A Panel Method Analysis of Flow Past a
Semi-Span Hydrofoil
by
Christopher Matthew Waaler
B.S., University of Illinois (1987)
Submitted to the Department of Ocean Engineering
in partial fulfillment of the requirements for the degree of
Master of Naval Architecture and Marine Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 1995
( Massachusetts Institute of Technology 1995. All rights reserved.
Author ...
(
.f
.e.a..e. . ...........
Department of Ocean Engineering
January
13, 1995
I1
N
. .n1
Uertitied by..
Justin E. Kerwin
Professor of Naval Architecture
Thesis Supervisor
Accepted by..
A. Dou....glas Carm.....chael.................
<
A. Douglas Carmichael
Chairman, Departmental Committee on Graduate Students
A Panel Method Analysis of Flow Past a Semi-Span
Hydrofoil
by
Christopher Matthew Waaler
Submitted to the Department of Ocean Engineering
on January 13, 1995, in partial fulfillment of the
requirements for the degree of
Master of Naval Architecture and Marine Engineering
Abstract
A low order, perturbation-potential based panel model is modified and applied to
the analysis of a semi-span hydrofoil attached to a half body enclosed in a water
tunnel. Effects of the body and tunnel walls are included by distributing panels on
their surfaces. An iterative wake alignment routine is employed which converges to
the steady state wake geometry. The angle of attack of the semi-span hydrofoil and
the length of the tunnel walls may be adjusted for comparative studies.
Results from measurements of the tip vortices from a physical model of the same
semi-span hydrofoil operating in a uniform flow field are presented. The techniques
of laser doppler velocimetry are used to map the vortex structures.
Numerical results of the panel method are shown to be convergent. Comparison is made between the first order approximation produced by the code and the
experimental results with favorable agreement.
Thesis Supervisor: Justin E. Kerwin
Title: Professor of Naval Architecture
Acknowledgments
The results of the experiment and the numerical code presented in this thesis are only
possible through the diligent efforts of a group of exceptionally talented people with
whom I have had the great pleasure to work. Professor Justin Kerwin's knowledgeable and friendly support throughout my graduate studies is very much appreciated.
Dr. Spyros Kinnas and Dr. Charles Mazel provided valuable insight to the various
difficulties encountered throughout my research. A very warm thank you must go
also to all the students of the Propeller Nuts group for their critical, yet encouraging
critique of my work throughout the various stages of its completion.
My greatest thanks, however, must go to Dr. David Keenan, who believed in me
throughout the numerous highs and lows. Our countless discussions heightened my
understanding of fluid dynamics and writing code. Without his constant vigilance
and guidance, the final product might never have been attained.
I extend my sincere appreciation to the staff of the Naval ROTC program at MIT.
Their flexibility and support made this goal a reality. Thanks also to Professors Alan
Brown and Jeff Reed for their constant guidance and support throughout.
Finally, I must express my deepest gratitude to my family, Jack, Virginia, Luke,
and Grant, for their continuous support throughout my education and to my fiance,
Elizabeth, for her unceasing encouragement and support during my stay at MIT.
Support for this project was provided by the Office of Naval Research, contract
number N00014-89-J-3194, under the supervision of Dr. Arthur M. Reed and with
enthusiastic assistance from Ms. Toby J. Ratcliffe.
Contents
1 Introduction
13
2 Mathematical Background
15
2.1
Green's
2 nd
Identity and perturbation flow . . . . . . . . . . . . . . .
2.1.1
Single Domain .
2.1.2
Multiple domains.
.
2.2 Application to a water tunnel ....
. . . . . . . . . . . . . . .
16
. . . . . . . . . . . . . . .
20
. . . . . . . . . . . . . . .
24
3 Numerical Formulation
3.1
16
26
Discretization of the Boundary Integral Equation
26
. . . . . . . . . . .
30
3.2
Foil geometry.
3.3
Body geometry.
31
3.4
Wall geometry .
32
3.5
Wake geometry.
32
.
.
.
.
.
38
4 Experimentation
4.1 Apparatus ................
4.1.1 The water tunnel ........
4.2
4.1.2
Laser Doppler Velocimeter .
4.1.3
Model description.
Experimental Procedure.
4.2.1
Test Conditions .........
4.2.2
Coordinate system.
4
.................
.................
.................
.................
.................
.................
.................
39
39
39
41
43
43
43
4.2.3
4.3
5
Velocity Measurements .....................
Experimental Results ..
.....................
Calculations
.
44
45
65
5.1
Convergence Studies
5.2
Comparison to experimental data ...................
...........................
65
69
6 Conclusions
89
7 Appendix
91
5
List of Figures
2-1 Basic geometrical relationships for Green's 2 nd Identity . . . . . ...
17
2-2 Geometrical relationships for Multi-domain case ............
21
3-1 Panel arrangementfor hydrofoil,Mr = 6, N, = 12 . . . . . . . . . . .
30
3-2 Panel arrangement on foil and body. NU = 8, Nb = 8, Nd = 8, Mt = 8,
ac=10 degrees
. . . . . . . . . . . . . . . .
.
31
. . . . . . ......
3-3 Interpolation scheme for aligning the wake panels ...........
35
3-4 Plan view of wake panelling showing contraction ratio of 30 degrees .
36
4-1
Schematic of tip-vortex experiment data flow ..............
39
4-2 Percent turbulence for LDV and hot-wire measurements in water tunnel 41
. . .
44
4-4 Streamwise velocity of inflow at X/Span =+1.06. (coarse grid). ... .
46
4-5 Vertical velocity of inflow at X/Span =+1.06. (coarse grid). .....
46
4-3 Velocity measurement grids for hydroplane in side configuration
4-6
Standard deviation of streamwise velocity of inflow at X/Span =+1.06.
47
(coarse grid) ..........................
4-7
Standard deviation of vertical velocity of inflow at X/Span =+1.06.
(coarse grid).
47
............................
4-8 Velocities from vertical traverse of test section at X/Span
(coarse grid).
=+1.06.
...............................
48
4-9 Velocities from spanwise traverse of test section at X/Span =+1.06.
(coarse grid).
................................
49
4-10 Standard deviations of velocities of vertical traverse of test section at
X/Span =+1.06 mm. (coarse grid) ....................
6
49
4-11 Streamwise velocity at X/Span
=-0.044.
Angle of attack is +5.36
degrees (side orientation/coarse grid) ...................
50
4-12 Streamwise velocity at X/Span =-0.34. Angle of attack is +5.36 degrees (side orientation/coarse grid)
...................
50
4-13 Streamwise velocity at X/Span =-0.66. Angle of attack is +5.36 degrees (side orientation/coarse grid) .
...................
51
4-14 Streamwise velocity at X/Span =-1.28. Angle of attack is +5.36 degrees (side orientation/coarse grid)
...................
51
4-15 Streamwise velocity at X/Span =-1.67. Angle of attack is +5.36 degrees (side orientation/coarse grid)
...................
52
4-16 Vertical velocity at X/Span =-0.044. Angle of attack is +5.36 degrees
(sideorientation/coarse
grid) .
..........
.. . ...
53
4-17 Vertical velocity at X/Span =-0.34. Angle of attack is +5.36 degrees
(side orientation/coarse grid). ......................
53
4-18 Vertical velocity at X/Span =-0.66. Angle of attack is +5.36 degrees
(side orientation/coarse grid). ......................
54
4-19 Vertical velocity at X/Span =-1.28. Angle of attack is +5.36 degrees
(side orientation/coarse grid). ......................
54
4-20 Vertical velocity at X/Span =-1.67. Angle of attack is +5.36 degrees
(side orientation/coarse grid)
.................
55
4-21 Spanwise velocity at X/Span =-0.34. Angle of attack is +5.36 degrees
(top orientation/coarse grid). ......................
55
4-22 Spanwise velocity at X/Span =-0.66. Angle of attack is +5.36 degrees
(top orientation/coarse
grid).
. . . . . . . . . . . . . . .
. . .
.
56
4-23 Spanwise velocity at X/Span =-1.28. Angle of attack is +5.36 degrees
(top orientation/coarse grid). ......................
56
4-24 Axial vorticity at X/Span =-0.044. Angle of attack is +5.36 degrees
(coarse
grid).
. . . . . . . . . . . . . . . .
.
. . . . . . ......
57
4-25 Axial vorticity at X/Span =-0.66. Angle of attack is +5.36 degrees
(coarse
grid).
. . . . . . . . . . . . . ..
7
. ............
57
4-26 Streamwise velocity at X/Span =-0.044. Angle of attack is -5.36 degrees (side orientation/coarse grid) ....................
59
4-27 Streamwise velocity at X/Span =-1.67. Angle of attack is -5.36 degrees
(side orientation/coarse grid)
...............
.
59
4-28 Vertical velocity at X/Span =-0.044. Angle of attack is -5.36 degrees
(side orientation/coarse grid). ......................
60
4-29 Vertical velocity at X/Span =-1.67. Angle of attack is -5.36 degrees
(side orientation/coarse
grid)
...............
4-30 Streamwise velocity at X/Span =-0.044.
... .
.
60
Angle of attack is +5.36
degrees (side orientation/fine grid). ...................
61
4-31 Streamwise velocity at X/Span =-0.34. Angle of attack is +5.36 degrees (side orientation/fine grid).
.....................
61
4-32 Streamwise velocity at X/Span =-0.66. Angle of attack is +5.36 degrees (side orientation/fine grid)..
...................
62
4-33 Vertical velocity at X/Span =-0.044. Angle of attack is +5.36 degrees
(side orientation/fine
grid) ..
. . . . . . . . . . . . . . . . . . . . . . .
62
4-34 Vertical velocity at X/Span =-0.34. Angle of attack is +5.36 degrees
(side orientation/fine grid) ......................
63
4-35 Vertical velocity at X/Span =-0.66. Angle of attack is +5.36 degrees
(side orientation/fine grid) ...................
63
4-36 Spanwise velocity at X/Span =-0.34. Angle of attack is +5.36 degrees
(top orientation/fine grid).)..........
.....
64
4-37 Spanwise velocity at X/Span =-0.66. Angle of attack is +5.36 degrees
(top orientation/fine grid)
...
......
...
64
5-1 Comparison of spanwise circulation distribution by increasing the number of spanwise panels
........................
..
.
66
5-2 Comparison of vertical position of wake sheet by increasing the number
of spanwise
panels
. . . . . . . . . . . .
8
......
67
5-3 Comparison of vertical position of wake sheet by increasing the number
of spanwise panels.
5-4
.........................
67
Comparison of vertical position of wake sheet by increasing the number
of spanwise panels .............................
5-5
Complete panelling scheme for 16 spanwise panels. (Downstream looking forward.)
5-6
68
...............................
69
Comparison of spanwise circulation distribution by increasing down-
stream tunnel length.
..........................
70
5-7 Comparison of spanwise wake shape by increasing downstream tunnel
length
5-8
70
Comparison of spanwise wake shape by increasing downstream tunnel
length
5-9
...................................
...................................
71
Comparison of spanwise wake shape by increasing downstream tunnel
length
...................................
71
5-10 Comparison of tip vortex position by increasing downstream tunnel
length
...................................
72
5-11 Comparison of root vortex position by increasing downstream tunnel
length
...................................
72
5-12 Calculated lift and drag coefficients.
..................
5-13 Computed streamwise velocity at X/Span=-0.044.
+5.36 degrees (coarse grid).
Angle of attack is
........................
5-14 Computed spanwise velocity at X/Span=-0.044.
+5.36 degrees (coarse grid).
73
75
Angle of attack is
.......................
5-15 Computed vertical velocity at X/Span=-0.044.
75
Angle of attack is
+5.36 degrees (coarse grid). .......................
5-16 Computed streamwise velocity at X/Span=-0.34.
+5.36 degrees (coarse grid). .....................
5-17 Computed spanwise velocity at X/Span=-0.34.
+5.36 degrees (coarse grid). ........................
9
76
Angle of attack is
...
76
Angle of attack is
77
5-18 Computed vertical velocity at X/Span=-0.34.
Angle of attack is +5.36
degrees (coarse grid) ............................
77
5-19 Computed streamwise velocity at X/Span=-0.66.
Angle of attack is
78
+5.36 degrees (coarse grid). .......................
5-20 Computed spanwise velocity at X/Span=-0.66.
Angle of attack is
79
+5.36 degrees (coarse grid). .......................
5-21 Computed vertical velocity at X/Span=-0.66.
Angle of attack is +5.36
79
degrees (coarse grid) ............................
5-22 Computed streamwise velocity at X/Span=-1.28.
Angle of attack is
80
+5.36 degrees (coarse grid). .......................
5-23 Computed spanwise velocity at X/Span=-1.28.
+5.36 degrees (coarse grid).
Angle of attack is
80
........................
5-24 Computed vertical velocity at X/Span=-1.28.
Angle of attack is +5.36
81
degrees (coarse grid) ............................
5-25 Computed streamwise velocity at X/Span=-1.67.
Angle of attack is
+5.36 degrees (coarse grid). .......................
81
5-26 Computed spanwise velocity at X/Span=-1.67.
Angle of attack is
82
+5.36 degrees (coarse grid). .......................
5-27 Computed vertical velocity at X/Span=-1.67.
Angle of attack is +5.36
degrees (coarse grid) ............................
82
5-28 Comparison of streamwise velocities across the surface of the hydrofoil
at 0 degree angle of attack ........................
83
5-29 Comparison of spanwise velocities across the surface of the hydrofoil
at 0 degree angle of attack ........................
83
5-30 Comparison of vertical velocities across the surface of the hydrofoil at
0 degree angle of attack
.........................
84
5-31 Comparison of streamwise velocities across the surface of the hydrofoil
at 0 degree angle of attack ........................
84
5-32 Experimental wake shape with panel code predicted wake shape superimposed. (X/Span=-0.044).
......................
10
85
5-33 Experimental wake shape with panel code predicted wake shape superimposed. (X/Span=-0.34)
........................
86
5-34 Experimental wake shape with panel code predicted wake shape super-
imposed. (X/Span=-0.66) ........................
86
5-35 Experimental wake shape with panel code predicted wake shape super-
imposed. (X/Span=-1.28) ........................
87
5-36 Experimental wake shape with panel code predicted wake shape super-
imposed. (X/Span=-1.67) ........................
87
7-1 Standard deviation of streamwise velocity at X/Span =-0.044. Angle
of attack is +5.36 degrees (side orientation/coarse grid).
.......
