A093 7 STEVENS INST OF TECH HOBOKEN NJ DAVIDSON LAB F/G 20/4A PARTIALLY CAVITATING HYDROFOIL IN A GUST.(U)NOV 80 A 5 PETERS T R GOODNAN, I P BRESLIN NOO0 -4-77-C-0122

UNCLASSIFIE ST-DL-O-2118 NL

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~'R-2118

DAVIDSONLABORATORY

REPORT SIT-DL-2118

November 1980

A PARTIALLY CAVITATING HYDROFOIL

IN A GUST

.S. Pe ters

T. R. 'oodman

J .. P. Brealin

Final ReportNovember 1978-November 1980

APPROVED FOR PUBLIC RELEASE;DISTRIBUTION UNLIMITED

Prepared forOffice of Naval Research800 N. Quincy StreetArlington, Virginia 22217

2 0

-,

STEVENS INSTITUTE OF TECHNOLOGYDAVIDSON LABORATORYCASTLE POINT STATIONHOBOKEN, NEW JERSEY

NOTICE

IN 1977 A REPORT ENTITLED "THEORY OF A PARTIALLY ANDINTERMITTENTLY CAVITATING HYDROFOIL IN A PERIODICTRAVELING GUST" BY JOHN P. BRESLIN AND THEODORE R.GCCDK'\N !.%S PUBLISHED, AND ONE OR MORE COPIES WERESENT TO YOU.

SUBSEQUENTLY, CERTAIN ERRORS IN THAT REPORT CAME TOLIGHT AND, AS A CONSEQUENCE, THE ENCLOSED REPORT WASPREPARED. THIS REPORT IS INTENDED T PER THEEARLIER ONE. THEREFORE, YOU ARE ADV TO DESTROYALL COPIES OF THE EARLIER REPORT AND SUBSTITUTE THEENCLOSED REPORT IN ITS STEAD.

I

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REPORT DOCUMENTATION PAGEREDISRCOS7 BEFORE COMPLETING FORM

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ABSTRACT (Continued)

numerical procedures to solve7 no attempt is made to solve them 0 her.0 Itis shown in an appendix that the earliest possible appearance of a non-vanishing second time derivative of the cross-sectional area of the cavityis to the order of the square ofthe reciprocal of the aspect ratio and thatto this order the pressures at distances large in comparison with the chordof the foil (but not the span) grow logarithmically. It may be inferredfrom this result that the pressures in the far field can be reduced by in-creasing the aspect ratio

I

~i

UNCLASSIFIEDSECURITY CLASSIICATION OF THIS PAGCE~lfb Date KnteRI0

STEVENS INSTITUTE OF TECHNOLOGYIDAVIDSON LABORATORYCASTLE POINT STATIONH4OBOKEN. NEW JERSEY

REPORT SIT-DL-80-21 18

November 1980

A PARTIALLY CA VITA TING HYDROFOIL

* IN A GUST

by

A. S. Peters

T. R. Goodman

J. P. Breslin

Performed Under

Office of Naval Research Contract Nooo14-77-C-0122Task Order NR-062-568

(DL Project 013)

APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED

Approved:.,

P.Se~n, Drco

R-2118

ABSTRACT

An analysis is made of a hydrofoil with partial cavitation in a

traveling gust. The problem is formulated in the time domain with

pressure (acceleration potential) as the fundamental dependent variable.

It is shown that, in the limit, as the aspect ratio of the foil becomes

large, the second time derivative of the cross-sectional area of the cavity

vanishes regardless of the shape of the gust. Integral equations are

presented through which the length of the cavity and the unsteady pressure

distribution on the foil can be determined. These equations are highly

nonlinear and require formidable numerical procedures to solve; no attempt

is made to solve them here. It is shown in an appendix that the earliest

possible appearance of a nonvanishing second time derivative of the

cross-sectional area of the cavity is to the order of the square of the

reciprocal of the aspect ratio and that to this order the pressures at

distances large in comparison with the chord of the foil (but not the

span) grow logarithmically. It may be inferred from this result that the

pressures in the far field can be reduced by increasing the aspect ratio

of the foil.

a

R-21 lP

NOMENCLATURE

A nondimensional cross-sectional area of the cavity

B denotes wetted portion of foil

C denotes cavity

c nondimensional local chord of the cavity

c C/E

F complex function whose real part is P

f in the appendix t/c (the same as h in the text)

h ZIci V/-T

1 unit vector in free stream direction

i unit vector along the chord of the foil

unit vector normal to free stream direction

± unit vector normal to

K a constant0K a constant

nondimensional local chord of foil

n ordinate of cavity surface

P acceleration potential

P nondimensional cavity pressurec

p nondimensional pressure

S semi-span

s nondimensional distance along span

t time

t nondimensional time

u perturbation velocity due to foil

U free stream velocity

u nondimensional perturbation velocity

v vector sum of free stream velocity and gust velocity

v unsteady velocity of fluid

v gust velocitygv nondimensional gust velocitygW denotes wake

x1 distance in free stream direction

x distance along chord of foil

x nondimensional distance along chord of foil

y distance normal to chord of foil

Jk ........ . ... .. Il II....... .. ... I.... II...... -........ ... ...... 1III1 .... . |'Il .... .. . . ...... 3' "

R-21 18

y nondimensional distance normal to chord of foil

z x + iy

a angle of attack

-y x/E

C small quantity inversely proportional to aspect ratio

mapped complex plane = + in

in imaginary coordinate in mapped plane

K 2 unknown functions of time

A dummy spacelike variable

nondimensional difference between ambient pressure and cavity pressure

V CA

nondimensional dummy chordwise coordinate in first section;also in second section real coordinate in mapped plane

p fluid density

o nondimensional dummy spanwise coordinate; also in third section

denotes dummy time

T t/e ; also in third section denotes t-x

* transformed acceleration potential

* nondimensional velocity potential

Wy/

( ) except where specifically defined, denotes first order in c;

similarly for( )2' )'39 ( )4

iii

R-2118

INTRODUCTION

Ship propeller blades are known to cavitate intermittently during

their traverse of regions of large hull wake fractions or low inflow

velocity. Such cavities are responsible for large pressure changes on

the hull with concurrent intense hull and sub-structural vibration. It

is known that the vibratory forces generated by cavitating propellers

are much larger than the forces generated by noncavitating propellers.

The reason for this is that the cavity behaves in the far field like a

source whose strength is proportional to the second time derivative of

the cavity volumel. It is also significant that cavity-induced forces

arising at the first few integer multiples of blade frequency are of the

same order as those at blade frequency. This is in sharp distinction to

the excitations induced by non-cavitating blade loadings which are in-

variably small at all higher multiples of blade frequency, as has been

pointed out by Breslin 1 . Thus, it is clear that the dynamics of unsteady

cavities on propeller blades must be understood and reduced to a calcula-

tion yielding their time-varying geometry before the pressure field (which

induces the hull vibration) can be predicted.

