Surface tension and free surface effects in steady two-dimensional cavity flow about slender bodies

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SURFACE TENSION AND FREE SURFACE EFFECTS IN STEADY TWO-DIMENSIONAL CAVITY FLOW ABOUT

SLENDER BODIES

by

Steven H. Schot, Ph.D.

HYDROMECHANICS LABORATORY

RESEARCH AND DEVELOPMENT REPORT APPLIED

MAThEMATICS

January 1962 _{Report 1566}

SURFACE TENSION AND FREE SURFACE EFFECTS IN STEADY TWO-DIMENSIONAL CAVITY FLOW ABOUT

SLENDER BODIES

by

n

TABLE OF CONTENTS

Page

ABSTRACT

i

INTRODUCTION.. 1

SECTION I - LINEARIZED CAVITY ThEORY WITh SURFACE TENSION ... 2 Solution for the Cavity Source Distribution .. 5

Cavity Shape 8

Slender Wedge 9

Flow Past Slender Bodies with Unspecified Detachment Point 10

Convex Shape with Multiple Detachment Points 12

SECTION II - SUPERCAVITATING FOIL BELOW A FREE SURFACE 12

Mapping of the Physical Plane 15

Solution of the Boundary-Value Problem 18

Flow over a Flat Plate 18

Cavity Shape for a Flat Plate 20

LIST OF FIGURES

Page

Figure 1 - Supercavitating Flow about a Slender Symmetric Body

at Zero Cavitation Number ₃

Figure 2 - Linearized Boundary Conditions for Supercavitating Flow

with Surface Tension ₄

Figure 3 - Effect of Surface Tension _{on Cavity Shape for Slender Wedge 9}

Figure 4 - Body with Unspecified Detachment Point ₁₀

Figure 5 - Convex Body with Two Detachment Points ₁₃ Figure 6 - Supercavitating Hydrofoil below a Free Surface 13

Figure 7 - Linearized Boundary Conditions for Supercavitating _Hydrofoil

below a Free Surface ₁₄

Figure 8 - Boundary Conditions in the Transformed Plane ₁₆

Figure 9 - Lift Coefficient for Flat Plate Hydrofoil below a Free Surface

NOTATION

a Location of detachment point

B Constant of integration in Betz' inversion formula

CD Drag coefficient

CL Lift coefficient

Pitching moment coefficient

c Body chord length

d Mapping parameter

E Transformed complex disturbance velocity, E = f. g

f

Complex disturbance velocity, f = u - iv

g

Function g()

= vY'J + C

h Depth of submergence

K Constant of integration

i Cavity length

m Strength of source distribution

Cavity pressure Free-stream pressure

T Surface tension

Free-stream velocity

u The x-component of the disturbance velocity y The si-component of the disturbance velocity

V Local disturbance velocity

z Coordinate along the axis of symmetry of the body

Coordinate perpendicular to the axis of symmetry of the body

Ye Semithickness of the cavity

Semithickness of the body

z Complex variable, z = z + iy

a Angle of attack

y Half angle of slender wedge

Complex variable, = + ill

77 Imaginary coordinate in the complex c-plane

K

A

E

p

o

Nondimensionalized surface tension parameter

Infinite series, A =

(_1)x21'

v=O (2v+1)2

Real coordinate in the complex ¿-plane

Fluid density

Cavitation number, a _{= (Pc, - P} 11J2

ABSTRACT

Tulin's linearized cavity flow theory for zero cavitation number is extended

to the following two cases: (1) a symmetric body in an unbounded stream with

sur-face tension acting across the cavity intersur-face, and (2) a hydrofoil submerged below a free surface. The first problem formulates into an integro-differential equation for the cavity source distribution which is solved by a perturbation method using the surface tension as a small parameter. The surface tension makes the cavity detach from the body abruptly with a discontinuity in slope. The treatment is extended to

the case where the detachment point is unspecified. In the second problem, the

linearized body-cavity-surface configuration is mapped into a half-plane and the

resulting problem is treated as a mixed analytic function problem. The lift coef-ficient for a flat plate at small angles of attack as a function of depth of submer-gence reduces to the known solution for infinite depth but overestimates the results

obtained from the nonlinear theory for the planing plate.

