UNIVERSITY OF MINNESOTA

ST. ANTHONY FALLS HYDRAULIC LABORATORY

Project Report No. 111

Vibration of Cavitating Hydrofoils

by CHARLES C. S. SONG

This research was carried out under the Naval Ship Systems Command Hydro­foil Exploratory Development Program SF 013 0201, administered by the Naval Ship Research and Development Center. Prepared under the Office of Naval Research Contract NOOO 14-67·A·0 1 13·0005.

OCTOBER 1969

MINNEAPOLIS, MINNESOTA

This document has been approved for public release and sale; its distribution is unlimited.

Reproduction in whole or in part is permitted

for any purpose of the United States Government

CONTENTS

List of Illustrations • • • • • • • • • • , ., . Table of Notation

(It • • (It • .. • (It • • (It (It • ~ (It (It

Abstract. • • • • • • • • • • • • (It • .. (It f • • • • •

I.

II.

III.

IV.

INTRODUCTION. • • • • • • • • • • • • • It (It • "

EXPERIMENTAL APPARATUS AND PROCEDURE. • , (It • •

EXPERIMENTAL RESULTS. • • (It ## • • • • • • • (It •

A. B.

Fully Developed Cavities ••• • • • • • (It

Partial Cavities. • • • • • • · , . , . . 1.

2.

Natural Frequency of Partial Cavities. • • • • • • " • • . . . Two Types of Flutter-Like Vibrations. • • • • • • • • • • (It (It

CONCLUSIONS • • • • • • • • • • • (It (It • • , (It •

List of References. • . . . • • • (It • • • (It • (It . . . Figures 1 through 20. • • • • • • • (It •. • • (It • (It , (It

-iii-

Page

iv

v

vii

1

3

4

6

7

8

11

13

1.5

LIST OF ILLUSTRATIONS

Figure

1 A view of the Two-Dimensional Test Section of the Free-Jet Water Tunnel

2 Cross-sectional Profiles of Test Bodies

3 Typical record of Free-Vibration Test

4 RMS values of Fluctuating Lift Coefficient for Fully Developed Cavities

5 RMS values of Fluctuating Drag Coefficient for Fully Developed Cavities

6 RMS values of Fluctuating Moment Coefficient for Fully Developed Cavities

7 Intensity of Cavity Pulsation as a function of Cavity Length for Ventilated Cavities

8 Autocorrelation of Fluctuating Lift of Non-Pulsating Cavity

9 Autocorrelation of Pulsating Cavity

10 Relative Intensity of Fluctuation for Short Cavities

11 Relative Intensity of Fluctuating Lift of a Stable Cavity

12 A Typical Record showing the Vibration predominantly due to Self­Excited Cavity Vibration

13 Natural Frequency of Short Cavities as a function of Cavity Length

14 Relative Intensity of Self-Excited Cavity Vibration as a function of Cavity Length

15 Typical records showing a Flutter-Like Vibration for a Symmetrical Wedge (Type 1)

16 Reduced Frequency of a Flutter-Like Vibration, Type 1

17 A typical record of a Flutter-Like Vibration (Type 2) for a Non­Symmetrical Wedge

18 Reduced Frequency of a Flutter-Like Vibration, Type 2

19 Amplitude ratio of Flutter-Like Vibrations

20 Comparison of Frequency Ratios

-iv~

TABLE OF NOTATION

AL ~ amplitude of lift

~ ~ amplitude of moment

A* ~ amplitude ratio

C ~ chord length of the foil

CD ~ drag coefficient

CL ~ lift coefficient

CM ~ moment coefficient

D ~ drag

f ~ frequency

. fo ~ fundamental frequency of the cavity vibration

fl ~ frequency of the first type hydro elastic vibration

f2 ~ frequency of the second type hydroelastic vibration

f l ' ~ first natural frequency of the system in air

f2' ~ second natural frequency of the system in air

k ~ reduced frequency of the cavity Vibration o

~ ~ average cavity length

L ~ lift

L' ~ fluctuating component of the lift

M ~ moment

~ ~ autocorrelation of lift fluctuation

t ~ time

U ~ ambient speed

a ~ angle of attack

p ~ density of fluid

a ~ cavitation number based on vapor pressure v

'T ~ delay time

w = angular speed of Vibrations

ABSTRACT

A primarily experimental research program has been carried out

using a free-jet water tunnel for the purpose of studying force and

moment fluctuations on cavitating two-dimensional hydrofoils. Both a

symmetrical wedge and a non-symmetrical wedge were tested for a wide

range of cavity lengths and several different elastic conditions.

Fluctuations in lift and moment were of primary concern in the experi­

ments.

It was revealed that the force and moment were quite steady if the

cavity was longer than two chords unless an excessive amount of ventila­

tion caused cavity pulsations. For a shorter cavity, however, the flow

was generally very unstable, and severe vibrations were noted. A cavity

of any length was found to be basically unstable and to oscillate at

a characteristic frequency which was primarily a function of the cavity

length. The vibrating cavity may cause an elastically supported foil to

vibrate severely when the cavity is short. The largest-amplitude vibra­

tion often occurred when the cavity length was approximately equal to

one chord.