92
7-2 Standard deviation of streamwise velocity at X/Span =-0.34. Angle
of attack is +5.36 degrees (side orientation/coarse grid).
.......
92
7-3 Standard deviation of streamwise velocity at X/Span =-0.66. Angle
of attack is +5.36 degrees (side orientation/coarse grid).
.......
93
7-4 Standard deviation of streamwise velocity at X/Span =-1.28. Angle
of attack is +5.36 degrees (side orientation/coarse grid).
.......
93
7-5 Standard deviation of streamwise velocity at X/Span =-1.67. Angle
of attack is +5.36 degrees (side orientation/coarse grid).
.......
94
7-6 Standard deviation of spanwise velocity at X/Span =-0.34. Angle of
attack is +5.36 degrees (top orientation/coarse grid). .........
94
7-7 Standard deviation of spanwise velocity at X/Span =-0.66. Angle of
attack is +5.36 degrees (top orientation/coarse grid). .........
95
7-8 Standard deviation of spanwise velocity at X/Span =-1.28. Angle of
attack is +5.36 degrees (top orientation/coarse grid). .........
95
7-9 Standard deviation of vertical velocity at X/Span =-0.044. Angle of
attack is +5.36 degrees (side orientation/coarse grid) .........
96
7-10 Standard deviation of vertical velocity at X/Span =-0.34. Angle of
attack is +5.36 degrees (side orientation/coarse grid) .........
11
96
7-11 Standard deviation of vertical velocity at X/Span =-0.66. Angle of
attack is +5.36 degrees (side orientation/coarse grid) ..........
97
7-12 Standard deviation of vertical velocity at X/Span =-1.28. Angle of
attack is +5.36 degrees (side orientation/coarse grid) ..........
97
7-13 Standard deviation of vertical velocity at X/Span =-1.67. Angle of
attack is +5.36 degrees (side orientation/coarse grid) ..........
98
7-14 Standard deviation of streamwise velocity at X/Span =-0.66. Angle
of attack is +5.36 degrees (side orientation/fine grid) ..........
98
7-15 Standard deviation of vertical velocity at X/Span =-0.044. Angle of
attack is +5.36 degrees (side orientation/fine grid).
..........
99
7-16 Standard deviation of vertical velocity at X/Span =-0.34. Angle of
attack is +5.36 degrees (side orientation/fine grid).
..........
99
7-17 Standard deviation of vertical velocity at X/Span =-0.66. Angle of
attack is +5.36 degrees (side orientation/fine grid).
..........
100
7-18 Standard deviation of spanwise velocity at X/Span =-0.66. Angle of
attack is +5.36 degrees (top orientation/fine grid)
12
............
100
Chapter 1
Introduction
Fluid behavior around hydrofoils and airfoils has interested naval architects and aerodynamicists for some time. Understanding and modelling the flow around a simple
hydrofoil has established fundamental principles which can then be applied to marine
propellers and helicopter rotors. The examination of flow past the single hydrofoil
has led to improved design of aircraft wings and surface ship and submarine control
surfaces.
In particular, the flow around the tip of a foil has been studied by engineers of
virtually all of these major industries.
A concentrated tip vortex is created as a
byproduct of the lift and the three dimensionality of a lifting surface geometry. It is
the tip vortex which generates the very powerful, spiraling horizontal columns of air
in the wake of large jet aircraft, inducing a dangerous roll moment on any trailing
aircraft. This problem becomes especially acute when, in the interest of efficiency,
numerous aircraft closely follow one another both landing and taking off from major
metropolitan airports, generally only minutes apart.
The rotor blades on a helicopter are also foils which closely follow one another
on the same flight path. Tip vortices generated from the individual blades create a
turbulent and swirling inflow for the subsequent blade which leads to unwanted noise
and vibration. Control of the tip vortices of helicopter blades would extend the life
of the entire rotor assembly.
Marine propellers suffer from the same problem and propeller designs are often
13
based upon eliminating blade tip cavitation.
Cavitation develops from the same
conditions which create tip vortices and leads to both propeller vibration and noise.
Cavitation also causes pitting and corrosion of the blade surface. All of these factors
contribute to a reduction of propeller efficiency and desirability.
Control surfaces used to stabilize surface ships in roll and used to control the
submerged submarine also have specific design criteria. A submerged control surface
must be designed to minimize the debilitating effects of cavitation similar to a marine
propeller, but must also be efficient and functionally similar to the aircraft wing.
In the aeronautics industry, there have been two approaches developed to tackle
the problem of wing tip vortices. One method is simply to detect the vortices in airport environments. Numerous sensors such as hot-wire anemometers, acoustic sensors
and laser-radar (LIDAR) have been installed in aircraft to detect and warn pilots of
impending danger. The other approach has been to add various configurations to
the wing tips in an attempt to eliminate or at least weaken the strength of the tip
vortices.
In this work, the tip vortices from a submerged control surface are detected by
a laser doppler velocimeter (LDV) and modelled by a code developed to predict the
effects of different tip geometries on the trailing tip vortices.
14
Chapter 2
Mathematical Background
In the last few decades, as the full potential of computers in this application has
been realized, there is increasing reliance on finite element analysis of complex flow
behavior.
For large Reynold's numbers (on the order of one million), the theory
of potential flow establishes a basic description of flow patterns and finite element
analyses, specifically panel methods, are extremely well suited to model the potential
flow behavior. There are many variations of panel methods which have been developed
and some are more successful than others in each of the various aerodynamic and
hydrodynamic applications. Kerwin et al. [10], Lee [13] and Hsin [6] have developed
robust and versatile perturbation potential panel methods which have been used in
the analysis of marine propellers. They have been shown to accurately predict fluid
behavior for highly skewed propellers, ducted propellers, and propellers encountering
unsteady inflow. In this work, Hsin's code has been adapted to analyz e the flow
around a semi-span hydrofoil attached to a half body. The associated theory, based
on Lamb [12], is discussed below.
15
2.1
2.1.1
Green's
2 d
Identity and perturbation flow
Single Domain
The foundation of panel methods is Green's
2 nd
Identity. However, to understand
the usefulness of the theorem, one must begin with the fundamental principles of
potential flow theory. Using this theory, flow is defined to be incompressible, inviscid,
and irrotational. This allows the continuity equation for conservation of mass to be
represented by the familiar differential equation of Laplace:
CV2.[ =
(2.1)
where
V = V,
(2.2)
V being the velocity vector of the fluid and 14is the total velocity potential of the
fluid.
Green's
2 nd
Identity states that for two arbitrary harmonic functions 0 and 9
defined inside a volume V which satisfy the Laplace equation (2.1) then
(0_5 - g )dS =
(2.3)
X 199
ds= a Jpds
(2.4)
or
where n is the unit vector normal to the surface S at point q and pointing into
the fluid domain V (See Figure 2-1). S is the surface which encloses the volume V.
Green's
2 nd
Identity can easily be proved by invoking the divergence theorem.
Let q represent the velocity potential of the fluid throughout the volume V and
make g = 1/r where r is the distance between a control point q in the fluid domain,
either on the surface S or within its boundaries, and p a fixed point located anywhere.
Examine the situation which arises if the point p is located in the fluid domain.
16
Idpoint P
tsideS
q
Controlpoint
on S
Field pointp
on S
Figure 2-1: Basic geometrical relationships for Green's
2 d
Identity
It is possible for the variable point q and the fixed point p to be collocated causing
r to become zero and g to be unbounded. To avoid this, Gauss' law is employed.
Consider a small sphere S, surrounding the point, with p at its center. Let the total
surface area be S, + S. Equation (2.4) may be rewritten
J(q)na( )dS + J(q)
a ()dS=
J
1 a() dSE+
/
o (q)dS. (2.5)
Gauss' law states
J
u. ndS
= 47rm(p)
(2.6)
or to be consistent
(q)dS, =
4 irm(p)
(2.7)
where m(p) is the strength of a source placed at the point p and u is the velocity
vector due to the potential at point p. Rather than considering the potential on the
surface S, due to the potential at p, consider at the potential at point p due to a
distributed potential on the surface S,.
Examination of Figure 2-1 shows that the normal vector to this surface S, is
17
identically opposite in direction to the radius vector thus
0()(q)
_
O(q)
An
(2.8)
ar
Substituting the above into Equation (2.7),
ar )dS, = -4 rxm(p).
|
(2.9)
Presuming that this sphere S, is hollow with a constant radius R and all the potential
d)(q)located on its surface, the &/Or term may be moved out of the integral and
a
JJ
(q)dS = - 4 7rm(p).
(2.10)
The potential in three dimensions of a source is given by
0(R) = 47rR = p.
(2.11)
Substituting this into Equation (2.10) for m gives
rf f
(q)dSE = -(4)
2Rop
(2.12)
and taking the integral of both sides from 0 to R results in
f f/)(q)dS = -(4r) 2 R 2 p.
(2.13)
Substituting this into the first term of Equation (2.5) and noting that
0
11
n((r
r2=
the term becomes
J (q) a ()dS
=4
p
which is independent of the radius of the small sphere S,.
18
(2.14)
Evaluating the first term on the right hand side of Equation (2.5) for a sphere of
a specified radius R,
1
/
When R --
0
(q)dS = 41rRb(q).
(2.15)
0, the term will vanish, leaving Equation (2.5) as
(/) 1= 1a(q) dS +
I
J(q) an )dS
If the field point p is located external to the surface S, the function
(2.16)
g is now finite
everywhere and no modification of Equation 2.3 is required. Thus
o0=
=
41f
1r
a(q)dS
+
n
f f
(q)-9 (r)dS
r(2.17)
(2.17)
This leaves the final possibility, that the fixed point p is located on the surface
S and again p and q could be collocated. In this case, S is indented by a small
hemisphere. Using the same logic as above with the small sphere, Equation 2.4 is
modified to give
1
4r/ ra=n dS +4rz fJn0d(q)&an()dS.
4rJran
r
2
1
2P-
c9
1
(2.18)
The 1/2 results from the surface area of a hemisphere being half that of a complete
sphere.
The three equations, (2.16), (2.17), and (2.18), allow the value of the potential
q to be calculated at any point p in or out of the fluid domain. It is determined by
the values of the potential (q) and ao(q)/an at the boundary of surface S. Closer
inspection reveals that the first term appears very similar to Equation (2.11). Thus
the first term represents the summation of source singularities on the surface S with
a strength of -90(q)/0an per unit area. The second term is similar to
407r(r2
)
(2.19)
which is the equation for a dipole in three dimensions with /t equal to the strength of
19
the dipole. In this case /, is equated to X(q), the dipole strength per unit area.
Equations (2.16), (2.17), and (2.18) may be combined into a general form
T4irp=J
p |-
a( dS+f$ (q)kn(!)dS
(2.20)
where when
T=1
p lies within the fluid domain V, but not on S; when
T=0
p lies outside of the fluid domain V, but not on S; and when
T=2
p lies on either face of the surface S.
The significance of Equation (2.20) is that if one of the two surface functions,
X
or
00/an is known at all points q on S, then using Equation (2.20) with T=1/2 for both
p and q located on the surface S, the complementary surface function may be obtained
because essentially there is only one unknown; qp and 0(q) are the same. Once both
surface functions are defined, the potential at any point p in the fluid domain may
be deduced. In other words, the Laplace differential equation has been restructured
through the use of Green's
2.1.2
2 nd
Identity to give a very useful integral equation.
Multiple domains
Equation (2.20) must be further refined to apply it to the problem of a semi-span
hydrofoil attached to a half body within a water tunnel because there are multiple
domains.
SB and S.
Consider the situation depicted in Figure 2-2. There are two surfaces,
The two surfaces may be thought of as parts of a larger surface S
which bounds the fluid domain labelled V. The single valued potential distributed
in the domain V, external to surface SB is defined as
. The single valued potential
distributed in the domain Vi, enclosed by SB is defined as 99. The potential at point p,
which is arbitrarily located in the fluid domain V, is a combination of the contributions
from all of the distributed single valued potential functions. The influence of the
20
d+ )dS,6
(2.2
Figure 2-2: Geometrical relationships for Multi-domain case
potential, 0, comes directly from Equation (2.20) setting T=1
4, p
--
rriTBo
S
a
(
&OanBo
r
Similarly, setting T=0, the contribution due to the inner surface p is
=Bfi-dS 'nBi
r rnBi
(-)dSo.
r
(2.22)
Because the potential is a scalar, the Equations (2.21) and (2.22) may be added,
carefully noting that the derivatives of the normal vectors for each face are in opposite
directions, O/lnBo = -/OnBi.
4srp= d
a
Therefore the sum is
ap )dS+J
n
(-)d pota
)dSB.
(2.23)
As the distance between a point q on S, and a point p near 8 B becomes infinite,
the function 5 goes to zero, leaving only the second integral from Equation (2.16).
If the surface of the infinite boundary is represented as 4rr 2 , then Equation (2.16)
21
becomes simply
op = o00.
(2.24)
Since the surface Soo is such a great distance away from SL3, CoO
is considered to be
the potential of the fluid, undisturbed by the presence of the body. In the present
formulation only the perturbation potential is determined; 0OOis set to zero.
Equation (2.23) may be rewritten
47rp
= JJK
0go(
a n)d SB + / K 2(b- )dSB
(2.25)
where
K --1r
and
(2.26)
0 1(r)
K2 = n
(2.27)
anBo r
are scalar "kernel" functions which are only dependent on the geometric relationship
between the surface potential point q and the field point p.
Only one of the two distributions of potential cpor 0 will have physical significance.
In this analysis cp is considered fictitious and is set to zero; Equation (2.25) becomes
47rp=
J 1(a
)dSB+JK
2
dSB
(2.28)
for a point not on the surface S.
Presuming that this general surface S is a physical body which might develop
a trailing wake, Equation (2.28) must be modified to take the wake surface into
account. The wake may be considered an infinitesimally thin closed surface. The
wake characteristically defines a discontinuity in potential from one face to the other
and the direction of the velocity vector will endure an instantaneous change. The
component of velocity normal to the wake surface is considered continuous, but equal
to zero
(a)upper -(
)lower
= 0
22
(2.29)
If
a
=
upper -
(2.30)
lower,
then applying Equation (2.20) to the wake with T=0 results in
f K /.qdSwake.
0=
(2.31)
2
Including the contributions from the wake, the potential at any point p in the
fluid domain V may be determined from
47rp = /J
Kl(
K2qdS +
o )dS, +//
KAqdSwake
2
(2.32)
K2 cdSB+ JJ K2 A^bdSwake
(2.33)
If the point p is on the surface of SB, T=1/2 and
27rp =
JJKi(
a
)dSB +
It is Equation (2.33) which is the basis of the numerical formulation of the solution.
