A hydrodynamic theory for cavitating delta wing hydrofoils

OCEANICS

INC.

A HYDRODYNANIC THEORY FOR CAVITATING

DELTA WING HYDROFOILS

by

Paul Kaplan, Theodore R. Goodman, and

C. C. Chen

Prepared for:

Office of Naval Research

Washington, D. C. 20360

under

Contract No. Nonr-4328(O0), NR 062-330

Distribution of this Document is Unlimited

Technical Report No. 67-33 December 1966

ARCHIEF

Lab. y. Schpbcwk.

A HYDRODYNANIC THEORY FOR CAVITATING

DELTA WING HYDROFOILS

by

Paul Kaplan, Theodore R. Goodman

and C. C. Chen

Prepared for:

Office of Naval Research

Washington, D. C. 20360

Prepared by: OCEANICS, Inc.

Technical Industrial Park

Plainview, New York 11803

Abstract

li

NoMenclature

iii

Introduction

-

...,

1

FormulationoftheProblern,...,,..,.,...,,

L

Solution of the Problem

12

Numerical Results and Discussion..,.

21

Conclusions,..

2t4

References

- _-

26

Appendix A

_..o

27

Appendix B

_{., . . . ... . .}

_{, . . . .}

32

The status of available hydrodynamìc theorjes of cavitating flow past delta wing hydrofoils is

re-viewed, and it is fOund that a.symptotic. results for limiting conditions of cavitation (i.e. large arid smal]. cavitation rniinbers) in earlier wrk by Tulin are incorrect.

The mathematical problem of conical flow for cavitated delta wings of fairly wide apex angles is recast with a new 'closure' condition replacing the previous cOndition

of no source-like flow in a transverse plane, The results

obtained for lift coefficient, for apex angles up to 600,

as a function of angle of attaák and cavitation number are

compared with available experimental data. The

compar-ison indicates good agreement of the new theory for delta wings

with ápex angles up to 450. Exthination and comparisons

of the slender-body theory of Cumberbatch and Wu shows the

limits f applicability of that theory to be restricted to

large angles of attack ( >17°) and small apex angles (<15°). Thus, differences in results obtained using different mathe-matical and physical models ax delineated., together with information on the respective regions of validit.y by virtue of comparison with experimental data.

A

A

a

111

NOMENCLATURE

Coefficient £n cavity area representàtion

Coeffiòients of the cornp].ex potential expansion

Nondimensional magnitude of the transverse velocity

on the. cavity Lift coefficient

'Vi_2

Coniete elliptic integral of the second kind,

Potenfi.a]. due to thikness

Term. in èxpression of cross-flow potential

Çaity thickness measuÑd from and normal to the

wing surface

K (k) _{Complete elliptic integral of the first kind} k _{Modulus of the elliptic integrals}

L Lift

L _{Nondimensiona]., width of the cavity measured in}

the transverse plane from the leading edge to the

inboard end of the cavity

n _{Normal direction} p PreS sure

Poe Préssure at. infinity Pressure on the cavity

S Cx) Cavity area

s Cx) _{Local foil span}

y Vértical velocity

w Transverse velocity

x,y,z Coordinate system defined by Figure 2

Z + i, transverse plane complex variable

a Angle of attack

8 Apex angle

y Parameter of elliptic integral of the third kind

= _{+ in, transformed plane complex variable}

X

=1-L

y _{w-iv, complex velOcity in the transverse plane}

fl(12,k) Complete elliptic integral of the third kind

p Fluid density

a Cavitation number

Perturbation velocity potential

Complex potential in the transverse plane

Two-dimensional perturbation potential in the transverse

plane

INTRODUCTI ON

The development of hydrofoil craft is presently oriented toward high speed operation, where cavitated flow

will occur. ThiS flow condition results from the reduòed local pressures associated with the high speed, and/or

ventilation from the atmosphere. For various situations

of

practical interest it appears that low aspect ratio hydro-foils will be used in such craft, especially in view of the

structural

advantages

of such

foils

One particularly im-portant class of low aspect ratio hydrofoil planfor'ms that is of interest is a delta wing

planform,

for which certain

theoretical hydrodynamic studies have been carried out by.

Thun

[i] and

by Cumberbatch and Wu [2).

Both theoretical studies are essentially based upon

a slender-body theory approach, but the particular physical

model as to cavity shape differs in each theory. _Tulin's

model assumes that the cavity covers only part of the foil,

a region adjacent to each of the leading edges

of

the delta wing, with a wetted region between the inner termini near

the wing centerline. The Cumberatth and Wu model considers

the cavity to envelop the entire upper surface of the wing, and to close at some point downstream of the foil trailing

edge. The range of applicability of each theory,

_and

the

degree of agreement With available experimental _{data, were not}

known. _{Only limited numerical evaluations of the}

while no numerical results of any extent were available

for the Tulin theory, except for asymptotic results in _two

limiting cases, viz., for very large and for very small

cavitation numbers.

