Theory of ventilating or cavitating flows about symmetric surface-piercing struts

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- JULI 1976

ARCHIEF

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DAVID W. TAYLOR

NAVAL SHIP RESEARCH AND DEVELOPMENT CENTER

Bethesda, Maryland 20034

THEORY OF VENTILATING OR CAVITATING FLOWS ABOUT

SYMMETRIC SURFACE-PIERCING STRUTS

by

B. Vim

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4. TITLE (and Subtitle)

THEORY OF VENTILATING OR CAVITATING FLOWS ABOUT SYMMETRIC SURFACE-PIERCING STRUTS

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David W. Taylor Naval Ship Research and Development Center

Bethesda, Maryland 20084

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Work Unit l-1520-001-40 Task Area Z-F424 21 001

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September 1975

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19. KEY WORDS (Continue on reverse aida if neceaeary and identify by block number)

Ventilating stru t Strut-foil system Cavitating strut Free surface Cavitation number

Froude number

20. ABSTRACT (Continue on rever.e aide If necea.ary and identify by block number)

Two stable states of high-speed strut flow are considered in three dimensions under linear boundary conditions. The strut is thin, symmetric, without angle of attack, and has base-ventilation or base cavitation without side cavitation. The strut and cavity are represented by

polynomial source distributions with unknown coefficients on the center plane. Coefficients

of the polynomial for the cavity are obtained as solutions for integral equations of cavity source.

UNCLASSIFIED

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(Block 20 continued)

using a least-squares method. All the singular double integrals in the present problem

are integrated in closed forms by using recurrence formulas. Thus, common errors due to numerical integration of the singular integrals are avoided. Approximate cavity

shape and cavity drag are obtained. Downwash is calculated at the hydrofoil plane. which is assumed to be located at the tip of and perpendicular to the strut.

Down-wash and drag are compared with the existing experimental results, and good agree-ment is found.

TABLE OF CONTENTS

Page

ABSTRACT

ADMINISTRATIVE INFORMATION

INTRODUCTION I

BOUNDARY VALUE PROBLEM 2

SOLUTION 6

DRAG AND DOWNWASH 9

SCHEME OF ITERATION 9

NUMERICAL RESULTS AND DISCUSSION

Il

CONCLUSIONS 15

APPENDIX A - INTEGRATIONS OF SINGULAR INTEGRALS 25

APPENDIX B - INTEGRATION OF LOGARITHMIC SINGULARITY 35

REFERENCES 37

LIST OF FIGURES

Strut Shapes 16

Cavity Shapes at 2. 136 Chords Behind the Trailing

Edge, Model 2 17

Cavity Shapes at 2. 136 Chords Behind the TrailingEdge 1 7

- u for Pressure Distribution of Ventilating-Wedge Strut,

k00.01415orF8.4l

18

- u for Cavitating Wedge 1 8

- u forVentilating-Strut Model 2 19

u for Cavitating-StrutModel 2 19

u Distributions of a Ventilating Parabolic Strut 20

u for Ventilating Parabolic Strut 20

Figure 10 - u for Cavitating Parabolic Strut 21

111 Figure 1 -Figure

2

-Figure

3

-Figure 4 Figure 5 Figure 6 Figure

7

-Figure

8

-Figure

9

-Page

Figure 11 - Drag Coefficients of Base-Vented or Base-Cavitating

Parabolic Strut, 2t/c = 0. 1 21

Figure 1 2 - Cavity Drag of Base-Vented Parabolic Strut 22

Figure 13 - Cavity Drag of Ventilating Parabolic Strut 22

Figure 14 - Cavity Drag of Ventilating Parabolic Strut,

Theory and Experiment 23

Figure 1 5 - Change in Streamline Angle near a 1 2-Percent

Parabolic Strut, dic = 24

Figure 16 - Change in Streamline Angle near a 1 2- Percent

NOTATION

a1 Coefficients of polynomials for a strut

b.1 Coefficients of a double polynomial

c Chord length

D Drag

D1 Projection of strut surface on the x-z plane

Df Projection of image strut surface on the x-z plane

d Submerged length of strut

g Acceleration of gravity

H The z-coordinate of streamline at infinity

h The z-coordinate of streamline at cavity

k0 = cg/U2

m Source strength

Atmospheric pressure

Cavity pressure

r(X, X1) The distance between two position vectors X and X1

S Projection of cavity surface on the x-z plane

U

TheveIocityatx--oo

u,v,w The x, y, z components of the perturbation velocity

x,y,z The right-handed cartesian coordinate system

a Cavitation number

SUBSCRIPTS

a Quantities for atmospheric pressure

C For cavity

f For free-surface image

L For leading edge

ABSTRACT

Two stable states of high-speed strut flow are considered in three dimensions under linear boundary conditions. The strut is thin, symmetric, without angle of attack, and has base-ventilation or base cavitation without side cavitation. The strut and cavity are represented by polynomial source distributions with unknown coefficients on the center plane. Coefficients of the polynomial for the cavity

are obtained as solutions for integral equations of cavity source, using a least-squares method. All the singular double integrals in the present problem are

integrated in closed forms by using recurrence formulas. Thus, common errors due to numerical integration of the singular integrals are avoided. Approximate cavity shape and cavity drag are obtained. Downwash is calculated at the hydrofoil plane, which is assumed to be located at the tip of and perpendicular to the strut. Downwash and drag are compared with the existing experimental

results, and good agreement is found.

ADMINISTRATIVE INFORMATION

The work reported herein was authorized and funded by the Naval Materials Command

under Task Area Z-F424 21 001, Work Unit l-1520-001. INTRODUCTION

Since the advent of hydrofoil boats, more knowledge of the flow about a ventilating strut has been needed. Unlike a hydrofoil, a two-dimensional approximation for a strut

cannot be used properly, although there have been some attempts' to solve the problem this way. Because of the free surface that the strut pierces, the problem is essentially

three-dimensional. In addition, the cavity shape behind the strut is a function of the strut speed.

Although many investigators have effectively solved free-streamline problems in two dimen-sions using complex variables, there is no such convenient mathematical tool in three

dimensions.

There have been several interesting experimental results for ventilating or cavitating

struts,2'3'4 especially when a foil is attached to the strut. The change of cavity shape with

'Thomsen, P., "A Comparison of Experimental Data with Simple Theoretical Results for Ventilation Flow," TRG, Inc., Report TRG-156-SR-1 (20 Aug 1962). A complete listing of references is given on pages 37-39.

