Incompressible Flow
Third Edition
Ronald L. Panton
WILEY
JOHN WILEY & SONS, INC.
Contents
Preface
xi
Preface to the Second Edition
Preface to the First Edition
1
Continuum Assumption
3
Fundamental Concepts, Definitions, and
Laws
3
Space and Time
5
Density, Velocity, and Internal Energy
Interface between Phases
10
Conclusions
12
Problems
14
1.3
1.4
1.5
1.6
Thermodynamics
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
3.1
3.2
3.3
3.4
3.5
3.6
3.7
7
4 Kinematics of Local Fluid Motion
4.1
4.2
4.3
4.4
4.5
15
Systems, Properties, and Processes
15
Independent Variables
16
Temperature and Entropy
17
Fundamental Equations of
Thermodynamics
18
Euler's Equation for Homogeneous
Functions
19
Gibbs-Duhem Equation
20
Intensive Forms of Basic Equations
20
Dimensions of Temperature and Entropy
Working Equations
21
Ideal Gas
22
Incompressible Substance
25
Conclusions
25
Problems
26
3 Vector Calculus and Index Notation
Index Notation Rules
28
Definition of Vectors and Tensors
30
Special Symbols and Isotropic Tensors
Direction Cosines and the Law of
Cosines
32
Algebra with Vectors
33
Symmetric and Antisymmetric Tensors
Algebra with Tensors
37
Vector Cross-Product
38
Alternative Definitions of Vectors and
Tensors
40
Principal Axes and Values
41
Derivative Operations on Vector Fields
Integral Formulas of Gauss and Stokes
Leibnitz's Theorem
47
Conclusions
48
Problems
49
3.10
3.11
3.12
3.13
3.14
Continuum Mechanics
1.1
1.2
2
3.8
3.9
XUl
4.6
4.7
4.8
4.9
4.10
21
5
27
5.6
5.7
5.8
31
5.9
5.10
5.11
35
5.12
51
Lagrangian Viewpoint
51
Eulerian Viewpoint
54
Substantial Derivative
56
Decomposition of Motion
57
Elementary Motions in a Linear Shear
Row
61
Proof of Vorticity Characteristics
64
Rate-of-Strain Characteristics
65
Rate of Expansion
66
Streamline Coordinates
67
Conclusions
69
Problems
69
Basic Laws
5.1
5.2
5.3
5.4
5.5
42
45
71
Continuity Equation
71
Momentum Equation
75
Surface Forces
76
Stress Tensor Derivation
76
Interpretation of the Stress Tensor
Components
78
Pressure and Viscous Stress Tensor
80
Differential Momentum Equation
81
Moment of Momentum, Angular Momentum,
and Symmetry of Ttj
85
Energy Equation
86
Mechanical and Thermal Energy
Equations
89
Energy Equation with Temperature as the
Dependent Variable
91
Second Law of Thermodynamics
91
vi
Contents
5.13
5.14
Integral Form of the Continuity Equation
92
Integral Form of the Momentum
Equation
94
Momentum Equation for a Deformable Particle
of Variable Mass
97
Energy Equation in Integral Form
100
Jump Equations at Interfaces
101
Conclusions
102
Problems
102
5.15
5.16
5.17
5.18
8.11
9 Compressible Flow
9.1
9.2
9.3
6 Newtonian Fluids and the Navier-Stokes
Equations
105
6.1
6.2
6.3
6.4
6.5
6.6
6.7
7
Newton's Viscosity Law
105
Molecular Model of Viscous Effects
108
Non-Newtonian Liquids
112
No-Slip Condition
115
Fourier's Heat Conduction Law
117
Navier-Stokes Equations
119
Conclusions
120
Problems
120
Some Incompressible Flow Patterns
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
121
Pressure-Driven Flow in a Slot
121
Mechanical Energy, Head Loss, and Bernoulli
Equations
126
Plane Couette Flow
132
Pressure-Driven Flow in a Slot with a Moving
Wall
133
Double Falling Film on a Wall
134
Outer Solution for Rotary Viscous
Coupling
137
Rayleigh Problem
138
Conclusions
143
Problems
144
9.4
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
8.10
145
Measurement and Dimensions