INTRODUCTION

In this report we extend Tulin's"2 linearized theory of steady two-dimensional cavity

flow in two directions. In the first part we treat the case of a supercavitating slender symmetric body with surface tension acting at the cavity interface, and in the second part we extend the linearized theory to the case of a hydrofoil below a free surface. For simplicity the cavitation number is assumed to be zero in both cases.

In the linearized theory, the surface tension problem formulates into an integro-differential equation for the source distribution representing the cavity. This equation is solved to the first

approximation by a perturbation method using the surface tension as a small parameter. Using this source distribution, we then reduce to quadratures the problem of finding the cavity shape. The results show that when the cavity forms at the base of the body, the body cavity juncture is no longer smooth, as in the case where surface tension is neglected, but the cavity springs

from the body with a discontinuity in slope reminiscent of capillary action. Details _{are worked}

out for the case of a slender wedge. In most physically realizable cases these effectsare too

small to be of practical interest as far as cavity shape is concerned.

The treatment is extended to body (or strut) shapes for which the point of detachment is

not known a priori. It is shown that postulating continuous slope at detachment leads to an

equation for the detachment point in terms of the body shape parameters which is_identical

with that obtained by Tulin,3 who neglected surface tension and assumed _{smooth detachment}

(i.e., continuous curvature) at the juncture. Details are worked out for polynomial strut shapes

of the sixth degree. A conjecture of Tulin3 is verified that there exist convex strut shapes for which more than one detachment point is possible. An explicit strut shape (polynomial shape of the fourth degree) exhibiting such a nonunique cavity flow is constructed.

In the second part of this report, the linearized theory is extended to the case of a hydrofoil below a free surface at zero cavitation number. This time the problem is treated as

a mapping problem. The use of the Schwarz-Christoffel transformation reduces the problem to

a mixed analytic function problem4 for a half-plane. The problem is then solved by the method used by Cohen, et al.,5 to solve the related problem of a hydrofoil placed asymmetrically

be-tween two rigid walls. For the case of a flat plate at small angles of attack, the lift coeffi-cient is computed as a function of depth of submergence. These values reduce to the correct linearized results obtained from Rayleigh's solution for the flat plate at infinite depth, but overestimate the results obtained from Green's nonlinear theory for the planing plate6 at two

chord lengths of submergence by from 8 to 20 percent for angles of attack from 4 deg to 12 deg,

respectively. Here the depth of submergence is interpreted as the thickness of the spray sheet

produced by the planing plate.

SECTION I - LINEARIZED CAVITY THEORY WITH SURFACE TENSION

Consider the steady two-dimensional irrotational inviscid flow about a slender symmet-.

neal body placed in an unbounded stream. Assume that the free-stream velocity (J _{is large}

po0 - PC

enough (i.e., the cavitation number a - is small enough) so that a steady vapor cavity

% p U002

forms behind the body. The flow is then said to be supercavitating. M.P. Tulin1 has developed a linearized theory for this type of flow (for zero as well as nonzero cavitation number) from which the cavity shape and drag can be calculated. We now extend this theory to the case where surface tension acts across the cavity interface. For simplicity, we consider only the case where the cavitation number is zero, i.e., the cavity is infinitely long.

We first treat blunt-based bodies, i.e., bodies for which the cavity forms at the base of the body. Let a coordinate system be chosen as shown in Figure 1.

