Permanent File Copy St. Anthony falls Hydraulic Laborator(

UNIVERSITY OF MINNESOTA

ST. ANTHONY FALLS HYDRAULIC LABORATORY

Technical Paper No. 49, Series B

Measurements of the Unsteady Force on Cavitating Hydrofoils

in a Free Jet

by

C. S. SONG

Prepared for OFFICE OF NAVAL RESEARCH

Department of the Navy Washington, D.C.

under Contract Nonr 710(24), Task NR 062-052

June 1964 Millneapolis, Minnesoto;

UNIVERSITY OF MINNESOTA

ST. ANTHONY FALLS HYDRAULIC LABORATORY

Technical Paper No, 49, Series B

Measurements of the Unsteady Force on Cavitating Hydrofoils

in a Free Jet

by C. S. SONG

Prepared for OFFICE OF NAVAL RESEARCH

Department of the Navy Washington, D.C.

under Contract Nonr 710(24), Task NR 062-052

June 1964 Minneapolis, Minnesota

Reproduction in whole or in part is permitted

for any purpose o~.tl1eU~ite~States Government ~. _ ..

'. '. ".' , ,

'·.:1' .,<" ..

-... :':

ABSTRACT

Experimental data concerning force and other related quantities as­

sociated with unsteady super cavitating flow about bodies tested in a free­

jet water tunnel are reported herein. Three types of unsteady flows were

studied -- those due to pitching oscillation, sudden ventilation, and sudden

angle of attack change. All test bodies were of such geometrical configura­

tion that separation could occur only at two or three fixed points.

The first type of unsteady flows was subdivided into three categories,

each having different characteristics. When a flat plate was oscillated

about a large mean angle of attack at small cavitation number and without

ventilation, the cavity pressure remained unchanged. When air was supplied

to the cavity for ventilation, the cavity pressure oscillated with the same

frequency as that of the body oscillation. When the plate was oscillated

about a small mean angle of attack, there was a change in the cavity config­

uration even without ventilation and the resulting flow was quite irregular.

The plate was oscillated at the maximum reduced frequency of 0.03.

The second phase of the experiment involved measurements of cavity

pressure, cavity length, and the force on the body following a sudden venti­

lation of an otherwise steady cavity. It was found that the change in cavity

length and the change in the force lagged behind the cavity pressure change.

Furthermore, the rate of cavity increase never exceeded the free-stream speed.

An attempt was also made to measure the response of the flow to a sud­

den angle of attack change. It was concluded that, due to the oscillatory

nature of the cavity, a faster angle of attack change than was attained in

the experiment is needed to obtain a useful unit function response curve.

iii

.CONTENTS

Page

Abstract • • iii List of Illustrations • • vii

I. INTRODUCTION 1

II. EXPERIMENTAL APPARATUS • 2

III. TEST RESULTS 4

A. Oscillating Foil • • • 4 B. Sudden Change ,in Cavity Pressure • • 12 C. Sudden Change in Angle of Attack 15

IV. CONCLUSIONS • 17

A. Oscillating Foil • 18 B. Sudden Ventilation • 18 C. Sudden Angle of Attack Change • • • 19

List of References 21 Figures 1 through 31 • 25

v

Figure

1

2

3

4

5

6

7

8

9

10

11

LIST OF ILLUSTRATIONS --

A View of the Free-Jet Tunnel with Foil and Dynamometer • • •

Steady Flow Cavity Length for a Flat Plate Hydrofoil . . ~ . Steady Flow Lift and Drag Coefficients, Flat Plate Wi th Supercavi ty • • • • • • • • • • • • • • • • • . . . . . Steady Flow Lift and Drag Coeff.icients, Flat Plate With and Without Cavity • • • • • • • • • • • • • • •

Typical Data for an Oscillating Plate • . . . . Record of Pulsating Cavity Behind an Oscillating Plate

. . . .

Effect of Vent.ilation on Instantaneous Lift Coefficient •

Effect of Ventilation on Instantaneous Drag Coefficient • • •

Phase Angle of Cavity Pressure Osc,illation . . . . . . . Effect of Ventilation on Lift Curve Slope and Cavitation Number Change During Osc.illation ••••••••••

Lift Coefficient as a Function of Instantaneous Angle of Attack and Instantaneous Cavitation Number • • • •

Page

25

26

27

28

29

30

31

32

33

34

35

12 Instantaneous Lift and Drag Coefficieni$as Influenced by Reduced Frequency • • • • • • • • • • • • • • • • • •• 36

13 Effect of Frequency on Lift Curve Slope and Rate of

14

15

16

17

Cavitation Number Change •••• • • • • • • •• 37

Steady State Lift Coefficients at Small Angle of Attack for a Flat Plate Hydrofoil • • • • • • • • . . . . . . Steady Lift Coefficient and Schematic Cavity Configuration for a = 60 • • • • • • • • • • . . . . . . . . Steady Lift Coefficient and Schematic Cavity Configuration for a = 40 • • • • • • • • • • . . . . . . . .

38

39

40

Steady Lift Coefficient and Schematic Cavity Configuration for a. = 20 • • • • • • • • • • • • • • •• 41

18 Typical Osc.illographic Record for Oscillat.ion About

19

Small Mean Angle of Attack • • • • • • • • • • • • • • • •• 42

Instantaneous Lift and Drag Coefficients at Small Angle of Attack and Small Cavitation Number • • •

vii

. . . . . . 43

Figure

20

21

22

23

'C::L 24

25

26

27

28

29

30

31

Instantaneous Lift and Drag Coefficients at Small Angle of Attack and Moderately Small Cavitation Number

Frequency Effect on Lift Curve Slope for Two Flow

• • •

Regmes • • • • • • • . • . • • . • • . • • • . • . • • • • •

Comparison of Forces on Steady Foil and Oscillating Foil at Small Angle of Attack • • • • • • • • • · . . . . Typical Records of Air Tank Pressure and Cavity Pressure for Sudden Ventilation, Normal Plate •

Cavity Length as a Function of Time with Sudden Ventilation • • • • • • • • • • • • • • • • • •

. . . . . . .

. . . . • • •

Cav.ity Length as a Function of Instantaneous Cavitation

Page

45

46

47

48

Number with Sudden Ventilation • • • • • • • • • • • • • • • 49

Correlation of Data for Sudden Ventilat.ion . . . . . . Typical Records of Lift, Drag, and Cavity Pressure Due to Sudden Ventilation • • • • • • • •• • • • • · . . . . Variatlion of Cavitation Number and Lift Coefficient After Sudden Ventilation • • • • • • • • • • • • • · . . . . Lift Coefficient as a Function of Cavitation Number with Sudden Ventilation • • • • • • • • • • • • • • •

Typical Records of Lift and Drag Change After Sudden Angle of Attack Change • • • • • • • • • • • • • • •

. . . .

. . . . The Zeroth Order Unit Step Funct.ion Response to Angular Rotation . • • . • • • • . • • . • . • . . • • • • • . · . .

viii

50

51

52

53

55

MEASUREMENTS OF THE UNSTEADY FORCE ON

CAVITATING HYDROFOILS IN A FREE JET

I. INTRODUCTION

The continuing development of hydrofoil craft and supercavi tating

propellers over the past several years has resulted in considerable advance­

ment in cavity flow theory. Nevertheless, due to its inherent difficulties,

the mechanism of unsteady cavity flows has not been completely understood.

It has become apparent for some time that good experimental data are urgently

needed.

In recent years, the st. Anthony Falls Hydraulic Laboratory has been

engaged in an experimental study of unsteady super cavitating flow problems

under the sponsorship of the Office of Naval Research of the United States

Navy. The types of unsteady super cavitating flows studied include flow pro­

duced by an oscillating foil, flow due to sudden air supply into the cavity,

and flow due to sudden angle of attack change. The first type of problem is

directly related to the problem of hydrofoils moving under a surface wave and

its solution is also useful in flutter analysis. The second type of problem

is applicable to hydrofoil boat control. When a hydrofoil is moving at. an

intermediate speed, which is too low for a natural super cavity to be formed

but too high to have fully wet ted flow, it may be necessary to create a super­

cavity by means of ventilation. Sometimes ventilation may also be desirable

* for noise reduction purposes, [lJ. Since a steady ventilated cavity flow

requires a constant ventilation rate, any change in ventilation rate will

result in an unsteady flow. This type of the unsteady flow is the concern

of the second phase of the experiments reported herein.

Provided the system is linear, the response of the flow to an arbitrary

change in hydrofoil characteristics may be synthesized using either the har­

monic responses or the unit step function response. To explore the possibil­

ity of using the latter method, a limited number of experimental data were

also obtained for the flow due to sudden change in the angle of attack.

* Numbers in brackets refer to the List of References on page 21.

2

The present report is based mainly on experimental data taken in the

free-jet tunnel prior to the su:mmer of 1963. A two-dimensional wedge and

flat plate were used as test bodies.

This research has been supported by the Office of Naval Research of

the United States Department of the Navy under Contract Nonr 710(24) Task

NR 062-052. The author wishes to acknowledge the contribution of Messrs.

John Almo and John Kolodnycki, who were responsible for the design and main­

tenance of various test facilities and Messrs. Paul T. Edstrom, R. A. Rao,

and Frank Yu-Fang Tsai, who were responsible for the experimental details.

'';LSpecial thanks is due to Prof. Edward Silberman for his helpful discussion

and guidance. The manuscript was prepared for printing by Mrs. Kathleen

Lagerberg.

II. EXPERIMENTAL APPARATUS

The experimental studies were conducted in the two-dimensional gravi­

ty-flow, free-jet water tunnel at the St. Anthony Falls Hydraulic Laboratory.

The tunnel is a free-falling, nonrecirculatingtype utilizing river water and

having a vertical test section. This tunnel has two principal features that

make it especially useful for the type of investigation reported herein.

First, very low cavitation numbers can be obtained without blockage; and se­

cond, large quantities of air can be added to a cavity without having to re­

move the air from the tunnel water subsequently. Furthermore, the test sec­

tion is completely transparent, except where a dynamometer is located, fa­

cilitating visual observation and photography. A front view of the tunnel

in operation is shown in Fig. 1.

The test section is designed to produce a rectangular jet 5 in. thick

and of variable width ranging from 6 to 12 inches. All the da ta reported

hereafter were taken in the 10-in. jet unless otherwise stated. The jet is

bounded by two solid walls and two free surfaces. The working velocity of

the jet is between 20 fps and 50 fps. .ArJ.y pressure between vapor pres'sure

and a pressure slightly lower than atmospheric pressure can be created at

the test section. The free-jet tunnel and the methods of velocity and am­

bient pressure measurement are descirbed in detail in Ref. [2J. Fifty-inch

mercury manometers were used for both average cavity pressure and average

:3

ambient pressure measurements and for measuring their differences. Elec­

tronic strain gage type pressure transducers made by Consolidated Electro­

dynamic Company were used for transient pressure measurements. Natural fre­

quencies of the transducers were ove,r 4 KC per second in air. Pressure trans­

ducers were always placed right at the point of interest to minimize the

dynamic effect of a fluid column.

No provision was made to determine the air content of the river water

used as te.st fluid. The effect of air content in the water is considered to

be of secondary importance for the type of experiment reported herein.

Two identical strain-gage type dynamometer units, specifically design­

ed for the experimental program, were mounted at opposite points on t~e test

section walls. The strain gages were mounted in pairs on sets of "lift"

beams and "drag" beams. A 5-in. span test body was suspended between these

units by means of retractable pins. Test bodies were so constructed that the

centerline of a wedge or the pressure side of a flat plate surface was aligned

with a pair of the drag beams. At non-zero attack angle, the true lift and

drag are related to the dynamometer reading by the following pair ofequa tions.

, , L = L co s a. - D sin a. , , D = D cos a. + L sin a.

(1)

, , where Land D are lift and drag, L and D are the dynamometer read-

ings, and a is the angle of attack.

A driving mechanism was attached to the dynamometer housing so that a

test body and the dynamometer could be rotated together while the water tun­

nel was in operation. The body-dynamometer unit could be oscillated at a

maximum amplitude of 15 degrees and at a maximum frequency of :3 cps at rela­

tively small amplitude. Alternately, a sudden change in angle of attack from

one fixed value to another could be produced. The dynamometer was designed

to measure a transient force of frequency up to 100 cps; the frequency of the

primary mode of vibration of the system in air with the2~in. flat plate test

body was determined to be approximately 170 cps. A detailed description of

the dynamometer may be found inRe~. [3J.

