A note on the effect of a slot on force characteristics of supercavitating flat plates

UNIVERSITY OF MINNESOTA

ST. ANTHONY FALLS HYDRAULIC LABORATORY

Technical Paper No. 28, Series A

A NOTE ON THE EFFECT OF A SLOT ON

FORCE CHARACTERISTICS OF

SUPERCAVITATING FLAT PLATES

by

C. S. Song

Prepared for Presentation at

Winter Annual Meeting and Energy Systems Exposition

American Society of Mechanical Engineers

New York, N. Y.

November 1966

June 1966

Minneapolis, Minnesota

$1.50 PER COPY

75C TO ASME MEMBERS

THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS,

Associate Professor of Civil Engineering,

St. Anthony Falls Hydraulic Laboratory,

University of Minnesota,

Minneapolis, Minn.

The force characteristics of a two-dimensional, supercavitating, flapped flat-plate

hydrofoil with a slot near the flop hinge were analyzed by means of o first-order

perturbation theory. The mathematical problem was shown to hove several physically

plausible solutions depending on the selection of boundary conditions. The lift

co-efficient was calculated for several boundary conditions.

Large discrepancies among

the different solutions were noted. In some instances the slot increased in lift, while

for most others the slot served to reduce the lift.

Contributed by the fluids Engineering Division for presentation at the Winter Annual Meeting and Energy Systems Exposition, New York, N. Y., November 27December 1, 1966, of The American Society of Mechanical Engineers. Manuscript received at ASME Headquarters, August 5, 1966.

Copies will be available until September1, 1967.

66-WA/ F E-38

The Society shall

not be responsible for

state-ments or opinions advanced in papers or in

dis-cussion at meetings of the Society or of its

Divisions or Sections, or printed in its publications. Discussio, is printed Only if the paper ¡s published

in an ASME jornal or Proceedings.

Released for general publication upon presentation

A Note on the Effect of a Slot on

Force

Characteristics of Supercavitating Flat Plates

C. S. SONG

A Note on the Effect of

_{a Slot on Force}

Characteristics of Supercavitating Flat Plates

C. S. SONG

NOMENC LATURE

A = an arbitrary real constant CL = lift coefficient CM = moment coefficient e = flap-chord ratio f = a mapping parameter = coefficients of expansion n integers

Q = leakage through slot S = slot dimension

U = flow speed at infinity

u = x-component of perturbation velocity V = total velocity vector

y = y-component of perturbation velocity W = u-iv; complex perturbation velocity x, y = physical plane coordinates

z = x + iy = complex coordinates = angle of attack = flap angle

6 = vertical offset of two corners of

slot

EL = effect of slot or leakage on lift coefficient

EM effect of slot or leakage on moment

coefficient

=

_+iT

transformed plane

= longitudinal coordinate of slot e = slot angle

2 = mapping parameters

p = density of fluid

r = variable of integration

INTRODUCTION

Leakage through a small gap at a flap hinge of a supercavitating flat plate has been observed during experiments on unsteady supercavitating

flows (i)) Although the leakage was too small

to substantially affect the unsteady force char-acteristics for this particular instance, it ¡cay

become important under other circumstances. As

will be seen later, a slot near the flap hinge

may greatly influence the flap effectiveness.

Ex-perimental data reported by Wade (2) which were

Underlined numbers in parentheses designate References at the end of the paper.

brought to the authors attention while the

manu-script was being prepared substantiated some of the present findings.

Recently, Oba studied this problem analyti-cally by means of a first-order perturbation the-ory (.3). _{11e assumed that the slot was very small} and that the leakage may be represented by a sink. The strength and the location of the sink were considered free parameters on which other physical quantities such as lift, drag, moment, and cavity

profiles depend. The strength of the sink, which

also corresponds to the amount of leakage through the opening, had to be estimated by seniiempirical

means. The mathematical model used by 'ba was,

in a sense, indeterminate.