The complete description of intermittent cavitation on propeller

blades requires a three-dimensional representation accounting for the mutual

interaction of cavitating and non-cavitating blade elements as well as the

mutual interaction of the blades. This is clearly a formidable task. As

a first step, we consider the blades to be hydrofoils acting independently

and we examine the limiting case of large aspect ratio. The problem is

thereby reduced to a single two-dimensional hydrofoil subjected to a

periodic gust pattern which is spatially stationary in hull-fixed coordi-

nates and, therefore, travels with the negative speed of the section in

foil-fixed coordinates.

In the past, there was, for an extensive period of time, a continuing

controversy about the possibility of solving unsteady cavity problems in

two dimensions because the existence of time-varying cavities of finite

length implies a time-varying source flow whose potential and non-convective

pressure fields do not vanish at large distance, but grow as log r. Birkhoff2 ,

Leehey 3 and others sought to invoke compressibility as a means for resolving

R-2118

this paradox. Benjamin 4 put the question to rest by pointing out that

such two-dimensional problems are artifacts of attempts to reduce three-

dimensional problems to manageable problems in two dimensions and that

they really represent the inner behavior of three-dimensional flows with

potentials and pressures vanishing as a three-dimensional source, i.e.,

as (r2 + s2)-l/2, where s is the spanwise coordinate. Thus, the logarithmic

character of the mathematical representation of two-dimensional time-varying

cavity flows must be included in order to match the inner expansion of

the outer field. Benjamin went on to assert that the rate of expansion

of the cavity section should be regarded as a parameter of the inner field

and could only be determined by resorting to the three-dimensional outer

field. With these concepts in mind, therefore, the problem will be set up

as a three-dimensional one and the aspect ratio will then be taken to be

large.

Several attempts to solve unsteady, partially cavitating hydrofoil

problems have been made, but all have fallen short of the goal set here,

principally because they considered only small disturbances on an estab-

lished steady cavity. In contrast, we must address flow patterns with

cavities whose size can grow and diminish over several orders of magni-

tude during one cycle. In one of the earliest assaults on the non-steadily

cavitating hydrofoil, Steinberg and Karp 6 studied the small disturbance

case, but they imposed a dubious condition, requiring that the velocity be

continuous across the foil-cavity wake while ignoring the cavity closure

condition altogether. Wang and Wu7 studied only small perturbations on an

established steady cavity. Unruh and Bass 8 presume that the cavity leading

edge can be assigned arbitrarily downstream of the foil leading edge, whereas,

many observations in water tunnels and a vacuum tank reveal that, on propel-

ler blades, cavitation almost invariably initiates at the leading edge.

Furthermore, they also imposed the, by now familiar, restriction to per-

turbations of an already existing steady cavity.

The three-dimensional problem posed by intermittent cavitation on

ship propellers has been addressed by Johnsson and S~ntvedt9 , NoordzijlOand,

more recently, by Kaplan, Bentson and Breslin 11. All of these have employed

quasi-steady strip theory utilizing the steady, partially cavitating, sec-

tion theory of Guerst1 with no consideration of unsteady flow mechanics.

An exception to this restriction is presented by Van Houten5 using anumerical procedure.

2

R-21 18

Despite this restriction, surprisingly good correlation with measurements

of hull pressures on models and on ships are reported in Reference 11.

The problem is posed on the pressure as the primary dependent vani-

able. Because the length and cross-sectional area of the cavity (both un-I

known a priori) may undergo large excursions during a cycle, it is not pos-

sible to solve the problem in the frequency domain. Instead, it must be

set up as an initial value problem in the time domain. As will be seen,

this results in a pair of nonlinear integro-differential equations (with

time as independent variable) which must be solved numerically. Inasmuch

as the input is periodic, the output must also be periodic and, even though

an initial value problem is posed, periodicity will be achieved by runningI

through several cycles of input allowing the transients introduced by the

starting conditions to decay. No numerical results are presented here,

however, and the primary result of the analysis is to show that, in two9

dimensions, the second time derivative of the cross-sectional area of the

cavity vanishes identically regardless of the shape of the gust. As a con-

sequence, neither the potential nor the pressure grow logarithmically at

large distance. In an appendix, it is then shown that only to the order9

of the square of the reciprocal of the aspect ratio is there a possibility

that the second time derivative of the cross-sectional area does not vanish.

The solution at this order allows for the possibility that the three-

dimensional flow behaves like a three-dimensional source in the far field.

The practical inference to be drawn from this result is that the vihratory

pressures in the far field will be reduced as the aspect ratio is increased.

Propeller blades of high aspect ratio are known to be more prone to cavitate

than propeller blades of low aspect ratio. So a compromise must be drawn

between low aspect ratio blades which cavitate less and high aspect ratio

blades which induce lower vibratory pressures in the far field even though

they are more prone to cavitate.

The authors are indebted to Program Director Mr. R. Cooper, Fluid

Dynamics Branch of the Office of Naval Research, for support of this work

under Contract N00014-77-C-0122, Task NR-062-568. The analysis has been

conducted under Davidson Laboratory Project 013.

3

R-2118

A 7UTIALLY CAVITATING FOIL

Our present interest is in the linear theory for a partially cavit-

ating planar foil which is immersed in an inviscid, incompressible, fluid

of constant density p. We suppose that the unsteady velocity of the fluid

is given by

V = V + U

where u denotes a perturbation velocity while

v = U i + v (Ut - x- -I g -i

i. =i cosa + sina

= -i sin + j cosa

x = x cosa + y sin

represents a uniform flow upon which is superposed a traveling cross gust

v g. We will take v/U and the angle of attack a to be of the same order

of small magnitude.

Hereafter, all terms will be rendered in dimensionless form by using

U; the semi-span S; and the constant density p as unit quantities. The

linearized velocity vector is then

v = i + [a + v (t-x)]j+ u

where i and i are unit vectors along the x and y axes respectively. We

also suppose that u is source free in the domain of the fluid and that u

is irrotational in the fluid domain minus the wake so that in the latter

domain there exists the velocity potential function 4(x,s,y,t).

Figure I (see following page) shows the area B of the foil and the

cavity surface whose height n is assumed to be small. The trailing edge

of the cavity is assumed to lie always in B; here x = c(s,t) and the foil

leading edge bound the area C under the cavity. The quantity c which ap-

pears in the equation for the trailing edge of the foil is supposed to be

small so that the foil is one of large aspect ratio. This foil with the

straight leading edge suffices for the exposition of the basic ideas.

Its wake W is defined by

L .. .. ... . .. . . .. . . . . .. . . ... . .... . . . m l' I . . .. .. . ." .. .. I 4-

R-21 18

y

AS

(O'1'0)

y~n~x~s~x c2~s~'N

W- y n 0; ,(s ; s 1. W

The velocity potential 4,(x,s,y,t) must satisfy

xx + ss + yy=0

in the domain of the fluid minus the wake and be such that

y (x'sO+,t) = -[ct+v 9(t-x)] cOst)<(s); -l<l

For the same domain, a linearization of the Euler momentum equations pro-

vides the first order acceleration potential

P(x~s~yt) 2* x+ t=-p + f4, +4,t + p

where p is the pressure. The pressure o[x,s,n(x,s,t),t] at the cavity

wall is required to be a function of t only. Since n is small, the linearized

version of this condition is

p(x,s,O+,t) = PC (t)

which leads to the condition

5

R-2118

for (x,s) in c, that is

0 < x < c(s,t)

-l<s< I

The cavity function n(x,s,t) is determined by using the linearized kine-

matic boundary condition for the surface

y = n(x,s,t).