Foiling the publication of these theoretical _papers,

two different sets of experiments _{were carried out on} ôavi-tating delta wings. _{Reichardt and Sattler [3] measured the}

forces acting on a series of delta wings with _{different apex}

angles, ranging from narrow to wide deltas. _{KiceniUk measured} the lift force on narrow delta wings. _{Both sets of}

experi-mental investigators made visual observations _{f the flow and}

cavity characteristics, and from these it could be concluded,

that the Cumbebatch-Wu model, was valid for very narrow delta

wings at large angles of attack, while _{the Tulin model was}

applicable to wings with larger apex angles. No detailed

comparison of the theoretically predicted _{forces with the}

ex-perimental results was adequately presented.

The first aim of this paper was to extend Tulin 's

wOrk in order to determine the _{hydrodynamic forces for the}

compiee range of cavitation numbers _{and apex angles. After}

accomplishing this, the second objective _{was to carry out}

further numerical evaluations of the Cumberbatch-Wu theory and

then to compare the results _{of the two theories with the}

available experimental data. In pursuing the first objective

-

.3-by Tulin did not follow from his theory, _{and in addition} it was also discovered that the _{zero source strength}

con-dition which he uses in developing _{his theory is}

incon-sistent with the conical flow assumption which he makes. It was therefore necessary to recast. the problem and resolve it completely before numerical _{results of any validity could}

be obtained, or any comparison with _{experiments could be made.}

This reworking of Tulin's model _{then became the primary} pur-pose of this paper.

A cavity closure condition is proposed to replace Tulin's zero source strength condition. _{The requirement of}

no source-like flow in the transverse plane is inconsistent

with the conical flow requirement _{which specifies that all}

transverse coordinates vary linearly with _{x (the longitudinal}

coordinate), while cross-sectional areas vary as x2. Simi-larly, conical flow requires that the total velocity potential

should be linear in x and the original _{Tulin model does not}

satisfy this condition. _{The capability of the newly formùlated}

model, with the cavity closure boundary _{condition, to satisfy}

these requirements will be demonstrated _{in the ensuing analysis.}

The problem considered here is _{that of the cavity flow}

past a delta wing placed _{at a small angle of attack in an}

other-wise uniform flow. _{Starting from the apex of the delta, the} flow forms a cavity _{on the top of the plate; the cavity grows as} the cavitation number is _{decreased. It is shown that}

> los Naturally, the flow is more conical near

the apex of the delta than near the trailing edge. The system of the wing and the cavity is _{assumed to be slender} so that slender-body theory may be used. _{A poténtial flow} model is assumed throughout the _{flow field. The boundary}

condition that the flow _{must be tangential to the boundary}

is applied both on the cavity and on the wetted part of the

wing. _{In addition, on the c.vity, the pressure must be constant.} it is only necessary to solve the transverse problem in one

transverse plane of the wing since the flow is assumed conical.

The problem is thus reduced _{to a two-dimensional boundary}

value problem with mixed boundary _{conditions. For the sake}

of convenience in calculating _{the complex velocity, the}

problem is transformed to anogher plane where the proper

solution can more readily be _{found. The transformation}

and solution, etc. provided by Tulin is still useful and will be used here. _{The lift on the delta wing may then be d}

termined by integrating _pressures0

FORMULATIOIJ OF THE PROBLEM

A schematic diagram of the _{cavitating flow past a}

slender delta wing placed at a small angle in a uniform

stream is shown in Figure (1). _{A conical flow model is} chosen for this cavity flow. _{Tulin's method is used}

for the initial analysis; _{however, the proper boundary}

is added and the subsequent analysis is different.

Assuming that the pressure at infinity is zero, the

linearized Bernoulli equation gives

as

p - pU

ax

where s is the perturbation velocity potential. From Equation (1),

L!5

constant on the cavity since the pressure there is

constant.. The delta wing length is chosen to be unity.

Thé kinematic boundary condition states that the flow

must be tangential to the wing and the cavity. Along the

bottom of the wing and on the wetted portion of the top of

the wing, the linearized boundary condition is

a_s:

-lia (2)

ay

where a is the angle of attack.

Accoriing to slender-body theory

_ES),

the potential of

a slender body placed at an angle of attack a to the uniform

stream is where and «x,y,z) = (y,z;x) + f(x) (3) 4' +4'

=0

yy zz poe IT 1 x-x f(x) = -

zL.

j S'(x1) &n 2j 4 _{ax o}

_{jx_x1f}

and S(x) is the cross-sectional area of the cavity at

section x, In order for .the flow to be conical it is

(1)

is discussed in Appendix A.

The first term in Equation (3),

(y,z;x) is the

potential of the cross-flow in the transverse plane, and

f(x) is the potential due to thickness.

For small angles of attack, the cavity thickness will also be small, and the boundary condition may be applied at the projection of the wing and cavity on the horizontal plane, i.e. along the top and bottom of a symmetrical slit

on the z-axis. Hence, by virtue of Green's theorem, the

cross-flow potential is

1/2

(y,z;x)

_{._}

4) (L±.

