The influence of gas nuclei sixe distribution on transient cavitation near inception

Project Report No. 107

The Influence of Gas Nuclei Size Distribution

on Transient Cavitation Near Inception

by

FRANK R. SCHIEBE

tj.S HYDRA

4

4°40014111Pit

This research was carried out under the Naval Ship Systems Command

General Hydromechanics Research Program SR 009 01 01, administered

by the Naval Ship Research cmd Development Center. Prepared under

the Office of Naval Research Contract Nonr 710 (67).

MAY 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

ABSTRACT

This report concerns research on the problem of transient cavita-tion on hydrodynamic bodies near incepcavita-tion condicavita-tions. A probabilistic model of the process is presented. The experimental apparatus for pro-viding the bubble size distribution, which is a necessary input to the probabilistic model, is described. An acoustic cavitation occurrence counting system that can be used to check on the output of the proba-bilistic model is also described. Experiments conducted on a half body in a 6-inch water tunnel test system achieved fair agreement between the pre-dictions of the probabilistic model and the measurements by the occurrence counting technique in some cases.

Page

Abstract

List of Illustrations vii

INTRODUCTION

A PROBABILISTIC MODEL OF TRANSIENT CAVITATION NEAR INCEPTION 2

THE DETERMINATION OF THE GAS BUBBLE SIZE DISTRIBUTION ₉

CAVITATION OCCURRENCE COUNTING 19

DISCUSSION OF THE EXPERIMENTAL RESULTS 21

List of References 26

Figures 1 through 18 31

Appendix - An Electronic Method of Determining the Bubble Nuclei

Size Distribution (with 2 accompanying figures

LIST OF ILLUSTRATIONS

Figure Page

1 Axisymmetric Half-Body with Streamlines . . . 31

2 Axisymmetric Half-Body with Isobars . . . 32

3 Critical Bubble Size ₃₃

4 Computed Probable Cavitation Occurrence Characteristics of an Axisymmetric Half-Body when Nuclei Follow the Stream Tubes

(Method I) 34

5 Computed Probable Cavitation Occurrence Characteristics of an

Axisymmetric Half-Body when Nuclei Cross Streamlines (Method II) 35

6 Comparison of Methods for the Axisymmetric Half-Body 36

7a Attenuation Function Templates ₃₇

7h Plot of Experimental Data 38

8 Measured Bubble Size Distribution, °/o Saturation 54.8

9 Schematic Diagram for Attenuation Measurements 40

10 Transmitter Construction ₄₁

11 Transmitter Element Connectors 41

12 2:1 Modified Ellipsoidal Headfort With Isobars . . 42

13 2:1 Modified Ellipsoidal Headform with Streamlines 43

14a,b Measured Cavitation Occurrence Characteristics for the 2:1

Modified Ellipsoidal Headform . . .

. ... . .. .

44

14c,d 45

14e,f ₄₆

15a,b Measured Cavitation Occurrence Characteristics for the

Half-Body 47

15c,d 48

16 Hysterisis Effect 49

17 Cavitation Inception Characteristics . 50

18a Comparison of Cavitation Occurrence Characteristics Measured

on a Half Body at 30 fps 51

18b Comparison of Cavitation Occurrence Characteristics Measured

on a Half Body at 30 fps 52

1-,A Bubble Sampling Probe 2-A Probe Calibration

I. -INTRODUCTION

Cavitation may occur in low pressure regions in the fluid on and near hydrodynamic forms when they move through a liquid. The many undesirable aspects of this phenomena are well known. The onset of cavitation represents

a limit to the power which can be economically applied to conventional

hy-draulic machinery, propellers, and sonar equipment. The severe collapse of the vaporous cavitation bubbles after they re-enter higher pressure regions

of the flow produces high intensity acoustical noise and undesirable erosion

of the surfaces where it occurs.

An important aspect of the study of cavitation phenomena is the

determination of the conditions which lead to incipient cavitation. It was the purpose of the research described herein to make a contribution to the problem of transient cavitation inception.

It has been postulated that cavitation bubbles grow from nuclei. The nuclei can be in the form of small stabilized gas bubbles, gas trapped in fissured particles, or in fissures in the boundary of the hydrodynamic body itself. On the other hand, recent theoretical work [1] has shown that

the tension at a solid-liquid interface may in some situations be low enough

so as to obviate the necessity of a nucleus from which a cavity would form. It has also been shown that some forms of blue-green algae, which are near-ly neutralnear-ly bouyant, contain tiny gas vacules in their cells [2].

In many water tunnels small bubbles are stabilized by the action

of dynamic processes in the tunnel flow and may be the primary source of cavitation bubbles. In prototype operation, however, it is likely that all of the above possibilities are superimposed with one or more processes achieving prime importance depending on the situation. There has been a lack of good correlation between the cavitation characteristics derived from model tests in various tunnels and the prototype, particularly near inception. Typical results are summarized in a recent ITTC study [3]. It

has been the concensus of many recent cavitation investigators that an in-complete knowledge of the type of the nucleus causing transient cavitation

in the various tunnel investigations is responsible for the lack of correlation.

2

If it is assumed that cavity bubbles grow from nuclei, and if it is recognized that nuclei are likely to be randomly distributed in the liquid, it is apparent that any rational investigation of the transient cavitation process must be conducted from a statistical point of view. This is especil.

ally true near inception conditions.

In the present research, the nucleus, which is considered of prime importance to the cavitation process near inception, is the gas bubble-sta-bilized by dynamic tunnel processes. A probabilistic model of this process is described in section II and experimental methods for the investigation of the model are described in sections III and IV. The results of this

re-search and their implications are discussed in section V.

This program Was sponsored by the Naval ship Research and Development Center under contract Nonr 710 (67), The author acknowledges the many

con-tributions Of Ni.. Thomas Rektor to the research and many discussions with his

colleagues, Dr. J hn M. Killen and Prof. John F. Ripken.

II. A PROBABTTTSTIC MODEL OF TRANSIENT CAVITATION NEAR INCEPTION

In developing the probabilistic model it is necessary to obtain three

types of information: the size distribution of the entrained bubbles the

hydrodynamic characteristics of the test body, and the thermodynamics and flow parameters of the liquid.

Th the case considered, the most important nuclei are considered to consist of small gas bubbles stabilized by dynamic processes in the water

tun-nel flow system. These bubbles are, presumably, homogeneously distributed throughout the flow field if averages are taken over small, but finite, fluid

volumes.

In section III an experimental procedure is described by which the size distribution of small bubbles entrained in the test section of a water tunnel may be measured. Data obtained by this means could be used as input information to the probabilistic model.

However, for the purpose of developing a probabilistic model by which cavitation calculations can be made, an assumed bubble size distribution was

postulated. The Rayleigh function was chosen for this purse and is expressed

ht.)

_

f(r/62)

(..r2/282)

rb- r

where r = bubble radius for upstream of the test body, 8 = radius of the most abundant bubble sizes,

P(ro,ar) =

probability of nuclei occurring in the range 2Ar centered at rip,

= lim n(ro)

N-KD N

) = number of bubbles per second flowing through the test section

in the size range 2 r centered at ro,

N = total number of bubbles per second flowing through the test

section.

This function was chosen because of its close approximation to the measured distribution to be discussed later in this report. It is

realis-tic in that it gives zero probability for bubbles of zero diameter. It

has the possibility of giving a relatively large probability of occurrence of very small bubbles while the probability of occurrence decreases quite rapidly as the bubble size increases, becoming asymptotic to the abscissa

at large bubble sizes.