Although mathematical indeterminancy is in-herent in potential flow with a singularity, there might be some practical means, such as the Kutta-Joukowski condition, to render a unique solution. The present paper reports a more rigorous treat-ment of a similar problem using a first-order

per-turbation theory. As will be seen later, the

problem does not have a unique solution unless an

additional boundary condition is assumed. For the

additional boundary condition, several possibili-ties have been considered, each of which

repre-sents a realistic physical condition. ba's

solu-tion is obtained as a special case by taking a limiting process whereby the slot is reduced to zero while the singularities are retained.

This paper follows the general procedures of linearized supercavitating flow theory on which numerous excellent papers have been published by

many investigators. No attempt will be made to

list all the related work except to mention that the present problem is somewhat similar to the biplane problem considered by Hsleh (Li.)

GENHEAL CONSIDERATIONS OF THE PROBLEM

Flow Patterns

The problem to be considered is that of a two-dimensional supercavitating flat plate with a trailing flap and a slot between the flap and

the main body. The cavity pressure is assumed to

be equal to the pressure at infinity so that the free streamlines separating from the leading and

y

E

_/

- U Sharp corner

--f.

(a) With Free -Jet

C

Stagnation streamline

(b) Limiting Cose (no free -jet)

Fig. i

Possible flow configurations with a singularity at

up-stream corner of slot

the trailing edges extend to downstream infinity. Since the pressure on the wetted side of the foil Is known to be higher than the cavity pressure, there will generally be a flow throngh the slot

from the wetted side to the cavity side. The flow

throngh the slot will form a free jet inside the cavity and may or may not rejoin the main stream

somewhere downstream. Sketches shown in Figs.1(a)

and 2(a) represent flow patterns which are

con-sidered possible. The only difference between the

two sketches Is the type of separation at the up-stream edge of slot E.

If the question of stability Is ignored, there nay be a type of flow which results in no leakage; the slot gap is sealed off by a free

streamline. Depending upon the separation

condi-tion, the flow pattern may resemble either Fig.l

(b) or Fig.2(b). It will be shown that the latter

type of flow is, mathematically, a limiting case of the type with free jet.

Boundary C onditions

The problem will be solved by using a well-known linear theory wherein the angle of attack,

a, and the flap angle, ß, are so small that there exists a perturbation velocity which is an order of magnitude smaller than the speed at infinity. The perturbation-velocity components will be ex-pressed by u and y which are related to the total

(a) Flow Diagram

Stagootion streomline

(b) Alternate Flow Diagram

Fig. 2

Possible flow configurations without a singularity at

upstream corner of slot

velocity through the following expression:

V =

+ u) + IiJv

(1)

f

where V is the total velocity vector and U is the

speed at infinity. The complex perturbation

ve-locity

W(z)u-iv

(2)

is known to be a regular analytic function of the position variable

z = X + jy

(3)

everywhere inside the flow region, but may have a finite number of weak singularities on the

bound-ary. The physical problem can, then, be reduced

to a boundary-value problem of complex analytic functions.

The boundary-value problem is very difficult to solve exactly because the boundary profile and

the boundary values as well, are nonlinear. The

most commonly used technique for simplifying the problem is to linearize the boundary profile and

the boundary values. The boundary profile is then

represented by one or more slits, generally placed

on the x-axis. It will soon become clear, as

shown in Fig.(a), that two separate slits are

required if leakage is to be considered, and a

(o) Linearized physical plane, zx+iy

(b) The transformed plane, e+i,7

Fig. 3

Linearization and mapping of boundary

single slit is sufficient if there is no leakage. All the pertinent dimensions, measured in terms

of the chord, are shown in Fig.3(a). Furthermore,

the leading stagnation point is assumed to

coin-cide with the leading edge of the foil. This

as-suinption is known to be a good approximation when

.and are very small, as in the case considered

here.

The following boundary conditions are

clear-ly necessary, but, as will b seen later, they

are not sufficient:

1 The free-surface boundary condition

re-quires a constant pressure on the free

stream-lines AB, B'C, and DPE. In terms of the

perturba-tion velocity and up to the first-order accuracy, it follows that u = O on

+ +

O<x<., y-O

jjj) X2<x, y=_ó

X1<X<.T,

yO

iv) 1<x, y-ô

2 On the solid surfaces AE and DC the flow

must satisfy the following approximate solid boundary conditions:

i.)

y = -

on

O ¿ x < X1,

y = 0

ii)

y = - (+ )

X2 < X

1,

y =

The equality sign at x i is the consequence of

the well-known Kutta-Joukowski condition at the trailing edge.

