Analytical prediction of wall effect on fully cavitating lifting foils, using nonlinear theory

by

Elwyn S. Baker

DEPARTMENT OF THE NAVY

NAVAL SHIP RESEARCH AND DEVELOPMENT

CENTER

BETHESDA, MD. 20034

ANALYTICAL PREDICTION OF WALL EFFECT ON FULLY CAVITATING LIFTING FOILS, USING

NONLINEAR THEORY

APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED

TABLE OF CONTENTS LIST OF FIGURES 11 Figure Figure 1 2 -Flow-Field Boundary

Mapping Planes- -Choked Flow in an Arbitrarily

Shaped Tunnel, From Reference 3

Page

5

Figure 3 - Mapping Planes--Nonchoked Flow in a Straight-Sided

Tunnel, From Reference 3 16

Figure 4 - Lift and Drag Coefficients versus Cavitation Number

at Various Angles 34

Figure 5 - Percentage Reduction of the Predicted Lift and

Drag Coefficients Due to Tunnel Wall Proximity for the

CaseH/L=4 .

40

Page

ABSTRACT. 1

ADMINISTRATIVE INFORMATION 1

INTRODUCTION 1

WALL EFFECT THEORY 6

CHOKED FLOWS 6

FINITE CAVITY FLOWS 15

Equation Set 15

Changes to Equations 21

EXPLANATION OF THE COMPUTER PROGRAM 31

DISCUSSION OF RESULTS 60 CONCLUS IONS 69 RECOMMENDAT IONS 70 AC KNOWLEDGMENTS .71 APPENDIX A - w INTEGRAL 73 0

APPENDIX B - INTERPOLATION FORMULA . 81

APPENDIX C - COMPARISON WITH 1955 THEORY OF WU 83

APPENDIX D - COMPARISON WITH LAROCK AND STREET THEORY 93

APPENDIX £ - PROGRAM NOTES AND LISTING 99

APPENDIX F - COMPUTED DATA POINTS 129

REFERENCES . 134

Page Figure 6 - Choked Cavitation Number versus Incidence Angle and

Tunnel Wall Spacing at Various Angles 42

Figure 7 - Free Streamline Location versus Tunnel Wall Spacing

Under Various Conditions 45

Figure 8 - Pressure Coefficient Distribution versus Tunnel Wall Spacing Incidence Angle = 10 Degrees, Cavitation

Number = 0.4 48

Figure 9 - Comparison of Pressure Coefficient Predictions from Wu's Theories and Larock and Street's Theory for the Flat Plate in an Infinite Stream at Incidence

Angle = 10 Degrees 49

Figure 10 - Comparison of Streamline Locations Flat Plate

Infinite Stream c'. = 10 Degrees Wu's 1969 Theory

Roshko (or Transition) Wake Model 50

Figure 11 - Comparison of Streamline Locations Flat Plate

Infinite Stream ci. = 10 Degrees Larock and Street

Theory Tulin Single-Spiral Vortex Wake Model 51

Figure 12 - Comparison of Streamline Locations Flat Plate

Infinite Stream ci. = 10 Degrees Wu's 1963 Theory

Wu 1963 Wake Model 52

Figure 13 - Lift Coefficient versus Cavitation Number for a Flat

Plate Hydrofoil at c = 15 Degrees, Infinite

Stream--Comparison of Various Theoretical Predictions 53

Figure 14 - Comparison of Predicted Streamline Locations with Experimental Data Circular Arc Hydrofoil,

2Y = 16 Degrees, c = 10 Degrees, H/L = 6,

a=0.26

54

Figure 15 - Comparison of Predicted Coefficients with

Experimental Data 55

Figure 16 - Sample Cross Plot for Cavitation Number 59

Figure 17 - Comparison of Mapping Planes between Wu's 1955 and

1969 Theories 85

Figure 18 - Sketch of Common Parameter Plane Cut Values for Various Theories for Flat-Plate Hydrofoil in

Infinite Stream 94

Figure 19 - A Mapping Conversion from Common Parameter Plane of Larock and Street Theory to that of Wu 1969

Theory 96

Figure 20 - Comparison of Common Parameter Plane Cut Values between Wu 1969 Theory and Larock and Street Theory--Flat

Plate, = 10 Degrees, Infinite Stream 98

Thicknesses of upstream stream above and below dividing streamline, respectively

General line integral Imaginary part of

NOTAT ION

b

C

C

K Parameter in Larock and Street theory C

L Arc length of foil; segment of real axis

LE Leading edge

in Natural logarithm of real positive quantity

log Natural logarithm of complex quantity, principal

value

M Image point used in mapping

m Algebraic parameter on axis

Pc Cavity pressure, not normalized

p Pressure at upstream infinity, not normalized

A Real constant scale factor used in complex

potential mapping

Al Power series coefficients

A,B,B',C,C',D,D',E,E' Image points used in conformal mapping

a,b,b',c,c' Algebraic parameters on axis

CL, CD Lift and drag coefficients based upon chord length

and upstream velocity

Pressure coefficient (P.-P,) / _p

ds Infinitesimal element of arc length

dx, dy Components of ds

d1, d2 Thickness of upper and lower downstream streams

e Base of natural logarithm system 2.71

f Complex potential f = + i

f, g General real functions

h Thickness of upstream stream

hD, hD, Heights of leading and trailing edges of foil in

unit tunnel

h0 or hL Height of chord centerline in unit tunnel

h1, h2

I Im

;

q Real magnitude of normalized velocity vector,

called "speed"

R General radius

S Total arc length of foil

SP Stagnation point

S(), S(')

Arc length as a function of parameter plane axis ,

transformed axis '

s Arc length

TE Trailing edge

t Common parameter plane in Larock and Street theory

tb Parameter in Larock and Street theory

U Upstream velocity magnitude--normalized on the

free-streamline speed in the nonchoked case only

U Speed at upstream infinity

u,v Component magnitudes of complex velocity w

V Downstream velocity magnitude--same normalization

as u

w Complex velocity, u - iv, or qe_10

x Real axis of physical coordinate plane

y Imaginary axis of physical coordinate plane

z Complex physical plane variable z = x + iy

Inclination angle of foil chord line with respect

to the horizontal direction

-Asymptotic inclination angle of curved wall at downstream infinity and upstream infinity

ct(s) Local wall inclination angle as function of arc

length s along the wall

Local wall inclination angle as function of

parameter plane axis.

s(s) Local inclination angle of foil tangent as a

function of arc length along the foil, measured with respect to the chord line, positive clockwise; characterizes shape of foil

Same quantities as functions of parameter plane

axis , transformed axis '

One-half the included angle of a circular arc

Normalized pressure coefficient (C + a)/(l + c)

Small quantity

c. Complex parameter plane variable = + Imaginary axis of complex parameter plane

0 _{Real angular direction of normalized velocity}

vector, positive counterclockwise, called direction

00, 01 Components of 0

Real axis of complex parameter plane

(x) Values of on axis as a function of transformed

variable X

Point on axis

IT _3.14

a _{Cavitation number (ç_P)/ p U2}

aCH Chokd cavitation number or blockage constant

a _{Cavitation number based on minimum tunnel wall}

wall

pressure

T Logarithm of inverse speed

T0* _{Nonsingular component of}

T0, .t1 Components of T

General contour integral

Velocity potential; general complex function Variable of integration corresponding to

trans-formed values of F, > 1

Point on axis

Angular parameter, corresponding to positions

along the axis from c to c'

Values of x corresponding to the positions of

a,b,b'on

Two-dimensional, irrotational stream function Square root of complex quantity, branch to be

determined

Square root of positive real quantity, sign positive Superscript, limiting value on boundary from the upper half plane

n a wh i is ne s m coc rat by is wh the on me ve

th'

x

V \i I Aa _{"b' "b} 11) ( )l/2 +

e

ABSTRACT

A Fortran computer program, compatable with the CDC 6000 series, has been developed to evaluate the effect of a

closed-jet tunnel-test section on the force coefficients of

arbi-trarily cambered, sharp-nosed, fully cavitating, two-dimensional

hydrofoil sections. The numerical solution to this prcblem has been obtained using the nonlinear theory developed by

T.Y. Wu in 1969. The theory employs an open-wake model, commonly called the Roshko or transition-wake model, to

represent the cavity downstream of the foil section. Results

of the program include predicted force coefficients,

stream-line shapes and pressure coefficient distributions. Some

comparisons are made with the predictions of two other nonlinear theories available for the infinite-stream case only.