If we introduce a disturbance velocity with u and y as x- and y-components,

respec-tively, then u and y must satisfy the following boundary conditions. Since the stròainline

fol-lows the body contour, we have on the body

yo

-U00 + u(x, y)

c<x<O

{1}

and assuming that the pressure difference i at the cavity interface is proportional to the

y : y (

Slender Body

T

[1 + (!/C)2]3/2

where T is the surface tension and p is the fluid density. Note that u and y are known in terms of the body shape on the body itself, but not on the cavity boundary. In fact, the location of the cavity boundary itself is not known a priori and must be solved for along with the rest of

the problem.

y

3

Figure 1 - Supercavitating Flow about a Slender Symmetric Body at Zero Cavitation Number

Under the usual linearizing assumptions, the body-cavity combination is replaced by a slit along the z-axis, as shown in Figure 2, and the boundary conditions, Equations [1] and

[2], now applied along the axis, become

v(x, O) = Uc, _O

c<z<O

[31

u(x, O)

-p (I

Q<<oo

[4]

Replacing the body-cavity configuration by a source-sink distribution of strength m(x)

along the z-axis, we introduce the velocity potential

X: -c x:O

((X, y) =

f m() log

I(z-

)2 + y2 d

with the following velocity components at any point of the flow field

a i

r

u(x, = = + a i.

m()yd

v(x, II)

= =

-òy 2

(x)2y2

On the r-axis these expressions reduce to

i

u(x,O)= J

x--c

v(x, O) = m(x)

where the improper integral, as well as all subsequent integrals of this type, must be inter-preted in its Cauchy principal value sense.

u(xo) :0 v(x,o): O v(x,o) Uy

III/II'II1

I _Slender f Body x=-c x:0 L Ty0"

u(x,o)- -

_pU Cavity

Figure 2 - Linearized Boundary Conditions forSupercavitating Flow

with Surface Tension

Applying the linearized boundary conditions along the x-axis yields an expression for

the known source-sink distribution along the body

m(x)=2Vj'(x)

c<x<O

and the unknown source-sink distribution along the cavity °

_{2V y0'()d}

00 m(e)d T in (x)

j.

00 +

i

-

2pU

c

nT

On introducing the nondimensionalized surface tension parameter = and letting

p U0 C

2V00y0'()d

f (x)= - I , we can write this equation as

J

-c

00

xm'(x)+ J

m()d

_X

-

1(x)

o

Once this equation is solved for the source-sink distribution along the cavity, the cavity shape, drag, etc., can be computed by integration.

SOLUTION FOR THE CAVITY SOURCE DISTRIBUTION

The integro-differential equation [5], which is related to the Prandtl integro-differential equation for airfoils of finite span,4 does not seem to have been inverted explicitly, hence we shall be content to find a first-order approximation to the source distribution in terms of the

small surface tension parameter K.

Thus, assuming a solution of the form

,n(x) m0(x) + K1n1(x) + K2m2(x) +

substituting it into the integro-ditferential equation, and equating like powers of K lead to the

following iterative system of integral equations for m0(x), m1(x), m2(x), .

5

°

_m1()d

¿

T

m2()d

Lff2 2Uc,, V' . y0'(t)dt

(xt)

The first approximation m1(x) is now obtained by solving

/

2Uy0'(t)dt

J

c

d. [7]

It may be noted that each of these equations is of the airfoil type and can be inverted explicitly.

Tulin1 solved the base problem by using the well-known Betz inversion formula7

m(x)= ff2[I

x-

+

BI

[6]

and by applying the juncture condition ₌ '(°) which ensures a body-cavity transition with continuous slope. This condition serves as a kind of Kutta-Joukowski condition and picks

out a unique flow from the class of flows defined by Equation [6]. Application of Tulin's

junc-ture condition, by evaluating the arbitrary constant B so that m0(x) does not have a singularity at z = O, yields the solution of the base problem

P

Again making use of the inversion formula, Equation [61, we obtain

The constant of integration is taken to be zero this time to ensure the correct type of singu-larity at the juncture.

Now, since -+ Oat = _0, oo, we have, on integrating by parts

-m(x)

=

c

x 2(100

1

\/1og-y(-c)

_x+c

_{f y0'(t)}

_x-t

X

C

,2

I

[ n'(e)1 d

ir

_mo()-j

d(

00

-

ir

₂ d

and employing the solution of the base problem given in Equation1171

00 . + 000

y0'(t)dtl

¿2VT(x)2

I

f

Id

L

y'(t) [

_(x)d

3v

fi

(e-t)(x-)2

dt

"00 ,oy',

[

2

x+ t

(t)]

IC log V'

x-t

(x-t)2

00 d

[v'og()

20 ° o d 7

dt

Thus the source distribution along the cavity, taking into account first-order effects in surface tension, is given by

m(x) = m0(x) + Km1(x)

-where X 2 U00 ° '(t) dt 2 ( C) log o

log ()

d}

+ + K

_{f y0"(t)}

x

L Vt(xt)

-

x+c

c

xt

[911

It may be observed that the second term introduces a singularity at the juncture of the type

(log x)//.