2rhomsen, P., "Cavity Shape and Drag in Ventilated Flow: Theory and Experiment." TRG, Inc., Report 156-SR-2 (Feb 1963).

3Hay, A.D., "Flow About Semi-Submerged Cylinders of Finite Length," Princeton University (1 Oct 1947). 4Spangler, P.K., "Performance and Correlation Studies of the BuShips Parent Hydrofoil at Speeds from 40 to 75 Knots," David Taylor Model Basin Report 2354 (Dec 1966).

speed is most interesting to observe. When speed is low, there is a small depression behind

the blunt-based strut on the free surface. This depression grows both in depth and length with added speed. Momentarily a ventilated cavity spreads along3 the entire strut, entrains

behind it, and stays there for a considerable range of speeds. However, at a certain higher

speed, the cavity length becomes longer near the deepest end of the strut than near the free

surface, where the cavity length becomes zero. This has been called the choked stage. At

this stage, cavity pressure is the vapor pressure, not the atmospheric pressure; that is, the

ventilation ceases, and the strut is fully cavitating.

The physical explanation of the phenomenon can be given as follows. A large quantity of air is sucked in at high speed by ventilating cavity entrainment5 through the cavity behind

the strut. The high airspeed reduces the pressure of the cavity wall, and the mouth of the

venting cavity becomes smaller at the free surfaces. With increasing air speed, the cavity wall

lapses into an unstable state until it is completely closed, cutting off ventilation and forming

a steady cavitating flow. Thus we may assume that there are two stable statesof the

high-speed strut flow; one is ventilating, and the other is cavitating.

in the present study, these two stable states of high-speed strut flow have bee!1

consid-ered in three dimensions under linear boundary conditions. The strut is thin, symmetric,

without angle of attack, and has base ventilation or base cavitation without side cavitation. The strut and cavity have been represented by polynomial source distributions with unknown

coefficients on the center plane. Coefficients of the polynomial for the cavity have been obtained as solutions of integral equations for cavity source distribution, using a least-squares method. All of the singular double integrals in the present problem have been integrated in

closed forms, using recurrence formulas. Thus, common errors due to numerical integration of the singular integrals have been avoided. The approximate cavity shape and cavity drag

are obtained, and the downwash is obtained at a hypothetical hydrofoil plane, assumed to

he located at the tip of and perpendicular to the strut. Downwash and drag have been compared with the existing experimental results, and good agreement has been obtained.

BOUNDARY VALUE PROBLEM

A vertical strut is located in a uniform flow, having velocity U. The mean free surface

is considered to be the z =O plane, where the origin O of the right-handed rectangular

coord-inates, O-xyz, is located at the point of intersection of the mean free surface with the trailing

edge of the strut. The x-direction is the same as U. For convenience, we assume a uniform

section shape for a symmetric strut

5Elata, C., "Choking of Strut-Ventilated Foil Cavities," Hydronautics Technii Report 605.2 (May 1967).

in

S(O<x <Xe, O>z>d, y0)

where we first assume that S is rectangular. In Equations (2) and (3). a condition m1 m2 = Ua0 on the trailing edge x = O is automatically satisfied. Although the cavity shape is not

known a priori, we have to assume the shape first and to formulate an iteration scheme. In the case of a supercavitating hydrofoil with cavity length longer than three chord lengths, the cavity drag is known to be quite insensitive to the cavity planform behind the foil.6 Thus a long cavity analysis will be tried first. The coefficients b1 will be determined from boundary

conditions. At the free surface, the linearized boundary condition for large u, i.e.. large

Froude number, will be used. Considering perturbation-velocity components (u. u. w), we

have on the free surface

u0 on

z0

(4)

This indicates that we have only to consider image-source distributions

6Widnall S.E., "Unsteady Loads of Supercavitating Hydrofoils of Finite Span," Journal of Ship Research. Vol. 10.

No. 2, PP. 107-118 (Jun 1966). y = 2

E

i + I N a1 Xl I (I)

i0

in XL <x <O and O > z > d, where coefficients a1 are given, and XL is the x-coordinate of the

straight leading edge. Then the linear source strength for the strut will be

m1 _{I dy}

(2)

1=0

in

Dl(xL <x <O, O>z>d, y0)

where U UI. The cavity behind the strut may be represented by a source distribution

r'

N M

on

on

where is the atmospheric pressure.

4

Df(xL<x<O, O<z<d. y=O)

and

= - a

N M

b1 x' z

S(O<x<x, O<z<d, y=O)

The boundary condition on the cavity is from the Bernoulli equation along the cavity

streamline

PC I oø

- + - q +gh - + -

U2 +gH

p 2 p

where P is cavity pressure P is pressure at infinity

h.H are the z-coordinates of the streamline at the cavity and at infinity, respectively

g is the acceleration of gravity

is the total speed at the cavity.

If we linearize the Bernoulli equation

P00

Uu - + gH - - - gh

p p At infinity P00 a

= --gH

_{p p} _5_

=, a1x1

m11 N

l0

Thus, on the vapor-cavity boundary, we have a nondimensionai expression

U

-= - k h/c

U 2 0

where Ga

P)/(1/2pU2) and k0

gc/U2, with a standard chord length c. where c is

taken to be unity.

On a ventilating cavity

P =P

_{c a}

hence

= - k0 h/c (9)

These boundary conditions will be applied on the y = O plane. Since g and _a are constants,

and P is also approximately constant, cia is a function of only U, and k0 is a function of U

and c. For a choked cavity of a strut foil on a high-speed hydrofoil boat, U = 0 (100 fps),

and both cia and k0 h/c are small, however, k0 h/c is much smaller than 0a For example. if we take as a vapor pressure of water7 at 70° F or 0.36 psia and the air pressure at approxi-mately 14.7 psia, then

P P

32.lg

U a

=hg

I U2

gh/U2

U pgh _u2

Thus when U is sufficiently large, and h is small, the gravity effect can be negligible on the cavity, as well as on the free surface, compared with the cavitation number. However, if the scale of a hydrofoil is very large, and the submergence is almost 32 chords, say, the

gravity effect on the cavity should be important even for reasonably high speed. For a ventilating cavity, only the gravity effect should be considered, as stated by

Equation (9). Since the steady gravity wavelength is

= 2ir/k0

7"Handbook of Chemistry and Physics," Edited by Weast, R.C., et aL, The Chemical Rubber Co., Cleveland. Ohio (1964) p. D-92.

the cavity will be affected by the free-surface waves, when X is small or k0 is large. In this case, the wave-number dependent, free-surface boundary condition au/ax + k0w = O has to

be used. However, the cavitation or ventilation does not take place unless the speed is great.