In principle, the potential flow about almost any three-dimensional

body can be evaluated by the method of Hess and Smith [4]. At present, this method requires a great deal of digital computer time and in general would be quite costly. For analytical ease in demonstrating the principles

involved, a simple Blasius-Fuhrmann half body defined by the stagnation

streamline resulting from placing a point source in a uniform, infinite

stream was selected. The streamlines and isobars for this body in dimension-less form are shown in Figs. 1 and 2, respectively. Since the blockage ratio in the experimental facility was very small, wall effects were neglected.

Under some ambient flow conditions a critical volume of superheated liquid exists around portions of the hydrodynamic body by virtue of the pressure and temperature of the water. The trajectories of a portion of the entrained gas bubbles are such that they will be carried into this crit-ical volume of liquid. These bubbles will expand vaporously, provided that they are large enough that the surface tension forces are overcome. The number of times per second that a bubble enters this region and expands

14,

vaporously can be defined as the cavitation occurrence rate. This rate is an indication of the severity of cavitation under the given test condition.

The stability theory for a gas bubble in a liquid has been available

for a number of years. Johnton and Hsieh

[6]

have related the operating va-por cavitation number and the minimum pressure coefficient encountered to the initial critical bubble size by the equation:

where W =

C .

p, min OU 2d

o -oi

= bubble Weber number,

P min_- Po _ C . _ minimum pressure p,m1h 1/2 p Uo2 P - P o v

a =

= vapor cavitation number

tJ

1/2 po2

U = mean test section velocity p = test water density

y = surface tension

Pplin=minimum pressure encountered by the nucleus Po = mean test section pressure

P = vapor pressure of the test water

and d = diameter of the smallest bubble which will cavitate.

This relationship defines the lower limit of the free stream bubble diameter which will cavitate under the imposed av and Cpolin. All smaller nuclei will not cavitate and all larger nuclei will cavitate. This form

is

particularly useful since it employs the initial nuclei sizes, which would be the sizes measured by any practical instrument system designed for the purpose.

It is evident that a cavitation zone can be defined in the fluid as that volume enclosed by the isobar equal to the negative of the operating vapor

cavitation number. The water enclosed by this boundary is subjected to a pres-sure less than vapor prespres-sure and is in a superheated condition. The degree of superheat varies with the local pressure coefficient and reaches a maximum on the body where the pressure coefficient is minimum.

2 a (8/d UO3/2 v v a --34s/ (1 8/a 10-112 coefficient .encountered, (2)

Nuclei entering this zone will encounter different minimum pressure

coefficients depending. upon their trajectories. It is convenient for the present purpose to subdivide this Superheated zone into several equal flow, axisymmetric stream tubes. The minimum pressure coefficient that a nucleus

entrained in a particular stream tube will encounter may vary somewhat across the tube, but for simplicity an average minimum pressure coefficient

is assigned to each stream tube. This value

is

taken as the value on the

centerline of each stream tube and can be determined either graphically from Figs. 1 and 2 or numerically

by

computation. The incremental discharge

of the stream tube for this purpose is taken to be &still°. The cross sec-tional area ratio between a stream tube of this size and the tunnel test

section may be expressed as a/A and the number of nuclei entrained in the stream tube far upstream of the test body is (a/A)N. As has been pre-viously stated, the nuclei .size distribution in each stream tube is considered to be the same as in the test section as a whole.

It remains to determine the number of nuclei of critical diameter,

dot, and larger, Which are present in an individual stream tube entering

the cavitation zone.

A simplifying assumption could be made that the nuclei do not cross stream lines. This assumption is employed in what is referred to here as Method I. In this method the total number of nuclei in the stream tube in the cavitation zones is set equal to the total number in the stream tube far upstream.

Based on the distribution and postulations previously stated, the number

of

nuclei per second flowing in each individual stream tube in size

range 2Ar centered at ro is:

ro+br

n(ro,&t)

Na/Af(r/82) exp(

_/Zo

) dr

₍₃₎

ro-tr

If the bubbles do not cross stream lines the number of cavitation events contributed by an individual stream tube can be calculated as each bubble larger than the critical size, roi , determined from equation (2) will cavitate. The number of cavitation events, then, for the assumed model is

6

n(r6Axm)= 2N a/A exp (- r20d252)

In general, a particle entrained in a flaw will respond to inertia in-cluding inertia of the apparent mass, drag, and pressure gradient. In the case

of entrained bubbles flowing around test bodies very large with respect to the entrained bubbles, both the pressure gradient and inertia terms may be neglected, and the gas bubbles can be considered to remain in the stream tubes they were originally entrained in far upstream and Method I is adequate.

As the test bodies become smaller with respect to the entrained bubbles, the pressure gradient term in the equation of motion becomes more important as the bubble approaches the body and the larger bubbles are deflected away.

A better approximation of the number of cavitation events is obtained by what is referred to in this report as Method Ii. Using this method, the bubble as it moves along its path is allowed to respond to the pressure

gradients in the flow field and to thus' migrate across stream lines. The

larger-sized bubbles will be deflected far enough away by the positive pretsUre gradients in front of the test body so that they will not encounter a low

enough pressure. to cavitate..

It was necessary, then, to establish for each stream tube an upper bubble size limit above which a cavitation occurrence would not result. The

trajectories of the bubbles can be obtained by application of the methods of Johnson and Hsieh [6]. The time history of an individual gas bubble of a given

size originating in a particular stream tube far upstream of the test body must be examined to determine whether or not it

will

cavitate.

Following the technique of Johnson and Hsieh, the vector equation of motion of the bubble is:

d ib ₁₈

RebCD3

dT RefR2 24-- (4 - 4b)

eCp

where u = axial fluid velocity,

v = radial fluid velocity,

(u2 v2)1/2/

/

'0,

2-

21/2

= 177b/U07 _Cub + vb )

= vector bubble velocity, ub = axial bubble veocity,

vb-= radial bubble velocity, Uo = free stream velocity,

T = Uot/h, t = time,

h = asymptotic body radius,

Ref = 2hU/v = Reynolds number

of

the body,

Reb = RefRli

R' = instantaneous bubble radius,

CD = bubble drag coefficient, and

VCp = pressure coefficient gradient.

The quantities u, v, and C are determined by the hydrodynamics of the particular body being considered, a simple axisymmetric half body in this

case. The instantaneous bubble size and drag are calculated and the equation of motion of the bubble is solved by a numerical process on a digital computer.

In this manner an upper bubble size limit, rou _{can be calculated} for each stream tube of interest. The number of cavitation events contribu-ted by a particular stream tube for the given bubble size distribution is

n(rot, rou) = 2N a/A [exp (-r2oz/25 2) exp (-r2 /252)] -

(6)

Summing up the contributions of all the stream tubes in the cavitation zone

will yield

the

cavitation

occurrence rate.

A specific numerical example which can be closely approximated experi-mentally was worked out. The following values were used as input data:

= .3125 inches = body radius at infinity

U0 = 30 fps = free stream velocity

e

_{= 70° F}

= water temperature

AIN°

= .01 = incremental discharge

Ho = 3 inches = test section radius

a/A = 2.17 x 10-4 _{= ratio of stream tube sectional area to the}

8

With these imposed conditions upper and lower critical bubble sizes were computed and are presented in Fig.

3.

At any cavitation number the upper and lower sizes can be determined from the curve for any particular stream tube. Equation (6), then, yields the contribution to the cavitation occurrence rate of each stream tube in terms of the total number of bubbles flowing through the test section from the postulated size distribution.