-

ì_)

£n

[(zzI)2

+

y2]

dz'

2ir J an an

Using the following transfOrmation suggested by conical

flow:

s s

Equation (5) becomes

$(y,z;x) s(x) s(x) _d + s(x) ₍₆₎

2

-

an 2

The linearized boundary

condition

_{on the top of the}

-7-1±Ei.: ,..a+!

(7)

U

Uan

Uay

ax

where .h is the thickness of the cavity. Substituting Equation (7) into Equation (6) leads to

(y,z;x)

g(x) + s(x) ₍₈₎

2

whe re

g(x)

L

s(x) £n

fi()

dZ _(g)

and ¡ (37,) is a harmonic function.

Consider the pressure boundary condition _{on the}

cavity, A < i _{Upon substituting Equations (3) and}

(8) into (1), this condition becomes

a u i a'

(10)

a

f'(x) + g'(x) + ß $ z

-òx 2 2

where B is the apex angle (s: Bx), l-A is the

non-dimen-sional width of the cavity, and _{a is the cavitation number}

which is definéd by

POD - P0 i pU

Now consider the boundary condition _{on the bottom of the}

(12)

ay

From Equation (10) and the assumption of conical flow it

follows that both f(x) + g'(x) and - must be constant

az

since i! is constant on the cavity. It is shown in Appendix

ax

A that, under certain assumptions regarding the wake,

f'(x) + g'(x) = _{A(znß - 3Ln2 + 3) (13)} 27r where 1 A 2 -1 3x

£1

et) or

(Be)

Returning now to Equation (10) this becomes a differential

equation for in terms of Since must be symmetric

with respect to , the solution is

9

--aU

,

z>0

az

aU < o (16)

where a U- is the unknown but constant magnitude of the

trahsverse velocity on the cavity. There are now two

unknown parameters in the problem: the transverse velocity

which is measured by a and the width of the cavity which

is measuxd by A In order to make the problem

deter-minate it is necessary to specify two cqnditions. One

con-dition is that the potential along te leading edge be

specified. _{Upon substituting Equation (13) into (15) and}

setting 1, this condition becomes

4(0,1) U(. - a) - A(&nß + 3 - 3Ln2)

B irß

The seòond condition requires that the cavity surface be zero at the wing leading edges and also at the inböard ends

i.e., according to Figure (3)

h(- A,x) = h(-1,x) 0 (18)

h( X,x) = h(l,x) = O

From Equation (7), we have

t_x

h

J

X+a

dx (19)

xl U

where x1 * _{is a. point on the leading edge (see Figure}

).

With the definitions s = , Equation (19) be come s

+ a]

As can be seen from Equation (20) the conditions _{h( t 1,x)} O

are satisfied identically. _{The two physical conditions uséd}

to calculate and

a are Equation (17) and

-A

_r

-i

-h(-A,x)xA

X+ci

4=o

J-1 U z

Now consider the lift on the delta wing. According

to linear theory, s/2 =

J

-.s/2

[p(0-,z)

-

p(O+,z)] dz (20) (21) (22)

Substituting Equation (1) into Equation (22), _{there.. results}

,s/2

r

i!

ji!

(0-,z) - i!

_{(0+,z)fdz (23)}

-s/2 _Lax ax

j

and using both thé conical transformation and the _fact

that q _{is symmetric with respect to z, Equation (23) becomes}

= - pU

L

C0-,)

_(2L) Thus, - (0+.,)] d dL

C

r

¡V LU

L

lpU

82(1)

-II-L = - pU

s:(l)

f [(O-,) - 3(O+,)] d

(25)

Introducing

-i

as the complex potential in

the transverse plane, Equation (25) becomes

L

- pU

52(1)

Re dZ

₍₂₆₎

-t

where the contour is taken along the real axis,

O, and

Z +

The potential

can be expanded in a Laurent series, and

takes the form

A A

A P.nZ +

_!

₊

4

₊

₍₂₇₎

o

z

In general the As's are complex.

However it is

shown in [6] that A0

_{a real quantity.}

_Upon

sub-2r dx

stituting Equation (27) into Equation (26), the lift becomes

- Re[i7rA

_{+ lirA1] = - Re(inA1)}

₍₂₈₎

The first term of this equation can be obtained by integrating

by parts leading to f thZdZ

_(ZLnZ)00

_-

dZ.