This linearized condition is

nx(x,s,t) + n t =a+ Vg (t-x) + y (X,s,O+,t),O < x < c(s,t); -1 < s < 1

subject to

n(O,s,t) = 0 -1 < s < I

and the closure condition

n[c(s,t) ,s,t] = 0 -1 < s < 1

Let us begin the determination of *(x,s,y,t) with the representation

(x,s,y,t) =- ff y , [f]ddaB+W [(E-X) 2 +(o-S) 2 + y2]3/2

41ff FB+W [(_x) 2 + (a-s) 2 + y2]1/2

1 +121 2

[0] = *t(c,,O+,t) -fl(,I,0-,t)

[y I = y (E,o,O+,t) - *y(E,o,O-,t)

This implies a representation for the acceleration potential P(x,s,y,t).

We have

ax at 1]/ [OirdcB+W [(-x) 2 + (-s) 2 +y2] 3 2

+ Iff ( (02d E2 d4r B+W [((-X)2 +(a-s)2 +y2] 3/2

6

R-2118

At the leading edge 0] = 0; therefore, since y/[ ]3/2 0 as C 00,

an integration by parts yields

a- a 1 -IB+W (-x)

2 + (-s)2 + ]++y23/2

However, we require the acceleration potential

P(x,s,y,t) = ,x + t

to be continuous in the wake and so

ax +PO ) 1 = 4Tf f Y[Pld~daB 2 + (a-S) 2 +y2] 3 / 2

Since we also require the normal velocity y to be continuous in

the wake, we have [cy = 0 there, from which

SI -I dd12ff 1/2 [ydyd,

2 B [(-x)2 + (s)2 +y211/ 2

1 (1a) 1 -. i [ydX daT, f f 17 f ( )

=f • Id ddd-I -1 o (Cx)2y+ (-s)2 + y2]/ 2

1 I (a) [dXddda-1~f [x2+(a-S)2+y2] 1 / 2 ' o [yd o

1 £(a) £ (a)

f f (X) f [4y]dX d~da 6-l o [(&_x)2+(aos)2 +y2] 3 / 2 &

Then

+-L + f ,4 dE daax It) 2=- T 'x ')-l [x2+(a-s)2+y2] I / 2 0 y

W f f T&[(E.x)2 + (-s)2 + y213/2 £" [,y]dA d~da

-1 o (Fx2 (7)2

+ l () (R-x) a ()da- o [(E-x)2 + (a-s)2 +y2 3/2 "t [ Y]dd

S7

R-21 18

and an integration by parts of the second term on the right-hand side of

the last equation shows

1 9.(o)

ax O 2-, t 2 )i=- Y 1/2 f y Jd do- [x 2 +(o-s) 2 +y 2 1 0

I Q.(-x)

+ -1 o [(-x)2 + (a-S)2 +y213/2 {a E y td-[}y

Downstream of the partial cavity and on B, we have [y = 0 by virtue of

the boundary conditions. Therefore

+ = (t 0.daax at 2 I t f 1/2 0dta

ax_ 7r -1 [x2 + (a-s)2 +y 2 ] 0 o

+ 1[ ( y 0 -) /yf - af t 0 1d X -[¢oy d~do .+ f [ (EX)2 + (OS)2 +y21/ at d

This can be expressed in terms of n(x,s,t) if we use the condition

nx (x,s,t) +n t = 4y (X,s,O+,t) +a +v g(t-x) = I which holds on C; and the

closure condition

n[c(s,t),s,t] = 0

with

n(O,s,t) = 0

We find

c(at) [ c c1 [10y dE f n td&= t J" n(Q,ot)dE=At(a,t)

0 0 0

where A is the cross-sectional area of the cavity. Also

C(at)

at & y y

t {nff(X ,at) + nt}dX -n,(E,o,t) -nt

Ik8

R-2118

c c(a, t)

M(,,a,t) = a'2" f nt(X,o,t)dX -2nt(F,o,t) -n (E,a,t)

Our development shows that the acceleration potential can be expressed

as

P(x,s,y,t) = (x,s,y,t) + *t(x,s,y,t)

If *A (at)da

= -I [x2 + (o-s)2 +y2] 1/2 tt

+ l. ft) (C-x) M( ,ot)d~da-f o [(E-x) 2 + (a-s) 2 +y21 3 /2

1 )

+ ,~f fy[P]d~da- o [(E-X)

2 + (a-s)2 +Y2]3/2

which is the same as

P(x,s,y,t) - . Att(s,t) RXn(x 2+y2 )

S I l -sln[ -s l+ V(a-s) 7 + xz + yZ']At (,t)daa lii

-1

1f c(at) (R -x) (s-a) M(ea-t)d~da+r 7- -1 o (&-x)2 +y2 [(s-a) 2 + (E-x)2 +y2] 112

a I(o) (s-a) (P]d~do

+ T,7-- o ( -x)2 +y 2 [(s-a)2+( -x)2+y2l1/ 2

We introduce the quantities:

x = = Cv ; c(a,t) - Ec(a,t)

y = C (a) = ch(a) A Ac2

tI

which implies a high aspect ratio wing and, because of the scaling of the

time, it also implies oscillations whose wave length is of the order of

the chord.

We now seek an asymptotic expansion for d(x,s,y,t) such that

9

6

R-2118

O(xOsIy~t) O (CYSCW, :T)= S ~ (C (,W,T )

k=l

whereVk~ ()

L .° k( = 0

Each *k is to be determined by equating terms of like order after the ex-

pansion has been substituted for * in the representation for P. However,

for the time being, let us forego a detailed development of the expansion

and be concerned in this paper with only the lowest order term O1 (x,s,y,t).

In the appendix, however, the higher order terms are examined.

Apart from the term

I4 A t(s,t) in(x 2 +y 2 ) A n(x2 +y 2)),

the lowest order approximation to P can be obtained by neglecting the com-

binations x2 +y 2 and (&-x)2 +y 2 which appear in the radicals in the last

expression for P. With respect to the non-integrated logarithmic term,

we need to consider

IAt In c2 (y 2 +w2 )

and hence

A (s,t) In c = (0 J [ydE .in ctt 21 at0 ly

Withotit loss of generality, we can take vl(g) -1 so that the order of

-- Altt(s,t) In c

is given by in e. Now this order cannot be matched with that of any other

term which appears in the expansion of P. We, therefore, conclude that we

are forced to set

A (s,t) - < s <I tt

With this, it follows that the first order approximation to P is given by

P1 (x,s,y,t) 1 x(x,s,y~t) + * t(x,s,y,t)

10L

R-2118

c 1 c( St) (&-x)MN(l,a,t)d~da

1 0 (- X)2 +

4w as f"- I( T 7 -'X)2 + y2

or

c(st) (C-x)M1(C,s,t)dCP (x,sy,t) = 27r

0 Q _x) 2 + y2

+ (S) y rPlld

o (-x) 2 + y2

We see from the last result that, with s as a parameter, Pl(x,s,y,t) isa two-dimensional harmonic function in the exterior of the cut along the

x-axis from 0 to I(s); and we also see that P1 vanishes as rx---y 4.