For example, for c = 1 foot and U = 10 fps

k0 = gc/U2 = 0.32172

or

X/c = 19.53

In such cases, when the- wavelength is large, the cavity condition

u/U = - gz/U2 = - k0z/c

and the free-surface condition on z = O

u= O

can be effectively utilized to solve the problem approximately.

For two states of flow phenomena of the strut flow, when one is ventilating and the other is cavitating, only the approximate boundary conditions for both the free and the

wetted surfaces have been considered as shown in Equations (8) and (9). We do not

con-sider the cause of the phenomena, nor the airflow inside the cavity in our problem.

Further-more. as the linearized model, neither the effect of spray nor tip has been considered here.

SOLUTION

We represent u in terms of a source distribution

u(x,y. z) =

(-jO jO

d

XL

ofc

m2 + m1 + d O m1f O XL 6 IC m a ax

(I )dxldzl

_r(X,X1) (10)

where

L

= {(x_x1)2 + y2 + (z_z1)2} 2

If we use the boundary conditions of Equations (8) or (9) on yO, Equation (IO) is a

singular integral equation to solve for m2 because the kernel is singular at X = X1.

Inserting Equations (2), (3), (5), and (6) into Equation (IO) we obtain

where A. =

(-B1 =

(- (lp

u(x,y.z)='

aA

f050

d XL

These singular integrals will be analytically evaluated in Appendix A. using the concept of finite parts of singular integrals.8

For a finite cavity we may include a closure condition Xc J(m1 + m2) dx = 0 (15) M d 0

+S$

_O

_XL)

-

ax d XC

+5f

)xi'z

00

(12) (13)

8Hadamard, J., "Lectures on Cauchy's Problem in Linear Partial Differential Equations," Yale University Press, Conn. (1928).

Ao=( JO

_{d XL}

d XC\ a Xc

J

₎

O _XL i

-

_r

dx dz

al

- - dx dz

(14) r

N i+l

Ixc

b..

=0

j+l

1= 1

for

j1,2...M

(16)

If Equations (il) through (14), (16), and either (8) or (9) are combined, then linear simul-taneous equations for b1 and Ga and k0 are obtained. Since the cavity planform shape will

be assumed a priori, _a (or k0) cannot be given together. Thus when 0a (or k0) is given first, the answer for and the cavity planform will be obtained by iteration. If the simultaneous

equations are solved in the normal way by a matrix inversion method, the number of simul-taneous equations must be the same as the number of field points on the cavity plus the

number M of closure-condition Equation (16). Solution of the equation b1. Ga and k0 de-pends on the location and number of field points and the number of terms in the cavity-source polynomial. Convergence of the solution can be numerically tested. 1f the number of terms is small, then the corresponding number of field points has to be small also. When the number of terms is given, the accuracy of the solution is only a function of the location of field points. To choose "better" field points, the collocation method9 was devised for airfoil

lifting-surface theory and was widely used. However, in the collocation method the compli-cated behavior of convergence is not well understood.10

In the present analysis, we use the least-squares method to solve the simultaneous

equations.1 I A similar technique has been used for airfoil lifting-surface theory.'2 In this

method, we can have a unique solution no matter how many equations we have for a given number of unknowns. Thus, we can choose as many field points as needed to increase

accuracy for a given number of polynomial terms in Equation (3). The numerical programing

is simple for the least-squares solution. Solutions for the cavity source strengths, i.e.,

converge quite rapidly when the number of field points is increased.

8

9Multhopp, H., "Methods for Calculating the Lift Distribution of Wings, (Subsonic Lifting-Surface Theory)," Aeronautical Research Council Report and Memoranda 2884, Her Majesty's Stationary Office, London (1955).

'0Lomar iP., "A Modified Muithopp Approach for Predicting Lifting Pressures and Camber Shape for Composite Planforms in Subsonic Flow," National Aeronautics and Space Administration Report TN D-4427 (Jul 1968).

''Hildebrand, F.B., "Introduction to Numerical Analysis," McGraw-Hill Book Company, New York (1956)

pp. 258-302.

1 2Tulinius, J.R., "Theoretical Prediction of Thick Wing and Pylon-Fan Pod Nacelle AerodynamicCharacteristics at Subcritical Speeds," North American Rockwell Report NA-71-447 (1971).

The drag of the strut can be determined by integrating the pressure multiplied by the strut slope on the projeclion of the strut surface. The pressure at the leading edge is singu-lar but it is integrable. The drag of the strut to be numerically evaluated can be written for

the ventilating and cavitating cases in the form

DRAG AND DOWN WASH

dxdz +

where y(0) = t.

When we consider a strut-foil system such as that on a hydrofoil boat, knowledge of the downwash on the hydrofoil due to the strut as well as the downwash due to the foil is important. The flow angle at the foil is significantly influenced by strut downwash, which

may induce cavitation on the pressure side of the wing. Downwash due to the ventilating or

cavitating strut is given by

w(x,y,O)

_JJ

(m,+m1f+m2 +m_2f

) -

dx1 dz1

az

D1+Df+S+Sf r(X,X1)

This equation can also be integrated analytically,' although it is a little complicated as is

shown in Appendix A.

SCHEME OF ITERATION

When U is very large, the cavity may be choked, so that the cavity length near the free

surface becomes small. The small cavity length may have considerable influence on the

pressure at the strut. However, since there is no simple way to predict the cavity length, we have to rely upon an iterative format. We will, then, construct a scheme similar to the

Newton-Raphson scheme.

3Grobner, W. and N. Hofreiter, "Integraltafel Wien und Innsbruck," Springer-Verlag (1949).

(17) (18)

dc

D I

Çf4udy

d O

pu2D, D

3)

U dx

{

°c

k - - for ventilating case

a -

for cavitating case

a

2t d 2t

M

We assume an extra source distribution E b z in D for the strut in addition to that

j.1 Oi i given in Equation (2)

--

a1x

+;'

_b0z? in D1 m1 N M (19) i-1 j=i

hoping that the extra source distribution will compensate for the effect of the assumed cavity length on the cavity boundary condition. The coefficients will be determined from the boundary condition. Because of the continuity of source strength at the trailing edge of the strut, we can write for the cavity source distribution

U =a0

+T'b..xuIzIi

inS

(20)

i=O

j0

where i starts from O since m1 (x = O) = m2 (x = O).

