Franc J-P. Fundamentals of Cavitation

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FUNDAMENTALS OF CAVITATION52

•

is the reference flow velocity at infinity. Within the framework of steady

potential flow theory, the local flow velocity U(s) and the local pressure

coefficient Cp(s) are connected by the B

ERNOULLI equation, which gives:

Us U Cps() ()=-

•

(3.56)

Concerning the initial conditions, it is convenient to inject the nucleus at the

abscissa s

where the pressure coefficient is zero i.e. where the pressure is equal to

the upstream reference pressure. In addition,

is usually chosen to be zero at this

location.

The low-pressure surface of a hydrofoil exhibits a minimum in the pressure

distribution. The dynamic behavior of a nucleus that slides down the low-pressure

surface is described by equation (3.53). It depends primarily upon its critical

pressure p

compared with the minimum pressure p

min

If the minimum pressure is lower than the critical pressure, the nucleus is

destabilized and begins to grow. Whereas the biggest nuclei grow slowly, the

smallest ones explode violently as soon as their critical pressure is reached. By

then, they are larger than their original size by several orders of magnitude.

On the whole, when nuclei of different critical pressures are destabilized, their

size evolution R(t) is practically independent of their initial size R

A particular case of practical interest, as it corresponds approximately to the case

of blades working near their design point, is obtained when the pressure on the

low-pressure side of a foil is practically uniform. If surface tension, viscosity and

gas effects are neglected, the radial velocity

tends to the limit value given by

expression (3.34), or, in non-dimensional form, by:

dR dt

()

•

@--

(3.57)

As the local liquid velocity is connected to the pressure coefficient by the

ERNOULLI equation (3.56), then, in this particular case of a constant pressure

coefficient:

31()

(3.58)

so that the bubble radius increases proportionally to the curvilinear distance s:

()

min

(3.59)

This result agrees fairly well with the experimental behavior of bubbles exploding

on a foil.

3 - THE DYNAMICS OF SPHERICAL BUBBLES 53

Finally, it should be noted that, from the discussion given in chapter 2 (§ 2.3.1), the

transit time Dt should be compared with the thermal time D t

required for heat

transfer. Generally, Dt is far larger than Dt

so that the gas transformation can be

considered isothermal and g must be made equal to unity in the R

AYLEIGH-PLESSET

equation. Therefore, the critical pressure p

for which the nucleus is destabilized is

that given by expression (2.4). However, as soon as the nucleus is destabilized, the

transformation should actually be considered as adiabatic since the volume of the

nucleus increases substantially and heat exchange is no longer significant.

3.7. STABILITY OF THE SPHERICAL INTERFACE

The large values of the velocity

and of the acceleration

˙˙

of the interface found

in the previous sections may affect the stability of the thus far assumed spherical

shape of the interface.

It is well-known that a horizontal plane interface with water below and air above,

with both at rest, is stable under normal conditions in a gravitational field. More

generally, consider a system of two fluids of different densities, separated by a

plane interface, and replace gravity by a given acceleration of the two fluids. The

AYLEIGH-TAYLOR linear stability criterion requires that the acceleration be directed

from the heaviest to the lightest fluid. If this criterion, valid for a plane interface, is

abruptly applied to a growing or collapsing bubble, it should be expected that the

interface remain stable if the liquid acceleration is directed inwards, i.e.:

RR R

˙˙ ˙

(3.60)

Therefore, it is expected that a negative value of

˙˙

as well as a small value of

will promote stability.

The question is to know how this stability criterion is modified in the case of a

spherical interface. The problem was considered by P

LESSET (1954), BIRKHOFF

(1954), PLESSET and MITCHELL (1955) and later by HSIEH (1965). Here, we follow the

linearized P

LESSET method. Details of the calculation can be found in PLESSET's

original paper. Only the main conclusions are presented here.

The liquid is assumed inviscid and the flow irrotational. The bubble interface is

supposed distorted from its spherical shape

rRt= ()

and the stability of this

distortion is analyzed. As any arbitrary function of the two spherical coordinates,

angles q and f, can be developed in a series of spherical harmonics, P

LESSET

assumed the following shape for the bubble interface:

(,, ,) () () (,)Rr t Rt a t Y

qf qf=+

(3.61)

where the amplitudes a

of each spherical harmonic Y

are small in comparison with

R(t), consistent with a linear analysis. The kinematical condition on the interface

FUNDAMENTALS OF CAVITATION54

together with the generalized BERNOULLI equation require that the amplitude

coefficients a

(t) satisfy the differential equation:

˙˙

()

˙˙

()()( )

nnnS

nn n

+---

-++

112

(3.62)