ADMINISTRATIVE INFORMATION

This project was authorized and funded under the General Hydro-mechanics Research Program and partially by the In-House Exploratory De-velopment program.

INTRODUCTION

The past decade has seen remarkable progress in the development of

naval hydrofoil craft. These ships have hulls of the planing boat form,

which in the cruise condition rise up out of the water. The ship weight

is then supported by fully wetted wings of the aerofoil type running

be-neath the water surface. These wings are characterized by a relatively

small thickness ratio when compared to aircraft wings, by low lift coefficients, and by high efficiencies as measured by the lift-to-drag

ratio. The speed with which such wings can move through water is limited by cavitation erosion of the material, the inception of which phenomenon

is characterized by low values of the dimensionless cavitation number,

P - Pc

1 2

p _U0

where Pc, the cavity pressure, is generally taken as the vapor pressure of

the fluid. At high speeds and low values of a, small vapor bubbles appear on the suction side of the hydrofoil, which upon being swept into a higher

pressure region of the flow, can collapse and damage the material of the wing.- The design of fully wetted wings is therefore characterized by an attempt to make the thickness and camber ratios as small as possible consistent with structural requirements so that the lift coefficient, and hence the pressure difference across the foil, which encourages the growth

of cavitation bubbles, is small.

An alternative approach to the design of hydrofoil wings is to

encourage the growth of these bubbles to the point where the entire suction

face of the hydrofoil is covered with a large bubble of fluid vapor. The

resulting design is referred to as being supercavitating or fully

cavi-tating. The leading and trailing edges of such foils are generally

sharpened to encourage the separation of the fluid at both points. The

most notable example of such.a foil is the surface-piercing forward foil

on the Canadian Ship BRAS D'OR (renamed FHE-400). These foils are

characterized by somewhat higher lift coefficients and lower lift-to-drag ratios than their fully wetted counterparts; however, they do not suffer cavitation damage at high speeds.

The section shapes-for supercavitating foils are generally tested in

-cavitation tunnels. In contrast to the case of fully wetted foils, for which the tunnel correction method is well established, there is no simple method of calculating the effect of the tunnel walls on measured force data

of supercavitating sections. Indeed, the calculation of two-dimensional

wall effect on fully-cavitating lifting hydrofoils using nonlinear theory has hitherto only been attempted for flat-plate hydrofoil sections in choked (minimum cavitation number) flows.1

The computational difficulties involved in considering the cambered

foil case invite the use of linearized theory. However, in several cases

the linearized theories have been shown to be in error;2 in particular the

D.K. and Z.L. Harrison, "Wall Effects in Cavity Flow," California Institute of Technology Hydrodynamics Laboratory Report 111.3 (Apr 1965).

2Wu, T.Y., "A Note on the Linear and Nonlinear Theories for Fully

Cavitated Hydrofoils," California Institute of Technology Hydrodynamics Laboratory Report 21-22 (Aug 1956).

1: ti p tJ n( b di U] ac i

1

cc iT di

linearizing assumption of small perturbation velocities is not valid near the leading edge of the foil, where, because of high loads, an accurate prediction of the free-streamline shape, which controls the thickness of

the foil, is necessary for good design. The report by Wu et al.3 offers a

nonlinear theoretical model of the problem; the purpose of this work has been to convert the model into a computer program to numerically evaluate

the wall effect on arbitrary two-dimensional hydrofoil shapes.

While a three-dimensional wall-effect theory would most certainly be preferable to the two-dimensional theory considered here, the mathematical difficulties involved in analyzing three-dimensional cavity flows (partic-ularly the difficulty of applying exact boundary conditions at an unknown cavity boundary location) have so far limited developments in this

direction. It is to be hoped that the two-dimensional theory will give

satisfactory results for flows over high-aspect-ratio hydrofoils. In

addition to two-dimensionality, the present theory assumes that the flow is irrotational and inviscid and that separation occurs from the sharp

leading and trailing edges of the foil. Therefore, the lift-and-drag

coefficients derived from it are hydrodynamic coefficients only and do not-include viscous effects on the flow, such as skin friction and the

diffusion of vorticity.

In real cavity flows the steady upstream flow degenerates into turbulence at some distance behind the obstacle; therefore, representation of the flow downstream by means of steady-state, discrete streamlines can

be only a poor approximation to the true wake flow. Fortunately, the

hydrodynamic forces on the obstacle appear to depend primarily on the up-stream flow, where the potential flow representation is much more reason-able.

Several wake models have been developed which approximate the

vis-cous wake by a perfect fluid

flow.

These can be divided into two classes:

(1) closed-wake models, in which the free streamlines join at some down-stream point to form a closed cavity, and (2) open-wake models, in which

3Wu, T.Y. et al., "Wall Effects in Cavity Flows," California Instutute of Technology Report E-lllA.5, Chapters 7 and 8 (Apr 1969).

3

n

the free streamlines remain separated until downstream infinity. Since

most potential flow theorems take their simplest forms when applied to simply connected regions, which requires that the flow be "enclosed"

within a single continuous contour forming the boundary of the flow region, it is natural in considering the wall effect problem to first attempt an

open-wake cavity representation. Then the walls and the hydrofoil-free

streamline boundary can both be considered as a part of a single boundary. The up- and down-stream streams can also be considered as part of this boundary since the direction of the flow in them is known and constant, as

it is along a flat wall.

3

In the open-wake model used by T.Y. Wu, he assumes that the wake is divided into two regions: a near wake immediately behind the obstacle

and a far wake further downstream. In the near wake the pressure remains

constant at its minimum value anywhere in the flow field; in the far wake

it gradually regains part (but not all) of the upstream pressure. The

point at which this transition to the far wake begins is that at which the free streamlines have become parallel to the walls and have the free

streamline (maximum) speed. These bOunding streamlines continue downstream

from this point parallel to the walls but lose part of their speed to

account for the increase in wake pressure. If the velocity of a fluid

element is expressed as a magnitude and a direction, it can now be seen that one component of the velocity is known everywhere along the boundary

shown in Figure 1. The velocity magnitude is known (constant). on the free

streamlines, and the direction of the flow is known on the foil, at up and

downstream infinity, and on the walls. This is a sufficient specification

of boundary values for a potential flow solution in a simply connected

region.

In prograimning this problem, reliance has been almost entirely on

the theory developed by Dr. Wu, and only those changes have been made which were necessary for the numerical calculation of the results and to apply

q=U p= p00 q= 1 _{LE (LEADING EDGE)}

8= ir-fl(s)

SP (STAGNATION POINT) B = . (S) TE (TRAILING EDGE) q= 1 q-V

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

d1 q=V 8 ci FAR WAKE P >P NEAR WAKE p=Pc

d - (c-a) (c-b) (c-b')

using the Schwartz-Christoffel transformation, where A is a real constant

(scale factor). To map the velocity field onto the same plane, the velocity is expressed as the logarithmic hodograph variable

= log--=

T +

i.O

where

t =

ln V/q, normalized on the free streamline velocity V, which

equals the downstream velocity for infinite cavity choked flows (the value

of V is not normalized). This variable is decomposed into two parts, w0

and c

w=w0+W1,

W0T0+ie0,

()1T1+iO1

4Holt, M., "Basic Developments in Fluid Dynamics," Vol. 2, Academic Press, New York (1968).

WALL EFFECT THEORY

CHOKED FLOWS

The theory of wall effect for arbitrarily shaped two-dimensional bodies in choked (minimum cavitation number) flows was given by Wu in

Reference 3, Chapter 7. (See also References 1 and 4 for more

infor-mation on this problem.) In this presentation the shape of the tunnel

walls may also be arbitrary. Since ideal choked flows are characterized

by infinitely long cavities, the choice of a wake model does not enter this problem; both the free streamlines extend to downstream infinity, and the speed along them remains constant at the upstream value.