However, the solution of Equation (2.33) is not unique without the implementation
of some boundary conditions on the surface and the trailing edge.
The kinematic or external Neumann boundary condition dictates the value of the
velocity at the body surface. It is this relation which uniquely relates the perturbation
potential to the total potential. The kinematic boundary condition for a solid surface
may be written
On
09
=0
(2.34)
where ·> is the total potential of the fluid and is equal to bp found from Equation
(2.33) added to qOO
On
-=--U
n.
(2.35)
In addition, the Kutta condition, which requires flow around the trailing edge to
be finite, must be imposed. Kerwin et al. [10] present an explicit non-linear pressure
Kutta condition. For the numerical formulation of the Kutta condition, the potential
jump in the wake is set equal to the total potential values at the trailing edge of the
23
hydrofoil giving:
Awake
= o()t.e.upper
- (1)t.e.lower = ctt.e.upper -
+ U
+t.e.lower
rt.e.
(2.36)
where rt.e. is the unit vector which bisects the hydrofoil angle of attack.
Thus Equation (2.33) becomes a Fredholm integral equation of the second kind
[13]. This leaves 0 as the sole remaining unknown. Converting the surface integrals
into summations, discussed in the next chapter, Equation (2.33) may be solved for q
by solving a linear system of equations.
2.2
Application to a water tunnel
The modelling of a body within an air or water tunnel is more complicated than
simply modelling a body travelling in an infinite domain. Specifically, the decision
arises whether to apply the boundary integral equations to the confines of the tunnel
or to apply them to the tunnel and the infinite boundary. In the latter case, the
entire tunnel assembly acts like a body moving through an infinite domain.
If the tunnel walls enclose the fluid domain, then the kinematic boundary condition
must be applied to the four longitudinal surfaces and the flow through the upstream
"endcap" is forced to equal the undisturbed inflow, U.
The difficult task lies in
defining the correct boundary condition for the downstream endcap, specifically where
it intersects the wake sheet. For a tunnel of finite length, panelling the downstream
cap adds another layer of difficulty as the panels must reflect the shape of the wake
and the associated tip vortex roll-up.
An alternative approach is to model the entire tunnel assembly as if it were placed
in a uniform stream. The infinite boundary, S,, remains as it was for the general case
as described in section 2.1.2. Near this infinite boundary the perturbation potential
becomes zero, as before, leaving the total fluid potential equal to that of the uniform
stream
atop =U
3n
24
where the inflow velocity, U, is given an arbitrary magnitude, but the direction of U
has been aligned with the longitudinal axis of the water tunnel.
Green's
2 nd
Identity must be applied to all surfaces, including both sides of the
tunnel walls. Similar to the wake sheet, the walls may be considered to be a very
thin closed surface, allowing the entire tunnel assembly to be a multi-component
closed surface, S3. Within the region of interest, inside the walls of the tunnel, the
contributions of the half body, the hydrofoil, and the inside surface of the tunnel
walls are summed as before using the kinematic boundary condition to specify the
source distribution. The same boundary condition is applied to the outer surface of
the tunnel walls, but the inflow runs parallel to these surfaces and the perturbation
flow is zero.
Specifying the inflow velocity in this manner satisfies not only the mathematical
dilemma of finding a non-singular solution, but it also satisfies a physical quandary
of preventing the flow velocity at the trailing edge of the tunnel walls from becoming
unbounded through the use of continuity. The fixed inflow vector ensures the criterion
for the Kutta condition is met.
The wake sheet also extends infinitely downstream, in reality rolling up into
a concentrated tip vortex.
The wake's influence on the hydrofoil beyond several
chordlengths becomes minimal and is ignored in this treatment of the near field effects.
Equation (2.33) may therefore be used to solve the potential flow problem of flow
past the semi-span hydrofoil and half body configuration within the confines of a
water tunnel.
25
Chapter 3
Numerical Formulation
There are numerous panel-based formulations which have been developed over the
years to address complex flow modelling.
Although they differ in the placement
and the category of the distributed singularities, they all rely on a discretization
of the various integral equations. Some methods are more adept at particular flow
patterns than others. In the case of the semi-span problem presented here, a low-order
perturbation potential method is utilized.
3.1
Discretization of the Boundary Integral Equa-
tion
Both Lee [13] and Hsin [6] elaborate on some of the advantages of using the perturbation potential method. Some of these advantages include:
* The accuracy of potential based methods have been found to be more accurate
than velocity based methods.
* Potential based methods are less affected by irregular panelling.
* Scalars such as potentials require one third the allocated memory of vectors.
* n. - V
is easily specified.
26
Hunt [7] gives some background on the process of discretization of integral equations such as Fredholm's Integral Equation of the
2
d Kind (Equation 2.33). Accord-
ing to Hunt, there are essentially three basic discretization errors which occur when
forming the numerical solution and all contribute in some way to the deviation from
the analytical solution:
* Approximation of the physical surfaces
* Discretization of the boundary conditions
* Discretization of the singularity distributions
The most intuitive source of error in the finite element modelling is derived from
the coarse approximation of a body's surface. Although it is possible to develop
higher order geometric shapes to cope with highly skewed and curved surfaces, planar
rectangular panels are used in this work on the semi-span hydrofoil and its surrounding
geometry.
There are two major sources of error due to the panelled approximation of the solid
surface, both of them relating to the imposed kinematic boundary condition. First,
the calculation of the unit vector normal to a panel surface, particularly in regions of
high curvature, does not reflect the direction normal to the true surface at all points.
Secondly, the kinematic boundary condition should be satisfied everywhere on a solid
surface. However, in the panel approximation, the boundary condition may only be
feasibly satisfied, in the interest of computer time and memory, at one point on each
panel, commonly the centroid of each panel. As the number of panels is increased,
these errors are diminished.
The selection of this singular point to satisfy the boundary condition leads to
two errors in the finite element solution. Because the point may not actually lie on
the surface being modelled due to the approximation of the surface with panels, the
true boundary condition for the exact shape in the problem may not be realized.
Additionally, because the boundary condition is only satisfied at one point on the
panel, there will develop a small amount of leakage past the panel into the "solid"
27
surface. These errors also may be minimized by increasing the number of panels and
carefully selecting the distribution of surface functions.
The surface functions attributed to individual panels may be represented by functions of any order, from piecewise constant to bicubic splines. In the present analysis,
the unknown surface function of Equation (2.33) is piecewise constant over the panel.
The function is made up of two parts: a function of unit strength, g(, '7), where c
and
are the local two dimensional coordinates on the panel, and a weighing factor,
w, which is of unknown magnitude such that
E wj s,9g
j(
)K(p,
q)dSj = b(p)
(3.1)
with Sj denoting the surface area of the individual panel and N representing the
total number of panels. The right hand side b(p) forms the boundary condition at a
particular point p on the surface S.
In the semi-span case, the unknown surface function is
(q). For each panel,
combining the kernel function K with the unit strength function g(C, V) gives a new
variable, Dij on the body or Wij on the wake. Intuitively, Dij and Wij represent the
induced potential at a field point Pi due to a panel with a unit strength potential
function distributed uniformly across its surface, Sj. Specifically for the perturbation
potential method used here, both Dij and Wij represent unit dipole distributions.
The source strengths are derived from the kinematic boundary condition.
Let N indicate the number of panels on the surface Sr and Nw panels on the
wake surface S,,k,.
Denote the weighting factor w as
X
for the body and Ab for the
wake. Substituting these new variables into Equation (3.1), the boundary integral
equation (2.33) may be discretized as
N
JK2 dS + JJK
2 AdSwake
Nw
EN Dij + E WijAO
j=1
(3.2)
j=1
where the 2rbp term of Equation (2.33) is incorporated into the first summation on
the right hand side.
28
The remaining term from Equation (2.33) becomes the boundary condition b(p)
b(p)= J f K( a
)dS. = Si3 (
j=
n
)
(3.3)
i = i,2, 3,...,N
This is the link between the total potential of the system and the perturbation potential. Recall the kinematic boundary condition (Equation 2.35) specifies 90/&nBo.
Sij is the induced potential at pi due to a unit source strength distributed on panel
S3.
Equations (3.2) and (3.3) may be represented by the following matrix equation
[A]{0} = [B] -
f]{IA0}
(3.4)
where the matrices [A], [B] and [W] are known as "influence coefficient" matrices.
The method used to calculate the influence coefficients was derived by Newman [15]
who determines the exact forms of
j
K2 dS2
for field points p near the panel and develops multipole expansions used when the field
points are at distances much greater than the panel dimensions. A similar approach
is used to calculate
fs KdS
for the source terms.
·.
Because these influence coefficient matrices are quite large
(NxN), an iterative block solver routine developed by Clark [3] is used to solve the
system of equations. With the influence coefficients now defined, the geometry of the
problem must be developed.
29
Figure 3-1: Panel arrangement for hydrofoil, Mr = 6, NC = 12
3.2
Foil geometry
The cartesian coordinate system for the computer model originates at the midpoint
of the root chord of the foil. The x-axis is aligned with the inflow velocity and the
y-axis is directed outboard. Since the starboard half of the configuration is modelled,
the resulting z-axis, by the right hand convention, points upward. The span of the
hydrofoil is chosen to be the fundamental length for the entire model.
The semi-span foil is generated by dividing the unit spanlength into Mr spanwise
panels. A half-cosine spacing scheme is used to concentrate panels at the tip for
better definition of the flow in that region.
The planform of the foil, unlike the physical model, is symmetric about the midchord line with the leading and trailing edge slopes equal to the ratio between the tip
and root chords. Also unlike the physical model which has a NACA 0012 shape at
the tip, the computer model uses a NACA 0018 chordwise profile throughout.
The
chordwise panels have full cosine spacing with the total number of panels, N, equal
to the sum of panels on the upper and lower surface of the foil (See Figure 3 - 1).
The foil's spanwise thickness distribution is linear and based on the ratio of the tip
30
z
I
rl
Figure 3-2: Panel arrangement on foil and body. N, = 8, Nb = 8, Nd = 8, Mt = 8,
a=10 degrees
and root chords. From these parameters, the coordinates of each panel node can be
determined uniquely.
The tip of the foil is a revolved section of a radius equal to the thickness of the
foil at its outermost point. Figure 3-1 shows the tip panelling scheme.
3.3
Body geometry
The body geometry is dependent upon the foil and wake geometries. Measurements
from the experimental model position the hydrofoil on the body and determine the
body dimensions relative to the span of the hydrofoil. The thickness distribution is
that of a NACA four digit airfoil [1] with the trailing edge thickness value set to zero
to create a symmetric surface about its centerline. The top and bottom of the body
are not modelled as rounded sections like the physical model, rather the body spans
the height of the tunnel.
Body panelling is shown in Figure 3-2. It is critical that the panel boundaries
of the foil exactly match those of the body, therefore the full cosine chordwise panel
spacing of the foil is reflected in the body's mid-chord section. The panel spacing
31
forward of the foil is half-cosine spaced to better represent the increased curvature of
the leading edge of the body. The panelling aft of the foil is driven by the half-cosine
streamwise panel spacing of the wake. In addition, the panels aft of the foil adjust
vertically as the wake sheet is iterated to align with the local flow velocity.
The body's spanwise panelling follows the contours of the foil and wake near
midspan, but gradually straighten to become parallel to the floor and ceiling of the
tunnel (See Figure 3-2). As the foil is positioned to various angles of attack, the panels
on the body must reflect the change. This code presumes that the foil is rotated on
an axis located at the 1/4 chord of the foil. As the foil rotates, a gap would appear at
the base of the foil due to the curvature of the body's thickness profile. The fairing
associated with the experimental model is not simulated in the computer code. It is
acknowledged that the gap that develops and the discontinuity which then occurs at
the root of the trailing edge of the foil is not given proper consideration in this first
approximation panel code. In this model, the root of the semi-span foil is adjusted
such that it always matches the four-digit contour of the body. Further modelling of
the gap and the resulting root vortex which develops is reserved for
future work.
3.4
Wall geometry
The panelling on the walls is derived from the foil, wake, and body panelling schemes.
The tunnel height is dictated by the span of the body and the tunnel test section
is square, but the remaining length dimensions may be entered by the user. This
allows flexibility to vary the length of the downstream section to observe the model's
limitations due to the finite tunnel walls.
3.5
Wake geometry
Unlike the preceding geometries, the wake sheet panels only consists of dipoles. Since
this work addresses the steady state problem, the wake sheet is modelled as semi-
32
infinite. To accomplish this, the panels of interest extend the length of the downstream
tunnel wall at which point an additional set of panels, set to be ten span lengths
long, are added to the wake sheet streamwise to simulate the effects of the starting
vortex. The panelling nearest the foil is half-cosine spaced to ensure that a panel size
mismatch does not occur at the foil's trailing edge where the foil panels are small.
The panels are distributed spanwise identically to those on the foil.
The position of the simulated wake sheet affects the wake's influence on the foil's
potential. If its position is altered, the circulation distribution on the foil will reflect
the change. The dipole strength in the wake is directly dependent upon the circulation
of the foil. Revising the wake's dipole strength alters its geometry as it must be aligned
with the local velocity vector to enforce no flow through the wake sheet stipulation.
Aligning the wake thus becomes an interative process.
As described in Chapter 2, the Kutta condition is applied numerically as
AO = oupper- ler
+ U · rte.
The strength of the wake sheet dipoles are adjusted iteratively until the pressures
along the upper trailing edge panels match those along the lower trailing edge. The
above equation becomes the initial guess at the potential differencebetween the trailing edge panels. Lee [13] then adds another term, K(ACp)te. K is a coefficient which
is determined by the iteration described below and (Cp)te
is the non-dimensional
pressure coefficient at the trailing edge panels. Adding this term results in the following equation for a single streamwise strip of wake panels.
(Ab)k = (upper _ lowerk + (U
)
Kn = K( n - l )
n
K(\Cp)-1)
(3.5)
where
(Cp)
(
(36)
nc-)_- (ACp)n
is iterated using the Newton-Raphson method until an equal pressure condition results
at the foil's trailing edge.
33
Kerwin and Greeley [5] developed a fast, yet simple method of wake alignment
which is employed in this code. Initially, the wake sheet is positioned such that it
bisects the angle of attack of the hydrofoil and extends downstream without curvature.