3 The boundary condition at infinity is

-o X1

u=vO at

x=-Conformal Mapping

The mixed boundary-value problem may be solved readily by using the technique of Cheng

and Rott (). This will require mapping the

lin-earized z-plane as shown in Fig..3(b). By means

of the Schwarz-Christoffel theorem the mapping equation is found to be

-z=2+(2-f)

(4)

where the mapping parameters are a et of

solu-tions of the following simultaneous equasolu-tions with

1 ' g 2 ' > O: -- - r) (i + f )

(5)

2

-

-

+ f £n

-_1)

f

(6)

-

₂

2

-X2 2 2

2 -

2

+ f Ln

An electronic computer was used to solve the si-multaneous equations for some specific

combina-tions of A 2' and

6.

It is important to observe that when the vertical offset ô is equal to zero, then equation

(5)

requires '

2 such that

lun 12 -

f)

ô

Under this special condition the mapping equation is reduced to

2

1=C

This is the correct result since when 6 = O, the slit DCB' also falls on the x-axis and the

linear-ized boundary degenerates to a single slit.

More-over, the requirement f

2 implies that the

free streamline DF degenerates to a point and the free jet is reduced to a single free streamline

ED, as shown in Figs.l(b) and 2(b). Thus the

no-leakage oase is obtained when 6 is set equal to

zero. It should be noted that the foregoing con-clusion does not accurately represent the physical condition as far as the leakage rate is ccncerned. If a more accurate representation of the leakage rate is desired, then the spacing between the two

boundary slits should reflect both and S. In

this case the quantity S in equations (4) to

(8)

should be replaced by a function of 5 and S.

Complex Perturbation Velocity

The general solution for the complex

pertur-ly

B

-X

A(O,O) E(X,,O) F D(A2, -8) C(i,-8)

u0

v:_(as.ß)

uQ

4 u O -e

bation velocity satisfying all the boundary condi-tions listed can be written in several different

forms (6). The following is the general solution

wherein the continuity of the solution at the trailing edge is indicated explicitly

+ ₊

'_ + A

. +

)f+aJ

_{T +}

i

-1

Here A is an arbitrary real constant.

Obviously, the unique determination of the solution requires an additional boundary

condi-tion. In the

authorrs

opinion, one of the follow-ing five boundary conditions may be selected, de-pending on the actual geometry of the foil and other amblent conditions:

1 Smooth separation at E (see Fig.2).

2 Smooth separation or reattachment at D

(see Fig.1).

The free jet must be parallel to the

main-stream, as x

+

.

# The leakage rate is given.

5 The closure condition should be satisfied.

The first boundary condition is equivalent to applying the Kutta-Joukowski condition at the upstream corner of the slot and is believed to be applicable when the slot is relatively large and

the corner is not rounded. The solution is then

W()

_%I

+ i)(

+ +

dT

(T + 1)(T

₊ T - C

-1

If the second boundary condition is chosen, then the solution is w(ç) + i)( + n

_{-c(c +}

(lob)

t@

+)f+

af

_{(T + l)(T}

+

T

-_T(T + ) __.L. -1

It is readily seen

that

both equations (lOa) and

(lob) give nonzero y at downstream infinity in the

free jet, =

-f,

indicating reattachment of the

leakage jet to the mainstream. This situation is

also implicit in Hsieli's solution cf the biplane

problem (4) , although it was not pointed out by

the author. Moreover, Wade's (2) experiment seems

to indicate the existence of this type of flow for

the specific body tested. If the free jet should

become parallel with the mainstream at downstrean infinity, the third boundary condition should be used and the solution is reduced to

f_T(T+ 1)(T +

dT

T+l

(T-C)(T+f)

-1

If the fourth boundary condition is chosen, the constant A must be determined by the following equation:

Q = Im W

dJ

¿'J_f

where Q is the leakage rate assumed to be known and the integral is taken around the boundary

en-closing the whole flow field except the singular

point = -f.