A summary of this theory follows, using the Wu's notation shown in

Figure 2. The complex potential plane f = + i is mapped onto the upper

half of the parameter plane = + i by

Figure 2 - Mapping Planes--Choked Flow in an Arbitrarily Shaped Tunnel, From Reference 3

7 ity f-PLANE A

L1'

IJh.=Vd1 B' B E U B A U' B' ' =-Uh=-Vd B'

Following the boundary of the flow region counterclockwise, one

sees that at every point one component of the _{velocity is known.}

On the free streamline the velocity magnitude is given as V; on the foil surface

its direction is specified by s(s); _{on the wall surface, by ce(s).}

Across

the upstream and downstream flows _{the direction of the velocity is uniform}

at angle c, and its magnitude is U _{and V, respectively.}

To correspond to this decomposition, _{the boundary conditions}

on

the velocity are divided _{as follows:}

parameter plane +

<-1

11=0

+

-l<<b,, b'<<1 11=0

+

b<<b' n=O

>+l

ii=0

w0 velocity 00 = ir ₊ = 0

T00

₀₁ 0 =

The superposition of _{and w1 gives a velocity field in terms of}

con-sistent with the- conditions along the physical flow boundary. _{Once the}

f-plane and the w-plane are expressed in terms of a common parameter plane

, they can be combined to give the physical

z-plane and the physical

parameters of the problem.

Wu gives the solution to this boundary value problem for as

= = log 2 1/2) 1/2 C + (C -1)

-(-

(2))l/2

1b' [cL() - 0()] d b _(c-C) 3/(-b) (b'-)

(2)

w1 velocity 01 = 0 = 0 = ct() -01 =.0 +co

-

r

l

I

_(c-C)

(2) 5 Flow Eng i pro and wher (1) _know

where w0(C) must be analytic (throughout _{the 'upper half plane}

except at

°°) and satisfy the previously described _{boundary conditions.}

More details

of the derivation of this _{expression will be found in Appendix A.}

Similarly,

The derivation of this may be sketched by considering the

Rieinann-Hubert problem in an upper half-plane, as summarized by Larock (l966). Supposing the imaginary part of some complex function to be known everywhere along the real axis E, the analytic function Q will be given in the upper half t-plane by

'm

Q()

=

S

d

+'

A.

-where = + ir, A. real. The statement of the problem w1 is

Where = + i01, and 00 is known from the solution of the w0

problem and depends only on the function () and not on the parameters b

and b'.

Suppose a function H() such as

H()

₌ i4-') (c-b) (c-b') (-(-l)) (4)

= /(-b) (b'-)

(2)

where the argument of the square root changes sign at the points where the

known part of becomes real or imaginary. Let

wl

- H()

5 ,,

Larock, B.E. and R.L. Street, A Nonlinear Theory for Fully Cavitated

Flows Past an Inclined Flat Plate," Stanford University Department of Civil Engineering Technical Report 64 (Jun 1966).

9 j =0 (3) L) [y, 01 = 0

>+1

<-1

n=0'

T1 = 0

-1<<b b'<<+l

=0

01 = o() -

b<<b'

on the half-plane boundary to give

Thus, it is seen that the imaginary part of Q() is known everywhere along

the axis. Hence, from Equation (3), the extension of Q into the positive half-plane may be expressed as

Im Q

Q()

=

f

d +2'A.

i j=0 1 b' ct() -d A. ?

=J

b

(-)

and the velocity field is given by

= H() Q()

=

- (-

(-b')(2-[c() - O0()J d

(-)

The power series term is eliminated by considering the behavior of w near

the stagnation point. Since Re w

= T =

log l/q, as q - 0 at the stag-nation point, T must -- c _{logarithmically there. This behavior is exhibited}

by the term of the velocity field expression and must not be changed by

the addition of T1.

Therefore, the complete solution to this Riemann-Hilbert mixed-boundary-value problem in the upper half t-plane, continuous at the transition points, is given as

plal as a cc

bow

Corn undc The by Im Q() Im Q() = = 0

-l>,

0

-l<<b,

[cL()

->+l

=0

b'<<+l

=0

Im Q(fl - -

I

b<<b'

n=0

V(F-b) (b'-) (21)

)l 2

= + i0() = w0() + q () = log (1 + 2 1/2

(21)1/2)l(

-')

(1'

1/2 +

(b')

_(2l)1

jbt

=

J

q:()

d L - __________ b

(-)

The first term may be thought of as the velocity field due to a flat

plate in an infinite stream at the ch'oked-flow conditions; the second term,

as a correction due to cambering the plate at angle ; the third term, as

a correction due to the presence of a wall at angle c.

That this expression for the velocity does indeed satisfy the given boundary conditions can be seen from the limit formula for Cauchy integrals. Consider the wall effect term w.

2 1/2 1 1 l))

jbt

-d b

(-)

under the conditions

= _{- 00(e)} when b < < b'

when-l<<b

orb' <<+1

01 = 0 when

II

> 1

The limit from the positive half-plane of a Cauchy-type integral is given

by Plemelj's formula as6'7

=7r i

-

I

\c)d

Jl )()p/i

[c() - 0()]

d 4.vp

c() d

J

L (5)

6Mushelishvili, N.I., "Singular Integral Equations," Noordhoff Groningen,

Holland, p. 237 (1946).

7Pogorzelski, W., "Integral Equations and Their Applications," Pergamon

where the term on the right is the Cauchy principal value (real valued, if

J. is the real axis).

Comparing this with the expression for w,, with L

defined as the segment of the real axis from b to b', we see that the

limit

of as

-F0, ii -' is

wl(o) = .

(b'-0)

(2)

Then taking the integrand function in Plemelj's formula as

- 0o

Va-b) (b'-)

(l_2)

and comparing with the Cauchy integral gives

= -b' [ct()

- O()]

d +-- 0-b)(b'-0)(l-02)

J

b

(o

which has the required imaginary part.

Furthermore in the range -1 < <

b, b' <

< +

1 the term

is purely imaginary; since the integral

VP 1b'

[c() -

oo()I

d

b ()

is real in this range, the expression for must be purely

imaginary,

satisfying the condition that T1 = 0. A similar argument shows that

= 0 for IEI > 1.

b'

Ic() - e0(fl

d

(o

j(-b) (b'-)

(l2)

Now with expressions being given for both the velocity

field and the

velocity potential in terms of a common parameter plane (?-plane), the

physical z-plane is derived from

or wh e foL upF anc th Th an

fr

and axi

it df = w, z = 0 at = -1 1 e° df

z()

=

_J

1

and the arc length of the foil as a function along the parameter plane axis becomes (since IdzI = dS)

S()

=

-J

) d n = (7)

where S(l) = S, the total arc length.

Using the same distance Equation (6) allows an expression to be found for hD, the tunnel position of the nose, by integrating along the

upper streamline. b / -Ia - -Ia0

i0() df

hD = d1 + Im _te 0 _{(zB_zD))}

= d1 ++

5

Im Le e

jd(8)

-1

Similar integral equations are given for the positions of the upper and lower walls with respect to the foil.

The distance Equation (6) may also be used to find the thickness of

the up and downstream streams. Let vertical distances be given by

= liii

[e0 (zz0)]

= Im

[e0+J

eW

--d]

Then, supposing that the streams up and down stream asymptotically approach

an angle of ct0, at the upstream infinity we have - a, and by definition from the Riemann-Hilbert problem w = ln V/U + ia0, so

a+c d -1 A

[rO

ii4 5

_{(-a)(-b)(-b')]}

Ah=Im

e

CO

_a-c

replacing w by its value in the limit gives

13

n>

0 (6)

a+c

[-ict0

(in-4j_+ icx0)

j

d = - A e a-c (c-a) (c-b) (c-b')

]

a+c = - A . 1

im[iiml

1 _d (a-b)(a-b')

cO j

(c-a)

]

tr a-c

= - A/U 1 un [log(a+c_a) - log (a_c_a)]

(b-a) (b'-a)

1

liii [iogf-!-'l= log (-1)

L

\-i]

(b-a)(b'-a)

Take the negative log branch to give the upstream thickness as

Air Air - A/U A d2 = + (b-a) (b'-a) h (b'-b)(b'-a) U(b-a)(b'-a) U(a-b)(b'-a)

The negative sign which would naturally appear in this expression were it not for the choice of logarithm sign expresses the sense of the boundary-a positive c-direction corresponds to boundary-a jump from the upper to the lower

wall. In a similar fashion, the thicknesses of the downstream streams are

r.

