=Poo+p00U2!1
,
Ei = 2E+e1,
( 8.1.43)
in zone IF,
-P'1V1' k
Ps=Pi
=Poo+p,,U2k2,E'2=2E-E2
(8.1.44)
2kes"'p.-.U2k3,E's=ES-2E
(8.1.45)
and in zone III',
P'P2P',Vi2C1
We denote by n 1, n2, n3, n1, n; and n3 the perpendiculars to OP,
respectively PQ, QR, OP, P'Q' and Q'R pointing towards the body.
In the framework of the linear theory they will have the coordinates
(E1,-I),(-E2r-1),(-E3,-1),(es, 1),(E'2,1) and (-E'3i1).
The resultant of the pressure is
R = Lo(£1pini + t2P2n2 + 13p3n3 + tlp/ini + t21/2ns + tsp3n3)
.
(8.1.46)
The lift is
p _ Poo 2 Lo
[6 (E' - E1) + 12(C'2 + E2) + t3(E3
- 6'3))
.
(8.1.47)
Taking into account El, c'2, r'3, and the relation t1 + t2 + t3 = 1 , we
deduce
CL =
(8.1.48)
In the framework of the linear theory, the drag
Rt = 2p ,Lo(E + t1E1 - t2E2 - t3e3)
(8.1.49)
vanishes, because from the projection of the polygonal contour ORQ'P'
on the Oy axis we deduce
E = titJl + t2bJ2 - tstJ3
(8.1.50)
whence, taking into account the expressions E'1, E12, E13 we get
E + ties - t2E2 - 1363 = 0.
The formula (8.1.39) is very important. It shows that cL has the same
expression like in the case of the wing consisting only of its skeleton
OR. The result is natural, because of the symmetry of the wing with
respect to OR.
294
THE SUPERSONIC STEADY FLOW
8.1.6
Validity Conditions
We shall investigate, for the flat plate, the limits of validity of the
linear theory.
Since the flow is defined by the formulas (8.1.25) in I and II (fig.
8.1.1) and by Vt = Ui, pt = p in the regions upstream of AOA'
and downstream of BQB', we find that pi < pil and pi < p,,. Hence,
in order to have a positive pressure all over the fluid (this is a physical
condition), we must have
(8.1.51)
Pi > 0,
this representing a first condition of validity. Taking into account the
expression of pf given in (8.1.25), we deduce, with the notation kX =
= M2e,
0<ryX<1.
(8.1.52)
A second condition must ensure us that the flow is everywhere supersonic. Since from (8.1.25) or from Bernoulli's integral it results
IVtlf'2 < (Vt1 12, we deduce that we must have ( 1)2 < (V ')2, whence
'7Pf'/pfl = (cl'2 < U2 [(i - k)2 + x,2]
.
(8.1.53)
Taking into account that in the framework of the steady linear theory
we have p = M2p (it results from (2.1.7)), we deduce the inequality
11+X
equivalent to
k)2
<M2 (1
+s2,
1 +yX < (1+X)(M2 - 2X + X2).
(8.1.54)
(8.1.55)
In the linear approximation with respect to e, this inequality becomes
1-M2<(M2-2--y)X.
(8.1.56)
Further we shall deal with the inequalities (8.1.52) and (8.1.56) when
the fluid is the air ('y = 1, 405). For M2 < 3,405, the two inequalities
are satisfied in the shaded region from the (M2, X) plane (fig. 8.1.6),
where
M2 - 1 2
f (Af 2) = 3,405 - M
(8.1.57)
295
GROUND AND TUNNEL EFFECTS
x+
0.711
............. ......... ........... ............
0
Fig. 8.1.6.
This result shows that the linear theory is valid only in the zone
1 < M2 < 2. For a certain value given to M2 in this zone, E is
subjected to the restriction
(M2
e<
- 1)3/2
M2(3,405 - M2)
(8.1.58)
The last ratio increases when M2 increases, hence the superior limit of
the angle of attack is increasing with M2.
Other results concerning this subject are given in [8.17] [8.32].
8.2
8.2.1
Ground and Tunnel Effects
The General Solution
We assume now that the wing lies between two parallel planes having
the equations
a
y=-2 and y=Z
(8.2.1)
In this case, the general solution (8.1.8) and (8.1.9)
ku(x, y) _ -F(x - ky) + G(x + ky)
v(x, y) = F(x - ky) + G(x + ky)
(8.2.2)
P(x, y) = -u(x, Y),
with the damping condition
lim F(x - ky) = lim
s--00
x--ooG(x + ky) = 0,
(8.2.3)
296
THE SUPERSONIC STEADY FLOW
has to satisfy besides the slipping conditions on the profile,
v(x, ±0) = h4 (0),
x E [0,1] ,
(8.2.4)
the slipping conditions on the planes
,-a) = 0,
v
v
(x, 2 1 = 0,
x E R.
(8.2.5)
We assumed that the equations of the profile are
y = ht(x)
(8.2.6)
In the sequel we have to calculate the velocity field in the domain
D-={(x,y):x+kyE[0,1),0>y>-2}.
(8.2.7)
We consider the natural number n such that nka < 1 < (n+ 1)ka and
the segments (0, ka), (ka, 2ka), ..., ((n - 1)ka, nka), (nka, 1) on the
chord of the profile which is the segment [0,1] from the Ox axis. We
consider the characteristic having the equation - kq = C E (0, ka).
2).
For n=-2 wehave
The characteristic having the equation x + ky = E
(- ka ka
2,
2
ka
2, t E
does not intersect the chord of the profile, i.e. the
segment y = 0, x E (0, 1]. We have, by virtue of condition (8.2.3)
G
ka) =
G(x + ky) = 0, t E
lim
ka, ka
(8.2.8)
From (8.2.5) it results
0=va)
2
and from (8.2.8) and (8.2.9)
F(f+
Z)=0,
2
)
t E(-2, 2j.
(8.2.9)
(8.2.10)
Utilizing the notations
G(x+ky) = G- (x + ky), F(x - ky) = F"(x-ky),(x,y) E D" (8.2.11)
GROUND AND TUNNEL EFFECTS
297
we have on the upper surface of the profile
F- (x) = 0,
x E (0, ka) .
(8.2.12)
From the previous equation and from the slipping condition on the profile
(8.2.4)
h'_ (z) = v(x, -0) = F- (x) + G` (x), x E (0,11
(8.2.13)
it follows that
G-(x) = h'_(x) x E (0, ka) .
(8.2.14)
Further one demonstrates by induction that for x E (jka, (j+1)ka) ,
j E N, j < n, we have:
G-(x) = h'_(x) +h'_(x-ka)+h'_(x-2ka)+...
... + h'_ (x - jka),
F- (x) _ -h'-(x - ka) - h' (x - 2ka) - ... - h'_ (x - jka) .
(8.2.15)
(8.2.16)
Indeed, for j = 0 the relations are true. Assuming that the relations
are true for j, we shall demonstrate that they are also true for j + 1.
Let us consider x E ((j + 1)ka, (j + 2)ka). From (8.2.5) we obtain:
0=v(x-k2,
-G'(x-ka)+F-(x).
Since x - ka E (lka, (l + 1)ka), by the induction hypothesis we have
F-(z) =-G- (x-ka)=-h! (x-ka)-h! (x-2ka)-...
(8.2.17)
...-h'(x-(j+1)ka).
From the slipping condition on the profile
h'_ (x) = v(x, -0) = p-(x) + C-(z)
and from (8.2.14) it follows:
G-(x) = h' (x) +h'_(x-ka)+h'_(x-2ka)+...+h'_(x- (j+1)ka),
(8.2.18)
whence we deduce by induction the validity of the relations (8.2.15) and
(8.2.16).
Taking into account that
P(x, y) = -u(x, y) = k [F(x - ky) - G(x + ky))
,
(8.2.19)
298
THE SUPERSONIC STEADY FLOW
we may calculate the velocity field on the lower surface of the profile
[h'_ (x) + 2h' (x - ka) + ... + 211- (x - jka)]
p(x, -0)
,
(8.2.20)
x E (jka.(j+1)ka)n[0,1].
Considering the domain
D= (x, y) : x - ky E [0,1), 05Y:5 Z
(8.2.21)
and utilizing the notations
G(x + ky) = G+(x + ky), F(x - ky) = F+(x - ky), (x, y) E D+,
(8.2.22)
we can prove, like in the previous case, that for x e (jkb, (j + 1))kb) fl
[0, 1) we have
G+(x) = -h+(x - kb) - h+(x - 2kb) -
... - h+(x - jkb),
(8.2.23)
F+(x) = h+(x)+h+(x-kb)+h+(x-2kb)+...+h+(x- jkb), (8.2.24)
whence it results
P(x, +0) = k [h+(x) + 24' (x - kb) + ... + 2h+(x - jkb)]
(8.2.25)
x E (jkb,(j+1)kb)n[0,1).
8.2.2
The Aerodynamic Coefficients
As we know from section 8.1, the aerodynamic coefficients are given
by the formulas
cL =
j
rt
[p(x, -0) - p(x, +0)) d x
(8.2.26)
cA( = J x [p(x, -0) - p(x, +0)1 d x.
..110
In the sequel we shall consider some examples.
For the profile in a free stream (a = oo, b = oo), it results the already
known formulas (8.1.23).
299
GROUND AND TUNNEL EFFECTS
For the thin profile in ground effects (b = oo), taking for example
1 > ka > 1/2, we deduce
1
CL =
j[14(x) + h'_ (x)] d x- k
jh..dx
(8.2.27)
1
2
1
cM= --
x[h+(x)+h'_(x)]dxo
k
f
1-ka
(x+ka)h'dx.
o
For the thin profile in a wind tunnel, taking for example 12: ka >
1>_kb>1/2,we get
CL = - I
J
1 [h+ (z) + h' (x)] d x - k
-2 JI
h+(x)d x,
k 0
1
CM = -
k1
f 1-ka h'_ (x)d x-
1
2
j1-ka
x (h'+ (x) + h'_ (x)] d x - 2
(x + ka)h'_ (x)d x-
-2 11-kb (x + kb)h+(x)d x.
k
(8.2.28)
For the flat plate with the angle of attack e, we have h+(x) _
= h_ (x) = -Ex, x E (0,1], whence
2E
E
L= k, CM=k7
(8.2.29)
for the profile in a free stream,
(2 - ka), cm = k (2 - k2a2) ,
CL =
(8.2.30)
k
for the profile in ground effects (12: k > 1/2) and
cL =
(3 - ka - kb),
cu= k (3 - k2a2 - k2b2) ,
for the profile in tunnel effects (1 > ka > 1/2, 12: kg > 1/2).
This subsection was written following the paper [8.8].
(8.2.31)
300
8.3
8.3.1
THE SUPERSONIC STEADY FLAW
The Three-Dimensional Wing
Subsonic and Supersonic Edges
In this subsection we present the general theory of the thin wing in a
supersonic stream. We shall utilize the coordinates (2.1.1) and the fields
(2.1.3). The free flow is by hypothesis supersonic. Like in the subsonic
case, we shall denote by
z = h(x, y) ± hi (x, y)
(8.3.1)
= er(x, y) ± hi (x, y)}
the equations of the upper and lower surfaces of the wing. The projection of the wing on the xOy plane will be the domain D, assumed to
be simple connected. On the boundary r of this domain we have:
hi (x, y) = 0.
We assume that r is smooth. Then there exist a point F where the
tangent to r makes with the direction of the stream at infinity the
angle of Mach p defined by the formulas (8.1.7) and a point A, where
the tangent to r makes with the direction of the stream at infinity the
angle -µ (fig. 8.3.1). The point of intersection of these tangents will
be considered the origin of the frame of reference. There also exist two
Fig. 8.3.1.
Fig. 8.3.2.
points B and E where the tangents are parallel to the direction of
the stream at infinity. As we know from the subsonic case, the points
B and E separate the boundary r in two portions: the leading edge
301
THE THREE-DIMENSIONAL WING
EFAB and the trailing edge BCDE (C and D are the points where
the tangents make the angles p respectively -p with the direction of
the free stream). Obviously, we assume here again that every parallel to
the direction of the stream at infinity intersects the edge r in at most
two points at a finite distance.
Definition. We name supersonic (subsonic) part of the leading or
trailing edge, the part for which the absolute value of the component
normal to the edge of the velocity of the free stream is greater (smaller)
then the sound velocity.
We shall prove that this definition is equivalent to the following one:
If in a certain point of the leading or trailing edge the angle of the tangent to the edge with the direction of the unperturbed stream is greater
(respectively smaller) than Mach's angle, then in that point the edge is
supersonic (respectively subsonic).
Indeed, from figure 8.3.2 it results that the component normal to the
edge of the velocity of the free stream in the generic point P has the
magnitude U sin p1. If this is greater than the velocity of the sound in
the unperturbed flow we have U sin p > c, whence
sin pl >
1
-
= sin p
Utilizing the second definition, it results that the edge FA1 A from
figure 8.3.1 is a supersonic leading edge, the edges AB and FE are
subsonic leading edges, the edges BC and DE are subsonic trailing
edges, and the edge CB'E'D is a supersonic trailing edge.
It is known from the theory of hyperbolic partial differential equations (see also the plane problem from 8.1 and 8.2) that the zones of
influence are the zones delimited by the characteristic lines. For example, in figure 8.3.1, the zone of influence of the subsonic leading edge
FE is FF'E'E, FP and EE' being parallel to OA.
Definition. We name wing with independent subsonic leading or
trailing edges, a wing for which the zones of influence of these edges are
disjoint.
It results therefore that a wing has independent subsonic leading
edges if the Mach lines AN and FP do not intersect in the domain
D and independent subsonic trailing edges if BY and EE' do not
intersect in D. For example the wing from figure 8.3.1 has dependent
subsonic leading edges and independent subsonic trailing edges and the
wing from figure 8.3.3 has only independent subsonic edges.
THE SUPERSONIC STEADY FLOW
302
Fig. 8.3.3.
8.3.2
The Representation of the General Solution
Like in the subsonic case, we shall replace the wing with a continuous
distribution of forces having the form f = (fl, 0, f) defined on D. We
shall see that we may determine such a structure of f, in order to satisfy
the boundary conditions. The perturbation of the pressure determined
in the uniform stream by the distribution f will be, according to the
formula (2.3.32),
P(x,y,z) = -T"
fi)a-+f(Tl)8
'ID
J
G(ro,yo,z)ddq,
(8.3 2)
where
G(zo, yo, z) =
H(zo - s) '
z0-9
s = k ya + za.,
zo=z-t,
yo = Y-17.
(8.3.3)
For the velocity field, from (2.3.12) and (2.3.34), it results
v(z, y, z) = 6(z) lID f (rl)II (zo)6(yo)ddi+ V ,
(8.3.4)
where
( X, y, z) =
2Tr
ff
fi
n) - f (t+
o
G(zo, yo, z)dc dTl
z2
J
(8.3.5)
Obviously, the perturbation is potential excepting the trace of the domain D in the uniform stream, where the first term from the expression
of v does not vanish.
303
THE THREE-DIMENSIONAL WING
For the component w from (2.3.12) and (2.3.34) it results:
fJ f (, n)H(xo)6(yo)de dt7+
w(x, y, z) = 6(z)
+ 2a 8z JJD [h (t, n) - f
rl) 0 +
y
G(xo, I, z)dtdo
z2 J
(8.3.6)
For w we also have the representation:
w(x, y, z) = -'ID'n)
2
8 G(xo, y, z)d
(8.3.7)
+2x 11 f(t,n)N(xo,yo,z)dt dtl,
D
where
N(xo, yo, z) = k2
G(xo, yo, z) +
8
xo+G
(xo, Y0,
[-yol-+-Z2
z)(8.3.8)
J
which results from (2.3.37) and the representation
w(x, y, z) = ' AD ft (t, n) 8z G(xo, yo, z)dF dn+
1(2_02 _
(k
2n
8x2
2
2) AD f (C,
r(
)
H -7.1
[1-0*077
d11 d drl ,
(8.3.9)
which results from (2.3.38). Each of these representations determine
an integral equation for the function f (x, y). All the known representations (Evvard (8.9], Ward (8.34), Krasilscicova (8.20], Heaslet and Lomax
(8.15], Homentcovschi (8.16] and Dragoq (8.7)) are found in the formulas
(8.3.6)-(8.3.9). Prolonging the functions fl and f with 0 in R2\\,
the above representations may be written as convolutions. For example,
p(x, y, z) and rp(x, y, z) have the following form:
p=-2A 8 fl*G+BZf*G
if h #G-f * y2+z2G)
7r
where the sign
x, y.
*
(8.3.10)
,
indicates the convolution relative to the variables
304
8.3.3
THE SUPERSONIC STEADY FLOW
The Influence Zones. The Domain Di
First of all we must notice that the perturbation my be represented
by integrals whose integrand contains the factor H(xo - 8). Indeed, for
(8.3.5) this is obvious. Taking into account the formulas (2.3.35) and
(2.3.36), it results that the assertion is also valid for (8.3.2) and (8.3.7).
In (8.3.9) we have
f 7
f zp H(r - s)
d-r = H(xo - s) J 2
T -s
.ll:
rdT-s
(8.3.11)
Since the above integrands contain the factor H(xo - s) we deduce
that for a point M(x, y, z) from the domain occupied by the fluid, the
integrals on D are in fact calculated only on the domain Dl where
we have:
xo > s
(8.3.12)
This inequality implies
4 < x (x - 4)2 > k2[(y
-
11)2
+ z2].
(8.3.13)
The points from D verifying these inequalities are situated between
the leading edge and the hyperbola C which has the equation
(x - )2 = k2[(y
-
q)2
+ z2)
(8.3.14)
and the branches to -oc because < x (fig. 8.3.4). The hyperbola
C (the variables are
and t) has the axis parallel to Ox. In fact,
C represents the intersection of the cone having the equation
(x - C)2 = k2 [(y - t1)2 + (Z -
()2)
with the plane C = 0. This is Mach's cone. It has the vertex in Al and
the axis parallel to Or. From the mechanical point of view, this result
represents a consequence of a fact known from the hyperbolic partial
differential equations theory [1.6] namely the fact that in Al one can
receive only the perturbations produced in the points belonging to the
interior of Mach's cone with the vertex in Al. When Al will be on the
wing (z = 0), the hyperbola C will be reduced to the half-lines
x-==±k(y-,).
These are the characteristics issuing from M (fig. 8.3.5).
(8.3.15)
THE THREE-DIMENSIONAI. WING
FIg. 8.3.4.
305
Fig. U.S.
We may easily explain why the points from D - D1 do not affect
the perturbation in M if we have in view (the significance of the fundamental solution) that the perturbation produced in a point Q E D
propagates only in the interior of Mach's cone with the vertex in Q.
The point 141 is in the interior of all the cones with the vertices in Dl
and in the exterior of all the cones with the vertices in D - D1.
It also results that in the fluid exterior to the envelope of the posterior
cones with the vertices on D, the perturbation is zero.
Hence we can give up the factor H(xo-s) in the integrals expressing
the perturbation if we replace the domain D by D1. Prolonging the
functions f, and f in the exterior of D with the value zero, for a
given i;
,
ij will vary between Y_ and Y+ defined by (8.3.14) through
kYf = ky ±
xo - kszs .
(8.3.16)
The vertex of the hyperbola C has the coordinates tI = y, = x - kjzj
(obtained for Y+ = Y_).
8.3.4
The Boundary Values of the Pressure
For the integrals having the form:
I (x, y, z)
=
JJ
rl) 8zG(xo, yo, z)d>; dry
(8.3.17)
306
THE SUPERSONIC STEADY FLOW
we have:
Jz-kJ=I
1 _ 8z
d
f
Y+
fy_
r1)
ro-s
(8.3
dt) .
Performing the change of variables rq -+ 8:
kn = ky - Jxo - k2 z2 cos 0
(8.3.19)
we obtain
jx_&.izi
I (x, y, z) =
k.
z
d/f
rR
!
f
y - rx.2 - k2z2 cos 9) d8 =
C
z -L-1-TI
_ -sign z !O f (x - kjzj, y)d©+
JJ
dC
oo
Of
Jo 8rr
noose
xo - k z
dO.
Hence,
I (x, y, f0) = F it f (x, y) .
(8.3.20)
Using this formula, from (8.3.2) we obtain
Ax, y, ±) = - 2I j
jD f1 (e, n)
G(xo, No, 0)d do f 2 f (z, y)
whence,
(8.3.21)
f (x, y) = F(x, y, +0) - P(x, y, -0) .
This result puts into evidence the significance of the function f.
8.3.5
The First Form of the Integral Equation
The simplest way to obtain the lifting surface equation relies on
the representation (8.3.7). Taking into account the derivation formula
(2.3.35), we may write the kernel (8.3.8) as follows:
N(xo, yo, z) =
-k2
H(xo - s) _ xo(yo - p2) H(xo - s)
xo - s
yo + z2 (xo - s2)312
(yo
x022
(8.3.22)
We notice that for z = ±0 it appears the singular line n = y. Detaching
from D the domain DE defined by y - e < n < y + E in Dl - D,
(fig. 8.3.5), it is possible to simplify by yo after putting z = ±0.
Performing this.operation we deduce
N(xo, yo, f0)
xO
H(xo
o
xo
so)
so
H(xo, yo) .
(8.3.23)
307
THE THREE-DIMENSIONAL WING
Adopting the definition
f-'0 11.
f- D2
(8.3.24)
,
we shall prove that
j = lim lID. f ( r1)N(xo, Uo, z)ddq = 0 .
(8.3.25)
Fore small enough, we may perform the replacement n = y in the
integrand. Hence,
rx
= C-0
0o
lirn
(
x-kIzl
ao
[f-" f (F, n)N(xo, yo, z)di7 I d _
c
v-c
d= 0
[f()N(zotz)jdfll
1
-
.
Using the form ula (8.3.20) from (8.3.7) we deduce:
w(x, y, +0) = T 2 f1(x, y) +
27r
JJ f(f, n)N(xo, yo)dtdo . (8.3.26)
D
Adding and subtracting the boundary conditions
w(x, y, f0) = h..(x, y) ± h1x(x, y) (x, y) E D ,
(8.3.27)
one obtains:
h71 f(f,n)N(xo,yo)dt do = h.(x,y),
ft (x, y) = -2h1x (x, y) ,
(x, y) E D.
(8.3.28)
(8.3.29)
The equation (8.3.28) is the lifting surface equation in the supersonic
stream. It can be also written as follows:
f(")
_
,
yo
o::ll
xox k
(8.3.30)
o
D1 representing the shaded domain from figure 8.3.5.
The analogy of this equation with equation (5.1.28) is obvious. The
equation (8.3.30) was given in [8.7). For the sake of simplicity we shall
name it the equation D.
308
THE SUPERSONIC STEADY FLOW
The Equation D in Coordinates on Characteristics
8.3.6
We know from (8.3.15) that if the current point M is on the wing
(x, y) E D, z = 0, the hyperbola C from (8.3.4) degenerates in the
characteristics MMl and MM2 (fig. 8.3.5) having the equations
t -- krt = x - ky respectively t; + kr' = x + ky ( and q are the
variables, x and y are the coordinates of the point M). Performing
the change of variables C, ,q - a, Q, x, y -+ a, b defined by the formulas
- kn = a
x-ky=a
t;+kn =Q
x+ky=b,
(8.3.31)
we deduce
2kdt; drt = da d#.
The characteristic MAi1 has the equation a = a and the characteristic
MM2, the equation Q = b. The domain Dl given in the old variables
by the inequalities
t-k, <x-ky, +krl<x+ky
(8.3.32)
will be characterized in the new variables by
a < a;
f3 < b .
(8.3.33)
The axes OA and OF will be the new axes of coordinates because
on OA we have /3 = 0 and a is variable, and on OF, a = 0
and 0 is variable. The coordinates of M with respect to the new axes
are (a, b) (fig. 8.3.6). Since the new axes are characteristic lines, the
coordinates a and /3 will be named coordinates on characteristics. In
these coordinates the equation (8.3.30) becomes
f(a,
k
2rr
Dl
a - a + b -,8
(a - a - (b - Q)(2
dadQ
= H(a b),
(a a)(b - /3)
(8.3.34)
where we denoted by f (a,,6) the unknown f in the new variables,
H(a, b) the function given by h=(x, y) in the new variables and Dl,
the domain D in the new variables, i.e. the shaded domain from figure
8.3.6 (a<a,#<b).
Obviously, the integrand has in M (a = a, Q = b) a strong singularity. We shall isolate therefore this point drawing the parallel Q = b - f
THE THREE-DIMENSIONAL WING
309
Fig. 8.3.6.
and denoting by D6 the parallelogram indicated on figure 8.3.6. We
adopt in (8.3.34) the definition
Jf
,
C0
(8.3.35)
if, -n. *
In this way, the non - integrable denominator from (8.3.34) does not
vanish (even if a = a). In Dl - D6 we may utilize the identities
a - a+b _
1
-a-(b- /3)]2 n-a b-
2
0
a
s
a- a -(b-Q)-
P-i
a-a+_/_-_
82
-2 8a8b In
_
I
a --a - VS --- I I '
(8.3.36)
such that the equation (8.3.34) may be also written as follows
k0
('Pv, f(a, Q)
dadf
T-,6 a-a- (b-Q)
it 8a
k 82
;r 8aOb
a --a
JID,Di
H(a, b)
,
a --a + V- da d,B = H(a, b)
(a, Q) In
I
a --a - ../b-/31
(8.3.37)
(8.3.38)
310
THE SUPERSONIC STEADY FLOW
in order to have weaker singularities. In [8.7) one gives another form for
equation (8.3.37).
The Plane Problem
8.3.7
We act like in 5.1.7. In order to have a plane-parallel flow, we must
consider the case when the conditions that determine the flow are iden-
tical in every plane parallel to xOz. We assume therefore that the
equations (8.3.1) have the form
z = h(x) ± h1(x),
(8.3.39)
and the domain D is rectangular, such that the span 2b tends to
infinity (b - oo). Hence we assume that D is characterized by the
inequalities 0 < x < 1, -b < y < b. These conditions imply ff = fi(t)
and f = f (t;). Noticing that
2 100 H(xo
+0o
Ed
xp
k
1.00
_
- u + k z) du
-k'z2 u
°-k I 2T
2
kH(x0 -
lID f()8!!
(y02
(8.3.40)
xo - k222 - u
o
= kH(xo - kizi)
du
ldv
v
10
= -H(xo - kjzi),
+z2E/dj?=
f [.1:(j2 E)cli}df
1
we deduce the representation (8.1.17).
(8.3.41)
=0,
THE THREE-DIMENSIONAL WING
311
8.3.8 The Equation of Heaslet and Lomax (the HL Equation)
This equation may be deduced from the representation (8.3.6). To
this aim we denote
L(x, y, z)
8z
(/
f f fx° Di
6z
.
!l
2
do =
82.+8
+
(8.3.42)
d
+z2
Y_ 747-77
the functions Y+, Y_ being defined in (8.3.16). Here the change of
variable (8.3.19) is not indicated, because the term that one obtains by
deriving the superior limit of the first integral becomes oo for z --+ 0.
We act therefore as follows:
r-k
L = limm
Y+
as J-oo
f (4, q)
x0z
a yp + z2
fY_
(8.3.43)
so Ll (x, y, z) + Twki,1 xot(x0, Y, 44,
where, after performing the change of variable 9 -, it : q - y = u, we
have:
V+cf(x-k[+ £ q)
L1=-k2Jim
c-+o lyt
-
e-
do
+ z2
+s
_ -kz2f (x - kizl, y) hm
t(xo, y z )
F,
f
ft
-ki f (x - kjzj, y),
e - u (u2 + z2) =
(8.3.44)
Y+
z
f (F, n)
Y_
x0 - a Y + z
d ry =
a(s)
8
as
a(:)
F(u, z)zdu,
+z
(8.3.45)
with the notations
a(z) _
x8 - k2z 2
,
F(u z) ,
M,u+y)
(8 . 3 . 46)
x - k u2 + z)
We deduce therefore
L(x, y, ±0) _ -ka f (x, y) + f xot(xo, y, ±0)d..
00
(8.3.47)
312
THE SUPERSONIC STEADY FLOW
For determining the function L(xo, y, t0) we notice that a'(+-0) =0.
It results that the derivative from (8.3.45) and the integral interchanges.
We have therefore:
l(xo, y, ±0) =
_ limo
sL m
8z
+* OIF
f
8uOz
8
zUM
+°
8.. J C
r
a
{ [F(a, z) + F(-a, z)] arctan
z-»f0
J
+a OF
f4
=
}-
U2
+ z-.f0
lim
a
z2 + a2
z
+a OF
arctan z -d u + yli
lim (F(a, z) + F(-a, z)(
u
F(u, z)d arctan
u
+0
f
+ Z2
8F
du=
z
87Z u2 + ..2 d u+
c
u2+z2du.
8u
But
lim
z* .O
lim
z
0
r+n OF
f
a
J
du=
z
J
z-.f0 8z
8Z u2 + z2
+° OF
8F (0,z): /+` du
lim
u
du
a 8u u2 + z2
_
- el 0 z
0
f7rBF
f u2 + z2 =
8z (
+c OF
r
(-a
j+d019F
)
+ Ja
u
8; u2 + z2
0, 0)
du=
du
(u, U)
u
where ao = a(0). Finally, taking the formula (D.3.7) into account, we
obtain:
t( x0, y, f0 ) _ + W OF ( 0 , 0) =
F(oo, 0)
- F(-ao, 0) +
ao
ao
r
'/ +°O OF (u, 0) d
1
8u
u±
u
(8.3.48)
F(u2 0)d u
-,-0
f
BF(0 , 0) .
Having in view the expression of F(u, z) (8.3.46), we deduce:
L ( x, y, f0 )
= -k rf (x, y) +
f
oo
x0
T
y
f(t, n)
do
x =k-2-y- d on d t
(8.3.49)
313
THE THREE-DIMENSIONAL WING
Hence,
w(x,y,±0) _ T 2ft(x,y)+ Zf(x,y)(8.3.50)
[*
T7r
v- s
f
xk'y yo
L771 d
such that, imposing the boundary conditions (8.3.27), we find:
f
kf (x, y) - ! J
xo [
T
ok2yo
24(x, y), (8.3.51)
d
ao J
I1 (x, y) = (h' -
(8.3.51) is the HL equation.
Integrating in (8.3.42), at first with respect to
respect to y, we obtain
w(x,y,±0) = T 2ft(x,y) -
oo
«
2zr
(8.3.52)
(x, y).
.
°
r-kh#o)
d2y
b0
and then with
[.! ro-k o
dl
.
8.3.9 The Deduction of HL Equation from D Equation
This deduction was performed by V. Iftimie. To this aim, we shall
extend the integral from (8.3.30) from the shaded domain Dl, in figure
8.3.5 to the infinite domain D&, situated between the characteristics
MMl and MM2 extended to infinity, putting f = 0 in the exterior
of the arc M1 M2. We denote by I the integral obtained in this way
and we perform the change of variables t, n -+ u, v:
u=x-Z;, v=y-y.
(8.3.53)
In the new variables, the domain Df, will be transformed into the
domain Du,, (fig. 8.3.7) determined by the angle Mi MMz. Denoting
g(u, v) = f (x - u, y - v), we deduce
f(x,y) = g(0,0).
(8.3.54)
THE SUPERSONIC STEADY FLOW
314
Fig. 8.3.7.
The HL equations result from the formulas
g(u'
v)
u
dudv
v2
u-k
_
d
=-k,rg(0,0)+or duTAOU,v
00
=
it
v)
J
u
u
(8.3.55)
k
akv
which have to be demonstrated (the demonstration was given by
V.1ftimie). For proving the first formula, we isolate the singularity by
two parallels to the Mu axis (fig. 8.3.7) at the distance e from this
DD
axis. We denote by D' and D" the domains composing
and by I' and I" the integrals I corresponding to these domains.
Using the definition (8.3.24) we get
I = limo(I, + 1") .
Putting v = tu, we deduce:
I'-
11
JfJJJJ
g(u,v)u
v2
dudk
u
_ r0O 1
=J. u[u
/k
oc
v
-lip uI,J
(8.3,58)
r
JJ
t2
g(u,tu)
1-k t
dt du.
u
1
dv'du
315
THE THREE-DIMENSIONAL WING
We expand g in a Taylor series:
g(u,tu) = g(u,0) + tug"(u,0) + t2u2m(u,tu).
and we introduce this series in the above integral.
Acting analogously for 1", we deduce:
- °° u F f-`
I"
9(u, v)
d vl d
v2 u -k
J
1-v-1g(u,tu)
kE u l Jl
t2 1- k t
u=
dt1du
whence,
f
00
1' + 1" =
g(u, 0)
[ JiY
U
Jibe
r
t2 1- k t
+Jits g°(u,0)LI=" t lack t
+
i m(u, tu)
uf
LJ4
+r
dt
1-k t
d
The factor which multiplies
L
J_
+J-I
t+!J_
1t
dt
du+
t2 1- k "t
t I-
m(u, tu)
1
t
t,du+
d t d u.
is zero if we replace in the second
integral t by -t. An elementary calculus gives
rk1-k t
dt
411
_r
dt
_ u -k E
t2 1- k t - J_ t2 1- k t
whence,
P+ I"
E
e
k2E2d u+
u
(8.3.57)
+fJ-u(j1+j-'U) m(u,tu) dtdu
When a -+ 0, the first integral is a Finite Part. Hence
2 /'O g(u, 0)
C JkE
U
1 9(u1 0) - 9(01 0)
u2 - k2E2d u
E
fkC
U
u2
k2E2+
316
THE SUPERSONIC STEADY FLOW
u u
+E g(0,0}
ke du+E foo 9( L+0)
u2 - k2F2 d u .
JEc
The first and the last integrals from the right hand member are finite.
When we consider the Finite Part, we neglect these integrals because
they h a v e the factor E t . One calculates the integral from the middle
and one obtains
FP 2
0o
E Jke
u2 - k2e2d u = -kirg(0, 0) .
g(u' 0)
U
Hence, from (8.3.57) we deduce:
r00
I= lim(I`+I") =-k7rg(0, 0)+J
o
oJ
I
m1(uk tut)
,
d t]d u. (8.3.58)
We set now
J= f 00 [1" 9(u,v)v
f
dUk
Jdu.
u
(8.3.59)
Using the same Taylor expansion we deduce:
J' _
J6
u v)
g('
[Ilk
u
d v=
- Js
v2v
Y
d tk
9(u. 0)
t2 VI
U
t +9u(u1 0)J°
f I m(u, tu)
+u f
1 - 00
0
J =Jr 9(u, v)
dv
u - kv
+gv (u,
Y
0)
J
dt
g(u, tu)
u
t2 1 - k t
t
latkt
-
+
d t,
u
dt
t 1-k t
f_1
-{-
t2 1-k t+
.. .
whence,
J' + J" =
2 g(u, 0)
u - k e + u{
m(u, tu)
U
43
1-kt
dt.
317
THE THREE-DIMENSIONAL WINC
When we consider the Finite Part (FP) we neglect the first term and
we get
li
m
(J' + J") = u
m(u, tu)
f
J= /
1-k t
u
Jo
[
JI
m(u, tu) d t d u.
1-k t
(8.3.60)
From (8.3:58) and (8.3.60) it results
I = -k7rg(0, 0) + J,
(8.3.61)
which demonstrates the first equality from (8.3.55).
For proving the second equality, we denote:
00
K
I
T0,
J
u
g(u,v)
kv
dudv= lim(K`+K"),
where (we may change the order of integration)
K' - r-E r r
J_O0
sr
fI
g(u, v)
d
v2 7u!=-=kFJug(u, v)
kv v2 u - u v
0o
v du = I',
(8.3.62)
K" = 100 [/
r
- Jkr
d ul d v =
ky v2 u- JC v
h
00 u
Jk
uy(u, v)
du] d v =
c
u
g(u,v)
i v2Vu- 72 -k v
dv du = l"
Hence K = I and the demonstration is finished. The HL equations
resulting from (8.3.27) are
kf (x, y) - a
-1
f
a'°
00
l vklvo!
co
p
o°
f(")
d? l d t = 2h.(x, U),
k Uo V5
o-n)k o
F--7
d
t
(8.3.63)
= 2h= (x, y)
These are the equations (10-21) and (10-22) from (8.151.
(8.3.64)
318
THE SUPERSONIC STEADY FLOW
8.3.10 The Equation of Homentcovschi (H Equation)
This equation results from the representation (8.3.9). Taking into
account (8.3.20), we deduce:
w(x, y, t0) = T 1h (x, y)+
+2
f
l rr
(k2 52 +J
2/
_
=o
Id
f H(T ^s)drdn,
L
.0
o
(8.3.65)
such that imposing the boundary conditions (8.3.27), we obtain the
equation
(k 2
ll
axe - r7y2)
ff f
n)
f
r:o
d T d dn = 2 (x,b)
J
H(T - so) J
(8.3.66)
which was given for the first time (in a different way) by D.Homentcovschi
(8.16].
We have to notice that after evaluating the interior integral with the
formula (8.3.11) one obtains the equation
!
..
(k2..02
_
vx-!::
02
aye) Jf,
f (E, n) In xa + 40
,
sod d r1= 2h (x, b)
(8.3.67)
In the variables on characteristics this equation becomes (8.3.38). However, Homentcovschi does not follow this way. Inspired probably by the
papers of Evvard (8.9], Ward (8.33] and Krasilscicova (8.20], he intro.
duces the unknown N, related to f by the equality
N(x, y) = 1-00 f (t, y)d t,
(8.3.68)
f (x, y) = N= (x, y) .
(8.3.69)
whence it results
We have seen in (5.4.5) that o function similar to N also intervenes in
the theory of low aspect wings in a subsonic stream.
Taking (8.3.69) into account, the integrand from (8.3.66) becomes:
N
:° H(T - so)dT
J
-
= 0
[NJ° H(T - so)dT
9[NH(x(, -4o) f°u
x2 -4
o
dT
a_
so)
0J
V0-
o
_
319
THE THREE-DIMENSIONAL WING
[N(f , n) In 'To + sp o - so] +
= H(xo - so) {
F(I,,,) 1.
0
0
In this way, after applying Green's formula, the equation (8.3.66) becomes
111I N(C, g) d i d n+
a_ 82
(kz
xo-8o
81l
+ POD N(t, Y)) In
sod
xo +
so
1
(8.3.70)
n} = 2h.(x,1!)
In variables on characteristics this equation becomes
k
a dQ
02
8
AD
l
- 0) +
V'r(a - C
v1_a --a
+ JeD; v(a' Q) In
a-a
+
VT--71
d ( - a) } = H(a, b).
(8.3.71)
From (8.3.68) it follows that N(£, n) vanishes on the leading edge,
hence
on EFAB.
N = 0,
(8.3.72)
Indeed, for every point
n) E EFAB, the integral (8.3.68) is calculated on the half -line having the applicate q upstream of the wing,
where f = 0. We also notice that the curvilinear integral from (8.3.71)
vanishes on the characteristics a = a and (3 = b (fig. 8.3.6), because
the logarithm vanishes. Hence, for a given M(a, b) , the curvilinear
integral from (8.3.71) will differ from zero only in those points of the
trailing edge belonging to 0DI. For example, for the wing from 8.3.1,
the curvilinear integral will differ from zero only for M situated in
the zones of influence of the subsonic trailing edges BC and DE,
respectively in the domains BCB' and EE'D. Utilizing the definition
(D.3.6) for (8.3.71) we also have the equivalent forms:
k
4ir
N(a) p)d a d.8
f JD1
(a - or)-111(b - 0)312
+
(8.3.73)
+_7r 8a JD a - a - (b, Q)
-d(A-a) =H(a,b)
320
THE SUPERSONIC STEADY FLOW
k "f
4;r J
N(a,A)da dA
7D) (a - c')3/2(b - 10)3/2
8
f
N(a,
A) d
(8.3.74)
+k 8b oD1 b-,C-(a-a) a-a (A - ) = H(a, b),
the curvilinear integral being different from zero only in those points of
the trailing edge where a < a and A < b (see the definition of D1 in
coordinates on characteristics (fig. 8.3.6).
The equation (8.3.71) is the equation of Homentcovschi (H equation). In the following subsection we shall present the solutions of this
equations as they were determined by the author himself.
The Theory of Integration of the H Equation
8.4
8.4.1
Abel's Equation
Abel's equation has the form
x
f"Jdx=h(Y)1
-x
a< y.
(8.4.1)
This was the first equation encountered in applications. Multiplying
(8.4.1) by (A - y)-1/2 and integrating with respect to y on the
interval (a, A) we obtain:
dy
J
rV
Ja
f(x) dx
y-x
rA h(y) dy
-JQ Vx --y
Changing the order of integration in the first member (fig. 8.4.1),
we have:
A
A
f (x)
J=
dy
(A - y) (y - x)
and then
f f(x)dx = I
fA
dx=
Jay
h
dy.
A
h(y)
dy
Deriving this relation and utilizing the definition of the Finite Part
(D.4.3), we deduce:
d
f (A) = a d A
f'\
hid
y
I
21r
,
{A
h(y)3/2 d y.
(8.4.2)
THE THEORY OF IN BGRATION OF THE X EQUATION
321
y
a
x
1)
Pt
8.4.1.
It results that the solution of the equation (8.4.1) is (8.4.2) and conversely, the solution of the integral equation (8.4.2), where h is the
unknown, is given by the formula (8.4.1).