The velocities tend to be more singular when calculated directly at the panel edges,
so the local velocities in the streamwise and vertical directions are calculated for the
centroids of each interior wake panel. The influence of the foil, half body, and walls
is obtained using the technique developed by Newman [15] (See Figure 3-3).
The influence of the wake sheet upon itself is more difficult to compute. Although
Newman's technique might be employed, there are some hazards to applying it to the
singular region about the tip vortex. Newman's method can calculate the velocities
induced by a sheet of dipoles at the edges of the panels because an artificial, analytically obtained value is substituted in the subroutine for this otherwise unbounded
point. However, unless the field point is positioned precisely on the panel's edge, an
erroneously high magnitude of the velocity normal to the plane of the panel will result. Determining the exact position of the panel boundaries when wake contraction
and wake alignment schemes are involved is particularly difficult. A constant dipole
distribution over a panel surface is equivalent, mathematically, to placing discrete
vortices on the four sides of the panel [13]. Since the wake sheet contains no source
singularities, the difficulty of calculating the velocity at a panel's edge is alle viated
by employing a vortex lattice velocity computation.
The lattice of the wake may be separated into three types of vortices. The first
vortex filament is located at the trailing edge of the foil and is oriented in the same
direction as the bound circulation on the foil, opposite of the starting vortex. Its
strength is equal to and counters the difference in potential between the upper and
lower trailing edge panels. It decreases in magnitude from the root to the tip of the
foil, consistent with the spanwise circulation distribution.
The second portion of the lattice is formed by the discrete trailing vortices. Each
trailing vortex is located at the streamwise edges of two adjacent strips of wake panels. Its strength is equated to the difference in vorticity of these two panel strips,
considered constant in the streamwise direction of potential flow. In the limit of in34
0
Tirectl
[
Interpolated velocities
A
Extrapolated velocities
..
_
_,U
_
..
w,·CV
_._..._2,V
Figure 3-3: Interpolation scheme for aligning the wake panels
finite spanwise panelling, the strength of the trailing vortices becomes the derivative
of the spanwise circulation distribution. By this methodology, the vortex associated
with the inboard edge of the wake sheet is cancelled by the equal and opposite vortex on the other side of the half body centerline. The derivative of the circulation
distribution should be zero at this midspan point.
The third vortex or tip vortex is a special case of the trailing vortices. Since it
does not lie between two panels, its strength is given by the magnitude of vorticity of
the panel located on the outboard edge of the wake only. Physically, it has the greatest strength, however, numerically, sometimes it is matched in size by the adjacent,
inboard vortex, particularly in the limit of spanwise panels.
The magnitude of these velocities are interpolated bilinearly to determine the interior panels' nodal velocities (See Figure 3-3). The velocities associated with the
inboard edge of the wake sheet, in contact with the tunnel wall, are linearly extrapolated from the adjacent two interior nodal points. The velocities induced on the tip
vortex line are calculated directly, as discussed above, and merged with the previously
calculated velocities of the interior panels.
Once the velocities at each of the wake panel nodes have been established, the
ratio of the local downwash velocity w to the local streamwise velocity u plus the
35
Transition Wake Region
-
Q
rw
Figure 3-4: Plan view of wake panelling showing contraction ratio of 30 degrees
undisturbed inflow U, is compared to the existing pitch angle of that particular
panel. If the two values differ, the pitch angle of the panel is adjusted to match
that of the local flow velocity. After all the wake panels have been adjusted in this
manner, the entire process of determining the influence coefficients of the geometry
is repeated and again the local velocities at the wake are compared to the pitch angle
of the individual panels. This iteration continues until the wake panels are aligned,
within tolerance, with the local velocity vector.
Experimentation has shown that the tip vortex tends to sweep inboard slightly
as it travels downstream (See Figure 3-4). Kerwin and Greeley [5] also developed
an empirical estimate at the tip vortex contraction angle.
They define the area
downstream of the trailing edge of the foil where the wake is contracting as the
transition wake region. Further downstream, where the spanwise width of the wake
is constant, is defined as the ultimate wake region. Kerwin and Greeley developed
an empirical equation which relates the ultimate wake spanwise position, r and the
streamwise length of the transition region, xrw
rw
3(rt
-
r)
tanSJ
(3.7)
where rt is the spanwise origin of the tip vortex on the hydrofoil and c5is the contraction angle which is estimated or obtained from experimental data. In this case,
36
the length of the transition region is estimated to be the distance to the furthest
point measured downstream by the LDV and r, is also be obtained directly from the
experimental results. The resulting contraction angle 6, for a 5.36 degree angle of
attack is approximately 12.4 degrees.
With these parameters defined, the shape of the tip vortex contraction may be
estimated as a cubic polynomial, defining q as the fractional distance between the
foil trailing edge and xrw and r(q) as the spanwise position of the tip vortex in the
transition region
r(q) = rt + (rt - r.)(-3q + 3q2 - q3 )
(3.8)
and
r = ru
q > 1.
The discrete trailing vortices associated with the interior wake panel edges are
uniformly distributed between the tip vortex and the half body profile as shown in
Figure 3-4. Note that the half-cosine spacing of the semi-span hydrofoil tends to
concentrate the wake panels near the tip, aiding in the simulation of the tip vortex
roll-up process.
With the discretization of the basic integral equation of Chapter 2 and the geometry developed here, a potential flow estimation of the fluid behavior around a
semi-span hydrofoil attached to a half body may be computed. Upon comparison
with the experimental results, the model may undergo further revision to more accu-
rately predict the flowpatterns.
An important limitation of this code is that the wake panels are constrained to
movement only in the vertical direction. They are not permitted to move spanwise,
thereby preventing the panels from rolling up. This was done purely for simplicity.
The detailed geometry of the tip vortex core is not modelled in this code, but future
subroutines which can predict the vortex roll up may easily be appended.
37
Chapter 4
Experimentation
As described in Chapter 1, there is much interest in the generation of tip vortices
and there have been numerous experiments conducted to study tip vortex behavior.
Some recent studies of tip vortices associated with a single hydrofoil include Takahashi
and McAlister [16], who used laser velocimetry to measure the airflow past a large,
vertically oriented, NACA 0015 airfoil [1]. Lezius [14] implemented an alternative
approach by using a water towing tank and 16-mm cameras to record the tip vortex
behavior of a NACA 0015 foil at several angles of attack. Still other experimenters,
such as Corsiglia [2], have used hot-wire anemometry to measure the flow field behind
a NACA foil.
The present study used a semi-span NACA 0018/0012 foil affixed to a half body
four digit NACA foil. A 3 Watt TSI argon-ion laser doppler velocimeter (LDV)
measured the fluid behavior within the variable pressure water tunnel at the Marine
Hydrodynamics Laboratory of the Massachusetts Institute of Technology (MIT). The
objective of the experiment was to map the tip vortex behavior behind the semi-span
hydrofoil.
38
SF
DATA POINT
v
Figure 4-1: Schematic of tip-vortex experiment data flow
4.1
Apparatus
4.1.1
The water tunnel
The MIT water tunnel is a fresh water, closed-loop system which uses a 75 hp motordriven impeller to produce steady flow of up to 30 feet per second through a 53-inch
long, square test section. A honeycomb mesh is placed inside the 5:1 contraction
section just upstream of the test section to regulate and condition the flow.
Each side of the test section is 20 inches in width and has four two-inch thick
plexiglass windows which may be removed for installation of various models and
equipment. The laser beams from the LDV penetrate one of these highly polished
windows to extract velocity data (See Figure 4-1).
4.1.2
Laser Doppler Velocimeter
The LDV uses a pair of blue and a pair of green beams which are oriented in the
horizontal and vertical planes with respect to the tunnel test section. The four separate beams are focused to a common point where the fluid velocity is measured. The
39
intersection of the beams creates a pattern of alternating intensities of laser light.
Particles naturally present in the tunnel water augmented by 1 -m diameter silicon
carbide particles entrained in the fluid scatters the light at a frequency which is proportional to its velocity in the direction normal to the fringes, in this case, the vertical
and streamwise directions. The scattered light is collected by photo-multipliers associated with each pair of laser beams. The MIT system is a "back-scatter" system,
meaning that the light is reflected back towards the source of the beams.
Digital Intelligent Flow Analyzers (IFA's) receive data from the photomultipliers
and process them so that they can be read by the 20 MHz 386 PC. The IFA's analyze
the flow and discard data which are invalid. Errors may arise due to spurious signals
in the processing system. The PC's data-reduction software keeps the laser focused
at a single point until the prescribed number of samples has been obtained, generally
on the order of 250-300. The ensemble is averaged and the standard deviations are
calculated. The output consists of average velocities and a relative measurement of
turbulence intensity for the given field point. A 1992 study of the MIT water tunnel
compared LDV turbulence data to that measured by hot wire anemometry for various
Reynold's numbers. The results are shown in Figure 4-2. The hot film probe data
were taken with a TSI conical hot film probe, model 1231W. Vertical and horizontal
lines of data were taken across the test section intersecting at the center, approximate
ly five cm downstream of where the trailing edge of the hydrofoil would be located.
Turbulence intensity in Figure 4-2 is defined as
TI = U [N-1
(ui - )2]
(4.1)
where U0 is the freestream velocity, U is the instantaneous measured streamwise
velocity, ii is the mean streamwise velocity, and N is the number of sampled points.
The current experiment operated at a Reynold's number of approximately 1.5x106 .
The LDV is mounted on a computer-controlled three-axis table. The table may be
positioned within five microns, however, the control system is only accurate to within
0.01 mm. The data collection system of the LDV is linked to the same computer
40
.1
'U
LDV
i
°
U
10
0
n
0
Hot Film Probe
C~
Hot Film Probe/LDV
,
,
I
,
,
,
1.0
,
I
,
,
1.5
,
,
I
2.0
Reynolds Number (in millions)
Figure 4-2: Percent turbulence for LDV and hot-wire measurements in water tunnel
as the table thus allowing the flow measurements to be automated.
The range of
the laser is limited due to laser-beam blockage, either by the edge of the plexiglass
windows or by of the semi-span foil itself.
To non-dimensionalize the LDV output, a differential pressure (DP) transducer is
located in the contraction section of the water tunnel. The DP cell tracks the mean
velocity through the test section and allows variation in the flow rate to be eliminated
from the LDV results.
Both the water tunnel and the LDV are described by Kerwin [9] and Coney [4] in
greater detail.
4.1.3
Model description
The half body and semi-span used in this experiment were built by Coney [4] specifically for use inside the MIT water tunnel. The plexiglass model was mounted to an
aluminum splitter plate which located the assembly 2 inches further into the center
of the test section and allowed the developed boundary layer along the tunnel wall
41
to pass underneath the half body. The splitter plate was also designed such that it
created an image plane effectively generating a symmetric configuration.
The half body profile matches a 4-digit NACA airfoil. It has a chord length of 16
inches and a span of 19.5 inches for ease of installation. The span of the half body did
not extend the full height of the tunnel test section as modelled by the panel code.
The maximum thickness to chord ratio is 0.2215. It has rounded corners of 1.75 inch
radii. The semi-span is mounted to the half body by a 1.5 inch diameter steel shaft
which is located at approximately the 28 percent chord point of the half body. This
shaft allowed the semi-span to be adjusted to various angles of attack.
The hydrofoil has a span length of 8.06 inches. The root profile is that of a NACA
0018 foil with a 7.5 inch chord. The nominal tip is described by a NACA 0012 foil
with a 4.905 inch chord. The actual tip is a half body of revolution whose diameter at
each chordwise position matches the NACA 0012 profile. The steel shaft is inserted
into the semi-span at the quarter chord point. The leading edge sweep angle is 9.23
degrees and the mean chord length is 6.2 inches.
There is fairing which matches the root profile of the semi-span foil, affixed to the
half body. The span of the fairing was fabricated such that it did not extend beyond
the maximum thickness of the half body. The fairing prevented a gap from forming
between the semi-span and the half body. The computer model does not reflect this
fixed angle fairing.
Turbulence stimulators 0.008 inches high were placed 0.1 inches apart spanwise
along the 20 percent chord position, top and bottom. The positioning of the turbulence inducers was derived using a boundary layer code that predicted a displacement
thickness of 0.008 inches at that location. No turbulence stimulators were placed on
the half body.
42
4.2
4.2.1
Experimental Procedure
Test Conditions
The freestream velocity was set at 24 feet/sec and the average water temperature
during the data collection was 80 degrees Fahrenheit, corresponding to a Reynold's
number of approximately 1.5x106 with Reynold's number defined as
Re =Umean
v
(4.2)
where U is the freestream velocity, cean is the mean chord length, and v is the
kinematic velocity of the water at 80 degrees. The splitter plate extends 7 inches
forward of the leading edge of the half body. The turbulent boundary layer thickness
due to the splitter plate is estimated using the th power law at 0.153 inches or about
73 percent of the fairing span.
4.2.2
Coordinate system
The coordinate directions of the laser control system are permanently established
with the exception of the selection of the origin. The origin was selected to be at the
outermost tip of the trailing edge of the semi-span hydrofoil. This allowed the laser
to be realigned reliably and easily by direct observation on a repeatable basis. The
x-axis is aligned opposite the freestream velocity vector. The y-axis is oriented such
that it points outboard from the origin, away from the root of the semi-span. Using
a right-hand convention, the z-axis is directed upward. A positive angle of attack is
defined with the leading edge displacement upward (positive z).
Unfortunately the computer model of the experiment developed depicting the
starboard side of the configuration and the direction of the y axis is reversed in the
global sense, but remains pointing outboard.
The z axis remains upward and the
positive x direction is downstream. All of the comparisons of the results in Chapter
5 are corrected for this discrepancy.
The angle of attack was measured by several devices. Initially, the laser marked
43
42.5mm
X=-263mm
fj
X=-136mm
X=-70mm
X=-9mm
..
/
X
Figure 4-3: Velocity measurement grids for hydroplane in side configuration
the coordinates of the leading edge and the trailing edge, maintaining the laser's y
position constant and the angle of attack was determined by the arctangent of the
difference of the two coordinates. Additionally, the leading and trailing edges were
measured relative to the tunnel walls using a caliper and the chord of the foil was
measured at the same spanwise location.
The two results were consistent.
Once
defined, the same angle of attack could be obtained repeatedly by a shaft encoder
affixed to the stainless steel shaft of the semi-span foil. Also, the lift, measured by a
dynamometer attached to the shaft, uniquely defines the angle of attack.