I v

.\

b+c I -ic

lnv + lcL0)

. 1 1 1

d --AImie

e

limi

L

b-c

(-a)(-b)(-b')

A it =

(b-a) (b-b')

wherein the positive logarithm sign has been used.

Thus, when the functions ct() and () are given, where is the real

axis of the common parameter ?-plane, a sufficient number of equations is given to determine all the physical parameters of the flow, including the

shape of the resulting hydrofoil, which may be found by calculating s()

Fl0 Tec TN tat (J u and (S fin cal the FIN the flo cas Ref rep Fig edg t Un fun ang vel ye 1 and

e

ea 1

and eliminating the parameter _{between that function and () to give}

s(s). _{However, if it is desired to specify s(s), then it is necessary to}

find some function s() such that _{the substitution 8(s())}

= () and the

calculation of s() results in a generated function which _{is identical to}

the original s(). _{This may be done by integral iteration.8}

FINITE CAVITY FLOWS

Equation Set

For the case of cavity flows with _{a finite cavity, we specialize.}

the method used by Wu in Chapter 7 of Reference 3 in solving the choked

flow case by adopting his boundary conditions _{used in the solution of the}

case of finite cavity flow over a flat plate with wall effect, _{Chapter 8,}

9,10

Reference 3. This corresponds to the use of the Roshko _{wake model to}

represent the finite cavity. _{The mapping planes then become those shown in} Figure 3. _{The physical coordinate system is centeredon the foil leading}

edge, with its x-axis along the chord. _{We choose the case of a straight}

tunnel with parallel walls at angle a to the chord of the foil, so that the

function a() reduces to a constant a. _{The upstream velocity becomes U at}

angle a; the downstream velocities, V _{at angle a. Having normalized the}

velocity field on the free streamline velocity, _{taken as unity, the}

velocities have the relation

U<V< 1

and the cavitation number is given by

1

a 1

U

8Wu, _{T.Y. and D.P. Wang, "An Approximate Scheme for the Theory of Cavity}

Flows Past Obstacles of Arbitrary Profile," California Institute of Technology Report 111-1 (Jul 1963).

9Roshko, _{A., "A New Hodograph for Free Streamline Theory," NACA} TN 3168 (1954).

T.Y., "A Free Streamline Theory for Two-Dimensional Fully Cavi-tated Hydrofoils," California Instutute of Technology HL Report 21-17

(Jul 1955).

z - plane

- plane

Figure 3 - Mapping Planes--Nonchoked Flow in a

Straight-Sided Tunnel, From Reference 3

f - plane

n -'

p.-r

I4 3 o 9

In the choked flow case the cavity extended downstream to infinity, and consequently the points C, C' at which the free streamlines became parallel to the walls coincided with the points B, B' at downstream

in-finity. For flows .with a finite cavity, the open-wake model represents

the real physical case of convex free streamlines meeting in a turbulent wake as streamlines extending to downstream infinity from the points at

which they become parallel to the walls (C, C'). The velocity along these

lines then varies in such a manner that the net pressure effect on the foil

ahead of them remains unchanged. Thus, for finite cavities the streamline parallel points C, C' are some finite distance behind the foil, marking

the transition from the near wake, where the free streamline speed is

unity, to the far wake, where the streamline velocity approaches V.

Conse-quently, as the boundary of the flow region is traced around the path

shown in Figure 1, we let = c at C and = c' at C'. The statement of

This may be written more concisely as

- _ - i e =

ii +

B()

T0

+C<<+c'

c'<<+l

r=O

17 +

T10

H _{the Riemann-Hilbert boundary value problem corresponding to this case}

be-comes

D

Location Segment Parameter Velocity + -4 _{Foil Forward of} Stagnation Point ED

n= 0

-<<- 1

0 = t + Upper Streamline DC

- 1<

< C ci ci Upper Wake CB

c<<b

0=c

-4 I) _{Upper Wall BA} b

<<

a

0=ch

Lower Wall AB' a < < b' 0= c

Lower Wake B'C' b' < < c'

0=ct

Lower Streamline C'D'

C' <

< 1

Foil Rearward of

Stagnation Point D'E'

1<

which, remembering the principle of superposition, used in the last w

section, can be decomposed into i

C

<-

1 - 1 < < c, c' < < + 1

C <

< C'

01=0

t0=0

T0 = 0 _{01 = a}

-01=0

By comparison with the choked-flow case we see that the statement of the Riemann-Hilbert problem would be the same if the parameters c and c'

replaced b and b', and the constant a replaced the function ct(E).

There-fore we replace b, b' with c, c' and adopt Wu's solution for the previous Riemann-Hilbert problem, giving

W

W + ()

= ln--+-i0 0 1 q 001 U w0 log [ 2 1/2] 1 2 1/2

[f1

()

d -

j()( -1)

1 =

_{+( -1)}

] jf ( -1) -2 1/2 (11) C r ) d C' a-00( 1/2 (12) 1

Wl =

[(c-c)

(c-c')

(2l)]

1/2

()

[-c)

(c'-)

(i2)]

C

The condition at the stagnation point becomes

Jc,

oo() d

ct a - d C

0,or a-

(13) c

-I

d c

_V(-c)

(c'-E)

(2)

A theoretical explanation of this condition, necessary because of the singu-lar behavior of w at infinity, may be found in Reference 7, pg. 654.

Now we apply the boundary conditions used in solving the flat-plate

t

case. The complex potential expression remains unchanged, as do the

algebraic stream thickness relations, see Reference 3, page 40. However,

1

a

- ..

we must now require that the downstream streams have equal velocities at

infinity, to correspond to a physical situation of a closed wake with a

constant under pressure, or

= T1(b')

= in _(i-)

The upstream velocity is calculated from

1

T1(a) = in

and m, which is the point in the c-plane corresponding to the minimum

velocity position on the wail, is calculated from the maximum point of the

curve.

T1(m) = maximum , c < m < c'

We choose our tunnel position parameter as hL, the height of the

midchord point of the foil, and require this to equal one-half in a tunnel

of unit height. Then hL = (hD + hD')/2, and hD and hD, may be determined by integrating the distance formula along the upper and lower streamlines.

There is now a sufficient number of equations to determine the

physical characteristics of the problem. The solution is expressed in

terms of five unknown algebraic parameters a, b,.b', c, c' and one unknown

function s()/S. For inverse problems these parameters are arbitrarily

chosen, and the foil shape is calculated from them; for direct problems it

is necessary to iterate for s()/S. The full equation set is

=+in

w=ifl-+iOW0+W1T.10

'S

= log 1/21 - _ + (;2l) iT

2(

I-i

J)

()

-

1

(-)

w) =

[(c

(c-c')

(21)]

jC?

-

d C

(-)

19 f 1) 3)

a-C, d -c) (c'-)

(2)

TJ(b)

= T1(b')

= in j T1(a) = in = in (1 + ci) df -A (c-a) (c-b) (c-b') fb'-a\ (a-b) = (U/V) Ib'_b) d2 = (U/V) (b'-b) A =- U (a-b)(b'-a) _{hD = hD,}

0() d

s() =

-s=-

-hD = di - A hD, = d2 - A

:1:

(.1

+ sin (0(c) -a) d

J1

_{(c-a) (c-b) (-b')} sin (0(c) -a) d (c-a) (c-b) (c-b')

T()

df d e d j e d + / d Arclength of foil df ds() = d(s()) = eT _d

-T()

_P()=l-q()2=

_p

1 +ci

Normalized Pres sure Coefficient He ye no mo ar

Si

pa ne th Ch w0 si 0

p

Lift = Drag = Lift Drag

CL=

p/2

U2 chord CD =

p/2

U2 chord

Here all distances are normalized on a tunnel height of unity; all velocities, on a free streamline velocity of 1.

Changes to Equations

The previously described equations as written provide an exact, nonlinear description of the wall-effect problem using the Roshko wake

model. However, for practical applications and computer applicability they

are unsatisfactory for two reasons: (1) the integral expressions for w are

singular; and

(2)

at the low-incidence angles of practical interest the

parameters c, b, a, b', c' approach each other and +1.. Therefore, it is necessary to investigate various transformations of the equations, with

the object of eliminating the more serious of these drawbacks.