145
Variables and Functions
147
Pi Theorem and Its Application
150
Pump or Blower Analysis: Use of Extra
Assumptions
153
Number of Primary Dimensions
157
Proof of Bridgman's Equation
159
Proof of the Pi Theorem
161
Dynamic Similarity
164
Similarity with Geometric Distortion
165
Nondimensional Formulation of Physical
Problems
168
175
Compressible Couette Flow: Adiabatic
Wall
175
Flow with Power Law Transport
Properties
178
Inviscid Compressible Waves: Speed of
Sound 180
Conclusions
186
Problems
187
10 Incompressible Flow
188
10.1
10.2
Characterization
188
Incompressible Flow as Low-Mach-Number
Flow with Adiabatic Walls
189
10.3 Nondimensional Problem Statement
191
10.4
Characteristics of Incompressible Flow
195
10.5
Splitting the Pressure into Kinetic and
Hydrostatic Parts
197
10.6 Mathematical Aspects of the Limit Process
M2 — 0
200
10.7
Invariance of Incompressible Flow Equations
under Unsteady Motion
201
10.8
Low-Mach-Number Flows with ConstantTemperature Walls
203
10.9
Energy Equation Paradox
206
10.10 Conclusions
208
Problems
208
11
11.1
8 Dimensional Analysis
Conclusions
173
Problems
173
Some Solutions of the Navier-Stokes
Equations
210
Pressure-Driven Flow in Tubes of Various Cross
Sections: Elliptical Tube
211
11.2
Flow in a Rectangular Tube
213
11.3
Channel with Longitudinal Ribs
216
11.4
Stokes's Oscillating Plate
218
11.5
Wall under an Oscillating Free Stream
221
11.6 Transient for a Stokes Oscillating Plate
224
11.7
Flow in a Slot with a Steady and Oscillating
Pressure Gradient
226
11.8 Decay of an Ideal Line Vortex (Oseen
Vortex)
230
11.9
Plane Stagnation-Point Flow (Hiemenz
Flow)
235
11.10 Burgers Vortex
241
11.11 Complete Solution for Rotary Coupling
242
Contents
11.12 Von Karman Viscous Pump
11.13 Conclusions
248
Problems
249
244
12 Streamfunctions and the Velocity
Potential
251
12.1
12.2
12.3
12.4
12.5
12.6
12.7
12.8
Streamlines
251
Streamfunction for Plane Flows
254
Flow in a Slot with Porous Walls
256
Streamlines and Streamsurfaces for a ThreeDimensional Flow
259
Vector Potential and the E2 Operator
262
Velocity Potential and the Unsteady Bernoulli
Equation
266
Flow Caused by a Sphere with Variable
Radius
267
Conclusions
269
Problems
270
13 Vorticity Dynamics
13.1
13.2
13.3
13.4
13.5
13.6
13.7
13.8
13.9
13.10
13.11
13.12
13.13
13.14
13.15
271
Vorticity
271
Kinematic Results Concerning Vorticity
272
Vorticity Equation
274
Vorticity Diffusion
275
Vorticity Intensification by Straining Vortex
Lines
277
Hill's Spherical Vortex
278
Production of Vorticity at a Stationary
Wall
280
Production of Vorticity at a Translating
Wall
282
Helmholtz's Laws for Inviscid Flow
284
Kelvin's Theorem
285
Inviscid Motion of Point Vortices
286
Reconnection of Vortex Lines
288
Development of Typical Vorticity
Distributions
288
Vortex Breakdown
294
Conclusions
298
Problems
299
14 Flows at Moderate Reynolds
Numbers
300
14.1
14.2
14.3
Some Unusual Flow Patterns
301
Entrance Flows
303
Entrance Flow into a Cascade of Plates:
Computer Solution by the StreamfunctionVorticity Method
305
14.4
14.5
14.6
14.7
14.8
14.9
15
15.1
15.2
15.3
15.4
15.5
15.6
15.7
15.8
15.9
16
16.1
16.2
16.3
16.4
16.5
17
17.1
17.2
17.3
17.4
17.5
Entrance Flow into a Cascade of Plates:
Pressure Solution
,313
Entrance Flow into a Cascade of Plates:
Results
316
Flow around a Circular Cylinder
320
Jeffrey-Hamel Flow in a Wedge
336
Limiting Cases for Re —» 0 and Re —»