Probable cavitation characteristics were constructed using the total number of bubbles flowing, N, and the most abundant sizeof bubbles, 6, as parameters. The cavitation occurrence characteristics as computed by the two methods described above are given in Figs. 4 and 5 and the two methods are

com-pared in Fig. 6.

If information regarding the hydrodynamics and thermodynamics of the flow situation, together with a knowledge of the size distribution of the bubble nuclei is available, the severity of cavitation expressed by an occurrence rate can be predicted by the statistical model just described.

The effect of increasing the bubble concentration while retaining the same relative size distribution is shown by Figs. 4a, 4b, and 5a. As might be expected, the occurrence rate increases with total number of nuclei flowing through the test section at a given flow condition expressed by the cavitation number

based on vapor pressure. Families of occurrence characteristics are thus

deter-mined with total nurriber of nuclei in the test water as a parameter.

Changes

in

occurrence rate resulting from variance

in

the relative bulb

ble eize distributions are shown by Figs. 4c and 5b for a constant ñwtber of bubbles flowing in the test section. As the most probable size is shifted to higher values, relatively more bubbles are above the lower critical size where

surface tension forces prohibit vaporous expansion. Figure 4c exhibits this

shift directly. Figure 5b combines this effect with the tendency for the larger-sized bubbles to be deflected away by the pressure field in front of the body. Figures 6a and 6b indicate the relative effect of taking the bubble trajectories into account for different size distributions. When the relative bubble sizes with respect to the body size are very small, the bubbles do not appreciably

cross streamlines. This is shown in Fig. 6a, where a comparison of the two

methods shows almost no difference. As the relative bubble sizes increase the importance of including the effect of the bubble trajectories becomes greater. In the example shown in Fig. 6b the error can be quite large.

The effect of bubble radius relative to the characteristic body

thickness is quite important i tié resultiiig trajectory of the bubbles.

In model tests at relatively small scales it would be expected that the

calculated occurrence rates

would

be affected to a large extent by the bubble trajectories.

Presumably the bubble sizes encountered in model tests are not too different from those encountered by the prototypes. It is therefore

likely that for most large, streamlined forms the much simplified method I may be used to estimate cavitation severity in prototype situations, even

though it may not be applicable to a model.

An alternate parameter which can be utilized as a cross parameter rather than the total number of nuclei flowing is the total free gas

con-tent of the test water. This parameter, usually expressed in parts per million by volume, is an integrated value of the bubble size distribution.

It has been used in the past

(5)

and it has been shown to have considerable significance in affecting the cavitation inception number. A simple re-lationship exists between the total number of nuclei flowing per second, the quantity of water flowing and the nuclei size distribution. In the example presented in this paper, the relationship is:

-2sTN83 73/2

6

x 10

T

U0A

where T is expressed in parts per million by volume. The calculated

magnitudes of the gas contents in the example discussed are less than

1 ppmv.

III. THE DETERMINATION OF THE GAS BUBBLE SIZE DISTRIBUTION

In the previous section a probabilistic model was developed utilizing for convenience an assumed gas bubble size distribution. This was useful for simplifying the model and revealing several aspects of the cavitation phenomena over which the size distribution has a good deal of control. In the model tests, however, it is necessary to have knowledge of the actual bubble size distribution and to use this in place of the assumed

distribu-tion. The appendix describes an electronic method which has been under

10

development for measuring bubble size distribution. However, the electronic method is not fully operational and the method which has been used in this research is based on acoustic principles [7]. A description of this method is repeated here for purposes of clarity and completeness.

In the acoustic method the amplitude attenuation of an acoustic tone burst is measured as it is propagated through the test water. The attenua-tion is measured as a funcattenua-tion of the carrier frequency of the tone burst

over the range of interest. It is well known that only bubbles at, or near resonance, contribute significantly to the sound attenuation. The resonance frequency of a bubble is given by the following equation:

= 1 3kPo

g

2uR0 P a

Where Ro = bubble radius,

fo = resonant frequency of the bubble,

k = ratio of specific heats of the bubble gas,

P0 = ambient pressure,

p = water density

g 1 1 1 i( 3kP,3

= g 4 rry fo -

517)

= a function arising from heat conduction and surface tension which is plotted as a function of ambient pressure and frequency in Fig. 3, Reference 7, and was defined in References [8,9], y = surface tension

a = a function plotted in References [8,9].

Bubble size measurements alone can have considerable value in applied tunnel operations; however, the sizes would be much more meaningful if associ-ated with a concentration and ultimately a number per unit volume in a given

size range. The measurement of air bubble concentration in water by means of its effect on acoustic attenuation involves the considerations discussed below.

The physical effect which is most readily measured is the decrease in amplitude of a pressure wave as it propagates through the bubbly mixture. In bubbly water, the reduction in pressure at any point at a distance away from

a mall transmitter, in the absence of standing waves, can be expressed in

terms of an attenuation factor, f3 , as:

p =

poxo.

exp _Xo)}

₍₉₎

where p = acoustic pressure at some distance, x, along the axis of the transmitter

where ip() = acoustic pressure at reference distance,

x0

from the acoustic sourbe.

If the pressure, p, is measured at a point in bubble-free water, the pressure

will.

be given as

and in bubbly water at the same point as

where

01 ,

= attenuation factors of bubble-free and bubbly water, respectively. The ratio

p2/p1

is then given by

P2/P1 =

e

1(32

1)(x xo)}

or ax 20 log

p2/pi

=

8.686 (01

-

02)(x -

x0)

where a attenuation per unit length. The values

xo

may be chosen arbi-trarily as very close to the source and therefore

ax2e.8.686 (01

-

02).

The factor a represents the attenuation due to the resonant and near-resonant bubbles in decibles per unit length. The attenuation arising from a number of bubbles, as measured above, has been shown to be a function of size, pressure, density, etc.., by the following general relationship [S]:

12

where A(w) = attenuation function

T(w) = resonant volume concentration function per radian per second w = angular frequency.

The function T(W) is the quantity to be measured in order to define

the size distribution of the entrained bubbles. Rewriting Eq. (10),

w,

a =

PA(w)

To(w) dw + f2A(w) T1 (w) dw +jr,A( ) T2(w)dw + .e. (11)

wl _W2

where wi, w2, w bounding frequencies of the small frequency intervals

under consideration.

In general, it is expected that the function T(w) would be a continu-ously varying function. As such, a unique solution would not be possible. It

is reasonable, however, when considering a small frequency interval, to assume

that T(w) is a constant in that interval. Equation (11) can then be written

i+1

where Ii

=fA(w)

dw

Wi

Since Ii needs to be evaluated for a relatively small number (n), of

frequency intervals the summation of Equation (12) becomes relatively

easy.

An expression for A(r) has been given in Ref. [8] and is reproduced here in its

essential elements. This involves the equation of motion for a single bubble of small radius compared to the wave length of the exciting pressure which

can be written: d2v _dv = exP p (jwt) m _{+ D} Kv dt2

where M = added mass of water surrounding the bubble, = p2/4re

D = energy dissipation,

(12)

size becomes: where and where -o o Ro = resonant radius

The velocity of propagation of a wave in a mixture can be written

in terms of a complex velocity of propagation Ref. [7]. -vN

pejet

K = compressibility of the bubble gas,

(g)

41T113 a

v = incremental volume Change.