Changes to w0(). First consider the expression for = + i 00

= _{+ in}

= log

+

(2i)

1/2]

_(2i)

1/2

(l

-8(e)

d - J1

I

Since all the physical quantities of interest in the problem are defined

on the boundaries of the flow field, which correspond to the real axis of the

parameter plane, Equation (11) is of primary interest when ii + 0

Further-more, it appears from the statement of the Riemann-Hilbert problem for

21 Ep

cos

()

cos

ds d ds - sin

₍₎

sn

p 2

w0() that w0 is purely imaginary over the range +

-l<<+l

n=O

Therefore, for < 1, i = 0+ 1 / 2\ log [

+.(2l)h/2]

= log +

(2l)

_{) = log (} +

On an Argand diagram it is seen that

and

i.e . . -1

log (e ) = 10 = 1 CO5

so. that for < 1, r =

log ( 2

l/2\

+(-l)

)=icos'

For the integral term, it is desirable from a numerical standpoint

to avoid the integration to infinity. Consider the change of variable11

= - sec (c'ir) = 0

lI >

1

then = - 1 '=0 leading edge

= * ' = 1/2 stagnation point

= + 1 '

= 1 trailing edge

2 1/2

The square-root coefficient (t -1) was defined in such a way that

,, 1/2

1

0

-(-l)

+ for complex ; therefore, as 11 -'- O', we must take the

negative real branch, if < -1, and the positive real branch, if > +1.

If j < 1, n = 0, we take the positive root. The term within the

1Abrainowitz, M. and LA. Stegun, "Handbook of Mathematical Functions," National Bureau of Standards, p. 889 (1964).

integ chang see ate as fc This Simi Car roo Ri e +

_e0

II

_{< 1 , real}

integral is always real and takes the plus root also. With this

change of variable the integral expression can be written as

-1 +

J

')

d'

I<l

(14) 0 +

see Appendix A. This integral is nonsingular but still difficult to

evalu-ate as -' ± 1. Therefore, we subtract out the near singularity at - + 1

as follows:

(sb') d' Cl

I)

+ (l) -

(l)

I

-I

d'

1 + cos ('ir) 1 + cos (q"ir)

1 (c') - (1) = I

d'+(l)

I J0 _{1 + cos}

(4'ii)

J 1 + cos ('ir) Cl (ct') - (l) (l) = I

d' +

1 + cos ('ii)

This gives an expression for 00 as

1

(t)

-

(1)

d'

₊

_(i)

_I<'

(15)

oo()

= cos() +

1 + cos ('r)

Similarly, we subtract out the near singularity at -1 and get

1

-=

cos'()

+

j

1 + cos (4'ir)

d' +

(0) F1<1

Care must be taken in numerically evaluating the integrand and the

square-root term because of the subtractions of nearly equal numbers. FrOm the

Riemann-Hilbert boundary conditions, the limits of 00 are

23

liiii

O()

= (1)

++

1

urn U = it .'- (0)

which are seen to be the correct limits of Equation (15).

When

+

such that > 1, r O, the conditions of the

Riemann-1-Lilbert problem state that must have a real part T0 and an imaginary

part equal to () or it + fit _{+ )\} =

T0()

+ or ) II>i

-'-\

)

/

Unfortunately the integral is now singular. We consider the loga.rithin term

first and as let

= -

sec ('ir)

/

l/2\

I

I

log i +

(2.i)

) -

log

\-

sec ('ir) +

sign('- 1/2)

(-I I l/2\\.

Re log + ç2_l)

₎₎₌

ln sec ('ir) + tan ('ir)

11

= in

_{tan (T--') I}

where we have remembered the sign convention that

2 1/2

( -1) - as - complex infinity

and used the trigonometric identity,

secO+tanO=tan(ir/4+U/2)

For the singular integral term we again make the change of variable

sec 'TT in the variable of integration and also in the limit value of

-3. _Ti

II

> 1

= -

sec (40trr)

and then substitute into the integral form of the w0 expression shown in

Equation (14) sec

)2)

(16) us i ho of

i tan however,

C1

1/2 (- sec

(oI)2_1)

01

cos (0tir)

using the same square-root sign convention. This integral is singular when

'V = Subtracting out the singularity in the conventional manner leaves

1 (c1t) d

Jcos ('ir

[.1

_{[(t)(qot)] d4'}

IJr

cos ('iT)

lo

11-L

L c°s (0'ir) dO

d'

cos (')

- cos 0 X=ii

r

/ 2 1,2 _\ 1 I 1 '1 ((sec

°o4

tan + 1 - sec 00

21og

₂

[(sec

o_l)

J

\(sec

₀₀₁₎

tan-i - 1 + sec 00

0

= ± cotan 0 [log(l) - log (-1)] = ± i cotan

Since for the real part T0 we are only interested in the real value

of the integral, we ignore this purely imaginary term and let

25 cos

1')

d'

1 + (- sec

cf'7T)

cos

+ (qo')

d'

J

₀₁

cos (qr) cos ("ir) 0 < 00 < 71

0<0<11

0 = (17) (18) =+ tan (07r) rm

H)4)

16) e £ in

Consequently the expression for is now

To.

)=

in tan (1r/4+ (Tr/2)o')ll + sin (q0tTr)

where the square-root sign convention is reflected in the sign of the sine

term. This integral is nonsingular but again involves a difficult limit

as

' '

during the integration.

Changes to w1(). Again we are interested in evaluating the expression for

on the boundaries of the flow field, where the physical flow quantities

are of interest and which correspond to the real axis of the parameter. plane.

[(c-c) (c-c') (2l)]

1/2

Consider first the range

+

-3.E; n-'-O

C<<C'

and the change of variable'1

.2x

= c + (c'-.c) sin (-f

which we make in both the square-root coefficient, using

x0

and in the

2 1/2

variable of integration, using X. The term {(l-c)(-c')( -1)] as

-3.

;, n -'-

becomes

C <

< C

whic 1/2 2 1/2 [1-c ] 26 [c

- 8()] d

(-)

h-c)

(t)

(2)

1 [(q:')-(q0')] d4'

cos ('ir) - cos ('ir) R C

'i', C

and c wher Re 1 1ü th I = tan

-d'

(19) cos c'7r 1 cos

-)

nie.

With this change of variable

cl-c R

2

2 2 2

(c-c) (c'-) = R (1-cos x) (1 + cos x) = R (1 - cos. _x)

2.2

= R sin _X and consequently 2 1/2 [(-c)(c'-)(1- )] =

where (x) = c + (c'-c) sin2 _x/2. The integral expression transforms similarly

'ct-c' Re = T1(X0) 2 )

Sjfl

x0 P4

-

2(x0) c' -c 2 (c-c) = R(1-cos x) sin X and (c'-) = R(l+cos x) fc'-c\ 2 ) sin x cl-c 2 x=o = _{. sin}

jXT

[_00(X))] d/dX

dx Ic'-c

[(x)-(x0)]sin

₂ (20)

C'

-c = 2 [cos X0 - cos x]

which results in an integral expression of the form iT

42(x)

$

a- ((x))] d 0 [cos x0 - cos x]

1/j2(x)

27 (21) [i - cos X - (1-cos x0)]

This' equation is singular when _{X =} x0 and is of the same form as the

inte-gral Equation (19). Therefore, the singularity can be subtracted out directly leaving

T1(X0) = sin

'C

a-00((x))

ct-Oo((x6))

cos - cos _x

Finally we multiply through the square-root coefficient to give

E/1(xo))

a-00 ((x))]

El-x0fl

(l+(X)) ('-(x)) 0

cosx0-cosx

[a- 00 [a-00((X))] /((1+o)

/(l-)

\

Tr

8i(0)

= (0-c)(0-c')

J

(l+(X))

0 ' (x)

Similarly for the range > 1 we have

T(t)

=

1

(0-c)(0-c')

$

[a-8

((x))]V(

0 (x)

sin2--sec (o') and

(x) = c + (c'-c)

=-2

Changes to a. When this change of variable is applied to the expression for a, Equation (13), it becomes

(22) (23) wh c Iig2 C

use

Chai hi where (24) To

\(o-1)

)

to

'(x)i

l-(X)

d (25)

which is nonsingular but again involves a difficult limit as x+ x0 during

the integration.

For the range

+

, -1 < < C, Ct < < +1 fl

+

0+ _{there is no}

whe

singularity to subtract, and -'

Thu:

-where T = T

+ T1.