If we assume a cavity shape y = x0 y"i (E C1 ') with unknown coefficients a and C1, integration of Equations (12) through (14) can be performed in closed form by approxi-mating x with several broken lines

x = a1 + 31z (21)

For determination of the unknown coefficients a and C1, we use the Newton-Raphson

method.1 i That is, when a is specified for a given strut (Equation (2)) the coefficients a and C1 are of such a nature that the calculated a has to be equal to the specified a, and the resulting extra strut source or the coefficient b0 has to be equal to zero. Thus, when a first approximation of a and C1, say, a' and c1', are given, the next approximation can be

obtained as follows from the Newton-Raphson scheme. When we solve simultaneous equations

(4 aCi)()

=

(T°°

ab ab a IO (22)

where a is the given cavitation number, and ci is the calculated value, a2

= a'

+

(23)

= C11 + C1

will be the next approximation. By continuing this process until a and h. are very close to the aiming values, a = 00, and b0 O.

In general, partial derivatives should be computed numerically. Accuracy is not crucial

for the iteration. When i and j of C1 and b0 are large, the computing time will be the main problem. However, even for a choice of small values of i and j, the calculation should be

useful for understanding the cavity shape and for estimating cavity drag.

NUMERICAL RESULTS AND DISCUSSION

We cannot, obviously, obtain a real solution of cavity flow by using the inviscid model.

Then what kind of model can best approximate the real solution? There are many kinds of

inviscid models for two-dimensional cavity flow such as the Riabuchinsky symmetric model,

the Roshko wake model, the TWin double or single spiral vortex model, etc. As long as the cavity length is fairly large, these models differ little in their effect on flow phenomena near

the foil. In general, to obtain the cavity length, the cavity-closure condition is applied for

closed-cavity models. For the three-dimensional cavity model, which we are dealing with,

the first approximation of the cavity planform influences the first approximation of the

solution. In this respect, we have tried several methods numerically to have a better first

approximation without considering the extra source on the strut shown in Equation (19). First we tried a rectangular planform with the closure condition of Equation (16) and found that the cavity shape did not agree with experimental results. Naturally the cavity length can be controlled both by the singularity planform of the cavity and the closure condition. For a strut flow in which the free-surface effect makes the flow significantly deviate from the two-dimensional flow, it has been found that the closure condition, in addition to the given singularity planform, unreasonably restricts the solution. Therefore, when we neglected the closure condition for the simultaneous equations for the cavity source,

results of the first approximation solution were quite good in both ventilating and cavitating cases. The cross sections of the cavity shapes in both the ventilating and cavitating cases,

approximately two chords behind the parabolic and wedge struts (Figure 1), are shown in Figures 2 and 3 for drafts of one chord and two chords. Although Models 3 and 4 of struts

differ in shape from the real parabolic strut, the physical quantities vary little. A closer approximation to the parabolic strut near the leading edge by Equation (I) is difficult. Therefore, we substitute Model 3 for the parabolic strut.

To relax the closure condition at the cavity end, we also considered a vertical line sink at x = x whose strength was the same as the total strut source per unit vertical length. This influenced the value of k0 only. When we compared the flows with the same k0 with and without the cavity-end sink, little difference was observed. However, either k0 or u were obtained as a solution, thus, to obtain higher values of k0 or u, we had to reduce the length

of the cavity planform and/or increase the strength of the cavity-end sink.

In the polynomial representation of cavity-source strength in Equation (6), sets of x and z terms (N,M) = (4,4), (4,5), (5,5), and (4,6) were tested. The result was that cavity shapes

were changed a small amount; however, drag was changed very little.

The distribution of u, the x-component of velocity on the struts shown in Figure 1 has been obtained and is shown in Figures 4 through 10. From Figures 4 through 10, the pres-sure distributions can be obtained immediately from the Bernoulli equation

PP

_a

-

U - pU2

- 2k0 h/c

together with Equation (9). The pressure on the wedge-shaped strut was normally greater than atmospheric pressure. The pressure was sensitive to the curvature of the strut surface, and

the minimum pressure became less than atmospheric pressure at a comparatively small

curva-ture of the strut surface as shown in Figures 4 through 10. For the approximate parabolic

strut (Model 3), the pressure was less than atmospheric over most of the strut surface. How-ever, it is believed that it would not cavitate unless the pressure should become lower than the vapor pressure. For a base-cavitating parabolic strut, the pressure on the after part of the

strut falls slightly below the cavity pressure (Figure 10). These findings may be a

manifesta-tion of free-surface and three-dimensional effects because although in the two-dimensional

cavity flow, we assume that the pressure in the flow is always larger than the cavity pressure, this is strictly an assumption and not a fact.14 Nondimensional velocity distributions for

ventilating struts at moderate speeds (small k0) were almost the same as for the corresponding cavitating struts at much higher speeds with u=0.(k0), except the trailing edge (Figures 4

through 10), although cavity shapes were considerably different.

The strut drag coefficients, obtained by integrating the pressure distribution multiplied by the slope of the strut surface, are shown in Figures 11 and 1 2. For a parabolic

two-dimensional foil with zero cavitation number,15 the drag coefficient is

14BllkhOff G. and E.H. Zarantondllo, "Jets, Wakes, and Cavities," Academic Press, Inc., New York (1957).

15Auslaender, J., "The Linearized Theory for Supercavitating Hydrofoils Operating at High Speeds Near a Free Surface," Journal of Ship Research, Vol. 6, No. 2, pp. 8-23 (Oct 1962).

irt2

CD

=

Of course this cannot be directly used for a parabolic strut because of three-dimensional and

free-surface effects. If a parabolic strut of finite-aspect ratio is located in a high-speed flow of an infinite medium, we have only to consider the three-dimensional effect. The drag can be

approximated by the well-known expression16

t2

C2

X

CO3

(-g)

-

_t 2 _{X+ I +CD2/(lrt2/c2)}

(7)

where CD3 and CD2 are three- and two-dimensional drag coefficients, respectively, and X is the

aspect ratio. The free-surface effect tends to reduce the drag; this can be seen from the com-puted pressure distribution near the free surface in Figures 4 through I 0. The free-surface

effect of a two-dimensional finite wedge entering the water surface was known to reduce the drag almost 45 percent, compared to the base-vented wedge in an infinite medium.t7 Thus we can expect a sizable free-surface effect which tends to reduce the cavity drag. However.

we may not be able to neglect spray drag which can be quite large according to many

experi-ments with nonventilating struts.