The steady state solution for u is known to be:

j

-Pewt

w2M K - 1\ + j

w2M

The compressibility of a large number, of bubbles of a single

2

[(a;

wo = the resonant frequency

4 3 0 = Ti=ra 3 T = volume concentration 3TZ2 (15) wo woM w

0"

2,R2D2

1

4(16)

(14)

1

= _

and in turn in terms of a complex compressibility where

7

= A - jB where A and B are the real and imaginary parts of

R.

,F;T:77

12

Real (1) = part C

Alp+

+ + 2 - 2 / (2Z

-S:777+

712) 247:772 (2z

1,447717)4-1 2 )1 A2 (17) (18) (2o) (22)

-Imag. part

(76

=/T

pA

li/(1

_A (19)

expressing the attenuation a in terms of the phase velocity (C and

ph converting to decibels gives

8.68 =

wpCphB in decibels unit length

For a mixture Of many bubble sizes in which the concentration per cycle is constant and equal to T(w) and the damping factor i8 the

com-pressibility becomes: . dZ R = 1 ₊ pw T(w) JiZ2 Z2

A

jB 2TrkP0(f) z1 Z2 - 1 jTIZ . where

Co = velocity of propagation of bubble free water

1 pw TOO

(

T1Z

)tan

A = +

+

pCo

27k%*

7

z2

-The phase velocity

jB -2

(.7j7E72

-

)tan-i

( 2Z

+44

-[

pw

T(w)

B - . 47k130(1) jpw

T(w)

arkPo*

8.68

pCphw2T(w)

[F

Nri

4iTkPo(f.) -41n 1.12) 2 2Awo) w 211 ( 2 o o w i2 ₂ co

2w 2)2

(1

-

+ 1

2 -

2(1L2-w wo o (up 2Z - 4

1

tan-1 (---7===-7) Z1

Substituting for

IBI

in Eq. (20) gives the working equation

(2A1

w2 2T1 --- 2

8.68

_PC w2T(w) _W ph _{o o} -

[tan-1(

2

2

Nri

417kP0(;) 1 2 2 w 2(Awo)2(w2 )(w2 ):) 2) wo wo2 wo \w 2 w 2 o o (23) --- (24)

If an effort is made to reduce Eq. (25) to the relationship for a

single bubble size given in Ref. [8], as the interval Awo-, 0 the neglected

(25)

In the range of bubble Sizes of particular interest the value of _Ti required in Eq. (23) varies from .05 to 0.1 (Ref. 10); consequently, the

magnitude of the logarithmic term is Stall compared to the Inverse tangent

term and is neglected for convenience. The term Z1 is now equal to

wo - Awo

w +w0

and Z2 -

and w

is now a mid value in the

inter-val

Z2 - Z1 The magnitude of B can now be written, neglecting small terms

in

the denominator, as:

W

16

logarithmic term must be included, thus showing the approximation becomes less valid for very small bandwidths (Small bubble site range).

The constant in Eq.

(25)

may now be written

8.68 C

pT(w)-_ph_ 2w

/ w

wo2 - 4.usE kP0(#) wo 2 [F

8,68 x., 1 x 1.48_x 105

T(w)

p( w\1

.o518

T(w) pe9]

o (27) 6 .g, wo

w2

41.1-sE

x 1.4 x 10 P ('-zi

w(5

F4(g)-O 1 o

r

112 where P'o is now in Atmospheres and

2 is Incorporated in the bracket term. w

The last equation permits evaluAion of the concentration, T(w), 511 terms of the applied frequency characteristics, the inherent damping and the observed attenuation.

The conversion of attenuation into concentration requires the formation of a sum, as in Eq. (12). This must in some manner be commensurate with the physical situation so as to approximate the form of the total measured attenu-ation pattern as a function of frequency. It has been found by experience that only bubbles resonant in the frequency interval being examined and the two adjacent intervals contribute significantly to the attenuation.

The quantity 2hw ) 2 w 21-1(-,a-w2 tan-1 wo w-02 4(21,-) 2 (1

_

wO -W2' w2 2 ( A 2 w 2') + 112 ' -- ..

2e9

w

--II +

w 2 2 2 2 w 0 o o o o o (28)

has been plotted in Fig. 7a for frequencies of 20,

50,

and 100 kilocycles. Values of 11 were taken from the experimentally determined values as compiled by Turner [11].

The units of

T(w)

are volume concentration per radian per second and therefore this value

to obtain the volume

presses the interval

must be Aultiplied by the Width of the

concentration,

Tqw).

The term 2/tw0

-I--0

of integration on the frequency scale.

frequency interval in Eq. (28)

ex-The value = 1

is a convenient midband identification. An example of the division of the

frequency scale of the measured data into a series of descrete bands is shown

by the heavy vertical lines on Fig. 7b. Each line is located at -7- equal to unity in its own interval and the lines are 2 -7-- _{apart. An arbitrary} value of 1/8 _{has been chosen for 7- ; therefore, the frequency value for each}

vertical line is 9/7 times the frequancy of the previous line. The 9/7 value derives from _{f1 x 9/8 f2 x} 7/8.

a P (g)

a a

The ordinate of Fig. 7h is plotted in units of

30f2

attenuation a

is in .db/cm and Pa is in psi; this mixed combination of

units gives a convenient constant of 30.

The value of the quantity _{F(-7--) when plotted on semi-logarithmic}

paper as in Fig. 7a may be used repeatedly like a template on succeeding determinations of concentration as will be described shortly.

On Fig. 7b the usual practice of plotting frequency on a logarith-mic scale has been followed. The linear length on Fig. 7h should be Chosen

so as to equal the length Of an equivalent ratio of(60 _{on Fig. 7a. The} length Z on Fig. 7a must equal the length Z on Fig. 7b.

From the basic nature of a logarithmic scale, that equal ratios cut off equal length at any position on the frequency scale, the templates of Fig. 7b may be superimposed at any point along the frequency scale of Fig.

7b. However, the variation of 71 with frequency will restrict this

move-ment to small distances near the particular value of

w in

-(13-. When the

first template from Fig. 7a is superimposed on the lowest frequency range (20 KC) of Fig. 7h, the tail of the template will contribute a value to the attenuation in the adjacent frequency interval. This must be subtracted before the next template is superimposed. In this case, the value is meas-ured as A' on the bottom of the diagram and is subtracted at the top as

18

shown on Fig. 7b. The contributions from the frequency intervals on both Sides of the interval of interest must be handled in this manner.

The loci of the peaks of the templates of Fig. 7a superimposed on Fig. 7h are the numbers related to the contribution of bubble concentration acting

independently on the attenuation in that particular interval. Division by the peak value of F( m--) in each interval and multiplication by the absolute

width of the interval, (26%), in cycles gives the total concentration of air

in that interval. The frequency limits of each interval may now be converted

into radii by Eq. (8).

An example of a size distribution obtained in this manner is shown in Fig. 8, where the frequency scale has been converted to bubble size by Eq. (8). Since in the final application it was desired to know how many bubbles in a given size range are present, T'(R) /mast be converted to number of bubbles.

If it is assumed that all of the bubbles in the size interval are exactly of

resonant size,

Ro' then

= T'(R)

where N' = number of bubbles in the size interval flowing through the test

section per second,

Q = water flow through the test section.

Since the measurement was made in the contraction approaching the test section, the sizes are modified by the pressure change, including the surface tension term to present the distribution as it is in the test section.