8.4.2
The Solution of the H Equation in the Domain of
Influence of the Supersonic tailing Edge
If M is in the zone of influence of the supersonic trailing edge
(fig. 8.4.2), as we noticed at the end of the section 8.3., the curvilinear
integral from the equations (8.3.73) and (8.3.74) disappears. We see in
fact on the figure 8.4.2 that on the arc BM2 we have a > a and on
the arc Ml E, /3 > b. It remains to integrate the equation:
k
4a
N(a, /3)d a d,6
JDI
(a - a)3/2(b -
= H(a, b) .
(8. 4 . 3)
/3)3/2
Denoting:
a = A(,B), the equation of the arc DEFA and
(8.4.4)
/3 = B(a), the equation of the contour FABC,
t h e equation (8 .4 . 3 ) may be wr itten as follows
k'b
Ni (a, fl)
4ir JBi,l (b -8)3/2 d
= H( at b) ,
(8. 4 . 5)
THE SUPERSONIC STEADY FLOW
322
Fig. 8.4.2.
where
r
Nt (a, Q) _
N(a' a)
A(p) (a - a)3/2
(8.4.6)
d c t.
Utilizing (8.4.2) and (8.4.1), we deduce from (8.4.5)
2
_
(a, Q)
J
°)
H(Nl
3d
(8.4.7)
br
In this way, the equation (8.4.6) becomes
_
1
27r
N(a,fl) da =
1A(8) (a - a)3/2
T11
H(a,b')db'.
(8.4.8)
$(°)
The equation (8.4.8) is similar to (8.4.2), so that
1V(a, A)
ra
d a'
JA(9)
a--a,
1 r H(a', b') dg.
Irk ,lg(a') 13
b'
For a = a, 6 = b we obtain
N(a,b) _
1
"
Irk IA(b)
da'
H(a',b')db,
b - b'
a --a' J Br(.')
b
(8.4.9)
1 If
r(ab)
H(a',b')da' db'
a - a' (b - d)
THE THEORY OF INTEGRATION OF TIIE H EQUATION
323
This is the solution in the zone of influence of the supersonic leading
edge.
8.4.3 The Solution in the Domains of Influence of the Subsonic Leading Edge
We assume that M(a, b) is situated in the zone of influence of the
subsonic leading edge AB (fig. 8.4.3). In this case, the curvilinear
integral also vanishes, because on BM2 we have a > a, and on
M1E, ,8 > b. The equation which has to be integrated is also (8.4.3).
Obviously, A(b) < a1 < a, B(a) < b < b1.
The equation (8.4.3) may be written as follows
k
47r
J
N(a, /3)
da
(b) (a - a)3/2 B(a) (b - 0)112
d p = H(a , b) ,
(8.4.10)
such that we denote
N2 ( a, b)
(
2 JB(a) (b
we obtain the integral equation
Fig. 8.4.3.
0Q/2 d
1
(8 .4 11)
.
324
THE SUPERSONIC STEADY FLOW
k
N2(a, b)
JA(b)
(a - a)3/2
da
= H (a, b )
( 8 . 4 . 12)
whose solution is
N2(a, b) =
-1
k
H(a'' b)
J
A(b)
a- a' d a
(8.4.13)
for a > A(b). From (8.4.11) we obtain for N(a,/3)
the following
integral equation
2 r r l B(o) (b
(IB 2 d 1 3 =
7rk IA'
(b)
Haas'
d a'
(8.4.14)
Utilizing again the solution of the equation (8.4.2), we deduce:
N(a, R) = irk
1
J B(a)
for a > A(b) and
N(a,6) -
d &'
H(O' b') d a ,
f(- b JA(bl) a - a'
> B(a). Putting a = a, Q = b, we obtain:
1
db'
JB(4) b - b'
(8.4.15)
H(a', b')d a' d b'
D]
Irk
fN)a='
H(a,b')dIrk
(0)
V'r(a,
- a) (b - b')
Dl being the shaded domain from figure 8.4.3 We notice that in this
case D1 is not the entire domain determined by the leading edge and
the characteristics issuing from M. From this domain one eliminates
the strip where b' < B(a). This result was obtained for the first time
in 1949, independently, by Evvard and Krasilscicova and it is called in
some books the theorem of Evvarrl and Krasilscicoua
The solution is obtained analogously when M is in the zone of
influence of the edge FE, with the difference that in this case one
eliminates from Dl a strip parallel to the 0/3 axis.
8.4.4 The Wing with Dependent Subsonic Leading Edges
and Independent Subsonic Trailing Edges
For a wing with dependent subsonic leading edges and independent
subsonic trailing edges (fig. 8.4.4) the solution in the domain bounded
TIE THEORY OF INTEGRATION OF THE H EQUATION
325
by the curve AHFA is given by the formula (8.4.9), the solution in
the domain bounded by ABH'HA - by the formula (8.4.15), and
the solution in the domain bounded by FHH"EF - by a formula
analogous to (8.4.15). The case when M is in the common zone of
influence HH'B'E'H"H of the subsonic leading edges is presented in
the sequel.
Fig. 8.4.4.
We notice at first that in this case the curvilinear integral from
(8.3.73) also vanishes, because on BM2 we have a > a, and on
All E'3 > b. Hence, we have to integrate the equation (8.4.3). Denoting
by R1, R2 the domains bounded by the curves FM"RF, respectively FMIt'M"F and by Q1, Q2 the domains bounded by the curves
AQA1', respectively AM'M'A, we notice that on Rl and Q, we
have N = 0 (it results from (8.3.68)). Hence, the domain of integration
from (8.4.3) may be prolonged to the domain bounded by MRFAQM,
i.e. to the domain
D2+R1+R2+QI +Q2,
where D2 is the shaded region from figure 8.4.4, i.e. the region bounded
by the contour Iii"'AMA! A.!"Al't'. The leading edge of this domain is
entirely supersonic, such that the solution of the equation (8.4.3) is given
by the formula (8.4.9)
N(a,b)
k
[Jf.+JJRi+fJR,+ff, +ff.
(8.4.16)
H(a', b')
d a' d b' .
(a - a)(b - b')
THE SUPERSONIC STEADY FLOW
326
Let us consider now that M belongs to the zone of influence of the
subsonic leading edge AB. From formula (8.4.15) we deduce:
1V(a'b)
irk IJJ
+
f+[
,
R, ,
(a-a)(b-b')
da'd
(8.4.17)
Similarly, taking into account that M belongs to the zone of influence
of the leading edge FE, we deduce:
N(a, b)
Tk-
f
ffDz +If +f fQz,
Q1
H(a', b')
(a - a) (b - b')
dadbl.
(8.4.18)
From the formulas (8.4.16) - (8.4.18) it results
N(a,b)
irk
ff
bi)da'db',
(8.4.19)
(aH(a',b')
this representing the solution in the common zone of influence of the
two subsonic leading edges.
We notice that the solutions from the previously considered domains
have the same form, differing only the domains of integration. But we
may establish a common rule for determining the domains of integration. They are bounded by the parallels to the characteristics issuing
from the points where the first parallels intersect the leading edge and
by the remaining portion of the leading edge. In the first case, when
M is in the zone of influence of the supersonic edge, the parallels to
the characteristics issuing from the points where the parallels from M
towards infinity upstream intersect the leading edge, do not intersect
any longer this edge.
8.4.5
The Wing with Dependent Subsonic Trailing Edges
We consider now a wing for which the parallels M'M", M"Mf V
intersect in a point P belonging to the interior of the domain D
(fig. 8.4.5). In this case, denoting by R2 the domain bounded by
FM'PM"F and by Q2 the domain bounded by AM'PMIVA, (the
domains R1 and Ql keeping the same definition) and performing the
same reasoning like in the previous subsection, we obtain successively
N(a,b)=1(D3+R1+R2+D4+Q2+ Q1)
N(a, b) = I(Ds + Ri + R2) ,
N(a, b) = I(D3 + Q2 + Q1) ,
327
THE THEORY OF INTEGRATION OF THE H EQUATION
R'%G
A
Fig. &4.6.
I representing the symbol for the integral appearing in (8.4.19).
It results the solution
N(a, b)
AD,
(a H(a,'
'){
b
d a' d b'(8.4.20)
kiDD.
H(a',b')
(a-a' (b-b')
dadb'
The result may be generalized for every wing having finite dimensions.
For example, for the wing from figure 8.4.6 the solution in the point
M(a, b) is
N(a,b)=I(D3-D4+Ds-Ds).
(8.4.21)
We stop here the presentation of the solutions in the zones of influence
of the leading edge.
8.4.6
The Solution in the Zone of Influence of the Subsonic
Edges under the Hypothesis that the Subsonic Leading
Edges are Independent
For the wing from figure 8.4.7 the solution is determined in the domain bounded by the curve ABB'E'EFA. It remains to determine the
THE SUPERSONIC STEADY FLOW
328
solution in the zones BCB'B and E'DEE'. To this aim we shall use
the method of Homentcovschi [8.16].
Fig. 8.4.7.
Let M(a, b) be situated in the last zone. In this case, the curvilinear
integral from (8.3.73) does not vanish. More precisely, it is zero on BM2,
where a > a, it is zero on M1M", where 6 > b, but it is not zero
on WE. We assume at first that N is known on the trailing edge.
Since M is in the zone of influence of the subsonic leading edge FE,
the double integral may be inverted with a formula similar to (8.4.15).
We have therefore
N(a ,b) =
fl
EM"
1
Irk AD, (a,b)
/a'-a
H(a',br)da'db'
(a ---a') -(b- b'
b'-p a'-a-(b'-p)
+
1
W2
d(!g-a)1
AD
'
8a'.
da'db'
(a-a')(b-b')
(8.4.22)
where D1 (a, b) is in the shaded domain from figure 8.4.7, i.e. the
domain A(b) < a' < a, b' < b. In front of the last integral we have the
329
THE THEORY OF INTEGRATION OF THE H EQUATION
sign + because we have changed the sense on WE to EM". On
WE we shall denote
N(a,10) = N(A(Q), Q) = N(3) ,
(8.4.23)
because the equation of the edge DEFA is a = A(13).
The curvilinear integral imposes to eliminate from D1 (a, b) the
points where b' < p. D1 from the second integral (8.4.22)is therefore
the shaded domain from figure 8.4.8. Denoting this integral by T and
interchanging the curvilinear integral and the integral on the domain we
get:
T=
--In,
a
8a'
a[,! a
da' -a')(b
_
)(b-b')
rb
[ a' - A(/3) /
N(Q) V
1
J
[I-A'((3),d0 1.
a'-A(l3)
b'-(3 a -A(Q)-(b'-0)J
+
bN(f3)[1-A'(A)]dfJ
2
db'
(b' -)3)(b - b') u - b',
da' a
A(b) 7a =' r?a'
1
,
(8.4.24)
where u=a'+A-A(/3).
Fig. 8.4.8.
330
THE SUPERSONIC STEADY FLOW
For calculating the interior integral we shall prove first that b < u,
i.e. that
b<a'+l3-A(13).
(8.4.25)
Denoting by P, R, S the points having the ordinates 3 namely, P
on the trailing edge, R on the la intersection with the direction of the
unperturbed stream and S on the boundary of the domain Dl (fig.
8.4.8), we have R(A(fl),,3), S(A(b), (3), 3 RM"S =3 SRM" = it, p
representing Mach's angle (the angle made by the characteristic lines
with the direction of the unperturbed stream). It results
PS = A(b) - A(#).
RS = SM" = b - f3,
Since M"P is an are on the trailing edge we have RS < PS whence
6 < A(b) + A - A(#) < a' +,0 - A(/3) ,
(8.4.26)
because a' > A(b).
Employing the substitution
b'-b+A+b-Qcose,
2
u=b+A+b-Qs
2
2
2
we obtain
b
db'
1
d0
2
b-# 0 cosO-s'
(b'_/3)(b--b') u-b'
the inequality b < u implying 1 < s. This integral has the form (B.6.1)
and the solution is given by (B.6.4).
One obtains:
Il =
(u -b (u--/B)
7r
7r
a'-A(A)][a'+Q-A(Q)-
(8 . 4 . 27 )
The following integral has the form
°
8(
1
1
a' -v J
8a'
da
'=-1
Q
1
2 JAM (a'-v)3/a
da'
a-a,
where v = b + A(fl) - Q < A(b). This inequality results from the first
inequality (8.4.26). Noticing that
1
ZA(b)
(a'-v)3/2
da'
_
2
fa - A(b)
a-a' a-v A(b)-v
331
THE THEORY OF INTEGRATION OF THE H EQUATION
we deduce
T=-
1- A'(#)
N(/3)
a - A(b) ) 6
f3-A(f3)+A(b)-b 13-A(a)+a-b
62
(8.4.28)
whence
N(a,b) _
H(a',b)da' dN +T
1
Irk
(8.4.29)
ADi(a,b) -vf(-a - a' (b - 6')
in T, N(fl) being unknown. The form (8.4.29) is given in [8.16].
We consider-that M tends to the position M" on the boundary.
It means that we make in (8.4.29) the substitution, a
integral is
1
1=
-
ll
H (a', b')d a' d b'
r( ab)
-
(a - a')(b - b')
where
L(a')
= jb
A(b). The first
L(a')d a'
I
irk JA(b) a - a'
H(al'b')db'
(8.4.30)
(8.4.31)
(a)
b = B(a) representing the equation of the edge FABC. Obviously,
this integral vanishes when a - A(b).
We shall perform in the expression of T the substitution 0 - t
13 - A(/3) = t and we shall denote N(Q) = N1(t). We also denote
a-A(b)=E, b-A(b)=c, b2-A(b2)=c2.
(8.4.32)
From the first inequality (8.4.26) it results 13 - A((3) > b - A(b) whence
C2>C.
Hence,
T,=
ff Ni(t)
1r
c
dt
t - C+E
We shall integrate by parts setting
u=N1(t), dv= 7tIm td(t-c)
- c+E
It results
V=
arctan rLe--f
=
The integrated term vanishes because t = c2 implies A = b2 and
.
N1(c2) = N(b2) = 0.
332
THE SUPERSONIC STEADY FLOW
r
We obtain therefore:
T=--J
and
2
f_m T = -- 2 f
Ni (t) arctan
Vt
e
cd t
(8.4.33)
C2
Ni(t)d t = Ni (c) = N(b).
In this way, passing to the limit in (8.4.29) we obtain
N(A(b),b) = N(b),
i.e. an identity. This means that for an arbitrary given N(b)
, the
function N(a, b) given by (8.4.25) is a solution of the integral equation
(8.3.73). One obtains an indetermination like in the subsonic case).
This indetermination exists in the zones of influence of the subsonic
edges EE'DE and BCB'B. In these zones the integral equation of
the problem is not sufficient for determining the solution.
Like in the subsonic case we remove this indetermination imposing
the Kutta-Joukovsky condition. Imposing a finite velocity on EE', it
results that the jump of the velocity on EE' is finite. Since the jump
of the component u is given by the jump of the pressure with the
changed sign, and the jump of the pressure by f (x, y), it results that
it is sufficient to impose for f to have finite values on the subsonic
trailing edge.
We have
f(x, y)=8xN(x,y)_
\8a+
bN(a'b)=(8a+gb)(I+T).
(8.4.34)
Performing in (8.4.30) the change of variable a' -- s : a - a' =
_ (a - A(b)]s and keeping the notation a - A(b) = s we deduce
I
I=
rkoJ
L(a - es) f =
2
L(a) - O(£3/2) _
a V - A(b)L(a) - O(e312) ,
whence
a
+
)
I
k
.1-AI(b)
=
a fk
Ate)) L(a) +
jb
(a)
H(a, b')d b' + 0(E)
b - b'
(8.4.35)
.
333
THE THEORY OF INTEGRATION OF THE H EQUATION
From (8.4.32) we deduce
T=- 2 J
7t
(b2-A(bz)
t- b+ A(b)
NfI (t) arctan
&-A(b)
a - A(b)
dt
whence,
s
a
as + ab
b:-A%)
1- Al(b)
T=
t- b+ A(b)
N' (t)
Jb
?r/e-
t-b+a
b-A(b)
V
d t + O(,/E .
But
t-+A
-b
(b)
t-b+a
t --b + A(b) =
1
t-b+A(b)+e
t-b+A(b)
[1 + O(e))
,
such that finally we have:
a +
:b) T (aa
N'(t)dt
1 - Al(b) f*2-A(b2)
7r f
It - b + A(b)
-A(b)
+O(f). (8.4.36)
We obtain therefore
f(x,3!) =
A'(b)
1
1
[k
b
B(a)
H(a, )
db,+
%lb
(8.4.37)
-A(b z)
LAb)
N(t)d t
1
t --b + A(b)
+ O(f) .
The function f (x, y) has finite values on the trailing edge (a -+
A(b), e -+ 0) when the square bracket vanishes i.e. when
Ni (t)d t
pb:-A(b,)
1 fb
t - b + A(b)
Jb-A(b)
k JB(A(b))
H(b(b), ) d b'
-G(b) .
(8.4.38)
This condition is an integral equation (of Abel type) for determining
the unknown N' (t). Using the notations (8.4.27) we may write this
equation as follows
j2 Ni (t)d t
(8.4.39)
J
v
where Gl (c) = C1 (b - A(b)) = G(b). We deduce
f'2
z
dc
C
(C3 N'(t)dt
-xJc
t-C
-
t
z
C, (C)
xdC.
334
THE SUPERSONIC STEADY FLOW
Changing the order of integration in the left hand member we get:
C2
NI(t)dt J
do
(c - x)(t - c)
- _ f C2 Gi(c)
dc.
c-x
s
.
1.
Since Nl(c2) = N1(b2 - A(b2)) = N(b2) = 0 (on the leading edge N
vanishes), we obtain
j Glt
f t --X d t.
1
N, (x)
7r
For x = b - A(b) we deduce
l Jbz-A(bz)
_A(b)
Gl
dt
(t)
t - b + A(b)
or, using the change of variable t
i3 : 3 - A(,3) = t,
N(b)
G(i3)[1
1
N(b) _ R
A'(/3)l d /3
VP - A(/3) - b + A(b)
rb
(8.4.40)
K(fl)[1 - A'(i3)ld/3
(3 - b + A(b) - A(I3)
I f'
bz
where
8
H(A(#), b')
db
k D(A(#)) VF-7
1
(8.4.41)
.
This formula determines N in every point of the subsonic trailing edge
ED. Replacing this expression in (8.4.28), we may find out T.
In the sequel we shall give an explicit expression of T. Changing
the order of integration (fig. 8.4.9), we deduce:
T= 12 --A
b
dJ3
,3 - A(/3) + a
b
Jbzb2
A (fl) + A(b) - b
a
_b. [1
K(E3')(1-A'(f)ldA'
-b+A (b) -A(f3)J
_-
(8.4.42)
b
_
2
a --A (b)
b
K(A')I (1Y)(1 - A'(Q'))d // ,
where, using the substitution p -+ t:,3 - A(0) = t,/3' - A(i3') = t',
IV) =
f
1
(t - c)(t'-
dt
(8.4.43)
(b - a)
THE THEORY OF INTEGRATION OF THE H EQUATION
335
0
b
be
I
0
be
Fig. 8.4.9.
From (8.4.26) It results b - A(b) < i - A(8) for every j6, hence for
_ P. This implies c < t'. The integral from (8.4.43) has the form
(B.6.11). It results
Ir
1=-
[a-A(b)]
-A(.')-b+a
whence
T=_1
K ) [1-A'(B)]d3
(8.4.44)
rb, /3-A(P)-b+a
Using this form of T, we express the solution in the domain bounded
by the curve EE'DE by means of the formula (8.4.29).
2. If M(a, b) is in the domain bounded by the curve BCB'B (fig.
8.4.10), then we shall utilize the integral equation (8.3.74). Obviously,
the curvilinear integral does not vanish on the arc BM'. Utilizing the
solution (8.4.15), we deduce:
N(a, b)
where
Tl
= plc
1 JJD1(.,b)
H(a', b')
a - a' b -
d a' d b' + Ti
(8.4.45)
a' d b'
=- 2 JJJD, ad-a(b-Y)
(8.4.46)
ef°
8b' f
FL- B(a)
N(°`)
1]d a
-a b'-a'-B(a)+a
THE SUPERSONIC STEADY FLOW
336
Taking into account that the equation of the edge BC is Q = B(a),
we denoted N(a, B(a)) = N(a). Dl (a, b) is the shaded from figure
(8.4.10), i.e. the domain bounded by MM', the parallels 0 = B(a) and
A = b and the leading edge included between these parallels. D] is the
domain bounded by the contour MPQM' imposed by the condition
a < a'.
Interchanging in (8.4.45) the curvilinear integral and the double in-
337
THE THEORY OF INTEGRATION OF THE H EQUATION
tegral we obtain
JaN(a)[1
T, =
T1
a
5-b'
- B'(a)]
az
f
(
a)
b - b'
do
({/_B(a)j
'
w-all
db' d1
I
(8.4.47)
where
w=b'+a-B(a).
The similarity of the expressions (8.4.47) and (8.4.24) is obvious. TI
may be obtained from T replacing b, /3 and A by a, a and B and
conversely. It results therefore
N(a)
1-B(a) da
T,=- Ir b-B(a)f \/a - B(a)
+ B(a) - a a - B(a) + b - a
az
(8.4.48)
and then
1
T1 = - -
r° Kl (a) (1 - B'(a))d a
aZ
where
R 1 (a )
I
a-B(a)-a+b
k A(B(.))
H(a B( )) d a '
Q
(8.4.49)
(8 . 4 . 50)
The solution in the domain bounded by BCB'B is (8.4.45) where
Tl is given by (8.4.49).
8.4.7
The Wing with Dependent Subsonic Trailing Edges
For a wing with subsonic dependent trailing edges (fig. 8.4.11) the
solution in the zone AHF is determined by the formula (8.4.9), the
solution in the zone ABPH by the formula (8.4.15), the solution
in the zone FHA'E by a formula analogous to (8.4.15), the solution
in the zone HF'IA' by the formula (8.4.19), the solution in the zone
BCE'IB by the formula (8.4.45), where T1 is (8.4.48), and the solution
in the zone E113'DE by the formula (8.4.29), where T is (8.4.44).
It remains to determine the solution in the zone IE'B'I, i.e. in the
common zone of influence of the subsonic trailing edges.
Noticing that the curvilinear integral does not vanish on BM' and
WE and utilizing in the first case the expression from (8.3.74), and in
338
THE SUPERSONIC STEADY FLOW
the second case the expression from (8.3.73), we deduce that the integral
equation has the following form:
N(a , Q)
k
TJ , (a -
k Of
+7r 8b
0)3/2(b - /3)3/2 a d p+
N( a)
al,a)
+k
f
ir 8a
AruE
0
- B(a)
(B'(a) - IId a
a-a b-a-B(a)+a+
a - A(#)
(8.4.51)
[1 - A'(Q)]d 0
b-(3 a-b-A(O)+#
the integral on BM' representing in fact the integral with respect to
a on the interval (a2, a) and the integral on WE representing the
integral with respect to 13 on the interval (b, b2).
03
b
b3
M'
a.b)
O-
Fig. 8.4.11.
In this case, the point M(a, b) is in the common zone of influence
of the subsonic leading edges AB and FE. Hence the double integral
may be inverted according to the formula (8.4.19), D2 representing
the shaded domain from 8.4.11, i.e. the domain bounded by the curve
THE THEORY OF CONICAL MOTIONS
339
Af AI"AI""A1I,` AI'A1.
N(a,b)=
1
%TA'
ff ,
n
V
H (a' )
da'db'+Tt+T,
(a - a')(b - b')
(8.4.52)
where Tt has the expression (8.4.43), and T, (8.4.27).
Setting Al AI"(a -+ A(b)), the integral on BM' vanishes (as we
can see on the figure), such that from (8.4.52) one obtains N(A(b), b) =
= N(b). Imposing the Kutta-Joukovsky condition, we deduce that T
has the form (8.4.43). Similarly we deduce that Tt has the expression
(8.4.49).
Now the problem is completely solved.
In the end, it is at pleasant duty for me to mention that for elaborating
this section I utilized especially Homentcovschi's paper [8.16] and the
license thesis of my former student Luminita Berechet [8.2].
8.5
8.5.1
The Theory of Conical Motions
Introduction
The theory of conical motions was initiated by Busemann in 1943,
18.31. It refers to wings bounded by conical surfaces with the vertex in the
origin of the system of coordinates, the body being placed downstream.
The surface of such a body is a smooth surface consisting of half-lines
issuing from the origin and leaning on a closed curve situated in the plane
x = 1(xi = L I). According to the boundary conditions the velocity is
constant along every half-line passing through the origin and belonging
to the boundary of the body. The hypothesis of conical flow leads to the
assumption that the velocity has everywhere in the fluid this property.
We have therefore
v(mx, my, mz) = v(z, y, z)
(8.5.1)
for every in . real and positive. It means that the velocity is a homogeneous function having the zero degree. Under this assumption the
equation of the potential becomes simpler, the unknowns depending not
on three but on two variables. After Busemann, many authors (Langerstrom [8.22]. Germain [8.11], Poritsky [8.28], Ward [8.34], Heaslet and
Loniax [8.15], Iacob [8.18], Carafoli [1.5] $.a.) have contributed decisively
to the development of this theory. In all this theory, which will be called
the classical theory, we make the hypothesis that the motion is conical.
340
THE SUPERSONIC STEADY FLOW
Starting from the lifting surface equation in a supersonic stream, one
may prove that if the wing is conical, then the solution of the integral
equation is conical. For the equation (8.3.30) this thing is done in (8.5],
and for equation (8.3.71), in [8.161. In the present subsection, utilizing
the solution from the previous subsection, we shall give the solution of
the conical motions by particularization. We shall also give the basic
elements of the classical method, because they may be obtained directly,
without knowing the solution of the lifting surface equation.
8.5.2
The Wing with Supersonic Leading Edges
We assume that the surface of the wing is a conical surface. From
z = h(x, y) it results that Az = h(ax, ay) and, with A = (l/x),
h(x ,y ) = xh (1 , x ) = xg (x )
(8 . 5 . 2)
We deduce therefore
hr =.9 (x)
- r9 \x)
and then
H(a, b) = F (a )
Fig. 8.5.1.
.
8.5.3)
341
THE THEORY OF CONICAL MOTIONS
We shall consider now a wing with supersonic leading edges (fig.
8.5.1). Denoting by b = mla the equation of the edge OA and by
b = m2a the equation of the edge OF, it is obvious that m1 < 0,
m2 < 0, because on OA we have a > 0 , b < 0, and on OF, a < 0 ,
b > 0. Since the entire domain D is only in the zone of influence of
the supersonic leading edge, for every M(a, b) the solution is given by
the formula (8.4.9) where D1(a,b) is the domain limited by the curve
OAMFO, and H(a',b') will be replaced by F(µ), where b' = µa'. We
have therefore to put b = ma and to replace the variables b and b`
by m and p.
For M in the zone OCD we shall denote N = N2
N2 (a, ma) = N21 + N22 + N23,
(8.5.4)
where
k 1., Y-f*Jo
N21(a,m) --
1
-
ad 'a'
\J°
k
+
N23 (a, m)
m F()
1
N22(a, m) =
aa')(al-c)}dµ, c=
(a
Pa,.
'dµ+
(a - a) (c - a)
r°O F(µ)
r °'d a'
1dµ ,
Fv
14lJco
oo
aa''-c))df
(a
(8.5.5)
N21 representing the integral on OAA', N22 on OA'MF' and N23
the integral on OF'F. Performing the calculations we find:
a
irk
N21 = -
µ
m+11
C,
arctan
it
m
-
VM--\
Jt
/I F(µ)d µ
N23 = - a /"'.W 2 /m+p
ak J7-71 arccot rr m -
N22
a
irk
f
o
µ
,
)F(P)d,
(8.5.6)
\m+µ '+' -'1F(µ)dµa.
2µf If - /I
it
It is obvious that
f(x,Y) = (Oa + 8b)N2
(8.5.7)
342
THE SUPERSONIC STEADY FLOW
is constant on every half-line issuing from the origin. The flow is conical.
If M(a, b) is in the zone OBC, then the solution is
kJmt F(-) Jco
N2(a, m)
(a
aa')(a' - c) )
dy
(8.5.8)
a
f
m
F(p) m+1dp
M17-p
A
and if M is in the zone ODE, then
Ns(a, na) -
1
wk
-a
8.5.3
(p)
z FlN
f.
V---14
a d a'
C
(Z
(a - a')(a' - c)
(8.5.9)
"'2 P((µ) m+{A
dµ,
Im
11
fl
V-
The Wing with a Supersonic Leading Edge and with
Another Subsonic Leading or Trailing Edge
Further we shall consider a wing having a supersonic leading edge (the
edge OB from figures 8.5.2) and another subsonic leading edge (fig.
8.5.2a)), or a subsonic trailing edge (the edge OE from fig. 8.5.2b)).
In this case, the solution is obtained with the formula (8.4.15) where
Dl is the domain limited by MFF'AM. For M belonging to the
interior of Mach's cone i.e. M in the zone OCEO, noticing that the
equation of the line FF' is a= d (it is obtained from the intersection
rn
of b' = b with b' = m2a') where d =
a, we deduce
N2 (a, m) w
i
m2
F'(µ) d
ak
mi
+.1, J0
1
+ ;k J,,,
µ
ad a'
(a - a')(a' - c)
d
7
F)dµJ
' F(p)d µ
rjA
(a
sa a'
+
+
(8.5.10)
a'd a'
Jd
(a- a) (c - a') '
c being defined in the previous section. The interior integrals are elementary.
THE THEORY OF CONICAL MOTIONS
343
Oa
b)
a)
Fig. 8.5.2.
For M in the zone OBC noticing that the intersection of the line
b' =pa' with b' = b has the abscissa c, we obtain
Ni(a,m)
- ak
a
ffl
dJ
,
(8.5.11)
rm
F(p) m + µd
/ml 7
8.5.4
(a-&)(a'-c)
L
The Wing with Subsonic Leading Edges
When the two leading edges are subsonic, it is difficult to utilize the
solution from the previous subsection. We shall use therefore Homentcovschi's idea concerning the direct integration of the equation (8.3.73).
Assuming that N(a, µa) has the form
N(a, µa) = aN(µ),
(8.5.12)
344
THE SUPERSONIC STEADY FLOW
the equation we have in view reduces to
k
a2N(µ)d a d µ
_
47r JJD (a - a)3/2(ma - Na)3/2 - F(m)
(8.5.13)
D being the shaded domain from figure 8.5.3. With the same notation
for c (= ma/p), the equation (8.5.13) may be written as follows
/°
, µ3/2
fmjV('2
4a
0
a2da
d is
(a - a)3/2(c - a)3/2 J
(8.5.14)
N(p) J.r`
+k
47
r
a2d a
{a - a)3/2 (c - a)3/2
µ'/2
Obviously, in the first integral a < c, and in the second, c < a.
The interior integrals are considered in Hadamard's Finite Part sense.
Taking into account the formula
da
ml
f
aa
d,,=-'2
a a
0 (a
f
a)3/2
d a,
(8.5.15)
given in (D.4.3), the equation (8.5.14) may be written as follows
k
a2do
I -N(p)(02
)dp+
3/2
f
a
a
(a
a)(c - a)
,ni
o
k,
14
(8.5 .16)
ml N(
492
cf
143/2
0
2
a2da
ld it
(a - a)(c - a) / µ
We notice that in the first case (c > a), we have
f
- 4ac In fc + f
- 34 (a + c) fac+ 3(a + c)2
8
f - fa
a2d a
(a - a)(c - a)
Jo
and in the second (c < a),
f V(a -a)(c-a)
a2d a
`
_
3
4(a+c) ac+
3(a + c)2 - 4ac
f- + f
In
8
.
Vc-
(8.5.17)
The results are the same if we put under logarithm (/ - . Performing the calculations, it results that the integral equation (8.5.16)
may be written as follows
k
T' N) In
-2 mµ(M
+14)2}dµ=F(m)
1
(8.5.18)
THE THEORY OF CONICAL MOTIONS
345
0,
(N
Fig. 8.5.3.
for ml < m < m2. We put N(p) = µNI(µ).
Denoting
H(m) =
k
.,
In
.IA
/./ + ///7;,
t%'
V''
d µ,
on the basis of the equation (8.5.18) we obtain the following differential
equation
m2H"(m) + mH'(m) - 4H(m) = -F(m).
(8.5.20)
The homogeneous equation has the linearly independent solutions v/
and l//. Hence, the general solution of the equation (8.5.20) is:
H(m) = 2cl frra --
202
+ Fo(m),
(8.5.21)
cl and CG2 representing constant which have to be determined, and Fo
representing a particular solution of the non-homogeneous equation.
From (8.5.19) and (8.5.21) we deduce the following integral equation
for Ni
k fm
Nl(N) In
/ + `dµ = 2c1 VG
+ Fo(m).
(8.5.22)
Deriving with respect to m we obtain
k
Ni (l+) d
7r,'µ-m
p = cl +
- + /o (m) ,
m
(8.5.23)
THE SUPERSONIC STEADY FLOW
346
which is the classical equation of the thin profiles. As we have already
observed, from the definition of N it results that Nl (N) vanishes on
the leading edge. The solution of the equation (8.5.23) which vanishes
for m = ml and m = m2 has the form (C.1.14) with a condition
having the form (C.1.13). Taking also into account (B.5.8) it results
' mZ
1
fFa(µ)
Nl(m) = -7r (m - ml)(m2 -
+c2
m1){m2 -
dp
N-m+
(m - ml)(m2 - m)
m mlm2
(8.5.24)
The condition (C.1.13) will give
x cl +
rm
C2
mlm2
Voµ
+f
m, ( nl)(m'l -P)
d y= 0
(8.5.25)
and will be useful for the determination of the constant cl, after determining the constant c2. In fact, the constant cl is of no interest.
The constant c2 which intervenes effectively in the solution (8.5.24)
will be determined imposing for the solution to verify the integral equation (8.5.18). This condition is necessary because the solution was determined after some derivations.
Writing the equation (8.5.18) as follows
2
T",
Nl (µ) d µ+
N1(l1) d
1
µ-m
2m
(8.5.26)
- 4m jrn, ' Nl (i)K(m, it)d /1 = - k
F( m)
M3/2
where K(m, µ) is the symmetric kernel
K(m, u) =
+
1
Inl%FM-VjAI'
(8.5.27)
we notice that the equations (8.5.24) and (8.5.26) will determine the un-
knowns Nl and c2. The replacement of Nl from (8.5.24) in equation
(8.5.26) leads to difficult calculations. It is necessary, for example, to
know the formulas for interchanging the FP (Finite Part) with PV
(Principal Value) and PV with FP (the formula of Poincar&Bertrand
[A.27]).
FLAT WINOS
347
The equation (8.5.26) may be also solved numerically using the
Gauss-type quadrature formulas (because N has the form
N1(µ) =
(8.5.28)
(µ - ml)(m2 - A)n(p),
n(p) representing the new unknown).
Flat Wings
8.6
The Angular Wing with Supersonic Leading Edges
8.6.1
For the flat wings having the angle of attack e we have F = -e.
The theory of the angular airfoil with supersonic leading edges may be
obtained from 8.5.2, putting F = -2. Since
m
6
m=-,
U= b,W, c=-a,
a
it
(8.6.1)
it results
°
ad a'
(a-a')(a'-c)
Jo
N21 (a, m)
a r 1 + m) arctan F-!A+aFT, ( 8.6.2)
-
k-v [ m -mli arctan
-
(8.6.3)
Similarly one obtains
N23
m)
2ac (M -m 2
arctan
(8.6.4)
We have also
a
L
a'd a'
(a-a')(c-a')
-a
Iv-
m + a rI + ml
1i)
V u 2`
VrM +
and then, from (8.5.5) or directly from (8.5.6),
N22(a,m)ael,I J' ffm-1
kzr
2 .
m in Vm- +f ddµ
1
+
µ/
i
i- V M
(8.6.5)
THE SUPERSONIC STEADY FLOW
348
In I we make the change of variable
In this way one obtains
l1 = x and we denote vM- = q.
I=I1+g212-2g13,
(8.6.6)
where
r4
11= J In x+qdx+ i:° In x+qdx=111+112,
12-
Ix-g1
ix - qj
v
2In
x
13= 11 d. +
f
x
Ix - q)
=131+132.
The integral 11, is elementary (it has an integrable singularity). One
obtains
(8.6.8)
111 = 2q In 2 .
The integral 121 has a strong singularity in x = 0. It must be
considered in the Finite Part sense. On the basis of the formula (D.2.2)
we have
12i =
f9ln(x+q)-1n(q-x)-2x/q + 2Inq
x2
q
Integrating by parts, one obtains
121 =
2(ln2q+
q
1).
(8.6.9)
Using (D.2.3) we deduce
(8.6.10)
131 = In q.
For calculating 112,122 and 132 we make the substitution x = 1/y
and we utilize the results (8.6.8) - (8.6.10). One obtains 13 = 0,
11
2
q( -+1
q
=2g1u2+2
In
(8.6.11)
2
12=2g11n2+1) +g1n2,
such that
N22(a, m) = -, /m-(2 2 + 1) .
(8.6.12)
349
FLAT WINGS
In this way, taking into account that m = b/a, the formula (8.5.4),
together with (8.6.3), (8.6.4) and (8.6.12) give
a
b
N2(a, b) _ -
- Zee (
-m
VM1
am
b
_n2-aarctan
v72
a
-b -
arctan
2
-
)!
-
ab)
v a (2 in 2 + 1) .
(8.6.13)
This is the solution when M is in the zone limited by the characteristics
OC and OD .
If M(a, b) is in the domain limited by OD and OE, then we use
(8.5.9). We deduce that
N3 (a, b)
k
£ 7n2 (m2a
- b),
(8.6.14)
and if M is in the zone OB, OC (fig. 8.5.1)
N1 (a, b) _ - k
8.6.2
b
mla
(8.6.15)
The Triangular Wing. The Calculation of the Aerodynamic Action
In order to obtain a finite action, it is necessary to consider a wing
having a finite area. We assume that in the physical plane it has the
triangular form from figure 8.6.1. In order to obtain a well determined
wing we must give the coordinates of the points A and F. Let mi be
the inclination of the line OA and ai the ordinate of the point A in
the frame of reference Oa,J and m2 the inclination of the line OF
and a2 the ordinate of the point F in the same frame of reference.
Then the equations of the edges OA and OF will be b = mia
respectively b = n2a, and the coordinates of the points A and F
respectively (ai, bi = ml al) and (a2i b2 = m2a2). The equation of the
line is
b = (m1 - m3)a1 + mia,
where
m3
_
7Tb2a2 - miai
a2 - a1
(8.6.16)
(8.6.17)
350
'rilE SUPERSONIC STEADY FLOW
F(a,.b,)
Ap
.e
.0
D(O.b4)
A(a,.b,)
Fig. 8.6.1.
Denoting by (a3, b3) the coordinates of the point C and by (a,,, b4)
the coordinates of the point D, we deduce
a3=1-m1,
b3=0; a4=0, b4=(m1-9n3)a1.
(8.6.18)
As we already know, the lift is given by the formula
=
L
- ffD(x.y) Bpi I d x, d yi = -PmU,2,,,Lo J1D(=.y) epO d x d y,
and the lift coefficient cL, by the formula
CL = 1 L
2P0U00A
,
(8.6.19)
where A is the area of the wing and Lo, the reference wing.
Taking (8.3.69) into account, passing to coordinates on characteris-
FLAT WINGS
351
tics and applying Green's formula, it results
CL = - A
rf
2 o JJD(= v)
2L
A
f
f (x, y)d x d y =
8 N(x,
y)d x d y
ax
(8.6.20)
A
IJD(a, b)
(8a + ab
)N(ab)da d b =
- - Ac /eD(a,bjM (b - a)
IA 1,
where
1=11 + 12 + 13, It = JOA+AC+cd"1 d(b - a),
(8.6.21)
12
13 = JoD+DF+F 3 d (b - a).
- JO C+CD+DO-2 d (b - a),
Taking (8.6.15) into account, it results
Nid(b-a)loA=0
N1d(b-a)IAC_N1d(b-a)
b=(ml -m3)a1+msn
-E(m3 - m1)(m3 -1) (a
k
m1
N1d(b-a)I co=Nid(b-a)lb_o-
- ai)da
kmlada,
such that
I1
= _E(m3 km1)(m3 - 1)
at)da - £'nl
3
ada =
To
S(m3 - m1)(m3 - 1) (a3 - at)2
k -m1
2
E -m1 a3
k
2
(8.6.22)
In the same way one calculates 13. Taking into account (8.6.14), one
352
THE SUPERSONIC STEADY FLOW
obtains
13
- k -m2 2
E
443
E(n3 - 1)(m2 - 1n3)
k --r n2
a!+
2
(8.6.23)
+Eata2(m3 - 1)(m1 - m3)
k -rn2
and the problem of calculating c1, is solved.
8.6.3
The Trapezoidal Wing with Subsonic Lateral Edges
We assume that the projection of the wing (which is flat and has
the angle of attack s) on the plane xOy is the isosceles trapezoid
ABEF from figure 8.6.2, having the bases 21, 2L and the height h
(dimensionless quantities). The direction of the unperturbed stream is
perpendicular to the bases.
Fig. 8.6.2.
We consider the case a <,u. The leading edges AB and FE are
subsonic and the edge AF is supersonic. We also assume that h is
such that the subsonic edges are independent (for the sake of simplicity).
Obviously, in this case the fluid motion is not conical. In the domain
D2 the solution may be obtained from (8.4.9), and in D1 and D3
from (8.4.15). For developing these solutions, we must characterize the
domains in coordinates on characteristics. We must specify at first the
FIAT WINGS
353
physical coordinates. We introduce the parameter m by the formula
0<
h
tangy =
< tan p
.