4.2.3
Velocity Measurements
The semi-span hydrofoil was set at a specific angle of attack, either positive or negative
5.36 degrees.
The ambient pressure in the tunnel was reduced to a point where
the tip vortex could easily be seen and its coordinates mapped out by the LDV.
Five streamwise locations were chosen to measure the region about the tip vortex,
specifically at 9 mm, 70 mm, 136 mm, 263 mm and 342.5 mm downstream from the
origin (see Figure 4-3). This corresponds to x/Span coordinates of 0.044, 0.34, 0.66,
44
1.28 and 1.67, respectively. At each of these locations a grid of data points, 100 mm
on a side, was measured by the LDV in a plane oriented normal to the longitudinal
axis of the tunnel (See Figure 4-3). Initially the grid consisted of a coarse spacing of
5 mm. This allowed the fine grid of 0.5 mm spacing to be accurately positioned to
obtain the maximum amount of information about the tip vortex.
Another grid of data was measured one half of the mean chord forward of the
foil to record the inflow profile. To supplement this measurement, a single line of
velocities was measured from the upper extreme to the lower extreme of the tunnel
at the one half span position, limited by the laser interference with the frame of the
tunnel. A line of data was also measured spanwise, at a height equal to the vertical
position of the foil.
The time it took for the velocity at each point to be recorded varied from 15 seconds to several minutes. The sample time corresponded to the amount of turbulence
in the region. The time necessary to complete one grid could thus vary from two to
more than six hours.
As noted above, only two components of velocity could be measured at any point.
To complete the study and obtain the third component, the entire assembly was
rotated 90 degrees and placed in the top of the tunnel. This allowed the LDV to
measure the y component of the velocity vector and use the x component to compare
with the original orientation for consistency between measurements.
4.3
Experimental Results
The baseline measurement of variations in the inflow velocity field is shown in Figures
4-4 and 4-5. The grid was measured one half of the mean chord length forward of the
foil. The standard deviations for the streamwise and spanwise velocities are presented
in Figures 4-6 and 4-7.
Figures 4-8 and 4-9 show the results of the two one-dimensional traverses across
the test section of the tunnel in the vertical and spanwise directions, respectively.
Both lines were taken one half of a mean chord length forward of the leading edge
45
Level vxn
F
E
D
C
B
A
9
8
7
6
5
4
3
2
1
1.0
c
n
0
0.5
0.0
1.14219
1.12937
1.11656
1.10375
1.09094
1.07812
1.06531
1.0525
1.03969
1.02688
1.01406
1.00125
0.988437
0.975625
0.962812
0.0
-1.0
-0.5
Y/Span
0.0
0.5
Figure 4-4: Streamwise velocity of inflow at X/Span =+1.06. (coarse grid).
Level vzn
1.0
a
0
0.5
0.0)
F
E
D
C
B
A
0.0079545E
0.005
0.0020454E
-0.0009090E
-0.0038636
-0.0068181E
9
-0.0097727'
8
7
6
5
4
3
2
1
-0.0127273
-0.0156818
-0.0186364
-0.0215909
-0.0245455
-0.0275
-0.0304545
-0.0334091
0.0
-1.0
-0.5
Y/Span
0.0
0.5
Figure 4-5: Vertical velocity of inflow at X/Span =+1.06. (coarse grid).
46
0.3
Level sdxn
F
E
D
C
B
A
9
8
7
6
5
4
3
2
1
0.2
C
0.1
a
c
Q.
0.0
-0.1
-1.0
-0.9
-0.8
-0.7
-0.6
-0.5
0.28233
0.264659
0.246989
0.229318
0.211648
0.193977
0.176307
0.158636
0.140966
0.123295
0.105625
0.0879546
0.0702841
0.0526136
0.0349432
-0.4
Y/Span
Figure 4-6: Standard deviation of streamwise velocity of inflow at X/Span =+1.06.
(coarse grid).
0.3
Level sdzn
F
E
D
C
B
A
9
8
7
6
5
4
3
2
1
0.2
C
0.1
a
0.0
-0.1
-1.0
-0.9
-0.8
Y/Span
-0.7
-0.6
-0.5
0.209205
0.195682
0.182159
0.168636
0.155114
0.141591
0.128068
0.114545
0.101023
0.0875
0.0739773
0.0604545
0.0469318
0.0334091
0.0198864
-0.4
Figure 4-7: Standard deviation of vertical velocity of inflow at X/Span
(coarse grid).
47
=+1.06.
_ ___
0.010
0.005
N
as
0.000
S
-0.005
0
a)
.I
-0.010
3ri
a)
-0.015
o
c
o
ci)
(D
o
cij
-0.020
-0.025
-0.5
0.0
0.5
Z/Span
Figure 4-8: Velocities from vertical traverse of test section at X/Span =+1.06. (coarse
grid).
of the hydrofoil. Note in the spanwise plot that the tunnel-wall boundary layer is
distinctly visible on the left side of Figures 4-4 and 4-9. The standard deviations
associated with these inflow velocities are shown in Figure 4-10.
The velocity data between the side orientation of the assembly and the top orientation were checked for consistency and merged to obtain all three components of
the fluid's velocity. All the values were normalized to the magnitude of the steady
stream velocity measured by the differential pressure cell. The freestream velocity as
measured by the anemometer was approximately 90 percent of that measured by the
LDV.
Figures 4-11 through 4-15 show the hydrofoil's wake deficit as it changes in the
downstream direction. Note the shape of the wake can be visualized through the wake
deficit. The tip vortex can be seen distinctly separating from the remainder of the
wake and moving slightly inboard and upward as it travels downstream. The upward
motion of the tip vortex in the near field will be checked as the wake sheet rolls up into
two concentrated tip vortices further downstream; the downwash associated with one
48
AA
tD
0
1.00
U.3
0.95
0.2
CD
.N
o
i0
E
o
0.90
0.1
_
0
,-4.
_
So
0.85
0.0
.n
a)
CD
E
o
N
CD
0.80
-0.1
0.75
-0.2
0.70
-1.0
0.0
-0.5
0.5
Y/Span
Figure 4-9: Velocities from spanwise traverse of test section at X/Span =+1.06.
(coarse grid).
a--
sdxn
0.025
0.020
C
._
),
V)
0.010
0.010
t--V
-0.5
Mt
- -A , I
I
0.0
,
I
0.5
Z/Span
Figure 4-10: Standard deviations of velocities of vertical traverse of test section at
X/Span =+1.06 mm. (coarse grid).
49
0.20
Level vx
F
0.10
C
1 12937
D
C
111656
110375
B
1 09094
A
1.07812
1.06531
9
0.00
8
7
6
5
U)
0.
Ri
114219
E
1 0525
1 03969
1 02688
1.01406
1 NN159
-0.10
3
2
0988437
0975625
1
0.962812
-0.20
-0.2
-0.1
0.0
Y/Span
0.1
0.2
0.3
Figure 4-11: Streamwise velocity at X/Span =-0.044. Angle of attack is +5.36 degrees
(side orientation/coarse grid).
0.20
Level vx
F
E
0.10
D
C
C
0.00
Ri
-0.10
114219
112937
111656
110375
B
1 09094
A
9
1.07812
8
1.0525
7
1 03969
6
1 02688
5
1 01406
1.06531
4
1 00125
3
0 988437
2
0.975625
1
0 962812
-0.20
-0.2
-0.1
0.0
0.1
0.2
0.3
Y/Span
Figure 4-12: Streamwise velocity at X/Span -- 0.34. Angle of attack is +5.36 degrees
(side orientation/coarse grid).
50
0.20
Level vx
F
0.10
C
0.00
U')
izzj
-0.10
1 14219
E
1.12937
D
C
1 11656
110375
B
1 09094
A
1 07812
9
8
7
6
106531
10525
1 03969
1 02688
5
1 01406
4
3
2
1 00125
0 988437
0 975625
1
0.962812
-0.20
-0.2
-0.1
0.0
Y/Span
0.1
0.2
0.3
Figure 4-13: Streamwise velocity at X/Span =-0.66. Angle of attack is +5.36 degrees
(side orientation/coarse grid).
0.20
Level vx
C
114219
112937
111656
110375
B
1 09094
A
1 07812
9
1 06531
8
1 0525
F
E
0.10
C
D
0.00
Ri
03
-0.10
7
1 03969
6
1 02688
5
1 01406
4
1 00125
3
0.988437
0 975625
0 962812
2
1
-0.20
-0.2
-0.1
0.0
0.1
0.2
0.3
Y/Span
Figure 4-14: Streamwise velocity at X/Span =-1.28. Angle of attack is +5.36 degrees
(side orientation/coarse grid).
51
0.15
Level vxn
0.10
F
E
D
C
B
A
9
8
7
6
5
4
3
2
1
0.05
a
a,
0.00
-0.05
0.961364
0.950909
0.940455
0.93
0.919546
0.909091
0.898637
0.888182
0.877727
0.867273
0.856818
0.846364
0.835909
0.825455
0.815
-0.10
-0.20
-0.10
0.00
Y/Span
Figure 4-15: Streamwise velocity at X/Span =-1.67. Angle of attack is +5.36 degrees
(side orientation/coarse grid).
concentrated tip vortex will influence the other [11]. The LDV measurements taken in
this survey were limited to a few spanlengths downstream which precluded observing
the fully developed path of the tip vortex. Takahashi and McAlister [16] also show
the vertical rise in the tip vortex in their LDV measurements of a NACA 0015 airfoil,
but their results also only take near field effects into account.
The true workings of the tip vortex are better seen by examining the vertical
and spanwise velocity components. Figures 4-16 through 4-20 map out the vertical
velocity of the tip vortex as it travels downstream and figures 4-21 through 4-23
show the spanwise component of velocity. As described above, the vertical velocity
was extracted with the foil in the side configuration and the spanwise velocity was
obtained using the top configuration. In the plots that follow, however, the grid has
been rotated so that all velocity components are shown for the side orientation.
After the vertical and spanwise velocities were merged, a numerical approximation
of the vorticity was computed using a five-point Eulerian scheme. Figures 4-24 to 4-25
show the results.
52
0.20
Level
0.10
C
co
0.00
a.
C)
Qzz
-0.10
vz
F
E
D
C
B
A
9
8
7
6
5
0.20983
0.17875
0.14767
0.116591
0.0855114
0.0544318
0.0233523
-0.0077272E
-0.0388068
-0.0698864
-0.100966
4
-n
3
2
1
-0.163125
-0.194205
-0.225284
32045
-0.20
-0.2
-0.1
0.0
0.1
0.2
0.3
Y/Span
Figure 4-16: Vertical velocity at X/Span =-0.044. Angle of attack is +5.36 degrees
(side orientation/coarse grid).
0.20
Level vz
0.10
C
c
aQ
F
E
0.245227
0.210455
n
n4
C
B
A
9
8
7
6
5
4
3
2
0.00
j;;
-0.10
1
oan
0.140909
0.106136
0.0713636
0.0365909
0.0018181E
-0.0329545
-0.0677273
-0.1025
-0.137273
-0.172045
-0.206818
-0.241591
-0.20
-0.2
-0.1
0.0
0.1
0.2
0.3
Y/Span
Figure 4-17: Vertical velocity at X/Span =-0.34. Angle of attack is +5.36 degrees
(side orientation/coarse grid).
53
0.20
Level
0.10
c
i3
£9
0.00
-0.10
vz
F
E
D
C
B
A
9
8
7
6
5
0.199261
0.167614
0.135966
0.104318
0.0726704
0.0410227
0.0093749E
-0.0222727
-0.0539205
-0.0855682
-0.117216
I
_n
-1A
3
2
1
-0.180511
-0.212159
-0.243807
A
-0.20
-0.2
-0.1
0.0
Y/Span
0.1
0.2
0.3
Figure 4-18: Vertical velocity at X/Span =-0.66. Angle of attack is +5.36 degrees
(side orientation/coarse grid).
0.20
Level vz
0.10
C
CI
C)
0.00
Ri
-0.10
F
E
0.277614
0.235227
U
U.1YZ41
C
B
A
9
8
7
6
5
4
3
2
1
0.150455
0.108068
0.0656818
0.0232954
-0.0190909
-0.0614773
-0.103864
-0.14625
-0.188636
-0.231023
-0.273409
-0.315795
-0.20
-0.2
-0.1
0.0
Y/Span
0.1
0.2
0.3
Figure 4-19: Vertical velocity at X/Span =-1.28. Angle of attack is +5.36 degrees
(side orientation/coarse grid).
54
0.15
Level vzn
F
E
D
C
B
A
9
8
7
6
5
4
3
2
1
0.10
0.05
C
c
0.00
i,
-0.05
0.196534
0.165795
0.135057
0.104318
0.0735795
0.0428409
0.0121023
-0.0186364
-0.049375
-0.0801136
-0.110852
-0.141591
-0.17233
-0.203068
-0.233807
-0.10
-0.20
-0.10
0.00
Y/Span
Figure 4-20: Vertical velocity at X/Span =-1.67. Angle of attack is +5.36 degrees
(side orientation/coarse grid).
0.20
Level vy
F
E
D
C
B
A
9
8
7
6
5
4
3
2
1
0.10
c
0.00
0.
Co
-0.10
0.373523
0.334318
0.295114
0.255909
0.216705
0.1775
0.138295
0.0990909
0.0598864
0.0206818
-0.0185227
-0.0577273
-0.0969318
-0.136136
-0.175341
-0.20
-0.2
-0.1
0.0
Y/Span
0.1
0.2
0.3
Figure 4-21: Spanwise velocity at X/Span =-0.34. Angle of attack is +5.36 degrees
(top orientation/coarse grid).
55
0.20
Level vy
F
E
D
C
B
A
9
8
7
6
5
4
3
2
1
0.10
C
0.00
-0.10
0.381932
0.3375
0.293068
0.248636
0.204205
0.159773
0.115341
0.0709091
0.0264773
-0.0179545
-0.0623863
-0.106818
-0.15125
-0.195682
-0.240114
-0.20
-0.2
-0.1
0.0
0.1
0.2
0.3
Y/Span
Figure 4-22: Spanwise velocity at X/Span =-0.66. Angle of attack is +5.36 degrees
(top orientation/coarse grid).
0.20
Level vy
F
E
D
C
B
A
9
8
7
6
5
4
3
2
1
0.10
a
0.00
z.
