Unsteady cavitation of oscillating hydrofoil

ri

Unsteady Cavitation of Oscillating Hydrofoil

Technische Hogschoi

Deift

Unsteady Cavitation of Oscillating Hydrofoil

1. Introduction

Cavitation erosion and bent trailing edge of marine

propeller blades are considered to be mainly caused by the

unsteady cavitation, which results from the operation of a propeller in a non-uniform velocity field behind a ship

hull. For example, cavitation erosion is usually not

ob-served on the blades of a propeller working in a uniform

flow, even though the propeller blades are covered by

extensive cavitation. But if a model propeller is tested in a non-uniform flow simulated in the cavitation tunnel after

List of symbols

a : pitching center of a wing measured from the

leading edge

a a parameter which expresses the pressure varia-tion

C(k) Theodorsen function CL :lift coefficient

CM : moment coefficient around the mid-chord (posi-tive when the leading edge goes down)

C,, : pressure coefficient

pressure coefficient for the pressure due to the

unsteady motion of the wing

C,,5 : pressure coefficient for the pressure due to the

steady motion of the wing D : diameter of a bubble

il : nondimensionalized diameter of a bubble

=D/(\/a/2X)

F(fl : functional expression for the pressure variation

¡j(2) _{Hankel function of the second kind (order n)}

k : reduced frequency w //2v I: chord length

P :static pressure at the wing surface

P0 static pressure in the uniform flow

H. Tanlbayashi*

N. Chiba**

Bent trailing edges and erosion, which are often seen on marine propel/er blades, are considered to be mainly due to unsteady cavitation caused by the non-uniformity of the velocity field behind ship's hull. Theoretical and experimental investigations have so far been made into the mechanism of unsteady cavitation and the method of their prevention, however, many problems still remain

unsolved because of the complicated nature of the phenomenon.

As an approach to the study of unsteady cavitation of marine propellers, we carried out cavitation tests on a 2-dimensional oscillating hydrofoil which presents a relatively simple cavitation phenomenon. This report gives an outline of the test results and

discussions.

Cavitation on the hydrofoil was observed by the use of a high-speed camera as well as a strobo-light. The behaviour of the cavities

lust before collapse was quite similar to cloud cavitation which is observed in propeller cavitation test in non-uniform flow. The motions of the cavitiesinception, growth and collapsewere analyzed in the light of theoretical calculation and useful data were

obtained.

the wake distribution measured on a ship model, a

cavita-tion test of 30 minutes is enough to cause the cavitacavita-tion

erosion or the bent trailing edge on its blades.

Theoretical and experimental researches have been made

to clear the mechanism of the unsteady cavitation or to

find out the method for the prevention of cavitation

damages, and some useful key informations have been obtained. However, as the problem is so complicated and

is associated with both the unsteady flow around a

propel-ler and the physical phenomena of inception, growth and

1v pressure inside a bubble

P(t) : time-dependent pressure around a bubble LPalX) pressure distribution of the wing

R radius of a bubble

R0 : initial radius of a bubble

t: thickness of a wing time

time when a bubble reached the leading edge

velocity of the uniform flow

x coordinate in the direction of the uniform flow coordinate of the bubble at the time t

X, X' nondimensionalized coordinate (X = x//1, li, X'

= xli')

s' : coordinate in the direction perpendicular to the

uniform flow

a : half-amplitude of pitching

X _{the length of the region where the pressure on}

the wing surface is negative

the local coordinate used in the region where

the pressure on the wing surface is negative

o cavitation number

surface tension

p density of water

w circular frequency of the oscillation

120 -1 Model Wing II N Gauge plates to measure litt and moment

Fig. 3 Wing section tested

collapse of cavitation bubbles, there remain many problems to be solved.

In the tests reported here, we simplified the unsteady

flow field of a propeller by that of an oscillating

two-dimensional wing and tried to measure the fluctuating

force and to observe the occurrence and collapse of cavita-tion bubbles.

Cavitation tests of an oscillating wing have already been

done by ito(1), who gave a heaving motion to a

three-dimensional wing and observed the cavitation near the blade tip by the use of the strobo-light. He reported that

the cloud cavitation, which is responsible to cavitation damage, was observed when the reduced frequency was

between 0.3 to 0.4.