Flutter-like vibrations were noted in the first and second natural

modes of the two-degree-of-freedom system. The frequency of these vibra­

tions was found to be practically independent of the cavity length. The

severest vibration in the first natural mode usually occurred when the

average cavity length was approximately equal to one chord, whereas vi­

bration in the second natural mode was found more likely to occur when

the cavity was very short.

-vii-

--- ._ ... _-_ .. ---

VIBRATION OF CAVITATING HYDROFOILS

I. INTRODUCTION

Cavitation has long been recognized as a serious problem for those

concerned with hydraulic machinery or structures which may be exposed to a

low-pressure field in a flowing liquid. Cavitation damage, cavitation

noise, and reduction of efficiency are frequently associated with cavitat­

ing bodies. Design of lifting bodies to operate at cavitating conditions

with acceptable performance has been under study by various investigators

. during the past two decades. A great impetus was given to research in

this field when linearized potential flow theory was first used by Tulin * .

[lJ to estimate the steady load on supercavitating bodies. Since then a

great· number of papers have been written on methods of designing and

evaluating hydrofoils with various sizes of steady cavities. Direct and

indi rect methods, linear and· nonlinear theories' have been developed, and

their validity has, in most cases, been tested by experiments.

For those who have observed actual cavity flows it should be quite

apparent that the cavitating flows for a range of Reynolds numbers which

commonly occur in practice are by no means steady. So-called natural or

ventilated steady cavity flows are steady only in an average sense. Fluc­

tuations of random nature are omnipresent in these flows, and in some

cases periodic oscillations of the flow may also exist. If the forces

due to fluctuating or oscillating flow become a significant fraction of

the average force, then the existence of unsteady forces must not be over­

looked in the design process. Vibrations of a hydrofoil may be classified

under several categories according to the basic mechanisms involved. The I

vibration may be primarily due to the oscillatory force generated by a

changing ambient flow field, such as in the case of a propeller in a wake

or a hydrofoil moving under surface waves. Another type of vibration,

primarily due to hydroelastic instability, is known as flutter.

Neither of the two types of vibrations just mentioned was a primary .

concern of this investigation; rather, it was undertaken to study forced

* Numbers in brackets refer to the List of References on page 13

2

vibrations and resonance of a cavitating hydrofoil wherein the primary

driving force is generated by the instability of the cavity itself. Sil­

berman and Song have reported [2J that a long ventilated cavity located

near a free surface may become unstable and pulsate violently at certain

characteristic frequencies. A large-amplitude oscillatory force was found

to accompany the cavity pUlsation. Song [3J attributed this type of vibra­

tion to the resonance of the air in the cavity and the accompanying waves

on the cavity surface. It was not the purpose of this investigation to re­

study this problem.

Another well-known type of vibration [4,5,6J occurs when the average

cavity length is approximately equal to one chord. Although the mechanism

of this phenomenon has not been completely understood, the suggestion of

Wade and Acosta [5J and Kaplan [6J that the vibration was of a forced type

due to the cavity vibration appears to be correct. A recent paper by Besch

[7J reinforces this view. Assuming that the vibration was forced vibration,

he calculated the phase angle between lift and moment in a simulated two­

degree-of-freedom linear system under a sinusoidal driving force. Good agree­

ment was indicated between the calculated phase angle and the measured values.

A research program has been conducted at the st. Anthony Falls Hydraulic

Laboratory under the sponsorship of the Naval Ship Research and Development

Center of the U.S. Navy for the purpose of studying the characteristics of

unsteady forces on two-dimensional bodies under various cavitating conditions.

Hydrofoils of simple cross sections were tested in a free-jet water tunnel '

o~er a wide range of cavity lengths and using several different elastic sup­

porting conditions. As reported by others [4,5,6,7J, severe vibrations were

found most likely to occur when the average cavity length was nearly equal to

the chord. However, the problem was found to be much more complicated than

one might be led to believe by the previously published papers. The exist­

ence of forced vibration the frequency of which is a function of the cavity

length, as reported by Besch, has been confirmed. Also revealed is the exist­

ence of two other types of vibrations whose amplitude could be even greater

than that of the forced vibration under certain circumstances. These vibra­

tions appear to be hydro elastic in nature, but their characteristics and

mechanisms have not been completely determined in this study. The limited

data presented here without theoretical backing can only serve as evidence

indicating the existence of hydroelastic instabilities which are similar,

but not identical in a classical sense, to flutter.

II. EXPERIMENTAL APPARATUS AND PROCEDURE

The experiment was carried out in the rectangular test section of

the free-jet water tunnel at the St. Anthony Falls BYdraulic Laboratory.

This tunnel is essentially a 30 in. diameter vertic13.1 pipe drawing water

from the upper pool of st. Anthony Falls and dischargingintb the lower

pool approximately 50 ft below. A rectangular free-falling jet 5 in. thick

and of variable width (up to 18 in.) is generated in the pipe by an orifice

located upstream of the test section. The ambient velocity and pressure of

the jet are controlled jointly by regulating the intake valve opening and

the amount of air permitted to ventilate the space between the jet and the

tunnel walls. A view of the test section is shown in Fig. 1.