-0.10
0.250398
0.208068
0.165739
0.123409
0.0810795
0.03875
-0.0035795E
-0.0459091
-0.0882386
-0.130568
-0.172898
-0.215227
-0.257557
-0.299886
-0.342216
-0.20
-0.2
-0.1
0.0
0.1
0.2
0.3
Y/Span
Figure 4-23: Spanwise velocity at X/Span =-1.28. Angle of attack is +5.36 degrees
(top orientation/coarse grid).
56
0.20
0.10
C
aCZ
CL
0.00
-0.10
-0.20
-0.1
0.0
0.1
0.2
0.3
0.4
Y/Span
Figure 4-24: Axial vorticity at X/Span =-0.044. Angle of attack is +5.36 degrees
(coarse grid).
.1-
0.20
Level
F
E
D
C
B
A
9
8
7
6
5
4
3
2
1
0.10
C
0. 0.00
-0.10
-0.20
r\
I<:D 1
IF
. . . . . . . . . . . . . . . . . . . . ..
-0.2
-0.1
0.0
Y/Span
0.1
0.2
w
42.6801
39.6113
36.5424
33.4735
30.4046
27.3358
24.2669
21.198
18.1291
15.0603
11.9914
8.9225
5.85363
2.78475
-0.284125
. . . . . .
0.3
Figure 4-25: Axial vorticity at X/Span =-0.66. Angle of attack is +5.36 degrees
(coarse grid).
57
During the second phase of testing there was insufficient time to shift the assembly
to the top orientation, therefore vorticities could not be calculated because spanwise
velocity components were not obtained. However, the hydrofoil was positioned to a
-5.36 degree angle of attack and a set of velocity profiles were measured. Figures 4-26
through 4-29 show some of the results. Note the symmetry of the model is reflected
when comparing these plots to figures of the positive angle of attack above.
For more precise validation of the computer code and to study the effects of vortex
core wandering, a detailed mapping of the tip vortex core was performed. At this level
of detail the effects of time averaging the samples are more significant. The vortex
core wanders in a random motion about a mean longitudinal line. This causes the
contour plots of the core size and geometry to appear larger and more symmetric than
might actually be the case for a snapshot in time. Figures 4-30 through 4-37 show
the detailed view of the tip vortex core using a fine grid of 0.5 mm spacing between
sampling points. Detailed mapping of the vortex was also attempted at the planes
furthest downstream, but the data rate was so reduced that it became infeasible to
obtain an accurate plot.
Standard deviations calculated for the velocities presented in this chapter are
included in the appendix.
58
0.10
Level vxn
F
E
D
C
B
A
9
8
7
6
5
4
3
2
1
0.05
C
co
a
V)
0.00
-0.05
-0.10
-0.10
0.00
1 14219
1.12937
1.11656
1.10375
1.09094
1.07812
1.06531
1.0525
1.03969
1.02688
1.01406
1.00125
0.988437
0.975625
0.962812
0.10
Y/Span
Figure 4-26: Streamwise velocity at X/Span =-0.044. Angle of attack is -5.36 degrees
(side orientation/coarse grid).
Level vxn
0.1
F
E
D
C
B
A
9
8
7
6
5
4
3
2
1
C
CO
accl~
0.0
1.14219
1.12937
1.11656
1.10375
1.09094
1.07812
1.06531
1.0525
1.03969
1.02688
1.01406
1.00125
0.988437
0.975625
0.962812
-0.1
-0.2
-0.1
0.0
Y/Span
Figure 4-27: Streamwise velocity at X/Span =-1.67. Angle of attack is -5.36 degrees
(side orientation/coarse grid).
59
0.10
Level vzn
F
E
D
C
B
A
9
8
7
6
5
4
3
2
1
0.05
C
0.00
CD
-0.05
0.215227
0.182273
0.149318
0.116364
0.0834091
0.0504545
0.0175
-0.0154546
-0.0484091
-0.0813636
-0.114318
-0.147273
-0.180227
-0.213182
-0.246136
-0.10
-0.10
0.00
0.10
Y/Span
Figure 4-28: Vertical velocity at X/Span =-0.044. Angle of attack is -5.36 degrees
(side orientation/coarse grid).
Level vzn
0.1
F
E
D
C
B
A
9
8
7
6
5
4
3
2
c.
C',
0.0
1
0.23925
0.2065
0.17375
0.141
0.10825
0.0755
0.04275
0.01
-0.02275
-0.0555
-0.08825
-0.121
-0.15375
-0.1865
-0.21925
-0.1
-0.2
-0.1
0.0
Y/Span
Figure 4-29: Vertical velocity at X/Span =-1.67. Angle of attack is -5.36 degrees
(side orientation/coarse grid).
60
Level vx
1.02688
F
1.0223
E
1.01773
D
1.01315
C
1.00857
1.004
A
0.999422
9
0.994846
8
0.99027
7
0.985693
6
0.981117
5
0.976541
4
0.971965
3
2
0.967388
1
0.962812
0.040
0.030
co
CL
Cl)
iz3
0.020
0.010
0.000
-0.050
-0.040
-0.030
-0.020
-0.010
0.000
Y/Span
Figure 4-30: Streamwise velocity at X/Span =-0.044. Angle of attack is +5.36 degrees
(side orientation/fine grid).
0.070
0.060
Level vx
F
E
D
C
B
A
0.050
0.040
9
C
cQ. 0.030
8
7
6
0.020
5
4
0.010
3
2
1
1.14219
1.12937
1.11656
1.10375
1.09094
1.07812
1.06531
1.0525
1.03969
1.02688
1.01406
1.00125
0.988437
0.975625
0.962812
0.000
-0.08
-0.06
-0.04
-0.02
Y/Span
Figure 4-31: Streamwise velocity at X/Span =-0.34. Angle of attack is +5.36 degrees
(side orientation/fine grid).
61
0.070
Level
0.060
F
E
D
C
B
A
9
8
7
6
5
4
3
2
1
0.050
0.040
D. 0.030
0.020
0.010
vx
1.14219
1.12937
1.11656
1.10375
1.09094
1.07812
1.06531
1.0525
1.03969
1.02688
1.01406
1.00125
0.988437
0.975625
0.962812
0.000
-0.12
-0.10
-0.08
-0.06
-0.04
-0.02
Y/Span
Figure 4-32: Streamwise velocity at X/Span =-0.66. Angle of attack is +5.36 degrees
(side orientation/fine grid).
Level vz
0.040
F
E
D
C
B
A
9
8
7
6
5
4
3
2
1
0.030
C
r:
Cd
a
C', 0.020
0.010
0.26892
0.226023
0.183125
0.140227
0.0973295
0.0544318
0.0115341
-0.0313636
-0.0742614
-0.117159
-0.160057
-0.202955
-0.245852
-0.28875
-0.331648
0.000
-0.050
-0.040
-0.030
-0.020
-0.010
0.000
Y/Span
Figure 4-33: Vertical velocity at X/Span =-0.044. Angle of attack is +5.36 degrees
(side orientation/fine grid).
62
Il
I
0.070
0.060
Level vz
F
E
D
C
B
A
9
8
7
6
5
4
3
2
1
0.050
0.040
as~~~1---~
f RAX
! 0.030
0.344886
0.297045
0.249205
0.201364
0.153523
0.105682
0.0578409
0.0099999E
-0.0378409
-0.0856818
-0.133523
-0.181364
-0.229205
-0.277045
-0.324886
0.020
0.010
0.000
~ , ~,
,~~
,, , ,
-0.08
,
I
\
-0.06
Y/Span
-0.04
I
I\,1 I
. . . .
-0.02
Figure 4-34: Vertical velocity at X/Span =-0.34. Angle of attack is +5.36 degrees
(side orientation/fine grid).
0.070
Level vz
0.060
F
E
D
C
B
A
9
8
7
6
5
4
3
2
1
0.050
0.040
X.
0.030
0.020
0.010
0.266534
0.225795
0.185057
0.144318
0.10358
0.0628409
0.0221023
-0.0186364
-0.059375
-0.100114
-0.140852
-0.181591
-0.22233
-0.263068
-0.303807
0.000
-0.12
-0.10
-0.08
-0.06
-0.04
-0.02
Y/Span
Figure 4-35: Vertical velocity at X/Span =-0.66. Angle of attack is +5.36 degrees
(side orientation/fine grid).
63
Level vy
F
E
D
C
B
A
9
8
7
6
5
4
3
2
1
10
C
C0
CI
5
0.192875
0.14975
0.106625
0.0635
0.020375
-0.02275
-0.065875
-0.109
-0.152125
-0.19525
-0.238375
-0.2815
-0.324625
-0.36775
-0.410875
0
-15
-10
-5
0
Y/Span
Figure 4-36: Spanwise velocity at X/Span =-0.34. Angle of attack is +5.36 degrees
(top orientation/fine grid).
15
Level vy
F
0.192875
E
0.14975
0.106625
D
0.0635
C
B
0.020375
A -0.02275
-0.065875
9
-0.109
8
-0.152125
7
-0.19525
6
-0.238375
5
4
-0.2815
3 -0.324625
-0.36775
2
1 -0.410875
10
C
0
-20
-15
-10
-5
Y/Span
Figure 4-37: Spanwise velocity at X/Span =-0.66. Angle of attack is +5.36 degrees
(top orientation/fine grid).
64
Chapter 5
Calculations
The results of the numerical model developed earlier are presented in this chapter.
The first section will focus on convergence studies and the second portion will compare
the numerical results to the experimental data.
5.1
Convergence Studies
There are essentially three criteria used to examine the convergence of the code.
The first involves the number of iterations used for the wake to become aligned with
the local velocity vector. Essentially, the program will continue iterating until each
individual wake panel pitch angle is within 0.1 percent of the pitch angle of the local
velocities. Once this is achieved, both the circulation distribution on the hydrofoil
and the position of the wake are in agreement. Presumably, this is a unique state
for a given angle of attack. All of the variations presented in this chapter met this
convergence criterion.
The other two tests include convergence by increasing the
number of panels and convergence by lengthening the downstream portion of the
tunnel.
The number of panels located spanwise on the hydrofoil was increased from four
to 16, with the same number of panels chordwise. The maximum number of panels
was chosen at 16 because of time limitations only. Figure 5-1 shows the spanwise
circulation distribution as the number of panels is increased. There is a significant
65
3.0
2.5
D
2.0
o
1.5
1.0
0.5
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Y/Span
Figure 5-1: Comparison of spanwise circulation distribution by increasing the number
of spanwise panels
improvement in the profile going from four to eight panels. The decrease in circulation
in the proximity of the root is due to the foil's interaction with the half body. Overall,
it appears that the circulation values readily converge.
Figures 5-2 through 5-4 show a spanwise cross section of the wake as it varies
downstream. When X/Span is 1.01 there is quite a discrepancy in the treatment of
the tip vortex. With increased panelling and cosine spacing the number of panels
located near the foil tip increases. The smaller tip panels are more susceptible to the
complex velocity field near the tip. The tip behavior exhibited with the higher number
of panels is due to the effects of crossflow around the tip and due to a tendency for
these panels to simulate the vortex roll-up.
As the wake progresses downstream, the errant tip behavior of Figure 5-2 begins
to straighten out. A crease appears to develop, though, between the two adjacent
outboard panels. The strength of the interior wake discrete vortices is equal to the
difference in circulation between two adjoining panels. The tip vortex strength is
equated to the circulation associated with the very last panel only. As the panels
66
X/Snan= .0 1
-
16Spanwise Panels
10SpanwisePanels
0
8 Spanwise
Panels
' '-----
4 Spanwise Panels
-0.02
Pa
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
Y/Span
Figure 5-2: Comparison of vertical position of wake sheet by increasing the number
of spanwise panels
X/Snnn=
0.02
------
16 SpanwisePanels
----O
)
0SpanwisePanels
-
8 SpanwisePanels
4 Spanwise
Panels
0.00
-0.02
e,
I
s>
-0.04
-0.06
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
Y/Span
Figure 5-3: Comparison of vertical position of wake sheet by increasing the number
of spanwise panels
67
YInnn--
4A)
0.04
i
0.02
0.00
4.
'o
0.
-0.02
>
-0.04
-0.06
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
Y/Span
Figure 5-4: Comparison of vertical position of wake sheet by increasing the number
of spanwise panels
become smaller near the tip, a situation occurs where the strength of the tip vortex is
more than the influence of the nearest vortices forcing the panel boundary downward
and creating the crease. Note also that the decrease in circulation distribution near
the root of the foil is reflected in the downstream wake geometry for the higher panel
case.
Recall that the panels are not free to move in the spanwise direction. The tip
vortex roll-up therefore most evident by examining the slope of the outboard edge of
the wake. At X/Span of 3.42, the outer edge is almost vertical as the wake attempts
to roll over on itself. As the number of panels increases the vertical position of the
outboard edge of the wake will rise.
Chapter two discussed concern that modelling the tunnel with a finite length
might lead to incorrect results. The effects of tunnel length are examined in Figures
5-6 through 5-9. For consistency, the eight spanwise panel case was selected and the
length of the tunnel downstream of the foil was increased from the baseline of three
spanlengths to a maximum of 12 spanlengths.
68
As a reminder, in addition to the
Figure 5-5: Complete panelling scheme for 16 spanwise panels. (Downstream looking
forward.)
length of the downstream portion of the tunnel, a final row of wake panels is attached
downstream, extending beyond the end of the tunnel (See Figure 5-5). These panels
are 10 spans long and are used to simulate the effects of the starting vortex. Figure 5-6
shows the convergence of the circulation distribution. The seeming discrepancy which
occurs between the eight and 12 span lengths is due to an unexplained premature
pruning of the results for the 12 panel case by the computer. Figures 5-7 through
5-9 show the convergence of the spanwise wake shape with increasing tunnel length.
With the exception of X/Span of 3.42, there is virtually no difference in the local
flow velocities of the wake as the tunnel length is increased. Examining the behavior
of the root vortex (Figure 5-10) and the tip vortex (Figure 5-11), it seems that the
increased tunnel length has little bearing on the geometry of the wake.
5.2
Comparison to experimental data
The output of the code is also validated by comparing it not only to the experimental
data, but also to what might be expected analytically from potential flow theory.
69
2.0
1.5
5:
02
1.0
0.5
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Y/Span
Figure 5-6: Comparison of spanwise circulation distribution by increasing downstream
tunnel length
X/Span=l.01
---A
-0.020
-0-----v-----
9
0.