4

III. TEST RESULTS

A. Oscillating Foil

A 2-in. flat plate and a I-in. wedge with 15 degree vertex angle were

used as test bodies in this phase of the experimental program. For reference

purposes and to check the functioning of the equipment, steady flow measure­

ments were first taken in a lO-in. jet.

Figure 2 shows the result of the cavity length measurement for the

flat plate at three different angles of attack. Here the cavity length is

<\.. defined as the distance between the leading edge of the plate and a point at

the rear end of the cavity where the white 'cloud of gas-water mixture appear­

ed to be thinnest. It was observed that the cavity length thus defined fluc­

tuated about an apparent mean value even when the flow was kept as steady as

possible. The values plotted in the figure are the estimated mean values

which may contain errors as high as one inch or even more. The cavitation

number is defined as

CI =

where Pro = ambient pressure surrounding free jet,

P = measured cavity pressure, c p = density of water, and

V = ambient velocity with test body removed.

(2)

An alternate definition of the cavitation number, CIv ' wherein the vapor

pressure is used in place of the measured cavity pressure is sometimes more

convenient than the definition given by Eq. (2).

Figure 3 shows the corresponding lift and drag coefficients using the

pressure side of the surface as reference area. Skin friction drag was esti­

mated and subtracted from the drag data before plotting. Corrections due to

the tunnel wall effect and other secondary flows were neglected. Theoretical

lift and drag coefficients by Silberman [4J for zero cavitation number, are

also indicated in the figure. The agreement between the theory and experi­

ment is good. To show the variation of CL and CD for a wider range of

5

cavitation numbers, another set o~ data including non-cavitating, partially

cavitating and super cavitating conditions, were taken at a = 10.50 and the

data are plotted in Fig. 4; the abscissa in this ~igure is cavity length,

(Note that thevalueso~ CL/2na and Cn/2nd2 ~or non-cavitating conditions

are very close to unity, the limiting value ~rom thin- air~oil theory, while

their limiting values ~or very long cavities are both approximately eq~al to

0.2, Silberman's theoretical value.)

(1) Oscillation at Large Angle o~ Attack

The 2-in. plate was oscillated between 13 and 21 degree angle o~

attack at various super cavitating flow conditions. Oscillographic records

of lift, drag, angle o~ attack, and cavity pressure were taken for each run.

Samples o~ the records are shown in Fig. 5.

Figure 5a is ~or a natural cavity; there was no external air supply to

the cavity except possibly ~or a small quantity that may have leaked in. It

is seen that a nearly per~ect harmonic oscillation o~ the plate was achieved,

although at low ~requency. The resulting li~t and drag curves were also near­

ly harmonic with no observable phase shi~t. As may be expected, the cavity

pressure remained almost constant and only slightly larger than the vapor

pressure. The high ~requency and small amplitude hash are believed to be due

to flow noise and other vibrations which areo~no concern to the current in­

vestigation.

Figure 5b shows similar data ~or a lightly ventilated cavity. A char­

acteristic o~ the ventilated cavity shown by the record is that the cavity

pressure oscillates about its mean value almost anti-symmetrically with the

angle o~ attack oscillation. With somewhat larger air supply, the cavity

would pulsate i~ the angle o~ attack were kept constant at 13 degrees (le~t

side o~ Fig. 5c). Pulsations disappear i~ the ~oil is oscillated (right side

o~ Fig. 5c). Finally, Fig. 6 shows that with large air supply rate, the cav­

ity about an oscillating ~oil may also pulsate. The dimensionless numbers

appearing in these and the ~ollowing ~igures are de~ined as ~ollo'Ws:

c = Li~t = lift coef~icient L ~ pV2CS

<~.

6

where

CD - Total drag - Skin friction = drag coefficient - ~ pV2CS

= air supply coefficient

wC k = 2V = reduced frequency

C = chord,

S = span,

w = angular speed of oscillation,

WA = weight rate of air flow at room condition, and

YA = specific weight of air at room condition.

Figure 7 shows the data for the instantaneous lift coefficient when

the plate oscillated between 13 and 21 degree attack angle at the reduced

frequency, k, of about 0.005 with variable air supply coefficient. The

corresponding data for the drag coefficient are shown in Fig. 8. The skin

friction was roughly estimated assuming turbulent boundary layer flow and was

a very small part of the total drag in any event. It is seen that the ratio

of the lift coefficient to the drag coefficient was approximately equal to

tan a. for all cases, as may be expected for the supercavitating flat plate.

Examination of the records shown in Figs. 5 and 6 for CA > 0 reveals

that the cavity pressure curves are not exactly anti-symmetric with respect

to the angle of attack curve. Theyare somewhat skewed with a tendency for

the maximum and the minimum cavity pressure to occur sometime after the min­

imum and the maximum angle of attack, respectively_ The phase difference

between the maximum cavity pressure and the minimum angle of atta'ck, ¢l' and that between the minimum cavity pressure and the maximum angle of attack,

¢2' for a constant k (0.0055) and nearly constant ambient conditions are

plotted as functions of CA there is a maximum value of

in Fig. 9. For the flow conditions

¢l' occurring near CA ~ 4 x 10-3. tested,

The data shown in Fig. 7 may be used to compute the average lift curve

slope, A CL/AfJ.' and the average rate of cavitation number change per unit

7

angle change C1cr I C1 a. The results, fer a censtant reduced frequency (k = 0.00.5.5) but variable air supply ceefficient, are shewn in Fig. 10. It is

readily seen that both C1 cLI C1a and C1 cr I C1 a increase with increasing air

supply ceefficient.

All the data fer CL in Fig. 7 cerrespending te a = 13.0 .0 and a =21

are pletted as a functien .of cr en a single graph in Fig. 11. The mean .0

value .of the steady flew lift ceefficient fer a :;:: 13 is estimated frem

Fig. 3a and shewn en the same graph as a selid line. It is interesting te

nete that the line representing the steady flew data agrees fairly well with

the unsteady flew data. The scatter .of the unsteady flew data is alse cem­

parable te that .of the steady flew data. It may new be hypethesized that,

at a very lew reduced frequency, altheugh the cavity pressure behind an .os­

cillating plate with a censtant ventilatien rate varies with time, the in­

stantaneeus lift ceefficient may be determined by using the steady flew re­

latien previded that the instantaneeus values .of attack angle and cavitatien

number are used.

An attempt was alse made te study the effect .of frequencyenferce and

cavity pressure. Fer this purpese, the test bedy was .oscillated at different

frequencies while the ventilatien rate was kept censtant. Figure 12 shews

the data fer lift and. drag ceefficients when the air supply ceefficient, CA'

was kept approximately equal to 6.3 x 10-3 .. The maximum reduced frequency

attainable in the apparatus was .only 0.032.. Fer ventilated cavities, the

amplitude .of cavity pressure and lift coefficient .oscillations were found to

depend en the frequency .of the foil .oscillation. The amplitude per unit at­

tack angle change for a constant air supply coefficient, CA ;::::; 6.3 x 10-3 ,

are pletted as functions of the reduced frequency in Fig. 13. It is seen that

both lift curve slepe and the rate of cavi tatien number change are decreasing

functions .of reduced frequency. Also, at moderately high reduced frequency,

K > 0.012, the amplitudes .of the cavitati.on number and the lift coefficient

escillatiens are almest independent .of the reduced frequency.

(2) Oscillation at Small Angle of Attack

The experiment described in the last sectien invelved ne majer change

in the cavity cenfiguratien; there always existed a super cavity with fixed

8

separation points. When a foil was oscillated about a small attack angle,

major changes in the cavity configuration took place during each cycle of

oscillation. All the data taken in this connection are for the 2-in. flat

plate with natural cavity.

For the purpose of comparison, steady flow data for five different

angles of attack were taken. The measured steady flow lift coefficient data

are summarized and plotted as a function of cavitation number in Fig. 14.

The cavitation number is based on vapor pressure in this and the following

figures since the cavities disappeared in some cases so that cavity pressure

was not always measurable.

When the foil was set at 8 degree angle of attack no unusual flow be­

havior was observed. The leading edge cavity first appeared when CJ was v

lowered to approximately 1.4. Further reduction of CJv was followed by in-

creased cavity length and lift coefficient and the latter reached its maxi­

mum value when the cavity became nearly as long as the chord. For smaller

angle of attacks, however, different phenomena were observed. The observed

flow characteristics are described for the cases of a = 60 , a = 40 , and o

a. = 2 •

a = 60

For the sake of clearness, the experimental data for a = 60 are singled out from Fig. 14 and replotted in Fig. 15. Schematic drawings of the cavity configura­tions corresponding to some representative data points are also included in the figure. It may be seen that, when the cavitation number was gradually decreased from a large va.1ue, a cavity first appeared at the leading edge (sketch "a"). Another cavity could also appear at the apex (see sketch "b") if CJ was fur­ther reduced. The flow was very unstab1Z within a considerable range of CJ wherein different cavity configurations could be -gbserved. The leading edge cavity fluctuated to the extent that the two cavities joined to form a single large cavity (sketch "e") for part of the time. The resulting flow looked somewhat like the oscillating cavity recently reported by Wade [5J. When the cavity fluctuated, the lift force also changed. It is seen in Fig. 15 that a single large cavity corresponded to the larger lift and the double cavity condition corresponded to the smaller lift. When CJ was reduced below 0.6 only a single super­cavity ~as obtained.

a = 40

The lift coefficient data for 4 degree angle of attack shown in Fig. 14 are replotted in Fig. 16. At this angle of attack, the apex cavity appeared first when cavitation number was at approximately 1.35, (sketch "a"). With a slight 'drop in the cavitation number, approximately at 1.17, the leading edge cavity also appeared, as usual, on the suction side (sketch "b"). When the cavitation number was decreased further, the apex cavity continued to grow while the leading edge cavi ty remained rela ti vely unchanged until a point in­dicated by sketch "c" is reached. At this point, the leading edge cavity switched from the suction side to the pressure side intermittently. As indicated by sketches "f" and "g", the suction side cavity was as­sociated with a higher lift than the pressure side cavi ty. This unstable condition occurred only in a narrow range of cavitation number, 0.25 <.. a < 0.35. At a somewhat lower a, the leading edge ca~ity re­mained on the pressure v side, but its size fluctuated (sketches "h" and "i 11) • It should be noted that the shifting of the leading edge cavity was not necessar­ily followed by a shift in lift force direction. Only when a was less than 0.16 did the pressure side cavity ¥esult in negative lift (sketches' "h" and "i").

o a = 2 The lift coefficient data for this case are shown in Fig. 17. At this small angle of attack, the apex cav­ity appeared earlier (a = 1.73) than in the previous cases as a was lower¥d. The apex cavity continued to grow as va was lowered. The leading edge cavity appeared firlt at the suction side when a reached 1.1 (sketch "b"). This point corresponded ¥oughly to the point of maximum lift coefficient. As may be ex­pected, the leading edge cavity disappeared when a was lowered slightly under 1.1. There was a consider~ able range of a wherein no leading edge cavity could be observe~. Finally, approximately at a = 0.5, the leading edge cavity reappeared, this timevon the pressure side (sketch "f"). By this time, the apex cavity had extended beyond the trailing edge of the plate. Further reduction of a beyond 0.5 re­suIted in the growth of both cavi tres. The leading edge pressure side cavity fluctuated causing the lift coefficient to change by as much as 0.1 for a given a. The lift coefficient always remained negative wKen a was less than 0.2. A double cavity condition with tIre leading edge cavity extend,ing to the mid chord and the apex cavity extending two or three chords beyond the trailing edge of the plate was observed (sketch "h"). At a = 0.14 a single supercavity con­dition as indicate~ by sketch "i" was realized. As

9

10

may be expected, the magnitude of the negative lift coefficient dropped slightly when crv was reduced to zero.

The existence of double cavities and the occurrence of the leading edge cav­

ity on the pressure side were also reported by Parkin [6J. Some of the more

important facts revealed concerning steady flow at small angle of attack are

recapitulated below.

(a) Contrary to the familiar trend, the inception cav­itation nU1l1ber of the apex cavity is a decreasing function of the attack angle.

(b) For a given angle of attack, the leading edge cav­it y may appear on the suction side or the pressure side depending on the cavitation nU1l1ber.