In order to measure the amplitude attenuation, a pulse system was

adopted. A tone burst of several cycles' duration is applied to a piezo-electric transducer at various carrier frequencies between 10 and 160 kc. The

peak-to-peak amplitude of the acoustic tone burst at a receiver hydrophone located a short distance from the transmitter is measured on an oscilloscope. The amplitude of the tone burst thus presented.is compared with and without

bubbles. The bubbles are eliminated from the tunnel prior to the test by in-creasing the pressure in the tunnel until all bubbles are driven into solution. The amplitude of the tone burst under these conditions gives the reference value which is then compared to the amplitude under the test condition. The increase

in attenuation can then be directly calculated from the ratio of the reference

amplitude and the amplitude uhder the test.condition.

The equipment used was assembled from available Laboratory components. The electronic equipment is shown schematically in Fig.

9.

Basically, this

equipment consists of an oscillator, a tone burst generator, and a wide band

amplifier. In addition to these standard commercial items, a transistor power amplifier was designed to drive the crystal stacks.

An example of the transmitting transducer construction is shown in

Figs. 10 and 11. It was constructed by stacking ceramic disk transducer elements so that they were excited in parallel. The elements were

three-fourths inch in diameter by 0.1 inch thick. The whole assembly was mounted in a lucite block so that it could be inserted in a port in the water tunnel. In order to cover the frequency range of interest, seven of these transducer

stacks composed of various numbers of elements were constructed. These

transducers were mounted upstream of the test section in the contraction

chamber. The tone burst then was propagated across the contraction cham-ber and picked up by a receiver element mounted on the opposite wall. The receiver element was constructed from a 1/4-inch-diameter-by-0.1-inch-thick ceramic disk. The received signal was amplified and fed to the oscilloscope.

IV.

CAVITATION OCCURRENCE COUNTING

In the previous two sections methods were described to measure the gas nuclei size distribution in terms of numbers of bubbles in given size ranges and to use this information in an analytical model to predict a probable cavitation occurrence rate, the number of times per second a bubble will encounter the hydrodynamic body in such a way as to expand vaporously and to collapse violently.

In this section a technique will be described which provides an experimental measurement of the cavitation occurrence rate (12). This

measurement is accomplished by sensing the acoustic pressure pulse resulting

from the first collapse of the cavity. An average count per unit time is

determined for each test section condition, defined by the vapor cavitation

20

In the experiments, the collapses were sensed by a small ceramic cy-linder transducer element mounted in a castor oil bath in the center of the

test body. The body was constructed of a plastic material. The acoustic impedance of the plastic and the castor oil assists in reducing the acoustic

energy losses in the transmission path to the transducer. The test body was mounted on a sting, designed for a high degree of acoustic isolation

from the remainder of the tunnel.

The transduced electrical signal was then fed through an

ampli-fier and examined. Preliminary photographs of an oscilloscope

presenta-tion at a tunnel condipresenta-tion very near inceppresenta-tion indicated several voltage spikes due to the multiple collapse of a single cavity and some additional

transducer ringing. Near the inception condition, eligible nuclei occurred only occasionally, and it was rather easy to distinguish one occurrence

from another. The time duration from the initial voltage spike of an

occurrence until the remaining signal due to ringing and multiple collapses from the occurrence had died out was determined as not more than 0.5 milli-seconds for these experiments.

This signal was then processed by a single shot multivibrator.

The multivibrator was triggered by the initial spike of an occurrence and blanked the circuit until all further collapses of the given cavity had

taken place. For these experiments, the reset or resolving time of the multivibrator was adjusted to 0.6 milliseconds. Thus, the output of the

multivibrator gave one pulse per cavitation occurrence, and this signal

was fed to an electronic counter. At a rate of 10 cavitation occurrences per second, an experimental determination of the probability of one

occurrence to follow another within the 0.6 millisecond blank time

was

about 0.4 per cent.

It has been suggested [13] that a correction for losses due to

occurrence coincidence similar to that used in a Geiger Mueller detection

system can also be utilized in this system. This correction is given as

.

= mil-mT

where n = true occurrence rate,

A = observed occurrence rate, and T = resolving time.

. The calculation Shows a 0.6 per cent probability for one occurrence to

follow another, which compares quite well With the experimental value of o.4 percent.

V. DISCUSSION OF THE 'EXPERIMENTAL RESULTS.

The objective of the research program was to attempt to establish

ex-perimentally relationships between the bubble nuclei content of the test

water and the cavitation process near inception. The previous sections outlined the experimental and analytical techniques which were developed or adapted for this purpose.

The main test facility consisted of a somewhat conventional form of small closed-jet water tunnel of 6 inches test section diameter. In addition to being smaller than most other tunnels, this tunnel differs from conventional forms in that it contains a gas separator unit installed just upstream of the

contraction. The gas separator unit is a tunnel reach filled with a packet of small flow tubes, each with a crown shaped to a sharp, inverted V in which a greatly thickened boundary layer develops. The large entrained gas bubbles which pass through these tubes manage to gravitate upward into this boundary layer within the length of the tube and are exposed to a relatively low

velocity and a weakened transporting sySteM. With the tube axis tilted down-ward in the direction of flow, the bubbles collecting in the crown of the tube will be subject to a gravitational-force component acting upstream along the top of the tube. With appropriate adjustment of the tube slope and the flow velocity, it has been established that it is practical to promote the collection and upstream movement of bubbles greater than some critical size depending on velocity passing through the tube. This apparently acts as a filter with a gradual cutoff characteristic with size. It is analogous to an electric filter in this respect. To further enhance the separation process the bubbles coalesce in the crown, increasing their size and gravitation

velocities. To promote general collection of the bubbles, the individual tubes are provided with holes near the upstream end. These holes permit the bubbles to gravitate from one tube to the next above. The bubbles pro-gress upward through the tube stack to the top of the tunnel conduit where they are collected and bled off to a vacuum control tank. Standard, gal-vanized, corrugated steel sheets were stacked to produce tube lengths of 3

feet at an angle of about 20 degrees with the horizontal in the desired arrangement.

22

Two test bodies have been used, the first being the 2:1 modified ellip-soidal body recommended by the ITTC Cavitation Committee for a recent tunnel

comparison study [3]. This body is shown in Fig. 12 along with the calculated isobars in the fluid, and in Fig. 13, with the streamlines in the fluid. The

potential flow computations were performed by courtesy of the U. S. Naval Ord-nance Test Station at Pasadena using the Douglas Aircraft Program [14]. The

second body being used is the standard half body whose profile was computed as previously explained by placing a simple source in a uniform flow. (See Figs. 1 and 2.) The physical diameters of each body was 0.625 inch. This

pro-duced,approximately a 1 per cent blockage area to the flow in the tunnel test

section.

The test body was positioned on the test section axis at a point 1.6 in-ches downstream of the test section wall boundary reference static pressure tap by a long sting rod which was supported by a faired strut system located

down-stream of the test section. Prior to installation the body was cleaned with

a solvent. The body was machined from a black "Delrin" plastic to facilitate

observations of cavitation. The profile of the body was examined by an optical comparator and no deviation from specifications could be detected with a factor

of 10 magnification. An attempt WAS made to control the nuclei grown in and ejected from fissures in the model surface by polishing and waxing to eliminate

porosity.

The number of nuclei entrained in the water tunnel under test conditions was controlled by the regulation of the total gas content of the test water.

The tunnel water was changed daily. This was found to be necessary to

obtain consistent data. If the tunnel was inactive and allowed to remain full for only a few days, it was observed that the occurrence rates were tu h lower and not stable, other conditions being equivalent. The nuclei were evidently shrinking and being stabilized in some nebulous manner with time. An effect such as illustrated in Figs. 4c and 5b would be the result of bubble

shrinkage. Bubble stabilization by organic skins or other mechanisms probably

contributed to the non-repeatability of data taken in "old' water.