CAVITY SHAPE

The cavity shape may now be found from the source-sink distribution. The shape for m0(x) was worked out by Tulin.1 Adding the effects of the m1(x) term, we have

¡

2 IT

f

LoI '0'(t)dt d

17+-

K

(fl_t)]

c

o

fy'(t)tan1 l/-idt

c

+

L

y0"(t) 00 (_1)2X2

A(x)= 42:-'

=0 (2v+1) C

log-4K 1 + tYo'(_

e)

[iog

i/'

tan

1/!

_{+ A ( i/f)]}

VA(V1]dt

+

J y0"(t)

[loa

l/

tan -

r;:--c

[101

SLENDER WEDGE

For the case of a slender wedge of chord length c and semiangle y, the integrals in the formulas for the source distribution and the cavity shape derived in the previous section can be

evaluated explicitly, and the results take a relatively simple form. Thus, if we let y0 =

for - c < z < O, then Equations [91 and [lo] become

and 4)'

y(x)

y0(0) = if [ITX L2 4 _y m(x) = -if +

-

(z +c) tan C C

-

log

-X X X+ C 4y I

',/tan_1

i/c_El

- I

log 773L z

Curves showing these cavity shapes near the body-cavity juncture for y =5 deg and for several

values of the surface tension are presented in Figure 3. For small values of x/c, the

approxi-mati on

f z'

-

y0(0)

2yX [if

V-'

i/

[log

i+ ]

+

77

[2

CJ _if3 C X

has been used.

9

X C

Figure 3 - Effect of Surface Tension _{on Cavity Shape for Slender Wedge}

tan

11/

)J+K

FLOW PAST SLENDER BODIES WITH UNSPECIFIED CAVITY DETACHMENT POINT

In the previous sections the point of detachment of the cavity was assumed to be at the

base of the body. It was shown that with surface tension present, the detachment is abrupt,

although the magnitude of these effects is too small in most practical cases to be of

impor-tance. In this section we investigate how, by making certain plausible assumptions about the

action of surface tension at detachment, the location of the detachment point may be determined in the case of certain physically realizable shapes which do not have a blunt base. Tulin3 has shown that in the case without surface tension, imposition of continuous curvature at detach-ment is just sufficient to allow determination of the detachdetach-ment point. Under these assumptions Tulin worked out the explicit location of the detachment point for body shapes of the form of third-degree polynomials with infinite cavities.

In treating the case with surface tension, we shall assume that detachment takes place at a point x = a at which yÇ(a) = y'(a); i.e., the slope of the cavity is continuous with the

slope of the body. As in Tulin's juncture condition, this requirement is imposed by eliminat-ing those linearized solutions for the cavity source distribution which have a seliminat-ingularity at

the juncture. It is seen that this leads to determination of the detachment point without any explicit assumptions about the curvature at the juncture. There is, of course, an implicit re-lation between the curvature and the shape of the cavity which is inherent in the expression

for the surface tension, Equation [211. We now determine this detachment point.