( c) Even when all the ambient flow conditions are kept constant, the leading edge cavity may not be steady. It appears that two alternate cavity con­figurations are possible for a given ambient flow condition.

Of course, it is doubtful that the ambient flow (jet) was actually maintained

constant and it is possible that the slight unsteadyness of the ambient flow

might be the cause of the cavity fluctuation. Further investigation concern­

ing this matter has not been carried out.

After the steady flow investigation was completed, the foil was os-o / 0 cillated between a = 2 and a = 8-1 2 at various frequencies and cavita-

tion nU1l1bers. All the tests were made under non-ventilated conditions. A

typical oscillographic record (0 = 0.163, k = 0.0119) is shown in Fig. 18. v

It is seen that the lift and drag curves show considerable irregularities

even though an almost harmonic angle of attack oscillation was achieved. A

comparison of lift and drag coefficient for several cycles of oscillations

indicated large differences from one cycle to the next. In fact, it was im­

possible to find two cycles showing identical results. Examples of two con­

secutive cycles showing the largest discrepancies were selected from Fig. 18

and plotted in Fig. 19. Similar data for two higher cavitation nU1l1bers are

shown in Fig. 20. As may be seen from Figs. 19 and 20, the lift and drag

curves corresponding to increasing a were almost always lower than those

corresponding to decreasing a.. Although part of this hysteresis could be

due to the characteristics of the dynamometer, the dynamometer hysteresis

11

alone is far too small to account for the large change seen in Fig. 19b.

That the trend shown in Fig. 19 is opposite to the trend shown in Fig. 7 (for

large a with ventilation) also indicates that the hysteresis was due to the

flow rather than to the dynamometer. It should be noted that zero lift coef­

ficient corresponded roughly to minimum drag coefficient.

Observation of Fig. 19 and many other sim.ilar graphs revealed that the

lift coefficient curves have three distinct slopes: minimumslope exists be-o o. 0 60 tween a. = 2 and a = 3 and maxJ.mum slope between a. = 3 and a ~ •

Unfortunately, a clear view of the cavity on the foil was blocked by the

dynamometer and no attempt was made to take motion pictures of the cavity.

However, the steady flow data as shown in Figs. 15, 16, and 17 suggest that

a full single cavity might exist for a > 60 and suction side double cavi­

ties might exist for 30 < a < 60 • The minimum slope might correspond to a

pressure side leading edge cavity or even to flow without a leading edge cav­

ity. Average lift curve slopes corresponding to a. > 60 and 30 < a < 60

for cr = 0.163 are plotted in Fig. 21 as functions of k. Slight effect v of k, especially for a single full cavity region, is indicated.

Finally, the performance of the oscillating foil is compared with that

of a stationary foil in Fig. 22. Figure 22a shows the average lift coeffici­

ent as a function of a. It should be noted that, due to the instability of

the cavity configuration discussed earlier, the steady flow curve as indi­

cated by solid lines has branches. Since the unsteady flow data lie nearer

to the lower branch of the steady flow data and, since, the lower branch in­

dicated shorter leading edge cavities, it is suggested that the oscillation

tended to shorten the cavity. The same data are plotted in a polar form in

Fig. 22b. It is clearly indicated that the oscillating foil at small a. is

substantially inferior to the stationary foil at the same CL.

B. Sudden Change in Cavity Pressure

It is well known that the cavity pressure of a natural cavity is gen­

erally only slightly larger than the vapor pressure. The pressure in a

steadily ventilated cavity is a variable depending on the air supply coeffi­

cient and other factors. When a natural cavity is suddenly ventilated, the

flow shortly after the ventilation begins is unsteady; cavity pressure and

cavity size increase while lift and drag decrease.

12

For the purposes of this experiment, a 28 0 ... cubic in. air tank was used

as the source of air supply. The air tank was placed near the test section

and connected to the rear end of the test body by a short pipe equipped with

a gate valve. A strain gage type pressure transducer was attached to the

air tank for the purpose of measuring instantaneous tank pressure.

(1) Cavity Length and Cavity pressure

A 3/l6-in. normal plate and a l-in. symmetrical wedge with 2.8 degree

vertex angle were tested for the purpose of studying the effect of sudden

CI.. ventilation on cavity length and cavity pressure. Since the presence of the

dynamometer would partially obstruct the view of the cavity near the foil,

it was found desirable to replace the dynamometer with a lucite block. The

variation of cavity size and cavity shape was recorded by means of motion

pictures. Instantaneous cavity pressure was measured using a strain gage

type pressure transducer placed on the tunnel wall approximately one inch

downstream of the trailing end of the test body. For the purpose of syn­

chronization, arbitrary time origins were marked simultaneously on the pres­

sure record and the movie film.

Figure 23 shows two typical records of the tank pressure and the cav­

ity pressure changes obtained for a 3/l6-in. normal plate operated at o"v ~

0.4. It is readily observed that the tank pressure reduced almost uniformly

during the initial stage indicating almost constant ventilation rate. On

the contrary, the cavity pressure increased very rapidly at the initial in­

stant and less rapidly thereafter. Record (a) also shows that the cavity

pressure may oscillate about a mean value indicating the beginning of pulsa­

tion as reported in Ref. [7J and [8J. After most of the air in the tank was

expended, the mean cavity pressure as well as the amplitude of the cavity

pressure oscillation gradually decreased until the cavity ceased to pulsate

and the original natural cavity was reproduced. For a larger initial tank

pressure or a larger valve opening, as represented by record (b), the cavity

eventually grew too long and split (0" ~ 0 for the split cavity).

The variation of cavity length due to sudden ventilation measured for

the 3/l6-in. normal plate at O"v ::::s 0.4 is shown in Fig. 24a. The four

curves at the left side of the graph are for the initial stage of ventilation.

13

It is seen that the cavity grew very slowly at the initial instant, but the

rate of growth increased very rapidly and maximum rate is reached within a

short time (of order of 0.05 second). As maybe expected, the rate of cav­

ity growth is an increasing function of air supply rate. The maximum rate

of cavity length increase obtained from the first curve in Fig. 24a is 14.4

fps, which is considerably less than the ambient water speed of 41 fps. To

obtain data for faster expansion rate, the 28 degree wedge of 1 in. length

equipped with a larger air pipe was tested at approximately the same crv •

The test result is shown in Fig. 24b. A maximum cavity growth rate of ap­

proximately 40 fps, very close to the water speed, was achieved. With the

present system, a growth rate greater than the water speed was not realized.

It is conjectured that the cavity length can not grow faster than the ambient

water speed unless the cavity pressure is raised above the ambient pressure

as in the case of rocket propulsion. Attention is called to the onset of an

instability at high ventilation rate as evidenced by the first curve in Fig.

24b.

Curve No. 5 in Fig. 24a is typical of the curves at a later stage of

ventilation when tank air is nearly exhausted. It was found that the curve

is almost independent of the initial ventilation rate.

To show the relationship between cavity length and the instantaneous

cavitation number, the first four curves in Fig. 24a and those in Fig. 24b

are replotted in Fig. 25 using cr as the independent variable. The accuracy

of data plotted in this figure depends critically on the synchronization of

the film and the pressure record. The possible error involved in the estima­

tion of, the time origin was about 0.01 second. Allowing for the uncertainties

involved, it is reasonably well established that the cavity length change

lagged behind the cavitation number change. The lag is more evident for high

ventilation rates.

An attempt will now be made to correlate these sudden ventilation

data. Since the time involved in the initial stage of ventilation is small,

the amount of air escaping from the cavity due to diffusion should be negli­

gibily small compared with the air retained in the cavity. Assuming a baro­

tropic flow and assuming that the cavity volume is proportional to the n-th

~'Q"

14

power of the cavity length, the law of conservation of air mass leads to

where Po = Pt = P = c j, = K =

Po 11K _ ptl/K

P l/K c

initial air tank pressure,

instantaneous air tank pressure,

instantaneous cavity pressure,

cavity length,

a gas constant, and

K, n = correlation constants.

Although there should be some frictional loss in the gas flow, for the sake

of simplicity the flow is as sumed to be isentropic and K is taken to be

1.4. Figure 26 shows the correlation of the· data from Fig. 24a plotted in

accordance with Eq. (3) on log-log paper. Fairly good correlation for cav­

ities longer than 10 in. is apparent. It should also be noted that, accord­

ing to the figure, the empirical exponent n is approximately equal to 1.6.

This value is very close to the well known theoretical value for long steady

cavities, n = 1.5.

The 2-in. flat plate foil was tested in a 10-in. jet at 11 degrees

angle of attack for the purpose of studying the effect of sudden ventilation

on lift and drag. Figure 27a is a typical record of lift, drag, cavity pres­

sure, and the air tank pressure for a relatively large ventilation rate. The

cavity pressure transducer was placed 6 in. downstream of the leading edge

of the foil in order to clear the dynamometer. For this reason, only the

pressure in a cavity longer than 6 in. could be measured and the record for

the initial instant of ventilation was missing. Figure 27b is a similar xe­

cord for a smaller ventilation rate where it was possible for the cavity to

pulsate.

The variation of the cavitation number and the lift coefficient with

time shortely after sudden ventilation are plotted in Figs. 28a and 28b, re­

spectively. Figure 29 is a cross plot of CL as a function of o. It seems

15

that, for high ventilation rate, the change of CL lagged behind the 0

change. Even allowing for a possible error of about 0.01 second in the deter­

mination of the time origin, the trend is clearly indicated. There is no

theory available at the present time to check this experimental result.

C. Sudden Change in Angle of Attack

For this experiment, the foil-dynamometer uni t was geared to a shaft

and a pulley from which a weight was suspended. The pulley was so designed

that upon releasing the locking mechanism the shaft would rotate under the

action of the weight and stop at a preassigned angle. A braking force was

provided by a mechanical stop. The speed of rotation could be changed by

changing the suspended weight. The angle of attack was calibrated against

the angular position of a shaft attached to a gear situated midway between

the pulley and the dynamometer.

The 2-in. flat plate hydrofoil was tested at natural super cavitating

conditions. The plate could be rotated from 14 degree to 7 degree attack

angle at the maximum angular speed of 40 degrees per second. Using a rela­

tively small weight, a quite long period of constant angular speed could be

obtained. On the other hand, a larger weight would result in a more extended

period of angular acceleration and a shorter period of constant speed.

Some oscillographic records of changing angle of attack and instanta­

neous forces are shown in Fig. 30. A sharp break in the slope of the angle

of attack curves at a = 70 indicates a large angular deceleration. The

maximum angular deceleration was estimated to be about 300 rad. per sec. 2

which corresponds to a linear deceleration of nearly one g at the leading

edge of the foil. However, since the period of the maximum deceleration was

of the same order of II!l3:gni tude as the natural period of the system (0.0059

second), the virtual force could not be registered. Most records, with a few

exceptions such as Fig. 30c, showed immediate response of lift and drag to the

sudden stop. The reason for the peculiarity of the lift and drag curve fol­

lowing the sudden stop in record (c) was not determined although the opposite·

trends shown by the lift and drag curves suggests that a change in the rel­

ative orientation of the foil and the dynamometer might be the reason. The

foil was attached to the dynamometer by means of three pins at each end of

16

t.he foil; loose pins might have permitted the foil to rotate with respect

to the dynamometer.

If the system is linear, it is of great practical importance to deter­

mine the unit step function response to a. sudden attack angle change. The

response of the system to an arbitrary attack angle change may, then, be ob­

tained by integration. Since the records show almost linear inputs, they are

considered ideal linear functions in the following analysis. It is well

known that the response to a zeroth order unit step function input is equal

~:,\ to the first derivative of the response function due to a linear function

--.: input. The instantaneous lift coefficient was computed at successive instants

0.01 second apart. The first derivative of the lift coefficient, dCL/dt,

was computed by the method of finite differences using 0.05 second intervals.

The reason for chosing a 0.05 second interval rather than 0.01 second for

the numerical differentiation was to minimize the effect of vibration at nat­

ural frequency. The resulting unit step response function

(4)

having the unit of per radian is plotted in Fig. 31. Symbols used in Eq. (4) have the following meanings.

Co(t) = zeroth order unit function response,

instantaneous lift coefficient for linear CL(o.., t) = Aa AT = rate of change of attack angle.

a change, and

For the purpose of comparison, the steady state lift curve slope for

a supercavitating flat plate, assuming

C - 2n sin a (1 + ) L - 4 + n sin a cr (5)

is also indicated in Fig. 31 for the terminal attack angles of 14 and 7 de­

grees. Equation (5) is the well known equation [9J for infinite fluid and

infinite cavity adjusted to take into account the first order effect of

cr. Theoretically, the response function at large t should agree with Eq.