(8.6.24)
It results therefore 0 < m < 1. The distances dl and d2 are defined
bu the formulas
d1=lk,d2=d1+h.
(8.6.25)
The equations of the straight lines AB and FE are respectively
-y=l+ m(x - dj),y = t + m(x - d,).
(8.6.26)
The equation of the wing (see figure 8.6.3) is
d12d2 )
x
(8.6.27)
the function being defined on the domain D + D1 + D2 + D3 from the
XOy plane:
d1 <x<d2
(8.6.28)
-yi(x) < y < y1(x),
where
(x - d1).
y1(x) = e1
(8.6.29)
k
Fig. 8.6.3.
It obviously results
H(x, y) = -c.
(8.6.30)
Passing to coordinates on characteristics we put
b-a
a+b
2
'y
2k
(8.6.31)
It results
H(a, b) = -e
(8.6.32)
354
THE SUPERSONIC STEADY FLOW
and the following equations for the sides of the domain D
AF:a+b=2d1,
BE:a+b=2d2
(8.6.33)
AB:a=2d1+mob,FE:b=2d1+moa,
where we denoted
mo
=
i+ m
(8 . 6 . 34)
For the vertices we deduce the following coordinates:
A = (2d1, 0), B = (d2 + kL, d2 - U),
(8.6.35)
F= (0,2d1),E= (d2 -kL,d2+kL).
Taking (8.4.9) into account, we deduce for the solution in D2
a
kir
N2(a,b)
da'
1-b
db'
f
a - a' J2d1 -a, %Fb - b'
(8.6.36)
=-1(a+b-2d1).
The solution in D1 is given by (8.4.15). Noticing that
A(b) = 2d1 - b,
B(a) = 2d1 - a,
(8.6.37)
it results
Nl
(a'
er
b)
krr
d b'
, -4
b
a
da'
b' Jet -V a - a'
(8.6.38)
Obviously, this expression may be obtained from (8.6.36), changing a
with b. Hence
N1(a,b) = -1(a + b - 2d1)
(8.6.39)
and an identical expression for N3(a, b) (because of the symmetry).
According to (8.6.20), for cL we have the expression
2
CL = - kA
(8.6.40)
where
I= POD N(a + b - 2dj)d (b - a)
-
(a + b - 2d1)d (b - a) = (1- m02)(d2 - kL)2.
AB+BE+EF
FLAT WINGS
355
The Trapezoidal Wing with Lateral Supersonic Edges
8.6.4
We consider an isoeoelea trapezium with the bases having the length
2L respectively 2t perpendicular on the direction of the stream at
infinity (fig. 8.6.4). We Introduce here again the parameter m defined
by the relation:
k-tana=Lht>tanw=k
(8.6.41)
F. 8.6.4.
Obviously, m > 1. Denoting by dl the distance from the small
basis to 0 and by d2 the distance corresponding to the great basis,
we obviously have d2 = dl + h, and for the c arte ian coordinates of the
vertices of the trapezium
A = (d,, t), B = (da, -L), F = (dl, t), E = (d2, L)
(8.6.42)
The coordinates on characteristics are obtained from the formulas,
a=x-ky, b=x+ky.
No ti c i ng
th a t
(8.6.43)
d2 = kL, we ded uce
A = (di + kt, dl - kt),
B = (2kL, 0)
F = (dl - kt, dl + kt),
E _ (0, 2kL)
(8.6.44)
356
THE SUPERSONIC STEADY FLOW
Using the notation
d1-k£
d1+2h-k8
k
2
k-1
m+1
(8.6.45)
and observing that from figure 8.6.4 we have the compatibility condition
ao < p, which implies
dl = tan ao < tan p =
1
dl > kt,
(8.6.46)
we deduce 0 < k < 1. Now, the equations of the sides of the trapezium
may be written as follows
AB :a+kb=2d2iFE: k-la +b=2d2,
(8.6.47)
BE:a+b=2d2,AF:a+b=2d1.
The entire leading edge is supersonic. The solution may be expressed
by means of the formula (8.4.9). To this aim it is necessary to specify
the functions a = A(#) and A = B(a). We have:
- on the edge BA,
b .- BI(a) =
-- on the edge AF,
2d2
a,
a = Bi 1(b) = 2d2 - kb,
(8.6.48)
b = B2(a) = 2d1- a,
a=
t
1
}=Li_b.
We must also observe that for determining the lift coefficient we do
not need N(a, b) on the entire wing, but only on the trailing edge BE.
Indeed, this may be expressed with the formula (8.6.20), and N on the
leading edge BAFE vanishes as we have already mentioned in formula
(8.3.68). The domains of influence are (fig. 8.6.3) :
D1 = ABA'A,
D2 = AA'A"A,
Do = AA"F"A,
D3 = FF"F'F, D4 = FF'EF
Hence, we shall put
I=11-+-12+I3+ 14,
(8.6.49)
FLAT WINGS
357
where
r
it = J
BA'
I. = r
JA"F"
14 =
JF"F.
12 = f
Ni (a, b)d (b - a),
JA'A"
No(a, b)d (b -- a). 13 = J
Ni (a, b)d (b - a)
"F'
N2 (a, b)d (b - a),
N3(a, b)d (b - a),
(8.6.50)
.
Using the formula (8.4.9) and the equations (8.6.47), we deduce
__ 8
N1 (a'b)IBA' =
N2 (a, b)IA'A#l
a
k
,l Bt I(b)
E
Tr -
Bi (b)
No= -h, N4=
-,-(I -
da' Ilrb
dillu
a -a' B,(u') vb - v 1,9A'
da'
a - a'
b
JB2(n')
_eb(l-k)
db'
b - b' L'A" - ... ,
)a
After elementary calculations we deduce:
E(I - k)
11 =
14
and finally,
k2
k.
f
d2-"-h
bdb
e(1 - k) (dl - kf)
= k2 Vk-
2
4Eh
= - E(1- k) (d1 -2kt)2 ' to =-k2
(k£ - h),
2'
kv-.
Chapter 9
The Steady Transonic Flow
The Equations of the Transonic Flow
9.1
9.1.1
The Presence of the Transonic Flow
We call transonic flow the flow which is subsonic in a domain of the
space and supersonic in the adjacent domain. One demonstrates (for the
potential flow - see [1.21] pp 517, 518) that the equality v = c comes
true in E2 only on curves separating the domains where the flow is
subsonic from the domains where the flow is supersonic, and in E3 on
the surfaces which separate such domains. The name of transonic flow
was introduced by Th. von
in 1947.
In the present paper the transonic flow has been encountered in
several situations.
At first, we have to mention the one-dimensional flow [1.11] §4.5.
The formula (4.5.8) which gives the variation of the velocity against
tile variation of the cross section indicates that, in the subsonic flow
(Al < 1). the velocity increases when the area decreases and decreases
when area increases (like ca in the incompressible fluid), while in the
supersonic flow (M > 1) the variations are produced in the same
sense. This circumstance leads to the conclusion that in a tube having
the shape from figure 9.1.1 the flow may become transonic. To this aim
it is sufficient for the upstream subsonic velocity to have the critical
value in the section of minimum area. Further since the area of the
section increases, the velocity also increases, remaining supersonic.
In the linearized theory we deduced for the aerodynamic action the
formulas (3.1.33) and (3.1.34) in the subsonic case and (8.1.9) in the
supersonic case. It is obvious that these formulas are not valid in the
vicinity of A! = 1. For the flat plate these formulas become (3.1.35)
and (8.1.22). The figures (3.1.3) and (8.1.3) are very suggestive.
In the cause of the subsonic flow with great velocity past thick bodies
like in figure 9.1.2. the flow may become transonic. Indeed, considering
THE STEADY 'TRANSONIC FLOW
360
Fig. 9.1.1.
the flow between the streamline which includes the boundary and a
neighbor streamline L, we shall find that the flow is like in a tube.
Since the domain between these lines narrows because of the body, it
Fig. 9.1.2.
follows that in the vicinity of the body the flow nay become supersonic.
The transition from the supersonic flow to the subsonic flow is performed
by a shock wave S according to the scheme described in 1.3.6. Until
S the flow is transonic. We shall deduce in the sequel the equations
which describe this flow. The flow with great subsonic velocity past
thick bodies is described by the scheme from 9.1.3.
->
V<C
y« r- Loctaau
Fig. 9.1.3.
Finally, in the supersonic flow, for great velocities, practically in the
hypersonic regime, it appears, as we noticed in 1.3.6, detachedor at-
361
THE EQUATIONS OF THE, TRANSONIC FLOW
tached shock waves (figure 1.3.5). Behind these waves the flow is transonic (it passes from the subsonic regime (A,12 < 1) to the supersonic
one (A12 > 1)).
As we could see, in modern aerodynamics the transonic regime is
frequent. So one explains the great number of papers devoted to this
subject in the last years. We mention especially the papers of Bauer,
Garabedian and Korn [9.1] devoted to the theory of minimum drag
wings. There are three dominant methods for studying the transonic
flow, namely:
1° the hodagmph medwd, suitable only for the plane steady jet flow
(see for example Ferrari and Tricomi [9.11], Manwell [9.30], [9.31] etc.);
2° direct analytical methods, based on the semi-linearized equation
of the potential. They lead to integral equations which may be solved
numerically;
3° numerical methods applied directly to the system of equations
which describes the fluid flow (we mention especially the finite elements
method).
In this chapter we present some direct analytic methods.
9.1.2
The Equation of the Potential
The reasoning based on the assumption that the independent variables x, y, z have the same role in the structure of gyp, (utilized for
deducing the equation (2.1.39)), is not valid for the flow in the vicinity
of M = 1. Indeed, in this vicinity M2 - 1 becomes itself a small
parameter. If, for example AI2 - 1 = O(E), then for V_: = 0(cp) it
results +p.y and W.'. = 0(E2). One imposes an analysis of the order of
magnitude of the perturbations depending on the geometry of the body
and the conditions which determine the flow (Mach's number Al, the
thickness and length parameters, the angle of attack, etc.). In fact, the
idea that the variables y and z do not behave like the variable x.
results from the special property of the Ox axis (which is parallel to
the direction on the unperturbed stream). We shall introduce therefore
the variables
y = u(E)y.
4 = V(C)Z,
(9.1.1)
expecting for vv(e), like for q(r) from the expansion
Or, Y. Z' `) = U (x + T (E),(x, J, <_) +
... ]
to be determined by comparing the orders of magnitude.
(9.1.2)
362
THE STEADY TRANSONIC FLOW
Coining back to the linearized theory 2.1, we notice that for Al =
= 1 a catastrophe is produced (it disappears terms from the equations).
The lift and moment coefficients become therefore infinite. But this
catastrophe has only a mathematical nature, not a physical one. It is
determined by the fact that in the vicinity of the value Af = 1, the
order of magnitude of all the first order derivatives is not the same (E).
One imposes therefore (9.1.1).
It results
0r=U(1+1150:+...),OS,=U1)v
+...,O:=Urpi
(9.1.3)
Orr = UtWrr,
Ory = UTJVP,,y,
0, = Ugv2 ;OW .
From (1.2.17) we deduce
c2 = c2 -
(7 - l )U2>);pr + 0(112) ,
(9.1.4)
and from (1.2.16) written explicitly as follows
(r.2¢r).o +(C?-0y)Oyy-20=.ysOv+...=0,
(9.1.5)
we deduce
[1 - M2 -(y + 1)(M2 -
(y +
+ [1 _ (y - 1)(Af
0(7)2)l(p=r+
(y - I)W=)v2 y-
(9.1.6)
- 2M2vrl2"o + 0(1j2v2) + ... = 0.
For a fixed Al , we see that the equation is consistent if ii --+ 0 when
q-'0,so
v2, t), 1- M2_ 71.
(9.1.7)
We introduce now the boundary condition. If
z = eh.(x, y)
(9.1.8)
is the equation of the perturbing surface, imposing the condition to be
a material surface i.e.
Eh=dOr + Eh,,Oy = &
which implies, taking into account (9.1.2)
Eh. =
(9.1.9)
THE EQUATIONS OF THE TRANSONIC FLOW
363
whence
E = 9V.
(9.1.10)
Taking (9.1.7) into account, we deduce
t)=E2'3
V=Et13
(9.1.11)
When M - I
we have to compare. in (9.1.6) the terms of order
immediately superior to those which gave (9.1.7). It results I - M2 =
= Kv2 whence
K-
_ hl2
1
.
(9.1.12)
.
K is called the parameter of the transonic similitude. In this way, the
first approximation from (9.1.6) (the dominant equation) is
[K - (7 +
rpyp + 4p: = 0.
(9.1.13)
This is the equation of the transonic flow (the equation of the potential).
It is elliptic if Cpl < K/(7 + 1) and hyperbolic if gyp= > K/(y + 1).
The relation V_ = If/(-y + 1) is verified on the surface where
V2 = c2. Indeed, using the notations (2.1.3), and taking (1.3.32) into
account, the condition V2 = c2 becomes
V2=c2=co-7 v;2=cam- 1 o (Vt2-U2).
(9.1.14)
Here, the dominant relation is
u2 1+2 u, } = c2 - (7 - 1)U2vu,
(9.1.15)
T
whence ii = K/(7 + 1).
Now it is clear that the non - linearity is necessary for making this
transition possible.
The first study of the transonic flow has been performed by von
Karmsui [9.26). By various methods the problem was investigated by
Ovsiannikov [9.531, Guderley [9.15), Cole & Messiter [9.6[ etc. Cole's
study from 1975 relying on the method of perturbations was continued
by the same author in 1978. In the last study one proves that if we
denote by s the thickness parameter and we set for the cross sections
y-Et13y, =-E113' ,
then the potential 0 has the following structure [9.54]:
4)(i',y,4;Al ,Q,b,b) = U[z+E2/39(x,V,z;K,A,B)+
(9.1.16)
+E413 t'2(x, y, Z; K, A, B) + ...,
,
364
TilE STEADY TRANSONIC FLOW
where K is the transonic parameter (9.1.12), A, the parameter of the
angle of attack = aft, and 13, the span parameter = 6e113.
For p one obtains the equation (9.1.13).
9.1.3
The System of Transonic Flow
It is rigorous to perform the asymptotic analysis on the system of
equations and not on the equation of the potential which has been obtained from the system by derivation with respect to the x, y, z coordinates. We present here such an analysis which was performed together
with professor A. Halanay in the years '80. We utilize the coordinates
y and r in the form (9.1.1) and we denote
E)
=v
\x' V(E) , +'7
(E)
)'
h(x, y,
E) = h (x,
(9.1.17)
It results
r"(X,j/,E)
= it , x,
h,
V(£)
(2-,
y(E)/ V(E)
and the boundary condition
Elts(x, y) 11 + u(x, y, Eh(x, y))] + Eh' ,(x, y)v(x, y, eh(x, y))
_
= w(x, y, Eh (x, y))
becomes
X.
(r, y,
c) [1 + u (1', lI, - h(x, P, E), E)] +
+sv(E)hV(x, N, 0V (x, /,
TU
_
S,fj,
E
_
v(E)-(x'v,
ll
v(s)h(x,p,E),EJ
The dominant term in the first member would be Ehx(x4, E) if 9
would not disturb. But for a small p we have
hr (x, y) E) =1t= (T,
vVE) )
=
hx (x, 0) + Y( h=y(x, 0) +
r)
...
From the physical conditions of the problem it results that le, (x, 0) 74 0.
365
THE EQUATIONS OF THE TRANSONIC FLOW
The condition (9.1.18) suggests that the right hand and member has
the order of c. Hence,
w (x,
v(E)
h(x, y, e) J = Eii (X,
h(x, b,
v{s)
E)>
E
(9.1.19)
We assume that this is valid in the entire domain occupied by the fluid,
i.e.:
i (x,y,',E) = ew(x,y,;F,E).
(9.1.20)
Taking (9.1.19) into account, from (9.1.18) we retain in the first
approximation, under the hypothesis that v(E) --+0
111 +u(x,y,0,e)] = ii (x,y,0,E)
(9.1.21)
Using the notations (2.1.3) the system which determines the perturbation produced by a fixed body in the uniform flow of a compressible
fluid characterized by M is determined (see (2.1.10) - (2.1.13)) by the
system
(9.1.22)
(1 + p)M2p = (1 +7M2p)p
M2p + (1 + yM2)div v = 0
(9.1.23)
(1+p)v+gradp=0
(9.1.24)
where
[(1+u
a
8
)8x+vp,....
(9.1.25)
We notice now that from the structure
p x,
(9.1.26)
v(E)
it results the formulas
Op
Ox
_ Op ap _
Ox ' 8g
Op Op
v(e) 8y ' 8~
I
1
Op
v(e) Oz '
(9.1.27)
which will be replaced in the projections of the equation (9.1.24) on the
axes of coordinates. In this way, the projection on Oz gives
r
here
Comparing
thu8x
8w
Op
+evVOw +e2vw
+v !=0.
(9.1.28)
the dominant terms we deduce that
P(x, y, =, E) =
v(E)
'lx,
, z, £)
1
(9.1.29)
THE STEADY TRANSONIC FLOW
366
and from (9.1.28) one retains
(1+p)(1+u)8 +-=0.
(9.1.30)
Analogously, from the projection of the equation (9.1.24) on the Oy
axis, it results
+v(E)v-+EV(E)w-J +E
v8/
=0,
(9.1.31)
From this equation it follows
v(x, y, z, E) = 6(x, T, z, E)
and then
(1 +;5)(1 + u)
(9.1.32)
e + 5i = 0.
(9.1.33)
At last, the projection of the equation (9.1.24) on the Ox axis gives
(1 + P)
}- EYi
1(1 + u)
GIV
zi + EvUI
v(E)
= 0,
whence we obtain
11(X,
Y(E)
u(x,
(9.1.34)
and then
89 = 0 .
(9.1.35)
+ P) Fx + 8x
The behaviour (9.1.34) determines for (9.1.33) and (9.1.30) the forms:
(1
8; + 8y =0,
(1+P)LW
(1+P)WV +-=0,
8x
(9.1.36)
and the boundary condition (9.1.21) determines the equality
K(x,y,E) = 10(x,9,0,E)
which implies
hx(x, y) = w(x, y, 0) .
(9.1.37)
Knowing that M2 = 1 constitutes a singularity, we shall consider
in (9.1.22) and (9.1.23) M2 = I + µ and we shall keep the dominant
367
THE EQUATIONS OF THE TRANSONIC FLOW
terms for a small p . Utilizing the previous results, the equation (9.1.22)
becomes
l+YU1a+£vu+evrI
[1+-1(1+µ)EPI
11
p
+EVWF
K
whence we deduce
4
7
v e Pox,
,
(9 . 1 . 38)
and then
89 = ap .
(9.1.39)
8x 8x
Having in view the damping condition at infinity for the perturbation,
from the last equation we deduce
p =P-
(9.1.40)
Taking the relation (9.1.38) into account, it results that the dominant
parts in the equations (9.1.35) and (9.1.36) are
8
+8 =0, 8 + =0, 8 F=0,
(9.1.41)
whence it results
u=-p,
ex
-=0, -=0.
(9.1.42)
Z
Finally, from the equation (9.1.23) written as follows
(1+µ){I1+ _u) 2ff +EVv
V
ax
+£VW
+
+[l+ry(1+µ)ipj[+2+v]
=0
ax
ay
6F
we obtain, if we have in view
(µ+vu+µv
+'Y(1+µ)vpe
(9.1.41)1,
+(1+µ)EV(v
+10 Lv) +
ax
8Y
ft
4rV
&0
0.
368
THE STEADY TRANSONIC FLOW
The dominant part is obtained from the linear terms. We may write
therefore
P
\µ+;u}
49V
+7vpax+uj( O-V +
E
E
v
v
az-
J
0,
(9.1.43)
and the residual equation
(-K+u) 2E +YpBx+ a +a =0.
Bxp
(9.1.44)
At last, from (9.1.43), (9.1.44) and (9.1.42) one obtains
K=
v(£) _ £1/3
try
r?x
1 - M2
£2/3
tr =0.
(9.1.45)
(9.1.4G
)
This equation, together with the equations (9.1.42) constitutes the gen-
eral system of equations of the steady transonic flow. In the x, V, z
space the equations (9.1.42) give the irrotational conditionof the velocity of coordinates (u, u, is). Introducing the potential jp(x, ji, -_,F) by
means of the formulas
u = c0*,
V = cpy,
w = Pz'
(9.1.47)
one obtains (9.1.13) from (9.1.46).
9.1.4 The Shock Equations
In the case of the flow with shock waves, from the integral form of
the equations of motion (9.1.42) and (9.1.46), written in the conservative
form (by means of the div operator),
vi+(-u)y=0, vas+(-u): =0
r
f Kii
I
(9.1.48)
- L+-'iP
2
+i%+urf=0,
1 :
369
THE PLANE FLOW
integrating on every domain which contains the shock surface and passing to the limit as usually, it results
[i1n+:
[IKu
-
- Qulny = 0, 9wf ns - QiiOnr = 0,
7+
2
(9.1.49)
421% + Ovlny + OwOny = 0,
where n=, ny, n1 are the coordinates of the normal to the shock surface,
i.e.
n;r = (d7jdz)
ny = (d-zdx)
ns = (dxd-y), .
(9.1.50)
If, for example, the parametric equations of the shock surface are
x = x(A1, A2),
11=
A2),
z = {ai, 2) ,
then from n = da1z x da3X, it results
n=
9.2
"2
8a1 812
"Z
8a1
dJ11da2i
... .
(9.1.51)
The Plane Flow
9.2.1
The Fundamental Solution
\Ve consider. like in Chapter 3, that an uniform stream, having the
Mach number M is perturbed by the presence of an infinite cylindrical
body, with the generatrices perpendicular on the direction of the stream
which coincides with the Ox axis. The Oy axis is in the section
perpendicular to the generatrix. Let
y=h*(x),
1xI <1
(9.2.1)
be the equations of the profile determined by the cross section. Our aim
is to determine the perturbation and the action of the fluid against the
profile.
The flow is obviously plane. The velocity will lie in the xOy
plane and will not depend on the variable z. Taking into account the
orientation of the axes, we deduce from (9.1.37) the boundary condition
v(x, ±0) = h' (x),
jxI < 1.
(9.2.2)
370
THE STEADY TRANSONIC FLOW
According to (9.1.42) and (9.1.46), the perturbation will be determined by the equations
il = -n
i y - vs = 0,
(9.2.3)
Kit= + ij = ( + 1) uit= .
(9.2.4)
Using the change of variables
y=fl?i, u'=VY u,
(9.2.5)
we reduce the system (9.2.4) to
NO 8
ay-ax=0' ax
a
86
NO
2
gay=kaxu,
9.2.6
)
where we denote ry + I = 2kK3/2. In the sequel we shall integrate the
system (9.2.6) without writing the marks *;' any longer.
The fundamental solution of this system is determined by the equations
au
ay
av
8u
a
8u
- ax = ld(x, Y), ax + ay = k ax u2 + m (x, y) .
(9.2.7)
For the Fourier transforms u and v one obtains the formulas
-ial£+ia2n+kala2F
tl =
a2
a2
,
(9.2.8)
where F = F[u2 ], a2 = a + a2. We take into account the formulas
F-'
a2,
-
i
_ 2a In ro - e, ro =
!LA
x2 + y2
(9.2.9)
F-1
[F]
4",
u2 * e
11
t d ri,
tt
where u2 * c is the convolution product and
r =
+ yo,
xo = x - C YO=Y-71-
(9.2.10)
From (9.2.8) it results
1 mx + ly
u(x, y) = 21r x2 + 2 +
k
8J
ax
xo
rJ)
r2
d
(9.2.11)
1 My-ex
v(x, y) = 2ir x2 + y2
u2
s
+ 2 ax
fJasr
*1)
d d il .
371
THE PLANE FLOW
Obviously, the last integrals are singular. They have the shape (E.3).
Isolating the singular point (x, y) with a circle having the radius a and
setting f = xo/r and respectively yo/r we deduce that the condition
(E.5) is satisfied and the integrals are singular. Also, from the formula
(E.10) we getr
I = Ox / f z
=
+7) r0 d d rl =
2U2
JJ.
J=a JIz
(n)df dn,
II2
(9.2.12)
or, performing the calculations,
=Jju2(,q)dxd,7_,ru2(x,y)
17y0r4x0J
I
(9.2.13)
U2(t,11)x2
=-ff
9.2.2
2dOdx
dn.
The General Solution
Replacing the profile by a perturbing distribution defined on the segment (--1, +1) (the chord of the profile), it results the following general
representation of the perturbation:
u(x, y) =
1
1
27r
40Y + m(Oxo
P-1
O2
-}- y 2
k
d t + 2n I ,
(9.2.14)
1
v(x, y) = 2a 1
+1 m(E)v - f(F)xn 1
0
-
k -
T2r
Taking into account that
xo + y2 = (xo - i y)(xo + i y),
xo + yo = (xo - i yo)(xo t i yo) ,
for complex velocity
w(z) = u(x, y) - i v(x, Y),
372
TUE STEADY TRANSONIC FLOW
it results the following formula:
W(y)
=
1
+i
2k L(-),
M(C) +i
2
(9.2.15)
where
ata
JJ
=
u2( F''1)L(z)
Z-(
1,
and
_=x+ir1,
In order to impose the conditions (9.2.2) we shall pass to the limit on
the segment [-1, +11. Using Plemelj's formulas we obtain from (9.2.15)
u(x, f0) - i v(x, f0) = T 2 [rrt(x) + i e(x)] +
(9.2.16)
1
+ 'IrJr+l m(O + i e(f)
xo
k 0 If r u2(t, 71) d
d +27r OxJf T x-
do
whence
k 0 f/ 2
+2;r 8a R su
xa
4
,dl;
r1`
(9.2.17)
' +1
v(x, ±0) = ± m(x)
- 2n 11 zn) d £-
49
-2 a_JJ
2d dn.
It results
e(x) = u(x, +0) - u(x, -0) = K (p(x, -0) - p(x, +0)1.
(9.2.18)
The function l(x) will determine therefore the lift.
Imposing the conditions (9.2.2) we shall obtain
m(x) = h+(x) - h' (x) twt h`(x),
lxj < 1
(9.2.19)
373
THE PLANE FLOW
' +t
w 8x fir ul{t,*))
J-t
dd dn+
o
(9.2.20)
C(Z), IxI < 1,
+h+(x)
where we have utilized the notation h+(x) = h+(x) + h'-(x).
The relation (9.2.19) determines the unknown m(z). Considering
that G(x) is known, the equation (9.2.20) determines f with the aid
of the formula (C.1.9)
t(x) -_ 1
Vi- a
l+z
7r
1-z
1
/;:E t h+(t) d t-
l+t 8
+1
l+xJ_
n
1
-t t - x
k
d
1-t 8t
)2+tp}
dt
t-x'
(9.2.21)
Noticing that according to the formulas (B.5.3) and (B.5.4) we have
f+l 1-t dl:
t-.z(1+
t-z
dt
1
(9.2.22)
-J
1-4 df ) a
1+4f-t t-z
+1
1
VEZ+ 1
and, after replacing (9.2.21) in (9.2.15), it results
W(Z)
+t
1
C
(9.2.23)
z- 1
1
tai z+
+l
1
1+ t h+(t)
1-t
dtk
k
t-z+2L(z)-wM(z),
where we denoted
M(z)
-M
z
+1J 1
1
1+trd fr
1
n
dCdr1
( - t)2 +V2
z
t d t Jf u ('*1)
(9.2.24)
dt
--:
the formula coinciding with (4.11) from [9.21].
TIIE STEADY TRANSONIC FLOW
374
In the sequel we shall deal with M(z). We may write
M(')
7ri
z + I f12 u2(f, q)
ltd
f+1
{
,
q
1-t8t (t-t)2+q2,
dt
t-z1dq
1
7n
x
1
z+1
+1
ILl (F, q)'
2
The last integral is calculated by Homentcovschi by means of the residue
theorem. It may be calculated elementary noticing that we have
q
1
1
1
t-z - 2i((-z) t - (-
(t-e)2+
rt
1
1
1
2i((-z) t-(+((-z)2+q2 t-z
and taking (B.5.1) into account. It results
_
M(z)
1
z+lIir
z
1
u2((,ri)8
1
1
I(1_
+
We write the first integral as follows
If u'(t,n) I
1
dt drt_
1
z
C1z s±i
u2(E,17)-u2(t,-17)
C-z
JJR2
r
1
_
and the second
8z 11.2 u2(t,q)( (1-z
(1 z}d(dq
1
}dtdn,
THE PLANE FLOW
375
=
Jf/f
u2(
((- z)2
JJR2
We have therefore
+00
M(z) = 2 If [u2(.7,) - u2(C, -t1)] K(z,C)dC d tj-00
(9.2.25)
1
+00u2(t, n)
-2rJ
-00
u2(C, -n)
((-z)2
d d tj,
where
K(z'
}
_
z
1
z+1
(+1
1
_
C - 1 L(S - z)2
1
(9.2.26)
(C - z)(C2
Taking also (9.2.13) and (9.2.14) into account, we have that
y02 -x02 +2ix°y°
1
(C - z)2 ,
H
L(z) = I - i J = -7ru2(x, y) - IL
(9.2.27)
' 'ddrj.
(9.2.28)
(C - Z)2
We replace the expressions (9.2.25) and (9.2.27) by (9.2.23) and we obtain the complex velocity. Separating the real part of the complex velocity, we obtain the integral equation of the problem. We have:
'j,
+1
u(x, y) =
--fl
1
h
1
l
4tr
l+th+(t)
t
x2 + y2
d
z+l 1- z+l
z
1t-z
VE
1
i t-; jdt+
(9.2.29)
+00
+2I(x,Y)-$ JJ
q)
[K(z,c)
-00
+K(,)]d drj+ !T[u2(1
yor4x°dC
dn
This equation, was given in a slightly different form by Homentcovschi
in [9.211.
376
THE STEADY TRANSONIC FLOW
9.2.3
The Lift Coefficient
From (9.2.18) we deduce the following formula for the lift coefficient
+1
1
c
+I V=x
1
fKl-te(x)dy
t+( )dt
' +t 1+ t
r
l+z .!_t
dx- ;2/
k
J
+t v=1
x
l+x
i-t
(9.2.30)
We have
J
R3u2(frn)5
[(t-t)2+
n21dfdtl
u2(f,8tn)t-(
'( 1-t--(
1)dtdn
2i1 JJ .2
1
-2i J2
(9.2.31)
u2(f.n)-u2(f,-n)df do
(t-()2
From (B.5.1), derivating, it results:
l+t
t
1
dt
1
1-t (t-f)2
it
1
(9.2.32)
f-1 (2_1
Using these results we obtain
cp
7K
1
2i
9.2.4
j
k
I-t
hf (t)d t(9.2.33)
rJ u2(f+n) - u2(f, -n) d f do
S-1
JJets
f-1
.
The Symmetric Wing
If the wing is symmetric, then the equations (9.2.1) have the form
y = ±h(x)
(9.2.34)
377
THE PLANE FLOW
From the boundary conditions (9.2.2) we have the relation
v(x, +0) = -v(x, -0)
which suggests that the solution of the system (9.2.6) has the property
v(x, y) = -v(x, -y) .
(9.2.35)
We easily check that if u(x, y), v(x, y) is a solution of the system
(9.2.6), then u(x, -y), -v(x, -y) also have this property. By virtue of
u(x, y) = u(x, -y),
the uniqueness theorem it results (9.2.35) whence u2(x, rj) = u2(t,
The integral equation (9.2.29) receives the form
+1
ff(t)
-To
d4+
(9.2.36)
+2
Z
if
dn-u2(x,y),
JJ00
given for the first time by Oswatitsch (9.491.
Obviously, the lift coefficient vanishes. The result is natural because
the angle of attack of the wing is zero.
9.2.5 The Solution in Real
We shall present a solution which is different from Hamentcovschi's
solution which utilizes the complex velocity. Using the formulas (3.1.19)
and (3.1.20), one obtains from the general solution (9.2.14):
u(x, ±0) =
±-t(x) + Zx I
(ssd s +
J
v(x,f0) = f2m(x) - 2R ,r+t
k I (X, ±0),
T7r
ds+ 2 J(x, ±0),
(9.2.37)
-1<s<1,
where
I(x, t0)
.2
J(x, ±0) = 2JJu
2- Z
0° + n2)2 d t d it
- iru2(x, f0)
,
dri, -00 < ,n < 00.
(9.2.38)
378
THE STEADY TRANSONIC FLOW
We obtain the formula (9.2.18), and from the boundary conditions (9.2.2)
m(x) = h- (x)
1
e(s)
JJ
s-x
7r
(9.2.39)
d s = h+(x) - kJ(z, 10),
w
1xI < 1.
(9.2.40)
By means of the formula (C.1.9) we deduce
r+
£(z) - -V1:11 x
l+x
n
+ t h+(t)d
1-tt-x
t+
(9.2.41)
1+ t J(t, f0)
+as 1+z
1-i t-x
1
dt
.
For obtaining the integral equation of the problem, we shall replace
m and t in (9.2.14)1. We shall denote by s the superposition variable
from the first integrals (9.2.14) and xt = x - s. Changing the order
of integration, we deduce
1+t 1&+(t)N(3, t)d t-
+1
u(x, y) = 27ra
J-1i
y
-27r
y T2
,
f
1
± t J(t, ±0)N(x, y, t)d t+
(9.2.42)
k
+2R I+1x2+ 2ds+2-I(x,y),
x1 r -- s,
1
I
1
where we denoted
N(x, y, t) =
f+i V1 -s
1+s
J
1
Since
1
x
I
ds
y2 s-t
(9.2.43)
1
fl is 1J +
s-t z-z t- s-z t-z s-x
1
1
1
1
1
1
(x-t)2+y2 s-t'
taking the formulas (B.5.3) and (B.5.4) into account, we deduce
N
n
x-z
(
1
t-I
'z-1+1
t
1
`+111.
z
-x
(9.2.44)
379
THE PLANE FLOW
Let us calculate nw the term
T =1
i + J(t, f0)N(x, y, t)d t .
t
i
(9.2.45)
To this aim we shall specify J(t, f0). Taking into account (9.2.13) and
the identity
2h7
1
t-C - t
(t-w +
1
we deduce
J(x, ±0) =
- JJ
rlLe2(F,
18
iT 3i
1
1+I
2i
1C-t C)dCdq=
f
_
and then
I
ldfdq=
v))!
IL
ifm
(9.2.46)
u2(Cn)-UU(E,-11)ded+l=
t-(
u2(E,11)
- U'(C, -WdC dq
( t-C)2
dil}
(t- u2(e,-17)dt
- C)2iy
l +t VI u2(F,17)
1-t
1
,
x
.
(9.2.47)
s
1
l
1
1
t -z z+l t--l+1
cit.
Changing the order of integration and denoting
IPTE
To(C, Z) = El N 1
1
t,
t (t i()2 t zd
(9.2.48)
we deduce
T
If
4Y
JJ*3
[ j2((, +1)
- U2(t, -n)1 [To((z)J4_
(9.2.49)
-To(CIT) FT 711
3+1 jddti
380
THE STEADY TRANSONIC FLOW
for specifying the function To we notice that
I
t
1
(t-C)2 j---Z
1
+[Z
1
_
1
(t
((Z
1
- ()2
1
1
1
z-( (z-S)2 t-C+(z-()2
Taking (B.5.1) into account , one obtains
1+1 -
A
To(C,z) _ (z - ()2
,r
-
1
-
z+1
n
(z - C)2
z
1
- 1
-
1+S
d
z-C dC C-1'
(9.2.50)
The term T which intervenes in the integral equation (9.2.42) has
therefore the form (9.2.49) where To It is given by the formula (9.2.50).
Using the expression (9.2.41) for the lift coefficient
cL =
JJt(x)dx
(9.2.51)
one obtains the formula
1
CL
k +i
ai
-VrJ1
FI+- ft+(t)dt+7Vn f
i+tJ(t,±0)dt.
1
(9.2.52)
The function J(t, f0) is given in (9.2.46). For calculating the last
term we change the order of integration and we take the formula (9.2.32)
into account. One obtains the formula (9.2.33).
9.2.6
The Symmetric Wing
We gave in (9.2.36) the equation of Oswatitsch for the symmetric
wing. We obtained this equation from the general theory presented in
(9.2.2). We present here the direct deduction based on the equation of
the potential
cpn + lpyy = lops
which may be obtained from (9.1.13) or (9.2.6).
(9.2.53)
381
THE PLANE FLOW
The fundamental solution is defined by the equation
Eu+EI,o, =k
49 E.3
+m6(x,y).
(9.2.54)
Applying the Fourier transform it results
(a2+a4)E=iaikF-m, F- 9e ],
whence
ia1F
al
m
aFI+ a
and then
-F rl
-k 2
ax
F
f
f
ll
- mF-1 [-
]
.
(9.2.55)
1
al +a2
-71+072
Applying the convolution theorem (A.6.13) and taking (A.7.11) into
account, we obtain
8
F-l
r. (_ '
[-'r
a14J
s
47r
(+
+ao
--4, f f E=(t,q)ln(xo+14)dCdn,
-00
such that
+*0
E
= + 7r JJ
-6. 2(4, n)
ax
f d f d n + 4n ln(x2 + y2).
(9.2.56)
-00
Replacing the wing with a continuous distribution of perturbation
sources defined on the chord of the profile (only in the symmetric case
the sources distribution is sufficient), from (9.2.56) it results, in the
domain occupied by the fluid, the following general representation of
the solution of the equation (9.2.53):
'P(z,y) = I
fl
m(f)ht{so+y2}dC +
j;
ffR?u2(t,n)
X0
PO +
y dt dn,
(9.2.57)
where in(k) is a function which has to be determined by the shape of
the profile, i.e. by the boundary condition
v(x, f0) = ±h'(x), x E [0, 1],
(9.2.58)
382
THE STEADY TRANSONIC FLOW
which is imposed by (9.2.34).
From (9.2.57) we deduce
1+1
0 + y2
(9.2.59)
f+1
v=SPY= 2rr
xo
y2
where
I = ff
po d 4' d ii
u2 (4, n) x2
(9.2.60)
.
The singular integral I has the shape (E.9). Applying the formula
(E.10) we deduce
1= fju2(f,rl)8x
(x o+
f2fR2
x2
-
u2(4, 11)
Jdt;dn-aru2(x,y)
o
2
/
(xd {
d *1- 7ru2(x, y)
+ y02)2
(9.2.61)
,
I=-2 ffzu2(t,,1)(xo
Hence we have the following representation of the solution
u(x, y) =
1
+1
/-1
xo
z0 + y2 d
2 f fR2 u2 v,
r+t
v(x, y) = T
!-1
- +y
2
rl)
(x o
z
d Cd +1-
k
d t- fJ uz
m(C) xo
2uz(x, y)
,
x2y2
n) (xo + y02)2 d
dn
(9.2.62)
which is also given in (9.2.14).
The boundary values of the first integrals are given in (3.1.19) and
(3.1.20). We obtain therefore
v(xi f 0) = f2rn(x) +
11
ir
u2(f,rl)
2
(xz + 2)2dt d>l.
o
(9.2.63)
383
THE THREE-DIMENSIONAL FLOW
Imposing the boundary conditions (9.2.58) we find
+oo
II
u2V,77)
22'°'1d d drl>
(ro+r1)
(9.2.64)
u2(x, y) = u2(x, -y)
(9.2.65)
whence we deduce
17) = u2(s, -+l)
Taking into account (9.2.63) and the previous relation, it results
m(x) = 2h'(x).
(9.2.66)
In this way we determine the distribution m. Coming back to (9.2.62),
we obtain the equation:
u(x, y) + k u2(x, y) +
2 11.2
2_ 2
(xUO+ y02)2
+i
d
d j?
(9.2.67)
rr -1
which coincides with (9.2.36).
9.3
9.3.1
The Three-Dimensional Flow
The Fundamental Solution
In the last 40 years, a great number of papers was devoted to the
steady transonic flow past thin bodies. Usually one assumes that the
flow is irrotational, the potential satisfying a non-linear equation having
the form (9.1.13). For deducing the integral equations of the problem,
we apply Green's formula to the equation of Poisson and we assume that
a vortices layer is present downstream the wing. Derivating, we obtain
the non-linear integral system for the components of the velocity (see
for example [9.36]).
In the case of the symmetric profiles the system reduces to a single
equation for the component u(x, y, z). In this sense, after the initial
paper of Oswatitsch [9.50] where one defines a principal value for the
singular integral which intervenes in the representation, it followed the
paper of Heaslet and Spreiter [9.17] where one gives a general representation which in the symmetric case reduces to an equation. The
384
THE STEADY TRANSONIC FLOW
representation is valid both for the flow with shock waves and the flow
without shock waves.
For the lifting wings the forms of Norstrud [9.41] and Nixon [9.39]
are available. In this case, the problem reduces to a system of two nonlinear integral equations. At last, we mention the paper of Ogana [9.471
where one shows how the integral equations depend on the definition
given to the principal value of the singular integrals.
A new point of view, belonging to D. Homentcovschi [9.20], [9.21]
and L. Dragog [9.8] ]9.9] does not assume that the flow is potential.
Utilizing the system of equations of motion it is necessary to assume the
existence of the vortices layer downstream. In the sequel we shall utilize
the method of fundamental solutions [9.8].
The system which determines the perturbation is (9.1.42) and (9.1.46).