Analytic Determination of sLet us turn to the determination of P1 =x ( x s t) +1 and~ (x s'y't).

For typographical convenience, we will drop the subscript and often suppress

the parameter s. With this understanding, the first problem we have to

solve is this: Find P(x,y,t) such that

P(x,y,t) = x(X,y,t) + *t(x,y,t)

is harmonic in the exterior of the slit along the x-axis from 0 to 9(s),

wh i le

P(x,O+,t) = Ox(x,O+,t) + t (x,0+,t)= 1j(t) 0 < x < c(s,t);

P y(x,O+,t) = yx(x,O+,t)+ yt(xO+,t) = 0 c(s,t) < x < E(s);

P y(xO-,t) *yx(xO-,t) + ytx(,O-,t) = 0 0 < x < Z(S)

We assume that, if P(x,y,t) possesses any singularities, then they

exist at only one or both of the points

(0,0) ; (c,o+)

where (0,O)denotes the leading edge of the foil and (c,O+) denotes the

11

gt

R-2118

moving trailing edge of the cavity. Let us suppose that the physically

acceptable solution for P(x,y,t) is the one with the mildest singularities

at (0,0) and (c,O+). With respect to the behavior in the neighborhood

of infinity, we impose the condition dictated by the three-dimensional

analysis, namely

P4*O

as /7 + y-

If z = x + iy, the function defined by

i=r z -c

z ct

8=(t) = V4/-c;

maps the exterior of the slit along the x-axis from 0 to 2Z into the quarter

plane Im 4 > 0, Re > 0 where Im denotes the imaginary part and Re denotes

the real part. For this mapping, the branch /'(-z-)/z is defined as shown:

V(,-z)lZ = jv'(2-z)/zj exp i(61-6)/2

y

z Z/"

z"' (i-z)

e L~ 2

L3 (2.,O)L3

FIGURE 2. PHYSICAL PLANE

12$

R-2118

(The quantity 8 = ,r(-c)/c), for example, is negative.) Then L1 is

mapped

A

M1

i M2 M 3

FIGURE 3. MAPPED PLANE

into M defined by

0 ; ->n>O ;

L2 is mapped into M2 defined by

n - 0 ; 0 < < V'

and L3 is mapped into M3 defined by

n- 0 ; /--< < W

The point at infinity in the z-plane is mapped into

Under this mapping P(x,y,t) is transformed into a function (A,n) which

is harmonic in C > 0; n > 0 with singularities at r = 0 and r = ,. The

function

- t)

vanishes along M1 while

a { -_ " t ) }an

vanishes along M2 and M3 - By using the theory of analytic continuation.

13

7.

R-2118

it follows that ID is the real part of a function of which, except for

poles, is analytic in the 4-plane. (The continuation shows that we cannot

admit branch point singularities.) If we admit only the mildest singular-

ities at 0 and C = ", we have

0 P(t) + cl(t)Re J+ K2 (t)Re C

or in terms of z

P(x,y,t) = (t) +r(t)Re 1 + k2(t)Re z _

z

In order to avoid excessive writing in the sequel, let us express the last

result as

P(x,y,t) = Re F(z,t)

We now have the problem of finding the velocity potential *(x,y,t)

which is harmonic in the xy-plane minus the positive x-axis; exhibits a

wake downstream along the x-axis from x = k to x = o; and which satisfies

x + t = Re F(z,t)

The general solution of this equation can be expressed as

0

O(x,y,t) = f Re F(a+z, a+t)da + O(x-t+a,y,a)a-t

where a is an arbitrary constant. We may suppose that O(x,y,O) 0 for

all (x,y). Then, if we take a = 0, we have

0

O(x,y,t) f Re F(a+z,a+t)da-t

This function is harmonic for 0 < ang z < 2n; and, if we write it in the

formx

O(x,y,t) f Re F(X+iy,X+t-x)dXx-t

it is not difficult to verify that it shows a downstream wake in the sense

that 4, and Oxsuffer a jump, provided t is sufficiently large, as z crosses

the downstream x-axis.

0 14

R-21 18

Determination of the Coefficients

The next step is to enforce the two boundary conditions which 4

must satisfy: the closure condition, and the condition on the regular

part of P at infinity.

The component 4y is

0 ay(x,y,t) f f 7 Re F(a+z,a+t)day -t

- f Re i F(o+z,cy+t)da

ft

-tatxf Re i F(a+z-t,a)da

o•

For 0 < x - Oi < Z, we must have

t-[[I+Vg (t-x)] = (x,O-,t) = f f Re i F(a-t+x-Oi,o)do

0

t-x t

a { f + f Re i F(a-t+x-Oi,o)do}o t-x

However, Re i F(o-t+x-Oi,o)= 0 for 0 < a - t+x - Oi < k or t-x < a < t + Z-x

and, consequently,we must have

at x

t-+v9(tx) f Re i F(a-t+x-Oi,a)d•0

Then, if we set t-x T , we see that for T > 0

9 T 0

or

f [c, +vg ( d)a - f Re i F(o-T,or)da (1)0 0

15S

R-2118

This is the first equation which relates the unknowns Ki, K and 8.2

For c(t) < x + Oi < Z, we must have

a t-[a+V (t-x)] = f f Re i F(o-t+x+Oi,u)du

0

t-x t= { f + f Re i F(o-t+x+Oi ,C)do}

0 t-x

We already know that

at-x-[a+v (t-x)] jxRe i F(a-t+x-Oi,a)da

go0

t-x= x f Re i F(a-t+x+Oi,a)da

0

for t-x > 0 because the space variable a-t+x, in the range

-(t-x) < a - t + x < 0

required in the last integrals lies ahead of the foil and F is continuous

there. Therefore

t0 f Re i F(c-t+x+OiY)do (2)

axt-x

c(t) < x <

is the second equation which relates the unknowns.

Equation (2) can be simplified. A change in the dummy variable of

integration, a-t+x=)=, gives

a ×

0 f f Re i F(+Oi,X+t-x)dX (2a)0

C < x <

Note that

Re i F(X+Oi,X+t-x) = 0

when

c(x+t6x) < x < z16

R-21 18

This inequality defines an upper limit for the integral in (2a). For

simplicity, suppose that

X = c(A+t-x)

has but one root Al for each t-x. Then

X,=C(X 1 +t-x)

and this root is a function of t-x so that we can write

X,= x 1 (t-x)

(The'root XA1 is given by the ordinate of the point of intersection of

and the line

X = a -t+ x

which passes through the point (t,x) with slope equal to 1.) Consequently,

equation (2a) becomes

A (t-x)

-x- f Re i F(X+Oi,A,+t-x)dX = 00

orc (x1+t-x)

a Re i F(X+Oi,X+t-x)dX = 0ax

0

and, if we introduce t -X=T, we have

c(x I +-[)

d f Re i F(X+0i,X+T)dX 0

Integration gives

f Re i F (A+O i, X+t)d X0

A 1(0)

- f Re i F(X+Oi, -)dX = cons't. K

0 17

R-2118

This can be written as

f Re i F[X+O1,X++XI-c(xI+r)]dX = K00

and if we set

t = + +T

we have

c(tC)f Re i F[X+Oi, +t"-c(t*)]dX = K (2b)

00

where t is arbitrary.