Although numerous experimental results (Reference 1 8) exist for high-speed strut flow. few data exist for symmetric base-vented struts. Therefore, we took a result for a base-vented

parabolic strut from Reference 19 where the friction-drag coefficient was taken to be 0.003. Comparisons between the computed cavity drag of a parabolic strut, i.e.. Model 3. and the residual drag determined from experiment19 are shown in Figures 11 and 12. The difference between the computed and measured drags may be mainly due to the spray drag.

16Glauert, H., "The Elements of Aerofoil and Airserew Theory," Cambridge University Press, London (1962). B., "Investigation of Gravity and Ventilation Effects in Water Entry of Thin Foils," Proceedings International Union of Theoretical and Applied Mechanics Symposium, Leningrad, USSR (Jun 1971) pp. 47 1-489.

18Rothblum, R.S. et al., "Ventilation, Cavitation, and Other Characteristics of High-Speed, Surface-Piercing Struts." NSRDC Report 3023 (Jul 1969).

19Aerojet-General Corporation, "Hydrodynamic Characteristics of Base-Vented and Super-Cavitating Struts for Hydrofoil Ships," Vols. I and 2 reported to U.S. Navy Bureau of Ships, Washington, D.C., Report 2796 (Aug 1964).

Although there are several empirical formulas for the spray drag of nonventilating

struts,2022 they differ a great deal. Furthermore, spray drag measurements for a base-vented strut do not seem to exist. In general, the spray drag for a high Froude-number flow is a function of the thickness-to-chord ratio, the nondimensional distance from the leading edge

to the point of maximum thickness, and the Reynolds number. When we consider existing spray drag measurements and their dependence on these parameters, the present result seems to be reasonable. The variation of drag coefficient with draft-to-chord ratio, shown in

Fig-ures 11 and 1 2, indicates a large three-dimensional free-surface effect for d/c < 2 and for

small k0.

We note here that the chord length c is only a small portion of the total strut-cavity length. For a parabolic strut, i.e., Model 3, of one-chord draft, the three-dimensional drag coefficients as a function of k0 and a are shown in Figure 13 together with results from

two-dimensional theory.23 This figure shows that CD is very nearly a linear function of k0 or u.

In fact, we have seen that the nondimensional velocity distributions on a strut do not change too much for different values of k0 or a in Figures 4 through 10. The calculated differences in drag coefficients come mainly from the direct contribution of k0 or u, represented in the second term of the right-hand side of Equation (1 7). This can be understood when we

com-pare curves of drag coefficients with the curve of

CD CDO

+-

_u

/ 2t'

2t 2

()2

) (-z-) CDO k0d or

+-(2t'2

2t

\c /

represented by dashed curves in Figure 13, where CDO is the value of CD for k0 = 0. In

Figure 13, another interesting fact is that the drag coefficient for a ventilating strut is very close to that for a cavitating strut at the same values of k0d/c and u. However, we note that u for the ventilating strut is approximately equal to 1 6 k0 for the cavitating strut at the same

speed. Therefore, when a strut suddenly switches to cavitation from ventilation, the drag

jumps to quite a large value. Another comparison between calculated and experimental

results is shown in Figure 14, where good agreement can be observed.

D. and J.P. Breslin, "Experimental Study of Spray Drag of Some Vertical Surface-Piercing Struts." Davidson Laboratory Report 1192 (Dec 1966).

21Hoerner, S.F., "Some Characteristics of Spray and Ventilation," Gibbs and Cox, Inc., Technical Report 15 (Sep 1953).

22Chapman, R.B., "Spray Drag of Surface-Piercing Struts," American Institute of Aeronautics and Astronautics Paper 72-605 (Jul 1972).

23Johnson, V.E., Jr. and S.E. Starley, "The Design of Base-Vented Struts for High-Speed Hydrofoil Systems," Pro-ceedings of the Hydrofoils and Air Cushion Vehicles Meeting, Washington, D.C., Sep 1962, published by Institute of the Aerospace Sciences (1962) pp. 35-57.

24Andrews, T.M., "Pendulum Tests of Two Cruise Foils and Various Struts," Dynamic Developments, Inc., Report Contract 2852(00), (Oct 1961).

The effect of the strut on downwash at the foil is easy to calculate and is shown in Fig-ures 1 5 and 16. The calculated downwashes agree with the experimental results fairly well,

although the strut cavity shape with a foil in place may be different from that without a foil. In fact, downwash does not seem to be too sensitive to the cavity shape.

The previously mentioned numerical results have all been first approximations, without successive iterations to determine different cavity planforms. In addition to this problem, we

may incorporate angle of attack, using a lifting-surface theory that is being considered for a supercavitating hydrofoil of finite span. This problem is physically and mathematically

inter-esting in itself and is related to the engineering-design problem of a high-speed, strut-foil

sys-tem of a hydrofoil boat. A computer program27 has been written as a future unit of the strut-foil system so that all the subroutines can be utilized for both the cavitating and lifting foil as well as the interference between the strut and the foil.

CONCLUSIONS

Flows of ventilating or cavitating struts have been analyzed numerically, using a

three-dimensional mathematical model. With a double-polynomial representation of the cavity

source, integral equations for the cavity source are solved, and various interesting physical phenomena are found. Cavity drag and downwash seem to agree well with existing

experi-ments. However, since the measured data are widely scattered and a large portion of the measured total drag is the friction drag which is difficult to estimate correctly or separate from other components, we need to be cautious in drawing quick conclusions.

The present analysis can be easily extended to a symmetric strut of any shape. For better prediction of cavity shape, an iterative scheme may be needed. However, for the prediction of strut pressure distribution or drag, the first approximation seems to be enough. and no iteration is necessary as long as the cavitation number or Froude number is obtained as a solution because the pressure on the strut is almost insensitive to both cavity shape and cavitation number. Comparison with the two-dimensional solution23 shows that the latter

considerably overestimates the drag. Numerical experiments with various cavity models have

been very helpful in determining a suitable cavity model and the source polynomial for this problem. Thus, the present method can be confidently applied to other three-dimensional

cavity problems.

25Altman, R. and C. Elata, "Effects of Ambient Conditions, the Gravity Field and Struts, on Flows Over Ventilated Hydrofoils," Hydronautics, Inc., Technical Report 605-1 (May 1967).