17

(5). The agreement is fairly good if the large ~plitude oscillation of the

data at the frequency of an order of 10 cps is overlooked. The same type of

oscillation was also found when a. was kept constant. This is believed to

be due to the oscillatory nature of the cavity length which may be observed

even in a steady state condition. In fact, it was reported by Song and Sil­

berman [lJ that the frequency spectrum of the pressure fluctuation near a

steady cavity always exhibited a peak in a low frequency range, f < 60 cps.

Furthermore, the correlation of the intensity of the pressure fluctuation and

the distance between the pressure transducer and the tail of the cavity en­

abled them to identify this peak with the cavity length fluctuation [1]. The

apparent time lag of the response curve of approximately 0.095 second as

shown in Fig. 31 was probably due to gaps between transmission gears rather

than being an indication of a hydrodynamic property.

The amplitude of an oscillatory force at approximately 10 cps is so

large that the usefulness of data shown in Fig. 31 is doubtful. Furthermore,

it seems that the response time of the flow is less than 0.1 sec and, hence,

a faster angular rotation (T < 0.1) is necessary for abetter result.

Although the present investigation does not concern flutter phenomenon,

it may be of interest to some readers to point out evidence of a finite ampli­

tude flutter oscillation of approximately 170 cps in Fig. 30. There were many

similar records. This frequency is of the same order of magni tude as the

natural frequency of the system in air. A sudden growth followed by a decay

of the oscillation, apparently excited by the sudden deceleration of the foil

as shown in the last record of Fig. 30 is noteworthy. A similar sudden burst

of vibration could also take place without apparent external excitation as

may be observed near the left hand edge of record (b) in the same figure.

IV. CONCLUSIONS

Unsteady super cavitating flows around a foil were generated by pitch­

ing oscillation, sudden ventilation, and sudden attack angle change. The

response of the flow to the specified changes in the ambient condition were

measured. As a result of the present investigation, the following conclusions

are offered.

18

A. Oscillating Foil

The following conclusions are based on the experimental data taken at

the reduced frequency less than 0.03. Depending on the ambient conditions,

three distinctively different flow characteristics were observed.

1. There was no observable unsteadyness effect when the foil was oscillated about a large mean angle of attack and small cavitation number so that a natural (non-ventilated) supercavity always ex­isted. This is to be expected at the relatively low frequency range tested.

2.

3·

When a foil with a ventilated cavity was oscil­lated about a large mean angle of attack, the cavity pressure as well as the lift and drag also oscillated at the same frequency as that of the foil oscillation, but with a phase difference. The amplitudes of oscillation for the cavitation number and for the lift and drag coefficients are increasing functions of the air supply coefficient and decreasing functions of the reduced frequency. Nevertheless, itwas found that the instantaneous lift and drag coefficients agreed with their steady flow counterparts based on the instantan­eous angle of attack and the instantaneous cavi­tation number. The phase shift of the cavity pressure oscillation is a function of the reduc­ed frequency. With a sufficient ventilation rate, two types of pressure pulsation, that of a pul­sating cavi~y and that due to the angular oscil­lation, may co-exist.

Naturally cavitating flow at small angle of at­tack (a < 60 ) for the flat plate was found to be very unstable. There existed two alternate cav­ity conditions for any given steady ambient con­dition. Consequently, force on the foil was also found to be unstable. For this reason, a rather erratic flow was observed when the foil was oscil­lated about a small mean angle of attack at small cavitation number. Furthermore, the instantan­eous lift coefficient at any instantaneous angle of attack was considerably lower than the cor­responding steady flow value.

B. Sudden Ventilation

When air was suddenly supplied to an otherwise non-ventilated cavity,

both the cavity pressure and the cavi tylength increased with time whereas

19

the lift force decreased with t:i.m.e. The change in cavity length and lift

force .appeared to lag behind the cavity pressure change. The rate of in­

crease of the cavity length increased with the air supply rate, but never

exceeded the free-stream speed.

C. Sudden Angle of Attack Change

The response of the lift coefficient to a zero order unit step func­

tion change in the angle of attack was estimated using the experimental data

for a nearly constant rate of angle change. To avoid the natural frequency

of the system, the numerical differentiation of the lift data had to be per­

formed using an interval not less than 0.05 second. Moreover, the period of

the cavity length vibration was of the order of 0.1 second, which was small

enough to interfere with the response function. The response time of the

system due to angle of attack change was apparently less than 0.1 second and,

hence, a faster rotation (T < 0.1 sec) is necessary for a better result.

The present report on pitching oscillations is l:i.m.ited to a very low

reduced frequency range. A new program concerned with a considerably high­

er frequency range is now in progress and a report of the result will follow

at a later date.

21

LIST OF REFERENCES ---

[lJ Song, C. S. and Silberman, E. Experimental Studies of Cavitation Noise in a Free-Jet Tunnel, University of Minnesota, St. Anthony Falls Hydraulic Laboratory, Technical Paper No. 33, Series B, July 1961.

[2J Silberman, E. and Ripken, J. F. The St. Anthony Falls Hydraulic Labor­atoryGravity-FlowFree-Jet Water Tunnel, University of Minnesota, St. Anthony Falls Hydraulic Laboratory, Technical Paper No. 24, Series B, August 1959.

[JJ Silberman, E. and Daugherty, R. H. A Dynamometer for the Two-Dimension­al, Free-Jet Water Tunnel Test Section, University of Minnesota, St. Anthony Falls Hydraulic Laboratory, Technical Paper No. 40, Series B, June 1962.

[4J Silberman, E. "Experimental Studies of Super cavitating Flow about Sim­ple Two-Dimensional Bodies in a Jet," Journal of Fluid Mechanics, Vol. 5, Part 3, 1959.

[5J Wade, R. B. Water Tunnel Observations on the Flow Past a Plane-Convex Hydrofoil, California Institute of Technology Report No. E-79-6, February 1964.

[6J Parkin, BlaineR. Experiments on Circular Arc and Flat Plate Hydrofoils in Non-Cavitating and Full Cavity Flows, California Institute of Technology, Hydrodynamic Laboratory, Report No. 47-6, February 1956.

[7J Silberman, E. and Song, C. S. "Instability of Ventilated Cavities," Journal of Ship Research, Vol. 5, No.1, June 1961.

[8J Song, C. S. "Pulsation of Two-Dimensional Cavities," Fourth Symposium on Naval Hydrodynamics, Washington, D. C., August 1962.

[9J Lamb, H. Hydrodynamics, Sixth Edition, Dover Press: April, 1945

EIQ:!!!i~2 (1 through 31)

Figo 1 - A view of the free-jet tunnel with foil and dynamometer (1 = free-jet; 2 = cavity; 3 - dynamometer; 4 = driving rod; 5 = air tank)

25

26

0(

VI "0

o .r:;

16

14

12

10

u 8 5 .r:; -;, c: Q)

.;:­'> c u 6

4

2

o

\

\

o

t· . 1

~~ \ ~ FI,. ~~ \ \ 2"-----;

1 \ ~ \ \ \ \ :\ \ o a = 7 0

o a = 100

.~ t\ \ 0\ !;,. a = 140

\ .\ ~ \ o \

"" "~ ~

\ "-

~ ,

'" ~ '" i'-.....

" "" .........

" '" ~. '-....... .... ~

..... ~ 0

~ ............... """-

~ ~ r-

............... 0

~

.04 .08 .12 .16 .20 .24

Cavitation number, (T

Fig. 2 - Steady flow cavity length for a flat plate hydrofoil

27

0 a = 7°

.45 A a = 10°

0 a = 14° 0

--- Th!!ory,by .40 Silberman 0

..J U

.... .35 c:

Q)

'u --Q)

0 .30 u

.... -:::i .25

.02 .04 .06 .08 .10 .12 .14 .16 .18 .20

Cavitation number I CT'

(a) Li ft coefficient

.12

. .--i:r" ---~ ~ L

~ A \...--Lr A -' A

~~ A. A

---~ .10

0 u

- .08 c::: Q)

'u -.... Q)

r+ b -0 u .06

0 n 0 ~

gI c ...

..I:L n "Ill

~-U -

0

.04

..0_ - .... ]lJ 0 'U

rrJ.. uu t+

.02

o o .02 .04 .06 .08 .. 10 .12 .14 .16 .18 .20

Cavitati on number. (1'

(b) Drag coefficient

Fig, 3 - Steady flow lift and drag coefficients, flat plate with supercavity

28

..J U

. -c: CD 'u ;;: ....

CD 0 u

-.... ::i

<f~t~_

Q

U

c: ., 'u .... .... ., 0 U

01 C ... 0

1.2

1.0

~ ~----0- 0 q )0 On

1O?c~ FI", L 2" ::> ~ . ~ ---

0

Cavity length--J ~

0 0.8

0

0.6

> 0

0.4 ) q

Partial cavity .... ~ Supercavity

0.2

o o .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

.20

.15

.10

.05

Cavity length in chords

(a) Lift coefficient

~~

o Q:,c~ ;>00 0

0

0

-0

Partial cavity .... ~ Supercavity ,

o .5

(

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Cavity length in chords

(b) Drag coefficient

Fig. 4 - Steady flow lift and drag coefficients, flat plate with and without cavity

5.0

t---

5.0

! il , i

, '

i

I \ ,

V=42.7fps k=O CT=0.053

(1= 13°

! II

i , !I . 1

: t : I

• I'

,

11

V= 44.0fps k:0.005 CT:O.l18 1=6"-10" (1=13°-21°

V=43.2fps k:0.005 CT=0.IOO-0.125 1 = II" -13" (1 = 13'l-.. 21 °

i

I I,

I

i I

"ll' Tl I r(

L

Ilrr~l~ r~ 111~ilTlf~~~ri~1~~rr

I

I I

I

n

V:42.7fps k=0.031 CT=0.143-0.157

~ = 11"- 13" a = 13°- 21°

Fig. 5 - Typical data for an oscillating plate

29

ill'l II i'li II I I· il

"

~ I 1I11

Ii III~ Ih iii I

II ~ It r\r~l(~'" n .I

II ,

II

30

q! '!"i --'~---- .

, ---"--I

I II !i !Ji !I I; i illJ.Wt1ttttttl+Ulll'

ID III i I

i I

i I ill! II II : 11~.i,~'(~'i~lii;ll~ll~! ~,~,w, tm"Wi ,W';,'II'IIIIIIIIIIIIIIIIIIIIIIII,

i!!lli lUll 'I jill 1,1 i I III

CA = 2.62 X 10-2, V = 38.8 fps , k = 0.006

a z= 16°......-.. 23.50, III plate , (Tv. = 0.586 \

• Fig. 6 - Record of pulsating cavity behind an oscillating plate

Notation

a Increasing Decreasing

13°

15°

17°

19°

21°

..J U

.55

.50

~ .45 '(3 .... -Q) o <..> .40 +­.... :.:::i

.35

.6.

0

<>

0

0

C = A

.300 . 8

.55

• • •

• •

(b)

1.40 X 10-3

~

c. ~<> ~

<> 0

o· t~

.12

(d)

CA= 6.30 X 10-3

..J U

.50

'E A5 . ~

<..> -.... Q)

2.40 -.... :.:::i

.35

<>

0

La.

.60

.55

.50

..J u

"i/' Q) .45 <..> --Q)

0 <..> +- .40 -..J

.35

til

<tt

• <> ~

• £::,.

.16 .20

til

• <

• <>

... 0

.6. ~

.30 .08 .12 .16 20

Cavitation number (j

(a)

CA = 0

Ii 0

<;Ii

~ A

.12

.24

A

o 41

......

•

.24

.16 .20 .24 .. 28 Cavitation number (j

..J U +-c: Q)

<..> .... -Q)

o

.55

.50

~5

<..> .40 +­.... :.:::i

.35

.55

(c)

C = 290 X 10-3 A .

til

/it .. 0 • ~ • <>

<>

o 4 <i ~

• .6.

.6.

.12 .16

(e)

31

til

e • <> 0

• A

.32 .36

til

~

• <>

• 0

~

.20 .24

I<

41 CA= 9.30 X 10-3

I;iI 0

.50

..J U

+-

~ !45 <..> :;:: -Q)

2 .40 +­-:.:::i

.35

.3~08

- • <> a.