We now turn our attention to the cavity surface, the equation for which is

y = n(x,s,t) 0 < x < c(s,t)

or, if we suppress s,

y = n(x,t) 0 < x < c(t)

and, as we have seen in the three-dimensional analysis, n(x,t) can be deter-

mined by using the linearized kinematic condition

nx(x,t) +nt(x,t) (xO+,t) + [c+v g(t-x)I 0 S x S c(t)

In terms of F this is

tnx(x,t) +n t(x,t) .- f Re i F(u-t+x+Oi,o)da + [a+V (t-x)]

t-x t

- f + Re i F(o-t+x+Oi,a)do} + [a+v (t-x)Io t-x 9

which, by virtue of the first condition (1), reduces tot

nx(x,t) +nt(x't) =.x f Re i F(a-t+x+Oi,a)do 0 : x < c(t)t-x

for t-x>0. Now n(x,t) must vanish at the leading edge, i.e.,

n(O,t) = 0

and the only solution of the differential equation for n(x,t) which satisfies

this condition is

18

R-2118 4

n(x,t) = - [(x-A)Re i F,(A+Oi, ,+t-x)dX0

The formula for the cross-sectional area A(s,t) determined by the

cavity surface and the foil has already been used in the three-dimensional janalysis. Here it is:

c(t)A(t) = f n(x,t)dx

0

C(t)cf [c(t) - X]Re i F[A+Oi ,X+t-c(t)]dX

0

The closure condition to be imposed is

n[c(t),t] = 0 4

This can be enforced by integrating

a tnx(x,t) +nt(x,t) =-x f Re i F(a-t+x+Oi,a)da

axt-x

x= a- f Re i F(X+Oi,X+t-x)dX

0

0 <_ x _ c(t)

Since n(O,t) = 0, we have

x xn(x,t) + f nt(E,t)d& = f Re i F(X+Oi,X+t-x)d1

0 0

or, using n[c(t),t] = 0,

c(t) d c(t) c(t)at n(C,t)d = t- f n(C,t)d&= f Re i F(X+Oi ,X+t-c)dX

o 0 0

which is the same as

c(t)A t(t) = f Re i F[X+Oi,X+t-c(t)]dX (3)

0

19

R-2118

This equation relates the rate of change of the cross-sectional area of

the cavity to the unknown parameters K1, K2 and a. However, Eqs. (2b)

and (3) imply that

A t = K°

from which

Att 0

(as we already know is required) and

A = K t + K0

If'a monotonic growth of the cavity cross-sectional area is precluded,

it follows that we must take

K = 00

It appears then that the first order approximation produces a constant

value for the cross-sectional area of the partial cavity. Equation (3)

then becomes

c(t)f Re i F[X+0i,X+t-c(t)]dX. = 0 (3a)

0

and we now have equations (1) and (3a) for the determination of K1, K 2 '

and a.

The last equation for the determination of the unknowns is

Llzl _*Re F (z,t) = 0

or

Re{ P(t) + + (t)/-} = 0 (4)

The equations (I), (3a) and (4) form a set for the determination of

Ki' K 2 and 8. The equations are linear in KI and K2 , but disconcertingly

nonlinear in 8. Two of them are highly nonlinear integral equations in t

and the ?arameter s. Their solution for given ,vg (t) and p(t) can only

be accomplished by using formidable numerical procedures.

20

R-2118

CONCLUDING REMARKS

The result 4x +4 t =Re F(z,t) presented on page 14, where F(z,t) is g

also given there, can be shown to subsume some classical results. For

example, if the dependence on time is eliminated, and KI and K 2 are eval-

uated from the kinematic condition that the vertical velocity is simply

a, then, together with the (steady) closure condition, the result can be

shown to reduce to that of Guerst 12.

As another example, take K, = 0 and write

+ x +t= '(t) + I/ K2 Re (1 - z)/ 1 / 2

As 6 *, take p - K2 =0 and we have

K t 2~i = -.Re(--/

which is the case of the unsteady, completely wetted, foil.

The analysis presented thus far is in a somewhat unsatisfactory

state inasmuch as it has been shown that Att = 0 so that no two-dimensional

source-like flow at distances large in comparison with the chord of the

foil is possible. It is clear that this result comes about because the

aspect ratio has been taken to be infinite. It is desirable to determine

to what order in aspect ratio a non-zero value of Art will occur. To this

end, we are devising a procedure for the methodical determination of

higher order asymptotic approximations. If c is the reciprocal of the

aspect ratio, and, if the orders of the asymptotic approximations are

characterized by 1, e, E2 PnE, E2...., we find that there are strong

indications that A tt remains equal to zero at least until K corresponds

to e2. Thus, it may be expected that, as the aspect ratio is increased,

the time variations of the cavity cross-sectional area and, hence, the 4

variation of the total volume of the cavity on the three-dimensional

foil) will diminish.

Finally, comparison of the formalism of this report with the two-

dimensional analysis of Van Houten shows that, whereas we have demonstrated

the necessity for taking Att = 0, thereby eliminating the source-like

21 .!

R-2118

behavior of the pressure at large distances, Van Houten has retained

this term in his work. We believe this to be improper, especially in

view of the requirement that the boundary condition at the cavity can

only be a statement that the difference between the cavity pressure and

the ambient pressure is given, whereas, if the pressure grows logarith-

mically at large distances, no ambient pressure can be defined.

The solution of the integral equations which Van Houten analyzes

numerically is equivalent to finding the acceleration potential

P"(x,s,y,t) = +x=1 A*

A- t t ( s ' t ) 9,nlx"-Y-

+ '--ai f-TZ n2la-sl A* (s,t)do47 s-1 l tt

I c(s't) (E - x)M Md2r (Ex) +yL

+I f~ [(P*dE2 7 0 (E-x) L + y,

wherec(s t)A t(s,t) = I! [ y ]dE

M a[C(ysft) w'dX -aty y

and P is subject to the cavity condition

P (x,s,O+,t) = 11(t) 0 < x < c(s,t); -l <s < I;

and the condition

P (xsO+,t) = 0

y

along the wetted surface if we take

(XS* ,-Ot) = - - Vg (t-x)

22

k

R-21 18

there. The representation for P shows that it may possess a logarithmic

singularity at infinity. However, if we admit such a singularity when

P *is determined by the function theory method used above, it turns out

again that A" (s ,t) must be zero in order to satisfy the boundary conditiont t

on the upper side of the foil downstream of the cavity and the closure

condition. Thus, it appears that the first order approximation A =t 0

can be regarded as a consequence of the imposed boundary conditions for

the two-dimensional approximating problem rather than a consequence

of matching in an asymptotic expansion.