The total dissolved gas content of the tunnel water was measured by a

conventional form of Van SlYke apparatus. The water samples were drawn from the tunnel just previous to and just after a test run. The lOcc sample was

withdrawn from a small bypass tube connecting a point just upstream of the

pump to a point in the test section. By this.arrangement a representative

tunnel sample could be introduced directly into the Van Slyke meter without

exposing it to the atmosphere. The flow in the bypass tube was interrupted

during the test run.

The tunnel water was first adjusted to the desired dissolved air con-tent. The deaeration consisted of subjecting the tunnel to a vacuum of such

intensity that strong cavitation was taking place in the test section. The resulting free gas then was separated from the water via the separator unit.

The duration of the vacuum determined the amount of gas removed. The unique

construction of the air-separator tunnel permitted reduction of the total dissolved air content from 20 ppm to

6

ppm in about 15 minutes.

Aeration of the tunnel was achieved by opening a stop cock in the tunnel test section while the tunnel was being operated at a selected

velocity. A sub-atmospheric condition existed at the test section and

air entered the stream via the stop cock. Aeration of the water by this means

could raise the total dissolved gas content of the tunnel water from 5 to

25

ppm in about 4 minutes. The usual procedure followed to establish tunnel

air content was first to deaerate to about 4 or 5 ppm and then to add air via the test section cock in increments of 15-or 30-seconds' duration. After

each addition a Van Slyke measurement was obtained.

After the total dissolved air content was established, the test

velocity was selected and set. The velocity Was inferred from a differential pressure or Venturi effect measured across the tunnel contraction.

It was determined early in the program that only steady state operating conditions would be attempted. Rapidly changing pressures in the test sec-tion for example, would be much too difficult to manage either experimentally or analytically.

The experimental procedure established was, therefore, one of very slow changes. After setting the dissolved gas content of the water and the test section velocity, the pressure, beginning with a high value, was reduced in small discrete steps. Ample time after each step was allowed to assure nuclei equilibrium in the tunnel. Previous experience has shown that incep-tion results are highly dependent on the rate of approach to cavitaincep-tion conditions.

24

A family of cavitation occurrence characteristics was determined using the ratio of the dissolved gas to the saturation value at atmospheric

pres-sure as parameter. For each member of the family, cavitation occurrences were counted (using the method of Part IV) aa a function of cavitation numbers. Cavitation occurrence characteristics for the ITTC standard test body are shown in Fig. 14 and for the half body in Fig. 15. Hysteresis effects were checked by comparing counts with increasing pressure against counts with de-creasing pressure at the same cavitation number. As may be seen, there is a very small difference. (See Fig. 16.)

The counting technique described herein makes possible a more detailed description of the cavitation phenomenon near inception than was possible

before. The responsibility for the judgment of when cavitation inception Occurs is removed from the tunnel operator and placed on the data

interpre-ter. It is possible for many arbitrary definitions of inception to be used

at this point. . One such definition, for example, could be that cavitation

inception occurs at a definite rate; for example, 10 occurrences per second. This definition has been arbitrarily used in determining the characteristics

of Fig. 17. These characteristics roughly correspond with earlier inception data obtained

in

the same tunnel by the traditional visual observation of

inception method, specified by the ITTC Committee for their recent study.

The amplitude of the collapse pulse is related to the quantity of

permanent gas enclosed in the cavity. This quantity is determined by the orig-inal sizes of the gas bubbles far upstream of the test body. Some discussion

of

the relationship

is

available in a semi-empirical form in papers by

Sirotyuk (15) and Khoroshev (16). Using these results a rather simple addition to the computer program determining the bubble trajectory could also produce the-amplitude of the collapse pulse. It would be possible to determine a statistical amplitude distribution of the collapse pulses in

this manner.

A preliminary study was conducted to observe experimentally the statistical amplitude distribution by varying the sensing level of the

elec-tronic circuitry. This produced additional information relative to the bubble size distribution far upstream of the test body. The occurrence characteris-tics of Fig. 18 illustrate data taken in this manner.

Figure 18 also serves as a comparison between probable occurrence rates computed by the method Outlined in section II based on bubble nuclei

distributions measured by the method outlined in section III. The filled symbols in Fig. 18 indicate the probable cavitation occurrence rates com-puted in this manner. The computed probable occurrence characteristic

should indicate one extreme: that of the sensory level set near zero volts.

The phenomenon of pseudo-cavitation is occurring simultaneously near the outer boundary of the superheated region of the fluid and will produce low

amplitude pulses which would be counted at low sensor trigger levels.

Several examples of the correlation of the probable with the measured cavitation occurrence rates are shown in Fig. 18. The correlation is

satisfactory in some cases and not satisfactory in others. It is believed

that the chief reason for this is that there is as yet insufficient

accuracy in the acoustic attenuation method for determining the bubble size distributions. Larger statistical samples, i.e., a larger number of data points, would improve the accuracy. Data collection via a high speed data acquisition system would be a definite improvement.

LIST OF REFERENCES

Peterson, Frank, Cavitation Origination at Liquid-Solid Interfaces, NSRDC Report 2799, September, 1968.

Ruttner, F., Fundamentals of Limnology, third edition, University of

Toronto Press, 1963.

Lindgren, H. and Johnson, C. A., "Cavitation Inception on Head Forms, ITTC Comparative Experiments, Appendix V," Report of the Cavitation

Committee to the 11th ITTC, Tokyo, Japan, 1966.

Hess, J. L. and Smith, A. M. O., Calculation of Non-Lifting Potential Flow About Arbitrary Three-Dimensional Bodies, Douglas Aircraft

Report No. E. S. 40622, March 1962.

Ripken, J. F.'and Killen, J. M., A Study of the Influence of Gas Nuclei on Scale Effects and Acoustic Noise for Incipient Cavitation in a Water Tunnel, University of Minnesota, St. Anthony Falls Hydraulic

Laboratory Technical Paper No. 27, Series B, September, 1959.

Johnson, V. E. Jr. and Hsieh, T., _{"The Influence of the Trajectories of}

Gas Nuclei on Cavitation Inception," Sixth Symposium on Naval

Hydrodynamics, Washington, D. C., September 28 - October 4, 1966. Killen, J. M. and Ripken, J. F., A Water Tunnel Air Content Meter,

University of Minnesota, St. Anthony Falls Hydraulic Laboratory, Project Report No. 70, February 1964.

Meyer, E. and Skudrzyk, E., On the Acoustic Properties of Gas Bubble Screens in Water, Translation by Charles Devin Jr., David Taylor

Model Basin Translation 285, November 1958.

Spitzer, L., Acoustic Properties of Gas Bubbles in_a Liquid, Office of

Scientific Research and Development, National Defense Research

Committee Div. 6, Sect. 6.1-Sr 20-918, 1943.

Devin, Charles Jr., Survey of Thermal, Radiation, and Viscous Damping of Air Bubbles in Water, David Taylor Model Basin Report 1329, August 1959.

Turner, W. R., Physics of Microbubbles, Vitro Laboratories Technical Note G 1654 01-2, Silver Springs, Maryland, August 1953.

Schiebe, F. R., "Cavitation Occurrence Counting - A New Technique in

Inception Research," ASME Cavitation Forum, New York, November

30, 1966.