I-<> • 0

A

~

.12 .16 .20 .24 Cavitation number (j

Fig. 7 - Effect of ventilation on instantaneous lift coefficient (k = 0.0055)

32

':t

Notation .2S

a Increasing Decreasing

.24 13° f:::.. A

15° 0 • 0.20 u -

17° <> • c Q)

'0 .16 ;.;:: -Q)

19° 0 • 0 0

g' .12

0 • ... 21° 0

.08

.O~S

.24 (b)

CA = 1.40 X 10-3

c u -c Q)

0 '+-'+-

Q) 0 0

Cl' a ..... 0

.20

.16

.l2

.08

.04 .OS

.24

(,jj

(,jj G <It

ra.

~ ~- ~

~ <+ ~

(j c:. f:::.. A

11.~

.12 .16 .20

(d )

CA= 6.30 X 10-3

o U

.... c

.20

Q) .16 '0 ;.;:: '+­

Q)

o o Cl' .12 a ..... o

.08

.04 .08

.. • 0

• <> <>

• 0 0

f:::.. ~ A.

.12 .16 .20 Cavitation number (J"

(a 1

CA = 0

ij

(]

<iI

~ A.

.12 .16

.24

~

o 4~

• •

.24

.20 .24 Cavitation number (J"

.24 (c )

CA = 2.90 X 10-3

.20 o

U -c .~ .16 o

'+­'+­Q)

o u Cl' .12 o ..... o

.08

.04 .OS

.24

;i

n -<> •

o 4

* 0

f:::..A

.12

(e )

CA= 9.30 X 10-3

(,jj

...

~

i

A.

.28 .32 .36

(,jj

(,jj

n p. .,. •

cJ' ~ ~

.16 .20 .24

(,jj

.20 o U

ot

-c Q)

'0 .16 '+­'+­Q)

o u

Cl' .12 a ..... o

.08

.04 .OS

• 0 • ~ • c· • u

A.

~

.12 .16 .2.0 .24 Cavitation number (J"

Figo 8 - Effect of ventilation on instantaneous drag coefficient (k == 0.0055)

80

70

6 0

<Il

~ 5 r ... 0> Q)

"'C

·U 0.0 4

-o

~ 0> 3 c « Q) UI

0

o 2 it

\ \

./

,/ °v

" ~

/'

33

k ::::0.0055

0.10< 0"<0.17

Phase angle

CPI maximum cavi ty pressure

CP2 minimum cavity pressure

~ ........ ~ .......... r------~

I--- ---~ <k ...- ---~ m-

'<.-00 2 3 4 5 6 7 8 9 10

Air Supply Coefficient. CAx Id

Fig. 9 - phase angle of cavity pressure oscillation

34

c: 0

""0 0 .... .... Q)

Q.

ts <l ....... b

<l

c: o ""0

1.6

1.4

e 1.2 .... Q)

Q. ~

ts <l 1.0 ~ <1

0.8

Q .1 '¢ 0.6

I

0.6

0.5

0.4

0.3

0.2

0.1

()

<> o

o

8 ~-

u

8 0

0

C 0 0

0 C

2 3 4 5 6 7

Air supply coefficient, CA X 103

(a) Lift curve slope

0

0

0

0

0 C

2 3 4 5 6 7

Air supply coefficient, CA X 103

(b) Rate of cavitation number change

8

8

Fig. 10 - Effect of ventilation on lift curve slope and cavitation

number change during oscillation (k = 0.0055)

0

~

IV

0

9 10

co 0

9 10

-' u ~ ....

c: CI)

:~ --CI)

a (,)

--.l

.60

.55

.50

.45

.40

.35

.30 o

0

o a = 13° o a = 21°

- Steady flow from fig. 3(0) 0 0 J 0

.04

0

0 0 ~

0

~ a=13°

V 0

/ /: ~ ,<

14 ru~ .08 .12 .16 .20 .24 .28 .32

Cavitation number, (j

Fig. 11 - Lift coefficient as a function of instantaneous angle of attack and instantanebus cavitation number (k == 0.0055)

35

.36 .40

36

...J u

-s:: G)

'0 ..... ..... G)

0 ~ -!!:: ...J

.~~~

...J u

-s:: G)

'0 ..... ..... G)

0 0 -..... ::i

...J u

-s:: G)

'0 ~ -G) 0 0

01-..... ::i

.50

.45

.40

.35

.30 .08

.50

.45

.40

.35

.30 .08

.50

.45

.40

.35

.30 .10

A

.12 .16

II [] (t ,.. ....

<Y <~~ (~

~ L1

.12 .16

MIl -

.... ~[I ~ ~---~

J ~" 10'

~ .14 .18

.20 .24

.22

Q U

'E G)

'0

.18

:E .14 G) o o

01

E .10 o

.06 .08

LI

~

• • 0 A

.12

(a) Frequency = 0.92 cps (k= 0.011)

.22

l1t h I~

Q

u .18

< ..-s::

c: • G) '0 ~ .14 .... G) 0

itf. 0

01 0 .10 .... 0

.06 .20 .24 .08 .12

(b) Frequency::: 2.08 cps (k = 0.026)

[I

• ~ ~

A

.22 .26

.22

o u .18

-s:: G)

'0 ::: .14 G)

o o

01

~ .10 o

.06 .10

II

~~ 'r;::-..-.... r"

• ~ .. r!I -~

.14

[I

IJ !. \J

<:Jt ... ..... <C or 4_

• 0 A

iJ

.16 .20

[I

.16 .20

~ II

• ~

~

A

.18 .22 Cavitation number. (1' Cavitation number , (1'

(c) Frequency = 2.56 cps (k= 0.032)

Fig. 12 - Instantaneous lift and drag coefficients as influenced by reduced

frequency (CA = ~.3 x 10-3, symbols as in Fig. 8)

.24

.24

.26

I .4r----.,---

1.:"'0---+-

1.21-----1-

I. I I----t

1.01----+

0.91----+

c: 0.81----+ o

'"C o .... ~ 0.71----+-0.

tj

<l '- 0.61----+_ b <l

0.51-----+--

0.4

0.3

0.2

0.1

.004

c: o

1.8

1.6

1.4

1.2

::c 1.0 o L.

.... Q) 0.

tj..().8

~ ...J

u <l0.6

0.4

0.2

0

0

A

0

A

0

.004 .008

0

0 () =0.40

0 () =0.35

A () = O. 28

CA = 6.2 X 10- 3

0 C

A ~

(a)

.012 .016 .020 .024 .028

( b)

0 0

.008 .012 .016 .020 .024 .028 .032

Reduced frequency, k

Figo 13 - Effect of frequency on lift curve slope and rate of . cavitation number change

.036

37

u

A~

.032 .036

.040

38

.1

.0

o .9

~-

o .8

o

.... 0 u ~ c: ., :~ 0 --., o u

.7

.6

.5

.4

0

0q

0 IJ

IJ 0 IJ

.3 n-.cO I:;,D.

o IJ

0

o .2 I:;,

IJ

l:;,iA I:;, <:

o .1 p <>0

¥:. <>0

0 2S'""" 0

C

<e> -0 .I

0

C

0 0 IJ

[

0

u IJ

IJ t.

0

I:;,

<¢.

0<>

- .

0

C IJ

0

IJ <e 0

0 IJ ) J IJ

IJ ~IJIJ IJ IJ IJ

C

IJ

<oAf:;. COl:;, I:;, I:;, I~ I:;,

\J I:;, 0 0 0 to < > "<> 0 00 C <>

<> <> 0 < <>

<>

~II~I V I. 2" .1

Notatian !!... (Degrees) ---0 8

IJ 6

I:;, 4

0 3

<> 2

.2 -0 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

Cavitation number rr

Fig~ 14 - Steady state lift coefficients at small angle of attack for a flat plate hydrofoil

0

I:;,

C

<>

2.6 2.8

I. I

1.0

0.9

o. 8

o. 7

o. 6

... u o. 5

-<: ., ·u

4 !E O. ., 8

.... :::io .3

o .2

o .1

0

-0 .1

-0 .2

~

o

0

0

0.2

39

~ (c)

0.-.l..~ (0 )

0

~- P 0

j, (b) . 00 0 0 ~

0 I 0--=

0

0

'1:

0

0 0

I" III

."~ ~ V~ 2"

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 Cavitation number (J"

Fig. 15 - Steady lift coeffi ci~nt qnd schemati c cavity configuration for a = 60

40

1.1

1.0

0.9

oJ <...>

0.8 -

0.7

0.6

...-0.5 c ., '0 :E ~ o. 4 o

.... :J

o. 3

o. 2

o. I

o

-0. 1

-

2 -0. 0

(b) ~

'\.

~ t:>

/:,. /:,. /:,.

/:,. ,L(G) /:,.

/:,. ~

~ , (c) ~

ld) c ~

c.-+

/:,. ~

~/:,. \: (fl I" III

~ .-4· tf£:j ~ V~ 2"

~ ~

~ "'----25.\ (i)

i~~~

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 Cavitation number (J'

Fig. 16 - Steady I ift coefficient and schematic cavity configuration for a = 40

-' u

c ., o

:;::

1.1

1.0

0.9

0.8

0.7

0.6

0.5

'to 0.4 o o

O. 3

o. 2

o. I

o

-0.1

~ 2 -0. 0

41

(b)

~

(0 )

< r <0 \ ~ ~

l I (e) <>

\ (d) ~ < <> ~

(e) ~

(f)

~

I" ( a= 2° Ili.50 I I <> V~I~I (9 )

<> ~-=-;, .,/

<: (h)

X~ ::::> p (.) '-~ ~

~ 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 l.8 2.0 2.2 2.4 2.6 2.8

Cavitation number cr

Fig. 17 - Steady lift coefficient and schematic cavity configuration fora = 20

,..-.;_ . .,.-...-.-

(Tv =0.163 • k:: 0.0119 • CA :: 0 2°~ a :S 8.5<>

/..;.;

Fig. 18 - Typical oscillographic record for oscillation about small mean angle of attack

{:" N

..J U

..J U

'E '" .<:;

:= '" 0 0

-:::i

43

0.07

Increasing a 0 0.2 t--

1 o. I

o

-0. I

0.2

O. I

0

-0.1

Decreasing a •

./ L V

/ /

VIr

0.06

'" u

-c

'" '0 E

f1Io

1/ l~

/ ,/ 0.05 '" 0

0

"" 0 ~ .) W~

~lf/ -

/ 0

0.04

~ K l>< ~ 7

,-

2 3 4 5 6 7 8 9 3 4 5 6 Angle of attack a, degrees

(a) IJv = 0.163 , k = 0.0119 , 'CA = 0

0.07

Increasing a 0 -

rJ 7' Decreasing a e) ~ )/ ~I " u

0.06

/ ~~ c

'" '0 !E 4\

..,,) /

7 8

) 'r! ~VJ II

/ ~ /

~ 0.05 o

"" o o (\ ~ /

,;/ /j ) V

V ./ V ~

2 3 4

V ~ ~~ l/

v 0.04

5 6 7 8 9 0.032 3 4 5 6 Angle of attack a , degrees

(b) IJv = 0.163 , k = 0.0119 , CA = 0

Fig. 19 - Instantaneous lift and drag coefficients at small angle of attack and small cavitation number

7 8

9

9

44

..J (.)

0.4

0.3

.~ 0.2 '" :E Q) o '"

o. I

I---Increasing a 0

Decreasing a •

~ I '/

/ II VI

~ ~

~

0.04

0'0.03

c: Q)

'u -Q) o '" 0.02

'" e Cl I

II

V ~

I j

~ f If/ If

~ [! o

I

/ [/ 0.0

..J (.)

0.4

0.3

~ 0.2 '" :E Q)

o '"

0.1

o

2 3

~

V -<

2 3

'"'-,

o 4 5 6 7 8 9 2 3 4 5 6 7 8

Angle of attack a, degrees

(a) <Tv = 0.272, k = 0.0014, CA = 0

Increasing a 0

0.05 r.

/ /

Decreasing a • po- / V

A if !

0.04

/1 I /

" (.)

.: 0.03 c: Q)

'u ::

V l

I Q)

o

" 0>

.!S 0.02

If Vj

~ t7

4

/; V

0.0 I If ~)1

5 6 7 8 9 2 3 4 5 6

Angle of attack a, degrees

(b) <Tv = 0.316 , k = 0.0015 , CA = 0

Fig. 20 - Instantaneous I ift and drag coefficients at small angle of attack and moderately small cavitation number

7 8

9

9

6.0

5.5

5.0

4.5

4.0

c: c 3.5

"C c ~

~

IV 0.