Two-dimensional bodies of relatively simple profile, as sketched in

Fig. 2, were tested. A symmetrical wedge (a) and a non-symmetrical wedge

(b) were the profiles most extensively tested, although the other two

types, (c) and (d), were also used to explore the significance of the lo­

cation and the manner of separation. In selecting the size of the test

bodies it was necessary to compromise between two conflicting requirements-­

they had to be large enough to facilitate cavity length measurement and yet

small enough for the wall effect to be insignificant. A jet-width-to-chord

ratio of 5 was considered acceptable, and hence a 2-1/2 in. chord was

selected to be used with a 12-1/2 in. wide jet.

A test body was supported at each end of its span on a tunnel wall

by a set of stainless steel plates and torsion bars such that the predom­

inant modes of vibration during experiments were pitching and heaving.

Strain gages were mounted on the lift and drag plates and the torsion bars.

Prior to each experimental run the spring constants were determined by a

static calibration and the natural mode and frequency of the system in air

were determined by a free-vibration test. The signals generated from the

strain gages during the experiments or calibrations were amplified using a

C.E.C. Type 1-127 carrier amplifier and recorded on a Precision Instrument -

FM tape recorder or on paper for subsequent analysis. A typical record of

free vibration in air due to a given combined initial displacement in the

---- --- -------------------------- ------------- ---- ----

3

4

direction of heave and rotation is shown in Fig. 3. The principal modes and

.frequencies are clearly separable from records of this kind. Since both the

coupling of fluctuating forces with the elastic system and the purely hydro­

dynamic instabilities were of interest, it was necessary to vary the system's

natural frequencies by changing the supporting membranes from time to time.

After a test body was secured in the test section at a desired angle of

attack, the tunnel was closed and a flow established. The ambient pressure

was then lowered step by step so that the average cavity length might in­

crease in small steps starting at zero. Cavity length was usually visible

through the transparent tunnel wall, and its average value in relation to

the chord length could be estimated by observation. The instantaneous force

and moment were recorded and analyzed using a Hewlett Packard type 300-A

wave analyzer Qr a Disa type 55A06 random signal indicator and correlator.

It was also possible to study the effect of ventilation by admitting air

into the cavity through a series of air supply valves attached to the tunnel

walls.

III. EXPERIMENTAL RESULTS

The characteristics of cavity fluctuations in the two regimes of the

flows, fully cavitating and partially cavitating, appeared to differ in some

fundamental ways. It is, therefore, convenient to discuss the experimental

results for the two regimes separately.

A~ Fully Developed Cavities

For flows With very small cavitation numbers it is well known that the

average forces and moments are linearly proportional to the cavitation

number, whereas the cavity length is inversely proportional to the second

power of the cavitation number. A small change in the cavitation number

near zero would thus give rise to a significant change in the appearance of

the flow, but cause very little change in the force and moment acting on the

foil. This quasi-steady type of argument is apparently valid for flows with

long, non-pulsating cavities, ventilated or otherwise. It was observed that

the length and width of a cavity fluctuate substantially as a rule, even

though the ambient flow condition is maintained as nearly constant as is

experimentally practicable. On the other hand, the fluctuating components

of the force and moment are generally very small. Some measured root mean

square values of the fluctuating lift, dl'ag, and moment on a symmetl'ical

wedge for cavities greater than or equal to 2_ chordsB,re shown in_ Figs. 4, 5, and 6. Here the RMS values of the force and moment coefficients are

defined as

whel'e 1, D, and M indicate lift, drag, and moment, respectively. As

.usual, the average and fluctuating values are designated by bars and

prL"l1eS, respectively. The symbols p, U, and C are respectively the

density of the fluid, the ambient speed, and the chord length of the foil.

It can be observed from these figures that the force and moment fluctua­

tions are small and practically independent of the cavity length. For the

flow conditions represented by the data shown,~\C1,)2 amounted to less

than 6 per cent of the corresponding average lift coefficient C1 •

Naturally, the result discussed in the preceding paragraph applies

only to non-pulsating cavities. When a cavity is excessively ventilated,

it may pUlsate violently [2J. Since the phenomenon of the pulsating cavity

is well known [2,3J, the subject will not be discussed further except to

mention that the intensity of the pulsation appears to vary inversely

with the cavity length, as shown in Fig. 7.

In an effort to study the detailed nature of the fluctuating forces

and moment for fully cavitating flows, statistical analysis of the data

was also carried out. When the cavity was long andnoh"'pulsating, the force

and moment fluctuations appeared random and the frequency spectrum was

neal'ly flat within the range investigated (1 Hz < f < 1 kHz). However,

certain prefel'red frequencies or time scales could be detected from

autocorrelation curves. Figure 8 shows the autocorrelation of the lift

fluctuation, 11" of a typical non-pulsating, fully developed cavity.

Here the autocol'relation is defined as

---_._-------._._-------_.

5

(1)

(2)

6

where L' is the fluctuating component of the lift and 1" is the delay

time. The scale factor of 64 on the abscissa is the ratio of the tape

speeds during analysis and recording. For the particular case shown in

this figure there are two characteristic frequencies, 14 Hz and 284Hz.