4

4

23

R-2118

REFERENCES

1. Breslin, J.P.: "A Theory for the Vibratory Forces on a Flat PlateArising from Intermittent Propeller Blade Cavitation" Symposium onHydrodynamics of Ship and Offshore Propulsion Systems, Paper No. 6,Session 3, Det Norske Veritas, H4vik, 1977

2. Birkhoff, G.: Hydrodynamics, Princeton University Press, p. 34, 1950

3. Leehey, P.: "Boundary Conditions for Unsteady Supercavitating Flows:Fourth Symposium on Naval Hydrodynanics. B.L. Silverstein, Editor.pp. 577-597. Office of Naval Research, ACR-92, 1962

4. Benjamin, T.B.: "Note on the Interpretation of Two-DimensionalTheories of Growing Cavities" J. Fluid Mech. Vol. 19, No. 1, pp. 137-144,1964

5. Van Houten, R.J.: "The Numerical Prediction of Unsteady Cavitationof High Aspect Ratio Hydrofoils" MIT, Department of Ocean Engineering,August 1979

6. Steinberg, H. and Karp, S. : "Unsteady Flow Past Partially CavitatedHydrofoils" Fourth Symposium on Naval Hydrodynamics. B.L. Silverstein,Editor. pp. 551-575. Office of Naval Research, ACR-92,1962

7. Wang, D.P. and Wu, T.Y.: "General Formulations of a PerturbationTheory for Unsteady Cavity Flows" Journal of Basic Engineering ASMETransactions, Series D, Vol. 87, No. 4, pp. 1006-1010, 1965

8. Unruh, J.F. and Bass, R.L.: "Doublet Lattice-Source Method for Cal-culating Unsteady Loads on Cavitating Hydrofoils" J. Hydronautics,Vol. 8, No. 4, pp. 146-153, October 1974

9. Johnsson, C.A. and Sintvedt, T.: "Propeller Excitation and Responseof 230,000 TDW Tankers" Ninth Symposium on Naval Hydrodynamics. R. Brardand A. Casera, Editors, Vol. 1, pp. 5111-G69, 00jffice of Naval Research,ACR-203,1972

10. Noordzij, L.: "Pressure Field Induced by a Cavitating Propeller"International Shipbuilding Progress, Vol. 23, No. 260, pp. 93-105, 1976

11. Kaplan, P., Bentson, J. and Breslin, J.P.: "Theoretical Analysis ofPropeller-Radiated Pressure and Blade Forces Due to Cavitation" PaperNo. 10, Symposium on Propeller-Induced Ship Vibration" Royal Inst.of Naval Architects, London, 10-14 December 1979

12. Guerst, J.A.: "Linearized Theory for Partially Cavitated Hydrofoils"International Shipbuilding Progress, Vol. 6, No. 60, pp. 369-384, 1959

24

I

R-2118

APPENDIX

HIGHER ORDER APPROXIMATIONS

FOR A CAVITATING FOIL

b

R-2118

We proceed to give a method for the theoretical determination of

higher order approximations for the cavitating foil. It is beyond the

scope of the present report to supply all of the details of the develop-

ment. It is expected that a fuller analysis will be produced later. Here,

our object is to find the order of approximation which gives the first

indication that Att #Owhere A is the cross-sectional area of the cavity.

We have found that the acceleration potential is

P(x,s,y,t) = x(X,s,y,t) + *t(x,s,y,t)

1 a a 1 c(C;t) y ]d~do

-1 0 [(Ex)2 + (os)2 +y2]1/ 2

+ I (a) y[P]d~do

4-l [(-x) 2 + (-s) 2 +y2]3/ 2

where

[41 = WE'at0) =(',uO+,t) - f(laO-,t)

In order to satisfy the boundary condition for the cavity, we must

satisfy

Ic(a t) [4 ]ddaP(x,s,O+,t)-i [P(x,s,t)] = - 4I f

for 0 < x < c(s,t) ; -1 < s < 1

In order to satisy the boundary condition for the wetted part of

the foil, we must satisfy

xy(X,s,O_,t) + ty(X,S,0±,t)

1 a a xt)y _I a2 1 t(O) /(x-) + (s-o) 2 [Pld~do= 2 ax at y 41T axas 0 (x-)(s-G)

An integration gives

CW

a 0y ( ,s,0,t)dC + at y( 's'O-+t)d

axA-

L A-l

R-2118

I!

2 fx

+ I (o) /,(x- ')2 +(s-a) 2 - I--- Pdd

- (x-) (s-) [P]dd

and if we solve this for 6

f 4,(C,s,O±,t)d& we findx

II

f y s;,0±,,t) dl - f [ y( ,s,t)1dC

xY x

x 1 (G) +________

+ f . f ( . )2 + (sa)2 I .} [P(,a,A+t-x)]d~dadAx-t -1 o s-0

In dimensional terms, the normal velocity along the wetted part of

the foil is required to be

-[ua + v (Ut-x)]

9

In dimensionless terms, this velocity is

v-[a + -Eu (St-sx)]

U

where S is the semi-span. For a foil of large aspect ratio, we can take

KS

where c is small and K is the chord length, so that the normal velocity can

be expressed as

V 6-{a + 9 [T (t-x)]}

U C

or

-[a + V (.-)]

In dimensional terms, the acceleration potential at the cavity wall is

P(t)

A-2w •

R-2118

In dimensionless terms, it is

p P(CU) _(t-P-- U= - = p(-

~I

In dimensional terms, the trailing edge of the foil is

x = Sk(s) = Sef(s)

and in dimensionless terms

x = Ef(s)

In dimensional terms, the trailing edge of the cavity is given by

x = c(s,t) = S C(s ) Sc(s,t)'U

and in dimensionless terms

x c c(s, fr.)= (4

Since the unknowns depend on the parameter- c, let us introduce

x - F-y ; - y C= jj t = ET

and assume the inner asymptotic expansions

O(x,s,y,t) = O(Ey,s,pIETj) = c' (vSit;e) = K cVK(E) 4K(V,S 'u,

')

c(s,-) - c ()c(s,T)E K1I

where

u ° (E)- - 0

K

In terms of the new variables, the boundary condition for the cavity is

(V sO ,T; ) - ['P(y,s,T;C)]

I c( ') (W (w,Tr;E)dwdo

-) f ' /4w Ty + ' - o [(s-a)2 +r2]1 2

A-3

R-21 18 4

where r2 = 2(-2

and 0< Y< c

This can be expanded into

1 ( s T ) kni -

0

+ CI. 1 a,T) 1L0-L Zn 2j0-sJ I Iduida4r as f -s 1

(A- 1)

a+a Ia 2 C 2iw-yi [l-j d0

la-sL.E2(-)2

+ 3 1c(U,-r) a-s 4

-1r 3s f a0-s C2 )2zn2lo-s( j'f? 1dwda

+ o(s2 ;UqsVWqy)