IV 3.0 0. o -If)

IV > ~

~

u 2. 5

2.0

" i I.

I. 0

o .5

~\

45

\ b, \ ~,

" 4.

" I '-..... .... l// ~ --_. ..,.",/

~ Double cavity

+-_...,c,-..,.,... .... ..",,,,;~

.... ....--Full single cavity

..n'" ,

V /'

", ,""'- " ...,

.002 .004 .006 .008 .010 ·012 .014 .016

Reduced frequency, k

Figo 21 - Frequency effect on lift curve slope for two flow regimes

46

.J u

.4

. 3

.,:.2 c (1l

U

'+­.... (1l

8.1 .,... .... ::::i

o

-.1

v---~'

.J U

c

...----. •

V v

/' A

V~ ~ :)

§ ~

VQ (a) Average lift coefficient

0

• k=O

)l 0 k=O.oOI7

0 k=0.0119

i [;" k=0.0176

() = 0.163

2 4 6 8 10 12

Angle of attack a. degrees

.4.-------.--------r-------r-------.------~

.3·~------+-------+-~~--+-------+-----~

.~ .Ir--------r----~~--~~--+_------~------~ u

.... .... Q)

o u

(b) Average polar diagram

• k=O

0 k=0.0017

0 k=0.0119

-,I~------4-----~~~~--~ [;" k=0.0176

Numbers denote a in degrees

.02 .04 .06 .08 ,I

Drag coefficient, CD

Figo 22 - Comparison of force~ on steady foil and oscillating foil at

small angle of attack

14

, ; l~jl I ,

I i

II

47

, , " i i , , , , , ,

, , i , Ii ,

II i ,

:3 " (a) Pulsating cavity , 16 normal plate, <Tv = 0.40

Initial tank pressure = 22.82 psia

Rate of air tank pressure decrease = 4.88 psi / sec

,

fill! I

i

II II I I I I I i

i'l I I I

I ,

I

, i , , , ! !

~, I . r ' , . I' ti'dilii! . I i ,

(b) :3 " Split cavity • 16 normal plate 1 O'"y =·0.40

Initial tank pressure = 23.75 psia

Rate of air tank pressure decrease = 7.61 psi/ sec

Fig. 23 - Typical records of air tank pressure and cavity pressure for sudden ventilation, normal plate

r!'\imlli~ ;;'

Jii "

48

28r-----.----.-----,-----,-----,----,-----,----------------, Run Initio I Tank

tank pressure no. pres~ure change PSIO psia/sec

24r-----r_----~~--r_----~~--r_~~r_--~

48.8 16.8

2 23.8 7.61 20

3 21. 8 5.36

11) 4 18.3 4.51 5 Shortening cavity CIl

..c: 16 (.)

.!::

..c: -QI ~ 12 ~ >--

If~~ (a)

3/16" normal plate

°o~--~~--~~--~~~~--~~--~~--~~--~--~~--~ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Time, sec

28 Initial

Run Tank

tank pressure no. pres:;ure c~ange

24 pSla pSla/sec

63 30.4

2 40 18.8 20 3 23 10.3

4 18.7 5.13 en CIl

..c: 16 (.)

~

L -QI ~ 12 CIl

>--> c u

8 (b)

28° wedge

°o~--~~--~----~----~----~--~~--~~~~----~--~ 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 Time, sec

Figo 24 - Cavity length as a function of time with sudden ventilation

III Q)

35

30

25

-fi 20 I::

-..c:: -C> 1::15 ~ >­-> o u 10

III Q)

..c::

5

35

30

25

g 20

..c:: ..,.. C>

. ~ 15

> o u 10

5

Notation Run no.

0 I

0 2 ~

0

~\<><> A 3

0 -0 4

c - Steady

C

o ~\<> CbA\ ~ 00

l~ A , .... ~ 0

~ 'bt 0 I~A ~ ..r>

[J ~ -0

0.1 0.2 0.3 0.4 Cavitation number (j

(0) 3/1611 normal plate

Notation Run no.

CCP 0 1

~ 0 2

~ A 3

~ -0 4

O/J. - Steady

0 C ~

0 \ t

OD L~ ~ A ~ A I""'!. - C

0.1 0.2 0.3 0.4 Cavitation number (j

(b) 28° wedge

Tank pre~ure

pSla

48.8

23.8

21.8

18.3 stote

0.5 0.6

Tonk pres~ure

pSla

63

40

23 I 8.7

state

0.5 0.6

Fig. 25 - Cavity length as a function of instantaneous cavitation number with sudden ventilation

49

50

~,~'-~--

VI Q)

.c: u £

.c -0> c: ~

>--.;;: c 0

100

80

60

40

20

10

8

6 !------t---:;;;l~/.___{r-~ I ~ Y. '- "Cavity volume oc ~1.6

4

2~-------+----4---+--4--~~~-+--------+-----~--~-r-+-+-+~

.02 .04 .06 .08 .I I I

I;K - ptiC

I piC c

.2

Fig. 26 - Correlation of data for sudden ventilation

.4 .6 .8 1.0

I

]11 IT' ',' '!"',: I r -nf rr '!! ! ' 'II I " ~ ! "II! ln I1:1!";'1 :il!'!lii III !Ii' ill II IIITIII I i'illl i:fj r:: 111 ,I , I,

II~,!I 1"Ui I'll

[I III i:11 i , Ii :!III,I!: Iii

I

I!II III Ii

I I i

II'

: i I i I

I ,I iii I,ii

II I~e~ ,I II :' I,: Illi : III I i Iii II!I I , I, I , , III!' ,I ,I 1 'pr $'4 , , , J ' ! ,

III I,' ,,~ II I i , I I

I ,1'1) i, ¢ , :' II

~V! I 1 I II, r Ii I !! I "i ,i

, I ,

~ ,

~ ~, ,

I~~ ,

tm , I Ii

Ir' I' I, I,i Iii " , I I , 'I

! ~ I ~ II ! I

I I I

I i "

, 1

, , I. I I I , I I

"

, I I ~ , II: I "

"

i!

I' I

'Iii 11 '1 I

il I I ! !, Ili l 1111'1 11,

11:1

1 ill :: Ii i 11111 II, J II II

i:, I I 1111 II , 1 :1 I

IIllull !:i,i,jlj :,ii,;1 :11 illll illl," i II 111ili II , " II I I I Ii

(a) a = 11 0, (Tv = 0.28 , Initial air tank pressure = 24.3 psia

fI [T;) 11 ~'lII!111 II"IT ITII!wr I, ., 'ii lii'i'!] III II ;T[lI~ TIII'I! nil i :11 , r II !I III II! I

II',: i III i! 'I' ", Ili;1 1

1'1 I

, " Ii I

i

Iii ,

I

I I

I ~

I i i i

! 1 1 ' ,

"

I' 1,1 ,:w I ! 'illilll il I

i ,

I ' i

I

II I

I I

'I I'" ' "

" I ! i " , ! ,

I' II I

I I I

I : I~ Itr I I ! I I Ii I

, I I I

I II

Ii il , I I

I , , Ii ~i I'

' ' W I

II "

i.

II .1, I I ,

' Iii I ! !I'

'I[ I ~ lilllill III" I I III ij: I III I III, I: Ii I~ lJji,1 Ilili

' , , Ilil Iii !I.I IT ,

' .. I I , I

i ' , I

I I I I ! , , , I i i ,I 'I' , Ii \ i I i i

,

I '.

Iii r i I

III I .,

W ,

I'

\ I 'I : !i

, I !

I ,

, , 'I i ,

I

I I'

(b) a = 11 0, (Tv = 0.28 , Initial air tank pressure = 14.3 psia

Fig. 27 - Typical records of lift, drag, and cavity pressure due to sudden ventilation

,

51

I II ,

. ,

, ;

1 II II

I ,

I

I 1

I I

,

, "

,. , , , ! : ' , , : \ I

, , , 'I

,

1

52

... Q)

.c E

.25

.20

g .15 c o +­o

of--

> 8 .10

.05

o o

.40

.35

i·3o <.> .... Q)

o <.>

:E .25 ...J

.20

H

0.04

D

A

0.04

Notation Initial tank

pressure

0 14.3 psi a

o· 24.3 " A 54.3 "

.6.0 A ~ (a)

0 0 ,

AD D 0

A 0 0 D

AA 0 D 0 D D ( i> A 0 D

L.l. A A 1. L AA

A ;;. A

A 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36

I

0 (b) 0

D 0

] D ( 0 Do

AA D 0 'L

AA D A D D

L.l. .6. 1.;>- U D A

A A A

.6. .6. ;;. .6. .6.

0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36

Time I seconds

Figo 28 - Variation of cavitation number and I ift coefficient after sudden ventilation

0040

J'.,.

0040

.45

Notation

.40

0

0

.35 ..J

b. U

-+-c Q)

.(3 .30 '+-'+-Q)

0 u

0 +-'+-

..J .25 O~

--....;

0

L

b. b. b. .20

Initial tonk pressure

psia

14.3

24.3

54.3 ~

~ Steady flow 0

~ 0

0 [ -""

0 ~ [ D U

c~ 0 0 b.

~ cP DO b. b.

b. b.

o 0 0 b. L

A

U I 0 b. b.~

b. 4 b. ,~

~ \\~e b.b.

.05 .10 .15

Cavitation number, (J

Figo 29 - Lift coefficient as a function of cavitation number with sudden ventilation

53

54

• iii:11I1111 ~1L.wU Jr" ", I '.1"\ .

ii i !III·' Ii . Ii " " ii' I

:.1 !I i , II, 'i r+ f , 111111111 I.ff ,.,

1:'[1 :.l.

, IIIIIII~ : I· I, . :', I II

IJ IlH9JI:1 ' II .+"; I III tk;.i ,.! I· jJ'

I, ;.[ , .. LJ, U: ,.1 L1i .. :1+ , /" ..... ,. )) !

(ala- = .103, tc = 14",...;..,7" , . V= 45.3 fps (b) (T = .247 , 2c= 5"-2", V= 40.6 fps

... ~

·(C) (T = .123, 9.c = 10",.... 4"; V = 42.0 fps (d)(T = ;085, I.e= 16" ...... 7" , V = 42.5 fps

·(e) (T = .054 , lc= 26"""12", V= 42.9 fps

Fig. 30 - Typical records of lift and drag change after sudden angle of attack c~ange

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

.~ 0.4 "0

~ ~

~ 0.2

o o

Q)

"' " o 0-

"' ~

" o :;: o "

2.0

~ 1.8

0-

2 "'

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

o o

55

2.0

(a) (b)

1.8 ~ (j" = 0.103

(j" = 0.247

r, f th I

a=7· 1.6

E-7~

a = 14· / Ir hi

H

10 (1 = 14·

1/\

R

\

1.4

1.2

1.0

0.8

0.6

Note:

0.4 .E...:.- Indicates steady I---

-V 0.1 0.2 0.3 0.4

(c)

(j" = 0.123

~ I

SEJ

a= 14· 1

0.5

2.0

1.8

1.6

1.4

0.6

(d)

(j" =0.085

0.2

o o

Q

\

state lift c.urve slope

0.1 0.2 0.3 0.4 0.5 0.6

2.0

(e) <;

1.8

(j" =0.054

(1= 7· f H).,-, 1.6

1.4 (1= 7·

f\ 1 '-- a= 14·

\ a=14·

(

1.2

1.0

1.2

1.0 \~

0.1 0.2 0.3 0.4

0.8

0.6

0.4

0.2

o o 0.1

'\

1\ ~

0.2 0.3

Time, seconds

0.8

0.6

0.4

0.2

o 0.4 o 0.1 0.2

Fig. 31 - The zeroth order unit step function response to angular rotation

0.3

0.7

0.4

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Organization

57

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1 Commander, US Naval Weapons Laboratory, Dahlgren, Virginia, Attn: Planning Department.

1 Commander, US Naval Ordnance Laboratory, White Oak, Maryland.

1 Commander, US Naval Weapons Laboratory,' Dahlgren, Virginia, Attn: Computation and Exterior Ballistics Laboratory (Dr. A. V. Hershey).