Thus

s(00') = Air

= Air

tan (ir/4+(ir/2)

e

sin (tTi) cos @'iT) dt

(i + a cos (t

i + b cos ('

i+ b' cos ('

which shows that

e((X)) dx

o

V l2(x)

-11 _dX

SO

where

(x) = c + (c'-c)

_{sin2 4- .}

This integral is nonsingular at both

limits and has no limit points.

The denominator of the expression has an

exact expression in terms of elliptic integrals, which, however, is not

used in the computer program.

Changes to s().

The expression for s in terms of

was

s()

(J

+

J)

eT

₍₇₎

29

+ A d

I (c-a)

(c-b) (-b

')I

1 /2\

log

(+(2l)

)+ T*

(26)

O

Itan (/4+(/2) a')!

e

sec (ir) taii

'ir d'

o

I(- sec (q'ir)-a) (- sec (4'ir)-b)

(- sec ('Tr) -

b') I

(27)

s()

+

+c211/2l)

e[TO*+Tl

4)

To convert this to the foil parameter

'

let

sec

ir

T(E) = t((q')) =

to give

t(')

=

5)

2)

to

ds(4')

-(i + a cos (q'ir)) (]. + b cos ('ir)) (i + b' cos (4'Tt))

These expressions are nonsingu].ar but again have a near singularity as a, b, b' at low incidence angles.

The normalized velocity magnitude on the foil is

_[T0*(t) +

q(') =

Icotan

(ir/4+(iT/2)

4')I e

and the normalized local pressure coefficient is

*AP@t)

= 1 - q2

+

1 + c

Changes to Algebraic Equations. Since values of between c and c' are

represented by angles

_x

such that

= c

(c'-c)

sin

it would seem logical to write the parameters, a, b, b' in the same manner.

Thus a _Xa b Xb b' - Xbt where 2 Xa a = c

+ (c'-c)

sin --- etc.

Then the algebraic equations transform into

2 1

XbXa\

fXa_Xb) /Xa4Xbt\

A = - (a-b)(b'-a) =

(c'-c)

sin ( 2 ) Sfl _I\ 2 sin 2 , Sin TI 0the B th

prog

foun curv negli trip to Si vo1v trip one-i inte: the are

func

of " 1 cad

12 .

Greville, T.N.E., Theory and Applications of Spline Functions, Academic Press, Inc., New York (1969).

31 (XaXbT\ U (b'-a\ U

Sifl\

₂ )

S1flt

2 / - V b'-b) V _fXbXb,\

fX1X

2 1 2 (XalXb) sin

(XX)

U(a_b)

2 d2 =

- v

'Xb+xb) _(xht-xb\

Sifl(

_{2 2}

1

Other material relating to calculation methods will be found in Appendices

B through D.

EXPLANATION OF THE COMPUTER PROGRAM

The problem described previously has been converted to a Fortran IV program for the CDC 6000 series digital computer system; the listing may be

found in Appendix B.

The use of discrete, digital representation for smooth mathematical curves always involves some loss of accuracy, which in general becomes more

negligible as the number of discrete points becomes larger. Because of the

triple integral in the expression for s@') in thi.s problem, it was decided

to store individual functions as tabular values, so that the functions

in-volved could be integrated successivel' without resorting to a cumbersome

triple integration. Therefore, a given function is stored internally as a one-dimensional array, and individual function values are developed by

interpolating from this array)2 The parameter integrations (integrals of

the form

C'

_{f() d}

g(0)

=

.i:

c-;0

for example)

are likewise accomplished by storing individual values of the integrand function f() as arrays of values and calculating a new integrand function

Th program is divided into main program sections and functions and _requirc

subroutines. The main section of the program contains a master program In part

loop which computes the values 0:

s(4,').

The heart of the iterative

requir

process is to assume some starting function s(') and to use this to calcu-

_previot

late a better value of s(i'). This process is repeated until the quantil

difference between successive s(4')-curves is less than some specified c

-amount. Then the foil shape, lift- and drag-coefficients, and cavitation proper number are calculated.

This problem involves six unknowns, namely, the algebraic values c,

b, a, b', c' and the normalized function s('). Similarly there are six

equations which relate them to the physical parameters of the problem: foil

shape (s/S); length of foil in a unit tunnel S, which is the inverse measure of the wall spacing; foil position in the tunnel h0; cavitation

number a which is directly related to the upstream velocity U; and V1

velocity of the downstream flows, which must be equal for the upper and

lower streams if the open-wake model is to b.e a valid approximation to the

real flow with a closed cavity. Examination of the Riemann-Hilbert mapping

shows that the parameters c + c' always occur in the order

plate

first

-1 < c < b < a < b'

< C' <

+1 _{of thc} During the programing, we will arbitrarily "associate" each of these

parameters with a physical flow quantity

c a or U b' - S a - h0 b - [requirement that V = V _] upper lower c' - ci.

The function

s(4i') is

associated with the foil shape $(s/S). The direct

specification of c, b, a, c'parameters and a function

sW)

will result

after a single calculation in the output of a hydrofoil shape which has some combination of the previously described physical parameters.

)ng

es e

require more iteration from the program to search for them simultaneously.

In particular the specification of a required shape for the foil (s/S)

requires iteration for the function s(c')/S, during each cycle of which the prev&ously described algebraic quantities must be iterated, if physical

quantities are to be given as well. Besides the requirement that the

c -' c' parameters occur in the given order, it is possible to deduce other

properties of their behavior from the free streamline model. In particular

aOin°°stream

c'c

choked flow--a =

_minimuinI

(00 _{cavity length) b-'.c}

_-b'--c'

S -'- maximum

ci.

+

0°

00 -

stream

h0

- 1/2

These relationships, as well as the curves of c, c', m for the flat-plate case, are used to select starting values of the parameters for the

first iteration. The order of the calculations performed during each cycle of the main program is:

a Keeping the starting value of c constant during the cycle, use

the starting s(') to generate (4,')

b Evaluate w0()

c Search for c' so that c equals the valUe required

d Evaluate w1(), based on these values of c

and c'

e Find a value of b consistent with the starting b' so that

't1(b) =

T1(b')

f Search for "a" such

that h0

= 1/2; calculate the arc length S

g Calculate a new s(')

h Adjust the new value of c, according to the value of in relation

to the desired value. If s(') has converged, generate foil

shape, lift

and

drag coefficients.

33 c, b, a, b', c' -'- +1 b, a, b' -'-m a (1/2)(b + b') C, C :h e

CL 1.4

1.3

1.2 1.1 1.0

0.9

0.8

0.7

0.6

0.5

0.4 0.3 0.0 01

Figure 4 - Lift and Drag Coefficients versus Cavitation Number

at Various Angles

02

03

0.4

05

a

Figure 4a - Lift Coefficient versus Cavitation Number, c'. = 20 Degrees

06

07

08

09

10

0.4

0.3 CD 0.2 0.1 0.0

LIFT COEFFICIENT VS CAVITATION NUMBER

INCIDENCE ANGLE = 200

I,

27=160

_{CIRCULAR ARC}

/

4,

H/L

10

,

V4.

FLAT

PLATE 6

AA'

20 10

H/L

-7,

oo

U.

I.

4

a

a

a

U

a

U

U

U

U

U

U

U

LAT

ATE 0.2 0.1

Figure 4b - Drag Coefficient versus Cavitation Number,

a = 20 Degrees 35 ;:i. DRAG COEFFICIENT I I VS I.

CAVITATION

I I NUMBER .1 INCIDENCE ANGLE = 20°

1427

CIRCULAR ARC

UI

aU

UUUUUUU

H/L

6 10 FLAT PLATE 20 U6

-10.