The

re-maining contour integral vanishes identically

_{and the}

inte-grated part becomes 2iriZ, which, since

_{= O, reduces to 2ir1.}

Thus, since A0 is real, the first term of Equation

(28)

complex velocity in the transverse piane and r

Equation (28) becomes, as also shown by Tulin,

.2

L

ridv

-- ipU s.h1) 2 dT

t0

Now the problem is to solve for the _{complex velocity,}

V,

in the transverse plane. _{This is a boundary value problem} with mixed boundary conditions _{on the upper side of the}

- axis and a simple constant boundary condition on the

lower side of - axis (see Figure (5).).

y >0 w = Ua

- 1<<-À

w -Ua

-Uct (30)

<o

V =

-Uct _IZI<l

SOLUTION OF ThE PROBLEM

The boundary value pröblem _{formed is exactly the same.}

as Tulin's [1). _{Therefore his method of solution is followed.} In order to find the complex velocity, _{it is more convenien.t}

to transform the problem _{on to the plane where}

i

-(1_Z2)7 =

+

jr

The boundary conditions in the _{- plane are (see Figure(6))}

-

13-w = Uâ

I1>0

w = -Ua

n co

V =

u.

ic

_<o

The complex velocity,

derived in. Ref, (i.] satisfies

these boundary conditions and is borrowed here:

v(ç)

iUcz

c(]._C2)h/Z

Ua(1c2)h/2

rdJco)2

dço

ir

(c(o))112

(ic2)1/2(c)

(32)

where c

V ix2.

_{After substitutìnp Equation (32) intO}

Equations (14) and (29), there xSuits

..

_.

_f°

+ !. uc

₍₃₃₎

where F

_o)

(1_2)

L=

TT..,,2j.

2,_

L - ir

(3L)

In Appendix B it is shown that L

jJ2s2()

C/(F;)__d

o

(l_F;02)J/2

o JF; (L-F; )

_r

I O

°

_dF; 2

,_

(1+c)E(k)+ 1(1+c)(4-c)K

°

_(1-.

2)

O 8 8 71 2

A :

-

\l+c (Ek)

- 1< + ...E..II(Y2,k)]

..ac}

3

- C

4 + 71 2 'l+c JE(k) -

K +

1+c

J

+ !(t.c2_].)fl(y2,k)

where K(k), E(k.) and fl(y2,k) are the complete elliptic integrals

of the first, secònd, and

third kind

_{respectively;}

k is

_the

_modulus

of

the

elliptic integrals; y2 is the parameter of the elliptic integrai of the third

kind.

In this

_{case, k2:}

.E..., y2:

Ç2

1+c

2

Substituting the expressions for the integrals into Equations

(33) and (34), we have ECk) +

1

1 [2 2 ' i+c

,±_

(3c2_L1)n(2,k)]

' 1+c

_J

(4 + c-3 C2 ) K

By virtue of Equation (32), Equation

(21) gives

_{a relationship}

- a71.

-15-E(k) + K -

-L-

u(12,k)

] L

1c

l+c + ECk) - f1+

._L:)x

+

_{___g(,r2,k;Ì.}

(37) 1+c 1+c

J

Solving

Equations (36)

and

(37)

simultaneously, there

results ari expression for the lift

in

terms of c with a as

parameter.

Numerical

calculation is needed for the

complete solution.

_{However, asymptotic values of lift}

can be obtained for the two limiting cases c -O (small

cavity) and e +1 (large cavity) without resorting to

numerical calculation.

'For the

case e -'O, the series expansion of the

complete elliptic integrals are obtained in the form

K(k)

= L (1+

L

+

L. c2

+

L

3 + E(k) 32 -

L

+ ..,

32 fl(y2k) = L (1+ _C +

L

z + 1 e3

+ ..)

2 16

After substituting these expressions into Equation (37),

and simplifying, there is obtained

a -

1cx (38)

By substituting Equation (38) together with the expanded

form for the expressions fôr K(k),

E(k)

and

ll(y2,k)

into

Equation (36), there results

2 !. 2 (1-2

£

2 16 +

L.

16 2

L 1TCL

(1+

c)

pU2s2(.l)

2

(39)

For vanishing cavity this reduces to the known solution

for the fully-wetted delta wing. A new

non-dimensional

quantity L,

the width of the cavity measured in the

transverse plane from the

wing

leading edge to the inboard

end of the cavity, is related to o as follows:

t = i - = 1

-Vi-c2

c2 (för c-' 0) (40) and therefore equation (39) becomes

L .

(1+ L) (41)

.pU2é2(l)

2 2

Thus, even a small amount of cavitation will increase the

lift above that found for the non-cavitating case. Similar results occur in case of small partial cavitation in two-dimensional flow, which is ascribed to effective increase in

camber due to cavity. This is also a possible explanation

of the effect in the present situation.

For c- 1, which represents a large cavity with both

inboard edges coming together, we have

K+øo

E(k)+

fl(y2k) + 2K

and therefore Equation (al.) gives

a (42)

Combining Equation (42) with the asymptotic values of K(k),

E(k)

and

fl(y2 _{k) and substituting in Equation (36), we obtain} L

-17-2 t4 21<

1

Care must be taken in the derivation and use of the results

of Equations (42) and (43), since we are working in the neighborhood where 1< is singular.