Consider the body shown in Figure 4. For convenience in representing the body shape,

we assume the body to be of unit length with leading edge at the origin. We aim to find the

detachment point x = a < i by equating the slope at the body-cavity juncture.

u00

y

A

Y0:y0()

Slender Body

(Wetted Portion Shaded)

Detachment Point

Figure 4 - Body with Unspecified Detachment Point

Y()

Cavity

X

The slope of the cavity can be obtained from the source distribution, Equation [9], which gives rise to the cavity

dy

2V

-Now, obviously, rn0(x) will not contribute to the determination of the detachment point, since

it has been chosen so as to automatically satisfy the juncture condition at x = a. Thus,

deter-mination of the detachment point must come from m1(x). _{On neglecting rn0(x) and translating}

the leading edge to the origin in Equation [.81, we have

2V dy 2V I O /atlog I a

fo(t)

xa

I m1 (x)= = J

y'(0) -

log + dt_{+ BI [11]} K dx

rr3,/xa

x

xa

xt

i o I

Note that the constant of integration B in Betz's inversion formula, Equation [61, is not neg-lected this time but is determined from the juncture condition.

For third-degree polynomial struts of the form (t) = a1 t -4- a2 2 + a3 t3 Equation [11] becomes m1(x) = 2

3x_a{1+4a2x+12

2_ a

a3x 4a3ax) log

-z

xa

A

(8

a2 24 a3 z - - a3 a) - Jx - a (8 a2 + 24 a3 z) log 11 a

xa

a

t tan

To avoid a singularity at z = a the terms in the braces must be made to vanish. Now the third

term in the braces vanishes at z= a (this is easily seen from the integral representation of

this term). The value of B may be chosen to cancel the second term in the braces at z=a.

The first term will vanish at z = a if a satisfies the condition

This is precisely Tulin's equation for the detachment point. Note that no explicit assumptions were made about the curvature at detachment in deriving this result.

Tulin did not work out the detachment condition for body shapes of higher than third

degree. Using the present analysis and setting up certain recursive formulas, we can easily extend the result to shapes of arbitrary order. Results have been worked out explicitly for

n= 6. This case is given by

64 128

+ -

512

a a =0

a1 + 4a2a+ 8a3a2

+ -

a4a + _-i-- a5a

21 6

Summing up the results of this and the preceding sections, we can describe the effect of the surface tension on the detachment point as follows: Whenever the body shape is such that smooth detachment is possible, the cavity separates from the body so that the slope at the transition is continuous. (This condition will in fact determine the location of the de-tachment point.) However, if the body shape is such that smooth dede-tachment is impossible (as in the case of certain blunt-based bodies), the cavity jumps from the body with a discon-tinuity in slope.

CONVEX SHAPE WITH MULTIPLE DETACHMENT POINTS

In his paper, Tuli'n raises the interesting question of whether there exist convex body shapes with two detachment points (both of which lead to a cavity which is convex with respect to the stream. Tulin showed that such nonunique cavity flows cannot exist for body shapes

with n 3 but conjectured that such a flow might occur for shapes with n > 3. Using the

pres-ent analysis and Equation [12], we found that such body shapes do indeed exist. A simple

example of such a shape for n= 4 is given by

y0 =

t(1t) (5t2-3t+2)

0 t i

This shape, plotted in Figure 5, has negative curvature everywhere and detachment points at

a = '/ and a=

SECTION II - SUPERCAVITATING FOIL BELOW A FREE SURFACE

Tulin and Burkart8 have developed a linearized theory for steady, two-dimensional

cav-ity flows about hydrofoil sections at zero cavitation number in an unbounded fluid. Their

meth-od consists of mapping the cavitating hydrofoil into an equivalent thin airfoil. They then

com-pute the lift, drag, and pitching moment of the hydrofoil in terms of certain characteristics of the equivalent airfoil. In this section, we modify the Tulin-Burkart theory to include the effects of the free surface. This is accomplished by mapping the hydrofoil-cavity-surface configuration

h

Figure 5 - Convex Body with Two Detachment Points

into a half-plane and then solving the problem by a method developed by Cohen, et al.,5 fora

related problem. In earlier investigations, Johnson9'10 dealt with the free surface problem by

using the airfoil mapping of Tulin and Burkhart and accounting for the free surface inan

ap-proximate way by locating a singularity judiciously in the stream.