-

4 20

H/L

---

00

U

0.4 0.3 CI,

00

0.1

02

0.3 0.4

05

06

0.7 0.8

09

-10

0.9

0.8

0.7 0.4 0.3 0.2

00

01

02

03

04

05

06

07

Figure 4c - Lift Coefficient versus Cavitation Number, c'. = 10 Degrees 36

08

09

0.05 0.04 0. LIFT COEFFICIENT VS I

CAVITATION

I I I NUMBER I INCIDENCE ANGLE I

I

= 100

ICIRCULAR

_2'y16°

ARC

H/L

/

20 10

FLAT

PLATE 6

'r'

20 (

A,

'.3

0.6 CL 0.5 0.14 0.13 0.12 0.11 0.10 0.09 CD

0.08

0.07 0.06

0.1 0.13 0.12 0.11 0.10 0.07 0.06 0.05 0.04

/

1

CIRCULAR ARC

27=160

FLAT PLATE

4jrH

_{4 /}

H/L

6 /

20/

4P00

INCIDENCE ANGLE = 10°

DRAG COEFFICIENT VS CAVITATION NUMBER

I

I I 1 I I

00

01

02

0.3 0.4 0.5 0.6

07

0.8

a

Figure 4d - Drag Coefficient versus Cavitation Number,

a = 10 Degrees 37

ARC

-o

09

0.09 CD 0.08

0.30 0.28 0.26 0.24 0.22 CL 0.20 0.18 0.16 0.14 0.12

/

H/L

INCIDENCE ANGLE = 50

LIFT COEFFICIENT VS CAVITATION NUMBER

00

01

a

Figure 4e - Lift Coefficient versus Cavitation Number,

c'. =

5 Degrees

38

02

03

CL 0.28 0.26 0.24 0.22 0.20 0.18 0.16 0.14 0.12 0.10 0.08

00

a

aa

a

a

a

a

a

a

a

a

a

a

a

ER

aa

p3

a

Figure 4f - Lift Coefficient versus Cavitation Number, a. = 3 Degrees H/L

3

6

10

20

Co

LIFT COEFFICIENT VS CAVITATION NUMBER

FLAT PLATE

INCIDENCE ANGLE 30

39

0.28

0.26

0.24

0.22

0.20

0.18

CL

0.16

0.14

0.12

0.10

0.08

00

0.1

02

03

U U -9% -8% -7% -6% -5% -3% -2% 0% +1% 0 01 40

rercentage Redaction of the Predicted Li ft and Drag Coefficients

Due to Tunnel Wall I roxiaiity for he Case lI/I. 1

02 03 04

Figare S - Percentage Reduction of the Predicted Lift an) Fr,

Due to Thenel lull Pronini to For the Cone Il/I.

-05 06 0.7 CAVITATION RUM Figure Sn - lift Co PLATE - PLATE o5' PLATE CIRCULAR 216 0_losARC

_J__

DS2 o = 4.2307 CL4 - C1 PERCENT LIFT I WALL EFFECT COEFFICIENT H/L4 I . ON I C1_, PLATE

_._L_.1I

PLATE a=30 I I CIRCULAR 2n165 a205ARC I I I I . PLATE I

'

/

0 OS2 0 0.2

041061

UPPEFI FOIL SURFACES NOT

°6L'

TO SCALE

0.6 0.7 08 1)9 10 11 1.2 1.3

41 PLATE PLATE . a PLATE 10

---I I

---,-15 ---PLATE -C0 _-C0

PERCENT WALL EFFECTON

-DRAGCOEFFCIENT H/L4 . -. - H--cD - ---. CIRCULAR ARC . I LOS 2 a4.2307' . . .-.1--. .

--.

.1

II

01 02 03 04 0.5 06 0.7 0.8 0.9 1.0 1.1 12 CAVITATION NUMBER Fig,r SI, - llrag Cofficicr,t

Figure 6 - Choked Cavitation Number versus Incidence Angle

and Tunnel Wall Spacing at Various Angles

INCIDENCE ANGLE (DEGREES) _{Figure 6a - Incidence Angie}

0.5 0.4

7

RCULAR 2 ARC ci 16° H/L = 8 PLATE

////777

CHOKED CAVITATION INCIDENCE I NUMBER I I (BLOCKAGE ANGLE I CONSTANT) VS I I I 0 _{:0 FLAT} CIRCULAR PLATE ARC 1969 DATA

tEXrERIET

ROSHKO FROM WAKE PARKIN 1956_j MODEL WU I I I I I 10 15 20

IIIIIIiIIiI

VITATION NUMBER (BLOCKAGE CONSTANT) VS TUNNEL SIZE I I I I I INCIDENCE ANGLE = 100

CIRCULAR ARC 27 = 16°

E 2 4 6 8 10

_H/L

0.6 0.5 0.4 0.3 0.2

0

CHOKED CAVITATION NUMBER

VS TUNNEL SIZE (BLOCKAGE CONSTANT)

\

\

I I 1 I

\

INCIDENCE ANGLE =20°

CIRCULAR ARC 27

16° FLAT PLATE 2 4 6 8 10 12 14 16 18 20

H/L

1.0

'(IL

0.9

0.8

0.7 0.6

0.5

Figure 7 - Free Streamline Location versus Tunnel Wall Spacing Under Various Conditions

01

02

03

04

05

06

07

08

XIL

09

Figure 7a - Flat Plate, c = 90 Degrees

45 1.0 11

12

[1

FLAT UPPER PLATE FREE PERPENDICULAR 1 STREAMLINE - CHOKED TO TUNNEL STREAM FLOW

I

THEORETICAL RAYLEIGHa=0 I I I WU19690=1.6 EXPERIMENTAL

---TUNNEL TEST NSRDC 1964

10"-12"TEST SECTION - 2" FLAT PLATE L/H =

0.1794

..'

0

6"

r4d

/

/

to

0.9

0.8

0.7

.0.30 0.28 0.26 0.24 0.22 0.20 0.18 0.16 Y/L 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.0 - 0.1-I

STREAMLINE ABOVE A CIRCULAR ARC HYDROFOIL

I I I

.1

I I I F CAVITATION NUMBFD

fl)

1°

INCIDENCE

-ANGLE = 100

H/L

_'Pp

TAil

411;

-A ..,,i,LIIUIIiWjwi 'I..

AUJIh1/II111'4WI/I1IIhilhifl)h.

7I1II

UU4W45øZ9.

11'

DIRECTION OF FLOW 00 0.9

10

Y/L

06

0.7 0.8 0.0 0 1 0.2 0.3 0.4 0.5

X/L

Figure 7b - Circular Ai'c Section, 2? = 16 Degrees, ci. = 10 Degrees, a= 0.2

I I I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

10

0.30 0.28 0.26 0.24 0.22 0.20 0.18 0.16

YIL

0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.0

IOU

H1I!Iii1HIHhII

IUIIIIIIIIIIIIIHH

STREAMLINE. ABOVE A FLAT PLATE HYDROFOIL INCIDENCE ANGLE a = 100

CAVITATION NO. a = 0.2

H/L

00

01

02

03

04

05

06

07

08

X/L

Figure 7c - Flat Plate, a = 10 Degrees, a = 0.2

47

k

H

CIRCULAR ARC

H/L = 20U

2)' = 16°

H/L=10

H/L =6

H/L = 4

H/L=20

H/L=10

FLAT PLATE

UH/L=6

SUSUUK1

111111111

NORMALIZED PRESSURE COEFFICIENT VS CHORDLENGTH

-CAVITATION NO. a = 0.4

INCIDENCE ANGLE a = 10°

-0.8 0.7 0.6 0.5

C +

a

1 +0

0.4 0.3 0.2 0.1 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6

0_i

08

09

10

x/L

Figure 8 - Pressure

Coefficient

Distribution versus

Tunnel

Wall Spacing Incidence Angle = 10 Degrees, Cavitation Number = 0.4

1.0'

0.9

0.8

0.7

0.6

C +a

0.5

0.4

0.3

0.2

0.10

02

04

06

08

10

XIL

Figure 9 - Comparison of Pressure Coefficient Predictions from Wu's Theories and Larock and Street's

Theory for the Flat Plate in an Infinite Stream at Incidence Angle = 10 Degrees

CL CL

;IL

X/L

1969-FLAT _ROSHKO PARALLEL.PLATE

NORMALIZED

PLATE

PRESSURE COEFFICIENT

_{a10° INFINITE STREAM}

WU 1963 WAKE

LAROCK & STREET - TULIN

MODEL

SING LESPI RAL VORTEX

WU

LINE.

a = 0.1 = 0.4 = 0.27234 = 0.000305 = 0.48749 = 0.000495 CL = 0.27665 CL = 0.27241 XSIL = 0.000324 )çIL = 0.000316 CL = 0.56532 CL = 0.49350 XIL = 0.000932 X5/L = 0.000724

\

N..

\

' ...,

S -.5.

-

SS--5. -5 .5.