In our tests, we gave the pitching motion around the mid-chord of the wing, because this mode of oscillation

caused a relatively large pressure change over the wing surface. Cavitation phenomena, i.e., inception, growth and collapse of bubbles, were observed not only by the

strobo-light but also by the high-speed photography. Since the

mode of the wing motion used here is only one case, it will

not be enough to apply these data directly to the cloud cavitation of a propeller, but will be still useful to promote

the study of unsteady cavitation. 2. Test method

2.1 Apparatus

Pitching motion was given to the wing by

vertically-oscillating four slender rods to which the model wing was connected at the corner points. The pitching center was

Driving motor

§

Fig. 4 Co-ordinate for theoretical calculation

easily changed by controlling the phase of the oscillation of

two frontal rods and two rear rods separately. The outline of the apparatus is shown in Figs. i and 2. The

vertically-oscillating motion was given by a 0.6 PS D-C motor set on the top of the measuring section through the

scotch-yoke, which converts the rotational motion of the motor into the reciprocal motion of the rods. To reproduce the two-dimensional flow around the test wing, four rods and

the end parts of the model wing were covered by the streamlined plates. The connection of the wing to the rods was done by pins through the connecting plates.

Lift and moment of the wing was measured by the use of

strain gauges stuck to the connecting plates.

2.2 Model wing

A rectangular wing was used in this test. The wi-tg

sec-tion is the blade section of the Troost-type propeller at

0.7 radius. (Fig. 3)

Span b = 120 mm (4.72")

Chord length

i

= 100 mm (3.94")

Thick-chord ratio t/i= 0.1 (constant over the span> The model wing is made of brass and connected to the

con-necting plate by the pins at one fourth and three fourth

chord-points from the leading edge.

2.3 Observation of cavitation

Cavitation was observed by the use of strobo-light

Fig. 1 Test apparatus Fig. 2 Test apparatus

75

Lead ing Trailing

which was synchronized with the pitching motion of the

wing. At the same time, to study the cavitation at each

phase of the pitching motion in detail, we used a high speed camera and film with high sensitivity as follows.

Camera: Fastax . .4000 frames/sec Film: Eastman Kodak 4-X -. . ASA 400

To cover the lack of distinction which is sometimes un-avoidable for the high-speed photography, we also used the standard camera, Zenza-Bronica, especially to record

the shape of the collapsing bubbles.

3. Calculation of pressure distribution, lift and moment of

wino by two-dimensional unsteady airfoil theory

The analytical methods to calculate

the flow field

around a two-dimensional wing oscillating in an ideal fluid

have been already reported by Küssner(2) _{and others.}

Even though these theories are all developed for a wing

oscillating without cavitation, these may be useful to check

the test results, as the theory for an oscillating wing with cavitation is still far beyond our knowledge. In the

follow-ing, such a theory by Küssner is applied for a wing with

pitching motion.

Fig. 4 shows the co-ordinate used in the following

theoretical calculation. The motion of the wing is defined

as follows.

y=(x_a).et

(1)

If we put x = - -i- cos O, the pressure distribution is

expres-sed as follows, J-(0) _.

_iwt

P _1/2pv2 e APa(0) O _J' 4a 1/2pv2

-

coi--C(k)2+ik(1

-

¡

+{cot--_4sinO(1_

k2asinOcos0

(3) where j.j( 2)( k)

C(k)

-H1(2)(k) + iH0-(k)

The lift and the moment coefficients around the mid-chord

are expressed as follows.

.2ak

CL =

[C(k)2+ik(1 -)} ik(1

i)] a

elWl

CM=---

[C(k)

4. Tests and results

Results of some preliminary tests showed that cavitation

occurred severely in case of the pitching motion around the mid-chord and this mode was most convenient for the

observation of cavitation. All the measurements were

(2)

therefore conducted for this mode of pitching.

4.1 Measurement of lift and moment

The measurements of the lift and moment of the wing

were conducted for the conditions shown in Table 1.