1 Superintendent, US Naval Academy, Annapolis, Maryland, Attn: Library.

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Organization

Superintendent, US Naval Postgraduate School, Monterey, California.

Commandant, US Coast Guard, 1300 E Street, NW, Washington, D.C.

Secretary, Ship Structure Commit tee, US Coast Guard Headquarters, 1300 E Street, NW, Washington, D.C.

Commander, Military Sea Transportation Service, Department of the Navy, Washington 25, D. C. .

Division of Ship Design, Maritime Administration, 441 G Street, NW, Washington 25, D. C.

Superintendent, US Merchant Marine Academy, Kings Point, Long Island, New York, Attn: Captain L. S. McCready.

Commanding Officer and Director, US Navy Mine Defense Laboratory, Panama City, Florida.

Com.manding Officer, NROTC and Naval Administrative Unit, Massachu­setts Institute of Technology~ Cambridge 39, Massachusetts.

Commander, Hdqs. US Army Transportation Research and Development Com­mand, Transportation Corps, Fort Eustis, Virginia, Attn: Marine Transportation Division.

Air Force Office of Scientific Research, Mechanics Division, Wash­ington 25, D. C.

Commander, Wright Air Development Division, Aircraft Laboratory, Wright-Patterson Air Force Base, Ohio, Attn: Mr. W. Mykytow, Dy­namics Branch.

Director of Research, Code RR, National Aeronautics and Space Admin­istration, 600 Independence Avenue, SW, Washington, D. C. 20546.

Director, Langley Research Center, Langley Station, Hampton, Virginia, Attn: Mr. I. E. Garrick.

Director, Langley Research Center, Langley Station, Hampton, Vir­ginia, Attn: Mr. D. J. Marten.

Director, Engineering Science Division, National Science Foundation, Washington, D. C.

Director, National Bureau of Standards, Washington 25, D. C., Attn: Mr. J. M. Franklin.

Dr. G. B. Schubauer, Fluid Mechanics Section, National Bureau of Standards, Washington 25, D. C.

Defense Documentation Center, CameronStatibn, Alexandria, Virginia.

Office of Technical Services, Department of Commerce, Washington 25, D. C.

Mr. Alfonso Alcadan 1., Director, Laboratorio Nacional de Hydraulica, Antiguo Camino .A Ancon, Casialla Postal, 682, Lima, Peru.

Mr. T. A. Duncan, Lycoming Division, AVCO Corporation, 1701 K Street, NW, Apartment 904, Washington, D. C.

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Baker Manufacturing Company, Evansville, Wisconsin.

Professor S. Siestrunck, BureauD'Analyse et de Recherche Appliquees, 47, Avenue Victor Cresson, Issy-les-Moulineaux, Seine, France.

Professor A. Acosta, California Institute of Technology, Pasadena 4, California.

Professor M. Plesset, California Institute of Technology, Pasadena 4, California.

Professor T. Y. Wu, California Institute of Technology, Pasadena 4, California.

Professor A. Powell, University of California, Los Angeles, Califor­nia.

Dr. Maurice L. Albertson, Professor of Civil Engineering, Colorado State University, Fort Collins, Colorado 8052l.

Professor J. E. Cermak, Colorado State University, Department of Civil Engineering, Fort Collins, Colorado.

Dr. Blaine R. Parkin, General Dynamics - Convair, P. O. Box 1950, San Diego 12, California.

Robert H. Over smith , Chief of ASW/Marine Sciences, Mail Zone 6-107, General ~amics - Convair, San Diego, California 92112.

Dr. Irving C. Statler, Head, Applied Mechanics Department, Cornell Aeronautical Laboratory, Inc., P. O. Box 235, Buffalo, New York l422l.

1 Mr. Richard P. White, Jr., Cornell Aeronautical Laboratory, 4455 Ge­nesee Street, Buffalo, New York.

1 Professor W. R. Sears, Graduate School of Aeronautical Engineering, Cornell University, Ithaca,New York.

1 Mr. George H. Pedersen, Curtiss-Wright Corporation, Wright Aero­nautical Division, Wood-Ridge, New Jersey, Location CC-l Engrg. Mezz.

1 Mr. G. Tedrew, Food Machinery Corporation, P. O. Box 367, San Jose, California.

1 General Applied Science Laboratory, Merrick and Stewart Avenues, Westbury, Long Island, New York, Attn: Dr. Frank Lane.

1 Mr. R. McCandliss, Electric Boat Division, General ~amics Corpora­tion, Groton, Connecticut.

1 Dr. A. S. Iberal1, President, General Technical Services, Inc., 2640 Whiton Road, Cleveland 18, Ohio.

1 Gibbs and Cox, Inc., 21 West Street, New York, New York 10006.

1 Mr. Eugene F. Baird, Chief of ~amic Analysis, Grumman Aircraft Engineering Corporation, Bethpage Long Island, New York.

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Organization Mr. Robert E. Bower, Chief, Advanced Development, Grumman Aircraft Engineering Corporation, Bethpage, Long Island, New York. Mr. William P. Carl, Grumman Aircraft Engineering Corporation, Beth­page, Long Island, New York. Grumman Aircraft Engineering Corporation, Research Department, Plant 25, Bethpage, Long Island, New York 11714, Attn: Mr. Kenneth Keen.

1 Dr. O. Grim, Hamburgische Schiffbau-Versuchsanstalt, Bramfelder Strasse 164, Hamburg 33, Germany. 1 Dr. H. W. Lerbs, Hamburgische Schiffbau-Versuchsanstalt, Bramfelder Strasse 164, Hamburg 33, Germany. 1 Dr. H. Schwanecke, Hamburgische Schiffbau-Versuchsanstalt, Bramfelder Strasse 164, Hamburg 33, Germany. 1 Professor G. P. Weinblum, Director, Institute for Schiffbau, Univer­sity of Hamburg, Berliner Tow 21, Hamburg, Germany. 1 Professor G. F. Carrier, Harvard University, Cambridge 38, Massachu­setts.

1 Dr. S. F. Hoerner, 148 Busteed Drive, Midland Park, New Jersey. 1 Mr. P. Eisenbert, President, Hydronautics, Incorporated, Pindell School Road, Howard County, Laurel, Maryland. 1 Professor Carl Prohaska, Hydro-og Aerodynamisk Laboratorium, Lyngby, Denmark.

1 Professor L. Landweber, Iowa Institute of Hydraulic Research, State University of Iowa, Iowa City, Iowa. 1 Professor H. Rouse, Iowa Institute of Hydraulic Research, state Uni­versity of Iowa, Iowa City, Iowa. 1 Professor S. Corrsin, Department of Mechanics, The Johns Hopkins University, Baltimore 18, Maryland. 2 Professor O. M. Phillips, Division of Mechanical Engineering, In­stitute for Cooperative Research, The Johns Hopkins University, Bal­timore 18, Maryland.

1 Mr. Bill East, Lockheed Aircraft Corporation, California Division, Hydrodynamics Research. Burbank, California. 1 Mr. R. W. Kermeen, Lockheed Missiles and Space Company, Department 8l-73/Bldg. 538, P. O. Box 504, Sunnyvale, California. 1 Department of Naval Arcbi tecture and Marine Engineering, Room 5-228, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cam­bridge, Massachusetts 02139. 1 Professor M. A. Abkowitz, Massachusetts. Institute of Technology, Cambridge 39, Massachusetts.

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Professor H. Ashley, Massachusetts Institute of Technology, Cambridge 39, Massachusetts.

Professor A. T. Ippen, Massachusetts Institute of Technology, Cam­bridge 39, Massachusetts.

Professor M. Landahl, Massachusetts Institute of Technology, Cam­bridge 39, Massachusetts.

Dr. H. Reichardt, Director, Max-Planck Institut fur Stromungsfor­schung, Bottingerstrasse 6-8, Gottingen, Germany.

Professor R. B. Couch, University of Michigan, Ann Arbor, Michigan.

Professor W. W. Willmarth, University of Michigan, Ann Arbor, Michi­gan.

Midwest Research Institute, 425 Volker Boulevard, Kansas City, Mis­souri, Attn: Library.

Director, St. Anthony Falls Hydraulic Laboratory, University of Min­nesota, Minneapolis 14, Minnesota.

Mr. C. S. Song, st. Anthony Falls Hydraulic Laboratory, University of Minnesota, Minneapolis 14, Minnesota.

Mr. J. M. Wetzel, St. Anthony Falls Hydraulic Laboratory, University of Minnesota, Minneapolis 14, Minnesota.

Head, Aerodynamics Division, National Physical Laboratory, Teddington, Middlesex, England.

Mr. A. Sil verleaf, National Physical Laboratory, Teddington, Middlesex, England.

The Aeronautical Library, National Research Council, Montreal Road, Ottawa 2, Canada.

Dr. J. B. Van Manen, Netherlands Ship Model Basin, Wageningen, The Netherlands.

Professor John J. Foody,. Chairman, Engineering Department, State University of New York, Maritime College, Bronx, New York· 10465.

Professor J. Keller, Institute of Mathematical Sciences, New York University, 25 Waverly Place, New York 3, New York.

Professor J. J. Stoker , Institute of Mathematical Sciences, New York University, 25 Waverly Place, New York 3, New York.

Dr. T. R. Goodrn?-n, Oceanics, Incorporated, Technical Industrial Park, Plainview, Long Island, New York.

Professor J. William Holl, Department of Aeronautical Engineering, The Pennsylvania State University, Ordnance Research Laboratory, P. O. Box 30, State College, Pennsylvania.

1 Dr. M. Sevik, Ordnance Research Laboratory, Pennsylvania State Uni­versity, Unive~sity Park, Pennsylvania.

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Organization Dr. George F. Wis1icenus, Garfield Thomas Water Tunnel, Ordnance Research Laboratory, The Pennsylvania State University, Post Office Box 30, state College, Pennsylvania 16801. Professor E. J. Brunelle, Princeton University, Princeton, New Jersey. Mr. David Wellinger, Hydrofoil Pro jects, Radio Corporation of America, Burlington, Massachusetts.

The RAND Corporation, 1700 Main Street, Santa Monica, California, 90406, Attn: Library.

Professor R. C. DiPrima, Depa;:-tment of Mathematics , Rensselaer Poly­technic Institute, Troy, New York. 1 Mr. L. M. White, U. S. Rubber Company, Research and Development De­partment, l.!Jayne, New Jersey. 1 Professor J. K. Lunde, Skipsmodelltanken, T'r,ondheim, Norway. 1 Editor, Applied Mechanics Review, Southwest Reseaech Institute, 8500 Culebra Road, San Antonio 6, Texas. 1 Dr. H. N. Abramson, Southwest Research Institute, 8500 Culebra Road, San Antonio 6, Texas. 1 Mr. G. Ransleben, Southwest Research Institute, 8500 Culebra Road, San Antonio 6, Texas.

1 Professor E. Y. Hsu, Stanford University, Stanford, Ca.lifornia. 1 Dr. Byrne Perry, Department of Civil Engineering, Stanford Uni versi ty, Stanford, California 94305. 1 Mr. J. P. Breslin, Stevens Institute of Technology, Davidson Labora­tory, Hoboken, New Jersey. 1 Mr. D. Savitsky, Stevens Institute of Technology, Davidson Laboratory, Hoboken, New Jersey.

1 Mr. S. Tsakonas, Stevens Institute of Technology, Davidson Laboratory, Hoboken, New Jersey.

1 Dr. Jack Kotik, Technical Research Group, Inc., Route 110, Melville, New York.

1 Dr. R. Tirnman, Department of Applied Mathematics, Technological Uni­versity, JUlianalaan, Delft, Holland. 1 Transportation Technical Research Institute, No. 1057-1 Chome, Mejiro­machi, Toshima-ku, Tokyo-to, Japan. 1 Dr. Grosse, Versuchsanstalt furWasserbau und Schiffbau, Schleusenin­sel im Tiergarten, Berlin, Germany. 1 Dr. S. Schuster, Director, Versuchsanstal t fur Wasserbau und Schiff­bau, Schleuseninsel im Tiergarten, Berlin, Germany.

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Organization

Technical Library, Webb Institute of Naval Architecture, Glen Cove, Long Island, New York.