26Huang, T.T., "Strut Induced Downwash," Hydronautics, Inc.. Technical Report 463-7 (Sep 1965). B. and L.M. Higgins, "Computer Program for a Ventilating or Cavitating Strut and the Users Manual," DTNSRDC Report 4754 (in publication).

y 0.8 0.6 0.4 0.2 0

Figure la - Ventilating Strut

z

Figure lb - Cavitating Strut

x/c

Figure Ic - Various Models Figure 1 - Strut Shapes

y 16 X X o -1.0 -0.8 -0.6 -0.4 -0.2

0.1 0.2 -= 0.0098 WEDGE 0.1 MODEL 2 PARABOLIC CHOKED CAVITY 157.5 KNOTS = 0.0146 MODEL 2 PARABOLIC I I I VENTILATING CAVITY 47.5 KNOTS 1I = 0.005 i/c

Figure 2 - Cavity Shapes at 2.136 Chords Behind tile Trailing Edge, Model 2

0.2 = 0.0116 WEDGE = 0.0075 PARABOLIC k0 = 0.0064 PARABOLIC I I I I 0.2 0.4 0.6 0.8 i/c

Figure 3 - Cavity Shapes at 2.136 Chords Behind the Trailing Edge

1.0 0.4 0.8 1.2 1.6 2.0 0.15 0.10 V C 0.05 o 0.15 0.1 V C 0.05 o

0 LJJ 0.1 > z o 0.2 w 0.1 0.3

LI

0.2 0.4 0.6 0.8 1.0 x/C

DISTANCE FROM LEADING EDGE

Figure 4 - u for Pressure Distribution of Ventilating-Wedge Strut, k0 = 0.01415 or F = 8.41 01 18 o = 0.01164 = 1 (DRAFT-CHORD RATIO) I I I I 0.2 0.4 0.6 0.8 x/C

Figure 5 - u for Cavitating Wedge

10 C = 0.05 u 0.1 z U(2t/c) = 0.95 C 0.2

-C 0.3

U(2tic) 2 u. 2 o

4

6 z = 0.0E 0.2 0.4 x/c dic 2 cj = 0.01406 t I 0.6 0.8

Figure 7 - u for Cavitating-Strut Model 2

1.0

0.2 0.4 0.6 0.8 1.0

x/c

dic = 2 k0 = 0.00585

0.2 0.2 u U(2t/c)

0.4

0.6

0.8

0.2 Z

-

= 0.05 C o

0.4

0.6

0.2 0.4 Z = 0.95 C

L

= 0.5 C d/c = i k0 = 0.00643 t t I I 0.2 0.4 0.6 0.8 1 0 xic

Figure 9 - u for Ventilating Parabolic Strut

20

x/c

Figure 8 - u Distributions of a Ventilating Parabolic Strut

0.6 0.8 10

u

0.2

0.6

0.2

C

Figure iO - u for Cavitating Parabolic Strut

0.4 C0 (2t/c)2 0.2 EXPERIMENT (REF. 19) WITH CF = 0.003 00 Li 0001 *

- o

0 I

L

dic

Figure 11 - Drag Coefficients of Base-Vented or

Base-Cavitating Parabolic Strut, 2t/c = 0.1

2 4 6

0.8

I I I

0.2 0.4 0.6

1.0 0.8 0.4 0.2 CD (2tJc)2 0.4 0.2 OEXPERIMENT (REF. 19) WITH CF = 0.003

Figure 1 2 - Cavity Drag of Base-Vented Parabolic Strut

/

//

TWO-DIMENSIONAL

/

//

THEORY _d (REF. 23) c

/

/

//

/

//

/

//

/

/

/

/

C0 C0 (J,k0dIC +

/

_/

/

(2t/c)2 - (2t/c)2 2tJc

/

//

_TING

/ /

CAV ITA G = 1

//

VENTILATIN THREE-DIMENSIONAL

//

THEORY (YIM)

//

_{= 3,VENTILATING, 3-D THEORY} 22

/

_/,4.-c

//_=

t

0.2 0.4 0.6 0.8 1.0 k0d/2t or ü/(2t/c)

Figure 1 3 - Cavity Drag of Ventilating Parabolic Strut Co 0.6 (2tIc)2 4 2 2t

=0.2

dIc C

0.03 0.025 0.02 0.015 0.01 0.005 o

/

/

WITH TURBULENCE FRICTION

/'

dic = 1, 2tIc = 0.167

/

/

I, r',

/

2t C = 0.21 = 0.167

=0.1

ç, ' 2tC EXPERIMENT (REF. 24) SYMBOL 2t/c dic c

o

0.21 2.005 0.21 1.365 D 0.21 0.683 5.86" 0.167 2.367

£

0.167 2.000 0.167 0.667 3" 0.167 1.333 I I 23 0.04 0.10 0.12 0.16 0 20 k0dic

o

3

4 o

1

o 2

3

= 1

X

= 0.25 THEORY i\..#

-/

C - - EXPERIMENT (REF. 23) 0.0 0.4 0.6 0.8 1.0 v/C

Figure 15 - Change in Streamline Angle near a 12-Percent Parabolic Strut, d/c =

24

0.2 0.4 0.6 0.8

y/c

Figure 16 - Change in Streamline Angle near a 12-Percent Wedge Strut, d/c = i

1

o

-2

where

= {(x_x1)2

+ y2 + (zz1)}

2

For the single integral of Equation (24), when we write

r(,1)

= (Az12 + 2Bz1 + C)2

we have

where

INTEGRATIONS OF SINGULAR INTEGRALS INTEGRALS IN INTEGRAL EQUATIONS

Equations (13) and (14) are composed of integrals of the type:

d XT _d =

xz - dx dz

-J

5XT

SS

a

a'

r O _XL O

=-: xz

dz1 +

Y(,A,d,j+ 1)

d APPENDIX A z? r(X,X1) A (A,B.C)

ai

_-

_{dx dz} ax1 r ix z

L

dx1dz1 dz1 =

J - z1

r Aj

(2j-1)

B

J

dz1

(j-1)

C zJ-2 dz1 j A r (24) XT r X1XL d (25) O

d _dz1 I Az1+B d + r I

Y(,A,d,l)

EEJ _-;-. =

log (

_/T

/

f o

All of the integrals appearing in Y(, À. d,j+l) may be evaluated by using available integral tables.13 1f we insert Equation (26) into Equation (25), we can obtain Y(X, À, d, 2) then if

we insert Y(.Ä,d, I) and

Y(,Ä,d,2) into Equation (25), we can obtain Y(,A,d,3).