-oo
340
Conclusions
344
Problems
345
Asymptotic Analysis Methods
347
Oscillation of a Gas Bubble in a Liquid
347
Order Symbols, Gauge Functions, and
Asymptotic Expansions
350
Inviscid Flow over a Wavy Wall
353
Nonuniform Expansions: Friedrich's
Problem
356
Matching Process: Van Dyke's Rule
358
Composite Expansions
363
Characteristics of Overlap Regions
365
Lagerstrom's Problems
370
Conclusions
374
Problems
374
Characteristics of High-Reynolds-Number
Flows
376
Physical Motivation
376
Inviscid Main Flows: Euler Equations
378
Pressure Changes in Steady Rows: Bernoulli
Equations
381
Boundary Layers
385
Conclusions
395
Problems
395
Kinematic Decomposition of Flow
Fields
396
General Approach
396
Helmholtz's Decomposition
397
Line Vortex and Vortex Sheet
398
Complex Lamellar Decomposition
401
Conclusions
404
Problems
404
18 Ideal Flows in a Plane
18.1
vii
405
Problem Formulation for Plane Ideal
Flows
406
viii
Contents
18.2
18.3
18.4
18.5
18.6
18.7
18.8
18.9
18.10
18.11
18.12
18.13
18.14
18.15
18.16
18.17
18.18
18.19
Simple Plane Flows
409
Line Source and Line Vortex
412
Flow over a Nose or a Cliff
414
Doublets
420
Cylinder in a Stream
422
Cylinder with Circulation in a Uniform
Stream
424
Lift and Drag on Two-Dimensional
Shapes
426
Magnus Effect
429
Conformal Transformations
431
Joukowski Transformation: Airfoil
Geometry
434
Kutta Condition
439
Flow over a Joukowski Airfoil: Airfoil
Lift
442
Numerical Method for Airfoils
448
Actual Airfoils
451
Schwarz-Christoffel Transformation
453
Diffuser or Contraction Flow
455
Gravity Waves in Liquids
460
Conclusions
465
Problems
466
20.3
20.4
20.5
20.6
20.7
20.8
20.9
20.10
20.11
20.12
20.13
20.14
20.15
20.16
20.17
20.18
20.19
20.20
20.21
20.22
19 Axisymmetric and Three-Dimensional
Ideal Flows
468
19.1
19.2
19.3
19.4
19.5
19.6
19.7
19.8
19.9
19.10
19.11
19.12
19.13
19.14
20
20.1
20.2
General Equations and Characteristics of
Three-Dimensional Ideal Flows
468
Swirling Flow Turned into an Annulus
470
Flow over a Weir
471
Point Source
473
Rankine Nose Shape
474
Experiments on the Nose Drag of Slender
Shapes
477
Flow from a Doublet
478
Flow over a Sphere
481
Kinetic Energy
483
Wake Drag of Bodies
484
Induced Drag: Drag due to Lift
486
Lifting Line Theory
490
Added Mass of Accelerating Bodies
491
Conclusions
496
Problems
496
Boundary Layers
498
Blasius Flow over a Flat Plate
Displacement Thickness
503
498
Von Karma'n Momentum Integral
505
Von Karman-Pohlhausen Approximate
Method
506
Falkner-Skan Similarity Solutions
508
Arbitrary Two-Dimensional Layers: CrankNicolson Difference Method
513
Vertical Velocity
521
Joukowski Airfoil Boundary Layer
524
Boundary Layer on a Bridge Piling
527
Boundary Layers Beginning at Infinity
530
Plane Boundary Layer Separation
535
Axisymmetric Boundary Layers
537
Jets
540
Far Wake of Nonlifting Bodies
543
Free Shear Layers
546
Unsteady and Erupting Boundary
Layers
548
Entrance Flow into a Cascade
551
Three-Dimensional Boundary Layers
553
Boundary Layer with a Constant Transverse
Pressure Gradient
557
Howarth's Stagnation Point
561
Three-Dimensional Separation
564
Conclusions
568
Problems
568
21 Flows at Low Reynolds Numbers
21.1
21.2
21.3
21.4
21.5
21.6
21.7
21.8
21.9
21.10
21.11
21.12
21.13
21.14
571
General Relations for Re - • 0: Stokes's
Equations
571
Global Equations for Stokes Flow
574
Streamfunction for Plane and Axisymmetric
Flows
577