Attention is now called to the possibility of

taking a limiting process wherein while

the leakage rate Is kept constant according to

equation (il). Equation (10) will then reduce to

W(C) = _îi(C

_f)

--f O (12)

s

-

r+ f

dT + A

-i

-f

Equation (12) implies that there is a simple pole

(source or sink) at the flap hinge. This is the

condition that ba has treated in his recent paper

(3). Of course this type of singularity at the

flap hinge could be eliminated by a proper choice

of the constant A. The result is

-f

O

dT

w(c) = [(a + )

f

+

cf

%T

(13)

-1

-f

which is the well-known solution of a flapped su-percavitating flat plate without a slot.

Finally, the closure condition which insures the continuity of the boundary curve, written in a complex form, is

(10)

+

i

ir

Lift and Moment

In terms of the perturbation velocity, the linearized expression for the lift coefficient is

X1 1

CL _2(J

+

J

) udc

o x2

Since dy = O on the linearized boundary and since

W is regular everywhere

in

the fluid, equation

(15) may be written

in terms

of the complex

veloc-ity potential as

CL -2

RefWdz

(16)

where the integration should be carried out in the counterclockwise direction on a circle having a radius approaching infinity.

The expression for the moment coefficient referred to the leading edge is

CM = -2

RefWzdz

(17)

Since W is given in an Integral form,

equa-tions (16) and

(17)

contain double integrations.

For the purpose of numerical calculations it is desirable to reduce the double integrals into

sin-gle Integrals. This can be done by changing the

order of integratIon and applying the residue

theorem. For example, if the complex perturbation

velocity is given by equation (loa), then, after

changing the order of integration,

it

is required CL' 4[(a +

)j

+

that the following integral be evaluated:

Re Ini

+ ')(c +

_dz + (15) (18)

Since the integrand does not contain a pole inside the contour, it is only necessary to find the

res-idue at the origin. To do this, it is first

nec-essary to expand the mapping equation, equation

(4) , into the following form:

-n/2 K

_f12 +

'ç-'

K21 z

₍₁₉₎

z n=O

Only the first few coefficients in equation (19)

are needed. Expanding the integrand of equation

into a series in l/ and using equation

,

it

can be readily seen that

RjJ1

+ + dz (14) + = ) [ + + 1 - + 2

-SOME NUMERICAL RESULTS

Flows With a Free Surface Across the Slot

Although the physical existence of such flows has not been demonstrated, there exist mathemati-cal solutions to which physimathemati-cal interpretation

may be given. This is the case when = O and the

mapping equation degenerates to equation (9) For this special case, the right-hand side of equation (11) can be shown to be identical to zero and hence, no fluid leaks through the slot.

It can also be shown that f =

2 and hence,

equa-tion (lac) is reduced to equaequa-tion (lob).

There-fore, two out of five previously given additional

boundary conditions lose their meaning. The

pur-pose of the present section is to study the ef-fect of the remaining three boundary conditions on the calculated lift and moment coefficients.

For the sake of compactness, different solu-tions correesponding to different additional boundary conditions will be designated by a super-script which identifies the boundary conditions given in the section on 'TComplex Perturbation

Velocity.i _{For example CL1 means the lift}

coef-ficient when the condition of smooth separation at E is assumed, whereas CM5 is the moment

coef-ficient when the closure condition is used. When

the first additional boundary condition is ap-plied, the lift coefficient is

(20)

fT(T _'%JÇ)

(r-2 (T

)(l -

r) (21)

And if the second additional boundary condition is applied, a similar equation resulting from

in-terchanging X1 andX2 in the integrand of equation

(21) is obtained. In general, the lift

coeffi-cient may be written in the following forrn

C =

C

+ , n = 1, 2, 5 (22)

where is the lift curve slope with respect to

attack angle and C is the lift curve slope with

respect to flap angle.