The time-average attitude of the wing was chosen as the pressure side of the wing to be parallel to the uniform flow. The measured lift and moment were expressed as the fluctuating components from the time-average values,

i.e.,

r-,

-i-'

iwt

LL LL e

-

,-LMLM

e

where CL and are complex values, i.e.,

Cj=

ICLIexp

iarg(cj)}

C

Iexp

iarg(C)}

Then the non-dimensionalized

lift and moment, CjI/a

and ICMI/a are compared with the calculated values, as

shown in Figs. 5 (1) and (2). In these figures, phase angles

for both lift and moment, arg (C1) and arg are also shown.

In case of the non-cavitating condition (the static sure in the uniform flow is equal to the atmospheric pres-sure), the measured CL I/a and CM I/a are a little smaller than those calculated, while the phase angles, arg(j) and arg(), are in good agreement.

In case of the cavitating condition, (u = 0.476),

mea-sured C'LI/a and ICMI/a become a little smaller than those in non-cavitatirig condition, but the phase angles were not affected by the occurrence of cavitation.

The reasons why the measured lift and moment are

smaller than those calculated have not yet been well

under-stood. The deviation from the two-dimensional flow

con-dition and the effect of viscosity (due to the effect of

viscosity, the pressure peak at the leading edge does not reach the calculated one) may be responsible for the

discrepancy. However, since the change of the lift and moment with the reduced frequency obtained by the cal-culation shows the same tendency as that measured, this method of calculation will be useful for discussion of the

test data.

4.2 Observation of cavitation

Cavitation observation tests were conducted under the same test conditions as the measurements of the lift and

moment, i.e., u = 0.476 and y = 5.65 m/s. But the

ampli-tude of the pitching motion was increased up to 9.17

Table i Test conditions

3 4.6 0-1.86 3.39 atmospheric 4.6 O-1.12 5.65 atmospheric 4.6 0-1.12 5.65 0.476 Half Amp. e(deg. I k vIm/si a

degrees in order to make the observation easier. The

rela-tion between the frequency of the pitching morela-tion and

the reduced frequency is shown in Table 2.

Under these test conditions, bubble cavitation was observed on the suction side of the wing which collapsed near the trailing edge and turned into a number of tiny,

foamy bubbles. This type of collapse of cavity is considered

to be mostly associated with the erosion of the propeller blades. We call this type of cavitation "cloud cavitation"

after van Manen.

Each phase of the cavitation, i.e., inception, growth and collapse phase, was recorded by the use of a

high-speed camera and the relation between the size of bubbles and the phase of the wing motion was analyzed.

The reduced frequency k is considered to be one of the

most important parameters representing the nature of the

unsteady cavitation. In our test results, it was observed that, with increase of the reduced frequency k, the number

of the bubbles and their size increased, and the bubbles

collapsed more severely to form cloud cavitation.

However, because our test range is so narrow (the tests

were conducted only for one wing section-shape and by the use of only one mode of pitching), it will be premature

to draw a conclusion about the effect of the reduced

frequency on cloud cavitation from our test data. 5. Discussion on unsteady cavitation

In the following, unsteady cavitation observed on the oscillating wing is discussed based on the results of the

theoretical studies already reported. The instantaneous

pressure distribution of the pitching wing is calculated

according to the unsteady two-dimensional wing theory. Since the cavitation observed in our tests was all bubble

cavitation, the growth and the collapse of bubbles are dis-cussed by applying the theory for a single spherical bubble.

5.1 Calculation of the pressure variation around a

bubble

In order to study the relation between the growth and collapse of a bubble and the pressure variation around it, the pressure variation along the path of a flowing bubble has to be calculated from the instantaneous pressure dis-tribution of the oscillating wing. The pressure disdis-tribution

around an oscillating wing can be divided into two parts,

i.e., a steady and an unsteady part. The steady part of the

pressure distribution can be calculated by use of Moriya's

method for a non-cavitating wing, while the unsteady

parts can be calculated by Eqs. (2) and (3).

We assumed that the velocity of moving bubbles is

almost the same as the uniform flow, based on the results of the analysis of the high-speed photographs of bubbles.

. 5

10

Test condit,on

Fig. 5(1) Results of measurement of lift

o_

.