-

-5-

.5-.5

-5-5-

1-a0

--.

a0.1

a0.4

-

-5-5.

'\

0.12 0.10 0.08

Y/L

0.0.6 0.04 0.02

0

0.12

FLAT PLATE a = 100 INFINITE STREAM

/

UPPER FREE STREAMLINE

100

IN EVALUATiON

- RAYLEIGH a = 0 CHOKE

WU 1969 ROSHKO WAKE

MODEL a0

03

.

04

UPPER FREE STREAMLINE

05

06

X/L

ACCUMULATED NUMERICAL ERROR

Figure 10 - Compari.son of Streamline Locations

Flat Plate Infinite

Stream

c = 10 Degrees

Wu's 1969 Theory Roshko (or Transition)

Wake Model a' 0

01

02

0

07

08

.09

10

11

FLATPLATE

UPPER FREE

a10°

STREAMLINE

INFINITESTREAM

ciO

o0.1

-J!I

= 0.4

a = cii

a=02

a=0.4

UU

o0

/

/

_/

LAROCK VORTEX RAYLEIGH

WAKE a & STREET = 0 MODEL CHOKED TUL

aO

N SINGLE SPIRAL

-

-A

\/aroo

----0 0.1

02

03

04

05

06

07

08

09

10 11

X/L

Figure 11 - Comparison of Streamline Locations

Flat Plate

Infinite

Stream

ct = 10 Degrees

Larock and Street Theory

Tulin

Single-Spital Vortex Wake Model

0.12 0.10 0.08

YIL

0.06 0.04 U, 0.02 0

u0l

FLAT PLATE UPPER a FREE = 100 STREAMLINE INFINITE STREAM a 0 U = 0.2-a = 0.4 !hhh15

.-a = 0.4

a0.2 p4

EXTRAPOLATED ci =

oØ'

RAYLEIGH 1963 WAKE a = 0 MODEL CHOKED

WU

a0

o

a-1O

---I.

01

02

03

04

05

06

07

0.8 0.9

X/L

Figure 12 - Comparison of Streamline Locations

Flat Plate Infinite Stream a. = 10 Degrees Wu's 1963 Theory Wu 1963 Wake Model 1.0 1.1 0.12 0.10 0.08

Y/L

0.06 0.04 0.02 0

0.60 0.55 0.40 0.35 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

CAVITATION NO. a

Figure 13 - Lift Coefficient versus Cavitation Number for a Flat

Plate Hydrofoil at c#. = 15 Degrees, Infinite

Stream--Comparison of Various Theoretical Predictions

53

LI FT COEFFICIENT VS. CAVITATION NO.

FLAT PLATE a=15° INFINITE STREAM _//

1 /

/

,/

I

/

1

I,,,

/

liv

/

-

a WU 1969 0.3670 0.3905 0.4188 0.4500 30 0.4831 0.5200 C.5564 0.5955 0.10 0.15 0.20 0.25 0.35 0.40 0.45 WU LAROCK 1963 & STREET 0.3685 0.3661 0.3964 0.3911 0.4288 0.4193 0.4652 0.4504 0.5053 0.4940 0.5489 0.5200 0.5957 0.5580 0.6452 0.5978 , /

f

//

0 1

/

1'

1"

//

,' ,, /,

/I

ri

,'

I,

/

,/:/'

,,

/

,.,

/1

-,

//

/

_/

/1'

1969 THEORY ROSHKO (OR TRANSITION)

WAKE MODEL

WU1963WAKEMODEL

& STREET TULIN SINGLE-SPIRAL

VORTEX WAKE MODEL

LINEARIZEDWAKEMODEL 1966 SECOND-ORDER REF. 31

---WU1963THEORY

WU

- - LAROCK

----SCHOT1971

- HSU

0.6 0.5 0.4 0.3

YIL.

0.2 0.1 0

0.1

I I STREAMLINES INCIDENCE ABOVE CAVITATION A 2y=16° ANGLE CIRCULAR = NO. 100 = 0.26 ARC H/L = HYDROFOIL 6

-PRESENT THEORY

02

04

06

08

10

12

14

XIL

Figure 14 - Comparison of Predicted Streamline Locations with Experimental Data Circular Arc Hydrofoil, 2? = 16 Degrees,

a = 10 Degrees, H/L = 6, a = 0.26

16

18

Figure 15 - Comparison of Predicted Coefficients with Experimental Data 1.2 1.1 1.0 C-)

z 0.9

w

C) U-w

0

C.) 0.8

U--j

0. 0. 0. 0..T 0 1 0.2 0.3 0.4 0.5 0.6 0 7 0 8 0.9 1.

CAVITATION NUMBER a

Figure iSa - Circular Arc Hydrofoil, 21 = 16 Degrees,

c = 10 and 20 Degrees

55

/ LIFT COEFFICIENT VS.

CAVITATION NO.

CIRCULAR ARC a =

100 & 20°

2'yl6°

Iv-/ /

/

/ / /

/ /

/

/1

/

a=20°

_/

_/

/

/

,.

U

CAV CHOKED TATION

a=20°

uuumiiim

NO.

U

/

/

r

//

/

y.

/

/.:

II.

/

/

_'A

rA _.R

/

I,

-THEORETICAL CURVES WU 1963 WAKE MODEL - IN WU 1969 ROSHKO WAKE MODEL WU 1969 ROSHKO WAKE MODEL

EXPERIMENTAL DATA 0 BASED ON MEASURED CAVITY PRESSURE Cl BASED ON FLUID VAPOR PRESSURE a1O° FINITE o e - PARKIN - HIL STREAM - INFINITE 6 a-20° -EXPERIMENT STREAM

-

-UUI

-/ CAVITATION

IiUUI

CHOKED

a=100

-'

NO.

URU

1.1 1.0 0.9 0.8

-I

C)

I-0.7 C.) L1 U. w

0

_'

0.6

I- U--J 0. 0. 0. 0.

01

0.2 0.3

04

05

06

07

08

0.9 1.0 1.1 CAVITATION NUMBER a

Figure 15b - Flat Plate Hydrofoil, c. = 10

and 20 Degrees

(*Data from TX. Wu)

56 1.2 1.1 1 .0 -J C)

I-zoc

Lii

0

U-LI. Ui

0

0

LI. -J 0.4 I I L I I

LIFT

PLATE

CAVITATION

COEFFICIENT a 100 NO. VS. & 20°

//jFLAT

0 = a 20° CHOKED a = 20° CAVITATION NO. 0

UI

//

a=lo°7V

/

_//

//

THEORETICAL CURVES L

1969 ROSHKO WAKE MODEL - INFINITE STREAM

1969 ROSHKO WAKE MODEL - H/L 8

-EXPERIMENTAL DATA - PARKIN EXPERIMENT

a1O° a2O°

-a a

-PRESSURE PRESSURE OBASEDONMEASURED CAVITY GBASEDONFLUID VAPOR WU WU

/

CAVITATION I CHOKED

a=10°

I I NO. I

-1.2 1.1 1.0

0

I.-0.9

0

U-LU

0

C.) 0.8 U--J 0.7 0.6 0.5 0.4

LIFTCOEFFICIENTVS.

CAVITATION NO.

FLAT PLATE a = 15°

-I

/0

I

/

/

_/

/

/0

O /

/

/

/

/

//

0 0/

/

/

/

0/

/

/

//

CAVITATION CHOKED

a=15°

NO.

/

/

/

THEORETICAL CURVE ROSHKO WAKE MODEL 1969 ROSHKO WAKE MODEL

1971 LINEARIZED-INFINITE 1971 LINEARIZED -EXPERIMENTAL DATA a=15° PRESSURE

0

ON FLUID ₀ PRESSURE WU1969 SCHOT GBASEOONMEASURED CAVITY aBASED VAPOR ---WU1963WAKEMODEL-INFINITESTREAM H/L - PARKIN - INFINITE - H/L = 8 (4 IN STREAM EXPERIMENT STREAM* 8* REF. TEXT)

-WU

/;"/

_----SCHOT

,,

/

,,

/

02

0.3

04

05

06

0.7 0.8

09

1.0 11 1.2 1.3 1.4 CAVITATION NUMBER a

Figure l5c - Flat Plate Hydrofoil, a = 15 Degrees

(*Data from T.Y. Wu)