The physical problem considered is illustrated in Figure 6. A hydrofoil section of shape y0 = y0(x) and chord length c is placed a distance h below a free surface in a steady

y Free Surface

xO

y:y0(x) X:C 13 Upper Cavity Lower Covity

y:y (y<o)

Figure 6 - Supercavitating Hydrofoil below a Free Surface

y>o)

inviscid stream. The flow is assumed to be such that an infinitely long cavity (o = O) springs

from the leading and trailing edges of the foil. It is assumed that the foil is shaped so that

the application of linearized theory becomes meaningful. Furthermore, wavemaking on the

free surface and cavity is neglected (infinite Froude number).

We now introduce a complex perturbation velocity function

1(2) = uiv

with perturbation velocity components u(x, y) and v(x, y) that are much smaller than U00.

Applying the usual linearized boundary conditions on the free surface and on the body-cavity

as represented by the positive z-axis, we have (compare with Equations [3] and [4])

A

(x :_,y :h)

' ,' ' , 'Ç

Here it is consistent with 4he linearization assumptions to satisfy the conditions on the upper

and lower sides of the foil-cavity combination on the two sides of the real axis instead. The linearized version of the above problem may then be depicted schematically in a complex 2-plane as shown in Figure 7.

y L u: O U:O X

v:1Jv

C u:O B

(x=c,y<o)

(x:oo ,y<o)

z - Plane

Figure 7 - Linearized Boundary Conditions for Supercavitating Hydrofoil

below a Free Surface

>0

,/ (x:c,yo)

u=O 00 y= h

u=O O<x<oo y=

V = U00 y0'

O<x<c

y = O

In addition to the conditions just stated, the perturbation velocity components are ex-pected to satisfy the following requirements:

Leading edge singularity at x= O, y= O (since the streamline splits into two parts at

the leading edge).

Tulin's juncture condition at the trailing edge x= c, y = 0 (this specification of

con-tinuous slope at the foil-cavity juncture serves as a Kutta condition and selects a unique

flow).

(C) Vanishing of the perturbation velocity components far upstream at r=

MAPPING OF THE PHYSICAL PLANE

The simplicity of the above boundary-value problem, which results by postulating an infinite cavity (simple-connectedness) and by applying linearized theory, may now be utilized by first mapping the configuration shown in Figure 7 into a simpler domain, say a half-plane,

and then solving the simplified boundary-value problem. Using the Schwarz-Christoffel

trans-formation, the 2-plane (with a branch cut along the positive real axis) is mapped onto the upper

half (including the real axis) of a ¿-p1ane, where = + jr, O. The explicit mapping

for-mula is

2 = +

log

(ti-h I ¿

d d [131

where.d is a parameter of the mapping and represents the distance of the image of D from the origin in the ¿J-plane. For convenience, we make the point 2 = i map into the point ¿J = c.

Then d may be obtained in terms of the geometric parameters c and h from

j=-iog

(i+)

[14]

It may be verified that as h - oo (and hence d ao), the mapping reduces to the square root

transformation

[151

which represents the equivalent airfoil transformation used by Tulin andBurkart8 and

Johnson.9''0 Instead of using the exact transformation, Equation [131, Johnson9'10 used

the much simpler transformation, Equation [151, and satisfied the free surface condition to a first approximation by concentrating the circulation of the equivalent airfoil at its center of pressure and locating a single image vortex forward and belowthe.1eding edge in the ¿J-plane. He then applied the equivalent airfoil concepts worked out by Tulin and Burkart8 to compute the force and moment coefficients for variously shared hydrofoils, with and

without camber. His results are in good agreement with experimental data.

SOLUTION OF THE BOUNDARYVALUE PROBLEM

The boundary conditions in the c-plane may now be stated as follows:

Re[f}

:o

-b -c

- Plone

Figure 8 Boundary Conditions in the Transformed Plane

Thus in the t-plane the boundary values in Re Lf} and 1m _f} are prescribed alternately in

ad-jacent intervals along the e-axis. Now if

f()

were transformed into some other analytic

func-tion

F ()

in such a way that it still had the proper behavior at (0, 0), ( e, 0), and at infinity,

but at the time the boundary conditions of

f()

(which at present alternato between Re tfl and

1m f} on the real C-axis) were transformed into conditions involving only Re LP} or only 1m LF

along the C-axis, then the problem of findingF (and hence _f) in the upper half-plane could be solved in a simple manner.