The lower frequency appears to be associated with the natural frequency of

the cavitating flow, although it was extremely weak and difficult to detect

for longer cavities. The second characteristic frequency is more readily

detectable and is identifiable as that related to a natural frequency of

the hydroelastic system. No new information could be obtained from cross­

correlations among lift, drag, and moment. The autocorrelation curve of a

pulsating cavity, obtainable by increasing the rate of air ventilation into

the same cavity which Fig. 8 represents, is shown in Fig. 9. Clearly, the

characteristic frequency of the pulsating cavity, 26 Hz in this case, is

unrelated to those of non-pUlsating cavities.

B. Partial Cavities

As compared to its fully developed counterpart, a partial cavity was

found to be considerably more unstable as a rule. Moreover, since part of

the surface profi.le on the cavity side of the foil may be wet, the hydrofoil

shape may complicate the problem. Experiments were carried out using mainly

the symmetrical wedge shown in Fig. 2(a) and the modified wedge shown in

Fig. 2(b). The break in the surface profile of the second wedge appeared

to cause vibration when the cavity was short; this will be discussed in more

detail later.

As has been observed by others [4,5,6,7J, very severe vibration occurs

quite frequently when the cavity is approximately equal to one chord. In

fact, this type of vibration is so common that it is rather difficult to

avoid. A typical plot of the relative RMS values of the fluctuations (lift,

drag, and moment) normalized with the respective maximum values, as shown in

Fig. 10, exhibits a peak intensity at ~/C ~ 1. The vibration at ~/C ~ 1

is not unavoidable, however. Using a relatively stiff system it was found

possible to have flow at ~/C ~ 1 without severe vibration, at least for

the case of a symmetrical wedge at a relatively small angle of attack.

Figure 11 shows that in this case the RMS values are relatively flat over

the entire region without an apparent peak at~/C = 1. Frequency spectrum

analysis of the fluotuation further revealed the concentration of energy at

a certain frequency whioh changed with the cavity .l~ngth.

Three types of instabilities were observed to exist in addition to

the ever-present random vibrations. Because of high energy concentrations

these vibrations are of practioal importance and should be studied in

detail. Further experimental and analytical evidenoe ooncerning the three

major instabilities is discussed in the following seotion.

1. Natural Frequency of Partial Cavities

Unlike the fully developed oavities, short cavities produoe rather

severe vibrations at certain frequencies. The intensity of these vibra­

tions is so large that they are readily recognizable even on unfiltered

oscillographic records. Figure 12 is typical of'such a record wherein a

large-amplitude vibration believed to be caused by the hydrodynamic in­

stability of the cavitating flow can be noted.' A more precise determina­

tion of the frequency associated With the vibration was obtained by means

of frequency spectrum analysis. In general, large and well-defined peaks

existed in the frequency spectrum of a short cavity because the three types

of instabilities mentioned previously could be made to occur at frequencies

sufficiently far apart.

As was reported by Besch [7J, the fundamental frequency of th~ vibra­

tion depends primarily on the cavity length, but not 'on the elasticity of

the spring system. Figure 13 shows the functional relationship between

the reduced frequency, k ~ and the dimensionless cavity length. Here o the reduced frequency is defined as

f C o k =-U o

where f is the fundamental frequency of the vibration. Included in this o

figure are also the data shown in Fig. 25 of Besch's report [7J and those

of Kaplan and Lehman [6J. The correlation is good considering the inac­

curacy involved in the cavity length measurement and the large differences

which existed among the three experimental set-ups, For example, the

natural frequencies of the elastic system used in this experiment are at

least ten times as great as in that of Besch, whereas the foil was only

7

8

about one-third as large as Besch's. Clearly, a cavity flow is basically

unsteady and oscillates at a characteristic frequency which is a function

of cavity length and flow speed.

The frequency spectrum analysis also provided a rough estimate of

the amplitude of the vibration at the natural frequency of the cavity_

Figure 14 shows the amplitude in volts as a function of the relative

cavity length. No attempt was made to relate the voltage output to the

real physical quantities because of the questionable accuracy involved in

this particular case. The figure is intended to show only the general

trend that the amplitude is largest when the average cavity length is

nearly equal to one.

2. Two Types of Flutter-Like Vibrations

As was noted by Besch [7J, the signals from lift and moment trans­

ducers often contained more than one frequency component of large-ampli­

tude oscillations. However, practically no attention has been given by

previous investigators to the significance of these vibrations. In most

cases they have not even recognized the differences between these phe­

nomena, which are primarily hydroelastic, and other types of vibration,

which are primarily hydrodynamic.

Typical oscillographic records taken for a symmetrical wedge for

three average cavity lengths~-t/C ~ 1/6, 1/2, and l--are shown in Fig. 15.

Nearly sinusoidal vibrations in the neighborhood of 315 Hz in both lift

and moment are clearly shown for all three cases. It is significant that

the reduced frequency, as shown in Fig. 16, is nearly independent of the

cavity length but appears to vary with the stiffness of the supporting

springs. Furthermore, the frequency of the vibration is of the same

order of magnitude as the system's coupled primary frequency, denoted

f l ' in Fig. 16, but an order of magnitude greater than the natural fre­

quency of the cavity previously discussed. The amplitude of the vibra­

tion, however, is strongly dependent on the cavity length. Generally, as

the cavity length increases, the amplitude first decreases to a minimum

at around mid-chord and then increases to a maximum at around one chord, as

clearly suggested by the records shown in Fig. 15. In some cases, however,

the amplitude decreases monotonically with the cavity length and fails to

9

have a severe vibration when the cavity length is equal to one chord. Rela­

tively stable flow is likely to oCQur for a symmetrical wedge a,t small attack

angles with relatively stiff torsional restraint. Indeed, for some experi­

ments it appeared that the critical flutter speed might have been obtained.