Equation (17) i.e used to find the expression for in terms of o. In order to solve for . , it is necessary

to find the potential at = 1, = O. _{The potential is}

defined as

¡ (t) - ¡(CD)

WdZ

(44)

CD

where the integra], is taken along the real axis (yO)

_and

_with

w the real part of the complex velocity v By expanding

Equation (32) in terms of (!) for large _{C it is found} that

L

B

}

hm

_v(c)

= B n (45)

where

B

-

. A, and ₌

Im()

irß

Rewriting Equation (44), we have

z

dz

(z) -

B].nz

where the logarithmic infinity of the potential is cancelled

by this method. Therefore

'r

¡(1)

= 1

/w)

_{- (46)}

3

After evaluation, Equation (46) becomes

(O,l): -2Bln2 +Blnc

+ /2B1n(l+

t12) +

!:La2. t2ln(l+V2)-'V'i]

4 +

_-4=

a I

c2_02)x_02_c

V2 ' l

:0

V1+

(c2-02)x2

F(0)dxd

(47)

An exact evaluation of Equation (47) requires numerical integration; however, the limiting values of

5..

for cO

and

c+l

can

be obtained. For the limit c-+0, Equation(47) becomes

(O,l) =

_2_

_{cicD (48)}

-19-Substituting Equation ('e8) into Equation (17), we have

eD U( - a) - (j.nß+3-31n2)

(9)

B B ti

and using the relation

um

A : O leads to

c+0

2.S.D

-

a (50)

2 3

From Equation (38), a: , and Equation (50) then becomes

aß e

which becomes in the limit

a p

v-ic

a .L4

»1

aß c

Examination of Equation (38) results in the following

ex-pression for a, viz.

a2

(52)

B

For the limiting condition where c+l, we have

¡(o,1) Ua(1n2-2) + G

2 21<

where G is a finite quantity, and thus (since K-,co)

um (O,l)

Ua(].n2-c+]. 2

(51)

(53)

(5L)

Substituting this expression into Equation (17), gives

Ucs(1n2- 2.) U( - -a) - (lnß+3-31n2) (55)

where 1 A = -4 1 and a = - air K

Upon Substitution, Equation (55) reduces to

= _{1(1+1n2+lnB) (56)}

14.

where 8 > 100 so that the two-cavity flow _{model proposed here}

is applicable, because for B 100, _{the quantity}

-

_0,

aß

which is physically unreasonable. _{This restriction might}

also be explained as due to the requirement _{of a different}

cavity configuration for the case of narrow delta wings

at low cavitation nuffibers, viz, _{the single cavity completely}

enveloping the wing upper surface, _{as proposed by Cumberbatch}

and Wu. _{Further discussion of this point will be given in} the next section of the paper.

To simplify the numerical calculation _{of Equation (47),}

the double integral is integrated _{once with respect to the}

variable x to give _{a single integrai, and then with c cose} Equation (47) becomes

-21-(O,1)

-231n2 + Bine -

Uczc

2

o

+

Uac2

f sine

/1-cose

lnR[-cos!

cose + cote sink (l+cose)]

2 \127r

ir/2

V l-c2cos2e

_L 2 2

-

_{(O+ir/2-6) [cosO sine/2 + cosO/2}

_{cote (l+coso)1}

_dO

₍₅₇₎

whe re

R

_{(l+4cos20)2-8cos2e+4V2cose sin(36/2)+82cose}

_{cose coso/2}

6 =tan

acose +2\'2cosO cosot2

sine +2'y2cose sine/2

Equations (36), (37) and (57.) are solved simuitaneöusiy

to give numerical values of lift coefficient

in

terms of a

with

as parameters0

_{The lift coefficient,}

_{CL, in the}

present report is defined with the tötal wing area as the

area reference, and thereby di:ffers from the lift coefficient

value defined by Tulin.

_{However the definition used herein}

is the saine as that used by Cumberbatch and Wu, as well as

jn both sets of experimental data, thereby.,

_{faòilitating}

comparisons between theory and experiment.

NUMERICAL RESULTS AND DISCUSSION

In order to illustrate the

range of validity of the

present theory, as well as the theory developed by

_Cuberbatch

arid Wu [2), comparisons will be made with the exterimental

presented herein are plots of the nondimensi..'na1 cavity width parameter £ vs. angle of attack a with cavitation index a as a parameter. Figure.7 shows the results for a delta wing

with :8 15°, ánd Figure 8 demonstrates the values for 8 = '15°.

While no experimental values are given for comparison, com-paring the results in these twO figures yields information on the relative sizes of the cavities for the two different

apex angles.