05 tO

Putting Eq. (7)_{into (2»}

V

s

Reduced frequer,gy k

Fig. 5(2) Results of measurement of moment

If the time when the bubble reached the leading edge of the wing is t, the position of the bubblex' on the wing

at the time lis expressed by

X' =

v(tt0)

-o

Mokon of wing yaO0 Moment CM='CMe coefficient .OlXang ,fl iniesured) 0.1 Xarg lCw) (cal. 1.5 20 P0(x) ei)(to+

X'+T

112pv2 t, 2.5 2.5

Introducing the non-dimensionalized co-ordinate as lollows, o 4.6' a3gm/s atm 4.6' 565m Is atm O 4.6' 5.65m/s 0.476 0 4.6' 3.39m/s atm 4.6 565m /s atm O 4.6' 565m Is 0.476

Table 2 Relation between frequency of pitching and reduced frequency Frequency n(lIs) Reduced frequency k 5.0 0.278 10.0 0.556 12.5 0.695 15.0 0.834 17.5 0.973 20.0 1.112 Thus,

_x,+-.

t = to + V IL 15 o ò Test conditions 15 lc o 2,0 Motion of w,ng 'I = 00e L ft coeff,c,ent C i.= Cc -0 05 1.0 1.5 Reduced f requengy k

- io o Leading edge k = 1.112 - 1.6 E E20 - mio-a' =43 Leading edge r4 k =0.6950 o o0 o O o 0.5 X Trailing edge LO E 20 -E n OE476 fia- I lo Leading edge (21 6=0.9732

tinguish the bubble. But the initial minimum size of a

bubble which begins to grow will be 0.01 0.03 mm

according to the equation of bubble dynamics

dis-cussed in the section 5.6.

5.3 Growth of bubbles

The cavitation observed in our tests is all so-called

bubble cavitation. The size of bubbles varies in a wide

range, but bubbles which collapsed into cloud cavitation

reached almost 20 mm in diameter and covered about 20%

of the chord. The bubbles in their growth looked like a

hem i-sphere and attached to the blade surface, as illustrated in Fig. 7.

Bubble

10 Trading edge

Fig. 6 Growth of bubbles and pressure variations around them

f20 f 10

Fig. 7 Shape of bubble

To analyze the motion of bubbles of this type of shape,

the equation for a single spherical bubble was simply

ap-plied, neglecting the mutual interaction of the bubbles.

Neglecting the surface tension, the equation of the

bubble motion is given as follows,

Rk+2=_.P_P(r)}

(12)

According to the results of our pressure measurements

of the partially cavitating wing, the pressure inside the

cavity was almost equal to the vapor pressure. So P, can be equalled toe, i.e.,

P=e

(13)

The independent variable t in Eq. (12) can be changed

to x' by the use of Eq. (16). Furthermore, if we use the

C,, and a defined by Leading edge (3i 6=0.8341 - 1.0 'a 61 X C Trailing edge n =0,476 ,en s o 35 1.0 rakg edge 5

x,=

(9)

x+-i-and rewriting C1 (x', ro + ) as C1(X', t0), we

ob-tain

C1(X'. t0)

= 1/2pv2 ewtoei2kX' Pa (X) (10)

So the pressure at a bubble is obtained by adding Eq.

(10) to the pressure of the steady part C3(X),

C(X',t0)=C3(X')+Ci(X',t0)

(11)

5.2 Inception of cavitation

Fig. 6 shows the change of the bubble size along the chord analyzed from the high-speed photographs and the

pressure variation around the bubble calculated. These

data show that the bubbles begin to grow a little down

stream of the position where the pressure coefficient plus the cavitation number is zero or

Ç, = -a. Ø

shown in

each figure of Fig. 6 is the phase angle of the pitching

motion of the wing at the time when the bubble reached

the leading edge. ( = O corresponds to the phase at which the incident angle of the wing is negative maximum.)

The reason why the visible cavitation inception point is

a little down stream of the point of C,, = -a may be

ex-plained by following three facts:

The actual minimum pressure will be a little larger than

that calculated (the lift and moment measured were a

little smaller than those calculated.)

The effect of the surface tension usually delays the

inception.