The effect of the slot on the lift

06

0.5

-

0.4

- 0.3

w 'n _0.2 0.1 o 0.3

02

-0.1

-0.2

-0.3

7

(o) Smooth Separation At Upstream Corner erO .4lIlI

PP4

0.5 (b) Smooth Reattochment

At Downs ream Corner

r,,,

0

___.s

0.4 0 01

02

0.3

04

05

06

07

0.8 Slot Size, SX2-X1 0 01

02

03

04

05

06

07

0.8

Slot Size. SX)

Fig. 4

Effect of slot oil lift coefficient at zero flap angle

the factor The numerical values of e and eg were

computed for the first two cases (n = 1, 2) and

n

= LO. ₍₂₃₎ the results are plotted in Figs.4 and 5.

c(o)

The moment coefficient about the leading edge

can also be expressed in the same form as the lift

where C(0) is the value of Cwhen there is no

coefficient. The factors representing the effect

slot. Stmilarly, the effect of the slot on the of the slot on the moment coefficient denoted

Lift coefficient when = O is represented by and respectively, are plotted in Figs.6 and

Cn

h

1 L__ (2k) As shown in PigsJl. and

5,

when a smooth

-0.1 -0.4 0.5 -0.6 -0.9 o 03 0.2 0.1 0.7 (b) Smooth reottochment ot downstream corner

A4

uviii

,i1

I\_

.i

_LIN PAM

e'0.50 0.40 0.25

Fig. 5 Effect of slot on lift coefficient at zero angle of attack

increase lift instead of decreasing it. This

phe-nomenon may be explained by the fact that admit-ting the singularity at point E is equivalent to assuming that the point E is a stagnation point. Such a flow might be created by providing a small

step such as a trim tab at E. In reality, the

separation and the reattachment conditions will probably be affected by the actual geometry of the

corners as well as other ambient

conditions.

Ob-viously the arbitrary

constant

A alone is not

suf-ficient to take care of all possible variations. Nevertheless, it may be of some value to determine

the contribution of the constant to the calculated lift coefficient.

Starting from equation (lo), the lift

coef-ficient may be calculated and the following equa-tion is obtained:

CL(A) - cL(A = o) = 2(J _+,J - 1) A

(25)

where CL(A) = lift coefficient for an arbitrary

A. Clearly the contribution is a linear function

of A and its coefficient is a simple function of the flap-chord ratio and the slot dimension.

It is also interesting to note that for any

given set of)1 andA2

A2 A5 7 A1 (26)

Here again the superscripts refer to the

particu-lar addItional boundary condition imposed. It

follows that

CLC5>C

_{L <L}

ifjÇ+jTl

It is also noteworthy that for a small flap-chord ratio and a narrow slot, the slot will af-fect the force owing to the flap to a greater ex-tent than the force owing to the main foil alone. This result agrees with Wade1s experimental result mentioned earlier.

Flows With a Free Jet Through the Slot

As stated previously, the vertical offset of the slot should be retained in the analysis if the effect of the leakage is to bè taken into

ac-count. As a result, a more complicated mapping f ornnila, equation (4), is required.

When the first additional boundary condition

(smooth separation at E) is applied, the lift

co-efficient is i C11

_-

2

[(a + B)J +

af

/ T(2 - T)

_{(Ti+i_2}

₊

- f)dr

(iT)(1T)

z 2

If other boundary conditions are applied, then the integrand in equation (27) should be modified.

When

6-t'

O, then equation

(27)

is reduced to

equa-tion (21), as expected.

Numerical integrations were carried out

as-suming e = 0.3 and 6 = S tan ® for O equal to O,

5,

10, and

15

deg. The calculated leakage-effect

factors and

corresponding to the first model are shown in Fig.8, while those

correspond-ing to the third model are plotted in Fig.9. It

may be observed that the slot may increase or de-crease the lift depending on whether the first or

the third additional boundary

condition

is used.