Curves of the lift coefficient CL as a

function

of a,.

for the various angles of attadk tested in [3), are given in Figures 9 - 12 for four different delta wings, viz. ß = 15°, 3Ó°, '15°, 60°. The experimental values obtained in [3] are

also plotted in these same figures, thereby allowing dixct

comparison between theory and expériment. Good agreement

between the present theory and the experiments of [3] is

ex-hibited for the configurations where B = 300 _{and '15°. For}

the case B 15 0 _{good agreement is only obtained for a < 60,}

and there is poor agreement for B 600. For a < .10 there

is poor agreement for all the configurations regardless of the

value of a or B At the same time, the possibility of errors

in the experimental data of [3] at such low cavitation numbers

must be considered, especially in view of the nature of the free-jet curvature in the water tunnel as well as the

inter-ference in the cavity pattern due to the sting support. Another

-23-600,

where agreement should not be expectéd in view _öf

the limits of ápplicability of slender-body _{theory to such}

a relatively wide wing, even for the case of non-cavitated

flow,

Similar plots are made for the values _{of CL} calcu-latéd from the Cumberbatch and Wu theory [2), _{and these are}

presented iñ Figures 13 and 14 for the configurations _with

15° and 30e. _{Comparisons with the data of [3] were also} made for the cases ß 450 _and 600, _{but the results are not} shown in any Of the figures. _{It is observed from ail of these} comparisons that there _{s hardly any agreement at ail between}

the Cumberbatch-Wu theory and this set of experimental data,

except for the case of the largest angle _{of attack (a > 17°)}

and then only for the case of thé _{smallest apex angle} confg-uration (B 15°).

In Figures 15 _{- 17, values of CL from the}

Cumberbatch-Wu theory are compared with _{data obtained in the California}

Institute of Technology water tunnel tests [4), for delta

wing configurations with ß 10°, 15°, and

3O

_The

experi-mental data were measured at large _{angles of attack (ranging} from 100 to 30°) and also for relatively _{low cavitation numbers,} It is found that there is fairly _{good agreement between the}

theory of [2] and this set of experiments for the smaller apex

angles, 8 _{10° and l5, while much poorer agreement is obtained} for the wider delta wing, _ß

300,

of a large single cavity enveloping the foil _{upper surface}

for the case of large angles of attaòk, with smàll cavitation numbers, and also for narrow (small 8) delta wing configurations. Similarly, the cavity observations 'and _{photgraphs prèsented}

in [3]

support

_{the two-caity model or ginally proposed by} Tulin

Ci]

and utilized in the present theoretical development

for wider delta wings.

While no extensive investigation has been carried

out, the small efféct. of cavitation on the lift coefficient

of a delta wing foil, for angles of attack _{up to 8° and}

cavitation numbers down to a = .18 ( as shown by the experi-ments in [3]), indicates that lift performance close _to

non-cavitdted conditions can be achieved by _{such foils. Thus delta}

wing hydrofoil configurations _{can provide úseful}

lift

_effects

at speeds up to 70 knots, while extensive cavitation is _present,

without any appreciable penalty due _{to that flow pattern.}

CONCLUS IONS

On the baSis of the numerical comparisons given _above, the conical flow model developed in this _{paper adequte1y} des-cribes the cavitating flow _{past delta wings, both qualitatively}

and quantitatively for moderate angles of _{attack (up to about}

12°) and for apex àrigles up to about 5°, For smaller apex

angles, e.g. _{8 = 15°, the valid angle of attack range is up} to 6°, while for ß '= 30° the angle of attack range extends up

-25-to about 17° as long as a

>

.10.

This particular theory is

not necessarily restricted by linearity with regard to angle

of attack, since nonlinear effects are present in-the. resülts,

especially for the smaller cavitation numbers in the range

of validity.

This conclusion may more readily be demonstrated

by plots of CL vs. angle of attack, with a as a parementer,

as shown in some of the graphs of experimental results in [3].

Prèsentation of the theory in this manner has been checked,

arid the results support this conclusion.

The Cuthberbatch-Wu theory [2] only appears tO be valid for

very narrow delta wings with large angles of attack and small

cavitation numbers.

These requirements are also necessary

in order to obtain the large single cavity that is the physical

model proposed in the analysis of [2].

The present investigation has delineated the regiOns

of validity for the available hydrodynainic theories applicabl

to cavitated delta wing hydrofoils.

Cons-idering the practical

application to a realist-ic

hydrofoil craft, the theory

devel-oped within this paper is most appropriate to the wider delta

wing configurations (B up to LIS°) that would serve as support

foils for high speed vehicles.

The possible small penalty in

lift effectiveness due to òavitation for such foils, compared

to the non-cavitated foil performance, indicates the prospect

of wider utility of such configurations jn future hydrofoil

craft developmènt.

RE FE REN CE S

i

Tulin, M. P.: "Supercavitating Flow Past Slender Delta

Wings", Journal o

Ship Research, Vol. 3, No. 3,

December 1959.

Cumberbatch, E. and Wu, T. Y.: "Cavity Flòw Past a

Slender Pointed Hydrofoil", Journal of Flu:jd Me ch.,

Vol. 11, 1961.

Reiôhardt, H. and Sattler, W.:

"Three-Corronent-Measuremsnts on Delta Wings with Cavitation", Fina].

Report Max-P].anck-Institut fur Stromungs forschung,

Gottingen, July 1962.

Kiceniuk, T.

"Supervèntilated Flow Past. Delta Wings",

Calif. Inst. of Teòh, Report Mo. E - l0l5, July l9L.