Very small bubbles were not analyzed because of the

difficulty to distinguish them on the film. If the

bubble size is smaller than 2 mm in diameter, on the film of the high-speed photographs, we could not

dis-s f20

n =0.476

f

10

10 =

Trailing edge Leading edge 61 9=0.2780

- O'o =42 _1.0

Trailing edge Leading edge

5 k =0.5560 na -42 - 1.0 e 0476 -O---0.5 0.5 10 f 20 f 10 f20 f 10

p - p0

= 1/2pv2

P0 - e

0=

1/2pv2

we obtain the pressure around the bubble

P

as follows,

P(x') = 1/2pv2 (C + o) +

1/pv2 (16)

Thus Eq. (12) is expressed as follows.

R

d2R 3(dR

)2 =-1(C+a)

(17)

dx'2 2

_dx'

2

In order to solve Eq. (17), the value of C,, in the right-hand side should be known. In case of the wing which is covered with steady sheet cavitation, it was shown by our

experiments that C1, = O. However, for the wing on which

the bubble cavitation is occurring, no methods have been

reported to estimate the pressure distribution. If we assume

C,, = G, then the right-hand side of Eq. (17) ¡s zero which

means the bubble will not begin to grow. Because C',,,

+ G < O is necessary for the growth of a bubble, we assumed in the following discussion that the pressure distribution of

the wing is not affected by the existence of the unsteady bubble cavitation. In our tests, O = 0.476, and e/}pv2

= 0.18, therefore P(x') = _pv2 (C,,, + 0.656) is obtained.

Thus if C,, < -0.656, the liquid suffers from the negative pressure. According to our observation, all the bubbles, which collapsed into cloud cavitation, passed through this

as

d D

-

a/2X

(22)

Eq. (21) can be expressed as follows,

+ f

()2

=

4F()

(23)

d

..d2

d2

The bubble size in growth, analyzed from the high-speed

photographs, was non-dimensionalized by the use of Eq.

(22) and plotted against the non-dimensionalized distance as shown in Fig. 9.

As shown in Fig. 9, aIl the growth curves obtained for

each reduced frequency k can be expressed by one mean

line. These mean lines for each k are shown in Fig. 10. Again, all the mean lines for each k can be expressed by one mean line.

In order to compare the mean growth curve with the solution of Eq. (23), we put

(24)

and integrated Eq. (23) by the following method.

d =

'4V1/

₂

i"

d

cj

=d+ci

As we found out in our preliminary calculations that

the solution is stable if ¿ 0.5 x 10', the calculation was done by the use of z = 0.5 x iO3. The initial values used in the calculation are,

=O

d0=0

d0 = 0.2, 0.02, 0.002, 0.0002

Fig. 11 shows the comparisons of the growth curve

obtained from Eq. (23) with that obtained by the

experi-ments. The bubbles which have the initial values d0 = 0.02

0.0002 became almost the same bubble size in their growth, except at the very early stage. Even for the bubble

with the initial diameter d0 = 0.2 the bubble size in its growth is not so far from that ford0= 0.02 0.0002. The

calculated curve of the bubble growth shows the same

tendency as that obtained by the experiments, but is about 2.5 times larger.

5.4 Effect of the surface tension on bubble growth One of the reasons of the quantitative discrepancy of the bubble growth curve between calculation and experiments may be the effect of the surface tension.

where

E =0 E=10

Fig. 8 Non-dimensional expression of pressure variation

negative pressure zone on the blade. The value of the

negative pressure was P = -887 kg/m2 when Cpmjn = 1.2.

Many papers have reported that the water can endure this level of the negative pressure for a relatively short time(6).

As seen from the Fig. 6,C,,+ O

'x

curve is very close to

a parabola, we used the parameters, shown in Fig. 8, to

express the pressure curve.

x'

= X -x0' (18)

where x0' isxcoordinate where C,,equals to -ci.