In terms of the new variables. the boundary condition for the wetted

surface is

f 0 (W,S,O±,T;c4dw = ++

Y Y1 a f(a) _____

+i - as }~-)[P(,G,a,+T-Y;E)]dwdadX

where

q c(-)

An expansion of this yields

OD1 c

Y Y4

+ E a f. ( Y) W,G,X+T--Y;c)]dwdadX

Y T - o

A-4I

R-21 18

IYf(a) ['P(W,S,X+T-Y;C)]dwdX

yT 0

+L -J f"f - Zn2Is-ai (X-w)[P(w,sX+T-y;E)]dwdadX

+ 1 f 0a

+4 f _T f o(E2 ;j,s,W) [W(w,a,X+T-y;c)]dwdadXY-T o

where

h(ca-s) ={j >

The expression (A-2) shows that the inner asymptotic expansion of

4)(x,s,y,t) should have the form 4)(x,s,y,t) = 4-)SypsjiE) =cf(ypstiT;E)

where

'(YV~,suTp,) = 4)1(YPsPiPT) + C 2 + C tn + 24+

Then for P(x,s,y,t), we have

P(x,S,Y,t) 4) (xps'y't) + 4)t (x,s~y,t) or

P = 4 (Yts*VT;E) +T

-P I(Yps,j~T) +EP 2

+ £29,nc -P + e2p + .

3 4

The function P(x,s,y,t) is a three-dimension potential function in2 22

the exterior of the foil. If we apply the operator +x + - t h

assumed asymptotic expansion, we find

A- 5

R-21 18

v K(E) [PKyy + PK~lp + 62 P KS

and, hence, each P K(Y's,ii1T) must satisfy

PY (y,5,'JT) + P H (Y'sPT)

in the exterior of the slit defined by

0 < y < f(s) ; '=0

For example, we have

P K + P =PJ 0 K =1,2,3

wh il1eIP +yy P =-p P .

The equations for the I 's can now be found by equating terms of

like order in (A-]) and (A-2).

Consider first In (A-i) there is no match for the term

2r +a f [ ,,]dw0

and, therefore, we must take

(.-+~- f [ 1 d. -- + '- A1 (s,T) =A (S,T) = 0

Here A is the first order approximation to the cavity area.

For the cavity, 4~must then satisfy the integral equation

I a c I (s ,t0P(T) [ P1 (y,s,T)] -L(2 T y T f Znjbw-yj [ , ]dw

0

and, for the wetted surface, it must satisfy

0 c 1 1Tf(s) (P 1 (w,s,X)]dwdX

f (WpSpO±,T)dw± f [0, (W,S,r)]dw + 'Tf f X + ~-Y 2Y 0 0 Y

from which

r A-64

R-2 118

T f(s) [Pl(w;s,X)]dadX

= 21T D f f X+ - T0 0

and1 11 .( Y ' s , O -+ ,r ) + I 1 T ( Y p s , O -± ,T )

f(s) [P (w,s,T) ]dwf(7a + (Y',s , T)]T Y 0

where, on the wetted surface,

= -[a + v (t- )]

As we have seen, these integral equations can be solved by first

finding the analytic function F(z,t) such that

Re F(z,t) = 1ix(x,s,y,t) + t(x'syt)

a, c I f(s) P[IP1dw+ ( y'-) f Zn/(y) 2 +j 2 [qudw + - foir aY o (W-y) 2 + 1a2

2I (ax -aL n -x + I 9" (s) y[P ]d21 x3 nV-.x + P d f (E-x)2 +'0 0

1 c(s,t) ( -x) M1 ( ,s,t)d I P(s) y[P 1 ]df J ( -x)2 + y2 + 27r T -- 70 0

The above representations and the integral equations imply that we must

have

ReF (x+Oi,t) = p(t)

for the cavity and

Re F (z, t) = 0ay

y=O_+

Iy(xs,0±,t) = -[a+v g (t-x)]

along the wetted surface while ReF (z,t) vanishes at infinity.

It has been shown that the first approximation to the cavity surface

y = n (x,s,t) can be found from

A-7

R-2118

nlx(X,s,t) + nit(x,s,t) = [bly(xSO,t)]

x= x" f Re i F(\+Oi, k+t-x)dX

0

xRe i F(x+Oi ,t) + f -i- Re i F(k+Oi ,X+t-x)dX

0

for 0 < x < c. From thisc-c

[1 (C-c,s,O,t)] = Re iF(c-E+Oi,t) - Cf ' Re i F(X+Oi,X+t-x) dX 4y 0 1X=C-

Rei F(c-E+Oi,t) _I aS ReE F(X+Oi X+t-c+c)dXatf e

if c is a sufficiently small positive quantity.

The closure condition

n(c,s,t) = 0

givesc cf nlt( 's't)dC f [bly (E,s,O,t)]d

00

orc

Alt = fRei F( X+Oi,X+t-c)dX

0

orc

A = - SRei FX+Oi,X+t-c)dX

0

but we have found that A tmust vanish. Therefore

L Rei F(c-c+Oi,t)L o qly( - , , t) = .-+o 1 - E:

which is infinite as can be seen from the explicit expression for F. On

the other hance

L [ly (c+cs,O,t] = 0

A-8

R-21 18

Consider second 42 For the determination of *2' we need to

cons ider the term

I* c

We leave c unexpanded and define it to be equal to c1. Indeed,

[,Ivanishes beyond c, so that such terms can make no contribution

to 02. A similar argument will hold for the higher orders. Since no

other term of order F_ Znc appears in (A-i), we must take

A = 4]dw =0

0

We see now that, even for the second order approximation, the second

derivative of the cross-sectional cavity area is zero.

With this, 2 must satisfy

'P2 Y S,O+,T) - -L [P2 (y~st)]= Wc1(s,

for the cavity.

For the wetted surface, 2 must satisfy

f (w,s,O±,T)dw T- f [0 (w, s,-)]dw

T If (a) ____________crd T..L f(S) [P2 (w,s, X))d(d X

from which

T f(s) [P (w,s,X)Idw~dX(YpspO±VT) 2 ~~( ,) +--

2p+2 0YS'r] 27r Dy 02

and hence

2p (YSOT)+ 2 T(YpSQ±,r)

7Yf 7s) [P (W,S,T)]dw

A- 9

R-21 18

These integral equations can be solved by first finding the potential

function

Re F2 (zt) = 2x(X,syt) + 2 t(x,s,y,t)

c v f(s) iJ[P 2 ]dw

2-r ""ay+ T n (-y) + 2 T[]d+" (W-y)z + P z0 0

c 9 (S) ysP 2[]dP

(x+ f kn/(-x)2 +y2 [2yd+0 0

Sc (U-x)M2 dE I 9,(s) y[P2] dE

=21T 27 (0x' y / ( -x) 2 + y2_

The function Re F2 (z,t) must vanish at infinity and the integral

equations show that it must be such that

Re F2 (x+Oi,t) = P2 (x,s,O+,t)

for 0 < x < c, while along the wetted surface

Re F2 (z,t) y=0 + = (2xy(X,s,O,t) + 2ty(X,s,O,t)

where d 2y(x,s,O,t) is an induced velocity on the wetted surface and

P2 (x,s,O+,t) is an induced acceleration potential at the cavity. The

latter quantities are to be determined by matching inner and outer expan-

sions.