Adams, M. C. and Sears, W. R.: "Slender-Body Theory

-Review and Extension", Journal of Aero Sci., Voi, 20,

February 1953.

Sacks, A. H.: "Aerodynamic Forces, Moments, and Stàbility

Derivatives for Slendè±' Bodies of General Cross

Section", NACA TN 3283, November ].95L,

Byrd, P.

F. and Friedman, M. D.: "Handbook of Elliptic

Integrals for Engineers and Physicists, Springe

f(x) - !_. -_

41T dx

-27-Appendix A

Evaluation of f(x) and g(x)

In this appendix the terms in the potential that

re-present the effects of three-dimensional flow (axial flow

dependence) are evaluated in detail. The function f(x) is given by 1ST(x?)ln2lx_xtßdxt - J S'(x')ln2Ix-x'Idx' co - JS?(x?)1n2fx_x?Idxt j (A ].)

where x is the point beyond which the flow is no longer

conical, and this is located somewhat ahead of the trailin,g edges

Let _{f(x) f1(x) + f2(x) + f3(x)} x where f1(x) - 3L-

_i-..

f S'(x')j.n2Ix-x'Idx' (A 2) thr dx b x U d ₁ C

(x')ln2Ix-x'

dx' (A 3) 14it dx x 471 dx X C

Representing the cavity cross-sectional area by S(x) = Ax2,

Equation (A2) becomes

X f1(x) -

Y-

A

f 1(x)

Upon carrying out the. integration it is found that

A i (1n2-1)x + xlnx (A 5.) 2ir _i J Similarly f2(x) =

A[-ln2

- x

ix -xln(x-x)], x>x

2ff

To evaluáte f3(x), the wake integral, Equation (AL&) is

recast in the form of a Stieltjes integral:

f (x) = u-. _ _{ln2(x'-x)dS (A 7)}

thr

_dx

The actual area behavior in the wake is not known, but the efféct of the wake integral on

the sölution over the wing must be

small for a low aspeöt ratio wing. It therefore behooves us to make a very simple assumption regarding. S in thé wake and the simplest

assumption is that S is

discont-inu-ous across thé point x and constant

thereafter, as shown in the sketch. Equation (A7) then becomes

f (x) U AS (A 8)

L&ir

-29-where S denotes the jump in S. The final expression for f(.x) is then r -' f(x)

A d-2xln2 +2x

_-xc

-xln(x-x) -xlnx

2ir j -J U AS

'½ x-x

The quantity g(x) is given by

i

g(x) = s(x) in s(x) (

()

4ir 2

Liax

where the integral can be related to A in the f011owing manner:

The identity

s/2 S(x)

J h(x,z)dz = _J h(x,z)dz

-s/2 -1

can be transformed into

i

S(x) !.2.

J

. (d

2

-lax

by utilizing the conical property that h is a linear function

of x. Thus from the definition of A it is seen that

1

A=-

_J

)d

2

-lax

It follows that Equation (A 10) can be written

g(x)

L

A xln (A 12)

2

Since x

is near the trailing edge of the wing,

arid assuming x<cl, the sum f(x) + g(x) becomes

f(x) + g(x)

L

A (xlnß -3xln2 +2x-l) -

S(l+x)

(A 13)

2n

where only linear terms in x are retained.

The quantityS is determined by the following

reason-ing.

The sum f(x) + g(x), according to the conical flow

assumption, must be proportional to x.

Therefore the constant

terms in Equation (A 13) must vanish, which leads to

o

Hence,

- 2A(l)2

(A iLl)

where the quantity (1)2 stands for Xc2

i.e.

-2A x2 = -2 (Area of cavity at trailing edge).

The constant area region after the point x

approximates the

cavity area where the flow is no longer conical.

The negative

sign and resultant negative cavity area may possibly be

ex-plained as due to the presence of

a re-entrant jet.

However,

the precise value of the cavity area in the downstream region

(i.e. the wake) will not materially affect the flow

_fieldöver

the wing itself.

Thus the physical explanation for such

or

-31

effeòts will have little bearing on the evaluation of the

forces acting on the foil and no further effort will be

devoted to this point.

Thus, Euation (A 13) becomes

f(x) + g(x)

A (xln8 -3x1n2 +3x)

(A 15)

2ir

f'(x) + g'(x)

L.

A (].nß -31n2 +3)

Appendix B

This appendix presents detailed evaluation of various

elliptic integrals appearing in the main text of the report.