C+u=-aF()

(19) where

F(0) =F(1)

= O

max.F()

= 1.0

Instead of the radius, we use the diameter of a bubble,

D=2R

(20)

Putting Eqs. (18) and (19) into (17), we obtain,

D

+-()2

=4--F()

(21)

0.25 0.5 0.5 0.25 0.5 0.25 (1) 6=1.112 (3 5=0.834 0.5 1 .0 5=0.278 =0,5560 00,6950 k '0. 9731 Mean Une ,_-' =1.112 ,,k 0.8341 0.5

Fig. 10 Growth of bubbles (mean line)

1 .0 05 025 05 0.25 0.2 5 o (4( 6=0.695 (6( k0.278 0.5 (2 k =0.973 0.5 cl

e P(t)

Since a = 0.476, Cpmjn = 0.8 1.2, we obtain R0 = 0.006 0.014 mm. 10

The vapor pressure decreases from e to (e - IR),

which results in increase of the cavitation number.

The static pressure increases from P(î) to P(t)

+ /R.

Both effects restrain the occurrence and growth of the cavi-tation.

As to the inception of cavitaiton, if the right-hand side

of Eq. (25) is negative, i.e.,

(26)

the bubble will not begin to grow. So, for the inception of If we include the surface tension term in Eq. (12), we cavitation the radius of the initial bubble should be larger

get than some limiting value, i.e.,

RR + =

I

p -- -p(t) }

₍₂₅₎

where the surface tension is 74.5 x i0 kg/rn at 15°C

The effect of the surface tension can be explained apparent-ly in two ways.

(27)

7

o 05 1 .0

6=0.556

Fig. 9 Growth of bubbles

05 1 .0 0.5 1,0 arr 1 .0 0.75 0.5 0.7 5 0.5 0.25

l4

Calculation

Without surface tension

Fig. 11 Comparison of growth of bubble between measure-ment and calculation

Thus the initial bubble diameter may have been 0.01 - 0.03 mm in our tests.

To check the effect of the surface tension on the growth

of a bubble, Eq. (25) was integrated for one bubble (D0

= 0.02 mm), which is observed at k = 0.973 and compared

with the mean curve obtained by the experiments, as

shown in Fig. 12.

By the inclusion of the effect of surface tension in

cal-culation, the calculated growth curve became very close to the mean growth curve experimentally obtained, but there remains a room to check for further examination.

Effect of the surface tension is a very important factor

when we consider the scale effect of cavitation. If the effect

of the surface tension is small, it is reasonable to assume

that the size of the bubble is proportional to the model

size. But, especially in model tests where the length scale is

usually very small in comparison with that of prototype,

the size of the cavity is very small, which means the cavita-tion is very sensitive to surface tension. Thus it is very important to use a model as large as possible for the tests of cavitation observation.

5.5 Collapse of bubbles

Two kinds of cavitation were observed which collapsed near the trailing edge of the wing: one collapsed into cloud cavitation while the other simply disappeared without

collapse.

The bubbles, which disappeared near the trailing edge, were usually small in size and the pressure around them was

also low. The bubbles shrank gradually into a shape very

different from the spherical shape.

On the other hand, the bubbles which collapsed into cloud cavitation, grew very rapidly and reached almost more than 20 mm in diameter, which collapsed due to the

sharp pressure increase near the trailing edge.

2.0

C

o

'L _tension)

4

Bubble diameter ' _{With surface} Calculation i ' _{tension I} OD2rr .0 S' V" Bubble d,ameter I Eoperinrent I e-0476

Fig. 12 Effect of surface tension on growth of bubble

To see the effect of the pressure gradient on the col-lapse of cavitation bubbles, we calculated the pressure gradient at the position where C,, = u s reached for both

types of cavitation at each reduced frequency. From these data, the maximum and the minimum pressure gra-dient within which cavitation bubble collapsed into cloud

cavitation were found, as shown in the Table 3. The critical

pressure gradient for the collapse of cavitation can be said to be dC/dX' = 2.4 in our tests, which correspond

to the change of the static pressure of cl"

= 2.2 x iO5 kg/m2s

This result does not always mean that the cloud

cavi-tation occurs anytime if this static pressure change is

attained. For example, in case of the steady cavitation,

this pressure gradient will be easily obtained by simply increasing the uniform velocity. However, the minimum

pressure is usually determined by Cpmin = - u in case of the

steady cavitation, the bubbles will not grow in large size

Table 3 Critical pressure gradient for the occurrence of cloud cavitation k 1.112 0.973 0.834 0.695 0.556 dCp/df Maximum 2.8 2.4 3.0 2.9 3.0

and, as the result, will not collapse as in the case of our

tests.