For simplicity, P

LESSET introduced the following change of variables:

at t

() ()

(3.63)

With the new variable a

, the differential equation (3.62) is simplified and becomes:

˙˙

()aa

nn n

Gt-=0

(3.64)

with

nRRRn n n

() ( )

˙˙ ˙

()()( )=++--++

112

(3.65)

If we neglect the influence of the coefficient

(/)

in equation (3.63), then the

stability of the shape of the interface depends simply on the growth rate of a

Stability requires that G

(t) be negative, otherwise an exponential growth of the

coefficient a

would be expected, leading to instability. In conclusion, the spherical

interface is stable if the following condition is met for all the spherical harmonics:

()

˙˙ ˙

()()( )21

112

nRRRn n n

++--++ <

(3.66)

This condition is not very different from the approximate condition (3.60) although

some new terms are present. It shows that negative values of

˙˙

associated with

relatively small values of the interface velocity

are favorable to stability. On the

contrary, positive values of

˙˙

and large values of the interface velocity will promote

instability. However, attention must be paid to the last term, which becomes large

for high order harmonics and small bubble radius. Thus, a general conclusion is not

straightforward.

As an example, consider the final stage of the R

AYLEIGH collapse under a constant

overpressure. Equation (3.18) is written:

112

from which one obtains:

23112 112

RG t n

nnn

( ) . ( )( )( )@-

-- + +

3 - THE DYNAMICS OF SPHERICAL BUBBLES 55

This expression, which is negative, shows that the spherical shape tends to be

stable in the final stages of the so-called R

AYLEIGH collapse.

However, in the real case of a bubble containing a non-condensable gas, the

phenomenon of regrowth will change the conclusion. Close to the end of collapse,

just before any rebound,

will reach large negative values with

˙˙

R > 0

. From the

stability argument, it appears that both conditions are favorable to the development

of instabilities.

On the contrary, at the end of the growth phase, before any collapse,

˙˙

R < 0

while

is positive and small, so that the spherical shape is likely to be stable.

In a series of experiments, J

OMNI et al. (1999) observed successive collapses and

rebounds of very small bubbles of radii of the order of a micrometer. This suggests

that the bubble interface was stable throughout its lifetime. In other cases

encountered in hydraulics, the explosion of nuclei moving along blades usually

appears to be stable, as their interface remains smooth and regular. However, their

subsequent collapse looks rather unstable, leading to the formation of several

smaller bubbles and resulting in a multi-peak acoustic signal. Of course, these last

observations are not easily comparable to the theoretical case analyzed here since

the actual flow near a blade is turbulent and the pressure field around the bubble is

obviously not uniform.

REFERENCES

BESANT W. –1859– Hydrostatics and Hydrodynamics. Cambridge University Press.

IRKHOFF G. –1955– Stability of spherical bubbles. Quart. Applied Math. 13, 451 sq.

OROTNIKOVA M.I. & SOLOUKIN R.I. –1964– Sov. Phys. Acoust. 10, 28-32.

OLE R.H. –1948– Underwater explosions. Princeton University Press.

RIGHTON D.G. & FFOWCS WILLIAMS J.E. –1969– Sound generation by turbulent

two-phase flow. J. Fluid Mech. 36, part 3, 585-603.

RANC J.P., AVELLAN F., BELAHADJI B., BILLARD J.Y., BRIANÇON-MARJOLLET L.,

RÉCHOU D., FRUMAN D.H., KARIMI A., KUENY J.L. & MICHEL J.M.

–1995– La cavitation. Mécanismes physiques et aspects industriels.

Presses Universitaires de Grenoble – Collection Grenoble Sciences, 580 p.

SIEH D.Y. –1965– Some analytical aspects of bubble dynamics. J. Basic Eng.

87, 991-1005.

OMNI F. –1997– Étude des phénomènes hydrodynamiques engendrés dans les

liquides diélectriques par un champ électrique très intense.

Thesis, Grenoble University (France).

FUNDAMENTALS OF CAVITATION56

JOMNI F., AITKEN F. & DENAT A. –1999– Dynamics of microscopic bubbles

generated by a corona discharge in insulating liquids: influence of pressure.

J. Electrostatics 47, 49-59.

ELLER J.B. & MIKSIS M. –1980– Bubble oscillations of large amplitude.

J. Acoust. Soc. Am. 68, 628-623.

ARLITZ U., ENGLISH V., SCHEFFCZYK C. & LAUTERBORN W. –1990–

Bifurcation structure of bubble oscillators. J. Acoust. Soc. Am. 88, 2 sq.

LESSET M.S. –1954– On the stability of fluid flows with spherical symmetry.