Because of the time limitation and other factors to be discussed later, no

precise determination of the flutter boundary was obtained. For the sym­

metrical wedge at a 5-degree attack angle and a one-chord cavity length, the

critical flutter speed, U/wC, appeared to be in the neighborhood of 0.08

(an order of magnitude smaller than that computed by Besch). If the phe­

nomenon could be shown to be classical flutter, then the critical flutter

,speed would have to be very sensitive to the attack angle, the surface

profile, and the point of separation.

Using the present elastic system it was not possible to obtain a

stable flow for the non-symmetrical wedge shown in Fig. 2(b) at a 5-degree

attack angle when the flow separated from the leo9.dingedge. Contrarily, a

limited number of experiments carried out for the type of foil shown in

Figs. 2(c) and 2(d), wherein separation was forced to occur at a small

distance downstream of the leading edge, failed to produce any vibration

of the type discussed herein. However, severe vibrations occurred as soon

as the separation point moved back to the leo9.ding edge while all other

variables were kept practically unchanged. This strong influence of the

separation point on the critical flutter speed has o9.lso been reported [8J

for the case of supercavitating hydrofoils.

A second type of vib:r.o9.tion was found to occur when the non-symmetrical

wedge, Fig. 2(b), was tested. It Wo9.S also a neo9.rly sinusoidal vibration,

o9.S shown in Fig. 17. As with the other flutter-like vibration, the fre­

quency was independent of the cavity length, but the amplitude varied

considerably with the cavity length. Fortunately, the two types of flut­

ter-like vibro9.tions 'W'ere readily distinguishable because the last type

vibrated at a higher frequency (about 500 Hz). Furthermore, the amplitude

varied in such a way that the maximum occurred when the cavity length was

approximately equal to 0.4 chord.

Figure 18 illustrates the fact that the reduced frequency of the

vibration is independent of the cavity length, but dependent on the stiff­

ness of the system (natural frequency in the second mode, f2" in this

10

case) • As might be observed from the typical records presented in Figs'.

15 and 17, the two types of vibrations are of different modes. This is

clearly seen in Fig. 19, where the amplitude ratio,

Cl\ A* = 1\r

is plotted for the two types of vibrations. Here AL and 1\r repre­

sent, respectively, the amplitude of the lift and the moment. It is

also interesting to compare the frequency ratio, f l /f2, of the two

types of vibration with that of the two fundamental frequencies of the

system in air, f l '/f2 ', as shown in Fig. 20. Data taken for the non­

symmetrical wedge for which both fl and f2 existed are shown as

circles in this figure. Included in the same figure also are similar

data for the symmetrical wedges of different masses. Since there was

(4)

no f 2-type vibration for the symmetrical wedges, the corresponding val­

ue of f2 for the non-symmetrical wedge was used to calculate

the ratio f l /f2• It is noteworthy that straight lines of slope one

can be drawn to represent the data in this figure.

All: the evidence discussed above strongly suggests that the vibra­

tions were flutter--self-excited vibration in a natural mode of the sys­

tem. The occurrence of these vibrations, however, is strongly influenced

by many factors, such as surface profile, angle of attack, and cavity

length. This is a complication due to the cavity reattachment, which is

not present in supercavitating flows. Analytical prediction of the crit­

ical flutter speed surely would be very difficult. It appears only

natural that Besch [7J has found no agreement between data and a theory

which was rough and at best represented only the flow about a flat plate.

It may also be of interest to point out that the natural frequencies of

the system used in Besch's water tunnel experiment were too close to the

natural frequencies of the cavities to permit easy identification of the

basically different phenomena. In fact, a careful inspection of the data

presented in Figs. 25 and 27 of Ref. [7J reveals the existence of flutter­

like vibration when the cavity was nearly one chord long.

IV. CONCLUSIONS

The following conclusions may be drawn from this research program:

1. Although the shape and size of a fully developed cavity may

fluctuate substantially, the fluctuating components of force

and moment are small unless an excessive amount of ventilation

causes pulsation.

2. A cavitating flow is basically unstable, and there exists a

characteristic frequency at which the force and moment may

oscillate. This characteristic frequency is primarily a func­

tion of the cavity length and the flow speed. The reduced

frequency of the cavity vibration correlates very well 'with

the relative cavity length. Data obtained from three dif­

ferent facilities agreed reasonably well among themselves.

3. Hydroelastic instabilities (flutter-like vibrations) were also

found to occur under partially cavitating conditions.

4. In the case of a symmetrical wedge, flutter-like vibration may

occur for all cavity lengths ranging from very small to nearly

equal to one. The mode of the flutter was found to coincide

quite closely with the first natural mode of the two-degrees­

of-freedom system. The maximum amplitude of the vibration

frequently occurred when the cavity length was approximately

equal to the chord.