If

we continue this process we can obtain Y(, À, d,j+l) for any integer j. For the double integral appearing in Equation (24), we write

F(,d.i+l,j+l)

jXT

xz

dx1dz1 O _XL where r2 = A1 x + 2B1 x1 + C1 A1 =1 B1 C1

= x2 + y2 + (zz1)2

Using Equations (25) and (26), Equation (27) can be written

F(,di+l.i+l)=J

![x1r_(2i_l)BiJ1''

dx1 1Xr

_(i_l)CiJ

_{r dx1 f} z?dz1 J x1=x and, for i = O, XT

F(,d, l,j+l)

=

J

log(x1 x+r)

z j1 dz1 (29) XL 26 (26)

If we integrate Equation (29) by parts, then

where y = A (x1 x)

6

B (x1 x)

with A and B the same as in Equation (25)

The integrals in F(,d, l,j+l) can be

linear function of z for

_k

and x, for later

X1- a, +b,z, so that the leading and the t

perpendicular to the mean free surface but plane. Then, making use of Equations (25)

written d (a2 + b2 z1 )z o

F(,d, l,j+l)

-

[-T log(x1 x+r)

_jd

...L1

z1 z

₊ 7(z1

z)+6 '

- dz1I

j+l

_1.

_{(z1 z)+y2}

_{{(z1 z)2 +y2}

} r J _{_j X=Xj} o r(X,X,1.)

integrated term by term. use. That is, we will use

railing edges or the cavity

I

(a1 +biz1)z? . dz1

are represented by any straight line on the y = O

through (27) and (30), Equation (24) can be

(30)

We will consider here a XL = a1 + b1 z and end are not necessarily

(31) +

jd

X

x1 z

dx1 dz1 O _XL

(b2\

k Y (, Ä, d, j+k+ I)

\

a)

ICk

(b

k

Y(,A11,d,j+k+l)

ai /

+ i F(,d,i,j+l)

where

and

Then

where

A1 (A'.B,C11),

i= 1,2

(33)

Integral (32) by the same method as used is evaluated for Equation (25). However, Integral

(27) is considerably different when XL and XT are functions of z. As in Equation (28). we

write XII1

F(,d,i+l,j+l)

r(2il)B1

_r dx1

k!(ik)!

Y(Ä11.di+k+l)=J

z

(x_a1_b1z1)2+y2+(z_z1)2

dz1 d I

Jz(Az

+2B'z1

+C.1)2

dz1 (32) o 28 (34) i-2 ₇ a2+bz

(i

l)C1J

dx1

J

f a1+bz1 z?dz1 r

F(,d, l,j+l)

jd

_(_

I)' log(a1 + bz1 x +r1)zdz1 (35) =

(a + bz1 x)2 + y2 + (z1 z)2

(36)

Integrating Equation (35) by parts

where

= b1A1

y=b1B1 +A1(b1z+a1x)

(38)

= B1 (b1z+a1x)

The integrals in the right-hand side of Equation (37) can be integrated term by term, using the

following integrations J z2 +y2 Ç

Zr

_dZ

if m2i+1

Ç

Z2 ±2i

(

y2Zr

ZrdZ+f

dZ

J Z2+y2

JZ2+y2

= ₊

± y2)ZrdZ

t

y1Zr

dZ (39)

J z2+y2

d j+1 I J

i+l

o

F(,d, 1,j+l) =

j)I I I Jog (a1 + bz1 - x + r1)

Ii+l

b1r

z1 z

(z1 z)2+y2

+

(z1 z)2+y2

ß(z1 z)2 +r(z1 z)+61

1 + dz1 (37) { (z1

z)2 +y2} r,

J

J

30

(40) with upper sign

when i is Çodd

lower sign even,

if m2i

=J(Z2(1_1)

-

_{Z2('-) y2} + . . .

y2))

rdZ

f yr

dZ

JZ2+y2

C I dZ .1 z2 +y2 Z2(i-2)+2 z2

2(i-1)+2

2(i-2)+2

2 + ± 2(i_1) 2i

+ - Iog(Z2+y2)

2

r

zm

Z2'

Z2'2"

dZ =

-

y2 + . . . ± Zy2

j

(Z2 +y2)

2(i-1)+1

2(i-2)+1

z ,2i_1 tan'

-y,

dZ

J

dZ f(Z2i±2i)ZJ dZ (Z2 +y2)r (Z2+y2)r

j

(Z+y)r

= $z2i_1_z2i_uY2 + . . ±y2U-)) dZ

where we use C dZ L

= log

J (Z2 +y2)r 2yp2

(71Z+1 r)2 +(72Z+ó2)2

í3

f

72Z+2

+ - (tan'

L

+ tan

\

Z

71Z+ój_r)

and

with upper sign

Çy21

li

dZ lower sign

.3 (Z2 y2)r and

j0,l.

1 z

(y1Z+1 r)2 +(y2Z+,)2

dZ = - log

J (Z2 +y2)r 2p2 Z2 + y2

+ - tan

(

-1

- +tan

y 72Z+52 Z

71Z+bi_r)

where

rAZ2+2BZ+C, 0CAy2

=

±VIT

2 + =

P2

-O

with the sign chosen by ß1 12 = By

= (02 + 4B2y2)2 = (Ay3 + B31) when i is { odd even

and B1

= ;;-

(Ayß1 Bß2) & = ,..L (Byß2 + Cß1) ¡ 2 p (43) 62 =

(By31 Cß2)

Thus F(,b,i,j+l) can be evaluated for any j, although it is a little complicated. The first

term in the right-hand side of Equation (28) can be evaluated from the following recurrence

relation

Jxrdx1

x_l

_(2m+l)

B1 1rdx1

(ml)C1 Jx2rdxi

(44)

r -

Ix

-- (m+2)A1

m+2

_{A1 J} (m+2)A1

x+

Jrdxl

1 A1

_{A1C1 B?}

A1x1 +B1 = 2 _2A1

lo(

+r)

where A1, B1, and C1 are the same as in Equation (28). Now we can evaluate F(X,d, 2,j+1) in Equation (28) by using Equation (44) and F(, d, I ,j+ I); then F(X, d, 3, j+ I) by using

Equation (44), F(,d, l,j+l), and F(X,d,2,j+l). By this process for F(,d,i+l,j+l) with

any integer i, we use Equation (44) and integrals F(,d,i1,j+l) and F(,d,i,j+l).

Although this process is quite complicated, it is far faster in computing time and more

accurate than evaluating the singular integral by a numerical scheme.