(27)

-0.25 0.30

-e-Il--

0.40 0.50 !ØIPII (( Smooth separOtion of upstream corner Ql 0.2 0.3 0.4 05 06 07 08 09 Slot Size, S-). 01 02 03 04 05 06 07

Slot Size, S.,-A 08

09

0.5 0.4 0.3 0.2

i

0. I o

i 0.I

e 0.2 0.3 0.4 -0.5 0.3 0.2 0. 06 o o

u-.'-

. 020

UUÒI

0.25 030

L',

(o) Soth i.palio 9ll s' cone 0) 02 S. 03 -0.) o 0.40

_A

-I

Au,'

Mill

uiria_

A,NA__

AWA

AW

If all the curves for ein Figs.8(a) and

9(a),

or those for e

in Figs.8(b) and 9(b) are plotted

on a single sheet, it will be noted that the

lift

takes extreme values when e = 0.

Since increasibg

e means increasing the vertical offset, the

dif-ferences between the two models become smaller as

the offset becomes larger.

Since the experimental flow conditions of

Wade (2) do not quite correspond to the conditions

considered herein, direct comparison of the data

and theory is not possible.

Nevertheless, an

in-direct comparison by a suitable interpretation of

the data and theory may be useful at this stage.

For this purpose, lift coefficients corresponding

to the mininauti cavitation number and c.

= 8

deg are

taken from Figs.t(.-1 and 7-1 of reference (2).

These values are divided by the factor i + e,

where

is the measured cavitation number, and the

results are regarded as experimental values at

zero cavitation number.

The leakage-effect factor

is then estimated from the data to be 0.165

for the 10 percent gap-clearance case.

0? 02 03 0.4 0.5

Slot Stz. Sr).

(b i Smooth r.oflochment at

downst,on can,,

Fig. 6

Effect of slot on moment coefficient at zero flap angle

.0.20 I.e 0.9 08 0.7 0.6 0.5 0.4 0.3 0,2 o. i 0.)

Since Wade observed a strong jet flowing

through the gap and striking a cavity surface, lt

is most appropriate to compare the data with the

first inde1 herein,

E.

Fig 8fb) which is

ap-propriate for this case gives e

= 0.125 for

e

= 8

deg and flap-chord ratio e = 0.3.

Since the

data point is for e

0.2, a correction factor

must be applied to the redicted value.

From Pig.

5(a) it is noted that 6L for e = 0.2 case is

1.37 times that of e = 0.3 case.

The predicted

value for e = 0.2 is 0.171 which is very close to

the measured value.

A satisfactory comparison for

is not possible for 10 percent gap because

the value is small compared with the experimental

scatter.

CONO LUS IONS

A problem concerning two-dimensional

super-cavitating flow about a flapped-plate hydrofoil

with a slot near the flap hinge was solved by

means of a first-order perturbation theory.

The

9

0.1 0.2 0.4 0.5 0.6 0.7 0.8

06 07 08

0.2e

o, 0, -02 e 020 Q30,25 0g

IIIi4

WVMLU

04 'f 03 02 0. 03 o

I..

(b S,nooth reottocnent al downstream corner 0I 02 03 04 05 06 Slot Size SXX1 o. i 0.4 o

o( Zero Flap Açle

L

4 -.

'uI.JO

25

r1

040

(o) SnooTh oeparotioe °'

uoel,.Om Corner

0'0

-I

Fig. 7 Effect of slot on moment coeffic ent at zero angle of attack

fluid was assumed to be incompressible and nonvis-cous, and the main cavity was assumed to be

in-finitely long. The free-streamline problem thus

formulated results in nonunique solutions unless an additional restriction is imposed on the

solu-tion. For the additional restriction there are several available choices, each of which has its

particular physical interpretation. Since the

fluid is assumed nonviscous, the question of the existence and stability of the real fluid flow is beyond the scope of this paper.

Lift and moment coefficients were calculated

for various possible solutions. The theoretical

results are summarized in the following:

O Ci 02 0.3 04 05 0.6 07 08

Slot Size, S '5,X,

(b(Zero d.ttock AsgI.

Fig. 8 Effect of leakage ori lift coefficient, assuming smooth

separation at upstream corner of slot

1 If the boundary condition is applied on a

straight line (a usual linearization procedure), the boundary representing the slot must consist of one free streamline, implying no leakage unless a singularity of first order is applied at tne

slot. To determine the leakage effect, it is nec-essary to place the linearized boundary on two offset lines.