Robinson, M. J., Discussion of Schiebe, F. R., "Cavitation Occurrence Counting - A New Technique in Inception Research," ASME Cavitation

Forum, New York, November

30, 1966.

Neldon, D. M. and Hoyt, J. W., Personal Communication October 3

1966.

Sirotyuk, M. G., "Effect of the Temperature and Gas Content of the

Liquid on Cavitation Processes," Soviet Physics - Acoustics,

Vol. 12, No. 1, July-September,

1966.

EhorosheV, G. A "Collapse Of Vapor-Air Cavitation Bubbles," Soviet Physics - AcouStics, Vol.

9,

No.

3,

January-March

_1964.

FIGURES

0.400

0.300

0.200

0.150

0.100

0.050

0.025

Dimensionless Distance, reference to Body Radius

Fig. 1

2.0

1)

0.4

0

_a

Dimensionless 'Distance, reference to Body Radius at Infinity

Fig. 2 - Axisymmetric Half-Body with Isobars

1.2

0.8

0.4

0

0.4

0.8

1 f;2' 1..6

2.0

.30

.28

.26

.24

.22

.20

0 9 10 20 40

N

\

;

Notes:

X \ \

4

\ \ \

X

x

N

\

5

\

\

\

\

N

\

6N

N

\

X

X N X X \ \

X

N

N X ,X \ i

NN X N.\\

NN x \ \

.,,a

\

N N x \

\

i x

,

x

\

----\

N

\N

\

N '

X\

NN x xs.

---.. --__.

---,,

NN

...6 ..,,, NI -____

---.

N N

N\

\

.,,-

---,

\ N N

60 80 100 120

Bubble Diameter, ft x

105

Fig. 3 - Critical Bubble Sizes

lower critical' d

upper critical, do,

Stream tube is indicated by number starting with

the central core as 1.

Each stream tube has an

incremental discharge of

2)

30 10

2)(10= N

4 x10 4x1C

.2

x10

IMethod

-.59 1.x 10 ft -4 _

1 0-4ft

Note: The population of the stream nuclei is assumed

to be described by the Rayleigh distribution

with most probable size 6 and total number N.

.2

Fig. 4 - Computed Probable Cavitation Occurrence Characteristics of an Axisymmetric Half-Body

when Nuclei Follow the 'Stream Tubes (Method q

Olt:Methodl

N=io

JucIei per seco

.3

(c) Cp,min,;

.4

.4, 'Method T x 110 16=2.1345: 2x104 It cx101 4x1b4 I I,

iIICp,min.

I '5x103 I

I I

1

2.5x1

i

t1 lillsi,\

.3

.4

.2

.3

.4

(a) 04

50 40

0 20

-8 2 o_ 10

5x 10

2.5x10

.3

(a)

2x10

3x10

_lx10-4ft

4

4x10 = N

C ,

min.

C ,

min.

Cavitation Number based on Vapor Pressure

Fig. 5 - Computed Probable Cavitation Occurrence Characteristics

of an Axisymmetric Half-Body when Nuclei Cross

Streamlines (Method II)

.4

N=1

x104

N

per secon

Note: The population of the stream nuclei is assumed

to be described by the Rayleigh distribution

with most probable size 8 and total number N.

50

N = 1 x 10

Nuclei

per second

-4

8 = 1 x 10

ft

40 -a) a) 30 u 30 20 0.) 0

8

a_ 10

Method I

,

4

Method II

N =

x 10

Nuclei

per seconi

Method I,

-4

= 2.845 x 10

ft

.2

.3

.2

.3

(a) (b)

Cavitation Number based on Vapor Pressure

4.0

3.0

2.0

1.0

.8

.6

.5

.4

.3

.2

WJW =

11/8 o = .106 20 KC

1.0

1 .4

1.0

104

w/wo

1.0

Fig. 7a - Attenuation Function Templates

wo

4 1/8

.13

100 KC

38

AO.

4111)--Fig. 7b

Plot of Experimental Data

3.0

2.0

1.0

g/Ml

a.°

a co -CI amo, 0.1 0

0.8

0.6

0.4

0.3

0.2

0.1

80 50 6 20 30

f (KC)

10

2

4

Bubble Diameter, in. x 103

o=0

0

NT =

1

.294

28,900 bubbles per/sec

2 4 6

Bubble Diameter, in. x

103

40

20

2

4

6

Bubble Diameter, in. x

103

2

4

6

Bubble Diameter, in. x 103

Fig. 8 - Measured Bubble 'Size Distributions, % Saturation

54.8 a = 0.290

NT =

155,600

bubbles per/sec

40 40 a = 0.322 O = 0.304 T =

141,160 bubbles per/sec

--= 135,500

bubbles per/sec

20

20

Power

Amplifier

Oscillator

Tone Burst

Gate

Fig. 9 - Schematic Diagram for Attenuation Measurements

WATER TUNNEL

_{CROSS SECTION}

Sync.

Osc i I I °scope

1 K

4--"vvvA,

5 mh

Amphenol Receptacle

110'4

WZ/I NIONIIMINIAMOOLNIS 4N.%k .

Illisk

MINS

I

Brass

Screws\

INIWIMINNIMMIWPS\ A s

& Plate

_0-Ring

Amphenol AN/MS 3102A-I4S-5P

Receptacle

Fig. 10 - Transmitter Construction

Fig. 11 - Transmitter Element Connectors

Lead ---t

t-- .00 'r

. Brass

Clevite P.Z.T 5 Ceramic

-.

Crystals

Dimensionless Distance, refz:rence to Maximum Body Radius

3.50

3.00

2.50

2.00

q/Uo

1.50

1.00

,.75

.50

.25

0

1.2

0.8

0.4

0

0.4

0.8

1.2

Dimensionless Distance, reference to Maximum Body

Radius

Fig. 13 - 2:1 Modified Ellipsoidal Headform with Streamlines

50

I 50.

a. 20 ft per sec

40

30

20

.7

.3

.4'

Cavitation Number Based on Vapor Pressure

b.

25 ft per. sec

54.2 % Sat.

IC

.

V.-Fig. 14 a, b - Measured Cavitation Occurrence Characteristics for the 2:1 Modified Ellipsoidal Headform

50 -o 4).

40

c). _tui 3 0 $.1 a 30

020

.-

> 10 12 40 30 20 10 0

Fig. 14 c, d - Measured Cavitation Occurrence Characteristics for the 2:1 Modified Ellipsoidal

Headform

.4

.5

.6

.7

.3

.4

.5

Cavitation Number Based on Vapor Pressure

c.

30 ft per sec

d.

50 -o o°

40

cu" a_ 3 0

30

oc 41) -u 3 ,

0

20 0 _4- 0 a co°1:3) 10

.4

.5

.6

Cavitation Number Based on Vapor Pressure

e. 40 ft per sec

f.

45 ft per sec

Fig. 14 e,

- Measured Cavitation Occurrence

50 -o SC 1.21 40 In 13

30

ce Ct.; L.21; t'u

0

20 0> 41)0) 0 I- II 50 40 30 20 10 0 531.0 4.3.6

% Sat. = 31.5

C

p,mm

Fig. 15 a, b - Measured Cavitation Occurrence Characteristics for the Half-Body

.3

.4

.2

.3

Cavitation Number Based on Vapor Pressure

a.

25 ft per sec

In _4- M 0 50

36.8

34.5

40.5

46,9

% Sot.. -= 33.3

.2

50 40- 30?

₂₀

.3

.4

.1

Cavitation Number 'Based on Vapor Pressure

59.2

51.4

.2

.3

A

c.