This means that in case of steady cavitation, the static

pressure does not fall below the vapor pressure. But in case

of unsteady cavitation, it falls below the vapour pressure

and becomes negative, so the' bubbles grow rapidly and

vio-lently, which results in the emission of high energy at the

collapse.

The position at which the bubbles collapse is just

be-40 E E 30 E 20 10 O 1,0 Trailing edge 0,5 Leoding edge Minimum 2.2 1.7 2.1 2.4 2.4

a

tween the point where the pressure increases again to

C,, = - u and the trailing edge, as shown in Fig. 13.

Fig. 14 shows the typical phase of bubble collapse,

photographed by the use of the single lens reflex camera,

i.e.,

The bubble is just before the beginning of growth. The

bubbles look like hemi-spherical and attached to the

wing surface.

The bubble reaches the maximum size.

The bubble begins to collapse. The downstream side of the bubble becomes unstable and disturbed.

The bubble continues to collapse. The shape of the

bubble is so deformed and so disturbed that ¡t is hard

to say that this is a bubble.

The bubble collapsed into cloud cavitation.

Judging from the phase (c), the pressure around the

bubble is not uniform. On the contrary, due to the pressure

increase towards the downstream side and towards the normal direction of the wing surface, the bubble collapses

into the shape just like a ball hit on one side, as

schemati-cally illustrated in Fig. 15. In fact, according to the

high-speed photographs, the bubble moving along the wing

sur-face collapsed into the shape of an ellipsoid with the long

axis perpendicular to the flow direction.

We checked the surface of the wing at the point where

the cloud cavitation was observed, but no erosion was

found against expectation. This may be because

(1) the material of the wing is brass, which is much

strong-er to the cavitation strong-erosion than the anti-corrosion

aluminum usually used for the model propellers

the static pressure level was so low that the energy

contained in the bubble was small and the test time was too short.

Flow direction

Fig. 13 Location of bubble collapse Disturbance

Fig. 14 Photographs of bubbles

- ce Cloud cavitation Range where bubbles Cellapse 1.0

Fig. 15 Sketch of bubble collapse

6. Concluding remarks

The test results and the following discussions based on the calculation can be summarized as follows:

The unsteady two-dimensional wing theory was found to be useful for discussing the results of these kinds of cavitation test. (Range of the reduced frequency k was

O'-'-1.86).

The cloud cavitation, which is usually observed on the

model propeller tested in the non-uniform flow, was simulated on the two-dimensional wing with pitching

motion.

The unsteady cavitation appeared at a little

down-stream of the point where the pressure, calculated by the theory neglecting the effect of the cavitation on

the pressure distribution, becomes the vapor pressure,

or Ç = o.

(1) Ito T., Study on the Unsteady Cavitation of the Marine

Propel-lers, Journal S.N.A.J.. Vol.111, 1962 (in Japanese>.

12) Küssner G. G., Zusammenfassender Bericht über den

insta-tionären Antrieb von Flügeln, Luftfahrtforschung, Teil 13

(1935>.

(3) Van Manen J. D., Fundamentals of Ship Resistance and

Propul-References

One of the necessary conditions for the occurrence of

the cloud cavitation seems to be the rapid growth of

the bubble and the steep increase of the pressure

around the bubble soon after the bubble reaches its maximum size. In our tests, the pressure gradient for

the occurrence of cloud cavitation was 2.2 kg/m2-s.

The shape of the bubble at the collapse stage is so complicated that it looks very difficult to calculate by

a simple theory.

According to our test results, the critical reduced

fre-quency, which is proposed for the occurrence of cloud

cavitation by Ito, was not recognized. Instead, the

motion of the bubbles seems to be explained by the pressure distribution calculated for the cavitation-free condition.

sion, Part B, NSMB.

14) Moriya T., Introduction to the Aerodynamics, Baifukan (in Japanese>.

(5) Plesset M. S., The Dynamics of Cavitation Bubbles, Journal of

Applied Mechanics, (1949-9).