1 Professor E. V. Lewis, Webb Institute of Naval Architecture, Glen Cove, Long Island, New York.

1 Mr. C. Wigley, Flat 103, 6-9 Charterhouse Square, London E. C. 1, England.

1 Coordinator of Research, Maritime Administration, 441 G Street, NW, Washington 25, D. C.

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Technical Paper No. 49, Series B St. Anthony Falls Hydraulic Laboratory

MEASUREMENTS OF THE UNSTEADY FORCE ON CAV­ITATING HYDROFOILS IN A FREE JET, by C. S. Song. June 1964. 55 pages incl. 31 illus. Contract Nonr 710(24).

Experimental data concerning force and other related quan­tities associated with unsteady super cavitating flow about bodies tested in a free-jet water tunnel are reported herein. Three types of unsteady flows were studied -- those due to pitching oscillation, sudden ventilation, and sudden angle of attack change. All test bodies were of such geometrical configuration that separation could occur only at two or three fixed points.

Available from St. Anthony Falls Hydraulic Laboratory, Univer sity of Minnesota, at $1. 7 5 per copy.

Technical Paper No. 49, Series B St. Anthony Falls Hydraulic Laboratory

MEASUREMENTS OF THE UNSTEADY FORCE ON CAV­ITATING HYDROFOILS IN' A FREE JET, by C. S. Song. June 1964. 55 pages incl. 31 illus. Contract Nonr 710(24).

Experimental data concerning force and other related quan­tities associated with unsteady super cavitating flow about bodies tested in a free-jet water tunnel are reported herein. Three types of unsteady flows were studied -- those due to pitching oscillation, sudden ventilation, and sudden angle of attack change. All test bodies were of such geometrical configuration that separation could occur only at two or three fixed points.

Available from St. Anthony Falls Hydraulic Laboratory, University of Minnesota, at $1. 75 per copy.

UNCLASSIFIED

1. Cavitation 2. Hydrofoil 3. Supercavitating Foil 4. Oscillating Foil 5. Free-Surface Effect

I. Title II. Song, C. S.

III. St. Anthony Falls Hydraulic Laboratory

IV. Contract No. 710(24)

UNCLASSIFIED

1. Cavitation 2. Hydrofoil 3. Supercavitating Foil 4. Oscillating Foil 5. Free-Surface Effect

I. Title II. Song, C. S.

III. St. Anthony Falls Hydraulic Laboratory

IV. Contract No. 710(24)

Technical Paper No. 49, Series B St. Anthony Falls Hydraulic Laboratory

MEASUREMENTS OF THE UNSTEADY FORCE ON CAV­ITATING HYDROFOILS IN A FREE JET, by C. S. Song. June 1964. 55 pages incl. 31 illus. Contract Nonr 710(24).

Experimental data concerning force and other related quan­tities associated with unsteady super cavitating flow about bodies tested in a free-jet water tunnel are reported herein. Three types of unsteady flows were studied -- those due to pitching oscillation, sudden ventilation, and sudden angle of attack change. All test bodies were of such geometrical configuration that separation could occur only at two or three fixed points.

Available from St. Anthony Falls Hydraulic Laboratory, University of Minnesota, at $1. 75 per copy.

Technical Paper No. 49, Series B St. Anthony Falls Hydraulic Laboratory

MEASUREMENTS OF THE UNSTEADY FORCE ON CAV­ITATING HYDROFOILS IN A FREE JET, by C. S. Song. June 1964. 55 pages incl. 31.illus. Contract Nonr 710(24).

Experimental data concerning force and other related quan­tities associated with unsteady super cavitating flow about bodies tested in a free-jet water tunnel are reported herein. Three types of unsteady flows were studied -- those due to pitching oscillation, sudden ventilation, and sudden angle of attack change. All test bodies were of such geometrical configuration that separation could occur only at two or three fixed points.

Available from St. Anthony Falls Hydraulic Laboratory, University of Minnesota, at $1. 75 per copy.

UNCLASSIFIED

1. Cavitation 2. Hydrofoil 3. Supercavitating Foil 4. Oscillating Foil 5. Free-Surface Effect

I. Title II. Song, C. S.

III. St. Anthony Falls Hydraulic Laboratory

IV. Contract No. 710(24)

UNCLASSIFIED

1. Cavitation 2. Hydrofoil 3. Supercavitating Foil 4. Oscillating Foil 5. Free-Surface Effect

I. Title II. Song, C. S.

Ill. St. Anthony Falls Hydraulic Laboratory

IV. Contract No. 710(24)

Technical Paper No. 49, Series B St. Anthony Falls Hydraulic Laboratory

MEASUREMENTS OF THE UNSTEADY FORCE ON CAV­ITATING HYDROFOILS IN A FREE JET, by C. S. Song. June 1964. 55 pages inc!. 31 illus. Contract Nonr 710(24).

Experimental data concerning force and other related quan­tities associated with unsteady super cavitating flow about bodies tested in a free-jet water tunnel are reported herein. Three types of unsteady flows were studied -- those due to pitching oscillation, sudden ventilation, and sudden angle of attack change. All test bodies were of such geometrical configuration that separation could occur only at two or three fixed points.

Available from St. Anthony Falls Hydraulic Laboratory, University of Minnesota, at $1. 75 per copy.

Technical Paper No. 49, Series B St. Anthony Falls Hydraulic Laboratory

MEASUREMENTS OF THE UNSTEADY FORCE ON CAV­ITATING HYDROFOILS IN A FREE JET, by C. S. Song. June 1964. 55 pages inc!. 31 illus. Contract Nonr 710(24).

Experimental data concerning force and other related quan­tities associated with unsteady super cavitating flow about bodies tested in a free-jet water tunnel are reported herein. Three types of unsteady flows were studied -- those due to pitching oscillation, sudden ventilation, and sudden angle of attack change. All test bodies were of such geometrical configuration that separation could occur only at two or three fixed points.

Available from St. Anthony Falls Hydraulic Laboratory, University of Minnesota, at $1.75 per copy.

I

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UNCLASSIFIED

1. Cavitation 2. Hydrofoil 3. Supercavitating Foil 4. Oscillating Foil 5. Free-Surface Effect

1. Title II. Song, C. S.

lll. St. Anthony Falls Hydraulic Laboratory

IV. Contract No. 710(24)

UNCLASSIFIED

1. Cavitation 2. Hydrofoil 3. Supercavitating Foil 4. Oscillating Foil 5. Free-Surface Effect

1. Title II. Song, C. S.

lll. St. Anthony Falls Hydraulic Laboratory

IV. Contract No. 710(24)

Technical Paper No. 49, Series B St. Anthony Falls Hydraulic Laboratory

MEASUREMENTS OF THE UNSTEADY FORCE ON CAV­ITATING HYDROFOILS IN A FREE JET, by C. S. Song. June 1964. 55 pages inc!. 31 illus. Contract Nonr 710(24).

Experimental data concerning force and other related quan­tities associated with unsteady supercavitating flow about bodies tested in a free-jet water tunnel are reported herein. Three types of unsteady flows were studied -- those due to pitching OSCillation, sudden ventilation, and sudden angle of attack change. All test bodies were of such geometrical configuration that separation could occur only at two or three fixed points.

Available from St. Anthony Falls Hydraulic Laboratory, University of Minnesota, at $1.75 per copy.

Technical Paper No. 49, Series B St. Anthony Falls Hydraulic Laboratory

MEASUREMENTS OF THE UNSTEADY FORCE ON CAV­ITATING HYDROFOILS IN A FREE JET, by C. S. Song. June 1964. 55 pages incl. 31 illus. Contract Nonr 710(24).

Experimental data concerning force and other related quan­tities associated with unsteady super cavitating flow about bodies tested in a free-jet water tunnel are reported herein. Three types of unsteady flows were studied -- those due to pitching oscillation, sudden ventilation, and sudden angle of attack change. All test bodies were of such geometrical configuration that separation could occur only at two or three fixed points.

Available from St. Anthony Falls Hydraulic Laboratory, University of Minnesota, at $1.75 per copy.

I

UNCLASSIFIED

1. CaVitation 2. Hydrofoil 3. Supercavitating Foil 4. Oscillating Foil 5. Free-Surface Effect

1. Title II. Song, C. S.

Ill. St. Anthony Falls Hydraulic Laboratory

IV. Contract No. 710(24)

UNCLASSIFIED

1. Cavitation 2. Hydrofoil 3. Supercavitating Foil 4. Oscillating Foil 5. Free-Surface Effect

1. Title II. Song, C. S.

Ill. St. Anthony Falls Hydraulic Laboratory

IV. Contract No. 710(24)

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Technical Paper No. 49, Series B St. Anthony Falls Hydraulic Laboratory

MEASUREMENTS OF THE UNSTEADY FORCE ON CAV­ITATING HYDROFOILS IN A FREE JET, by C. S. Song. June 1964. 55 pages incl. 31 illus. Contract Nonr 710(24).

Experimental data concerning force and other related quan­tities associated with unsteady super cavitating flow about bodies tested in a free-jet water tunnel are reported herein. Three types of unsteady flows were studied -- those due to pitching oscillation, sudden ventilation, and sudden angle of attack change. All test bodies were of such geometrical configuration that separation could occur only at two or three fixed points.

Available from St. Anthony Falls Hydraulic Laboratory, University of Minnesota, at $1.75 per copy.

Technical Paper No. 49, Series B St. Anthony Falls Hydraulic Laboratory

MEASUREMENTS OF THE UNSTEADY FORCE ON CAV­ITATING HYDROFOILS IN A FREE JET, by C. S. Song. June 1964. 55 pages incl. 31 illus. Contract Nonr 710 (24).

Experimental data concerning force and other related quan­tities associated with unsteady super cavitating flow about bodies tested in a free-jet water tunnel are reported herein. Three types of unsteady flows were studied -- those due to pitching oscillation, sudden ventilation, and sudden angle of attack change. All test bodies were of such geometrical configuration that separation could occur only at two or three fixed points.

Available from St. Anthony Falls Hydraulic Laboratory, University of Minnesota, at $1. 75 per copy.

UNCLASSIFIED

1. Cavitation 2. Hydrofoil 3. Supercavitating Foil 4. Oscillating Foil 5. Free-Surface Effect

I. Title II. Song, C. S.

Ill. St. Anthony Falls Hydraulic Laboratory

IV. Contract No. 710(24)

UNCLASSIFIED

1. Cavitation 2. Hydrofoil 3. Supercavitating Foil 4. Oscillating Foil 5. Free-Surface Effect

I. Title II. Song, C. S.

III. St. Anthony Falls Hydraulic Laboratory

IV. Contract No. 710(24)

Technical Paper No. 49, Series B St. Anthony Falls Hydraulic Laboratory

MEASUREMENTS OF THE UNSTEADY FORCE ON CAV­ITATING HYDROFOILS IN A FREE JET, by C. S. Song. June 1964. 55 pages incl. 31 illus. Contract Nonr 710(24).

Experimental data concerning force and other related quan­tities associated with unsteady super cavitating flow about bodies tested in a free-jet water tunnel are reported herein. Three types of unsteady flows were studied -- those due to pitching oscillation, sudden ventilation, and sudden angle of attack change. All test bodies were of such geometrical configuration that separation could occur only at two or three fixed points.

Available from St. Anthony Falls Hydraulic Laboratory, University of Minnesota, at $1. 75 per copy.

Technical Paper No. 49, Series B St. Anthony Falls Hydraulic Laboratory

MEASUREMENTS OF THE UNSTEADY FORCE ON CAV­ITATING HYDROFOILS IN A FREE JET, by C. S. Song. June 1964. 55 pages incl. 31 illus. Contract Nonr 710(24).

Experimental data concerning force and other related quan­tities associated with unsteady super cavitating flow about bodies tested in a free-jet water tunnel are reported herein. Three types of unsteady flows were studied -- those due to pitching oscillation, sudden ventilation, and sudden angle of attack change. All test bodies were of such geometrical configuration that separation could occur only at two or three fixed points.

Available from St. Anthony Falls Hydraulic Laboratory, University of Minnesota, at $1. 75 per copy.

UNCLASSIFIED

1. CaVitation 2. Hydrofoil 3. Supercavitating Foil 4. Oscillating Foil 5. Free-Surface Effect

I. Title II. Song, C. S.

III. St. Anthony Falls Hydraulic Laboratory

IV. Contract No. 710(24)

UNCLASSIFIED

1. Cavitation 2. Hydrofoil 3. Supercavitating Foil 4. Oscillating Foil 5. Free-Surface Effect

1. Title II. Song, C. S.

III. St. Anthony Falls Hydraulic Laboratory

IV. Contract No. 710(24)