5. Flutter at the second natural mode of the two-degrees-of-freedom

system may also occur when a non-symmetrical wedge is used as

the test body. The maximum amplitude occurred when the'cavity

reattachment occurred near the point of an abrupt surface pro­

fi le change.

6. The frequency ratio of the two flutter-like vibrations, f l /f2,

was found to be linearly related to the ratio of the first two

natural frequencies measured in air, f l t /f2t •

7. The critical flutter speed of the partially cavitating flows

appeared to be very low. It was very difficult to determine

11

12

the critical flutter speed because it appeared to be sensitive to

the surface profile. the angle of attack, and the point of separa­

tion.

8. The existing theory is not accurate enough to predict flutter of

partially cavitating foils. A theory taking into account the

separation and the reattachment condition with considerable pre­

cision is probably needed.

9. Further experiments using a system permitting greater and more

precise variation of the hydroelastic and hydrodynamic variables

are needed for complete understanding of the problem.

-- -- -- -- -- -- -- -- --

13

LIST OF REFERENCES

[lJ Tulin, M. P., Steady Two-Dimensional Cavity Flows about Slender BodieS 2

David Taylor Model Basin Report 834, 1953.

[2J Silberman, E. and Song, C. S., "Instability of Ventilated Cavities," Journal of Ship Research, Vol. 5, No.1, June 1961.

[3J Song, C. S., "Pulsation of Ventilated Cavities," Journal of Ship Research, Vol. 5, No.4, March 1962.

[4J Meijer, M. C., "Some Experiments on Partly Cavitating Hydrofoils," International Shipbuilding Progress, Vol. 6, No. 60, 1959.

[5J Wade, R. B. and Acosta, A. J., "Experimental Observations on the Flow Past a Plano-Convex Hydrofoil," J. Eng. for Power, ASME, No. 65-FE3, 1965.

[6J

[7J Besch, Peter K., Flutter and Cavity-Induced Oscillation of a Two­Degree-of-Freedom Hydrofoil in Two-Dimensional Cavitating Flow,

. Naval Ship Research and Development Center, Department of the Navy, Report 3000, April 1969.

[8J Song, C. S. and Almo, John, An Experimental Study of the Hydroelastic Instability of Supercavitating Eydrofoils, University of Minnesota, st. Anthony Falls Hydraulic Laboratory Project Report No. 89, February 1967.

EIQQB:m§ (1 through 20)

1. Viewing Window

2. Air Intake for Tunnel Pressure Control

3. Plastic Housing for Spring System

4. Spring System

5. Venl'ilation Valve

Fig. 1 - A View of the Two-Dimensional Test Section of the Free-Jet Water Tunnel

17

18

(a) Simple Wedge

tE-.j . ~----I

(cf Stepped Wedge

r- 11i +-1-1/2"----1

I~ J

(b) Modified Wedge

cl=::-. ------,1 ~ 111 -)oi+-~ -1-1/211~

(d) Notched Wedge

Fig_ 2 - Cross-sectional Profiles of Test Bodies

19

~ '-"--~"-'i

t == 0.1 sec

(a) Lift and moment vibration

, '~

'" ~

I,Ji

1111

I

- .1 ,1 I .M. LI. lid

'T I Iii'

r ,. t! " i I I". M .. I-... I

I """----.. - ..• ~---.. -.. - .. - •• - .. ---j , t == 0.1 sec

(b) Predominantly I ift vibration

Fig. 3 - Typical record of Free-Vibration Test

8 x

tt ... .....

s:: .~ u E

CD 0

U ..... .... . -..J

0) . s:: .-.....

0 j ..... u j

u. .... 0

V')

~ IX

5 '---~-~---~I - r -~

;Lr= t =:j o a = 7-1/20

4r ~~ o a = 8-1/20 a c-';

3r 0 0

2 0 0 0 0 0

0 0 0 0 0

0 0

0 8

0Q' I I I I I I 2 4 6 :8' 10 12 1~

Cavity length in Chords, .t/C

Fig o 4 - RMS Values of Fluctuating Lift Coefficient for Fully Developed Cavities

--I:

16

N o

0 0

x

li ... ......

c Q)

u 4: ......

Q) o

U 0> o ....

Q

0> C

"'-o :::l ...... U :::l

u.. ...... o

V')

~ ~

-----­,:--~"""

100 I -~-----------------,

008

006

0.4

I

0.2i

00

o

o

0 0

8 0

0 0

2 4

o

o

o

o

o

8

o a=7-1/2°

C a = 8-1/20

-"----', _______ ~ _____ __i_

6 - 8 10 12 14 Cavity length in Chords, ~/C

Figo 5 - RMS Values of Fluctuating Drag Coefficient for Fully Developed Cavities

16

N I-'

'I

0 0051 0

x

o.J ~ .. ....

s:: .~ 0.3" u tE

Q) 0

u .... s:: Q)

E 0 0.2~ ~ 0) c .... 0 :::l .... U :::l

LI- 0.1 t-~

0

Vl

~ 0:::

01 0

a = 7-1/2 0

a = 8-1/20

0 0 0

0 O· 0

0 [Q 0

0

0 0 DO 0

0

I I I ! I I I 2 4 6 8 10 12 14

Cavity length in Chords, l/C

Fig. 6 - RMS Values of Fluctuating Moment Coefficient for Fully Developed Cavities

16

!\) !\)

8 .-

>< ...J

U <1

... c:; 0 .-.... e I/)

:l I:l..