Performing the change of variables
u`=
,
(9.3.1)
and omitting the marks * and A, the system becomes
uy-v==0 U,-w2=O
(9.3.2)
us + vy + w. = k(u2)s,
where k has the same significance like in (9.2.6). We shall see further that employing a fundamental solution similar to the fundamental
solution of the system
uy-uz=eo(x,y,z), us-ws=0
(9.3.3)
u. +vy+w2 = k(u2)=+mb(x,y,z)
we may satisfy all the conditions of the problem. This solution will
be determined in the manner described in 2.3. Applying the Fourier
transform, solving the algebraic system just obtained and considering
the inverse Fourier transform, on the basis of the formulas from appendix
385
THE THREE-DIMENSIONAL FLOW
A, we obtain:
u(x,y,z)
)r
41r (m&x + t
v(x,y,z) - 4w 8z
k
4w 8y
r
82
_ M 49
-4w 8
r,
8zsuz
r
8
as
8z
alai
1
4w 8xOy
w(z, y, z)
82
4w
8 I_ m 8 1+
t
_
__
us
r,
k
2
02
2
1
(r} ^18z [ate, - 4 8x8zu * r,
1
(9.3.4)
where r =
+ y + z and,
-.
I 8, C) dv ,
(9.3.5)
u
r Iffits I z - 41
with the notation dv = d£ d, d(. This integral is called the aonvolntion
of the functions u2 and 1/r. TaIdng M = 0, from the formulas
(2.3.11), (2.2.6) and (2.3.27) it results
us
al l = -.f`1
8
1
J i alas
[ala2J
=
(9.3.6)
4Ir 8y
r
j.
4w y2 + z2 11
+r
and a similar formula. In fact one obtains the following form of the
fundamental solution:
u(2,y,z)_-4w (mf +e ) r 1
v(x, y, z) = 4w
K
jr02
w(z,y,z) =
8
r 8z
[y2+z2 (i. +
r
Io,
r-
-m
1
4
t
8
+ 4w 8a
r), r 4w 8xjo'
y
+z
(1
x)
+w
k 8K°
47r 8x,
(9.3.7)
386
THE STEADY TRANSONIC FLOW
where we denoted
lo= au2*1,
r Jo= ay
TX
au2#1,
Ko= aZ
r
r
(9.3.8)
P. and m being constants.
9.3.2
The Study of the Singular Integrals
The integral (9.3.5) has a weak (integrable) singularity. The integral
exists, (it is convergent) (u2 is zero far away) and it may be derived
(the convolution, if it exists may be derived (A.3.7)), such that we have
I° = u2 * -
-u2 * Ix13 = J u2(4) 14 - I3dv
-r=
and similar expressions for J0 and Iio. Since the integral has the form
(E.3), it is convergent. With the notation
E -- xl,
x
f __ IC
has the form (E.9) and may be derived according to the formula
(E.10). For calculating the last term one utilizes the spherical coordinates with the center in the point having the vector of position x:
1°
-x=sin9c s
rl - y=sinOsinV
( -z=cos9.
One obtains
Jfces(n,x)dn =
Ir
whence it follows the formula
8xlo = J ul()FZ (::,1) dv - 43 u2(x) _
(9.3.9)
U2(t)2=02
0
"°dv - 4xu2(x)
3
.
THE THREE-DIMENSIONAL FLOW
387
Analogously one demonstrates that
J a to = 1,
49X
u2(,) Ix?o lsdv,
(9.3.10)
K = ±Ko =JR3 u2(F)Ixxo{Isdv,
where xo = x - t, yo = y- rt, zo =z-(. The integrals we have obtained
are convergent if u2 satisfies Holder's condition and if its behaviour at
infinity is u2(C) = O(ItI-`) with I > 1.
9.3.3
The General Solution
Denoting by D the projection of the wing on the xOz plane and by
y = h(x, z) ± hl(x, z),
(x, z) E D
(9.3.11)
the equations of the wing (which is assumed to be thin), we shall be
able to satisfy the conditions of the problem with a continuous superposition of solutions having the form (9.3.7), defined on D. It results the
following general representation:
(1R)
u(x,y,z) =-4a JJ [M(C 0ax
dt dC
(9.3.12)
-4 J(x,y,z),
v(x,y,z) = 47r
11D
[e(C) ax
-
(R)
d d(-
{2:
47rf
(9.3.13)
k
ir- Ax' Y, z),
u !(x, Y, Z)
(R) +
T" AD
f
(9.3.14)
1
D
K
I y +:.p
Y
dt d(,
THE STEADY TRANSONIC FLOW
zo=z-(, R- xo+y2+zo.
(9.3.15)
Taking the formulas (5.1.16), (5.1.18) and (5.1.24) into account, it
results
u(x,10,z) =±t(x,z)+4Ao,, m(E,()
D
v(x,±0, z) = f1m(z,
z) + 4-r
no dEdC- 4x1(x't0,Z)
(9.3.16)
1.
(z((,C)2
(i + x0)d{ d(-
- 4 J(x, ±0, z)
(9.3.17)
where
Ro= zo+zo
(9.3.18)
and the mark * indicates the Finite Part like in (5.1.24). From (9.1.42)
and (9.3.16) we deduce the significance of the function
t(x, z) : t(x, z) = p(x, -0, z) - p(x, +0, Z).
(9.3.19)
Hence, t(x, z) gives the jump of the pressure. This function will be
utilized for calculating the aerodynamic action.
From the expression of v(x, ±0, z) and from the boundary condition
v(x, f0, z) = h'(x,z) ± h' (x, z) (x, z) E D,
(9.3.20)
where the mark "prime" indicates the derivative with respect to the x
variable, it results after subtracting and adding
m(x, z) = 2h' (x, z),
4ir,1Dt(
(9.3.21)
(1+)dd(+
+ 2k s u2(4) (xo + q2
+
Z02
T,,-, d v = h'(x, z), (x, z) E D.
(9.3.22)
389
THE THREE-DIMENSIONAL FLOW
The formula (9.3.21) determines directly the unknown m(x, z). In
the equation for t(x, z) it intervenes the values of u2 in R3. They are
obtained from (9.3.12) after replacing m by (9.3.21).
We deduce
u(x)- 3u2(x) +
k
4I
U2(4)2xix MI5 -o d v(9.3.23)
-41T f t((,()R3d(dq
2,1
D
f Dh'(k,()R3dCd(
Hence, for determining the unknown f(x, z) on D we have to solve the
system consisting of the equations (9.3.22) and (9.3.23) where u(x) is
defined on R3. For u(x, f0, z) we shall utilize the values (9.3.16). The
mathematical problem is extremely difficult and there are not known
any attempts for solving it.
For the symmetric wing (h = 0), the solution is obtained for t = 0
and u(x, y, z) = u(x, -y, z) if
k
u(x2)- u2(x) + 4 /
2x x05 d v =
U
3
(9.3.24)
3
= 21r JDhi((,()Rd(d
9.3.4
Flows with Shock Waves
In the case of the flow with shock waves, the general solution has also
the form (9.3.12)-(9.3.14). We can see it in the simplest way if we utilise
the notion of Fburier transform for bounded domains, introduced by
D.Homentcovschi 19.191. Indeed, in the fluid domain D the equations
vx-uy=0, wz-uz=0,
(9.3.25)
uz + vy + w: = k(u2). ,
with the notations from (9.3.2) have to be satisfied. On the shock waves
E one imposes the relations
OvOnz
- OuOny = 0,
llwOn: -
0,
(9.3.26)
uOnz + [lvjny + Uu'Qn: = kjJu21nz,
THE STEADY TRANSONIC FLOW
390
deduced from (9.1.49), and on the borders S+ (upper surface) and S_
(lower surface), the conditions (9.3.20).
Applying the Fourier transform for bounded domains, we shall utilize
the formulas of the type (A.8.1). From (9.3.25) we deduce
-iaiv+ia2u=S1+T1i
-iaiw+ia3u=S2+T2,
(9.3.27)
-ialu - ia2v - ia3w+kialu2 = S3 +T3i
where, taking into account that on S+ we have n = (0, 1, 0), and on
S_, n = (0, -1, 0)
(vnz - uny)e' a'xd a =
S1 =
s++s_
I Juleiaxda = -JDt(x,z)e'("'+"'s)da,
_-
D
S2 = J
S3=
; +s_
s++s_
(wns - un=)e' a'Zd a = 0,
(9.3.28)
f(urn+vny+wn;-ku2nr)e'c.'da=
=f OvOe1a xda _
fom(x,z)e'(",+"':)da.
t(x, z) and m(x, z) having the signification from (9.3.19) and (9.3.21).
The integrals
Ti =
J (OvOn. - Julnw)
T3 =
JE
T2 = f£ (OwOn.=
(Duin. - Ovlny + Owonz
- Juln=)
et°r'ada
- klu2Onz)
vanish because of the relations (9.3.26). Hence, the system (9.3.27)
reduces to
-ia1v+ia2u=S1
-ialw+ia3u=0
-i&- ia2i;-ia3w= S3-kialu2,
(9.3.29)
391
THE THREE-DIMENSIONAL FLOW
which has the solution
u= `ia2sas al Ss +
i a2 S3
V=
op, +
10 S1+ka
t
W=
where a2 = ari + d
u2,
3s
u,
(9.3.30)
a S1 + ka y-
i
1
+ &23.
Considering the inverse Fourier transform and utilizdng the formulas
(A.6.9) we obtain
1
[83]
- k-t-r-I [a,]
(9.3.31)
Cf2
f is a2]
+8
-k.Ozox jr8a
1
us
8°
-
u2
1
i-k_
8
air
By direct calculations, we deduce
.F-1
f
(2703 OY J22
=
t la {atE+wt)d dC
L_ L
f
e '(a1x+a*v+o*z)d a
da]dt ds
fD[(211)3mnhJP.3
47s
f
a
_
O R ()dd(.
(9.3.32)
where, with the notation
R= V;i.
,
(9.3.33)
THE STEADY TRANSONIC FLOW
392
we utilized the formula (A.7.10). From (2.3.11) and (9.3.27), it also
results
a 'r- j (
I_ 1 8 1-
1
iata2
t?z
1
47r
4ir Oz J
oo V2.2
ao
+ y2 + z-2
_
zd x
1
f
dx
(x3 + y2 + z2)3/2
1
z
4-x y= + z2 (1 +
x)
r
(9.3.34)
'On the basis of this formula we deduce
_
1
8 jI
(27r)3 Vz st3
I
e't(alx-+a,y+ass)da=
P
i0la2
OZ .3
(2n)3
1L
4;r
i
ala2
dct]d d( =
'y2+z2 (1+)ded(.
(9.3.35)
At last, taking into account the definition of the convolution product,
it results
u2
i
u2(t) d
(9.3.36)
a2
4:r a3 Ix - f 1
With these formulas and with the similar ones it is not difficult to
see that in (9.3.31) we have just the solution (9.3.12)-(9.3.14).
9.4
9.4.1
The Lifting Line Theory
The Velocity Field
The lifting line theory in the transonic flow is studied in (9.55) and
[9.$). In the last reference, it is obtained, as it is natural to do, from the
Lifting surface theory. This method is also utilized herein.
We shall deduce the equations of the lifting line theory from the
lifting surface equations using the assumptions 10,V,3" (Prandtl's
393
THE LIFTING LINE THEORY
hypotheses) from 6.1. Hence we shall take hl = 0 and we shall consider
that the unknown is the circulation
C(c) = +
(9.4.1)
e(4, ()d
C(±c)=0
(9 .4.2)
and we shall utilize the formula
tim
=
f
()k(x, y, z, 4, ()d e d ( =
fJ
s-(()-O-s+(()
(9.4.3)
}r.
C(C)k(x, y, z, 0, <)d
2c representing the span of the wing on the direction of the Oz axis.
On the basis of the first hypothesis (hi = 0), from (9.3.21) it results
m = 0 and the representation
u(x, y, z) = T"
fJ
C) R3
d d(
T-
I (x,
y, z)
v(x,y,z) =- 1 ID
f t
4sr
(1+
j)]ded(-
- J(x,y,z),
y
(i+
az y2 + s02
R) ,did( -
I(x,y,z)
a ly +
AD f
0
ev,oa[
1 ffD
w(x, y, z) = 4z
(9.4.4)
or, using also the formulas (9.4.1) and (9.4.3):
u(x, y, z) =
1
4zr
+`
J
k
C(() R3 d _4 I (x, y, z) ,
1
v(x, y, z) _ -
c
C(S)
d
I
- 4?r
-C
C`(C) yz
zo
(1
+
+C
w(x,y,z)-:i!. f
C(S) 2+x2
y
o
Ri) d(
4 J(x, y, z)
,
k
(1+R )dC- K(x,y,z),
1
(9.4.5)
THE STEADY TRANSONIC FLOW
394
where we denoted
x2 +y2+; 2.
R.1 =
(9.4.6)
The Integral Equations
9.4.2
Using the identity
I(1+xo\^8 xo+Ro
Ro J
8z
(9.4.7)
rozo
where Ro is (9.3.18), we deduce for (9.3.22)
JJn
1+'0
r(io
RD / d
d
td (=
8z
AD e(c o
X0 0ze
td C =
FC(C)Zd
-d
Using the calculations (6.1.11) - (6.1.13) we deduce in the sequel
T
-j
C'() d C az ff4;)s)flzOd t d ( =
e
+(1) t(co )d(
10
Je
' +` C(() d
Lc
zp
L(()
e
a(z-S)d(
(9.4.8)
2 , _+(_) t(F, z) d
J=-(z)
xo
Substituting this in (9.3.22), multiplying the obtained equation by
x - x_ (z)
Vx+(z) - x
and, integrating with respect to x on the interval (x- (z), x+ (z)), one
obtains the following equation:
C(z) =
a(2)
'
'((z d(+
is u2(C)S(=, F, n, ()d v+ H(z) , (9.4.9)
THE LIF"TINC LINE THEORY
395
where
x - x_(z)
j_+(`)
3xogdx
S(z, , , C) = flV x+(z) - x (xo +, 2 + 4)512 ,
(9.4.10)
H(z) = 2
s+(_
- x-(Zh (x, z)dx .
x
77
The equation (9.3.23) for the unknown C(z) is
u(z) - 3 u2(x) +
4J [
y +`
-^ 4
2X.2
U
-
x
2
y
2
zo
is
dv
(9.4.11)
C(C)d C
, (x2 + y2 + 40)3/2
We have therefore the equations (9.4.9) and (9.4.11) for the unknowns C(z) and u(x, y, z), C being defined on [-c, +c], and
it, on R3. The continuation of this reasoning may be found in (9.8).
Obviously, we have to consider only numerical solutions.
Chapter 10
The Unsteady Flow
10.1
10.1.1
The Oscillatory Profile in a Subsonic Stream
The Statement of the Problem
As we have already mentioned (see Chapter 2), the general problem
of aerodynamics, i.e. the problem concerning the determination of the
perturbation produced by an arbitrary moving body, in a fluid whose
state is known, is very difficult. A presentation of this subject can be
found in the papers of Kiissner [10.37], [10.38), [10.39]. We consider
in this chapter, the particular case of the wing which is oscillating harmonically in an uniform stream. The problem is important in the flutter
theory. For the incompressible fluid the plane problem was investigated
by Theodorsen (10.75). For the compressible, subsonic fluid in a plane
flow, the problem was studied at first by Possio [10.59) and then by Dietze [10.14) and Haskind 110.30). These authors used the potential of
accelerations i1', replacing the body by a doublets distribution defined
on the chord of the profile. The integral equation which was obtained
made the object of many researches [10.22], [10.23]. In [1.1], [1.3], [1.8],
[1.18] one may find syntheses and references.
Considering that the replacement of the wing by a doublets distribution has no physical justification, in [10.15] we deduced the integral
equation starting from the idea that the wing must be replaced according
to Cauchy's principle by a distribution of forces. In the same paper, we
gave a solution for the integral equation for small values of the frequency
(w K 1). The present' section is written on the basis of this paper.
10.1.2
The Fundamental Solution
We utilize the dimensionless variables x, y, z, t introduced by (2.1.1),
and for the velocity field V 1 and pressure P1 resulting from the super-
398
THE UNSTEADY FLOW
position of the perturbation over the basic flow, we set
V1 =UO +V), Pi = P.+p,0U;pP.
(10.1.1)
Like in the previous sections Umi is the velocity of the unperturbed
stream, and p,,. and pa, the pressure and the density in that stream.
Imposing the condition that these fields verify the equations of motion,
it results (see the equations (2.1.26) and (2.1.27)) the system
oV /at + 9V/Ox + grad P = F,
A2(aP/at + OP/ax) + div V = 0,
(10.1.2)
lim (V, P) = 0,
ixl
where F is the force density assumed to be small (the system is the
result of a linearization), and M is Mach's number for the unperturbed
stream.
If the uniform flow of the fluid is perturbed by harmonic forces having
the form
f Cos (iwt), f Bill (iwt),
applied in the origin of the axes of coordinates, then we shall put in
(10.1.2)
F=
(10.1.3)
f5(x)ek'Ji
.
Solving the system, we shall obtain that the real part of the solution
is determined by the force f eos(iwt) , and the imaginary part by
f Bin(iwt)
.
The perturbation produced by (10.1.3) will obviously have the shape
V = v(x)ei"' P = P(x)e"
(10.1.4)
Replacing in 10.1.2 one obtains the system
iwv + av/ax + grad p = f 5(x)
M2(iwp + ap/ax) + div v = 0,
,
(10.1.5)
lim (v, p) = 0.
Ixi-. Q
The solutions of this system are given in 2.4 (they are the solutions of
the system (2.4.3)).
399
THE OSCILLATORY PROFILE IN A SUBSONIC STREAM
In the case of the subsonic, two-dimensional flow, assuming that f
has the form (0, f) , it results from (2.4.6)
(10.1.6)
P(x,y) _ -fly Go(x,y),
and from (2.4.16) and (2.4.21)
v(x, y) = f e-"' [2iwG - f32G= + w21 x
G(7-, y) d 71
.
(10.1.7)
00
We denoted
Go (x, y)
- 4I
G (x, y) =
4if3
)
H
°
( k/ x2 + My2) e»
H(2) (k
(10.1.8)
x2 + /32y 2) e1
where
k=wM, 0 =w/$2, a=kM
(10.1.9)
Hoe) being Hankel's function.
10.1.3 The Integral Equation
Assuming that the perturbation of the fluid is determined by an oscillatory wing having the equation
y = h(x)e'`'t , 1x1:51,
(10.1.10)
from (2.1.29) it results the following boundary condition
v(x, 0) = h(x) + iwh(x) = H(x), jx) < 1,
(10.1.11)
Let us replace the action of this wing against the fluid by the action of
a continuous forces distribution having the form (0, f) defined on the
interval (-1,+i] . From (10.1.6) and (10.1.7) it results the following
general representation of the perturbation
P(x, y) = -
J
+1
v(x,y) = f f(t)e
11
f (t)
Go (xo, y) d
r.,xo(2LG(xo,y)
,
(10.1.12)
- f32GX(xo,y)+
(10.1.13)
+w2J
G(r,y)dr]d,
THE UNSTEADY FLOW
400
where xo = x - f
.
The function f has to be determined from the
condition (10.1.11).
For obtaining the limit values p(x, 0) and v(x, 0) , we shall take
into account that Hankel's functions H0(2)(u) and H12)(u) satisfy the
relation
(10.1.14)
Hoe) (u) =
(u) ,
du
and have the following asymptotic behaviour for small values of the
argument [1.16], [1.40]:
H421(u) . I - 1n u , r = 1 -
(10.1.15)
?l 2(C- 1n 2) ,
Hi2l() _i u ,
(10.1.16)
where C(= In y) is Euler's constant (= 0.577215) . We deduce
y
P(x, y) =
i::' f
+ p2y2)
(k
exp (iaxo)
o + 2y
d£
.
When we calculate p(x, 0) we observe that, because of the presence of the factor y, the product y f+1 vanishes excepting the vicinity
(x - e, x + t) where the integrand becomes infinite (for t = x ). If a is
(iaxo) may be apsmall enough, in this vicinity the function f
proximated by its value in the middle of the interval (we assume that
f (C) is continuous), i.e. with f (x). It results
rte
,
f (x)
P(x, ±0) =
y 1 Hill(k xo + [32y2)
vlli
4i
x+y
_e
-+ t :
and, performing the substitution
into account,
dt
+C
AX, f0) _
f (x)Y--+O
limy
2w
J
-x = t , and taking (10.1.16)
t2 + /32y2
1 f (x).
2
(10.1.17)
Hence we obtain the significance of the function f (x) . We have
P(-T' + 0) - P(x, - 0) = f (x) .
(10.1.18)
The component v(x, y) is the sum of three terms vi , V2, V3, which
are represented as follows
+i
v1 (x, y) =
J
f
j f We-
G(xo,
y)d
""r° Hoe) (kx2 +/32y2)ei0xo d
401
THE OSCILLATORY PROFILE IN A SUBSONIC STREAM
r+1
1
VI (X'0) =
H(21
4iQ 1-1 f (
(kjxol)e0'0 dE,
(10.1.19)
the singularity from j 2) being integrable. We have also,
v2 (s, Y)-
J
+1
f(e)e-I° 8 G(xo, II) d t =
k
4iQ
+i
f
1
(k
xq + Qom)
x
-07
0
°d t+
f+1
+ 4Q3
fl f
(k
and taking (10.1.16) into account
1
k
V2 (x, 0)
Q J-1
f
(klxol) f!l e i2-0 d E
OHo (k1xo1)e
f+' f(t)-
Odd,
[JZ0G(r,y)dr]
0
00
1
V3 (r,0)
(10.1.20)
I
From
one obtains
,
,
(f)e-6-0 H(2)
+403 f-+1
v3(x,U)=
xo + Q2y2)e'D*° d
dt,
1 f(t)e- '-O [JXOJ;42)(kfrI)ev1d1] dt,
_!_ f
where one performs the change of variable r -+ u : for = u and one
takes into account the formula
J
J
Hoe) ( M [u1) eh'd u
=
2 n 1 Mf
I
,
(10 . 1 . 21)
given in [1.2[. One obtains
V3(X,0)
26;
If(4)e!
0 [! 1M6
+ Zr
o
(M1u[)ej"duJd
J
(10.1.22)
After all, from (10.1.13) it results
I
v(x,0) = T f(C)Ni(xo)dC+J
11
f(F2(xo)dt,
(10.1.23)
402
THE UNSTEADY FLOW
where we denoted
'
Nk(xo) = nk(xo) e'
nl ( xo ) =
4 pM
k''X0
,
k=1,2
(10.1.24)
wr0
Hi2)(k (xoI) } o_ e
n 2 ( Xo ) = A`-'A HO(2) (kl xol)e °''z0
fQC)
4
- 2W-
In
1+,3
(10.1.25)
M
H2)(MIuI)eudu.
Imposing the boundary condition 10.1.11 it results the following singular integral equation
r
I
i
f
d
H(x) , IxI < 1
(10.1.26)
where
N(xo) = N,(xo) + N2(xo)
(10.1.27)
In fact the kernel N depends on the variable xo and on the parameters
M and w. We deduce therefore that the kernel has the following explicit
expression (a = kM)
N(xo,
_iw
M'w)
e_
-- --Hi2)(klxol) Ixol etQTp +
°r
QHQ2)(klxol)e' O-
e'i'' 0 In
(10.1.28)
The equation (10.1.26) is Possio's equation [10.591. There are a lot of
papers devoted to the kernel (10.1.28) in the literature 110.5], [10.22),
[10.301.
10.1.4
Considerations on the Kernel
Taking (10.1.16) and (10.1.16) into account, we deduce
N(xo, M, 0) = urn N(xo, M, w) =
2
n
403
THE OSCILLATORY PROFILE IN A SUBSONIC STREAM
which is the kernel for the steady flow. With this kernel the integral
equation (10.1.26) reduces to (3.1.15).
The kernel for the incompressible fluid is obtained from (10.1.28)
calculating
(10.1.29)
N(xo, 0, w) Mhm.(xo, M, w) .
Integrating by parts we transform the formula (10.1.28) into
et '° {Mi42)(kIxoI)
N(xo. Al, w) =
- iHi2)(k(xol) (X(, +
4A
+ t!ft e-6'0 lim Hoe) (Maul)
4
- W4M e-6'0-
Utilizing now the relations (10.1.15) and (10.1.16) the notations (1.16)
00
J;
Ci(z) =lnyz+
u-Idu =lnryz+-1)"(
JJ
Si(z) =
1sill u
u
n)t
,
n-1
°O
du = > (-1)r (
n=O
2n+1
+ 1)(2n + 1)! '
(10.1.30)
called, the first, integral cosine, and the second integral sine, we obtain
N(xo' 0, W)
2
2 + Si(wxo), }
.
(10.1.31)
This is the kernel for the incompressible fluid.
One demonstrates in (10.15] that for small values of the frequency
(w a 1), the integral equation (10.1.26) has the form
a
xd
w
+
mQJ_11f(t)(In (lx-t(+r))dt=2H(x),
(10.1.32)
where r is a constant. This kind of equations are solved in (A.16). We
leave to the reader the task of writing explicitly the solution.
In (10.22) one shows that the general kernel (10.1.28) has the form
N(xo,Af,w) = Ao(xo)+Ai(xo,M,w)In(lxol)+A2(xo,M,w), (10.1.33)
404
THE UNSTEADY FLOW
where
Ao =
_0
2,
A,=- 2 CBI(xc,M,w)e ,mxo
(10.1.34)
A2
-iwxo
2. B2 (xo, M, w)e
,
A, and A2 being analytic functions with respect to x0.
10.2
10.2.1
The Oscillatory Surface in a Subsonic Stream
The General Solution
The problem presented in this subsection was studied in many papers
(10.86], (10.87), (10.45], (10.35], (10.83] where the integral equation was
obtained by means of the potential of accelerations, replacing the wing
by a distribution of doublets. A slightly different investigation was given
in (10.12]. We studied this problem in (10.6] utilizing the fundamental
solutions method which will be presented in the sequel.
The problem is the following-, an uniform stream having the velocity
the pressure p and density po , is perturbed by a surface,
oscillating according to one of the laws
z = ho(x, y) coo (wt), z = ho sin (wt), (x, y) E D.
(10.2.1)
One requires to determine the perturbation. One utilizes the dimensionless variables introduced in (2.1.1) and the notations (10.1.1). The
problem is simplified if we replace the laws (10.2.1) by
z = ho(x, y)e"" ,
(x, y) E D .
(10.2.2)
In this case the real part of the solution will give the perturbation produced by (10.2.1) and the imaginary part the perturbation produced by
(10.2.1b). The boundary condition (2.1.20) and the linearized system
(10.1.2) lead to solutions having the form (10.1.4) where the functions
v and p are determined by the system (10.1.5) and by the boundary
conditions
w(x, y, 0) =
8
ho(x, y) + iwho(x, y) = H(x, y), (x, Y) E D. (10.2.3)
The solution of the system (10.1.5) under the assumptions that f =
_ (0, 0, f) and the unperturbed stream is subsonic (M < 1) is obtained
from (2.4.7) and (2.4.17) as follows
a
(10.2.4)
P( x,Y,z _ - f 8z 0o(x,Y,z),
405
TILE OSCILLATORY SURFACE IN A SUBSONIC STREAM
w(x, y, z) = f e
i"x
l
-2 . ) G + (w2
[(2i
) f G(T, Yz) d r
00
J
where
1 exp [io(x - M RI )l
(10.2.5)
R,
x +
RI(x,y,z) =
(y T .
As we already know, the formulas (10.2.4) define the perturbation
produced in the uniform stream by the force (0, 0, f) exp(iwt) applied
in the origin of the axes of coordinates. Replacing the wing with a
continuous distribution of such forces, defined on the domain D, we
obtain the following general representation of the perturbation
P(x,y,z) _
w(x, y, z) =
J J°
f(f,rr)
a8 G
(10.2.6)
G(xo, yo, z)+
f (t, -T) e-'"' [(2iw - 021)
ox
0-2
(10.2.7)
+(w2-a-y2)
where, as usually, xo = x-t, yo = y-7) .The function f is the unknown.
10.2.2
The Integral Equation
In order to determine the unknown f , we shall impose the conditions
(10.2.3). At first we shall prove that if f (x, y) is a continuous function,
then
Zlim0JJ f(E,rr)a Go(xo,yo,z)dfdi' = r2f(x,y),
(10.2.8)
D
Indeed, we have
j f fl) iwM + Ro j )q aik(Mx°-RO) d e d >)
rJ°
a
P(x, y, f0) =
4Z
J
z = 0, the integrand
will be zero excepting the point Q(x, y) E D. Denoting by DE the disk
where Ro = Rt (xo, yo, z) . We notice that if we
/set
having the center Q and the radius E and assuming that t is small
406
THE UNSTEADY FWW
enough in order to approximate f t)) with f (x, y) (this is possible if
f is continuous) and the exponential with the unity, it results
P(x, y, ±0)
4lymo
Jf 4 (ic?vf +
)ddr
.
Performing the change of variables t, q -* r, 0:
-x = Qrcos0, r/- y = rsin0,
0<r<e,0<0<27r,
we deduce
P(x, y, ±0) = 2p f (x, y) zluuo z
i
(iwM +
r .8 z } r2+
r
z2 =
/!
=fif(x,y),
(10.2.9)
and then
P(x, y, +0) - p(x, Y. -0) = f (x, y) ,
(10.2.10)
The formula (10.2.9) proves (10.2.8) and (10.2.10) gives the significance
of the function f (x, y) .
In (10.2.7) we may interchange the limit and the derivations (with
respect to x and y). It results therefore
w(x,y,0) =
nj
' fj f
26 -,62
x
(,)enw [n+n2]ddtl,
(10.2.11)
eR ,
(10.2.12)
oo e(1iEl=d,,
F00
E
=j
R,
d
dT
,02X = (xo R,. =
=J
T2 +
MR), R = xo + /yo
ya
(10.2.13)
407
THE OSCILLATORY SURFACE IN A SUBSONIC STREAM
In E1 we perform the substitution r
A:
r + AIRT = /32A ,
(10.2.14)
Using the notation Iyot = r, we deduce
dr
dA
7,\T+:+
vrr2
-+#2r2
(10.2.15)
7,
and then
r
El =
00
a-WA
Mr/ A +r
a wA
+r
00
da = fo
Mr
lp
dA
A
a-iWA
A +r
o
r
M
lo
= Ko(wr) - 2 [Io(wr) - Lo(wr)] - f
dA
dµ,
el + µ2
(10.2.16)
Ko and to being Bessel functions, and Lo the Struve function [1.16].
The expression of El may be derived taking into account that 02/0y2 =
c72/c')r2 . Using the relations between Bessel's functions and their derivar
tives, we deduce
-w2
rtir/$
1 + µze''wrs`dµ .
0
In E2 we make the notation iIyoj = u and then the change of variables
r--# B . r=usinh9.
(10.2.17)
Observing that 82/8y2 = (3282/&u2 , we deduce
8
e&oRX
=7
W2 1
/32 u2
+ j92 T3 e x + iwM R2 e-x - yo R2 e wx K2-
/'s0 (r - MR,)2efr-Mx.)
dr
J0
Rr
where
/ 2X = ao - MX, R =
xo + u2 ,
Rr =
r2 -+U2.
408
THE UNSTEADY FLOW
Performing inside the integral the change of variable T --- A::
r - MR. = Q2A ,
(10.2.18)
we notice that the formula (10.2.15) remains valid because it depends
only on A,12. One obtains the identity
1
I
F2+ (j4
s0 {r -
dT
R
yo Jo
=
(10.2.19)
1
z/Q'
A2+yoe'"'xdA,
Y6
-Mlyoi/$
Taking these results into account we obtain
xoe ,x
n(xo,yo)=ni+n2=-Q2 R +iw R-xoM
R K1(wr)
2
w,x
#2y02e
-
III (wr) - L1(wr))
+W2
-w2 jJ11/1)
1 + µ2ek`'r0dp
f ar s
A,
A2 + r2e d
(10.2.20)
where, for the sake of simplicity we maintain the notation r = 1yol, and
X and R are given in (10.2.13). Obviously the line yo = 0 is singular.
Employing this form of the expression nl + n2 in (10.2.11) and
imposing the boundary condition (10.2.3) we get the following integral
equation
4a jf f
n)N(xo, ?ro)d d r) = H(x, y) , (x, y) E D,
(10.2.21)
where
N(xo, yo) = e-"n(xo, yo)
(10.2.22)
is the kernel.
Since w 4z 1 we have the asymptotic expansions
Ki(wr) = 1-, +O(w), Il(wr) = O(w), Ll(wr) =
W
7r
+O(w) (10.2.23)
we deduce
limn=-4(1+...),
t/0
R
(10.2.24)
409
THE OSCILLATORY SURFACE IN A SUBSONIC STREAM
i.e. just the kernel from the steady case.
In the case of the incompressible fluid one obtains the kernel setting
M = 0 in (10.2.20). From (10.2.13) it results
X=r0, R= xo+yo,
(10.2.25)
whence,
n(xo,yo) =-XoeR +iw y-'e4'0 - iKI(wr)2
-2w
(I1(wr)
- LI (wr')) + Uf
;07 J
0 so
,\2
+ r2eWX d A,
(10.2.26)
This is the kernel for the ineompressible fluid
10.2.3
Other Expressions of the Kernel Function
Because of the Bessel functions, the expression (10.2.20) is considered
to be complicated. However it does not contain divergent integrals. The
idea concerning the introduction of these functions may be found in
[10.86] and it was also taken into consideration in (10.17J. We obtain
another expression of the kernel function, starting from the fundamental
solution (2.4.13). Indeed, with this one, the component w(x, y, z) is
written as follows
w(x,?!,z) = a
fj f (.,?1)n(xo,yo,z)d dn,
where
2
,2
n(xo,yo,z) = a
f
=o
000
R1T
dr
(10.2.27)
( 10.2.28 )
with the notation
Rir = Vr2 + /32(y02 + z2) .
(10.2.29)
The passage to the limit and the derivation with respect to z do not
interchange (only the passage to the limit and the derivations with respect to y interchange; for this reason we gave the expression (2.4.23)).
Therefore we shall derivate at first with respect to z. The expression
410
THE UNSTEADY FLOW
(10.2.28) may be found at Watldns [10.87], Williams [10.89], Dowell and
others. Performing the change of variable r --t A :
,Q2A = T
- MRir,
(10.2.30)
we deduce as we have already seen,
dT
dA
'
A' +s'
s2yp+z2.
(10.2.31)
Hence,
E(xoyo, a)
110
eMR,)
R11.
0o
/X1
dr=J
dA
el"
(10.2.32)
A+
oo
where
32X1 = xo - M xo + p2s2.
(10.2.33)
Derivating we obtain
8E _ _
8z
ek'X,
Me
R /X+
(10.2.34)
iwX1
:-.0 8z2
dA
I+s
xo +
E
fim
Ma
ewa
JA1
0
dA
W (A2 + y02)3/2
where
R
V xo+/32Uo, A2X1
=xo--MR.
(10.2.35)
Imposing the boundary condition (10.2.3), we get the integral equation
(10.2.21), where
M ew'X1
R Xl +
n(xo, yo)
x,
J-oo (A2 + g2)3/2
d A,
(10.2.36)
Obviously the line yo = 0 is singular. This expression of n(xa, yo) is
simpler than the expression (10.2.20), but here the integral is no longer
convergent. We must therefore consider the Finite Part. The expression (10.2.36) is utilized by Ueda and Dowell (10.80), Ando [10.3], etc.
Utilizing the identity
1
X?+yo
Mxo + R
M
xo+yo -x0 -X1,
(10.2.37)
411
THE OSCILLATORY SURFACE IN A SUBSONIC STREAM
one obtains two other expressions for n(xo, yo) .
For the incompressible fluid (M = 0), we deduce
eirra
1XO
n(xo.ieo)=- J
(,\2 + y02)3/2
To
d a.
(10.2-38)
If the unperturbed flow has the sound velocity, we obtain the kernel
considering M - 0. Since
Al
lim X = lim x° -
At--1
Xo +
yo
02
AM-1
= xo - yo _X1
(10.2.39)
2r
it results
xl,
xo 2+ 2 e
a nw
N(xo, yo) = -e
a
'
' + J oo (A2 + y 2)3/2
dA
1
.
J
(10.2.40)
This is the kernel for the sonic flow.
One may verify the similarity of the representations (10.2.20) and
(10.2.40), if we employ the relation
eiA
X
e"
J1 =
Ix j,\2 + yo)3/2 d J1 + Kl (wr) +
r
(10.2.41)
+i 2 r ill (wr) - Li (wr)J
which can be easily proved if we take into account the formulas
1 °°
( t 2 CD s 2r ) 3 / 2
d t = r KI wr) ,
(10.2.42)
sill Wt
Joy
ir W
(t2 + ,.2)s/2 d t = - 2 r
[11
- Ll (fir)]
given in the tables dedicated to Fourier transforms [1.16].
From the integral representation of the function Ko we deduce the
identity
rX
ei'.3(r-MRjT)
1-,0 (t2 + r2)3/2
dT
2Ko(w
yo + z2)
jf
oo
R1,.
d T'
(10.2.43)
with the notation (10.2.25). Putting this in 10.2.24) and deriving according to the formula
8z2
82
X32 8
s 8s '
(10.2.44)
THE UNSTEADY FLOW
412
we obtain to the limit
2w
-iWxo
K t (wr) -
/x
(iWN1 +
eo(r-rlRa.)
Q2
dT
Ix
(10.2.45)
where
r2 + 012
Ror =
(10.2.46)
This is another expression of the kernel.
The Structure of the Kernel
10.2.4
The identity (10.2.41) makes possible to deduce from (10.2.32) the
structure of the kernel in the vicinity of the singular line yo = 0. Indeed
for a small r we have (1.40]
1
K1(wr) =
1+
wr
2
In r + ...
,
Ii (wr) = wr + ... , Lt(wr) =
2
a
(10.2.47)
+ ... ,
such that
X
eiWX
sadA+iI
- Jo (ACos
Jo
+ yo) /2
J'00 (A + yo)
3/2dA=
+72 -
tr
(A2
tn AdA+
3/0)3/2
+
2
+ 2 IAr + ... ,
(10.2.48)
the points representing series of integer powers of r.
If X > 0 the last two integrals have no singularities. Indeed, using
the expansion formulas for
LA
d a .. .
(A2 + yo2)v+1/2
given by Ueda (10.81], we deduce
X
°T
(A
10
)3/2 d
x sinwA
J0
00
(A2 + r2)3/2
A =E (
(2n)1 2n
n=0
00 (_1)nw"+i
d A - nL-.o
(2n + 1)! 12n+1,
(10.2.49)
413
THE OSCILLATORY SURFACE IN A SUBSONIC STREAM
where
=1rX (A2 + r2)3/2 d A =
Atm
Im
Xm-I
1))(,-3 X2 + r2 - (m - 1)(m - 3)J,.-4.
(M
X +*
(10.2.50)
Using the notations
Jm=
o
0
+r2dA=X--i(X2+r2)312-(m-1)r2Jm-2,
Am/A2
(10.2.51)
we may calculate the coefficients I,,, step by step, noticing that
IX
1
r2=rs
+
0=
11 = VTr+--r-l -r.
The formulas (10.2.48) and (10.2.49) give the structure of the kernel.
10.2.5
The Sonic Flow
If the velocity of the stream equals the sound velocity (U. = ca ),
then one obtains the solution of the problem from (10.2.6), (10.2.7) and
(10.2.12) setting M --# 1. We have shown in (10.2.27), that in this case
there are perturbations only in the x > 0 zone. When we shall pass to
limit we shall consider therefore x0 > 0. We have
limp Go (xo, Spin, z) =
exp
4
[iW
Jim M2x0 _MR,1
z0exp-iw
2
+2xo
=
i - Go
(10.2.52)
Z2
X02
lime G(x0, y0, z) =
4
lim X = lim
M-.1
M-.1
exp I
-i.
w
J
z0-MR =
hml E2 = 110 exp
T-
u.
2(
(xo
2
= Gl ,
- 14),
)T
(10.2.53)
414
THE UNSTEADY FLOW
From (10.2.16) it obviously results lim E1 = 0 . Hence, the representation obtained from (10.2.6), (10.2.7) and (10.2.52), gives the perturbation in the sonic flow and (10.2.12) gives the kernel of the integral
equation. This is
s
r
nl+n3 = oexp 1 2 (x0 -
zo
92
J
L
10.2.6
iW
1
2 (T -
y02
)1
dT .
(10.2.54)
The Plane Flow
Acting like in 5.1 we obtain the formulas for the plane flow from the
formulas which characterize the three - dimensional flow. We assume
that in the equation of the perturbing surface (10.2.2), ho(x, y) has the
form ho(x), i.e. it has the same form for every y. With a suitable
choice of the reference length Lo, the domain D will be rectangular
with -1 < x < 1, -b < y < b. In (10.2.6) and (10.2.7) we shall have
f = f (r;) . Considering b - oo we obtain
+00
go (X0,
z) =
J
Go(xo, yo, z)d r1=
00
eik(Mxo-Ro)
1
=97r ED
dr1=
Ro
e
QHoe)(k Vxo+ 62z2)=
00
G(xo, yo, z)d r1=
g(xo, z) = f
xo +
Q
e(xo, z) = r-00 (E1 + E2)d r1= roo g(r, z)d r ,
+00 02
81/2G(xo, yo, z)d q =
co
+oo 02
J
G(xo, yo, z)d r1= 0 .
ao
X7
(10.2.55)
22
(10.2.56)
Using these results we obtain from (10.2.6) and (10.2.7) the representations (10.1.12) and (10.1.13).
Taking into account the formulas of Fresnel
1 sin x2d x = fo
0
00
oos x2d x
YY
OSCILLATORY PROFILE IN A SUPERSONIC STREAM
we deduce
415
r+ e-In"'du= Vjj(1-i),
/
.