EVALUATION OF DOWNWASH INTEGRAL The downwash w $5

al

-

= -

m - - dx1 dz1

u

òz r

has a general term

12

Jfxizi

a i

a

dx1dz1

When we exchange (x. x1) with (z, z1), they become exactly the same as the integral already evaluated for the integral equation. As was mentioned earlier, the evaluation of integrals on

y = O was a little simpler than on y $ O for the integral equation; however, for the down wash. we need integrals for y * O.

APPENDIX B

INTEGRATION OF LOGARITHMIC SINGULARITY

The numerical integration of a logarithmic singularity which arises in the pressure

inte-gration for drag, using the Simpson rule, is performed in the following way. We consider four values in an interval (o, x) where the logarithmic singularity is located at, say, x O

y(x1), y(x2), y(x3). y(x4)

We assume a function

y(x) = a1 + a2x + a3x2 + a4 log x (46)

Then we can obtain

y(x) { y(x1) - a4 log (x1)} 1(x) + a4 log x

where

(xx2)(xx3)

(xx1)(xx3)

(xx1)(xx2)

- (x

x2)(x1

x3) '

2 -

_{(x2 x1)(x2 -X3)}

(X3

x1)(x3

-X2) and

_3

y11(x4)+y(x4) 35 log x4

-'

log (x1)Q (x4)

Therefore, the integration from x = O to x = x1

X Jy (x)dx - a4 log (x1) } L1 (x) + a4 x (log x - I) o (48) a4 - (47)

where

(X

-3(xx )(xx )

, etc.

6(x1 x2)(x1 -X3) 3 2

This can be nicely incorporated with the integration from the Simpson rule, since the Simpson

rule has also the same form for uneven intervals.

i i

(xx2)3

(xx2)2(xx3)

}

L1(x)=

REFERENCES

Thomsen, P., "A Comparison of Experimental Data with Simple Theoretical Results

for Ventilation Flow," TRG. Inc., Report TRG-156-SR-1 (20 Aug 1962).

Thomsen, P., "Cavity Shape and Drag in Ventilated Flow: Theory and Experiment," TRG, Inc., Report 156-SR-2 (Feb 1963).

Hay, AD., "Flow About Semi-Submerged Cylinders of Finite Length," Princeton

University (1 Oct 1947).

Spangler, P.K., "Performance and Correlation Studies of the BuShips Parent

Hydro-foil at Speeds from 40 to 75 Knots." David Taylor Model Basin Report 2354 (Dec 1966).

Elata, C., "Choking of Strut-Ventilated Foil Cavities," Hydronautics, Inc., Technical Report 605-2 (May 1967).

Widnall, S.E., "Unsteady Loads of Supercavitating Hydrofoils of Finite Span,"

Journal of Ship Research. Vol. lO, No. 2. pp. 107-118 (Jun 1966).

"Handbook of Chemistry and Physics," Edited by Weast, R.C., et al., The Chemical Rubber Co., Cleveland. Ohio (1964) p. D-92.

Hadamard, J., "Lectures on Cauchy's Problem in Linear Partial Differential Equations," Yale University Press, Conn. (1928).

Multhopp, H., "Methods for Calculating the Lift Distribution of Wings, (Subsonic Lifting-Surface Theory)," Aeronautical Research Council Report and Memoranda 2884, Her Majesty's Stationary Office, London (1955).

Lamar, J.P., "A Modified Muithopp Approach for Predicting Lifting Pressures and Camber Shape for Composite Planforms in Subsonic Flow," National Aeronautics and Space

Administration Report TN D-4427 (Jul 1968).

Il.

Hildebrand, F.B., "Introduction to Numerical Analysis," McGraw-Hill Book Company,

New York (1956) pp. 258-302.

Tulinius, J.R., "Theoretical Prediction of Thick Wing and Pylon-Fan Pod Nacelle Aerodynamic Characteristics at Subcritical Speeds," North American Rockwell Report NA-71-447 (1971).

Grobner, W. and N. Hofreiter, "Integraltafel Wien und Innsbruck," Springer-Verlag (1949).

14. Birkhoff, G. and E.H. Zarantonello, "Jets, Wakes, and Cavities," Academic Press, Inc., New York (1957).

1 5. Auslaender, J., "The Linearized Theory for Supercavitating Hydrofoils Operating at

High Speeds Near a Free Surface," Journal of Ship Research, Vol. 6, No. 2, pp. 8-23 (Oct 1962).

Glauert, H., "The Elements of Aerofoil and Airscrew Theory," Cambridge University Press, London (1962).

Yim, B., "Investigation of Gravity and Ventilation Effects in Water Entry of Thin

Foils," Proceedings International Union of Theoretical and Applied Mechanics Symposium,

Leningrad. USSR (Jun 1971) pp. 471-489.

Rothblum, R.S. et al., "Ventilation, Cavitation, and Other Characteristics of

High-Speed, Surface-Piercing Struts," NSRDC Report 3023 (Jul 1969).

Aerojet-General Corporation, "Hydrodynamic Characteristics of Base-Vented and Super-Cavitating Struts for Hydrofoil Ships," Vols. I and 2 reported to U.S. Navy Bureau of

Ships, Washington, D.C., Report 2796 (Aug 1964).

Savitsky, D. and J.P. Breslin, "Experimental Study of Spray Drag of Some Vertical Surface-Piercing Struts," Davidson Laboratory Report 1192 (Dec 1966).

Hoerner, S.F., "Some Characteristics of Spray and Ventilation," Gibbs and Cox, Inc., Technical Report 15 (Sep 1953).

Chapman, R.B.. "Spray Drag of Surface-Piercing Struts," American Institute of Aeronautics and Astronautics Paper 72-605 (Jul 1972).

Johnson, V.E., Jr. and S.E. Starley, "The Design of Base-Vented Struts for

High-Speed Hydrofoil Systems," Proceedings of the Hydrofoils and Air Cushion Vehicles Meeting,

Washington, D.C., Sep 1962, published by Institute of the Aerospace Sciences (1962)

pp. 35-57.

Andrews, T.M., "Pendulum Tests of Two Cruise Foils and Various Struts,"

Dynamic Developments, Inc., Report Contract 2852(00), (Oct 1961).

Altman, R. and C. Elata, "Effects of Ambient Conditions, the Gravity Field and

Struts, on Flows Over Ventilated Hydrofoils," Hydronautics, Inc., Technical Report 605-1

Huang, T.T., "Strut Induced Downwash," Hydronautics, Inc., Technical Report

463-7 (Sep 1965).

Yim, B. and L.M. Higgins, "Computer Program for a Ventilating or Cavitating Strut and the Users Manual," DTNSRDC Report 4754 (in publication).

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