.... "-.,.. ...J

"-0

>.. .t I/) c:; CLl ....

..E'

12r-------~1--------~1------~,--------r-1----~

10 0 - -

0 Two-Dimensional Wedge

8 - ,-

0 0

0 6 - -

0

0

4 - 0 -

0

2 !- -

o ~------_I~----~_~I------_~I------_~I------~ o 0.1 0.2 0.3 0.4 0.5

Inverse of C~vity Length, CIl,

Fig. 7 .. Intensity of Cavity Pulsation as a function of Cavity Length for Ventilated Cavities

23

24-

100 ----

f ::; 284 Hz

o O~----------~--~1~-~'--------~----~2--------------~3 t)( 104/64 sec

1.0

0.5

f = 14 Hz

o o 3 2 ,. x 1 0 /64 sec

3

Fig. 8 - Autqcorrelation of FluctlJatlng. Lift of Non-Pulsating Cavity , ",

Ule = 2, a = 12°~ h = 0.690) v

u I

i o.

o II t-

0 0 -+

II

co -.0 • • 0 0

+ +

'lilt ~ 0 ~ • • • 0 0 0

+ + I

rr

'lilt -.c) co 0 • • 0 0

0 0 0 -I I • I

25

....

0

~ -.0

o o II

b :>

.. o ~ ,.... II c

o

.~ u.

26

lA--.--~--~------~--------T-~----~------~

1.2 ~~ tao·~

C . • u ....

Q)

.. 1.0 0

~ 0

0 ~

ti°.a

A .. 0

; 0 ::l 0.6 A ~ V) 0 ~ Q)

.~ 1; 0.4 Q) 0 Crl: B 0

0.2

Moment

0.4 0.8 1.2 1,,6

Cavity length in Chords, tiC

Fig_ 10 - Relative Intensity of -Fluctuation for Short Cavities

27

1.6

0

1.4

~1.2 0

tf . ~ 1.0

II)

Q) ;:)

o o o o 0

> '" ~

0~8 . Q)

.2: .... 0 Q)

~

0.6

0.4 0.8 1.2 2.0 . Cavity length in Chords, tiC

Fig. 11 - Relative Intensity of Fluctuating Lift of 'a Stable Cavity

0.6

-u 1- 0.5 ,-0 .::>

, ;:.... u c .. ::>

g- 0.4 L. '-

""0 II) u ::>

""0 II)

a::: 0.3

0.2

0.10•2

o

004

o o

o 00

0.6

o SAF-

-0 Besch In

D kaplan and Lehman 16]

0

0 0

0 0

0 0 B 0

BOO

o

0.8 1.0

Cavity length in Chords, .tIC

0

o o

L2

o 0° o

o 1..4

o

Fig. 13 - Natural Frequency of Short Cavities as a function of Cavity LenQth

1.6 N '-0

'" .... o > c

(J) -0 :J ....

0.2

~ 0 0 1 <

00•2 004 006

" "

0 0 8 LO 1.2 104

Average cavity length in Chords, tiC

1'..6 1.8

Fig. 14 - Relative Intensity of Self-Excited Cavity Vibration as a function of Cavity Length

2.0

\..V o

o '-0. r-

. ....-...0 'I I:

~.-~

....­r-

...0 ~

-I ,.;

--

l(") .,..

...0 'I I:

~·-o ....

31

, ...... , .... ljiilllllUJWnflm'M

~~!! ~ I ~~~)! '1:\ :~ Iii ~ II if '~ I ill iii W ~i I

~=-:t_. __ .. _ ... ~ ~ ______ ._~---I t :::: 001 sec

(a) ~/c:::: 1/6

k--t :::: D. 1 sec

(b)~/c::::1/2

~------- . ---~----l t :::: .0.1 sec

(c) ~ / c :::: 1

Fig. 15 - Typical records show ing a FI utter-Like Vibration for a Symmetri cal Wedge (Type 1) .

32

6- f I

1 ::: 365 Hz

0 f I 1 == 310 Hz

3.0 0 f I = 255 Hz 1

2,,5

U I .::::> ...:::-

~ II 6 0

-::I. 2.0' 0 0 0 ...

0 >-" 0 u 0 c: Q) ::J 0" Q) ... 1.5 ....