(10.2.57)
o0
and then, with (10.2.54)
+00
=
it
-oo
(ni + rn)d q _
+ i)
Trw
OX-0
iw
exp(xo)
- wi2
JL
Zo
iw
exp(2 r)
d7
r
(10.2.58)
This is the kernel of the two-dimensional sonic flow.
10.3
10.3.1
The Theory of the Oscillatory Profile in a Supersonic
Stream
The General Solution
One considers that Carrick and Rubinow 110.241 have investigated
for the first time this problem. They have utilized the method of the
pulsating sources potential. The presentation which follows relies on the
fundamental solutions method [10.17]. The integral equation is solved
explicitly. One obtains finite formulas for the lift and moment coefficients.
One utilizes the dimensionless variables (x, y) defined like in (2.1.1)
and the representations (2.1.3) for the velocity and the pressure, in order
to deduce that in (2.4.8), (2.4.18) and (2.4.21) the perturbation pressure
and velocity determined by the action of an oscillatory force having the
form felt, applied in the origin of the axes of coordinates, have the
form (10.1.4), where
P(x,y) = _(f V)Go(x,y),
v(x, y)
=f
(10.3.1)
1e-k.% [G(r, y)d r] +
(10.3.2)
r]
+f2e-k''x [2k4G + k2Cr + w2 f C(r, y)d r]
Utilizing the notations W = w1k2 , v = wAf and H for the function of
THE UNSTEADY FLOW
416
Heaviside, in (10.3.1) and (10.3.2) we have
Go(x, y) = H(x - klyi)so(x, y),
(10.3.3)
G(x, y) = H(x - klyl)g(x, y),
where
a = vM
2kgo(x, y) = Jo
(''- k2y2) e`ia"
2k9(x, y) = Jo (vVx2 -
k
)
(10.3.4)
e-k:,x.
As it is shown in (2.4.22) the perturbation (10.3.1) and (10.3.2) is
zero outside Mach's dihedron with the edge on the Oz axis and the
opening 21,t . This is the fundamental solution of the problem.
Let us consider now that the uniform flow having the Mach number
M, is perturbed by the presence of a profile whose equation is
y = h(x)exp(k it) .
(10.3.5)
Taking the origin of the reference frame on the leading edge and the
length of the chord as reference length L0 , the function h(x) will be
defined on the interval [0,1] . Replacing the profile by a forces distribution (0, f)(t)e''t defined on [0,1] , one obtains the following general
representation of the perturbation
P(x,y)
_-/
f
o
+1
i
f (f )e
iWxo
L2iwG(xo, y)+
(10.3.6)
-G(xo, Y) + W2 f G(r, y)d rJ d
00
.
J
Taking into account the definition of the function H(xo - kIWI) and
the formulas (A.3.15), with the notation
X = x - kIyI,
(10.3.7)
417
OSCILLA'T'ORY PROFILE IN A SUPERSONIC STREAM
we deduce
jf()Go(xoiY)d t _
= H(X)
y f t f (E)9o(xo, y)d = H(X) Jo, f W
`""°G(xo, y)d t = H(X) f f
9o(xo, y)d
,
°g(xo, y)d t ,
o
0
f
(xo, y)d =
=f1
f
f
J=
X
f' f
A
[H(x) I f (t)9o(xo y)d
(t)e-'.."`og(xo,
ff(t)-iW8 ax (H(xo-kI yI)9(xa, y)ld
y)d(xo - kjyj)d4 + H(X) f f (f )e-k"19. (xo, y)dk,
o
:o
= H(X) f to9(r,y)dr.
G(r,y)dr,
kM
00
(10.3.8)
Hence the solution (10.3.6) is
P=O, u=0, ifX<0,
(10.3.9)
X
P(x,y) =
-Jo
u(x,y) = fo f
if X > 0,
(e)e-iwxonl(xo,y)dt+ Zf(x)exp(ialy)),
ifX > 0,
(10.3.10)
where
xo
n i (xo, y) = 2iwg(xo, y) + k29=(xo, y) + a'Z
f 9(r, y)d r .
The formulas (10.3.9) show that the perturbation produced by the
profile propagates only in the interior of Mach's angle with the vertex in
0, and (10.3.10) that in a point M(x, y) from the interior of this angle
one receives only the perturbation produced by the segment OMo (fig.
10.3.1).
THE UNSTEADY FLOW
418
Fig. 10.3. 1.
The Integral Equation and Its Solution
10.3.2
For 0 < x < 1 we deduce
v(x, 0)
p(x, +0) - p(x, -0) = f (z)
(10.3.11)
= if (x) + 2 I f (t)N(xo)d t,
(10.3.12)
where
N(xo) = e-"
=
ym
n1(xo, y) _
s
k
(Jo(vzu) + iMJ1(vxo)J e
+ke
J
Jo(v-r)e rd z ,
0
(10.3.13)
Imposing the boundary condition (10.1.11) one obtains the following
integral equation
k f (x) + J f (t)N(xo)d t = 2H(z) ,
0 < x < 1.
(10.3.14)
0
This is a Volternz type integral equation of first order. We solve it
using the Laplace transform. One knows (see for example (1.32)) that
the Laplace transform of a certain function g(x) is the function g(p),
defined by the operator
£(g) = jg(x)e_Pxdz
(10.3.15)
419
OSCILLATORY PROFILE IN A SUPERSONIC STREAM
where p can be a complex number(p = pi + ip2) whose real part is
positive.
[JX]
Applying the operator G in (10.3.14) we obtain
kf + fo
aPxdx= 2A(p) .
Here we shall change the order of integration. In figure 10.3.2 we observe
that the domain of integration is D (for a given x , C goes from 0 to
x). But D can be also covered integrating at first with respect to x
to oo and then with respect to . We have therefore
from
k7+
f (t;) [f°°e_P0tN(xo)dx]d C= 2R(p)
.
(10.3.16)
XAK
0
F'ig. 10.3.2.
Using the change of variable z -+ u : x - = u, we deduce from
(10.3.16)
(k+R)7=2f1.
(10.3.17)
In order to determine the transformation R we shall utilize the formula
1
(10.3.18)
G(Ja(vx)1
=
which may be found in the tables with Laplace transforms (1.161, [1.32],
11.331. So, using the notations
Nl (x)
=
Jo(vx)e-i.x
, N2(x) = Ji (vx)e ix" ,
(10.3.19)
N3(x) = e-16'" I = J0(vr)e W-dr,
0
THE UNSTEADY FLOW
420
with a = vM , we obtain
00
C(Nl} = f Jo( vx )a- (p+")xd x =
(p + ia} + v2 p + is +
00
£(N3) =
(p+ ia)2 + v'j'
v
1
G(N2) =
1
e-pie-"[f
dxJ
(p + ia) + v2v2 '
Jo(pT)e-a d r ]
0
=
(10.3.20)
°O
fc*
Jo(vr)e`+E [je
ir-"'i"dxI dr
r00 e-(P+",)udu =
Jo(t r)e_u/M dT
0
0
1
1
(p+ivM) +v P+iw
With the notations
iwM '
Pt =
M+1'
UJAf
Af-1'
(10.3.21)
it results
1
(p + ivM) + v =
(P + P1)(P + p2)
k(k+N) [(P+iw) (P+Pt)(P+P2), =k(P+Pi)(P+p2),
A=
+k[p+ivM-Vll»PI)(P+P2),+
P11
+
)(P + P2)
w2
1
+ kP+iw}
(10.3.22)
and then
OSCILLATORY PROFILE IN A SUPERSONIC STREAM
not
g
_
k
_ p+P2-P2+iw
p+iw
k + N
(p + p1)(p + p2)
M1)1/2(p+
421
(p + P1)(p + p2)
M+M1)-1/2_
(p+
M
1w+1)
M -1/2
iw
M - I (p
iwM
1/2
(p+M+1)
M
whence
kf = H+ g.
(10.3.23)
From the tables with Laplace transforms [10.581 it results g = L-1(g)
and then with the convolution theorem
x
kf (x) = 2 f H (xo)g()d
= 2H(x) - 21-w f H(xo) [Jo(ve) + iMJ1(v4)1 a `t d t.
0
(10.3.24)
This is the solution of the integral equation (10.3.14). It was given in
[10.171.
10.3.3
Formulas for the Lift and Moment Coefficients
The lift and moment coefficients have the form
CL = cLexp(iwt), CM = cmexp(iwt),
(10.3.25)
where, because of the formula (10.3.11),
CL = -2
f
1
1
f (x)d x, cM = -2 fo x f (x)d x .
(10.3.26)
0
We considered that the length of the chord Lo is the reference length
and we defined
CL =
P
' CAt =
(1/2)poU*20Lo
(1/2)poU.2Lo '
P being the lift and M the moment on the direction Oz z.
(10.3.27)
422
THE UNSTEADY FLOW
Utilizing ((10.3.24) we find
4iv
f1
kAl 0
CL =
-4
CM =
-411 xH(x)dx +
J0
H(x)dx +
1
4iv
e 'a`d
(Jo(vl) - iMJI
,
f 1 G(t) (Jo(vt) - iMJ1(ve)J a '&(d
0
(10.3.28)
where
J
1
it x H(x - )dx
H(x - t)d x,
1
E
(10.3.29)
.
The coefficients (10.3.28) may be calculated numerically on a computer.
Another method consists in approximating the function h(x) by polynomials whence one deduces that CL and cAf may be expressed by means
of the terms having the form
f" (M, a)
=10
"Jo(vl;)e'°(d
1
(10.3.30)
9n(M,a)=
frJi(z)e'd7
Taking into account that Jj(z) = -Jo(-) and integrating by parts one
obtains that
vg, _-Jo(v)eis+nfi_1-isf", n=1,2,...,
(10.3.31)
vgo = -Jo(v)e is + 1- lab.
These formulas show that g" may be expressed in terms of f,,,. Integrating f by parts, we deduce
laf" = -Jo(v)e-' + of"-1 - v
ivM j '
f
1
0
F"Jlv )e-'f d
,
(10.3.32)
_ -Ji(v)e+ of+
(10.3.33)
+(n - 1)11
0
Substituting (10.3.33) in (10.3.32) we find for f" an expression which
contains the last term from (10.3.33). This may be eliminated with the
423
OSCILLATORY PROFILE IN A SUPERSONIC STREAM
aid of the relation (10.3.32) where n was replaced by n - 1. After all
one obtains
na
+ (n
a
Jo(v)e-i. - 1 Ji(P)e ia+
1)
(10.3.34)
1)2A-2
+ 1(1 - 2n)fn-1
This formula shows that all the terms fn may be expressed by means
of fo. This result was given for the first time by Schwartz in [10.70]. In
the same paper one gives the following expansion for fo
00
[,
fo=e
n=O
IMM1:Jn(a)+iJn.f.1(a)w".
(10.3.35)
2"ni(2n + 11
In [10.701 one gives tables with the numerical values of fo, with eight
exact decimals, for 1 < M < 10 and 0 < a < 5. In [10.33] one gives
the numerical values of the functions fn for n = 0,... , 11.
10.3.4
The Flat Plate
For the flat plate having the angle of attack -E (h = -ex) we deduce
H = -2E (1 + iu;x) such that it results
s
CL = -
E
[ 2 f2 + iw(2 + iw) fi - (1 + 2iw -
s
,
2
2e
CM = -
)f of
(10.3.36)
z
[f3+f2_2+2)f1 - 2(iw - 3 )fo]
w2
These formulas are sufficient if we utilize the tables for
, f1fo, f2' fs
For w -+ 0 one obtains the well known formulas of Ackeret
4e
2E
CL = k , cM= k .
Obviously, cL and cm may be expressed only by means of fo if one
utilizes (10.3.34).
Noticing that
f = fn + ifn ,
(10.3.37)
where
f1
f, = J
1
fn = -f eJo(4)sin(a)de,
424
THE UNSTEADY FLOW
we deduce from (10.3.36)
CL = c'L + icL , cAf = (.! + icM
,
(10.3.38)
If the equation of the plate has the form
y = -ex coswt = Re [-ex exp(iwt)J ,
(10.3.39)
then
CL = crL cos wt - CIL sin wt,
(10.3.40)
CAS =
cos wt - ciAf sin wt,
these formulas give the variation of the lift and moment coefficients
versus the time. For example, for w = 7r and M = 2, we obtain
CL = e(-9.2060 cos art + 11.8941 sin zrt)
,
(10.3.41)
CM = e(-6.8779 cos in + 17.5209 sin irt)
10.3.5
,
The Oscillatory Profile in the Sonic Flow
We are interested in the behaviour of the formulas of Nl and N2
when M 1 (k -- 0). It results v -+ oo such that we shall utilize the
well known asymptotic expressions
Jo(z) =
F2
Coo
7r
(z - )+0(.-,),
4
Jl
(z) =
(10.3.42)
r2z ooe (z - 34) + O(z-1)
for great values of z. In this way, we deduce
[Jo(vxo) + iMJI (vxo)J exp(-iaxo)
NJ =
_
k
1
=w xo
[(1- iM) cos axo cos vxo + (M - i) sin axo sin vxo-
-i(1 - iM) sin axo cos vxo + i(M - i) oos axo sin vxoJ ,
(10.3.43)
425
OSCILLATORY PROFILE IN A SUPERSONIC STREAM
and analogously
N2(x) = k
_
10"
Jo(vre)e'du
1-i
coe0(M-1)u+isinW(M- 1)ud u +I,
2fJo
Mu
(10.3.44)
where, with the change of variable u -+ t : u = (M -1)t, we have
_
cosw(M+1)u-isin'(M+1)u duI _ 2l+i
two
Mu
(10.3.45)
xo
l+i M-1 IMexp(-iwt)dt
2 xrw
ft
M Jo
Taking into account that we also have
xo
f
M -1 gyp( Ld)dt
slim Ja
- Jo
)d
t = w (1 +
it results that
,
No(xo) = J m1 N(xo)
=(i+1)
we
r 1xoexp(2iwT O ) wi
L
YYY
iw
2 Jexp(2 u)
I
du
,
(10.3.46)
he. exactly (10.2.54).
The integral equation (10.2.14) reduces to
fo f (4)No(x - 4)d4 - 2H(x)
(10.3.47)
This is also a Volterra-type equation of first kind. 'Lbt integrating it we
shall use again the Laplace transform. Applying this transformation we
deduce
(1 + i)% 7 = 2$b0,
where
§ o'=
(P+ 1w
2)
1f2
iwf
+ 2 (P+
iw
2
}`
(10.3.48)
1/2
(10.3.49)
426
THE UNSTEADY FLOW
From tables (see for example [10.581) we have that
G
p+iw/2,=exp(-)G[vii=-2exp
(-)
(10.3.50)
such that we obtain
go
2 Rx(i``' -
x)exp (
i2
x)
(10.3.51)
and using the convolution theorem, from (10.3.48) we deduce
After determining 1(x), the lift and moment coefficients result from
(10.3.25) and (10.3.26). We shall give calculation formulas in 10.5.2
when we shall consider again this problem.
10.4
10.4.1
The Theory of the Oscillatory Wing in a Supersonic
Stream
The General Solution
The theory of the oscillatory wing in a supersonic stream, was conceived according to the model of the theory in the subsonic stream. The
papers of Kussner [10.37J, [10.38] represent the starting point of this
theory. We mention then, the study of Garrick and Rubinow [10.25]
where the potential of the pulsating source is determined, the paper of
Miles where one considers the symmetric arrow - like wing, having the
leading edges outside Mach's cone [10.53, the paper of Nelson for the
triangular wing [10.57], etc. But the fundamental work in this domain
is the paper of Watkins and Berman [10.85). Here one may find for the
first time the integral equation of the problem and various forms of the
kernel. The method is similar to the method from the subsonic case.
From the potential of accelerations of a pulsating source, one obtains,
deriving with respect to z the potential of accelerations of a pulsating
doublet. The potential of the flow is obtained superposing the doublet
potentials. The boundary condition gives the integral equation of the
OSCILLATORY WING IN A SUPERSONIC STREAM
427
problem. In the following papers, due to Ashley, Windall and Landahl
[10:4], Landahl [10.44), Stark [10.72], Harder and Rodden [10.29], Ueda
and Dowell [10.81] the theory was developed and numerical methods for
the integrations of the equation of Watkins and Berman were given.
We shall indicate in this subsection how one may also solve this
problem by means of the fundamental solutions method. Assuming that
the equation of the wing is (10.2.2), we shall use distributions having
the shape
fe".,c
= (0, 0, f)e' .
(10.4.1)
Utilizing (2.4.9)we deduce that the perturbation of the pressure determined by such a force applied in the generic point (t, n, 0) is given by
the formula
(10.4.2)
p(x, y, z) = f azGo(xo, yo, z).
For the component w, it results from (2.4.13)
w(x, y, z) = - f e "" `0H(xo)b(1M)6(z)+
82
+f
:o
8z2a-""`O,
co
(10.4.3)
G(r,yo,z)dr,
and from (2.4.23)
fe",,xa[(2iw+k28x) G(xo,yo,z)+
w(x,y,z) =
8z
02)
+
(10.4.4)
IZO
G(7-, yo, z) d r ] ,
00
where we denoted
Go(xo, yo, z)
G(xo, 3fo, z)
= 2A
=
H(xS
s)
H(xSo
s)
cos (LS)e-iaxo ,
(10.4.5)
cos (vS)e-o"
1
k= M -1, 1=w/k2, v=OM, a=vM,
s=k yo+x , S= xo-s , ST= 'r -3 ,
(10.4.6)
H being the function of Heaviside.
One may prove, taking into account the formulas (2.3.35) and (2.3.36),
that the perturbation given by (10.4.2)-(10.4.5) vanishes in the exterior
428
THE UNSTEADY FLOW
of Mach's cone with the vertex in the point ((, i, 0) and with the axis on
the direction of the unperturbed stream (the Ox axis). Using a forces
distribution having the form (10.4.1), applied on the domain D - the
projection of the wing on the rOy plane, the perturbation will be given
by the formulas
p(x, y, z) =1
JD
w(x, y, z) = -6(z)
+21-
f (C OF Go(xo, yo, z) d d rl,
J JD
f (t,
i7)e-'
°H(xo)b(yo)d
(10.4.7)
d q+
(10.4.8)
f f f(e,q)e
D
w(x, y, z) = 2 .
wh ere
nl (xo, yo, z) _
IL f (t, q)e-'"" °n2(xo, yo, z)d t d i
82 f.0
z2
H(T - s)
0o
co$ (vSr) e
S
r
(10.4.9)
,
dr=
(10.4.10)
82
_ z2 H(xo - s) J
n2(xo, yo, z) = (2iw + k2 40)
+[w2-
\
10.4.2
02
2
Cos (VSr)
H(xS- s)
r,
cos (vS)e"'Ww'+
)H(xn-s) r
a
°sr)e-OrdT,
Cos
Sr
(10.4.11)
The Boundary Values of the Pressure
They may be obtained writing
p(x, y, z) =
o - s)
Cos (vS) a ;
, d t d q,
27r C7z
S (10.4.12)
and noticing that because of the presence of the factor H (xo - a), the integrand differs from zero only in the domain DI defined by the inequality
xo > s for a given M(x, y, z). This inequality is equivalent to
(t -x)2-k2(q-y)2> k2`z2,
429
OSCILLATORY WING IN A SUPERSONIC STREAM
which are solved in 8.3.3. Denoting by M'(x, y, z) the projection of the
point M on the xOy plane and X = t - x, Y = rI - y we deduce that
Dl is the foregoing branch of the hyperbola X2 - k2Y2 = k2z2 (fig.
8.3.4). When M' is in D, the hyperbola degenerates into the half-lines
X = ±kY (fig. 8.3.5)
Since the function f is defined only on D, we shall prolong it in
the outer region taking it equal to zero. It follows that in the perturbed
region from the fluid we have
I
f (C, n)Go(.To, yo, z) d d q=
Qi
l
J
ly- f()
cos (vk
(Y+-,)(,-(Y+
- 77)(9- Y_)
(10.4.13)
where
Yf = y
xok-2
With the change of variable q - 0:
2(Y++Y_)- 2(Y++Y_)cos0 = y-
- z2 .
(10.4.14)
P-
ok-2 - z2cos0, (10.4.15)
we deduce
21rk JO
cos
a-iwco 110 f (C y
-
xpk-2
z2cos0) .
(- z) 2sin 0d BJ d
whence, if f (x, y) is a continuous function,
P(x, y, ±0) _ _]
az I = Tf(x, Y)
(10.4.16)
P(x, y, +0) - P(x, y, -0) = f (x, y) .
(10.4.17)
and then
Hence, like in the previous sections, f represents the jump of the pressure on the wing.
430
THE UNSTEADY FLOW
10.4.3 The Boundary Values of the Velocity. The Integral
Equation
For z 0 0 the first term from (10.4.8) vanishes (5(z) = 0). It
has to be considered in the same way in the limit values for z - ±0.
The remaining term is the kernel given by Watkins and Berman [10.85).
Elementary calculations give
0
az
_ k2 z e
82
8s
22
8
k4z2 82
- k2s (1 _ k2z2
-s2-)T. + 2 8s2 .
(10.4.18)
In the cited papers one considers that the terms which contain the factor z2, vanish when z - 0. But this is not always true (see for exampie (3.1.20)). This is true when the factors which multiply z2 remain
bounded when passing to the limit. In the following we shall see that
for (10.4.10) the form obtained under this assumption is correct.
Hence we shall consider the kernel
Cos(1S')e 'dTJ
82
nI(xo, yo, z) °-`
8 {H(xo_s)
(10.4.19)
s
The derivation is performed according to the formula (A.3.15), but we
have to take care that for s = xo the integrand is unbounded. We
eliminate this inconvenient writing
(20Cos('1r2 -s')e '''rd7=
T -8
is
0 e-0 - e+
/s
T
-s
V7-r- S
Js
d T + CO°
TO
1
e' d-r+
dr
T -8
12
(10.4.20)
After all
k2
ni (xo, yo, z) =
s
H(xo - s)1,
(10.4.21)
s ) e "°TdT ,
(10 .4 .22 )
where
0
8s,
=o Cos
(
r
r
the derivation being possible if we utilize the equality (10.4.20), but
we have no interest to do it. The integral may be calculated with the
OSCILLATORY WING IN A SUPERSONIC STREAM
431
substitution r - A : r = scoshA. Deriving one obtains
1
- -rpcos(vxo-3 )e
S
O
V xp
zp
-.%1
-r
e'"''r
s
d
dr [sin (v
r2 - s2)] d r-
esin(v r2 - s2)d r
(10.4.23)
.
We integrate by parts in the second term from the right hand side.
Passing to the limit in (10.4.21) we notice that, like in the steady case,
it appears the singular line yo = 0. After eliminating from D the
domain D, defined by the inequalities y - E < tj < y+E we shall put in
the remaining domain z = 0. One obtains the following singular kernel
It 1(ro,yo) = lim nt(xo,I/o,z) _
Yo
+
ff
e-V"o sin (vS) + M
Cos(,S)e_k-xo+
u)
-H(x
J
1X0
e iur sin (vSS)d r =
= H (xo - u)n(xo, yo) ,
(10.4.24)
where, u = kIyol and
S=
x0-u2,
Sr =
Jr2-u2.
(10.4.25)
For w = 0 one obtains (8.3.23). This will be the kernel of the integral
equation.
In the sequel we shall give a demonstration where the terms which
contain tht factor :2 are not neglected. As we have already noticed, acting in the classical manner, we have to calculate the limits for z --+ f0
of some kernels which contain derivatives with respect to this variable
(see Nlangler [10.52] for the subsonic steady flow, Heaslet and Loomax
for the supersonic steady flow, Watkins, Runyan and Woolston [10.86}
for the oscillatory subsonic flow, Watkins and Berman for the supersonic
flow, etc.).
Since generally, for performing this calculation we have to evaluate
at first the derivatives, the passage to the limit becomes difficult. In
order to avoid this, we gave other expressions to the component w
432
THE UNSTEADY FLOW
((2.3.29), (2.3.37), (2.4.23)). In the general solutions built on the basis
of these expressions it appears only the derivatives with respect to y.
The passage to the limit interchanges with these derivatives. In the
actual case from (10.4.11) we obtain
n2(xo, yo) = line
n2(xo, yo, z) _
2-.o
_ (2iw + k2)
H(x u) cos (vS)e-'"'%O+
(10.4.26)
cos vS,. ecWrd
T,
82
+(w2 - 8y2 )H(xo - u)
S,)
where u = k1yoI .
Since we have 02/0y2 = k282/c9u2 , with the notation
J_
cos(vST)e-,,tdT,
49
au
(10.4.27)
r
u
we deduce
1,2
-y2 H(xo - u)J =
k2 82 H(xo - u)J = k2 0 H(xo - u)J
(10.4.28)
where J, calculated like 1 ,is
J = - xo caos (ys)e-'ixo
S
u
-!
e
MU
sin (vS)(10.4.29)
2°
W
Mu j
a w''r sin (vST)d r.
For determining H(xo - u)J we take (2.3.35) into account. In this way,
from (10.4.25) one obtains rigorously (10.4.24).
If the equation of the oscillatory surface is
z = h(x, y)e'
,
(x, y) E D
then one imposes the boundary condition
w(x, y, 0) = 8 h(x, y) + iwh(x, y) _- G(x, y), (x, y) E D
(10.4.30)
One obtains the following integral equation
J
1n,
f
n)e`"n(xo, yo) d 4 d n = 21rG(x, y),
(10.4.31)
DI being the domain marked in figure 8.3.5 (the domain where xo > u).
OSCILLATORY WING IN A SUPERSONIC STREAM
433
Other Expressions of the Ker e1
10.4.4
We have
rro
L =1 e-'
sin (vST )d r = 2i (L_ - L+) .
(10.4.32)
u
where we denoted
Tn
=
(10.4.33)
U
In L+ we perform the substitutions r --- A::
T
MS, = kiA.
(10.4.34)
Taking into account that T is positive in both cases we deduce
kT=-uA+uM
1+A2,
(10.4.35)
such that
LT- = tyo(
f/
1
1+A'
e-'+Iyolad A
(10.4.36)
e-i:wlyola d A.
(10.4.37)
whence
L
= -- j
(yo'
2i
(=o+MS)/ku
AtA
z o-AjS)/ku
1+A
-1
)
Since, on the other side,
(x +MS)/ku
eWroad A =
'
4yo
jxo_MS)/ku
sin (PS)
.
(10.4.38)
from (10.4.24) we deduce
Cos (VS) a-iUX0- iw
n(Xo' yo)
yo
S
2Iyo)
=o+MS)/ku
L0MS)/kU
A
e-i&+lYOI-%
dA.
V1--+-A2
(10.4.39)
This is the kernel given by Watkins [10.851. Obviously for w = 0 one
obtains the steady kernel.
434
THE UNSTEADY FLOW
Performing the change of variable A -> v : Jyp)a = v and and integrating by parts, we deduce
' _IYoEa
(x04-MS) f ku
a
2iyol
=-
1+A
.Y+
1
y2+X+
2y0
-
d.1=
iirJ
JX*
ye-n.rv
+L dt
-
2-y6'
X-
-
e--iwx_
+
TY0127X2
c- iwv
1#
1
e-'wx+
e
-d v'
2
x_ {Up + v2)312
(10.4.40)
where
k2X,=xo± MS.
(10.4.41)
Observing now that
2e-0'0 coss(vS)=e-+e-"'x+
(10.4.42)
utilizing (10.4.40) and the identities
xo
yQ + .X+
- SX+ = xo yo + X? + SX_ = Myc'
(10.4.43)
,
we obtain for n(x0, yo) the following form given by Harder and Rodden
[10.20)
2YL(xo, IM)
a-iwx..
e--W,x}
M
e-Iwv
.
V7=`
0+
1j,x
+2
(YO-1
+172)3/2
(10.4.44)
Another form of the kernel is obtained if one utilizes the identities
_
yO+X2
Mxo:F S
xp+yo
_
Al
xe+X
(10.4.45)
One obtains the relation
L12
2n.(ro,rlo)
S
a-ivx«
a swx
o+X_ + xo+X+)
X..
a-iwv
{3lo+tr2)3/2dv,
(10.4.46)
utilized by Ueda and Dowell [10.811 for obtaining the numerical solution
of the integral equation.
OSCILLATORY WINC IN A SUPERSONIC STREAM
435
One obtains the sonic limit at once from (10.4.44), or (10.4.46) notic-
ing that
lim
A!-l
X_=-12 (xOlxo/
-X,
limN--1X=00.
(10.4.47)
One obtains the following kernel
H(ro)
2
2
12g + yo
(xu, Ilo)
+e
iWx
la
+ J/
oo
e'""
(Jp + t')3/2d vJ
(10.4.48)
which coincides with (10.4.26).
A New Form
10.4.5
We utilize the formuhas (see for example 11.30], pp. 406, 422) with
real parameter
O° cos,
cos
"i
(pr)dT = ko(u v2 -12),
(10.4.49)
°° cos (vSr )
i-Sr
sin (pr)d r = 0.
For p = 0 we obtain the identity
r°
ST)e-wTd r. _ Ko(wlyol)
T
u
00 Cos(VST)e
- f.0
rrrdT,
(10.4.50)
T
as follows
2
fi Cw2 - k2
0' f
J
rn Cos (VS_)
u
a lord r =
r
1 Iw--k2ou2/
[Ko(-ku)- f
T7r
.J=f
T
J
In the last part we derive without any difficulty. Deriving, the kernel
(10.4.25) becomes
n2(xo, yo) = H(xo - u)n(xo, yo)
n(a'o,yo) _
r
- wM
kkt(Au)
u
T
J=°
. Cos(yST)e_Ord
-k.2
To
ST
sin(vST)e-`
S2
r,
TdT-
(10.4.51)
'TIE UNSTEADY FLOW
436
where u = kJy I . This is the new form of the kernel. Having in view
the behaviour of lit , for small values of the argument, this is
1
-
22
(I,, Iyul + r1) ,
r, = in 2 + -y
2
(10.4.52)
,
yo
-y being Euler's constant. An additive constant r2 also appears from
the two integrals (10.4.51).
The kernel of the integral equation in the case Al = 1 is obtained
from (10.4.51). We have
illl A2 I\1u) =
1yu1Rt(wIyoI)
Denoting
rao
It = 2iJ
Sr -
7)]d T
ao
I2-2i f exp(-i (MSr+r)1S
1117-
we have.
°O sin (yS1)
xp
r
2
_rd T = It + I2.
But
It
_
1
exp
2l
-
1
21
nhinl It =
f.0
[1'A12S2 - r2
Al ST + r
r2
exp iw
AI
(IT
S; =
dr
Aft 2
r2 - k2y + T
T2
,
- k2y0
li
10"exp Pww (r - TA) ] T L.
One obtains after all (10.6.13).
10.4.6
The Plane Problem
such that in the repIn this case, the density f q) becomes f
resentation (10.4.7)-(10.4.9) we can calculate the integral with respect
437
OSCILLATORY WING IN A SUPERSONIC STREAM
to r l. We have
90 =
f
00
Co(xo,Uo,z)d9=
+oo
1
J
= 27
H(xo - k r2 + z2)
cos[v xo-k (r +x
- k (r + z)
zo
)J
dr.
(10.4.53)
Because of the presence of the function H, the integrand differs from
zero only for xo > k r + z . This inequality implies zo > 0 and
k2r2 < x2 k2z2, whence xo > kizl and -c < kr < c, where c =
xo - k z2 . After all
-
oos{v
2-e'H(xo-klzl)f
90
-
C
-k r dr.
r
(10.4.54)
Utilizing the formula 11.16]
` Cos (p
c- x-x)
(10.4.55)
gxdx= 2Ja(cjp2+g2),
it results
go =
H(xo - klzl)Jo(v xo - k2z2
(10.4.56)
and analogously
9
Gd q = gH(xo - klzl)Jo(v
roo
+00
e
=
J
Edrt =
ao
1 jr-a° H(r - kjzj)Jo(v
_ H(xo - kjzj)
2k
k2z2)e'
xo
Jo(v
0,
(10.4.57)
r2 - k2z2)e''O'd r =
r2 - k2z2)e rdr.
fk1Z1
(10.4.58)
For obtaining the results from 10.3 we have to consider the chord of
the profile on the Ox axis (0 < x < Lo) and to take Lo as reference
length. Observing that
j
+oc &2
00
OY
2E(xo,yo,z)dq=-J
+00
02
8
Edt =-AEI
00=0,
438
THE UNSTEADY FLOW
we deduce
1
p(x,z) = - I f(e)e sodt,
w(x, z) =
f
(10.4.59)
1
f we-k""°
[(2iw+k2)g+,2c]d
which is exactly the solution (10.3.6). We obtain too
+00
n(xo) =
J
n2(xo, yo)d tl =
+m G(xo,
= 2iw
yo, 0) d >1 + k2
J00
+w2J
a
TX
+x G(xo,
y , 0)d n+
100
G(r,yo,0)dil = 2a(xo)Jo(z'xo)eui""`0+
00
+H(xo)no(xo),
(10.4.60)
where
no(xo) =
T-
T
[Jo(vxo) + 121 Jt(aod
e1 .lo(vr)ed T ,
o
(10.4.61)
i.e. (10.3.13).
10.5
10.5.1
The Oscillatory Profile in a Sonic Stream
The General Solution. The Integral Equation
We proved in (10.2.54) that there exists the limit of the subsonic
solution for M / 1, and in (10.2.45) that there exists the limit of the
supersonic solution for M \ 1, and in addition, the two limits coincide.
We shall prove now that there exists also the solution for M = 1, and
this one coincides with the two limits. It will result therefore that the
flow is continuous to the passage past the sonic barrier, unlike the case
of the steady flow. We shall consider therefore the oscillatory profile
(fig. 10.5.1) of the equation
y = h(x)exp(iwt),
(10.5.1)
439
OSCILLATORY PROFILE IN A SONIC STREAM
Yt
0x
I
Fig. 10.5.1.
which perturbs the uniform flow which has the velocity Ua, = cm
(M = 1). With the notations (10.1.4), the boundary condition (2.1.27)
gives
v(z, t0) = h'(x) + iwh(z), 0 < x < 1.
(10.5.2)
The fundamental solution in the two-dimensional sonic flow is given
by the formulas (2.4.32)-(2.4.34). If the profile is reduced to the skeleton
like in figure 10.5.1 it is sufficient to replace it by a forces distribution
having the form
(0, f)exp(iwt).
(10.5.3)
It results therefore
P(x, y) = --f
a Go,
(10.5.4)
v(x, y) = f ei`''°` [(21&G + w2 T G(r, y)d rj
o
where
r
Go(x, U) = H(x)
exp I _ !(X
+s
)J E H(z)So(x, p) ,
(10.5.5)
1
G(a, Y) = H(x) -OM
[(x -
)}
H(x)9(z, v) ,
with the notation 2f/ = 1.
A continuous superposition of forces having the shape (10.5.3) on
the segment [0, 11, will give the perturbation
i
P(X, U)
I
v(x, p) =
J0
1
f (() gpGo(xo, y)df ,
f (C)e^'"'° [(2iWG(Xo, y) + w2
Jzo
0
G(r, y)d rJ dl;
440
THE UNSTEADY FLOW
Taking the significance of the function H(x) into account. it results that
for x < 0 we have
P=O, v = O,
(10.5.7)
and for x > 0
f 2fV)ryyo(ro.y)d
P(x,y)
To
fr
v(x, t!) =
(10.5.8)
(2iw9(xo y) + w2 f 9(T, y)d r d ti
fWe-ten
I
.
0
(10.5.9)
This is the general solution of the problem. It was given in (10.18.
Performing the change of variable F -« u : u = y'/xo one obtains
°G
iw1Z Jy2
P(x. ,) =
f (x -
y-'
)exp
;r.
[_!f + u)]
ll
Ou-
and then
P(x, ±O) _ T-iwfl f (x) f exp
-'2 It
`
= +,l f (x )
(10.5.10)
It results therefore
p(x, -0) - p(x, +0) = f (x)
(10.5.11)
.
For 0 < x < 1 we deduce
v(x,±0) =
jj
f(Z;)e-'"`°n(xo)dy.
(10.5.12)
ro
27 [
exp(2 xo) - 2
(
}
J exp 'T)
1
.
(10.5.13)
Imposing the boundary condition (10.5.2) it results the following integral
equation
jf()n(xo)d
A(x)
(10.5.14)
where A(x) = h'(r) + iwh(x). The kernel (10.5.13) coincides with
(10.2.54) and (10.3.46).
441
OSCILLATORY PROFILE IN A SONIC STREAM
10.5.2
Some Formulas for the Lift and Moment Coefficients
Taking into iweount that
1-i
=
I
D- 2
(10.5.15)
4 fa-w
the solution (10.3.52) may be written as follows
3(x) = 2iwft
fA
)exp(- Z l;)d
10.5.16)
-2fl J e 12
ex p(-
2 t)u
,
the sign * indicating the Finite Part (Appendix D). Denoting
f
F = Ir A(xo)eXp(- +t)dt
A(,)
2
xo
o
P(_!2xo)df ,
(10.5.17)
with the definition formula (D.4.2), we deduce
1z
o
5.7r
[A(')
l so gyp(
_ iw
2
]
xo)d=
dx '
(10.5.18)
and then
-J
exp(
- 2 )d t - iw F+2df .
(10. 5 . 19)
After all the Jforinula (10.5.16) becomes
1(x) = ti
(ii' + dx
(10.5.20)
The lift and moment coefficients are given by the formulas (10.3.29)
with (10.3.30). Utilizing (10.5.20) we find
cL - SS2 J
1 A(1 - 0 + WB(t)
0
P(- 2
(10.5.21)
c,tt = 80
ri
A{1-
) + iwD(>;)
2 )d
where we denoted
I
1
B(4) _ I A(xo)dx, D(t) = J xA(xo)dx.
(10.5.22)
442
THE UNSTEADY FLOW
Approximating the function h(x) by polynomials, we deduce that the
integrals from (10.5.21) have the form
1
f(
2exp(-2 )d
,
it = 1, 2, ...
(10.5.23)
Integrating by parts we obtain
-.2 exp(-12)+
2n
1W
n=1,2,...
iW
(10.5.24)
It results that all the integrals from (10.5.21) may be expressed as functions of
Io =
J
exp(-
)
Vq
[c(/) - iS(
=2
11
7r
)]
,
(10.5.25)
where C(x) and S(x) are the integrals of Fresnel [1.30):
C(x) =
fS(x) =
1
2.
j
(10.5.26)
-
the notation z = irx2/2.
In the case of the flat plate having the angle of attack c (h = -E.r. )
one obtains
CG = -4f k(1 + iw)
CM
[2exP(_) +(I+ iW)I01
=-aQEK-iw+3tw+e P(12)+(2i
4)i].
+1
(10.5.27)
10.6 The Three-Dimensional Sonic Flow
10.6.1
The General Solution
In the three-dimensional sonic flow the fundamental solution is (2.4.36)
and (2.4.37). A force having the shape (0, 0, f)exp(iwt) , applied in the
origin, will produce the perturbation
P = H(x)p, W = H(x)w,
(10.6.1)
where
P(x,y) _ -f8 go,
w(x, y) = f e'w,x [2iw9 + (w2
-
z) /
JJJ
(10.6.2)
g(r, y, z)d Tl
J
TIIE THREE-DIMENSIONAL SONIC FLOW
443
with the notations
1
go (x, y, z) =
, zy)_
9(x,
4;rrexp
1
.rr
(-
2r.' 2
e`priw
IL 2
(10.6.3)
y2+z
z-
x
The perturbation produced by a distribution of forces having the
form
(0, 0, f (s, r1))cxp(iwt) ,
defined on the domain D (the projection of the wing on the xOy plane),
will be characterized by the formulas
P(x,y,z)
w(x, y, z) =
8
J JD
f (E,
r?}e-"'"`" 1(2iwg(xo,
+(wl - ay ,)J
10.6.2
yo, z)+
(10.6.4)
9(r,yo,z)drJdedr.
The Integral Equation
Assuming that D is such that every parallel to the span (the Oy
axis) intersects the boundary OD in at most two points and denoting
by y_ and y+ the ordinates of these points (fig. 10.6.1), we obtain
110
With the change of variables
x-
s1=y+l:I v;
f
V}}"'d'1 dC
(10.6.5)
r1) -p u, v :
zJdudv
(10.6.6)
THE UNSTEADY FLOW
444
Fig. 10.6.1.
we deduce
P(x,y,±O) =
47r
iw
lim x
lim
fp
tE) f(c 17)
+ yo + ~
zo
2
/
d
dn
22
Z
47r :-.to 2
(X0
I
xo
7- (F)
exp
f (x - -, y + I= Iv)
I
U
(f)-SI/I=I
exp
iw
z2
2
u
+u(v2+1)]}dudv
f /
=
exp [ - Z uv2)dv] du.
Utilizing the integrals of Fresnel
00
I
cosx2dx =2 f
0
20
Cost
dt=
°sinr2dx = 1
f s'n dt = 2
2
(10.6.7)
we deduce
P(x,y,f0)_T2f(x,y)
(10.6.8)
Hence
P(-T, y, -0) - P(x, y, +0) = f (x, y) .
(10.6.9)
445
THE THREE-DIMENSIONAL SONIC FLOW
Assuming that the equation of the oscillatory surface is
z = h(x, y)exp(iwt),
(10.6.10)
we obtain the boundary condition
w(x, y, ±0) = h'(x, y) + iwh(x, y) _- A(x, y) , (x, y) E D.