3 Spanlengths
4 Spanlengths
8 Spanlengths
12 Spanlengths
-0.030
0
U
3
*CT
.e -0.040
-0.050
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
Y/Span
Figure 5-7: Comparison of spanwise wake shape by increasing downstream tunnel
length
70
X/Span=2.22
0.020
----{o-
3 Span lengths
4 Span lengths
8Span lengths
12Span lengths
-
0.010
0.000
-0.010
o
e,
-0.020
-0.030
-0.040
-0.050
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
Y/Span
Figure 5-8: Comparison of spanwise wake shape by increasing downstream tunnel
length
X/Span=3.42
0.04
0
3 Spanlengths
4 Span lengths
8 Span lengths
12Span lengths
-
-..-<00.02
-
0.00
0
0
-0.02
-0.04
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
Y/Span
Figure 5-9: Comparison of spanwise wake shape by increasing downstream tunnel
length
71
, n,,
U.UUU
r
3 Spanlengths
4 Span lengths
V---O- 8 Spanlengths
-
-0.010
-0.020
-0.030
o
o
-0.040
-0.050
o
.!a
o -0.060
U-a-0.070
-0.080
-0.090
-0.100
,,-
,
,,I
1
,
,,,I
2
...,
.
3
I,,
4
5
6
7
8
X/Span
Figure 5-10: Comparison of tip vortex position by increasing downstream tunnel
length
,\
n
U.Zu
0.15
-
-
-
3 Spanlengths
4 Spanlengths
8 Spanlengths
12 Spanlengths
9
0.10
0.05
0
"o
Q)
0.00
0
*a
.9
0
-
-0.05
-
CW
-0.10
-
-0.15
-
-0.20
,
,I I . . I. .
1
. . . . I . . . . I .,. . . I .
2
3
4
5
,I .
6
. . I ,I I I I I
7
8
X/Span
Figure 5-11: Comparison of root vortex position by increasing downstream tunnel
length
72
0.5
a
.(
O
m
0.0
Wd
0
-0.5
-10
-5
0
5
10
Angle of Attack (degrees)
Figure 5-12: Calculated lift and drag coefficients
This latter, often ignored comparison is essential to the development of an accurate
code. To be useful, a code must be able to handle the rigors of the analytical limits of
the problem. A program which can only predict the results of the given experimental
conditions is self-defeating. There would be no reason to rely on a code if experimental
results are available.
The non-dimensional lift coefficient for an elliptical wing is given by
CL = 1 + 2/A
(5.1)
where CL is the total lift coefficient, a is the angle of attack, and A is the aspect ratio
of the wing,
A = 4(span)
(5.2)
7rco
The lift coefficient for the elliptical wing is shown against the lift coeficient calculated
by the code in Figure 5-12. The difference may be attributed partly to the presence
of the halfbody and partly due to the fact that the semi-span is not elliptical. The
computed drag coefficient is also presented for reference.
73
For an elliptical wing in an infinite domain, the sectional lift coefficient may be
regarded as equal to the total lift coefficient CL [8] and the spanwise circulation
distribution is
c(y)
r(y) = -CLU
2
(5.3)
where c(y) is the chord length. This equation is only valid for the elliptical wing, but
it gives a reasonable estimate of the maximum value of circulation for the semi-span,
which when non-dimensionalized by the freestream velocity and the span length, is
2.53.
Turning next to the experimental data, the circulation distribution was measured
by Coney for the same model [4], however he calculated values for o = 7.5 degrees.
His maximum measured value of circulation at the root of the semi-span equates to
a value near 2.9. Although the spanwise circulation was not measured in the present
testing, an approximate estimate of the the total circulation can be obtained by
assuming that all of the wake's free vorticity is concentrated in the tip vortex at the
X/Span = 1.67 point. The velocity data from this grid may be numerically integrated
to give a total circulation value of 3.625 when non-dimensionalised as above.
With the circulation data verified, the velocity output is examined. An important
difference between the computer model and the actual experiment is that the free
vorticity in the real wake will quickly roll up into the concentrated tip vortex. In
the model, the vorticity remains constant streamwise and only varies in the spanwise
direction, reflecting the circulation on the foil. The velocities due to a line vortex
comes from Biot-Savart
V
4 fsf
dl
(5.4)
If the vortex filament is sufficiently long enough the velocities induced by the vortex
are similar to that of a semi-infinite two-dimensional vortex [8]
w(y) = 4ry
(5.5)
the
4om
equation
above,
estimates
the
of
velocities
in the proximity of the tip vortex
From the equation above, estimates of the velocities in the proximity of the tip vortex
74
-
-
Level VX
0.20
\_/>\~4
0.10
C:
cf)
CZ
CL1
0.00
i>~~~~~
F
1 45812
E
1 41785
D
1.37757
C
1 3373
B
1 29702
A
1 25675
9
1 21648
8
7
11762
113592
6
1 09565
5
1 05537
4
1,0151
0 974825
0 93455
0 894275
3
2
-0.10
1
I
-0.2
I
.l, I .
-0.1
.,,
l
.
.
0.0
.1111i
..
..1
0.1
.
.
0.2
. . . .
0.3
Y/Span
Figure 5-13: Computed streamwise velocity at X/Span=-0.044.
+5.36 degrees (coarse grid).
Angle of attack is
0.20
375
35
025
2
375
0.10
C
55
725
CZ)
aa
M
9
075
0.00
25
425
225
05
-0.10
3875
-0.2
-0.1
0.0
0.1
0.2
0.3
Y/Span
Figure 5-14: Computed spanwise velocity at X/Span=-0.044. Angle of attack is
+5.36 degrees (coarse grid).
75
0.20
49996
5
iC
c
0.10
5
o.
CL)
5
5
0.00
5
95
05
-0.10
5
-0.2
-0.1
0.1
0.0
0.2
0.3
Y/Span
Figure 5-15: Computed vertical velocity at X/Span-0.044.
degrees (coarse grid).
Angle of attack is +5.36
0.2
31
43
54
65
76
0.1
C
375
988
CI
1
213
325
438
Rie
0.0
55
363
775
-0.1
388
-0.2
-0.1
0.0
0.1
0.2
0.3
Y/Span
Figure 5-16: Computed streamwise velocity at X/Span=-0.34.
+5.36 degrees (coarse grid).
76
Angle of attack is
0.2
9813
3625
7438
125
50625
1125
7312
35
9687
5875
2062
325
4437
3625
6812
0.1
cC
0.0
-0.1
-0.2
-0.1
0.0
0.1
0.2
0.3
Y/Span
Figure 5-17: Computed spanwise velocity at X/Span=-0.34. Angle of attack is +5.36
degrees (coarse grid).
0.2
75
5
25
0.1
75
C
5
25
C,
Wz
Fm
75
0.0
5001
7499
25
5
75
-0.1
-0.2
-0.1
0.0
0.1
0.2
0.3
Y/Span
Figure 5-18: Computed vertical velocity at X/Span=-0.34.
degrees (coarse grid).
77
Angle of attack is +5.36
0.2
0.1
Q.
C',
0.0
-0.1
-0.2
-0.1
0.0
0.1
0.2
0.3
Y/Span
Figure 5-19: Computed streamwise velocity at X/Span=-0.66.
+5.36 degrees (coarse grid).
Angle of attack is
may be estimated, assuming that the strength of the vortex is equal to the F determined above. It is at this point where a discrepency lies between the panel code and
the experimental data, causing the induced velocities to be roughly 10 times smaller
than the measured values. It is postulated that the velocities shown in Figures 5-13
through 5-27 are only due to the tip vortex of the wake, instead of the sum of the
wake vortices.
Since the strength of each vortex is fixed in the streamwise direction,
the tip vortex never increases in strength, as the actual tip vortex does, to induce
higher velocites.
As another comparison, the panel code calculated the velocities across the surface
of the semi-span. Unlike the velocities calculated above, the surface velocities were
calculated using a central difference formula to differentiate the potential of the hydrofoil panels. The potentials are only known at the panel centroids giving Figures
These surface velocities are compared to
5-28, 5-29, and 5-30 an irregular outline.
the two dimensional values presented by Abbot and Von Doenhoff [1] in Figure 5-31.
The maximum velocity across the semi-span is higher due to the half body reducing
78
0.2
3688
1375
9063
575
4438
2125
9813
75
5188
2875
)5625
175
4063
6375
8688
0.1
C
co
a
C
0.0
-0.1
-0.2
-0.1
0.1
0.0
0.2
0.3
Y/Span
Figure 5-20: Computed spanwise velocity at X/Span=-0.66.
degrees (coarse grid).
Angle of attack is +5.36
0.2
3062
3125
3187
825
33125
1625
6563
15
6438
1375
5313
125
'188
1125
,062
0.1
C
a
U)
Cu~
0.0
-0.1
-0.2
-0.1
0.1
0.0
0.2
0.3
Y/Span
Figure 5-21: Computed vertical velocity at X/Span=-0.66.
degrees (coarse grid).
79
Angle of attack is +5.36
0.2
0.1
rC
co
0.0
-0.1
-0.2
-0.1
0.0
0.1
0.2
0. 3
Y/Span
Figure 5-22: Computed streamwise velocity at X/Span=-1.213. Angle of attack is
+5.36 degrees (coarse grid).
0.2
5313
36251
30624
975
0.1
437
3125
4812
65
8187
9875
562
125
494
662
131
C
iZi
0.0
-0.1
-0.2
-0.1
0.0
0.1
0.2
0.3
Y/Span
Figure 5-23: Computed spanwise velocity at X/Span=-1.28.
degrees (coarse grid).
80
Angle of attack is +5.36
0.2
938
875
813
75
687
625
5625
5
0.1
C
Co
4375
0.0
625
6875
75
813
875
938
-0.1
-0.2
-0.1
0.1
0.0
0.2
0.3
Y/Span
Figure 5-24: Computed vertical velocity at X/Span=-1.28.
degrees (coarse grid).
Angle of attack is +5.36
0.2
0.1
eC
CU
0.0
-0.1
-0.2
-0.1
0.1
0.0
0.2
0.3
Y/Span
Figure 5-25: Computed str eamwise velocity at X/Span=-1.67.
+5.36 degrees (coarse grid).
81
Angle of attack is
0.2
3187
375
956248
'25
0.1
4063
Ca
)875
0.
688
45
1312
8125
4938
175
3563
5375
219
0.0
-0.1
-0.2
-0.1
0.0
Y/Span
0.1
0.2
Figure 5-26: Computed spanwise velocity at X/Span=-1.67.
degrees (coarse grid).
0.3
Angle of attack is +5.36
0.2
aC.
8688
2375
5063
975
3438
7125
31875
55
1812
8125
438
0.1
a)
izzJ
0.0
)75
7063
3375
9688
-0.1
-0.2
-0.1
0.0
0.1
0.2
0.3
Y/Span
Figure 5-27: Computed vertical velocity at X/Span=-1.67.
degrees (coarse grid).
82
Angle of attack is +5.36
1.0
0.9
0.8
eC,
C,,
0.7
3:
0.6
0.5
0.4
0.3
-0.2
0.0
0.2
0.4
0.6
X/SPan
Figure 5-28: Comparison of streamwise velocities across the surface of the hydrofoil
at 0 degree angle of attack
1.0
0.9
0.8
C
CL
C')
0.7
0.6
0.5
0.4
0.3
-0.2
0.0
0.2
0.4
0.6
X/SPan
Figure 5-29: Comparison of spanwise velocities across the surface of the hydrofoil at
0 degree angle of attack
83
1.0
0.9
0.8
aa
CL
0.7
a)
0.6
0.5
0.4
0.3
-0.2
0.0
0.2
0.4
0.6
X/SPan
Figure 5-30: Comparison of vertical velocities across the surface of the hydrofoil at 0
degree angle of attack
n
,
Z1.
1.5
a8
:D
1.0
0.5
0.0
0
20
40
60
80
100
Percent chord
Figure 5-31: Comparison cofstreamwise velocities across the surface of the hydrofoil
at 0 degree angle of attack
84
0.20
Level vx
F
E
D
C
B
A
9
8
7
6
5
4
3
2
1
0.10
C
0.00
-0.10
1.14219
1.12937
1.11656
1.10375
1.09094
1.07812
1.06531
1.0525
1.03969
1.02688
1.01406
1.00125
0.988437
0.975625
0.962812
-0.20
-0.2
-0.1
0.0
0.1
0.2
0.3
Y/Span
Figure 5-32: Experimental wake shape with panel code predicted wake shape superimposed. (X/Span=-0.044)
the tunnel's cross-sectional area at that point. The resulting velocity may be thought
of as a product of the effects the two joining hydrofoils.
The shape of the wake predicted by the panel code is quite similar to that observed in the experiment. Although this experiment focused on the wake behavior
nearest the tip vortex, Coney's work included measurements of the complete wake.
He used a negative 7.5 degree angle of attack and did not present detailed plots of the
coordinates of the measured wake, but the overall shape of his baseline case resembles the output of the panel code. The position of the computer-generated wake is
superimposed on the graphs of experimental data in Figures 5-32 through 5-36. The
location of the tip vortex, represented by the outboard edge of the wake, is reasonably
close to the experimental data.
The panel code predicts the circulation distribution, wake shape and wake position
reasonably well. Still, the error in calculating the induced velocities is not trivial. It
would appear to be localized to the routine which computes the local velocity vector
for a given field point, however, it could just as easily be a larger error which pervades
85
0.20
Level vx
F
E
D
C
B
A
0.10
c
9
8
7
0.00
zj
iz
6
5
4
-0.10
3
2
1
1.14219
1.12937
1.11656
1.10375
1.09094
1.07812
1.06531
1.0525
1.03969
1.02688
1.01406
1.00125
0.988437
0.975625
0.962812
-0.20
-0.2
-0. 1
0.0
0.1
0.2
0.3
Y/Span
Figure 5-33: Experimental wake shape with panel code predicted wake shape super-
imposed. (X/Span=-0.34)
0.20
Level vx
F
1.14219
E
1.12937
1.11656
D
1.10375
C
B
1.09094
1.07812
A
9
1.06531
8
1.0525
7
1.03969
1.02688
6
1.01406
5
4
1.00125
3
0.988437
0.975625
2
1
0.962812
0.10
C 0.00
a
CO
-0.10
-0.20
-0.2
-0.1
0.0
0.1
0.2
0.3
Y/Span
Figure 5-34: Experimental wake shape with panel code predicted wake shape super-
imposed. (X/Span=-0.66)
86
0.20
Level vx
1.14219
F
1.12937
E
1.11656
D
1.10375
C
1.09094
B
1.07812
A
1.06531
9
1.0525
8
1.03969
7
1.02688
6
1.01406
5
1.00125
4
0.988437
3
2
0.975625
1
0.962812
0.10
C
0.00
C)
CO
i;:
-0.10
-0.20
-0.2
-0. 1
0.1
0.0
0.2
0.3
Y/Span
Figure 5-35: Experimental wake shape with panel code predicted wake shape super-
imposed. (X/Span=-1.28)
0.15
Level vxn
1.14219
F
1.12937
E
1.11656
D
1.10375
C
1.09094
B
1.07812
A
1.06531
9
1.0525
8
1.03969
7
1.02688
6
1.01406
5
1.00125
4
0.988437
3
2
0.975625
0.962812
1
0.10
0.05
C
CL
U)
0.00
F4
-0.05
-0.10
-0.20
-0.10
0.00
Y/Span
Figure 5-36: Experimental wake shape with panel code predicted wake shape super-
imposed. (X/Span=-1.67)
87
the entire wake alignment procedure. The panel code presented here is a rudimentary,
first approximation. Further revision will be required to ensure that the program can
be exercised to its limits.