The transformation from

f()

into

F() is

achieved by multiplying

f()

by a function which has the following properties: (a) it is alternately real and imaginary on segments of

the real axis partitioned at C= - c and at C = O and (b) on multiplication by f(), it will yield

a function with the proper character in the sense of the conditions (a), (b), and (c) on page 15.

Such a ftmction is g() = + c. Then

71 Im(fJ:U,y _{Re (f1 :0} d o

I?ef}=

O -00 < < - C

7?0

1m fI = - U00 y0'

c<C< o

77=0

RefI=

O

o << 00

p(

V[u(C, 0) -iv(C,

_{-C- C}

0)]

V[v(C, 0) + iu«, O)]

I e

if

____

r C+c

and hence Re {E( is known along the entire real axis

(.1-C

VU y

00 0 Re !F'( )l =

O

[u(C, 0)- iv(C, 0)1

0 <<

The imaginary part of E() can now be found along the entire real axis by application

of the Hubert transform to Re F( ₎ along the real axis

i

_{Re 1F()}

i

_I''Eiiii

U00 y0'(t)

dt

¡mP()I=-_

J

dt=_;.

IP't+c

I

t_

-w--

c

In this manner F() and, hence, u and y are known along the entire C-axis for _{any given foil}

shape y0(t). Now in the linearized theory this knowledge of u and y along the axis is

suf-ficient to compute the forces on the foil. The lift coefsuf-ficient in the línearized theory_{is given}

by integrating the horizontal perturbation velocity component u0 over the foil

CL=-

_{f u0dx}

Similarly, the drag coefficient and the pitching moment are given by i is CD = - 2 U 000

i

17 i u0 y0'(x)dx u0 x dx -00

-c<C< O

-c<C<

O

( o<C<

00

j- 00 <C<

C [16]

In these expressions dx along the foil, as computed from the mapping formula, Equation [13],

h e

FLOW OVER A FLAT PLATE

The case of a cavitating flat plate at an angle of attack is particularly interesting since

the exact solution is known6 at infinite depth and, in a certain sense, near the free surface.

Thus it provides a means of comparing forces.

For the flat plate y0'(t) =, and the expression for E() along the foil can be obtained

explicitly from Equation [1611. IIence

P()

_L

i

27

c

t+ C

=UoQ

where the constant K is determined so that f(oc) = 0. The horizontal perturbation velocity

component u0 over the foil is then

u0

= u(,

O) = U00

and the lift coefficient becomes CL =

---=5-_iog (i+

hence 2 ha rrd

fc

C__ il'Z1T

dt.- (Ja

2

d-

d [181

where c/d is given in terms of h by Equation [1411.

As c/d -+ 0, h - and the result should reduce to CL for a flat plate in an unbounded

stream. For c/d « i

c 1

,c2

1

c3

-I.- l-I --(-I +.

dJ

2

kd/

3\d)

[2011

*The integrals appearing inEquations[is] and [19] may be evaluated by making use of the appendix of Thun' s report'

Making a similar approximation in the expression for the lift coefficient gives

This value of the lift coefficient agrees with that obtained by Rayleigh for a flat plate at a

small angle of attack in an unbounded fluid

2nsina

L4+sjna -

2

For small values of h (corresponding to shallow submergence) no simple approximation seems

possible, and the lift coefficient must be computed from its parametric representation, Equations {141 and [191. Graphs of this lift coefficient versus depth of submergence for selected angles of

attack are shown in Figure 9. The corresponding values obtained from Green's nonlinear theory for the planing plate as taken from Reference 9 are also presented. It is seen that the values

of CL obtained from the linearized theory consistently overestimate those obtained from the

Depth of Submergence in Chord Lengths

Figure 9 - Lift Coefficient for Flat Plate Hydrofoil below a Free Surface

for Various Angles of Attack

19 w o L) - _0.2 0.t O .5 s. s. 5. r 8° - 4° I I I I I I I I I a I _{2 3} 4 5 6 7 ₈ 9 lO 0.6 Linearized Theory

Greens Nonlinear Theory for

0.5 _{Planing Plate} 't -I L) 0.4 s' s' a r 12° C w _'t 't 0.3

nonlinearized theory. As may be expected, this discrepancy gets worse as the angle of attack increases. At two chord lengths of submergence, the percentage difference over the exact values is about 7 percent for a = 4 deg, 17 percent for a = S deg, and 26 percent for a = 12 deg.