2 The freedom of choice of the potential

flow solutions to represent a prctical flow

prob-lem is limited. For example, 1f a smooth

separa-tion at one of the corners of the opening is as-sumed, the perturbation velocity of the leaking fluid as it approaches downstream infinity cannot

vanish.

3 The choice of the additional boundary

con-dition may have a surprisingly large effect on the

calculated lift coefficient. For example, whereas

the opening is generally considered to reduce the lift, it may actually increase the lift if a sin-gularity is admitted at the upstream corner of the

slot. This boundary-condition effect seems to

be-come less important as the vertical offset of the slot is increased.

L Because of the large discrepancies in the

calculated lift coefficient using different

addi-07 08 0.5 0.4 w 'f 0.3 0.2 'n 08 07 0.6 03 0.2

3

_a, 0.I a, 02 -0.3

tional boundary conditions, it can be conjectured that for a practical problem, the effect of a slot may depend very much on the geometrical

con-dition of the slot. For example, the rounded

cor-ners, the relative thickness of the foil as com-pared with the width of the slot, the inclination of the hole, and so on, might greatly influence the flow.

5 A small slot near the flap hinge will

af-fect the force due to the flap to a greater extent than the force due to the main foil, especially

when the flap-chord ratio is small. That is to

say, even if the slot has little effect on the total lift, it may greatly affect the flap

effi-ciency.

6 Experimental work is needed to resolve the

uncertainties raised by the present analytical

work. Exploration of the possibility of using a slot for the purpose of lift angmentation as well as experimental study of existence and stability of the flow patterns is desirable.

An indirect comparison with an experimental result currently available indicates good agree-ment between the data and one of the theoretical models.

ACKNOWLEDGMENT

This research has been partially supported by the Office of Naval Research of the U. S. De-partment of the Navy under Contract Nonr 710(24),

Task NR 062-052. The author is grateful to

Mes-srs. Alwin C. H. Young and Young T. Shen for their assistance on numerical computations and computer programming.

REFERENCES

1 C. S. Song, °Supercavltating Flat-Plate

with an Oscillating Flap at Zero Cavitation Num-ber,' University of Minnesota, St. Anthony Falls Hydraulic Laboratory, Technical Paper No. 52, Se-ries B, November 1965.

2 R. B. Wade, 'Experimental Study of the

Effect of a Gap Clearance on the Performance of a Fully Cavitatirig Flat Plate With and Without a Flap,' California Institute of Technology, Karman

Laboratory, Report No. E-133.3, November 1965.

3 R. ba, "Performance of Supercavitating

Hydrofoils with Flaps, with Special Reference to Leakage and Optimization of Flap Design,' Univer-sity of Minnesota, St. Anthony Falls Hydraulic Laboratory, Technical Paper No. 51, Series B, May 1965.

4 T. Esieh, "Linearized Theory for a

Super-cavitating Biplane Operating at Zero Cavitation

04 0.3 0.2 -0.3

-04

-05 o

Ial Zero Flop Argie

0) 02

03

04

05

S'ot Size, S.A2-À1

b) Zero Attack Argle

-

80

06 07 08

Fig. 9

Effect of leakage on lift coefficient, assuming zero

perturbation at infinity

Number Near a Free Surface," Hydronautics, Inc., Technical Report 463-3, October 1964.

5 H. K. Cheng and N. Rott, "Generalization

cf the Inversion Formula of Thin Airfoil Theory," Journal of Rational Mechanics and Analysis, vol.

3, no. 3, Indiana University, Bloomington, md.,

1954.

6 C. S. Song, "A Quasi-Linear and Linear

Theory for Non-Separated and Separated Two-Dimen-sional, Incompressible, Irrotational Flow About Lifting Bodies," University of Minnesota, St. Anthony Falls Hydraulic Laboratory, Technical Paper No. 43, Series B, May 1963.

11

r

-0.5 04 03 ti) O2 w o o 0, I

-02