(10.6.11)
It results the following integral equation
2n
lID f( t, ri)N(xo, j) d t d 0 = A(x, y),
(x, y) E D ,
(10.6.12)
where
N(xo, yo) = amp
[iw(
2x o
y1 +
(10.6.13)
\W2
1
02
'O=p iw
yl
dr
2
We have also obtained this kernel from the kernel corresponding to the
subsonic flow (10.2.58). On this formula we cannot observe yet the
singular part of N We shall calculate therefore the last term. Using the
formulas
I
0
zo
J
exp
2
Ocein
coo
(r -
(ax- z )dx _ {2Ko(2v)
\l LT
(10.6.14)
2K0(11p0D)-1°O exp
f2
(r
}] dr
The last integral may be derived with respect to y interchanging the
derivation with the integration. One obtains
b
Texp ( 2r0)8'r' lrexp (
"4 exp
2
In this way we deduce the final form of the kernel
(.r
N(xo,yo)=-1 ! K1(wlllol)+ 2
_)1]J 7d 1,
2
(10.6.15)
446
THE UNSTEADY FLOW
The principal part is in the first term. Taking into account that for
small values of the argument we have
+zlnz+...,
K,(z)=
(10.6.16)
it results
N(xo,yo)=----w2(lnlyol+r)+....
(10.6.17)
A
The singularity has therefore the same order like in all the other spatial
problems.
10.6.3
The Plane Problem
We remind that one obtains the solution of the plane problem if we
assume that D is a rectangle having the dimensions Lo and bL0 and
we consider b --+ oo. Moreover we assume that every section with a
plane parallel to xOz determines the same profile, hence in (10.6.10) h
depends only on x. Taking Lo as a reference length, the domain D
will be defined by 0 < x < 1, -b < y < b. Considering b oo we
have to obtain
+00
n(xo) _ j
N(xo,yo)dri,
(10.6.18)
o0
where n(ro) must be (10.5.13), and N(xn, yo) (10.5.15).
Indeed, utilizing the representation 11.30]
!OD
J
(t2
C} ``'dt,
(10.6.19)
-2)3/2
we deduce
rc+Oht(_u)du=2
w)+'Kt(wlyol)dyl_2w
00
o
lyol
u
0
=2'(r0°coswt-1+Idt-7rw
Jo
t2
zrw+2J
o
t
J" coswtdt
t.
+2J,
t-
2 16 f00dt
+ 10
7rw
2
=
447
THE THREE-DIMENSIONAL. SONIC FLOW
Utilizing Fresnel's fortnulas (10.6.7) and integrating by parts, we obtain
exp
T2
011171d"
T
=21 cxp(i"T) [f
(1-i)
2
7.ro=
Pxp(
`
2
IIIJII
-10
4:
u2JduJ
tw
expTol+
T)dr
+ iw J exp (t2
=27ri+
27-
vfT
- tw
I
exp (t2 T)
vfT-1
(1-i)[ 2 exp(t-xo)-iwJrexp 2TdTr
7x=o
`\'2
o
We deduce therefore
-'.w [
ri(x0)=(1+i
1
yrr-o- x
p
o
iw
2x
o1 2
ex
p iw ) d T 1
of
2T sfJ
'
(10.6.20)
i.e. just (10.5.13).
10.6.4
Other Forms of the Kernel
From (2.4.13) and (2.4.35) it results the following representation of
the component w of the velocity
w(X ,
, z) = 5(`) JJ f(. rt)H(xo)6(yo) d d i+
t
f
.{.i-
(10.6.21)
n
Jf
where
2
11(x0. yo.
a'2 J
exp
2 (Ir _ yQ+z2/J
T
LT
(10.6.22)
One may demonstrate that this integral is convergent. Denoting r =
iyol and performing the change of variable r --+ A:
7.
T - - = 2A
dT
-r
dA
(10.6.23)
77 7
448
THE UNSTEADY FLOW
one obtains
n(xo,yo,z) =
8 J-.
d.1,
+r
(10.6.24)
where
X=2(TO-xo).
(10.6.25)
Observing that
02 _1 1_z29+za -2
r
Z2
(
r2) th
r2 5r2
we deduce that the line r = 0 is singular. Eliminating a vicinity of this
line we obtain
lim
82
_
10
r Or
O 0z2
whence
n(xo, yo, z) =:-to
lim n(xo, ?b, z) _
2
fA
(1u.o.lo)
eWA
00pt2+r2)3/2da.
This kernel was obtained in (10.2.36) as a limit of the subsonic kernel
and in (10.4.48) as a limit of the supersonic kernel (Ueda and Dowell
[10.80]). This fact proves that the oscillatory perturbation is continuous
to the passage of the sonic barrier.
f0, the first term from (10.6.17)
Passing to the limit, when z
vanishes because 6(f0) = 0.
Chapter 11
The Theory of Slender Bodies
11.1
11.1.1
The Linear Equations and Their Fundamental Solutions
The Boundary Condition. The Linear Equations
In this chapter we study the aerodynamics in the presence of slender
bodies (fig. 11.1.1). The axis of the body is considered the Ox axis
and the Oz axis lies in the plane determined by the velocity of the
unperturbed stream V. and by the axis of the body. We denote by
a the angle of attack of the stream and we assume that a = e, where
e characterizes the thickness of the body. We employ the cylindrical
coordinates x, r and 0 which are related to the cartesian coordinates
x, y, z by the formulas
x=x,y=rcosO,z=rsinO
(11.1.1)
xER,rE (0,oo),0E (0,21r).
Fig. 11.1.1.
The equation of the body has the form
r = h(x,0) = eh(x,0).
(11.1.2)
450
THE THI3OKY OF SIZNDER BODIES
Denoting by i, j, k the versors of the Ox, Oy and Oz axes and
i,., ie (fig. 11.1.2), we shall have the following formulas
i,. = jcos0+ksin8
j = ircos9--ipsin0
(11.1.3)
ie=--jsind+kcos8
k=i,.sin8+iscosO.
Fig. 11.1.2.
The velocity of the unperturbed stream is
V,, - U,,,(i cosa + ksina) = UU(i + ak) + 0(a2).
(11.1.4)
Denoting by
V1 = Uvv,
P1 = Poc + P-U«,P,
P1 = Poo(1 + P)
(11.1.5)
the fields which characterize the perturbed flow and using the cylindrical
coordinates
V=Ui+Vi,.+Wi$
(11.1.6)
v=Ui+I ,. +Wie,
we deduce
U=1+u, V =asin8+v, W =acoa8+w.
(11.1.7)
THE LINEAR EQUATIONS AND THEIR FUNDAMENTAL SOLUTIONS
451
Obviously, the perturbed flow will be steady because the conditions
which determines it do not vary time. On the boundary we shall impose
the condition
V grad F = 0,
(11.1.8)
where F = Eh(x, 0) - r. Taking into account that we have
gradF=
5s+
Vii.+Tiei
(11.1.9)
from (11.1.8) we deduce the condition
(1 + u)F as + (a cos 0 + w) r L = a sin 0 + v,
(11.1.10)
which must be satisfied when r = h. Comparing the orders of magnitude
we deduce:
v(x, h, 0) = ev(x, h, 0)
.
(11.1.11)
We assume that this structure is valid everywhere in the fluid. We have
therefore
v(x,r,9)=e'v(x,r,0).
(11.1.12)
From (11.1.10) and (11.1.11) we deduce the condition
v(x,r,0) +asin0 = hz(x,0),
(11.1.13)
which will be imposed for r = h. In fact, this condition must be imposed
for r = 0, but here v is not defined.
In cylindrical coordinates the equations of motion are (1.11]:
AP [Up.. + Vp. + (W/r)pg)+
+(1 + ryM2p)(U,+Vr + (1/r)V + (1/r)We] = o
(11.1.14}
(1+p)(UU=+VU.+(W/r)Ue]+p,, = 0
(I + p)[UV + vv. + (W/r)Ve - (W2/r)] +p. = 0
(11.1.15)
(1 + p)[U11=+VW.+(W/r)We+VW/r]+(1/r)pe =0,
(11.1.17)
(11.1.16)
where U, V, W will be replaced by (11.1.7), and P. =00/8x'...
With the reasonings from 2.1, the equation (11.1.16) gives
p(x,r,9)=E (x,r,0), v=+pr=0,
(11.1.18)
the equation (11.1.17)
w(x, r, 9) = sw(x, r, 9),
rwz + pe = 0 ,
(11.1.19)
452
THE THEORY OF SLENDER BODIES
and the equation (11.1.15)
7t(r,t,B)=SYf{e,r,B),
71r+p. =0.
(11.1.20)
Keeping the terms having the order of E, from (11.1.14) we deduce
AM2pr+u,.+vr+(1/r)v+(1/r)wo
0.
(11.1.21)
One observes that in the linearized system o does not intervene.
The system coincides with the system for cr = 0. It. is the system
(2.1.32) in cylindrical coordinate:.
One may also obtain the equation of the potential. Indeed. from
(11.1.18) -- (11.1.20) it results:
yr - ur = 0, rw;r - uy = 0.
P = --u.
(11.1.22)
The last two equations prove the existence of the function V(:r, r, 0), a.I.
7r. _ (j.,
V = 4'r i
to = (1/r);pe ,
(11.1.23)
and the equation (11.1.21) gives
(1 - Al `)
11.1.2
a"-
10
t1:r.2 +
r Or
r
a(pl
C Or
1
+
j92 = 0.
(11.1.24)
Fundamental Solutions
We shall utilize for the solution of the system (2.3.4) the intrinsic
form (2.3.8), (2.3.12) which will be written in cylindrical coordinates.
From the equality
ft: + f29 + f3k = f1 + frtr + fOZO,
(11.1.25)
taking (11.1.3) into account , we deduce
f2 = frcus0 - fosin0,
fr = f2cas0+ f3 sill 0
(11.1.26)
fi = fr S1118 - fB COS B,
f0 = -12 sin 04- f 1 cos 0 .
Writing the inner product in cylindrical coordinates, in the subsonic
ease and Taking (11.1.26) into ac( -ount, from (2.3.4) it results
r) _
-
I
+fri0) ` )
1
(11.1.27)
1
THE LINEAR EQUATIONS AND THEIR FUNDAMENTAL SOLUTIONS
453
where
R, =
x2 +#2r,2,
(11.1.213)
From (2.3.13) we deduce
_
1
47r
_
/
dx
fr J x
(x2 + #2x2)3/2
2r
f=
R, -
(11.1.29)
1
4:s
fr
Rl -
fr 1 + x
r(R,)
Taking into account the expression of the distribution 6(x) in cylindrical coordinates [A.7J, [A.10). we obtain
yr
=
f'.
L. a
H(x)d(r)
21rr
+ 4r, 8r
fr a
1
x
1
(11.1.30)
R, } 4;r 8r r
In the supersonic flow we have
P(x,r) _ -i-. (fr_8x +fr
where
E(x. r) _
i
l
J E(x ,r),
(11.1.31)
H( x-kr)
x- -k- r
(11.1.32)
Since
a f" H(r-'-r) d r =H (x- kr) 0
8r
oo T- k r
dT
=
,r r- - k r
2:E (x,r),
r
from (2.3.13) it results
p(x, r) = 1 (f= - x f,.) E(x, r)
(f..
19
+ r fr - r fr5T) E.
(11.1.33)
(11.1.34)
Taking into account the formula (2.3.35), p and Vr will be:
x
Vr
_
- fr
H(x - A-r) r
H(x)b(r)
27rr
x
81
+
27r
h
8
+
1kr,
x
r2
fr-
(11.1.35)
1
fr 5T x -k r
These formulas show that the perturbation propagates only in the
interior of the cone x = kr.
-r
454
THE THEORY 0FSLENDER BODES
11.2
The Slender Body in a Subsonic Stream
11.2.1
The Solution of the Problem
In the case of slender bodies of revolution, the equation (11.1.2) has
the form
r=h(r). 0 < x < 1.
(11.2.1)
Considering that the unperturbed flow has the angle of attack z in the
xOz plane, we deduce that. this plane will be the plane of symmetry of
the flow. We shall replace the body with a distribution of forces defined
on 10.11
with f2 = 0. From (11.1 26) we deduce fr = f sin0 (we
denoted f3 = f). Taking (11.1.27) (11.1.30) into account, we deduce
that perturbation produced by this distribution may be represented by
means of the formulas
p(.r, r, 0) .
" { f,
Rd
Jx + f (l;) Sin A
J
47r-
(11.2.2)
W(arrtd)=
xj
1
'r(X, r, O) =d(l)
R
sin 0
d
4irr
sin 0
,-TI
'
x0
(1+ R)f(E)d
Jhit(0
0
)
(11.2.3)
sill 0 y rl
(1 + x0) d
4 it 8r Jo
r
R
where
R= 4+Ir'r2.
.T.ox-
Imposing the boundary condition (11.1.13), we notice that d(h) = 0
because h does not vanish for 0 < x < 1. Separating the variables we
obtain the following integral equations:
f
f
0(I)d
41rh'(x)
(11.2.4)
(11.2.5)
for O < x < 1 and r = h(x).
In order to solve the first equation we shall utilize the identity
v
fir
1
R)
1
8 ((xol _ 1
5-x
iI_
((xo
R} ra(RI
(11.2.6)
455
THE SLENDER BODY IN A SUBSONIC STREAM
Integrating by parts, we obtain:
x - 1
47rh(x)h'(x) = ft(l)
x -1) + Q h
(11.2.7)
-fi (0)
r2
+
= - [fl(1) + fr(o)] [1 + 0(h2)]
h
where
_ t
I=f'
xo
h
xo+
fi(:;)dt.
(11.2.8)
For calculating the principal part of this integral, we notice that we
have
xo
+
lio' [i + 0(h2)]
h
(11.2.9)
excepting the vicinity of the point £ = x where xp = 0. In [1.1] one
utilizes this approximation on the entire interval (0, 1). Correctly, the
integral I must be written as follows
(/
I=
+ / 7+1
J
(11.2.10)
In the first and last integrals we may utilize the approximation (11.2.9).
For ,i small enough, in the second integral one may replace fl (t) by
f ' (x). We obtain therefore
I = lro
[ r:o
fi (t)d t -
J:+n fi (t)d C]
[1 + 0(h2)] +
1
+ f (x) lim
+''
,j- .0
q
dt
p
+h
We calculate the last integral with the substitution x - = u and
we observe that it vanishes. Hence,
I = [fl (x) - fi(0) - ft(1) + fl(x)J [1 + 0(h2)] .
(11.2.11)
Neglecting the terms of order h2 with respect to 1, from (11.2.8) and
(11.2.11) we obtain
fi(x) = -27rh(x)h'(x) = -S'(x),
(11.2.12)
with S(x) = 7rh2(x) the area of the crass section a of the body in the
point having the abscissa x.
456
THE THEORY OF SLENDER BODIES
Deriving in (11.2.5), we find
j' (1 + R)
f(E)d F + f32h2
"' f
J. 1T4
-4iah2 .
(11.2.13)
Calculating the integrals by means of the formula (11.2.10), we have
J1(1+ O)f(:)dF=21 f(t)dF[1+0(h2)]
Hence, neglecting 0(h2) with respect to 1, we obtain
f(e)d t = -2Tah2 = -2nS(x)
10
or, deriving,
f (x) = -2aS'(x) .
(11.2.14)
For the profile with zero angle of attack (a = 0) we deduce f = 0
whence
N(x,r) _ -4 0,
xo+
r
dF.
(11.2.15)
This representation of the potential is known in the literature [1.11,
(1.38]. We have also, from (11.2.2) (for r # 0),
(11.2.16)
P(x, I') = -4r, Vr = SFr
11.2.2
The Calculus of Lift and Moment Coefficients
We shall calculate at first the pressure for r = h. It can be obtained
from (11.2.2), (11.2.12) and (11.214). Utilizing the identity (11.2.6) and
the calculations (11.2.8) - (11.2.11), we deduce
jf(0
.
d
(R)
(11.2.17)
[1 + O(e2)]
.
Hence,
sin 0
P(x, h(x), e) = pi (x, h(x)) - a Wh(x) S`(x),
(11.2.18)
157
THE SLENDER BODY IN A SUBSONIC STREAM
where
r fl (E)' (1?)
P1 (X. h(.r)) _ - 1
J
1r,,iid E
(11.2.19)
1f1
4r
:rOS'(E)d E
(%2+132,12)3/'2
Taking into account that fr = O(V2), we calculate the principal
part of pl as follows:
I
4;rP1(x, h(x))
W) r_I,d
fl (S)3
_
11(1)
(x11-)
f1(0)
x'-+f32h2
-j3 h=
9 r,
+J fl(E)
I-E
S'(1) + S'(0)
+f
d
z*F7
+ fi(x) lim
r.
1-x
T
J0 x -
-
01 S"(r)
Er
>
r0+th`
+O(Fd)-
- S"(E)d
x-E
x
t
J
S"(x.)
E--x
1
+ S(x) In
1-c
r.
+ O(c4) ,
the principal part which was written being O(_2).
The lift coefficient may be calculated with the formula
cL=
(11.2.21)
- As
where S is the surface of the body, and n, the outer normal and with
the notation F = r - h(x), given by the formula
grad F - -h'i + i,
(11.2.22)
n =
Egrad FI
V1-1+ h'2(a)
Taking into account the element of area on the surface S, and the
relation (11.2.13), we obtain
r1
A
CL =- / / ,r p(x, h(x), O)h(x) sin 0d rd 0 =
o
=(I Es(1)-s(0)j.
(11.2.23)
458
THE THEORY OF SLENDER BODIES
For the drag coefficient one obtains
CD =- Jjpn id a =
jj(x,
1
h(x), O)h(x)W (x)d xd 0 =
(11.2.24)
pl(x, h(x))S'(x)d x = O(E4) .
The drag coefficient does not depend on a.
At last, the moment coefficients are the scalar components of the
product
-2 /f xxpnda,
S
where x = xi + h(x)ir. Taking into account (11.2.2) and (11.2.22), we
obtain
cr =0, cs=0,
(x + hh')h(x)p(x, h(x), 0) sin Od x d 0 =
f1I
-2a /
o
1
Lx +'
(11.2.25)
S'(x)d x = O(E3) .
J
Obviously, for a = 0 one obtains cy = 0.
Neglecting the term (S')2, for the moment coefficient c, we obtain
the approximate value
ri
c, = -2a J xS'(x)d x = 2aV,
0
b ecause S(1) = 0 and the term
of the body.
11.3
11.3.1
j
S(x)dx represents the volume V
The Thin Body in a Supersonic Stream
The General Solution
In this subsection we consider the same problem like in the previous
subsection, but now the unperturbed (free) flow is supersonic (M > 1).
459
THE THIN BODY IN A SUPERSONIC STREAM
titre consider again that in this case the xOz plane is a symmetry
plane, such that we have f; = 0 whence fr = f sin 0. The fundamental solution is (11.1.31), (11.1.34). and the corresponding potential
is (11.1.33). For a continuous superposition of forces on the segment
[1).11. the perturbation will be given by
1
+f w sin ©J E(.,ro, r)d
P(x. r, 0) = - T f
p(x, r, 0) =
[f=w
2-r J0
- rl f (5) sin 0] E(r(j, r)d
i
X
t'r(X, r, 0) = (5(1') in0
2'r
JU
.E(xa)d
,r
+
sin0
ff
U
1
err f0 roI
0Y
.
l - J)F(xor)d
r
9r,
(11.3.1)
The derivatives may interchange with the integrals and. taking (A.3.15)
into account, we have for example
f 1 f W E(xo, r)d t= ax f
_
a
x-kr
H(.r-kr)
o
f
x
1
f
E(xo, r)(1 _
FL-r
H(x-k'')r3
4.
`I
f (4) ==
zo
k-12-7-11
Hence, the solution (11.3.1) maybe written as follows
_
2r TX o
=
r-kr
1
r
_
yr
x-kr
8
1
P
x
sin 0_
8
d42z Or
x2 - k r(4)
dE-
in
xo - k 2 rI
sin 0
2rrr
f()cI+ 2 r
b(r)
+siI 0
27tr
1
Of
sin 0
8
f V)
dC
xo - k2r2
o
r-kr
xQ
2,rr Jo
1
r
r-kr
fx (50)
ro
r-
d
jr-kr
x
xo
xo-k'r
k2r2
d
(11.3.2)
460
THE THEORY OF SLENDER BODIES
valid for x > kr and
p=W=vr=0 forx<kr.
(11.3.3)
Imposing the boundary condition (11.1.13) we shall notice again that
6(h) = 0, because h # 0. Separating the variables we deduce the
following integral equations
8r 0
1
Cr
r
x0 _
Vd t)
rah
= 27rh'(x)
x-kr
a
x0
ar} Jo
xo - k ref (t)dtIrah = -27rah(x) .
(11.3.4)
(11.3.5)
In order to put into evidence the principal part of the integrals for
r = h = O(e), we must notice that we cannot derive directly the integrals, because the integrands become infinite for { = r - kr. For
avoiding this situation, we shall perform the change of variable 4 -- u:
x - a=krcoshu.
(11.3.6)
One obtains
a
of=-kr
a'
8
fr (S)
xo - k r
d = 8r
x
r
x -k r
x
f:(0)
fx(0)
x- k r
,
-kJ
f. (x
UCh
fi(x - kr cosh u)cosh u d u
o
_
_r
1
r f'(0) [1 + O(ff)]
- krcashu)du =
rx-kr
1
0
-r
0
xo
'
xo - k r f (t)d
-kr
f=(t) [1 + o(e2)] d.
(11.3.7)
Neglecting O(e) with respect to 1, we deduce that the principal
part of the integral from (11.3.4) is - f=(x)/r. Hence, from (11.3.4) we
deduce
ff(x) = -2whh' = -S(x),
(11.3.8)
S(x) giving the area of the cross section of the body in the point x.
TILE THIN BODY IN A SUPERSONIC STREAM
461
Similarly we obtain
0
r-Ar
xO
Or Jo
t-tr
1
r
X0
r \/x - 1
1
r2
z-kr
0
1f(01d --1
x -
0
+
f(0)
:c2
f(c do :c3 - k2r2
r
ra
u
f'(Od
(11.3.9)
Neglecting O(r-2) with respect to 1 we find at first
1
(7
z-kr
xp
.r
f,-Ir
f
f (E)d
-r-kr
rx`t
-rI
1
x-kr
f ()d + r
f (0)
fz_kr
f'(e)d
r1-jZkr
Ja
f(4)d4.
The equation (11.3.5) becomes
S kr
kr f (x - kr) +
0
f
-2-xnrh.
(11.3.10)
where we put r = h. In the left hand part of the equality, we expand
into series for obtaining the principal part. We have
r
kr f (:x) + 0(r2) +
kr f (x) + 0(r2) = -2 rorh .
f
0
Hence,
s
J
f
--2riS(x)
whence
f (x) = -2aS'(x),
(11.3.11)
the analogy with the subsonic case being obvious.
11.3.2
The Pressure on the Body. The Lift and Moment
Coefficients
In order to put into evidence the principal part and the expression of
the pressure, we notice that utilizing the change of variable (11.3.6), we
TILE THEORY OF SLENDER BODIES,
462
deduce
x-kr
8
o -krx
f=(t)
Ox
x -k r
r-kr
+f=(x)
xo-k r
o
f=(a)
o-k r
0
x
+f:(x)arccosh kr
__
rs
-
x
x - f.,
t
to
2s
= fs(x)ln kr
xo -kr
Jo
f'(C) - fr(x)d4+
df
f
ff(t)d t
r=-kr
-k r
+
f=(0)
+
f1(0)
d t=
-J
+
Jo
r f.'W - f.'W d t +
x-
d t+fx(x) In
x + x- k r e
kr
fi(x) - fY(0dt.
x-t
Taking into account the calculation performed at (11.3.7), we deduce
p(x, h, 0) = pi (x, h) - a sin 0 S'(x),
(11.3.12)
where
pi (x, h) = -
S"(x)
21r
kh _
2x
1
S"(x) - S"(4)
27r fo
X-
d
(11.3.13)
The lift, drag and moment coefficients may be calculated with the
formulas (11.2.23) - (11.2.25). We have therefore, like in the subsonic
case,
CL =a [S(1) - S(0)1 = 0(e3)
CD =Ji p,(x,h)S'(x)dx=0(e4)
4
(11.3.14)
Gr=az=0,
cy = - Cr f '
0
Ix +
2(x) J
S'(x)d x = O(s3) .
THE, THIN BODY IN A SUPERSONIC S1'IREAAI
11.3.3
463
The wing at zero angle of attack
For a = 0, we deduce f = 0 whence
P('r., r) =
9(r,
r)
t'r(x, 7.)
a
1
S'()
/'x-rrr
2;, Tx Jo
zo ` J. r d
=-kr S
-o
xo - kr'
1 of r-kr
2
j
S (e)d
o
d
S'(S)dS
.4-k r'
x-kr
1
2rr J0
Sit ( )d
k2r2
(11.3.15)
This is the solution between the wave from the louring edge and the
wave from the trailing edge i.e. in the region denoted by II, where
a > kr (fig. 11.3.1). In I (:c < kr) p = yr = 0, and in IN the solution is
p(x, r) _
1
oz-kr
2r,
2irr
(11.3.16)
z-kr
1
Vr(x,I-) -- -
S"()d t
xo-k-r'2
0
dc.
x _ 13-1
1A
w
1
Fig. 11.:3.1.
This explicit manner of representing the solution is also valid in the
general case.
11.3.4
Applications
At first we shall consider the thin body having the shape of a cone
(fig.
11.3.2) of equation r = ex, 0 < t < 1. We deduce S = 1re2x2
THE THEORY OF SLENDER BODIES
464
whence
2
CL = JrE
at cy = -
21u E2
3
ti
,
CD = -7rE' ln(k/2).
(11.3.17)
Fig. 11.3.3.
Fig. 11.3.2.
For cy and CD we have retained only the principal part. If the
angle of attack is zero, then in II we have
d _d C = 62 arccosh j _
z kr
p(x, r) = E2
E2 In
yr (x, r) = -
xo - k"jr
Jo
x+ x2-k r
x-kr
EZ
r
(11.3.18)
kr
o
x0
xkr-d t _ -8
2
:r - k r
r
We notice that on the radius vectors r = cx, c < 1/k, the pressure
and the velocity are constants. The flow is conical.
The second example
the double cone from figure
is
11.3.3
Since
Ex,
h(x) =
0 < x < 1/2
(11.3.19)
r(1-x), 1/2<x<1,
it results that S' has a discontinuity in the point x = 1/2. We suggest
to the reader to establish the formulas of derivation for
fW
dta
rx-kr
xo-k r
and to write the solution.
' arJ10
xo -k'r
d
(11.3.20)
Appendix A
Fourier Transform and Notions of the Theory of
Distributions
A.1
The Fourier Transform of Functions
The following definitions will be given in R3. Their expressions in
R1, R2 or RI will be easily deduced.
The Fourier transform of a function f : R3 --+ R is denoted .F[f],
or f and it is defined by the formula
(A.1.1)
F[f](a) = f(a) = j3
where
x = (x, y, z),
a = (al, a2, 03)
(A.1.2)
a.x=alx+a2y+a3z,
dx =dxdydz.
The operation F is called the Fourier transform. We notice that i
f (x) is absolutely integrable on R3, he. if
3 (f )d x < oo,
(A.1.3)
then the Fourier transform exists. Moreover, if f satisfies certain
conditions of regularity, for example, if f E C0(R3), or if f is piecewise
smooth with respect to every variable [A.10], then f may be obtained
from f using the following inversion formula
(2;r)3 f (x) _ (2-,,)3-F-1 [j(x) f
a=
(A.1.4)
_ F[fl(-x) _ [f (-a)J(x) ,
where ci a = d a1 d 02 d a3. The last two expressions show that the
inverse of a Fourier transform may be obtained by a direct transformation.
466
FOURIER TRANSFORM. THEORY OF DISTRIBUTIONS
One proves (Lebesque's theorem [A.9], (A.51) that j tends to zero
when ICkI -+ .x}.
The condition (A.1.3) is restrictive enough. For example, the function f = 1 does not verify this condition. The theory of distributions
allows to enlarge the class of the functions which admit a Fourier transform.
Notions of the Theory of Distributions
A.2 The Spaces D and S
The theory of distributions relies on the notion of space of the test
functions. The space V = D(R3) consists of the set of functions
W(x) infinitely derivable, with compact support in R3. We say that a
sequence of functions {V,} from D, converges to a function S' E D,
if for every multi-index k = (k1, k2, k3), k1, k2, k3 being non-negative
integers, we have:
DkW11 rya Dkcp,
(A.2.1)
where, with the notation IkI = k1 + k2 + k3, we have put
Dk
-0jkj
= --,-
k,
(A.2.2)
The set D endowed with this convergence law is a linear space-
Ale may we that the Fourier transform of a function from D is
not a function from D. Indeed, the Fourier transform is an analytical
function [A.12), but the support of an analytical function which cannot
be compact. There exists another space which is invariant to the Fourier
transform and plays a basic role in the definition of the Fourier transform
of a distribution. This is the space S = S(R3), i.e. the space of the
rapidly decreasing test functions. We call rapidly decreasing function
an infinitely derivable function V(z) in R3 which, together with its
derivatives decreases when Ixl - oo faster then every power of Exl`t.
This means that
(A.2.3)
IxkD'v(x)i < CL-j,
for every nmlti-indices k and 1. We denoted xk = xklyk'2zka. The
convergence in S is defined in the following manner: the sequence of
467
DISTRIBUTIONS
functions {V } from S converges to the function cp E S if for every
multi-indices k and 1 , we have:
X,
xkD'pn(X)
xkDrv(x).
(A.2.4)
The set of the rapidly decreasing functions, endowed with this convergence law. is a linear space. The convergence in D implies the conver-
gence in S. We have D C S. but S does not coincide with D; For
example. the function exp. (-x2) E S, but
A.3
D.
Distributions
The distributions are linear and continuous functionals on D or on
S. This means that a distribution f detennines the correspondence
between a test function V and a number denoted by (f, q) and we
have:
(A.3.1)
(f,Anpi +A 2) = an (f,Sit) +A2(f,p2) ,
for every two real or complex numbers an, A2 and that if P>s I V in
D (or S), then
(f, °Pn) - (f, (P)
.
(A.3.2)
One denotes by D' the set of the distributions defined on D and by
S' the set of the distributions defined on S. S' is the space of the
temperate distributions. Obviously, S' C D'.
We say that the distribution f is equal to zero in the domain f2
(it is denoted by f = 0) if (f, gyp) = 0 for every p from D (or S)
with the support in 11. The distributions fn and f2 are equal in n
if fl - f2 = 0 in 11. i.e. if
(f1, 0 = (f2. (P)
,
(A.3.3)
for every v E D.
We call the support of the distribution f and we denote it by supp f
the set of the points which have a vicinity where j is not equal to zero.
Every locally integrable function f (x) defines a distribution by
means of the functional
U. P) = j f (x)4%(x)d x .
(A.3.4)
A distribution f is it regular function-type distribution if it may have
the form (A.3.4). Every other distribution is singular. The best known
468
FOURIER TRANSFORM. THEORY OF DISTRIBUTIONS
example of singular distribution is the distribution of Dirac. The distribution of Dirac with the support in 4 is denoted by bo(x) or b(x and is defined by the functional
(b(x -
V, (x)) = rW .
(A.3.5)
Formally, it may be written as follows
J
v(x)6(x - 01 x = pW .
(A.3.6)
If rn(x) is an infuiitely derivable function, then mv is infinitely
the distribution f and
the function m., denoted by tnf, by means of the formula
derivable and we may define the product
rnv)
(inf, 4,)
(A.3.7)
It results
m(x)b(x - ) =
(A-3.8)
whence,
ni(x)b(x) = 0,
if rn(0) = 0.
(A.3.9)
For the existence (A.3.8) the continuity of the function is sufficient in
in t;.
We shall consider the three-factor c and the homothety cx =
c3z). If h(x) is a locally integrable function, then h(cz)
is also locally integrable, such that the distribution corresponding to
h(ex) is defined by the functional
_ (cix.
(h((x), 4"(x))
=f
h.(cx)p(x)d x
Setting cm = and observing that the integration limits interchange
when c; is negative, we deduce, with notation lei = (clc-2cy)
IcC (h(cz), 5'(x)) - j
h(E),p(k/c)d
_ (h(x). 4p(x/c))
For o distribution f (x), one defines the distribution f (cx) by means
of the formula
ICI (f (cx) s'(x)) = (f (x), cp(xl (:))
.
(A.3.10)
The definition may be extended to a non-singular linear transformation.
As a particular case, it results:
Ic16(cI x. C 2Y, car) = b(x ah z) .
(A.3.11)
469
DISTRIBUTIONS
f
The derivative of the distribution
determined by the formula
is denoted by
f'
and is
(A.3.12)
(f', V) _ - (f, ) .
Defining the function of Heaviside H(x) by the formula
x<0
H{r.) = 10,
1,
(A.3.13)
x > 0,
it results
00
(x)dx=5P(0)
i
0
Taking into account (A.3.5) we deduce
H'(x) = 6(x).
(A.3.14)
Let us establish now the formula
(m(x)H(x)' = m(0)6(x) + in'(x)H(x)
for a function E C'(R). We have
((mH)'. y,) =
= m(0M0) + f
- (mH. vp')
x
(A.3.15)
r
x=
m'(x)cp(x)d x = (m(0)6(x) + m'(r)H(x), v')
.
0
On the basis of (A.3.8) and (A.3.15) we deduce that the solution of
the differential equation
Lu = 6(x),
(A.3.16)
where Lu = u("')(x) + O, (X)u('"-I)(x) + ... + am(x)u(x). is
u(x) = H(x)v(x),
(A.3.17)
where v verifies the equation Lv = 0 and the conditions
v(O) = v'(0) ...
= v,(m-2)(0) =
1
0,
We call the direct product of the distributions f(x) E D'(R") and
g(y) E D'(Rm) the distribution f (x) g(y) E D'(R"+') defined by the
functional
(f (x) g(y), r'(x, y)) = (f (x), (g(y), W(x, y)))
,
(A.3.18)
470
FOURIER TRANSFORM. THEORY OF DISTRIBUTIONS
where V E D(R"+'"). The definition is valid also and when D' is
replaced by S. The direct product is commutative, i.e. f (x) g(y) _
= g(y) f (x). As an application, we shall prove that in R2, we have
(A.3.19)
b(x) = a(x) - a(y)
Indeed, applying the definition, we have:
(OX), AX, y)) = 4'(0, 0)
.
But we have also,
(a(x) a(y), Ox, y)) = (a(x), (a(y), (x, y))) _
= (6(x),;?(x,0)) = V(0,0)
A.4 The Convolution. Fundamental Solutions
For two functions mi(x) and m2(x) absolutely integrable in R3,
we define the convolution m(x) = (ini * in2)(z) by the formula
(ml * MAX) =
=1
3
J pa
"11)mx - t)d : =
m( m2(x - t)dt = (in2 * ni)(x)
The function m(x) is absolutely integrable, such that it generates
a function-type distribution (A.3.4). Writing explicitly that functional,
we are determined (A.51 to define the convolution 0= fi * f2, of two
distributions fi and f2 by means of the formula
(fi * f2, ') = (fi (x) f2(O, V(x + f)) ,
(A.4.1)
fi(x) f2(4) representing the direct product of the distributions fi
and f2. The convolution (A.4.1) exists if, for example, one of the
distributions have a compact support (A.5j. The equality f j *f2 = f2 *fi
results from the commutativity of the direct product.
As an application, we shall calculate b * f. We have
(6 * f, P) = (6(x) . f W,'?(x + 0) = (f (0, (6(x),'P(x + i)))
_(fW"PW)
THE CONVOLUTION. FUNDAMENTAL SOLUTIONS
Hence,
a*f=f*6=f.
471
(A.4.2)
We shall establish now the formula
D(fi*f2)=Df,*f2=fi*Df2,
(A.4.3)
the operator D being defined in (A.2.2). We have
(D(f1 * f2), Sp) = (-1)Ikl (fh * ff, DW) _
= (_ 1)lkl (fi (x) - ME), Dpp(z E)) _
(-1)Ikl (f2(E), (fi(x), Dip(x + i))) _
_ (f2(E), (Dfh(x), D& + F))) = (Dfi * f2, sp) ,
and the result proves the first equality from (A.4.3). The second equality
results from the commutativity of the convolution.
We shall consider now the linear differential operator of order m,
M
L = E a*Dk ,
Ikl-O
where ak are constant. We call fundamental solution of this operator,
the distribution t E D' which verify the equation
LE = 6(c).
(A.4.4)
Obviously, the fundamental solution is defined with the approximation
of an arbitrary solution of the homogeneous equation. We have the
following theor_mr. If f e D' and there exists in ?Y the convolution
f s E, then the solution of the equation
Lu = f
(A.4.5)
exists in D', is given by the formula
u = f *.C
(A.4.6)
and it is unique in the set of distributions from D', for which there
exists the convolution (A.4.6).
Indeed, utilizing (A.4.2) and (A.4.3), we deduce
Lu=L(f*E)
a1Dk(f*E)=f*LE= f$6= f.
lkI-O
472
FOURIER TRANSFORM. THEORY OF DISTRIBUTIONS
In order to demonstrate the uniqueness, we denote by u another so.
lution. Obviously, no = u - u satisfies the homogeneous equation
Luo = 0, such that we have:
uo=uo*6=uo*Le=L(uo*e)=(Luo)*e=0.
A.5 The Fourier Transform of the Functions from S
Since the functions from S are absolutely integrable, they have
Fourier transforms F[ pj defined by (A.1.1) and these are bounded continuous functions on R3. Integrating by parts and taking into account
(A.2.3), we deduce:
-F(VACt) = (-ial)FIVJ(a).
(A.5.1)
On the basis of the same condition, we may derive inside the integral,
such that it results
(O/Oai).('PJ(a) = .F(ixtipj(a).
(A.5.2)
By recurrence, it results:
.F(Dk(pJ(a) = (-(a)k.F(V](a),
f((ix)'4 j(a),
(A.5.3)
(A.5.4)
for every multi - indices k and L. The last relation show that F(VI is
infinitely derivable.
We shall prove in the following F(cpJ E S. To this aim we replace
cp with (ix)rcp tin (A.5.3) and taking into account (A.5.4), we deduce
F(Dk((ix)`p)J(a) = (-ia)k.F[(zx)l ,](a) =
= (-i)IkjakDI,F[Wj(a)
whence
]akD'F('j(a)I <_ fx$ IDk(xt4o)IdZ < 00,
i.e. Y[w] satisfies (A.2.3).
Taking into account that the Fourier transform F(spj is integrable
and continuously differentiable, we deduce that cp may be obtained
from cp sing a formula like (A.1.4).
THE FOURICR TRANSFORM OF THE TEMPERATE DISTRIBUTIONS
473
A.6 The Fourier Transform of the Temperate Distributions
The Fourier transform of the distribution
and it is defined by formula
f = .F[ f l
f E S' is denoted by
(f, 0) = (f. y i) ,
for every V E S. Since
(A.6.1)
E S. it results [A.121 that the Fourier
transform of a temperate distribution is also it temperate distribution.
For f E S' we define the inverse Fourier transform F--I, by the
formula(27r)'-F-' [f (x)1 = -F[f (-x)] .
(A.6.2)
We shall prove that
f
j:-- I [.[fl]
(A.6.3)
.
Indeed, taking (A.3.10) into account, we deduce for c = -1
(27r)3 (,F-1[.[f]h
(.F[.F[f](--c )l,
F-'
(-a)) = (2r)3
(27r)3
(f,.Fp,-' [.,III)
_ (f )
Similarly we prove the second equality.
We shall establish now the operations (A.5.3) and (A.5.4) for temperate distributions. We have:
(F[f=], } = (f=, ) _ - (f, a,.F[pl) =
[iaip]) _ (F[f].-iniy,) _
and we deduce,
whence
(_iaiJ,)
.F[f] =-iaif,
(A.6.4)
T[grad f) = -i a f , F[div f ] = -i a f
(A.6.5)
.F[rotfl=-iax f.
From (A.6.4) we obtain by recurrence a similar formula to (A.5.3).
Analogously, we have:
(
0 _771fl, V(a)
Ylfb
f..F
an.,
474
FOURIER TRANSFORM. THEORY OF DISTRIBUTIONS
(f, i x.F[d) = (i xf, Y[w)) = (f [i xf ], io)
hence,
(A.6.6)
.F[i:rf)
and, by recurrence, a similar formula to (A.5.4).
From the definition (A.6.2), it results that relations similar to (A.5.3)
and (A.5.4) are available for the inverse transformation too. Hence,
f-1
(i x)" f (x)
ia)!f](x) = D'f(x).
(A.6.7)
(A.6.8)
We shall frequently utilize relations of the form
.F-t[--t«tf1 = f:, F-t[cto2f) _ -f=y.
(A.6.9)
We shall prove in the following the formula
(A.6.10)
17-t(f(ca))(x) = ICIf 1 cl
c representing a three-factor. Performing the change of variable a =
= cf3, we have for a function
lip\alJ (x)=,l
(
cxd A = lclw(cx).
lei
For a distribution f we shall utilize the formula (A.3.10). We have
lei (F-t [f(Ca)1 P)
(f(a),F-tso(a/c))
lei (f(ca), -F- 1,P)
_
(f(x), 0(cx)) = (f (x/c), w)
whence (A.6.10).
For the direct product f (x - g(y)) we have:
(A.6.11)
F[f(x) - 9(y)] = Y[f)(a) - F[9)(0) ,
and for the convolution, if one of the distributions has a compact sup-
port, we deduce
.Fff * 9) _ F[f).F[9] .