35 ft per sec

d.

40 ft per sec

Fig. 16 - Hysteresis Effect

2:1 Modified Ellipsoidal Body

(ITTC standard form)

Velocity 30 fps

50 C

Decreasing

Pressure Increas ng Pressure 40

%

Sat. =

50.

15 30 ac

020

.-.5

%

Sat. =

34.0

a I-) 10 0

0.3

0.4

0.5

0.6

0.7

50

KEY:

2:1

Modified Ellipsoidal Body

(ITTC Standard Form)

p,min

ci

Defined at 10 Cavitat on Occurrencles per Second

5/8 in. Dia. Body

Total Gas Content

% of Saturated Value

A 60

050

040

Fig. 17 - Cavitation Inception Characteristics

45

20 25 30 35

40

60

0 301 20 10 0 1111 Calculated

Probable

Rate

3.5m

Discriminator

4.1Settingtving

4.8

To Sat. = 42.1

.28

.32

.24

(b)

Cavitation Number

(c)

.28

24

Fig. 18a - Comparison of Cavitation Occurrence Characteristics Measured

on a Half Body at 30 fps

(d)

2.4

.28

(a)

.32

.24

50 40

30

20 10

R e

3.45 v

Di

riminator

_tting

% Sat. = 7.8

.20

.24

.24

.28

.20

.24

.28

.20

.24

(e)

(f)

Cavitation Number

(9) (h)

Fig,18b---Comparison-oKavitation-Oc-currente7Ch-ortit-tiffrittia7-WaRired on a Half-BIOdy at 3071ps

.28

APPENDIX

AN ELECTRONIC METHOD OF DETERMINING THE BUBBLE NUCLEI SIZE DISTRIBUTION

An alternate method of determining the size distribution of the

bubble nuclei entrained in a water tunnel was investigated. This method

was based on the phenomena utilized in the commercially available Coulter Counter. The resistance to electrical current in a conducting fluid

flowing through an orifice is affected by the presence of a non-conducting object in the fluid as it passes through the orifice. If such an orifice is treated as a resistor conducting a constant electric current, a vol-tage pulse, Ae, is measured, the amplitude of which depends upon the size of the particle passing through. This relation is given by:

Po V

Le - (A-1)

A2 ( 1 - r2/R2)

where. I = electric current, po = water conductivity,

V = bubble volume, .A = aperature area,

r = bubble radius, R = aperature radius.

The design criteria, in order to obtain a pulse of appreciable ampli-tude, require an orifice of diameter not more than an order of magnitude larger and thickness not greater than the diameter of the particle of interest. This type of system has worked well in analyzing sizes of solid particles and many investigators have utilized commercially-available equip-ment for this purpose [A-1].

When applied to the case of gas bubbles in water, a difficulty exists. The bubble responds to a changing pressure field by altering its size. The measurement of the original size of the bubble is complicated then, by a

number of factors.

411

$tO

W4WP P P.We 1174 MAMA

41ANEW.A16-W41

PIOW02

211e-.4W/W1011NPVei

MI rIWIIWMCIIMIONISMIWNbWB

WI.

//f.

-"I"P.4.41W111019...401101PPr.0#05" '4111PP".4p1111.1/017

PW40/.40.10/IPM-IMIPIPIPZIOPM

/IAN& 401,10,' MYLAR WASHER KEY:

Fig. 1-A - Bubble Sampling Probe

CONSTANT CURRENT

yelit

A-2

A probe was designed and calibrated in an attempt to overcome these difficulties. The configuration is shown in Fig. (A-1). Water with the en-trained bubble is drawn into the probe such that the velocity in the

in-take is the same as the ambient velocity. The probe was designed so that

it could be placed into the contraction chamber of the water tunnel

up-stream of the test section. The probe contains a contraction which is

equivalent to that of the water tunnel. An entrained bubble will then ex-perience a pressure change similar to that exex-perienced as it passes from the

contraction chamber into the test section. Its size should therefore be

some-what equivalent to some-what it would be in the tunnel test section had it not

been sampled.

The bubble passes through

an

aperture which is hydraulically part

of a smooth pipe the electrical resistance of which is given by the term in

equation (A-1) multiplied by the current.

From an electrical viewpoint, the passage represents three electrodes. The resistors separate the center electrode from the two ground .potential end electrodes. The first resistor consists of a one-mil.-thick mylar washer

separating the first two electrodes. This arrangement produces a resistance of about 5000 ohms when ordinary tap water is used. Small bubbles produce fairly large changes in this value; these changes in turn produce measurable

voltage pulses on the second electrode, which is being driven with a constant current. The amplitude of the voltage pulse is directly proportional to the particle volume. The second resistance consists of a plastic tube 0.375 inches long which produces a resistance of about two megohms in parallel with the first. Small bubbles passing through this aperature do not change its resistance enough to produce any measurable pulse.

A bubble flowing through this sampler will then produce a single

vol-tage pulse whose amplitude is proportional to its size. This information then

yields a size distribution of the bubbles entrained in the sample. A calibra-tion of the probe was determined by dropping single glass beads which had been measured under a microscope through the probe. The reeulting pulse was captured

1.6

1.4

1.2

.4

.2

.05

Bead Volume, Cubic Millimeters

Fig. 2-A - Probe Calibration

5

A-3

The instrument has not as yet been used under test conditions in a

water tunnel. One problem to be expected is that the sample drawn is a very small fraction of the total flow and thus the statistical sample is

smaller than desirable. The size of the probe is fixed by the expected bubble sizes and cannot be enlarged. Long time samples must therefore be taken and tunnel conditions may shift somewhat as a function of time. The effect of the pipe walls upon the bubble size has also been neglected. This will undoubtedly have some effect upon the accuracy of the probe.

LIST OF APPENDIX REFERENCES

Irani, R. R., "Evaluation of Partidle Size Distributions Obtained from Electrolytic Resistivity Changes," Analytical Chemistry, Vol. 32, No.

9,

August 1960.

Copies Organization

40 Commander, Naval Ship Research and Development Center Washington, D. C. 20007, Attn:

Code L41 (39)

Code 513 (1)

2 Commanding Officer, Naval Ship Research and Development Laboratory, Annapolis, Maryland 21402, Attn: Library

2 _{Commanding Officer, Naval Ship Research and Development Laboratory,}

Panama City, Florida 32402, Attn: Library

8 Commander, Naval Ship Systems Command, Department of the Navy, Washington, D. C. 20360, Attn: Code 0342 (1) DEF Code 037 (1) ADF Code 00V (1) F Code 2052 (3) Code PMS 81 (1) ABCDF Code 03412 (1)

*20 _{Director, Defense Documentation Center, 5010 Duke Street, Alexandria,} Virginia 22314

Chief of Naval Research, Department of the Navy,

_Washington,

D. C.

20360, Attn: Mr. Ralph D. Cooper, Code 438

1 Director, Office of Naval Research Branch Office, 495 Summer Street, Boston, Massachusetts 02210

1 _{Director, Office of Naval Research Branch Office, 219 S. Dearborn}

Street, Chicago, Illinois 60604

1 Office of Naval Research Resident Representative, 207 West 24th Street, New York, New York 10011

1 Chief Scientist, Office of Naval Research Branch Office, 1030 East Green Street, Pasadena, California 91101

1 Director, Office of Naval Research Branch Office, 50 Fell Street, San Francisco, California 94102

3 _{Director, Naval Research Laboratory, Washington, D. C., 20390,}