88
Chapter 6
Conclusions
A perturbation potential panel code has been modified for use on a semi-span hydrofoil
attached to a half body within a water tunnel.
In doing so, the panel code has
been revised to address problems not normally encountered in its common usage as a
propeller code. Specifically, panelling the model walls in an attempt to gain additional
insight into their effects on the semi-span geometry is a complicated problem even
in potential flow. The addition of viscous effects and a developing boundary layer
on the walls introduces another dimension of complexity which has received much
attention from aerodynamic researchers. The code has also implemented a revised
method of modelling and aligning the wake, combining both a vortex lattice routine
for wake-induced velocities and relying on differentiation of the panel potentials to
calculate the effects of the semi-span, half body and tunnel. After several iterations of
the process, an independent prediction of the steady state wake geometry is obtained.
Several features could be added to this code to enhance its accuracy. Coupling
it to a viscous model is an obvious, but hardly a simple choice. In addition to the
boundary layer development along the tunnel walls and hydrofoils, the effects within
the wake sheet would also make a significant difference in the results. The tip vortex
separation would also be effected. Another clear choice would be to better simulate
the tip vortex roll up. This becomes especially difficult to panel as the they must
change in size and direction in three dimensions without penetrating one another. A
code which addresses this difficulty is currently under development at MIT and could
89
easily be attached to this modified panel code. The gap at the root of the semi-span
foil is also not treated in the present program. Much of the work that was done to
simulate the effects of the clearance gap for ducted propellers could be utilized here.
Finally, treatment of the random wandering of the tip vortex may be added for more
pre cise predictions of the tip vortex behavior.
The output of the modified panel code gives encouraging results. While there remains an error in the velocity calculation, computation of the circulation distribution
and the wake geometry is validated by the experimental data. With implementation
of the above features, this relatively simple code, based on fundamental principles
of potential flow, could be used to predict various semi-span geometries without expending the time and money to build scale models for water tunnel testing.
This
code may also be used as a cross check for one of the more complex Reynolds averaged Navier-Stokes (RANS) codes which have been developed in recent years such as
"DTNS3D", used at the David Taylor Model Basin [17]. The treatment given here
leaves open the opportunity to advance this panel code to an even more useful and
accurate predictor of tip vortex behavior.
90
Chapter 7
Appendix
This appendix contains plots of the standard deviations associated with the LDV
velocity measurements discussed in Chapter 4.
91
0.20
Level sdx
F
E
D
C
B
A
9
8
7
6
5
4
3
2
1
0.10
0.00
co
Ri
-0.10
0.0917614
0.0871591
0.0825568
0.0779545
0.0733523
0.06875
0.0641477
0.0595455
0.0549432
0.0503409
0.0457386
0.0411364
0.0365341
0.0319318
0.0273295
-0.20
-0.2
-0.1
0.0
0.1
0.2
0.3
Y/Span
Figure 7-1: Standard deviation of streamwise velocity at X/Span =-0.044. Angle of
attack is +5.36 degrees (side orientation/coarse grid).
0.20
Level sdx
F
E
D
C
B
A
9
8
7
6
5
4
3
2
1
0.10
C
0.00
co
(a.
Col
R
F,
-0.10
0.0713636
0.0681818
0.065
0.0618182
0.0586364
0.0554545
0.0522727
0.0490909
0.0459091
0.0427273
0.0395455
0.0363636
0.0331818
0.03
0.0268182
-0.20
-0.2
-0.1
0.0
0.1
0.2
0.3
Y/Span
Figure 7-2: Standard deviation of streamwise velocity at X/Span =-0.34. Angle of
attack is +5.36 degrees (side orientation/coarse grid).
92
0.20
Level sdx
0.10
C
0.00
0.
o3
rN
-0.10
F
0 0567045
E
D
00543182
0 0519318
C
B
0 0495455
0 0471591
A
9
0 0447727
0 0423864
8
0.04
7
6
5
4
0 0376136
0 0352273
0 0328409
0.0304545
3
0 0280682
2
0 0256818
1
0 0232955
-0.20
-0.2
-0.1
0.0
0.1
0.2
0.3
Y/Span
Figure 7-3: Standard deviation of streamwise velocity at X/Span =-0.66. Angle of
attack is +5.36 degrees (side orientation/coarse grid).
0.20
Level sdx
F
E
0.10
C
Co
D
C
B
A
9
8
7
0.00
U)
6
5
4
-0.10
0 0508523
0.0489773
0 0471023
0.0452273
0 0433523
0 0414773
0 0396023
0 0377273
0.0358523
0 0339773
0 0321023
0 0302273
2
0.0283523
0 0264773
1
0 0246023
3
-0.20
-0.2
-0.1
0.0
0.1
0.2
0.3
Y/Span
Figure 7-4: Standard deviation of streamwise velocity at X/Span =-1.28. Angle of
attack is +5.36 degrees (side orientation/coarse grid).
93
0.15
Level sdxn
F
E
D
C
B
A
9
8
7
6
5
4
3
2
1
0.10
0.05
c:
0.00
Ci,
CD
Fn
-0.05
0.0702841
0.0669318
0.0635795
0.0602273
0.056875
0.0535227
0.0501705
0.0468182
0.0434659
0.0401136
0.0367614
0.0334091
0.0300568
0.0267045
0.0233523
-0.10
-0.20
-0.10
0.00
Y/Span
Figure 7-5: Standard deviation of streamwise velocity at X/Span =-1.67. Angle of
attack is +5.36 degrees (side orientation/coarse grid).
50
40
Level sdy
30
F
E
D
C
B
A
9
8
7
6
5
4
3
2
1
20
10
C
0
Ct
QI
-10
-20
-30
0.088625
0.08325
0.077875
0.0725
0.067125
0.06175
0.056375
0.051
0.045625
0.04025
0.034875
0.0295
0.024125
0.01875
0.013375
-40
-50
-40
-20
0
20
40
60
Y/Span
Figure 7-6: Standard deviation of spanwise velocity at X/Span =-0.34. Angle of
attack is +5.36 degrees (top orientation/coarse grid).
94
50
40
Level sdy
30
F
E
D
C
B
A
9
8
7
6
5
4
3
2
20
10
cO
0
0-
-10
-20
-30
1
0 096125
0.09025
0.084375
0 0785
0 072625
0 06675
0.060875
0 055
0 049125
0 04325
0.037375
0 0315
0 025625
0.01975
0 013875
-40
-50
-40
-20
0
20
40
60
Y/Span
Figure 7-7: Standard deviation of spanwise velocity at X/Span =-0.66. Angle of
attack is +5.36 degreess (top orientation/coarse grid).
"Ie__
Du
_ _
40
Level sdy
30
F
E
D
C
B
A
9
8
7
6
5
4
3
2
1
20
10
C
CU
ca
0
U
-10
-20
-30
0 098
0 092
0 086
0.08
0074
0.068
0062
0 056
005
0.044
0.038
0.032
0 026
0 02
0014
-40
-50
-40
-20
0
20
40
60
Y/Span
Figure 7-8: Standard deviation of spanwise velocity at X/Span =-1.28. Angle of
attack is +5.36 degrees (top orientation/coarse grid).
95
0.20
Level sdz
F
E
D
C
B
A
9
8
7
6
5
4
3
2
1
0.10
a
0.00
a.
C)
-0.10
0.0728977
0.0685227
0.0641477
0.0597727
0.0553977
0.0510227
0.0466477
0.0422727
0.0378977
0.0335227
0.0291477
0.0247727
0.0203977
0.0160227
0.0116477
-0.20
-0.2
-0.1
0.0
Y/Span
0.1
0.2
0.3
Figure 7-9: Standard deviation of vertical velocity at X/Span =-0.044. Angle of
attack is +5.36 degrees (side orientation/coarse grid).
0.20
Level sdz
F
E
D
C
B
A
9
8
7
6
5
4
3
2
1
0.10
c
0.00
Q.
-0.10
0.0584091
0.055
0.0515909
0.0481818
0.0447727
0.0413636
0.0379545
0.0345455
0.0311364
0.0277273
0.0243182
0.0209091
0.0175
0.0140909
0.0106818
-0.20
-0.2
-0.1
0.0
Y/Span
0.1
0.2
0.3
Figure 7-10: Standard deviation of vertical velocity at X/Span =-0.34.
attack is +5.36 degrees (side orientation/coarse grid).
96
Angle of
0.20
Level sdz
F
E
D
C
B
A
9
8
7
6
5
4
3
2
1
0.10
C
0.00
a
co
-0.10
0.0567045
0.0534091
0.0501136
0.0468182
0.0435227
0.0402273
0.0369318
0.0336364
0.0303409
0.0270455
0.02375
0.0204545
0.0171591
0.0138636
0.0105682
-0.20
-0.2
-0.1
0.0
0.1
0.2
0.3
Y/Span
Figure 7-11: Standard deviation of vertical velocity at X/Span =-0.66.
attack is +5.36 degrees (side orientation/coarse grid).
Angle of
0.20
Level sdz
F
E
D
C
B
A
9
8
7
6
0.10
0.00
a.
C
5
4
3
2
1
-0.10
0.0746023
0.0701136
0.065625
0.0611364
0.0566477
0.0521591
0.0476705
0.0431818
0.0386932
0.0342045
0.0297159
0.0252273
0.0207386
0.01625
0.0117614
-0.20
-0.2
-0.1
0.0
0.1
0.2
0.3
Y/Span
Figure 7-12: Standard deviation of vertical velocity at X/Span =-1.28.
attack is +5.36 degrees (side orientation/coarse grid).
97
Angle of
0.15
Level sdzn
F
E
D
C
B
A
9
8
7
6
5
4
3
2
1
0.10
0.05
rC
0.00
0.
CL)
i
-0.05
0 178636
0 167273
0155909
0 144545
0 133182
0.121818
0110455
0 0990909
00877273
0.0763636
0.065
0 0536364
0 0422727
0 0309091
0 0195455
-0.10
-0.20
-0.10
0.00
Y/Span
Figure 7-13: Standard deviation of vertical velocity at X/Span
-1.67.
A]ngle of
attack is +5.36 degrees (side orientation/coarse grid).
0.070
Level sdx
0.060
0.050
0.040
C
. 0.030
0.020
0.010
F
E
D
C
B
A
9
8
7
6
5
4
3
2
0.0577841
0 0555682
0.0533523
0 0511364
0 0489205
00467045
0 0444886
0.0422727
0.0400568
0.0378409
0 035625
0 0334091
0 0311932
0.0289773
1
0 0267614
0.000
-0.12
-0.10
-0.08
-0.06
-0.04
-0.02
Y/Span
Figure 7-14: Standard deviation of streamwise velocity at X/Span =-0.66. Angle of
attack is +5.36 degrees (side orientation/fine
98
grid).
_
_
IQ
~~~~~~~~~~3/~~~~~~~
0.040
Level sdz
F
E
D
C
B
A
9
8
7
6
5
4
3
2
1
0.030
£~
C
0i--
~
~' ~-
~A
9
~ '~
0.020
~
6¢
~
7i~~~~~~
8~~~~
I
0.010
0.000
_
·
._._.____...__
-0.050
_ __71_·
· ^
-0.040
-0.030
Y/Span
·
-0.020
·
_____
1
·
,.
-0.010
.
.
0 103125
0 0971591
0.0911932
0 0852273
0 0792614
0.0732954
00673295
0.0613636
0 0553977
0 0494318
0 0434659
00375
0 0315341
0.0255682
0 0196023
·
0.000
Figure 7-15: Standard deviation of vertical velocity at X/Span --0.044.
attack is +5.36 degrees (side orientation/fine grid).
Angle of
0.070
0.060
Level sdz
F
E
D
C
B
A
0.050
0.040
9
8
7
6
5
4
3
2
C-
i
0.030
0.020
0.010
1
0 116477
0109318
0 102159
0 095
0 0878409
0 0806818
0 0735227
0 0663636
0 0592045
0.0520455
0 0448864
0 0377273
0 0305682
0.0234091
0 01625
0.000
-0.08
-0.06
Y/Span
-0.04
-0.02
Figure 7-16: Standard deviation of vertical velocity at X/Span
attack is +5.36 degrees (side orientation/fine grid).
99
=-0.34. Angle of
0.070
Level sdz
0.060
F
E
D
C
B
A
0.050
0.040
9
a
8
0.030
7
6
5
0.020
4
3
2
0.010
1
0.141307
0.132614
0.12392
0.115227
0.106534
0.0978409
0.0891477
0.0804545
0.0717614
0.0630682
0.054375
0.0456818
0.0369886
0.0282955
0.0196023
0.000
-0.12
-0.10
-0.08
-0.06
-0.04
-0.02
Y/Span
Figure 7-17: Standard deviation of vertical velocity at X/Span =-0.66. Angle of
attack is +5.36 degrees (side orientation/fine grid).
15
Level sdy
F
E
D
10
C
B
A
9
8
7
C
co
C,)
6
5
5
4
3
2
1
0.190188
0.178375
0.166563
0.15475
0.142938
0.131125
0.119313
0.1075
0.0956875
0.083875
0.0720625
0.06025
0.0484375
0.036625
0.0248125
0
-20
-15
Y/Span
-10
-5
Figure 7-18: Standard deviation of spanwise velocity at X/Span =-0.66. Angle of
attack is +5.36 degrees (top orientation/fine grid).
100
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102