From (A.6.11) it results
f-I
[f (a) 4(/3)) = f (x) . g(y) .
(A.6.12)
(A.6.13)
THE CALCULUS OF SOME INVERSE FOURIER TRANSFORMS
475
A.7 The Calculus of Some Inverse Fburier TSransforms
We shall calculate at first the Fourier transform of the distribution
of Dirac. For cp E S we have
(716(x - 4)], Sp) _ (a(z - ), r) ='P(A)
(A.7.1)
(el a-4, SP)
,
Hence
.F[a(x - a; )] = exp (i a ), b(a: - F) _ '-t [exp {i a }j ,
whence, for
(A.7.2)
= 0,
7A =1,
_ 1 [1] _
(A.7.3)
It results therefore
F[1](a) = (2a)3d(a) .
(A.7.4)
Hence the temperate distribution I has a Fourier transform.
Denoting by F2 the Fourier transform in R2 and by f'1 the
transformation in R1, we have on the basis of the formula (A.6.13):
.Fz 1 [1/i a] _ .F_ [l/i a1](x) '
t [1]{3l)
(A.7.5)
Applying the Fourier transform to the equation
du/dx = b(x),
(A.7.6)
we deduce -i o1i = 1 whence:
U = -X1[1/ialj(x).
(A.7.7)
The equation (A.7.6) has the form (A.3.16) and its solution has the form
(A.3.17). It results
s=X(x).
(A.7.8)
From (A.7.2) - (A.7.8) we deduce
T1 [1/i all = -H(x)6(y).
(A.7.9)
For the determination of the fundamental solution of the equation
of Laplace we need the following results (demonstrated, for example, in
[A.12], §97, [A. 131, §6.6, or [A.71, §5.3).
I
= 4x [ ['
n = 3,
(A.7.10)
476
FOURIER TRANSFORM. THEORY OF DISTRIBUTIONS
YV [i'i'_}
n=2,
2 (ln lxl + C),
(A.7.11)
C being a constant determined in [A.13], but without any importance
here. Taking into account (A.6.10) with c = (,3,1,1.) we deduce, on
the basis of the formulas (A.7.10) and (A.7.11):
1
Jrz 1
_
1
#2aI+a2+a3
- -]
I FP fl202
t
1
+
4n
(A.7.12)
(J +z )
= - 2n (ln x'- + 02y= + C - In (i) ,
2]
2J
i
1
(A.7.13)
,6
FP representing the symbol for the Finite Part (Appendix E), and p,
a positive constant.
Also, for determining the fundamental solution of the wave equation
we need the formulas
f-1 sinalalt
6(at - Iii),
4,rat
lal
T-I sin aloe lt1 H(at - l xI )
I
Ial
1
21r
a t- -- Ixl
n = 2,
(A.7.15)
demonstrated, for example (A.7.14) in [A.5] Chapter 2, §3.4, and (A.7.15)
in (A.12) §9.7. We shall prove in the sequel the following formulas
1
_ 1 H(x - k f2 + z2)
1
[-k2a"i + a2 + a3 J
F2 1
-k2
2a
x-k
(y
+ a2, = 2k H(x
+ s)
kl>!i).
,
(A.7.16)
(A.7.17)
In order to prove (A.7.16), one considers the following partial differential
equation:
k2u." - uyy - u=, = 6(x) .
(A.7.18)
Applying the Fourier transform, we deduce (-k2a? + az + a3)i = 1,
whence,
u
[_ k2al2 +a2+a321
1
.
(A.7.19)
It results therefore that the first member from (A.7.16) is just the
solution of the equation (A.7.18). We shall determine in a different
manner this solution, namely applying the Fourier transform only with
THE CALCULUS OF SOME INVERSE FOURIER TRANSFORMS
477
From (A.7.18) one obtains the
respect to the variables y and
equation
(k2d2/dx2 + w2)ta = d(x) ,
(A.7.20)
where w =
+ , and u represents the two-dimensional transformation. The equation (A.7.20) is like (A.3.16), and its solution has the
form (A.3.17). We deduce
u.-
H(x.) sin(wx/k)
k
w
Utilizing (A.7.15) one obtains (A.7.16).
Acting similarly, in the two-dimensional case, we find:
u=
ii
H(x) sin(a2x/k)
k
(Y2
t
being the notation for the Fourier transform with respect to the
variable y. Utilizing the formula
F1 sinXax11 7r
(A.7.21)
I(Inl>aa ,
0
given for example in (A.1], p.202, we obtain (A.7.17).
In the unsteady aerodynamics we shall meet the following type of
formulas
co alc tl
Mat
ct2
47r Ix
1 - cosalaltl
U2
- Iml),
(A.7.22)
a t - Ix+
Ix()
2r
= 3,
(at +
(A.7.23)
IxI
n=2.
We shall prove these formulas (following an idea suggested by V.Iftiinie)
using the results concerning the Cauchy problem for the non-homogeneous
wave equation. To this aiin we shall denote:
v(t, a) =
l - cosalalt
Ck2
We deduce
u.(0, a) = 0,
u((0, a) = 0 ,
irtc(t, a) = a' arsa.lalt = -(i2a2ii(t. a) + (L2.
478
FOURIER TRANSFORM. T'IIEORY OF DISTRIBUTIONS
Applying the operator F-', we deduce:
11(0, x) = 0,
ut(0. x) = 0
utt = a2Aat + u26(x)
.
For determining u we have therefore to solve a Cauchy problem for
the non-homogeneous wave equation. The solution of this problem in
the three- and hi-dimensional cases is given by Poisson's formula [A.12].
Utilizing this formula, we find (A.7.22) and (A.7.23).
At last, replacing x by x. - t in the formulas (A.7.9), (A.7.14),
(A.7.15), (A.7.22) and (A.7.23), we obtain in the three-dimensional Case,
with the notation R =
F-I
(c - t) + y' +
= 5(x - t) - a(t,.z)
Ceinit. I
sin aIctIt einat
f-I
_
I a5(at. - R),
)'
= TI-rat
(a1
n247rR
1 - c)sajait
iaIt - II(at
(A.7.24)
R)
and in the two-dimensional case, with the notation I#
F-a ei,«t]
f_1
Il
I
b(x - t) - (5(71),
I
ial
JI
27
(1 - cosa'aHt
dolt
I
rte
_
H(at - R)
a' '' --V
H(at -- 7)
2-r
In
at +
n."2t2 - f2
R
(A.7.25)
In [1.101 we may find direct. demonstrations of the formulas (A.7.25).
A.8
The Fourier Transform in Bounded Domains
In this last part, we return to the Fourier transform of the functions
and we give, following Homentcovschi's idea [A.61, the transformation
formulas in case that the function f (x) is defined on a bounded domain
D. We assume that D is bounded by a surface S which closes a domain
D' and by a surface of discontinuity E . We prolong f in DY, giving
479
THE FOURIER TRANSFORM IN BOUNDED DOMAINS
it the value zero. We make the same thing in the domain D° which
closes E. Applying the flux-divergence formula we obtain:
fxID=fx6:'
=fTIR3
rrfn
- JS
fie'
f,,n
fnleia'xda- J'; Of llnleiaxda.
(A.8.1)
F(grad f[D = -ic"f - 1 fne'o
F[div fJD = -ice f F[rot f]D = -ia x
where (A =f+ -f-.
J
xda- r
f nei°t'xda - J QfIne'a-xda
f f x ne`a'xda- J
s
DfOne`a'xda,
E
(A.8.2)
Appendix B
Cauchy-type Integrals. Dirichlet's Problem for
the Half-Plane. The Calculus of Some Integrals
B.1
Cauchy-type Integrals
We consider in the z = x + i y complex plane a smooth curve r, i.e.
a curve which has the parametric equations
x = x(8), y = y(s), sl
a
82,
(8.1.1)
where x(s) and y(a) are continuously differentiable functions, whose
derivatives do not vanish simultaneously in the same point. The curve
r may be closed or open; if it is dosed, then z(81) = z(82); if it is
open, then we assume z'(sl) = z'(sl + 0) and z'(s2) = z'(32 - 0).
By definition the positive sense on r is the sense corresponding to the
increase of the parameter s.
The smooth curve is obviously rectifiable, such that we may consider
as parameter 8 the length of the arc measured from sl (= 0) to 82(= 1).
In this case, we obviously have zi2 + y'2 = 1.
Let f(t) be a complex function depending on the complex variable
1, defined on r and Riemann integrable. The integral
F(z) = 2Ai
J
t (t) d t
(B.1.2)
is called Cauchy-type integral. As we lmow from the books of complex
analysis, the function F(z) is holomorphic in the interior of the contour
r. if r is at a finite distance, then F(z) behaves at infinity like 1z4'1.
We shall investigate, in the following, what happens with the integral
(B.1.2) if z = to E r. In this case, the integrand has obviously a nonintegrable singularity in to and generally the integral has no sense.
There exists however a large class of functions (we are not interested here
in the largest class) for which we may give a definition to the integral,
482
CAUCIIY-TYPE INTECRALS
namely the class of the functions which satisfy the so called Holder's
condition.
We say that the function f (t) satisfies Holder's condition on r if
there exist two positive constants (different from zero) A and µ(µ < 1),
such that, for every two points tl and t2 E r sa we have
(B.1.3)
If (ti) - f(t )I < Ajtl - t2VY.
Obviously, the functions f which satisfy Holder's condition are
continuous on r. If u = 1, the functions satisfy Lipschitz's condition.
B.2 The Principal Value in Cauchy's Sense
We shall give now the definition that we have mentioned before. We
assume at first that to does not coincide with any extremity of the arc
r (if it is open). We consider the are of circle with the center in to
and the radius s which cuts the curve F in two points tl and t2 and
(t) d t
we denote by - the are t1t2. If for a -+ 0 the integral
Jr-y tf to
has a finite limit, then this limit will be called the principal value in
Cauchy's sense. We denote
f(t)
Iifot--f(t) dt - frto
t - d t.
Y
to
(B.2.
t.
The principal value is it distribution (A.12], [A.14].
We shall prove in the sequel the following theorem: "If f (t) satisfies
Holder's condition in the vicinity of the point to, then the limit (B.2.1)
exists and it is unique. For the proof we shall write:
f
f(t) dt=
tto
fr_1,
f(t)-f(to)(I t+f(tu) f t - to
r1F t dt
- to
(B.2.2)
Having (B.1.3) in view, the limit of the first integral from the right
hand member exists and it equals the usual improper integral on r.
The last integral is calculated as follows:
f
tt
-7 t
o
= 111(t - t)la` + ln(l. - to)It = In u
- to+
+ ln(tl - to) - 111(t2 - to) .
Diu tl - to = Iti - tolein, t2 - to = It2 - tole'' and Itl - toI = It2 - tot.
483
PLEMELJ'S FORMULAS
It results In(tt - to) - ln(t.2 - to) = i (a - 0). Passing to limit, when
E-0, ct - W=ir,weget
dt
Inn
_
dt
e- 0
= In
b - to
a-t0
+ i a = In
b- to
to - a
(B.2.3)
Since the last integral from (B.2.2) has a well determined limit, the
theorem is demonstrated.
The case when to coincides with one of the extremities of the are
F. depends on the behaviour of the function f in that point (,See
for extunple [A.27], §29--32). If t9 coincides with an extremity and
f (to) = 0, we are in the previously considered case., because we may
extend arbitrarily the contour t beyond to, setting f = 0 on the
extension.
B.3
Plemelj's Formulas
We shall investigate the behaviour of the Cauchy-type integral in the
vicinity of the curve F. To this aim we shall give at first the following
definition [A.271: we say that F(z) is continuously prolongable on r
in to (different from the extremities) at left (right), if F(z) tends to
a well determined limit F+(to)(F_(to)) when z - to on every path
situated at left (right).
With this definition we may give the following fundamental theorem :
verifies or, t Holder's condition. then F(z) is continuously
If f (t)
prulongable on rat left and at right, excepting the extremities where
f (to) 76 0 and
Fi. (to) =
t2f(to) + 2ri I tf (t ndt.
(13.3. 1)
formulas. They have been
The formulas (B.3.1) are called
given in 1908 [A.29]. Their demonstration may be found for example in
[A.18], [A.271.
B.4
The Dirichlet's Problem for the Half-Plane
tine shall solve in the. sequel the following problem: We seek for the
function
(B.4.1)
F(z) = u(x y) + i v(x. y).
484
CAI;CIIY-TYPE, INTEGRALS
holornorphic in the half-plane y > 0 and continuously prolongablct on
the Ox, axis. which reduces at infinity, to an imaginary constant i C
(C may have the value zero) and whasc ir_al part is imposed on the above
mentioned axis, i.e.
u(x,O) = f(x),
(13.4.2)
where f is a function with a compact support compact which satisfies
Holders condition.
At first we have to mention that there exists a single function with
the above mentioned properties. Indeed, assuming that there exists two
functions F1 and r2 with these properties. their difference F =
= F1 - F2 is holotnorphic in the superior half -plane and it vanishes at
infinity. The real part of the function F is therefore harmonic in the
half-plane y > 0. zero on the boundary y = O and zero at infinity.
According to the maximum principle for the harmonic functions, the
real part of the function is identical zero. F reduces therefore to an
imaginary constant which is zero because F is zero at infinity. We
shall prove that the function
F(z)
7i
f }a
t
(t) d t + i C
(13.4.3)
CO
satisfies the conditions of the problem and it is therefore the solution we
are looking for. Indeed. the function F(z) defined by (13.4.3) is holomorphic in the superior half-plane because it is a Cauchy-type integral
and it is continuously prolongable on the real axis. At infinity it reduces
to the constant i C because, if we denote by (a, b) the support of
f(x),wehave
I-M
t
f (I
f () dt
(t) dt
/-
tl
dt.
The integral is therefore zero at infinity.
Passing to the limit with
z
x a point from the real axis, and
using Plemelj's formulas we obtain:
11(.x..0)+11'(x,0)= f(.1;)+
11 1
J+x
-x
., tf(t)dt+1C.
Taking the real part of this relation we obtain (B.4.2). Hence the problem is solved.
The real part of the solution (13.4.3), i.e.
UI(X. Y)
y
,or
fa+OC
f(t)
(t - T)2 + ry2
dt
(B.4.4)
THE CALCULUS OF CERTAIN INTEGRALS IN THE COMPLEX PLANE. 485
determines the harmonic function in the half-pl.uie y > 0, vanishing
at infinity, continuously prolongable on the Or axis and satisfying the
condition (13.4.2). This is the solution of Dirichlet's problem for the
half-plane concerning the harmonic function u.
B.5
The Calculus of Certain Integrals in the Complex
Plane
At first we shall prove that
G
1
a tit
t.
a
(B.5.1)
b - tt
where the determination of the radical is the positive one for z = r > b,
and (a, b) is an interval on the real axis.
Indeed, with the mentioned determination, we have
x<a. b<x
(B.5.2)
a<:x<b.
rb6
After all.
r<n. b<x
().
Re(i
r-a
b.....
z
a<x<b.
Since the real part of the holomorphic function i 1/: - n is known on
G
the real axis, the function will be determined by the formula (13.4.3).
We have therefore
Urb-t t..
i
a
-zzz-
b
-t--a dt
-ya'
+C.
Considering z oc it result., C = 1 and then the formula (B.5.1).
Analogously we deduce that
'
f
it
1
b
t
a
at
dt
=
-l + JfT b
(8.5.3)
486
CAUCHY-TYPE INTEGRALS
Passing to the limit and taking into account Plemelj's formulas;
(B.5.1) and (B.5.3) we obtain that:
'b fl -a dt
i t- x-
b
:r
1j
V
a t-
t
a
-
b=-t dt
(B-5-4)
-1
We must notice now that the above considered function
i
z-b
Y
YYY
reduces to an imaginary constant at infinity (this is i). If we want to
apply the same procedure for the function
(z - a)(z - b), we must
consider the expression
(z-a)(z-b)-z.
To this aim we have
x>b, x<a
n,
Re [i ( (z - a)(z - b) - z)} =
-(x-a)(b-x), a<x<b
whence
(z - a)(z - b) - z =
+- r
b
(.t
a)(b - t)
d t + C.
The integral becomes zero to the infinite. This imply
a+b
2
We have therefore
1
(t - a)(b
7r,"b
t-z
- t) d t = V"(-z-- a) (z - b) - z +
a
+b
(B.5.5)
and, passing to the limit and applying Plemelj's formulas,
rrt x
1
L
(t - a)(b - t)
d t = -x +
a
+
b
z
(B.5.6)
The following example is studied starting from the expression
i
(z - a)(z - b) '
which vanishes at infinity. Hence, the constant C will be zero. One
obtains the integrals
rb
V
a
(b - )(t - a)
t tz-
a)(; - b)
(B.5.7)
THE CALCULUS OF CERTAIN INTEGRALS t\ '1'flE COMvLI;Y PLANE 487
6
dt =
1
ir Ja f h - t) (t - a) t -
(B.5.8)
Xt) .
We consider now two intervals (a, b), (c, d) on the real axis. The
method can be extended to an arbitrary number of intervals, or even
disjoint arcs from the complex phuie, in the last case the integral being
solved in another way in [A.18], p.88. The result is useful in the study
of the grids of profiles. Denoting
Q(z) = (z - a)(-- - b)(z - c)(z - d),
(B.5.9)
for P(z) and R(z) arbitrary polynomials, we have
P(..)
n<x<b
- i li'(z)
Q(z)
C<x<d.
The function from square brackets is holomorphic in the half-plane y >
0. Utilizing the formula (B.4.3), we deduce
- R( _
V Q(=)
t
-r J
d tz
PQ(t)I
t
a t4
+C.
-Q(t)I
(B.5.10)
Taking into account. the behaviour of the integrals for z - oc, we
deduce that at; infinity we must. have the expansion
P(z)
Q(Z)
R(z) C+ a2+....
(B.5.11)
This formula allows to determine R(Z) when P(z) is known. For
example, for
P(z) = (z - b)(z - d)
it results R = 0 and C = 1. and for
P(z) = z(z - b)(z - d)
it results
n+c-b-d
2
CAUCIIY-TYPE INTEGRALS
488
After all, we have the results
(b - t))("_t)
(1_
(d - t)
°
1
dt
(t - b)(d - ')
t) d t
rt
1
(t.
t -z
(B.5.12)
-
(z - b)(z - d}
-
(z-a)(z-c)
r:
(b - t)(d - t)
1
rr,
_
td t
t-
1
+%r
l
d
1
(t - b)(d - t)
td t
(t-a)(t-c) t -,
(B.5.13)
a+c-b-d
(z-b)(z-d)
Passing to the limit for z
x E ((L, b), we apply Plemelj's formulas to
the first integrals. One obtains:
jd /(t-b)(d-t),t) dt
(b-t)(d-t) d t
(r- -C.) t-x
(t-a)(C-t} t-x}7f
6
1
q
b
tt
JG
(b - t)(d - t) td t
(t - b) (d - t) td t
(tc) t-x
1
(t-a)(c-t) t -
-1
x+
-x- a+c -2 b - d
(B.5.14)
and for z - x E (c,d), employing Plemelj's formulas for the integrals
on this interval, we obtain from (1.5.12) and (13.5.13):
(t-b)(d-t)
dt _ -1
1' J (t-b)(d-t)
(t-a)(t- c)
s,f J(t_a)(c-t)t_-x +r,
t
_
t-xdt-
1
ti
1
(t-a)(c-t) t-x + r
7r
1
(b-t)(d-t) dt
b
'
`t
(t-a)(t-c) t-x
(b-t)(d-t) tdt
a+c - b - d
2
489
t:LAUERT'S INTEGRAL
B.6
Glaucrt's Integral. Its Generalization and Some
Applications
We shall calculate the integral
1
_n
f -do
e ne
f
_
27r f r
cos e - 8
os ne
cos 9 - s
do,
(B.6 .1)
where s is a real number. For -1 < s < +1 the integral was calculated
by Glauert in [3.19]. The general case is considered in [5.38].
The method consists in passing to the complex variable. Denoting
exp (i 0) and noticing that we have
1
+" sin no
d8=0,
, cos B - s
because the integrand is an even function, it results
1
27r
++
f
x
-J=1_tz2-2sz+1
e ne
oos9-sd
1
9
/'
z"d z
iri
(B.6.2)
z"d z
in
where
lsl=1
(z - 0Z - Y)
1
a=s+ s2-1, Q=s -
82-1.
The last integral is calculated with the residue theorem.
Since aj3 = 1, It results that the poles of the integrand are situated
either one in the interior and the other in the exterior of the circle z = 1,
or both of them on the circle.
In the case s < -1, the pole z = a is interior and according to the
residue theorem, we have:
I = 2tr1
"
Q'"
1
= (3 +
S - 1)"
MQ
(B.6.3)
and for s > 1, the pole z = j3 is interior and it results
I.
1)"
(B.6.4)
At last, in the case -1 < 3 < 1, using the substitution s = cosa, we
have a = exp (i o), 0 = exp (-i a) and from the semi-residue theorem
[A.221. p.320 (the poles are situated on the integration path)
f (z) d z = ai f (zo),
z - :w
(B.6.5)
490
CAUCIIY-TYPE INTEGRALS
we deduce
In =
an
/i"
sin na
a-,Q
jj-rt
sin a,
Hence, Glauert's formula is
1 r*
rr ffff0
Cos no
Sill I
cos 0 - cos a d 0
sin o
n=0,1,2
(B.6.6)
Glauert's formula has many applications in aerodynamics. For example, using the substitutions
t=c+ecosO, x=c+ecoso,
(B.6.7)
2c=a+b, 2r.=b-a,
(B.6.8)
where
one obtains the formulas (8.5.4), (8.5.6), (8.5.8) and other similar ones,
like for example
Ft- t
Jt/---dt=ir(x+e).
(B.6.9)
b
Using the substitutions
t=c+ecosO, x=r.+es
(8.6.10)
and the formulas (B.6.3) and (13.6.4) we also obtain
l b t-a dt
71'
o
b-t t-x
4b-Xx
1,
l x-a
x - b
x<a
a<x<b
x>b
b-x - 1
1
7rI'.
b
t dt
t-a t-x
I va - x
1
1
n fL
a
dt
(b--t)(t-a) t - x
1
(a - x)(b - r,)
0.
1
Vf(x
etc.
x<a
491
OTHER INTEGRALS
B.7
Other Integrals
In the boundary elements methods one may encounter the following
integrals
cosne+i sin no
1k -1
(B.7.1)
for k = 1, 2. They are also calculated with the residue theorem putting
z = ei0. It results
sin8=(z-z) ,oos8(z+z),
21
Ii
12 -
_
z"d z
2
i b + Jc I:(z - zi)(z - z2) '
4i
(i
z"f 1d z
rJi:i_1 (z - zt)2(z
- z2)2
where, with the notation r = a - b zi
_
-a+r
b- ic'
z2
_
,
(B.7.2)
, we have.
-a-r
b- is
(B.7.3)
Obviously, J--1z21 = 1, such that either a root is in the interior of the
unit circle and the other in the exterior or both of them are on the circle.
We are interested in the case when a2 > b2 + c2 and a > 0. In
these conditions, the root z1 is in the interior of the unit circle, such
that, utilizing the residue theorem, one obtains:
71= F(r), 12 = G(r),
(B.7.4)
where
F(r) =
27r(-1)" (b+ic\" G(r) = 2w(-1)" a+nr (b+ic)"
r3
a+r
r
r ,
M IC)
(B.7.5)
Separating the real part from the imaginary one, we obtain a series of
492
CAUCIIY-TYPE INTEGRALS
integrals, like, for example
2a
d0
r
A a+bcos0+csin0
I
'"
a+bcos0+csiuO
o
f-
f
I
cos 0
r
d0
sin 0
cos 20
a+bcos0+csin9
'
=- 2:rr
b
c
d0=-'';r
r a+r'
rr+bcos0+csin0
'T
Jo
2.
d 0 = 2;.
b-'
r
(B.7.6)
d0
27ra.
Jo (a+bcos0+csin9)2 - r=s
.3T
Jci
(a+bcos0+csin0)2
o
(a+bcos0+csin0)2
'
d0=-rbi
sin 0
2:r
J0
- c2
cos 20
(a+bcos9+csin9)2
c
-;
a}
2r r
2+r1
r2 \d0
b2
- c2
(a+r)2.
The case a2 > b2 + c2 and a < 0 is studied in the same manner
(in this situation, z2 i3 in the interior of the circle). One obtains
Il = F(-r). 12 = G(-r).
(8.7.7)
If n2 < b' + c2, the two roots are on the unit circle, such that
the integrals are calculated by means of the semi-residue theorem. One
obtains
21, = F(r) + F(-r), 212 = G(r) + G(-r).
(B.7.8)
At last, if a2 = b2 + c2, the roots coincide and they are situated on
the unit circle. The integral will be calculated using the Finite Part (see
Appendix D).
If a = -s, b = 1. e = 0 and k = 1, one obtains the re cults from the
previous section.
_
Appendix C
Singular Integral Equations
C.1
The Thin Profile Equation
The thin profile equation in a free stream has the form
1
'b
f (t) d t= h(x),
7rt-x
a < x< b,
(C.1.1)
where 1(t) is the unknown, and h(x) is a given function. It is sufficient to assume that f satisfies Holder's condition on the interval
(a, b) for deducing the existence of the principal value from (C.1.1). The
solution of the equation also depends on the behaviour imposed to the
unknown at the extremities of the interval [a, b]. From the expression of
the solution it follows that h also has to satisfy Holder's condition on
the interval (a, b). The equation was pointed out by Birnbaum in 1923
[3.5) and solved for the first time by Sohngen in 1939 [A.34]. An ample
study devoted to this equation is due to Schroder [A.33]. The solution was found again, with different. methods by many other researchers
(Weissinger [3.49], Cheng [A.30], Homentcovschi [A.19], Carabineanu
[A.151 etc.). In the sequel we start from the method of C. lacob [A.20].
For solving the equation (C.1.1) we have to determine the function
F(z) = u(x, y) + i v(x, y), z = x+iy,
(C.1.2)
holomorphic in the superior half-plane y > 0, vanishing at infinity and
continuously prolongable on the real axis, with the conditions
u(x, 0) = 0,
for x E (-oo, a) U (b, oo)
(C.1.3)
v(x,0) = -h(x),
for x E (a,b).
In order to ensure the uniqueness of F(z) we have to impose its behaviour
in a and b. The function F may be bounded or not in these points.
494
S1N ULAR IN1 CRAIG l:QU VI`IONS
We w.,sume. for example that F is bounded in b, and it behaves in the
vicinity of a like
F(_)=0<cr<1.
(C.1.4)
where F` is bounded
We must notice that if we have determined such a function and if
we denote by f (x) the boundary values of a on (a, b), then It results
that the function f is just the solution of the equation (C.1.1). Indeed,
the function F(:), holomorphic in half-plane y > 0 and vanishing to
infinity, whose real part on Ox is
u =0,
for :t E (-nc,a)U(b,x)
(C.1.5)
it = f (.r.),
for :r E (a, b),
is given by the formula (8.4.3). i.e.
F(z) =
i
r
I
(t)
t
-
dt
(C.1.6)
.
Utilizing Pleanelj's formulas %,.-c obtain for ar< :r. < b
it (X, +0) + i v(r. +0) = AX) +
J t, t !
cl
(C.1.7)
t..
Separating the imaginary part and taking (C. 1.3) into account.. Ave obtain
(C.1.1) for v
We shall solve in the sequel the problem (C.1.2) - (C.1.4). Having
(C.1.4) in view, we shall consider the function (8.5.2). With the determination of the radical precised there and with the conditions (C.1.3),
we deduc c:
Re
_ tt 1
rE
0..
[r(z) Y-bJ --
T---a
b-x
a) U (b, ;,c)
x E (I a, b)
Hence, we may utilize the formula (13.4.3). It results
a
a1tV)
Cit.
tt
rri
F(`) z-Q`-1
the constant C vanishing because I' must v:uaish for z -- x. We
deduce
rz --b
t
(C. 1.8)
495
TIIE THIN PROFILE EQUATION
Applying Plemelj's formulas and separating the real part we obtain
Ax)
1
n
bx-
a
a s
_ta
b
t
IL (1)
t, - x
d t.
(C.1.9)
This is the solution of the integral equation (C.1.1) which satisfies the
boundedness condition in x = b.
The solution which is bounded in a and unbounded in b may be
-b
obtained starting from the function
We obtain
f(x)=-
'
W
r-a
b
6-x f
7r
.
4 - a
c
v
(C.1.1O)
(txat.
If we wish to obtain an unbounded solution the both end-points,
- a)(z - b), we have
using the function
Re [F(z)
e
(z - a)(z - b)j =
x E (-oo, a) U (b, oo)
ol
(x-a)(b-x.), xE(a,b)
h(x)
whence
!
f (x)
'b
r
(x-a)(b-x) Ja
(
1
i (t)
d t+
t - a)( b - t ) t-x
a
(C.1.11)
C
jr - a)(b - ar)
At last, for the solution bounded in the both end-points, we deduce
b
(z - a)(z - b) 1.
(t -a)(b -
t)
(C.1.12)
t t,
When z is great enough, this becomes
1
F(")
_
i
(z - a) (z - b)
ri
(1-a)"2 (i_)"jb
/b
Jn
h(t)
(t - a)(b - t)
dt
1- t/
_
-
h(t)(1+...)(lt
496
SINGULAR INTEGRAL EQUATIONS
such that imposing the zero value for F at infinity we have
f
h(t)
d t = 0.
(t - a) (b - t)
(C.1.13)
If this condition is satisfied, the solution of the equation (C.1.1) (bounded
at both end-points), is obtained passing to the limit in (C.1.12). Taking
into account the determination of the radical, we find
f(x)=-1R (a-a)(b-x)Ja
b
h(t)
(t - a)(b - t)
t dt
-r
(0.1.14)
Because of the restriction (C.1.13), this solution cannot be utilized in
aerodynamics.
C.2 The Generalized Equation of Thin Profiles
This has the form:
b
- Ja
(tad t +
f (t)K(t - x)d t = h(x) a < x < b,
(C.2.1)
n
where K is a non-singular kernel. We assume that f (x), K(x) and
h(x) satisfy Holder's condition on (a,b).
This type of equations are encountered in the theory of thin profiles
in ground effects, in the theory of grids of thin profiles, in the theory
of grids of thin wings etc. From the mathematical point of view, the
equations (C.1.5) are extensively studied in (A.27). In order to ensure
the uniqueness of the solution we have to impose the behaviour of the
unknown f at the extremities of the intagration interval. The method
of investigation consists In their regularization, i.e. they are reduced to
Fredholm-type equations for which existence and uniqueness theorems
are available. For the equation (C.2.1) this may be performed utilizing
the solution (C.1.1). If we are interested by the solution of the equation
(C.2.1) which vanishes in the trailing edge b, we shall utilize (C.1.9).
Assuming that the last two terms in (C.2.1) are known and knowing
that a singular integral interchanges with a non-singular integral (A.27),
we obtain the following Frcdholm-type integral equation
f(x) +
! f f (C)M(t, r)dt = H(x), a < x < b,
(C.2.2)
497
THE GENERALIZED EQUATION OF THIN PROFILES
where
b =x f
z) _ -1
t wa k(t - L)dt'
aQ
x
b
-1 bz [
t-a
_x- s,
W
t- x
t
(C.2.3)
h(t)
-C--t t - x
dt.
Sometimes we may specify the kernel M . For example, in the case
of the thin profile in ground effects, we have:
k{t -) =
(t
-)l
+
(C.2.4)
mz .
The integrals
t
1_
a
t-t
dt
)
(C.2.5)
b-t (t-t)2+m2 t-z
a
may be calculated with the residue theorem [A.20], or considering the
integral
_
=
f
b-a
m
dt
b-t (t-4)2+m t-x
8
and noticing that if we denote S = t + i m, we have:
6
t
dt
a
Fa
dt
V=t
t (t-()
l
-
(dt
f
n VLb_at
Utilizing (B.5.1) and (B.5.4) it results
1+iJ=z*C
C-b
)b
(C.2.6)
t-z
(C.2.7)
whence
1- a
J
2
1
S-a+
1
z-C-b x-S
x1 C
C-
-a
-b
c -b
(C.2.9)
498
SINGULAR INTEGRAL EQUATIONS
and con."juently,
b= x
1
1ll(f r) =
- a+
1
1
C/z:).
C.3 The Third Equation
We shall consider the equation [A.1G]:
-
b
1
J
f (t)(In it - xj + ro)d t = g(x) ,
(C.3.1)
defined on the interval (a, b), where f is the unknown and r o is a
constant. We intend to determine the general solution and to investi.
gate the conditions which are necessary for the solution to satisfv the
restrictions
f(a) = f(b) = 0.
(C.3.2)
With the change of variables t, x -. 0, a defined by the relations
t = c + e cos 9, x = c + e cosa ,
where
c=
a+b
-2'
e= b-a
2
,
(C.3.3)
(C.3.4)
the equation (C.3.1) becomes
1
-
f
F(9)(lnelcos0-cosaI +1'o)csinOd0=G(o)7T
With the notation
(t - a)(b - t) f (t),
(C.3.5)
'fF'(0)(lncjcm0-cosOl+l'o)d0=G(a),
(C.3.6)
F'(0) = eF(O) sin 9 =
this may be written as follows
1
where we have
1 coe m9 cos crier .
In 2l cos O - cos al = -2
m>1
(C.3.7)
499
THE TN1R1) EQUATION
The function F''(6) may be prolonged on the interval (-ir,O) such
that the result is an even function on (-pr, +7r) and it may be therefore
expanded into a trigonometric series, with the aid of the even functions
F' (9) = ao +
an cos no,
(C.3.8)
F` (9) cos nod 0,
(C.3.9)
n>1
where
an =
ao
r
ao =
I f,P(O)dOi f F(9)e sin 9d 9= 1Jaf f (t)d t.
7r
o
Analogously, the function G(a) may be expanded into the series
G(a) = bo + E bn aog no,
(C.3.10)
n>1
where
b,, = -
rx G(o) coo nod a = - 21 " G,(0,) I sin nad Q ,
0
9(x)
dx..
(C.3.11)
4 1o
(x a) (b - x)
a
Substituting these series into the integral equation (C.3.6), it results
bo
G(a)da = 1
A
r*
ao+E an coo no x
0
n>I
x
n>1
(_2!cosmOcosmo+F)d9=
(C.3.12)
m>_ 1 m
-aor +
n>1
i
a. cos no,
where we denoted
r = ro + In e = ro + I n (b 4 a) .
( C . 3 . 13)
Identifying the coefficients we find:
I
bo = -aor, bn = + an .
(C.3.14)
500
SIN(:ULAIt INTEGRAL EQUATIONS
In the sequel. we have
a) (b - t) g,(t)d
T
t..
(t 1 u)(b - 1 1 ( l (
fb
x
77
sin0
1
tt
r Jot COS a -, Cclti 0
_
1
sin a
r
0 COS a - C
ci
sin0
(C(4))
cos a - cos 0
(
(g)d0=
sin n0 d0=
7 cos(n + 1)0 - cos(n - 1)0 d
O
cos or - Cos 0
n
or, utilizing Glauert's integral (B.6.6),
i
(t.-a)(b-t),(t)dt
%b
ra
t -- x
sngit - 1)a
Stn(n + 1)a
n>1
n>1
-E a,, cosna+-F'(a)+tatt =
V12!1
(x - a)(b - x)f(x) - bo/r,
= --
(C.3.15)
where bo is (C.3.11). From (C.3.15) it results
f (s) =
-
1
rb y/ (t - (z)(G -
1
(x, - (1) (b
t-r
X) JJ4
t)9
(t)d t,(C.3.16)
_
1
r1'
1
(x - a)(b - x)
6
g{t)
dt.
(t -- -a) (b - t)
This is the first form of the general solution of the equation (C.3.I).
-
T11F,ru1RD EQUATION
501
One obtains another form if one utilizes the identity
(t - a) (b - t)
(x - a)(b - x)
(x -- a)(b - x)
(t - a)(b - t)
(C . 3 . 17)
(x-t)(x+t-a-b)
(r. - a)(b - x)(t - a)(b - t)
Substituting the first ratio in (0.3.16), it results the final solution
f(x)
(z
g'(t)
dt +
-a)(b-x.) (t-a
(b-t) t-x
a
1
1
1
-a)(b-a) J
g(t)1
I'
J
[(.
+ t - a - b)g'(t)-
(C.3.18)
dt
(t-«)(b--t)
We notice that from the relations (C.3.9), (C.3.14) and (0.3.11) it
results
1
Jn
bf(t)dt=TJ
b
-a)(b-r)clx,
(
(C.3.19)
which is an useful relation in applications.
We shall determine in the sequel the conditions which have to be
satisfied by g, such that the relations (C.3.2) are satisfied by the solution
(C.3.1). At first we notice that when the parameter r vanishes, a
necessary condition for the existence of the solution (C.3.18) is
b
L
g(t)
d t = 0.
(t - a)(b - t.)
(0.3.20)
The last integral from (C.3.18) defines for x real a polynomial of
the first degree. f vanishes in a, is this polynomial has the root a
(the first term from (0.3.18) vanishes for x = a). In this case, the last
integral has the order of (x - a), while the denominator of the fraction
has the order of (x - a) 1/2. Hence, we must have
-",)(b
f' [(t - b)g'(t)
grt)J
(t
- t) = 0.
(C.3.21)
502
SINGULAR INTEGRAL EQUATIONS
Analogously, f (b) = 0 implies
fb
[(t - u)9'(t) - 9(t) I
(t - u)(b - t)
0.
(C.3.22)
Imposing the both conditions, subtracting and adding, we obtain
91 (t)
L
b
J
[(t
a
9rt - - tad
- c)9 (t)
(C.3.23)
(t-a)(b-t)dt __ n,
(
t/b
)1
t)
0.
(C.3.24)
This is the answer for the proposed problem.
C.4 The Forth Equation
At least in nragnetoaerodynarnics [3.9] [1.9) p.208, in the theory of
oscillatory wings [10.15) and in the theory of the wing in fluids with
chemical reactions [3.10) [3.24], it intervenes the following singular integral equation [A.16)
fb
1
f(t)dt+- j
f(t)(lnt-x)+ro)dt=h(x), a<x<b,
(C.4.1)
where f is the unknown, ro an arbitrary constant and It, a given
function. The solution is unique if one imposes the behaviour of f in
one of the end-points of the interval. We shall impose the boundedness
of f sa in b.
Denoting
1b
g(x) = -,- f f(t)(ha it - xI + ro)d t,
(C.4.2)
a
we shall notice that
g'(x) =
r
ad t .
t
So, the equation (C.4.1) is reduced to the differential equation
(C.4.3)
a
(C.4.4)
whose general solution is
9(x) =
Aee('-`) +go(x),
(C.4.5)
503
THE FOIri'H EQUATION
where A is an undetermined constant, c is (C.3.4), and
(C.4.6)
The unknown f may- be found from the equation (C.4.2) which
is exactly the equation (C.3.1). Its solution is therefore (C.3.18) where
g will be replaced by (C.4.5). For specifying it, one observes that
performing the substitutions (C.3.3), denoting by lo(w), Il (w), .. .
Bessel's functions of imaginary argument and taking into account that
II(w) = Il (w), one obtains
(
CW(t
T1
)
a)(b -- t)
dt
rr" e
a
,
x
b
T2
'" °d 6 = 7rIo(M)
(t - a)(b - t)
dt
(c+ecosO)e 06Od0
(C.4.7)
o
= irc1o (w) + rrel i (w) .
From the expansion
exp(0cos9) = lo(Z3) +
(C.4.8)
n>l
and from Glauert's formula (B.G.6) we deduce
e`'(t-c)
b
dt
1
'
-x
-
2-,r
e
Sill it (T
n>1
smo
ejjr
ewcoo0
cos0-cosad8
(C.4.9)
-
2-,r
V-r(b
E 1,, (0) sin laor
- x)(x - a) ,>I
where w = ew. For T; and x we have the parametric representation
(C.4.9) and (C.3.3).
We deduce therefore
f(.x)=-'A (x-a)(G-x)7'3+
x
[w(x_2c)_]Ti+_
17
7,-
(x-a)(G-x)
x
--a)
(x
- (b - x)
(C.4. 10)
504
SINGULAR INTEGRAL EQUATIONS
fo being obtained from f by replacing g with go. In aerodynamics
f represents the jump of the pressure and we need therefore the integrals
j
rb
b
f x f (x)d x
f (x)d x,
(C.4.11)
4
which may be easily calculated without utilizing (C.4.9). Directly, the
first integral is obtained from (C.3.18).
In the sequel we shall determine the constant A. Imposing the
condition f (b) = 0, we have from (C.3.22) and (C.4.5)
A [ (aw
+ ,) Tt - wT2]
(C.4.12)
= GO,
where
b
Go
[(t - a)9o(t) - Tgo(t)
(t
u)(b
t)
(C.4.13)
Now the problem is solved.
C.5
The Fifth Equation
In the theory of oscillatory wings [10.20) it intervenes the following
integral equation (A.16[:
b(tf(t))2dt+
2
Jf(t)(lnIt-x[+I'o)dt=h(x), a < x < b,
b
(C.5.1)
which has to be solved together with the following conditions
f(a) = f(b) = 0.
(C.5.2)
The sign "' from (C.5.1) shows that. one considers the Finite Part.
Introducing the notation (C.4.2) and taking into account the formula
(D.3.6), from (C.4.3) we deduce
. b
o
(t - x)2 d t ,
(C.5.3)
a
such that the integral equation (C.5.1) reduces to the following differential equation
g" - w2g = h.