Internal Flows: Plane
579
Internal Flows: Three-Dimensional and
Axisymmetric
587
Local Flows: Plane (Moffatt Vortices)
591
Local Flows: Axisymmetric
596
External Flow: Sphere in a Uniform
Stream
599
Composite Expansion for Flow over a
Sphere
604
Stokes Flow near a Circular Cylinder
605
Axisymmetric Particles
606
Oseen's Equations
608
Interference Effects
609
Conclusions
611
Problems
611
Contents
22 Lubrication Approximation
22.1
22.2
22.3
22.4
22.5
22.6
22.7
613
25.3
Basic Characteristics: Channel Flow
613
Flow in a Channel with a Porous Wall
616
Reynolds Equation for Bearing Theory
618
Slipper Pad Bearing
620
Squeeze-Film Lubrication: Viscous
Adhesion
622
Journal Bearing
623
Conclusions
627
Problems
627
23 Surface Tension Effects
629
23.1
23.2
23.3
23.4
23.5
Interface Concepts and Laws
629
Statics: Plane Interfaces
636
Statics: Cylindrical Interfaces
639
Statics: Attached Bubbles and Drops
641
Constant-Tension Rows: Bubble in an Infinite
Stream
643
23.6
Constant-Tension Flows: Capillary
Waves
646
23.7
Moving Contact Lines
648
23.8
Constant-Tension Flows: Coating Flows
651
23.9
Marangoni Flow
656
23.10 Conclusions
665
Problems
665
24 Introduction to Microflows
24.1
24.2
24.3
24.4
24.5
24.6
24.7
24.8
24.9
24.10
24.11
667
Molecules
667
Continuum Description
669
Compressible Flows in Long Channels
Simple Solutions with Slip
673
Gases
676
Couette Flow in Gases
680
Poiseuille Row in Gases
682
Gas Row over a Sphere
687
Liquid Rows in Tubes and Channels
Liquid Rows near Walls
692
Conclusions
697
25 Introduction to Stability and
Transition
698
25.1
25.2
Linear Stability and Normal Modes as
Perturbations
699
Kelvin-Helmholtz Inviscid Shear Layer
Instability
700
670
690
25.4
25.5
25.6
25.7
25.8
25.9
25.10
25.11
25.12
25.13
25.14
25.15
25.16
ix
Stability Problem for Nearly Parallel Viscous
Rows
704
Orr-Sommerfeld Equation
707
Inviscid Stability of Nearly Parallel
Rows
708
Viscous Stability of Nearly Parallel
Rows
709
Experiments on Blasius Boundary
Layers
712
Transition, Secondary Instability, and
Bypass
714
Spatially Developing Open Rows
719
Transition in Free Shear Rows
719
Poiseuille and Plane Couette Rows
721
Inviscid Instability of Rows with Curved
Streamlines
723
Taylor Instability of Couette Row
725
Stability of Regions of Concentrated
Vorticity
727
Other Instabilities: Taylor, Curved Pipe,
Capillary Jets, and Gortler
728
Conclusions
731
26 Introduction to Turbulent Flows
732
26.1
26.2
26.3
26.4
26.5
26.6
26.7
26.8
26.9
26.10
Types of Turbulent Flows
732
Characteristics of Turbulent Rows
733
Reynolds Decomposition
736
Reynolds Stress
737
Free Turbulence: Plane Shear Layers
740
Free Turbulence: Turbulent Jet
741
Bifurcating and Blooming Jets
747
Correlations of Ructuations
747
Mean and Turbulent Kinetic Energy
750
Energy Cascade: Kolmogorov Scales and Taylor
Microscale
752
26.11 Wall Turbulence: Channel Flow
Analysis
756
26.12 Wall Layers: Experiments and Empirical
Correlation
764
26.13 Turbulent Structures
768
26.14 Conclusions
773
A Properties of Fluids
775
B Differential Operations in Cylindrical and
Spherical Coordinates
776
x
Contents
Basic Equations in Rectangular,
Cylindrical, and Spherical
Coordinates
781
D Streamfunction Relations in Rectangular,
Cylindrical, and Spherical
Coordinates
786
E Computer Code for Entrance Flow into a
Cascade
790
F Computer Code for Boundary Layer
Analysis
793
References
796
Index
813