The evaluation of the integral

F()

=

J \j

dx,

in terms öf tabulated elliptic integrals is given by the following:

In Ref. (7], Equation 25L&.20, the general substitution is given:

j(t-c1) (b-t)

j

dt (b-c1(o-d) a2 V (a-t)(t-d) where

a>by>c1>d

(b-d)(t-c ) (b-c )(a-d) .sn2u = , k2 (b-c1)(t-d) (a-c1)(b-d) (lambda) 2 _b-c 0< 2 1 <k2 -c1)(b-d) - b-d

g=

/(b_d)(yTcl) aiu

= sin1

V

(b-d1)(-d)

In the present case

a = 1, b = c, y e, g c1 O,. d -1 u (1 cn2u du (l-c*2sn2u)2 . snu1 = sin+

sn2u (l+c)x (1+x)c

x

:cf

-33

In Ref [7], Equation 254.10 gives

tin K -c 2sn2u)m

j

dt:g

)

111

du c /(a-t)(b-t)(t-c1)(t_d) (l_2Sfl2U)m (b-c1)d

where C1a12 = - e-- _{and all of the other syffibols}

(b-d)

are the same as in the first example _{of this appendix. Thus} k2 =

2

-

<k , = 1,

1+c

Therefore, after simplificatiön

dx

V1+c

_fE(k)

_{- K +} C:

o 1-x2

_1+c

j

which is the desired result,

The evaluation of the integral

Pc

JxF(x)dx

=

/x(c_x)

o o

can be decomposed into a sum of two integrals, viz.

g:

1+C 2

V l+c

X2 dx o X-V (l-x)(l+x)(c-x) (B 1) dx

where

and similarly

fC

;:

dx

1V3

-3V2 +3fl(y2,k) _K1

°

_{V(1_x)(l+x)x(c.x)}

1+c

_¡(B

₃₎ C j

1/x(C_x)

2

1+c

C

x2dx

=

°

_{V(l_x)(l+x)x(c_x)}

_vi

(k) + K

-where

V3

i

(l+c) K -

(5+3c)(y2,k) +

_{(4+c) V}

2 2 -

2.

Upon substituting Equations (B 2) and (B 3)

_{intoEquation (B 1)}

and simplifying there results

(1+c) E (k) +

_{(l+ò)('-c) K}

! c2_l)H(y2,k)

2 L.

J

which is the second desired result.

(B

L)

K -211(y2,k)

(.B 2)

4+c

T1(y2,k)

Fig. i Cavitation on a delta wing

j y;v:

Fig.

2 Cavitating flow past a delta wing and the coordinate system

o

S (X),CAVITY AREA

Xu

Fig.

3 Closure conditions in transverse Z- plane

X

Fig.

4 Schematic diagram of the wing in X-Z plane

X

Iv

v(Z)W-

iv

v()

=wiv

Fig.

______B,

e -

v:-U0jx

o(

Lh? s B A - -

V: U00oc

I

+1

Fig. 5 Boundary-value problem in transverse

Z -plane

w=U,Z

6 Boundary-value problem in transformed

C. -plane

--

I.o

_o=.02

CO 5

0:,Q7

L

ø =15

o=.ZO

o =30

0F.40

o.60

o. =. 80

o=LO

o

2

4.

8

lO

12 14 16

a, deg.

Fig.

7 Nondimensional width of the cavity vs. angle of attack,

8

,

deg.

Fig.

8 Nondimensional width of the cavity vs. angle of attack,

8

0.3

0.2

0.I

-e

F,

Ç

-e -e

0.I

0.2

-e

-

-a.

experiment [3]

theory

-e -e -e o

---17.5

0.3

0.4

Fig.

9 Compärison- of lift coefficient from present theory with

experimental values in [3], B

150

14.5

-

_-2.O

-CL

0.7

0.6

0.5

0.4

0.3

0.2

0.l

/

a,

Fig.

lo

Comparison of lift coefficient from present theory with

experimental values in [3],

300

e xpe riment

theory

/

.1

/

/

/

- 14.5 7.5

-/

_'df

_-

-- ø-5.5 T

0I

0.2

0.3

0.4

11.5

[3]

/

CL

experiment [3]

theory

o.

Fig. 11 Comparison of lift coefficient from present theory with

experimental values in [3], 8

CL

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.i

'I

r

-'I.

6

,Íp

-,

r,

,,,,1/

/ /

experiment [3]

theory

17.5

/

oc.

..-14.5

,

/

,

_,

,

,

,,e. 5

uo-5. 5 0.1

0.2

'.3

_0.4

o.

Fig. 12 Comparison of lift coefficient from présent theory with,

experimental va-lues in [3], ß

600

- 11.5

o.

Fig. 13 Comparison of Cumberbatch-Wu theory with experimental values

in [3], ß

= 15°

experiment [3]

theory [2]

ß 30°

experiment [3]

theory [2]

CL

0.1

0.2

0.3

0.4

Fig. lL Comparison of Cumberbatòh-Wu theory with experimental values

CL

0.5-

0.4-

0.3-0.2

0.I-o

o

30°

-ø 0

0

G,.

30°

25°

20°

theory [2]

i i 0.1

0.2

0.3

0.4

a.

Fig. 15 Comparison of umberbatOh-1u theory with experimental values

ß. 15°

experiment

[tê]

theory [2]

CL

0.1 _Q.2

_0.3

_0.4

u.

Fig. 16 Comparison of Curnberbatch-Wu theory with experimental values

o.

Fig. 17 Comparison of Cuthberbatch-Wu theory with experimental values

in [ti],

8

30°

experiment

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