(C.5.4)
505
TILE FIFTH EQUATION
The homogeneous equation has the solution
g = Aocosh w(x - c) + Bosinh w(x - c) .
Applying Lagrange's method of variation of constants, one obtains for
the solution of the differential equation (C.5.4),
g = Acosh w(s: - c) + Bsinh w(x - c) + go(x) ,
(C.5.5)
where
h(E)siuhw(x - E)dt;
r(l(x)
(C.5.6)
The solution of the integral equation (C.5.1) may be obtained from
(C.3.18) where one replaces (C.5.5). For specifying it we notice that
setting
t = c+eu,
u E f-I,+1J,
we deduce (t - a)(b - t) = e2(1 - u2) and then
sinhw(t - c) d t = r+1 sinh (Foil)
So
n
J
(t - a)(b - t)
1
d u = 0,
(1 -- uu
the integrand being an odd function,
r6 Coshw(t - c)
(t - a)(b - t)
n
b tcosh w(t - c)
C1
(t - a) (b - t)
Ja
ar.
=
Cosh Pu}
_rdu=glo(w)
fJ
J1 - u2
1
dt
(C.5.7)
Yt
J
Sl
/b
(c+eu)
cosh (Cu)
1-u
tsinhw(t - c)
(t-a)(b-t)
du=crrlo(w)
1 usinh (Ou)
dt=e
f-i
1-u
d u = eirl1( ,.r ) .
Also, using the change (C.3.3) and taking into account (C.4.8), Clauert's
formula (B.6.6) and the relations
(-1)' I,,(w),
(C.5.8)
SINGULAR INTEGRAL EQUATIONS
506
we deduce
?,s
b
sinh w(t - c)
dt
- a)(b - t)
r' sinh (iv cos 9)
e
o
cos0-cosy
d8
sina
(b - x)(x - a)
f[
1
sinner
=E
[1. (0)
e n>1
TC
t-x
-
Eli
n>1
cashw(t - c)
ner
(C.5.9)
dt
(t-a)(b-t) t-x
n
sin nay.
(b - x) (x - a) n>1[1 -
These formulas, together with (C.3.3), give the parametric representation of the integrals T., TT and the variable x.
With these results, the solution of the equation (C.5.1) is
f (x) _
+
(x - a)(b - x)(ATT + BTT)+
(x - a)(b -
x)I A(wS1 \\
11,Co)
+wB (x-2c)Co+C1I I,
L
(C.5.10)
where fo is obtained from (C.3.18), replacing g(t) by go(t) given
by (C.5.6). As we have already mentioned in the case of the equation
(C.4.1), in aerodynamics we need the coefficients (C.4.11). They may
be obtained easier utilizing the form (C.3.18).
For determining the constants A and B we impose the conditions
(C.3.23) and (C.3.24) where g' has the expression (C.5.5). One obtains
the system
wCOA=Gm,
(C.5.11)
w(C1- cCo)A+ (wS1 -- r-1Co)B = G2,
507
THE FIFTH EQUATION
where
''
C1
-L
9-'o(t)
dt
(t-a)(b-t),
(C.5.12)
rb
G2 = -
JJ
[(t - 09' W) -
l
TYO(t)J
(t.- a)(b - t)
Taking into account (C.5.7), we deduce
G1
A- w7rlj1(D)
B,
r-110(w)[
(C.5.13)
In many applications we encounter the situation when h = -c (the
case of the flat plates with the angle of attack e). In this situation, from
(C.5.6) it results
w290(r) = e[1 - cosh w(X - c)),
such that
G1 = 0, W2G2 = Frr[wl1(i)
- r-`lo(ay) + r-.1[
whence,
A=0;
13= 4
[1+zr11(0)-IQ(W)1
.
(C.5.14)
Appendix D
The Finite Part
D.1
Introductory Notions
The notion of "Finite Part" of a improper integral has been introduced by Hadamard in 1923 [A.39], in order to give a significance to
the divergent integrals which appear in applications and to utilize them.
Hadamard studied integrals having the form
b
f
Ja
fW
d
(b-X)*,+1/lr,
(D.1.1)
where n = 1,2,3,.... There exists however many integrands with
non-integrable singularities which appear in applications especially in
aerodynamics. It exists therefore different manners for treating this
problem. In the subsonic aerodynamics one utilizes especially the definition of Mangler [A.45], but we had not the possibility to read this
paper. A less cited contribution, but very adequate to aerodynamics
belongs to Ch. Fox [A.37]. Here, the notion of "Finite Part" appears
like a natural extension of the concept of "Principal Value" in Cauchy's
sense. We shall present in the sequel some results of this author. For the
integrals having the shape (D.1.1) we shall utilize the paper of Heaslet
and Lomax [A.44]. These ones appear in the supersonic aerodynamics.
Important results concerning the notion may be found in the papers
of Kutt [A.42) and Kaya and Erdogan [A.41]. At lasst, the theory of
distributions give an unitary method for the study of this notion [A.5].
D.2 The First Integral
We shall consider at first the integral
Il -
fa
dx,
n=0,1.....
(D.2.1)
510
THE FINITE PART
If f admits derivatives up to the order n + 1 in the origin, then
we may write
=Jara
[f(x) -
(dx
I
yt
fl')(0) + ft'?(o)}
d+1+
[.rx)-E
o
ft,,(0)]
-, xi-^ f(i)(0) a
i - it
=
i=0
r=0
=
dx
it
14)
xla
+ n!
In
0
The integrated part for x = 0 becomes infinite. Neglecting these
infinite constants, one obtains the so called "Finite Part" of the integral
1. Hence, indicating by an "asterisk" the Finite Part, we have:
f(X)
xn{i(Ix =
in
[1(x)
-
"- t
JO
dx
.n+1 +
im0
"
f(i)(O)
(D.2.2)
a'
f In) (0)
In a.
i-0
For f = 1, it results
Td'x =lna,I-!+i
J
»
1
it a
(n.> 1).
(D.2.3)
D.3 Integrals with Singularities in an Interval
We shall consider the integrals having the form
12
rb
I( X)
+idx,
11 =0,1,...,
(D.3.1)
where a < u < b. For n;-- 0, we consider the "Principal Value" of the
integral in Cauchy's sense.
,(J G-E +
c-» u \ ,
x-u d x = lien
lb
1L f (x)
r1
f (?) d x
xu
.
(D.3.2)
511
I? TECRALS WITH SI`CULARITIES IN AX INTERVAL
«Vc know from (B.1.2) that this limit exists if f satisfies Holder's
condition in the interval (a, 6).
Let us derive now (D.3.2) with respect to the variable u. In the
right hand part, the derivation is performed according to the derivation
formula for integrals containing the variable in the limits. We have
therefore
d(,(
b
d r =1im
U
C-0
d u Jn X - u
f 0x)
t. s+
b
f (X)
Ju+E) (:r - fl):.
-
d x-
-f(u-0 _ f(u+=)
or, expanding into a Taylor series the functions f (u - E), f (it + E),
,
+
x (xatd x
Tit J
en o
(.1:
L+J (z )2J
(D .3.3)
If the limit from the right hand part exists. we denote it by
Ja
T.
f 0-)
(T-u)2
(D.3.4)
d X.
and we have
f
d x def
(x - u)2
lim
s-'0 l
f u-e + fb l
C a
d rh f(x) dA, J
du a X - It
a
f (X) d
(:C - u)2
2--1
x-
(Xf(T)
-.11)2 dx.
(D.3.5)
E
(D.3.6)
The limit (D.3.5) defines the Finite Part of the integral from the left
hand part. The Finite Part is a distribution {A.14]. Ex. 11. One proves
[A.37{ that if there exists f'(x) on (a, 6) and this function satisfies
holder's condition, then the limit from (D.3.5) exists. We notice that
this theorem constitutes the extension of the theorem of existence of the
limit (D.3.2).
We indicate now how one may reduce the calculation of the integral
(D.3.1) to the calculation of an integral with a weaker singularity. We
consider the case n = 1. Hence, we demonstrate that in the same
conditions like above (f' is defined and satisfies Holder's condition on
(a, b) ), we have
f(T)
,
(x-n)2
f(u)
f(6)
a-u 6-it
f'
, :r - u
THE FINITE PART
512
for every of from (a, b). Indeed, employing for the left hand side
member the definition (D.3.5) and integrating by parts, we obtain
r f ()
a
+
ii
(x-u)2
fb
t+
U -C
d x = lim
e-'0
CL
I-
+
Zb
+t )
a
f
2f (u)}
XU d a - c
f (a)
f f (x), d x+
a; x - rt
-
f (b) +
a-u b-u
f.b
t(x)
f da
X-u
.
The extension of the definition (D.3.5) and theorem (D.3.7) to an
arbitrary value of n , is performed in (A.37]. In the same paper one
gives the respective definitions in the complex plane and also Plemelj's
formulas for integrals having the form
F(z)
7ri I
21
(t
1(t +I d t
(D.3.8)
.
Utilizing (D.3.6) we may calculate (by derivation) the Finite Part
when we know the Principal Value. So, from (D.4.1) it results the integral often used in Appendix C,
1
7r
(t-a)(b-t)dt=-i, u<s<b.
(t - X)2
(D.3.9)
From (B.5.3) we deduce
1`/b
ia
dt
1
0
(b - t) (t -a) (t - x)2
(D.3.10)
,
and from (B.6.9),
t
1
it Ja
t
a
b - t (t
x)2
dt = 1,
(D.3.11)
and the sequence of examples may be enlarged (see also (A.41]).
We also observe that we have
day
(x - u)2
_
1
a - it
- b - it
1
which may he obtained considering f = 1 in (D.3.7).
(D.3.12)
513
HADAMARD-TYPE: INTEGRALS
D.4
Hadamard-Type Integrals
We shall consider the integrals having the form
I3=Ja
a
(s x)'+tf2dx,
n=1,2,...
(D.4.1)
which intervene in the supersonic flow. As we have already seen in
(D.2.1), the basic idea in the definition of the Finite Part consists in
leaving apart the infinite values from the structure of the integrals. In
order to see the significance of the integral denoted by (D.4.1) in the
case n = 1, we shall observe that
d
ds
f'
78
{ (s
s(x)x
d. = sic dds
- e)
f +J
a -C 8
a
f (x)
s-xJ dxJ
a8
.
If f is continuous in a, the first term from the right hand part becomes
infinite such that we must leave it apart. We shall consider by definition
B(x)xl d x = ds.
Ja a
I
s(x) d x
(D.4.2)
or
like in (D.3.6). For f = 1 we deduce
dx
a
(s-x)3/2
-
2
s-a
(D.4.4)
Let. us prove now that if f is continuous in s and admits bounded
first order derivatives on [a, s), then we may give a formula for calculating the member from the right hand part of (D.4.3). Indeed, we
have
d
ds
f f(x))s f(a)dlmf(x)-f(a)+
[1(x)-f(s)dx.
ifs s - x Ja as
,/ -x
3-x J
(D.4.5)
But, with the above hypotheses, on the basis of Lagrange's formula
f(x) = f(s)+(x-s)f`[s+9(x-s)),
THE FINITE PART
514
we deduce that the limit from (D.4.5) is zero. Hence,
d
f(x) dx=
ds /
s-x
is
f(x)-f(s)dx+
,/Rx
dx
R-x
J =
!( s ),.
dx
--1JNf(x)-f(s)dx- f(s) f
2 a
(s - x)3/'
(s - x)3/2
(D.4.6)
2
If we also utilize (D.4.4), then (D.4.3) becomes:
' AX)
(s - x)3/2
d x=
fa f(x) - f(s) dx-
2f(s)
s --a
(s - x)3/2
(D.4.7)
'
this representing the calculation formula for the Finite Part. It is similar
with the formula (D.3.5).
From (D.4.5) it also results a derivation formula, in fact, the formula
analogous to (D.3.7), which reduces the calculation of the integral with
a strong singularity to the calculation of an integral whose singularity is
weaker with an unity. In the case we had in view (n = 1), the weaker
singularity will be integrable. Indeed, taking (D.4.4) into account, we
have:
ds
f" f (r) - f (s)
fa
s-x
d x=
d
ds
f (x) d x--
s-x
d
(D.4.8)
x - 77=7=
f (_)
lax V-9 - i
Noticing now that 8/8R = -8/8x, integrating by parts and taking into
account (D.4.6), we obtain:
1f(z) - f(s)
8s
s-x
dx
S^x,
d
a
-1NIf(x)-f(s)]dx
(7s l
a
f (g) Z
N
dx
797-77
x
)dx=
+f(a)-f(s)+
s-a
+f8 f'(x) dx.
s -x.
(D.4.9)
515
GENERALIZATION
Equating the first members from (D.4.8) and (D.4.9) according to the
formula (D.4.5), we obtain:
d
dx_ f(a) +1fs fI (x) dx,.
8-x
8-at a s - x
f(.T)
as
ds
(D.4.10)
This is the formula of reduction to a weaker singularity, analogous with
(D.3.7). It also proves that the term from the right hand part of the
equality (D.4.2) is finite. Analogously one obtains
d
ds
x(x)8dx =
I'
-
f (b)s
fx(x)sdx, .
+b
(D.4.11)
The generalization of the definition (D.4.2) is performed as follows
I's 88"
an [ f(x)
s-x ]d, =
dn
(D.4.12)
d8" Q s-x
if f is continuous in a, n times derivable and with the derivative of
order n bounded in (a, s) (A.40J.
Generalization
D.5
In the theory of oscillatory wings one may encounter integrals having
he shape (D.4.1) where f depends also of 8. We shall establish for
these ones the derivation formula
(i
rb AX'S) dx = - f (b, 8) + J/'b fs (x, 8) + fi (x, 8)
{D.5.1)
x x-e
b-s s
Vime
from which one obtains (D.4.11) in case that f does not depend on s.
d4
Indeed, we have
d
dsJ, xx-s
fdx 1
ds[rf(x's
L
+f(8,s)
x fss's)dr.+
X
JJ,I
-
(D.5.2)
sJ
But,
d bf(--,
cly
y
s,s)dx=\sf(z,8)x
fs8,d)+
z
(D.5.3)
+
/'b 8 [f(x1s)_f(ss)1dx8x-8
J
THE FINITE PART
516
If f (x, s) is continuous in x = s and admits the derivative
f=
bounded in the interval Is, b], then, from
f(x,s) = f(s,s) + (x - s)f [s,s+d(x - s)]
we deduce that the limit from (D.5.3) is zero. After all, changing 8/8s
by -a/Ox and integrating by parts, we obtain
f
f (x, s
r
d s J,b
+ Jb ff(x,s)
f"[f(x.s) - f (s,
s) d
- f(s,s))sdx =
-f(x,s) -
x-s
+
_
g s'
v
ad x+
x
fb f=(x's)dx+
777-
x-s
J3
Xd-f(s,4) j xs
b fi(x, s) + fr(x, 9)
f (b, s) - f (s, s)
+le
b-s
1
s)1849
s
b
e
x
dx-8sr(s'8)
x-s
Replacing in (D.5.2) and taking into account that
d
s
TS
dx
x-s
b - s'
we deduce (D.5.1).
The established formula (D.5.1) shows that the first member is finite.
Hence, we may set by definition
r O f(x ss
'
d
ds= ds,f f
(D.5.4)
d x,
for every function f (x, s) continuous in the point (s, s), derivable,
with bounded partial derivatives in x and a, where s < x < b. In the
general case we shall define
I
n f(x,8)dx..
dsb
f
(x,s)dx.
(D.5.5)
Appendix E
Singular Multiple Integrals
In the euclidean space with n dimensions E, one considers a
domain D bounded or not ( D may coincide with
and a function
If
F(,c) defined on D. We denote t = (fi, ... , r;,,), a = (XI, ... ,
1) is unbounded we shall assuine that F tends to zero when Itl -+ 00
in a certain iuanner which will be precised in the sequel. We admit that
there exists a point Q(r) in D, such that in every vicinity D. (having
the diameter E) F is unbounded. while, in D - DE, F is bounded
mica integrable in the usual seas. Then we set
E-U D-D,
IIDFdC=Urn
If this limits exists, it is finite and does not depend on the shape of
D,, then the integral will be convergent. Otherwise the integral will be
diverrgeiit.
We consider now a divergent integral. If there exists a certain shape
of D, (for example, sphere or cube) for which the limit exists and is finite
(hence it is unique for every sequence of spheres or cubes contracting
towards Q). then the integral will be called semi-convergent. The limit
(which depends on the shape of DE) will be called principal value of the
integral.
Utilizing spherical coordinates one demonstrates that the integrals
having the form
JD''
f
(E.2)
where r = IC - xi, and
is bounded in D, are convergent for
a < n and divergent for a > n. The case a = n will be investigated
separately.
We shall consider the integral having the form
V(X) = JD
(X, M)
where m =
r T,
(E.3)
SINGULAR MIXTIPI.E IN1'E<;RALS
518
where f will be called the characteristic, u will be called the density
and x the pole of the integral. The ratio K(x1) = f /r" will be
called kernel. All the integrals that we utilize in this book have the form
(E.3). The first who studied this type of integrals was Tricomi [A.351
who gives some results in the case a = 2. An ample presentation, which
will guide us in the sequel, of the theory of integrals having the forth
(E.3), may be found in the books of Mihlin (A.251 and [A.26]. In this
presentation. D, will be spheri cal
the convergence of the integral
will be investigated with respect to this form.
We assume that:
satisfies Holder condition in D; if D is unbounded, we
10
assume that u(t;) = 0(It;l-t), k > 0. Holder's condition means that
there exists two positive constants A and a, 0 < a < 1, such that for
every two points 1;1 and 42, front D sa we have
ju(f1)
-
Ajf 1
- 21' ;
(E.4)
2° The characteristic f (x, in) is hounded and for a fixed x it
is continuous in in. Under these assumptions, we have the following
theorem:
The necessary and sufficient condition for the existence of the inteyrnl
(E.3) is to have
'
is f(x,rn)dS=0,
(E.5)
where S is the surface of the unit sphere centered in x.
In order to make the demonstration, we isolate the pole with a sphere
included in D, having the radius 6 and the center in x. Obviously,
f
(r, n)d,
1)
1
Litre
<K6
r
JII--Ui
u(h)f
r"
(u(E) - u(x)J f (T.t-) d o + u(x) l t o J
t<r<6
f (rnm)
The first two integrals from the right hand part of the equality are
absolutely convergent. In the third one utilizes spherical coordinates
relative to the pole x. Since d = r" 1 d rd S, we obtain
J
Cr"
f (x, err)
iss
f (x, rn) J
= In
r
f f (x. rn)d S.
F fs
It results the necessary and sufficient condition (E.5).
519
SINGULAR MULTIPLE INTEGRALS
if this condition is satisfied, the integral (E.3) may be represented
as follows
L v.(o
r"
(E.6)
r
L - nj u(h)f (x,m)(it + inh Mu(d) - u(x)]f (x,m)dt.
r
As an application, we shall consider the integral:
f
0
J p 1(c,n)
(E.7)
1
Ox Rcwhere
I? is (5.1.11). With the change of variables
q - y = sin 0, the condition (E.5) gives
-x=
008 0,
21r
X (19
I2-.T
0
=
Ro
J0
cos 8d 9 = 0.
(E.8)
Hence, the integral (E.7) exists.
The second important theorem that we utilize (for the transonic flow)
is the following:
If, in addition to the hypotheses 10 and 20, we assume that
grad.K(:r. ) = O(r-"° 1), then the singular integral (E.3) (as a function
of x) satisfies Holder's condition, with the same exponent like u, in
every domain which is bounded, closed and included in D.
The theorem was proved for the first time by Giraud in 1934, and
for n = 1 by Privalov in 1916. As we can see in (A.251-and (A.261, the
demonstration is not simple.
The last theorem refers to the derivation of the integrals with weak
singularities having the shape
v(x) =
JD
u(h)f
(E.9)
which lead to integrals of the type (E.3). Like in the previous theory,
D may be a domain bounded or not of the space E,,, or it may coincide with the entire space. W e assume that the function f (x, m) is
continuous and bounded together with its first derivatives (the first order
derivatives with resped to the cartesian codrdiriates of the points x
and na). We also assume that u(i:) satisfies Holder's condition and
at infinity (if D is unbounded)
o('t;-'J), t > 1. Under these
520
SINGULAR MULTIPLE INTEGRALS
assumptions, there exist the first derivatives of the integral (E.9), and
they are given by the formula
axk = ID
xk
[1t]
d
, - u(x)
j
s
f(x, m)
cos(n,
xk)d S,
(E.10)
where, like above, S is the surface of the unit sphere centered in x, and
n, the outer normal to the sphere. Olviously, the first integral from the
right hand part of the equality is singular.
Appendix F
Gauss-Type Quadrature Formulas
F.1
General Theorems
This appendix relies on the paper [A.49] and it is completed with
some results due to Monegato [A.52). Gauss-type quadrature formulas
give exact evaluations for the integrals of polynomial functions, multiplied by a weight function w. In aerodynamics it also meet integrals
with singularities. Approximating an arbitrary function (according to
Weierstras's theorem) by a polynomial function, we may utilize these
evaluations. Practically, the approximation is performed by a Lagrangetype interpolation formula.
We shall consider in the sequel that to : 1-1, +11 - R is a positive
integrable function.
THEOREM 1. We have the exact evaluation
+i
1
n
Af (x.)
f (x)w(:c)d x
(F.1.1)
0=1
if.
10 f is a polynomial of degree < 2n -- 1;
2e the points x = x,,, a = 1, n are the n zeros of the polynomial
P,,(x) of degree n from the orthogonal system of the weight w(x) on
[-1, +1j
w(x)P,(x)Pj(x)dx = 0,
i 0j;
(F.1.2)
30 Using the notation
Wt) = fP(x)!.Q3dx,
(F.1.3)
the coefficients A. are given by the formulas
Qn(2a)
4a = Pn(xa) .
(F.1.4)
CAUSS-TYPE QUADRATURE FORMULAS
522
Proof. Taking into account that f is a polynomial of degree 2n-1,
and P is a polynomial of degree n, we may write
f
n
a,
E 27 -xa + Fn_1,
T-
(F.i.5)
where Fi_1 is a polynomial of degree < n - I.
We determine the coefficients a° multiplying (F.1.5) with x - x°
and putting x = xa. It results
(F.1.6)
as = f(z°)/PP(xa)
whence
n
__
f
P.
f(xa)
=1 Pn(X-)(Z - xa)
+ A,
Since
dx =
J- 11 n( ) )(x
)
( )`vn(xa)
a)
=
and because F,,-, may be written as a linear combination of P0,.. . , Pi_1,
such that
+1
Pn(x)FF-1(x)w(x)dx = 0,
we deduce (F.1.1).
THEOREM 2. We have the following exact evaluation:
rt
J- t
x; d x = E A° f (x°
n
f (x)
a=1
.
(F.1
(F.1.7)
x
where j = 1, 2, ..., if
10 f is a polynomial of degree < 2n;
2° the points x = x°t a = T-, n are those defined in Theorem 1;
3° the points t = t,, j = 1;2,..., are the zeros of the function Qn
defined by (F.1.3);
40 the coefficients A are defined by (F.1.4).
523
GENERAL THEOREMS
Prof. Reasoning like before, we have:
f P_E ac. +Fn,
X
X0,
A-1
where the degree of F, < n. It results that as has the form (F.1.6).
Setting F = (x - tj)Fn_1(x) + A, where the degree of Fs_1 < n - I,
we deduce
f
P'
L-i
whence
+1
P"(x)f(x0)w(x)
LI
f (xa
Pn(x.)(x-xQ)(x-tj)dx__
'(+1 P,(x)w(x)
P., (X-)
A
P (xQ)(x - x.) + (x -
1
A. (:r.) _
QXQ - tj
xQ - tj
[_jdx=
1
1
f (zz)Qn(t5)
1n(xa)(xa-tj)
_ A. f (x,)
xQtj
because Q,, (t,) = 0.
THEOREM 3. We have the exact evaluation
I
(f(x)
(F.1.9)
t
where the "asterisk" is for the Finite Part (I).3.5), if.-
10 f is a polynomial of degree < 2n + 1;
20 the points x = x.., a = 1, n are those defined in Theorem 1;
30 The points t = tj, j = 1, 2.... are those defined in Theorem 1;
4e the coefficients An are given by (F.1.4), and
A = Qgn(tj)
Pn(tj)
(F.i.10)
Proof. We shall notice at first that, on the basis of the definition
(D.3.G), we have for t E (-1, +1),
t0adx,
(F.1.11)
524
GAUSS-TYPE QUAUR .ATURE FORMULAS
whence
t
Q;,(tJ) =
jPn(x)(')2dz.
(F.1.12)
We shall write m above
is
PA
f - P" LF-
(F.1.13)
)x-)
being a polynomial of degree :5 n - I and B, C, constants.
For x = tj it results
I(ti)
C
F&
f(3.-*)
(F.1.14)
0-1 Fnxa)(tj - xa)
P"(tj)
Since
_
1
(x - xn)(x - tj)2
2
1
1
tj)2 x
(rct
1
x,.
1
1
(z _tj)2 x-tj + tj -xo (x-tj)2
Taking into account that
0 and that Ave have (F.1.12),G
we deduce
1+`
P"(x)f(x.)w(x)
f-1 Pn(x.)(x - X.)(x - t j)2
= f(x.)
(x.-tj)2( x-x
P (x.) (' +1 P"(x)w(x)
1 A
1
dx=
- r. 2- t3/ d x+
' +1
+J-1
(F.I.15)
dx
ty-xo
(x-ti)2
f(xa)
Q.(xrx) + Q;z(tj)
P.(x0) (x,, - tj)2 tj - Xw
Having in view the definitions of the Principal Value and Finite Part,
it results
t
T'I
x - tj
fl+' P.(r)u'(x)dx
x - tj
4n (tj) = 0
Utilizing (F.1.14) and (F.1.10), we deduce
1Pn(xQ)(tj
+!
C
+
_1
P"(x)(xu,(x)
- tj)
2dx = AI(tj) -
y
1{XQ}'^Ln(tj)
- xa)
525
FORMULAS OF INTEREST IN AERODYNAMICS
whence it results (F.1.9).
The integrals having the form
wx
( )n :
+1
1 f
(x) (x
n>2
'
(F.1.16)
1
are studied in [A.49] and [A.52].
F.2
Formulas of Interest in Aerodynamics
It is well known that on the interval (-1, +1] Jacobi's polynomials
(x) constitute an orthogonal basis with the weight function w(x) _
(1 - x)°(1 + x)' ( see, for example, (A.561). Obviously, the zeros xo
and t) from the theorem--, from F.1 do not depend on the factor of
normalization of the polynomial P,,. One can simplify this factor in
A,,
from (F.1.4) and A from (F.1.10). We shall utilize therefore
Jacobi's polynomials without the constant factor.
1°. For the weight w(x) = (1-x2)-1/2, Jacobi's polynomials reduce
to Chebyshev's polynomials
x = cos9.
TT(x) = cosn9,
(F.2.1)
F o r 0 < 0 < jr, the polynomials T. (n = 1, 2, ...) vanish when
9° - 2a-1 w ,
n
2a-lir ' a = Tn.
xQ_
- oos
2
n
Utilizing the notation t = coax
2
and Glauert's integral (B.6.6), we
deduce:
sin or
(F.2.3)
Qn(t) = x sin or
with the zero
a,=? , tj=cos
r(x) _ -
Since
1
-1-,n
n
(F.2.4)
sinn9
sin9
dTn
sing dO
from (F.1.4) it results that A. = 7r/n. Since
n(t))
V
(F.2.2)
d sin na
- -slam do sino'
= nir cos ja
cf
sill2
526
GAUSS-TYPE QUADRATURE FORMULAS
from (F.1.10) it results that A = -n7r/(1 - t).
We deduce therefore the following formulas:
tt
411
f(x) dx = T Ef(xn),
it
1-x
0
r
J-
i
+I
+1
f (x)
dx
7r c- f (x, )
y:"]X-tj
dz
f(x)
(x
_ tj)2
1-Z
J-1
(F.2.5)
1
n i-.
(F.2.6)
n.Lt
f (v)
tt7r
(xa - t;)2
1 --
t=f(tj)
(F.2.7)
for j = -, n- 1, where .r are given by (F.2.2) and tj, by (F.2.4).
211. For the weight function w(x) = (1- x2)1/2, Jacobi's polynomials
reduce to Chebyshev's polynomials of second order
sin(
e1)9
sin
Cyr
+
e"
I'
1
x=
xa - coy re[+ 1 '
(F.2.8)
a = 1, nrt .
(F.2.9)
Using the notation t = cos a and Glauert's integral (B.6.6), we deduce:
-ircos(n j- I)a
(P.2.10)
whence
2, j=1,ri+1.
tj=cvs't
(F.2.11)
1
Utilizing the formulas given in the theorems F.1 for A,, and A, on the
basis of Glauert's integral, we obtain:
- n4-1 (1 -" xa),
A = -r (n + 1)
.
Hence, the following quadrature formulas are established:
n
+'
1-1`.f(x)dx=n+1E(I-
+1
xt
f(rn),
(P.2.12)
es=1
f(r.)
x -tj
dx=
7r
I - xp f( :ra ),
-t;
(F.2.13)
527
FORMULAS OF INTEREST IN AERODYNAMICS
+1
1- xa
ir
f (x)
1-z (x-tj)2dx=n+1
J-1
(xQ-tj)2f(xo)
awl
(F.2.14)
-ir(n + 1)f (tj),
where j = 1, n + I, xa are given by (F.2.9) and tj by (F.2.11).
We notice that the numbers tj given by (F.2.4) are the zeros of the
and the numbers ti given in (F.2.11) are the zeros
of the polynomials Tn i(t).
x)1/2(1 + x)''/2, Jacobi's
3°. For the weight function w(x)
polynomials
polynomials are [A.56].
Pn(x)
=
with the zeros
sinj(2n + 1)8/2]
_
2aar
x = cos 0,
,
a=1-,n.
(F.2.15)
(F.2.16)
With the aid of Glauert's integral, we deduce
/2j'
-,'[(2n
Qn(t) =
cos(
CO6C
/Z)
whence
9=l .
(F.2.17)
Utilizing (F.1.4) and (F.1.10), it results:
A. =
2n
2n+1(1-xa),
A
_ 2n+1x
1+tj 2'
Hence, we established the following formulas:
+1
J-
1
r+i
. +1
_0
2w
fin
1f(x)dx 2n+1`
a-i
J
- x f (x)
_
(1-x.)f(x.),
27r
+x x-tjdx 2n+1 a-i -a
1-z
f(x)
-I V 1+x (x - tj)2
n
(F.2.18)
(F.2.19)
13
dx=
_
(F.2.20)
j 2f(tj)
2n+1r(Q-tj)2f(x.)- +
528
GAUSS-TYPE QUADRATURE FORMULAS
j = 1 n,
for
being given by the formula (F.2.16) and tj by
x,,
(F.2.17).
40. For the weight function w(.r.) = (1 - x)'"112(1 + r)h/2, Jacobi's
polynomials are
'n(s) with the zeros
cos((2n + 1)9/2J
cos(0/2)
2a -1
r
x = cos9,
o= ln.
2n+1T,
(F.2.21)
(F.2.22)
Utilizing Glauert's integral, we deduce:
Qn(t) _
sin[(2n+ 1)Q/2]
s in(a/2)
,
t=CADS a,
polynomials which have the zeros
t'-
(F.2.23)
2n+1'
On the basis of the formulas (F.1.4) and (F.I.10) we obtain
2'7r
A"2n+1(1+xn),
2n + I
An=-1-t,
ar
such that one establishes the following formula,
+1
J
1+x
1
'
n
L(1 +x,,)f(F.2.24)
f (x)dx = 2n + 1
27
J1 + x f (x)
21r
1-xx-tjdx=2n+l
1 + x0
1+x
ar 2n+1 it
1-t; ?f(tJ)
s"-tjf(n),
(F.2.25)
1
f
1--x {x-tJ)2
(F.2.26)
tar
n
2n+I
owl
(ca-tf)-f(x°)'
for j = 1, n, the zeros xa being given by the formula (F.2.22) and tj
by the formula (F.2.23).
We have to notice the relations between the formulas from 30 and
40. The points x,, from 3° coincide with the points t1 from 40, and
x,,, from 40 with t,, from 3°.
529
TILE MODIFIED MONECATO'S FORMULA
F.3
The Modified Monegato's Formula
It is preferable sometimes to replace the formula (F.1.9) which contains the numbers f (t;) by the formula given by Monegato (A.52] p.
279.
t1,(r)
f(r)
dx=>
(F.3.1)
(x - t)2
J-1
where
Q1. (Z.) - Q. (t) - Q/n(t)(x° - t)
Pn(x°)(x° W
(F.3.2)
,
the formula containing only the numbers f (x°). The formula (F.3.1) is
exact, i.e. Rn (f) = 0. if f (x) is a polynomial of degree n - 1.
In fact, in applications one utilizes not the formula (F.3.1) [5.10),
[6.5], but another one which may be obtained as follows.
In (F.3.1) we isolate the term corresponding to a = j and we pass
to the limit,
+1
t -.
We find
n,
ut(x) (x f(x) dx = E wa' (xj)f(x°)+
- xj)2
°=1
+wj(xj)f(xj) + Rn(f),
(F.3.3)
.
where the mark
at F, means. that one excepts the term corresponding to a = j. The factor w'°(xj) is obtained from (F.3.2). Using
the rule of I'Hospital u J(xi) we find that
q., (xj)
(F.3.4)
In applications one utilizes (F.3.3) for w(x) = (1-x2)1/2. The numbers xo are given therefore by (F.2.9). From an elementary calculation
it results
Qn(s'a) = -7r(-1)° , Q'n(xj) = 0,
+ 1)(-1)°
X'2
,
Qn(xj) = 70 + 1)2 01
s
1-x12
530
GAUSS-TYNE QUADRATURE FORMULAS
whence
f(x)
+'
1
_-2
E [1 (-1)j]
(xa -
0=1
2f(:rn) -
n2
1f(xr)+ R,,.
(F.3.5)
This is the modified formula.
From
f d_
+t
w(x)
1.-t
uwa(t)f (xo) + Rn(f)
a
(F.3.6)
a=1
where (A.50J page 275,
ww(t)
Rn (xo)
Qn(T )
- Q.t (t)
(F.3.7)
,
isolating in E the term for which a = j and passing to limit t --+ xj,
one obtains the formula
ry1
-t
Tl
u
w(x) f (x) d x =
x - xj
',)f (.r.)+
owl
xa)
(F.3.8)
f(x
'
For the weight function w(x) = (1 - x2)1/2 we find:
+t
1
n
1 - x2 1(x) cl x =
amt
n+1
2 f(
X( -xj x° )
(F.3.9)
F.4 A Useful Formula
We shall establish the following series expansion:
= -2E(j + 1)Uj(y)Uj(ri)
1
(n
X1)2
1
(F.4.1)
531
A USEFUL FORMULA
where U (x) are Chebyshev's polynomials (F.2.8). The series is divergent, but it has a first Cesaro finite sum:
n
(tl
-2
1 y)2
I
U + 1)Ui (y)Ui(q) .
(F.4.2)
Indeed. setting q = ooeO and y = oosa on the basis of Glauert's
integral, we deduce
r+1Ua(q)dq=-aoos(n+1)o,
(F.4.3)
and deriving and taking (D.3.6) into account, we obtain the formula
J
(q - y)z
d q = -w(n + 1)U*(y).
(F.4.4)
1
Expanding now the function (q - y)'2 on the interval [-1, +11 in a
Chebvshev-tvpc polynomial series, we have:
1
aJ(y)Uitq).
(F.4.5)
(q - y)" =
Taking into account (P.4.4), and the orthogonality condition
2!
UJ(1)Uk(q)dq =
10 JiAk
J7r
,1 = k,
(F.4.6)
we obtain:
a,(y) = -2(j + 1)UJ(y).
Replacing it in (F.4.5), we obtain the formula (F.4.1).
(F.4.7)
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(A.11) V. Vladimirov a.o.: Recueil de problemes d'equations de physique
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(A.20J D. Homentcovschi: Func;ii complexe cu aplica;ii in §tiinjd ,si
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Index
Abel's equation, 320
Acceleration
as material derivative , 2
of the particle, I
potential, 9, 397, 404
Ackeret's formulas, 288, 423
Acoustics, equation of, 29
Aerodynamic action
drag, 171, 172
lift, 76, 170, 172
moment, 76, 170
gyration, 171, 172
pitching, 171, 172
rolling, 171, 172
Arrow
of airfoil, 83
shaped wing, 156
Attached shock waves, 361
Complex
velocity, 71, 75, 120, 371, 375
Compressibility effects, 86, 148, 231
Conduction law, 13
Cone
body's shape, 463
double. 464
Conical flow, 339-347, 464
Conservation laws, 11
Convolution, 33, 370, 470
Critical velocity, 6, 359
Delta wing, 156
Descartes, folium, 18
Detached shock waves, 19, 360
Distributions, theory of, 465.479
Doublet
Bernoulli's integral, 5-7
Bessel's functions, 407, 503
Biplane, 92
density, 266
flow induced by, 148
potentials, 426
Drag coefficient co, 181, 203, 204, 213,
229, 231, 241, 292, 458, 462,
Body, theory of slender, 449-464
R":rndarv conditions
Duhamel principle, 57
airfoils in tandem, 1:,Y
airfoils parallel to the undisturbed
stream, 93
grids of profile, 99
ground effects, 83
material surface, 24
rest state, 28
tunnel efects, 88
uniform motion, 25, 70
Boundary conditions (nonlinear case),
464
Energy, 3, 4
Enthalpy, 15
Entropy, 4
Euler
-Lagrange criterion, 24
constant, 400
equation, 3
formulas, 146
theorem, 2
111, 112
Flutter, 397
Caloric equation, 4
Cauchy
integral, 71
principal value, 482, 510
problem, 61
Characteristics
coordinates on, 308
variables on, 318
Chebyshev polynomials, 80, 525, 526
Circulation, 125, 126, 274, 393
Clapeyron's equation, 4
Collocation method, 115, 231, 279
Complex
potential, 121, 146
Flux-divergence formula, 479
Forces, continuous distribution of, 73
Fourier, transform, 465-479
Fredholm integral equation, 209
Fundamental matrices, 61
Fundamental solutions
equation of potential
fluid at rest, 42
M = I oscillatory, 34
M = 1 unsteady, 41
subsonic oscillatory, 33
subsonic steady, 31
subsonic unsteady, 37
supersonic oscillatory, 34
INDEX
572
Fundamental solutions
equation of potential
supersonic steady, 31
supersonic unsteady, 38
oscillatory system
pressure formulae, 50
velocity formulae, 52
steady system
general form, 45
plane subsonic, 46
plane supersonic, 48
3-D subsonic, 47
3-D supersonic, 48
unsteady system, 57
Causs-type quadrature formulas, 521530
Clauert
integral, 489
method, 210
Gothic wing, 156
Green function, 142, 143
Ground effect, 82, 86, 136, 184, 238,
299
Hadamard
finite part, 509
type integrals, 513
harmonic forces, 398
Heat, specific c,,, c.,, 4
Helmholtz's equation, 5, 33
Henkel's function, 33
Homentropic motion, 4
Huygens' principle, 42
Ideal fluid, 1
Instable shock, 18
Integral equations
steady subsonic flow
lifting line, 201, 209, 219
lifting surface, 165, 167, 187
airfoils in tandem, 103
grids of profiles, 99
ground effect, 84
parallel airfoils, 94
tunnel efect, 90
thin profile, 72
steady supersonic flow
lifting surface, 307, 313, 320
Integral equations
steady transonic flow
lifting line, 395
lifting surface, 389
unsteady flow
sonic profile, 440
sonic wing, 445
subsonic profile, 402
subsonic wing, 408
supersonic profile, 418
supersonic wing. 432
Invariant, 16
Irrotational flow
condition, 368
definition, 5
equation of, 109
Isentropic motion, 4
Joukovaky profile, 86
Lagrange
-Cauchy's theorem, 5
interpolation, 223
variation of constants, 218
Leading edge, 62, 156, 301
subsonic, 301
supersonic. 301
Lift coefficient, 129
Lift coefficient CL, 76- 78, 81, 82, 85,
86,90, 96, 100, 104, 107, 171,
180, 183, 193, 195, 202, 204,
213, 229, 231, 237, 241, 287,
288, 291, 293, 298, 299, 350,
354, 357, 380, 421, 423, 441,
457, 462, 464
Mach
angle, 40
cone, 31
dihedron, 32
number, 8, 19
Moment coefficient cm, 76-78, 81, 82,
85, 86, 90, 96, 100, 104, 107,
287, 288, 291, 298, 299, 421,
423, 441
Moment coefficient cs, 203, 204, 229,
231, 241, 458, 462
Moment coefficients c1, cv, 171, 181,
183, 19.3, 195, 2(r2, 229, 231,
237, 241, 458, 462. 464
INDEX
573
Plemelj's formulae, 483
Prandtl's theory, 197-205
Prandtl-Mayer fan, 289
Source
mass, 43
Pressure coefficient Cp, 120, 129, 148
Swallow wing, 156
Rhombic wing, 156
Thermodynamics, equation of, 3
Trailing edge, 156. 301
Shock waves
Hugoniot's equation, 15
jump equations, 13
Prandtl's formula, 16
shock polar, 18
Sonic barrier, 438, 448
Sonic circle, 18, 19
potential, 36
Kutta-Joukovski, 81, 165, 275,332
subsonic, 301
supersonic, 301
Trapezoidal wing, 156
Vortex, 5, 71
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