"'tJ Q) U ::J

"'tJ Q) ~

1 .. 0 -L \

50

0.5

0.4 0.6 0.8 LO Cavity length in Chords, t Ie

Fig. 16 - Reduced Frequency of a Flutter-I ike Vibration, Type 1

...c -0-,

-I 0

1.0 C")

...c I c :::;! .;....1.0 -0

0

1.0

...c -0

-I 0

-0 C")

...c I c

:::;! '--0 0

1.0

----

-

-----

-

i-oIl .. ~--· .-~------~ t :::: 0 0 1 sec

(a) "/ c :::: 1/5

(b) "/ c :::: 2/5

I-I--------i t :::: 0.1 sec

t :::: O. 1 sec

Fig. 17 - A Typical Record of a Flutter-like Vibration (Type 2) for a Non-Symmetrical Wedge

33

34

405~~----~----~-T------~--------~------'-------~

6. f11 = 650 Hz

o fll = 610Hz 4.0 o fll = 575 Hz

305 0 6 6, . 6,

0 0,: I :::> 0 0

..... C'l 0 0 0

II 3.0 0 C'l

.~

.. >. 0 c CI) :::l

~ 205 ..... "'0

CI) 0

~ :::l

--L "'0 CI)

0::

2.0 \ ~ 50

1.5

0.2 0.4 0.6 0.8 1.0

Cavity length in Chords, t / c .

Fig. 18 - Reduced Frequency of a Flutter-Like Vibration, Type 2

12.5~i ------.------,-------r------~----~------I

10

-JI ~ « « u II

'" 7.5

« .~ .... c .... Q)

5~ "'C :::> .... a.. E «

I 2.5

00

6

0

0.2

6 Type 2 v ibrati on

0

o Type 1 vibration

0.4 0.6 Cavity length in Chords r

0.8 tiC

6

0

1.,0

Figo 19 - Amplitude ratio of Flutter-I ike Vibrations

0

1.2

'v) \.n

N O•8

'--......

c:: 0

I .--0 "- 0.7

1 -8 > ...... 0 on CI) a... >-..... 0 0.6 ~

CI) ..s:: .... ...... 0

0 0.5 '.0: 0

0=::

>-U c:: Q) ::)

0" CI) ...

0.40 •2 u..

0

.D.

/ 0 //

0.3 0.4 0.5 0.6

Ratio of Two Fundamental Frequencies of System in Air

Fig. 20 - Comparison of Frequency Ratios

-'

-brass

0.7 f Ilf I 1 2

::oJ

0.8

\...V 0'..

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Unclassifie<3, Security Classification

DOCUMENT CONTROl.. DATA· R&D (Security dasslflcation of title, body of abstract and Indexing annotation must be enterl;>d wh~n the ov<>rall report Is dasslfiedJ

1. ORIGINATING ACTIVITY (Corporatl;> author) ~". R~PORT S~CURITY CI-ASSIFICATION

St. Anthony Falls Hydraulio Laboratory Unclassified University of Minnesota 2b. GROUP

3. R~PORT TITI-.~

VIBRATION OF CAVITATING HYDROFOILS

4. P~SCRIPT1Vf;: NOTf;:S (Type of report and.inplu~lve dif4tes)

Final Report - October 1, 1967 to June 30, 1969 5. AUTHORIS) (First naml', midr1lfJ Inlital, IM'tnamr»

Charles C. S. Song

6. RIOPORT r;>ATf;: 7a. TOTAl-. NO. OF PAGIOS 17b. NO. O~ Rf;:PS

October 1969 36 8a. CONTRAcT OR GRANT NO. 9a. O'RIGINATOR'S RIOPORT NUM13IOR(S)

Nb001~-67-A-0113-0005 b. PROJIOCT NO. Project Report No. 111 SF 013 02 01

c. 9b. OTHIOR REPORT NO(S) (Any other numbers that may be assigned thi I' report) ,

d.

10. PISTRI13UTION STATf;:Mf;:NT

,This document has been approved for public release and sale; its distribution is unlimited.

11. SUPPI-~MIONTARY NOTf;:s 12. SPONSORING MII-ITARY ACTIVITY

Naval Ship Research and Development Center Washington, D. C. 20007

13. A13STRACT

A primarily experimental research program has been carried out using a free-jet water tunnel for the purpose of studying force and moment fluctuations on cavitating two-dimensional hydrofoils. Both a sYmmetrical wedge and· a non-symmetrical wedge were tested for a wide 'range of cavity lengths and several different elastic conditions. Fluctuations in lift and moment were of primary concern in the experiments.

It was revealed that the force and moment were quite steady if the cavity was longer than two chords unless an excessive amount of ventilation caused cavity pUlsations. For a shorter cavity, however, the flow was generally very unstable, and severe vibrations were noted. A cavity of any length was found to be basically unstable and to oscillate at a characteristic fr€quency which was primarily a function of the cavity length. The vibrating cavity may cause an elastically supported foil to vibrate severely when the cavity is short. The largest-amplitude vibration often occurred when the cavity length was approxi-mately equal to one chord. Flutter-like vibrations were observed in the first and second natural modes of the two-degree-of-freedom system. The frequency of these vibrations Was found to be practically independent of the cavity length. The severest vibration in the first natural mode usually occurred when the average velocity length was approximately equal to a chord, whereas vibration in the second natural mode was found more likely to occur when the cavity was very short.

DO ,F~oR~6514 73 (PAGE 1) . Unclassified

SIN 0101-807-6811 Security Classification A- 31408

Unclassified Security Classification

14. LINK A LIN K B l.INK C K gy WORDS

ROLE WT ROLE WT ROL e: WT

Hydrofoil, cavitating foils

Flutter, vibrating foils

Vibrating cavities, hydroelasticity

Instability of cavities

Unclassified SIN 0101-807-6821 Security Classification A-31409