Consider next 3: The component 3 is to be determined by equating terms

of order 62 in E in (A-l) and (A-2); but first we need to consider the terms

c i"+ - ) f E2 n [1 e Idw

Because the term of order e2 (zn0)2 cannot be matched in (A-1), it

follows that we must have

c

A = =0

0

A-IO- 4I,

R-2118

We can now see that 43 must satisfy

P (Y'S o+PT) 1 [P3(Y,S,-r)]3 2 3p3(y,s,O+,t') - -P 3 ys,)

c1

f Zn~w-yj [ 3 ] dw0

Cl

0c1

1 32 )2- 8 ' s' f (W-Y)2 [¢ I dw

0

for the cavity, i.e., 0 < y < pl; p = 0+.

For the remainder of the foil 03 must satisfy

f 4 31 (W,s,O±,T)dwY

c s T f(s)

-+ 203 f f [P3 (ws,

'W)]dwdd

Y 0 0

1 1 f()-4r- f .23 f f h(o-s) (X+y-T-w) [P1 (w,a,X)]dwdad

o -1 o

from which

3 f(s) [P3(),s,X)]dwd€311 ' '0+ T =+ 31,1 ' 2ny" 0 X,+Y-T-(W

I T a 3 1 f (c)

+- T 3 f f h (oy- s)[ P I(w ,7;)d d,7 d X

0 1 0

and

3,4Y 5,0± 4) +031T (YVSO±pT)

+A + ( ,,T)

A-11

R-2118

a -s) [P3( ,sX)]dw

2Tray Y-W

1 a3 1 f (o)+* - f f h(o-s) [P1(w,a,X)]dwdo

These integral equations, which express the boundary conditions,

can be solved by first finding the potential function

P3(ys,T) =Re F3(zt)c1

=- a- + f nV(w-y)2 +Z [,I d

+ L f(s) i[P3]dw 1 + c 1+ L I -( _ ) + -- 2 a F [t ]d w

_iT(aT a a2 1d

8ff ay + -) ' I {(w-y) 2 + 21 [0 Idw

0

1 a2 f(s)

0

where z = Y + i. From the integral equations, we find that Re F3 (Z,T)

must satisfy

Re F3(y+Oi,T) = P3 (Y,s,O+,r) for the cavity and

Re F3 (z ,-) =0 3 y (Y'sO 'T) + 0 3p,(y,s,OaT)

for the remainder of the foil; while F3(z,T) must have a pole at infinity.

If

c2, a f [01dw

is not zero, 04 remains to be determined from the equations for order c2

and there is a coupling between the approximations of third and fourth

order. In the solution for ReF 3, the induced quantities 0 3p(ys,O,T)

A-12ij. .5

and 3(y~,O+,) ar at irstR-21I 18 IandP 3YSO+,) ae a fistunknown. Ultimately they are to be found

by matching inner and outer expansions.Finll, lt s cnsder4): Along the cavity interval 4~must

satisfy

P (Y,s,cj+,t) - P4 2 [ 4 (YIsIT)]

r Clf X j- j (41(, ,)d

0

+ "s' n cl-l C dd12.2 as f-si f []d4 da-l o

+ay 1 2cl

1 330- s 1 33 1 1 S (w-y)2 [l + in l-i da

Along the wetted surface, 4 must satisfy

*4 (Y,S,O±,T)

= ± I[~()]- ~ tf~s [P4 (w ,s,X)]ddX

0 0

+ I 93 f~a 3 Is-al + II!k Lzn2s--0 ol~sa 2 n25 P1 (w,a,X)]dwdadX

fro wic + h(a-s) {nj)X+y-T-wj+l)

0) (YvsO±,T) "44jT(y'S,O±,T)

= ,* 33 1 3f(S) [P (w,S,T)]dw

0

1 33 1 f (a) 3I-l+~L. n2sa+ -~~f f 3 s-al+ Ia 2 1s-a lF7 S 0 )- [P 1(W,a,T)]dwdaL+h(a-s) (inly-wi +11

A- 13

R-21 18

In accordance with these integral equations, the function

Re F (zT ) (--+ a-) f YnV'(w-y) 2 + - [I+]dw f(s)[

27r 3y a 4 7 f (wy) z +r0 0

= ~*£ny fi2 a 1 f 7 ) p(P 4 ]dw

n i r y a(_t f [ ]dw+ T f y) y

27r 77YTZ+=) 7Tf[1 4 [ 414I0 W 4

must. satisfy

Re F4(y+Oi,) =

P (y,s,O+,T)4 1c 1

- ..± .2L n21o-sl f [41 ]dcoda

-.i as --1 0-s o~

c

+ .f1 (w-y) 2 ZnjW-YI [€1 1dw0

as I1 O C-S ( (_y)12 +Zn 2fa-sj} [4lQdwda

for the cavity, and

a Re F (zT)j

=4P (YS,O±,T) + *4 pT(y,s,O±,T)

J' f 1jT -L :0 +4I2. Zn 2js-a1S -1 o S [P Idwda

h(a-s) (Znly-wj+ 1} Jfor the wetted part of the foil.

As we have seen, the function P4 . 04y + 0 T must satisfy the Poisson

A-14

..

R-21 18

equation

P + =-P4yy 4PjIJ Iss

where

c

P Y,,P1)f 9,n Ay-w~) + jj t4 dw(y,s,p,) ( -_ a+T) n ( -) [lji

I fs) [IIP1 ]dw+ r 0 +P2

It can be shown that if S4 is an appropriate particular solution of the

Poisson equation, then

P4 = S 4 + Re F4 (z,T)

The potential function Re F4 possesses a logarithmic singularity at

infinity, namely,

ci

f [4]dw . kn/Yl7+1JP2

provided

c1

A4 = ] [(,]dw

does not vanish. So far there is no apparent reason for supposing that

A =0.4,TT

We conclude from our analysis that, if A(s,t) is the cross-sectional

cavity area and if Att does not vanish, then it must be of order equal to

or higher than c2.

A-15

JM

R-2118 1

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")

IJ 3v (w,s,O±,tr)dw

C1 Tf(s)

= 1 f( 31(W'S'T)Idw +-L f f [P3 (w,sX)JdwdXY 0 0

1T ;3 I f (a)- -- s7 f f h(a-s) -(A+Y-T-Wu) [P,(w,a,A)jdwdadA

0 o

om which

( y ~ , O ± r ) = ± 4. [ ~ ~ y, , )] 3~ f (s ) [P 3 (w ,s aX) J d w d XLv 3 i 21r 3y 0 0 +T-

4+ .-- -- s f h(a-s) [Pl (w,7;) 3fd-d~d0 - o

*314 $SO±,T) +-t-P (Y ,s,O±,T)

-1 (- +

A-I11

DAT