The pitching moment for the flat plate has not been computed since the integrations

in-volved were considered too formidable.

CAVITY SHAPE FOR A FLAT PLATE

The cavity shape for a flat plate may be obtained from the linearized boundary conditions

on the cavity v(,O) =

dy(x,y>0)

v(x,0)

y(x, y<0) = dx (J dy(x, y<O) v(x, 0) dx The shape is 1 (L i U

For a flat plate, the vertical component of the perturbation velocity on the e-axis is given from

Equation 18I1 by - UDO

ai

- UDO ai

¡ v(t,

[í

1]

1]

upper cavity lower cavity upper cavity lower cavity

_c<< O

[21]

Hence, making use of these expressions and Equation [17] and performing_{the integrations in}

Equation [21], we get the following representation of the cavity shape in _{terms of the parameter}

h

-- +log

_{i d}

ah

I

c

_(,/i+v+c)2

dlog

[¼i

-

-c(1)

k +

d log 1 d

ahíC'

(f+/_c)2

-

J + d log

(i

-

)+ /-c) +

(d ) log

-c

- d

log

[¼'-c(

\

-

1_i)

21

-c<< O

[22]

0<

here again, c, d, and h are related by Equation [14].

For deep submergence, i.e., large values of h and d, using Equations _{[151 and [201, and}

eliminating the parameter , Equations [22] reduce to the expressions for the câvity shape

be-hind a flat plate in an unbounded stream, as given by Tulin and Burkart.8

- '/ log (1 +

2-

_{0< x upper cavity}

x +

2x

+

0<x<1

x - + %

log-1 2 2x

+ 1 < X lower cavity

(1-

_d ah rid

-a

- %(1+

2/)

=

-ax

-

a{x+

(1 - 2)

REFERENCES

Tulin, M.P., "Steady Two-Dimensional Cavity Flows about Slender Bodies," David

Taylor Model Basin Report 834 (May 1953).

Parkin, B.R., "Linearized Theory of Cavity Flow in Two Dimensions," Rand

Corporation Report P-1745 (1959).

Tulin, M.?., "New Developments in the Theory of Supercavitating Flows," Proceedings Second Symposium on Naval Hydrodynamics (1958). In press.

Muskhelishvili, N.I., "Singular Integral Equations," P. Noordhoff, Groningen, pp. 275-281

(1953).

Cohen, H., et al., "Wall Effects in Cavitating Hydrofoil Flow," Journal Ship Research, Vol. 1, No. 3, pp. 31-39 (1957).

Mime-Thomson, L.M., "Theoretical Hydrodynamics," Fourth Edition, The Macmillan Company, New York (1960), pp. 323-326.

.7. Schmeidler, W., "Integraigleichungen mit Anwendungen in Physik und Technik," Second

Edition, Akademische Verlagsgesellschaft, Leipzig (1955), p. 51.

Tulin, M.?. and Burkart, M.P., "Linearized Theory for Flows about Lifting Foils at

Zero Cavitation Number,' David Taylor Model Basin Report C-638 (Feb 1955).

Johnson, V.E., Jr., "Theoretical and Experimental Investigation of Arbitrary Aspect

Ratio, Supercavitating Hydrofoils Operating Near the Free Water Surface," National Advisory Committee for Aeronautics RM L57116 (1957).

Johnson, V.E., Jr., "The Influence of Depth of Submersion, Aspect Ratio, and Thickness

on Supercavitating Hydrofoils Operating at Zero Cavitation Number," Proceedings Second Symposium on Naval Hydrodynamics (1958). In press.

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