J. Appl. Phys. 25, 96-98.

LESSET M.S. & MITCHELL T.P. –1955– On the stability of the spherical shape of

a vapor cavity in a liquid. Quart. Appl. Math. 13, 419-430.

PLESSET M.S. & PROSPERETTI A. –1977– Bubble dynamics and cavitation.

Ann. Rev. Fluid Mech. 9, 145-164.

ORITSKY H. –1952– The collapse or growth of a spherical bubble or cavity in

a viscous fluid. Proc. 1

US Nat. Congress of Appl. Mech, 813 sq.

AYLEIGH (Lord) -1917- The pressure developed in a liquid during the collapse of

a spherical cavity. Phil. Mag. 34, 94 sq.

SHIMA A. & FUJIWARA T. –1980– The collapse of bubbles in compressible hydraulic

oils. J. Acoust. Soc. Am. 68(5) 1509-1515.

HU S.S. –1952– Note on the collapse of a spherical cavity in a viscous

incompressible fluid. Proc. of the 1

US Nat. Congress of Appl. Mech.

MEREKA P., BIRENIR B. & BANERJEE S. –1987– Regular and chaotic bubble

oscillations in periodically driven pressure fields. Phys. Fluids 30, 11 sq.

ZABABAKHIN E.I. –1960– The collapse of bubbles in a viscous liquid.

PMM 24(6) 1129-1131.

4. BUBBLES

IN A

NON-SYMMETRICAL ENVIRONMENT

4.1. INTRODUCTION

In industrial situations, the rather ideal conditions considered in chapter 3 are often

far from being met. In particular, spherical symmetry can be disturbed by gravity,

pressure gradients, flow turbulence, etc.

In this chapter, we emphasize inertial effects and consider two main cases in which

departure from sphericity is expected:

— the case of a bubble, which moves in a static liquid and then explodes or

collapses (§ 4.2),

— the case of the explosion or collapse in the vicinity of a solid wall or of a free

surface (§ 4.3).

In both cases, the lack of spherical symmetry results from a non-uniform pressure

distribution in the liquid domain around the interface, while the gas pressure and

the vapor pressure inside the bubble are practically uniform. In general, the higher

pressure region outside causes a re-entrant jet to appear inside the bubble.

The last section of this chapter is dedicated to the case of a spherical bubble that

crosses a region of pressure gradient. The problem is to estimate the path of the

bubble, taking into account the effects of gravity and viscous drag, and then to

explain the screen effect by which bubbles can be deflected from the streamlines

of the base flow as they approach a solid body.

4.2. MOTION OF A SPHERICAL BUBBLE IN A LIQUID AT REST

4.2.1. TRANSLATION OF A SOLID SPHERE IN A LIQUID AT REST

We consider first the case of a sphere whose motion is controlled by an external

force

in a non-viscous liquid at rest (fig. 4.1). Gravity is ignored and the flow

due to the moving sphere is assumed irrotational. If the sphere moves with

velocity

Wt()

, liquid resistance is given by (see e.g. BATCHELOR 1967):

RMW

(4.1)

FUNDAMENTALS OF CAVITATION58

where M is the virtual mass (or added mass) of the immersed body, equal to half of

the mass of the displaced fluid:

MR=

(4.2)

4.1

Translation of a solid sphere

in a liquid at rest

The equation of motion of the sphere of

mass m is:

()

mMWF+=

(4.3)

The effect of the fluid on the movement of the

sphere under the action of the applied force

is the same as if the mass of the sphere were increased by one-half the mass of the

displaced fluid. Note that equation (4.3) is an illustration of d'A

LEMBERT's paradox,

according to which the drag of a moving body in a non-viscous fluid is zero if the

translation velocity is constant.

4.2.2. TRANSLATION WITH SIMULTANEOUS VOLUME VARIATIONS

Suppose now that the sphere is deformable and its radius R is variable due to

external effects. Deviations from symmetry in the pressure distribution will result

from translation of the sphere and variation of its volume. The translation is

considered linear and the translation velocity is W(t).

The velocity potential results from the superposition of the effects of volume

variation and translation and takes the classic form:

=- -

cos

(4.4)

The spherical components of the absolute velocity

Vgrad=j

are:

cos

sin

(4.5)

It is simple to check that the boundary condition on the sphere is fulfilled as the

normal velocity is equal to

cosRW+q



4 - BUBBLES IN A NON-SYMMETRICAL ENVIRONMENT 59

The pressure is known from the generalized BERNOULLI equation:

∂

++=

•

rrt

Wgrad

in which the term

∂∂j /t

is the time derivative in the frame of reference moving

with the bubble and

∂∂-jj/.t W grad

is the time derivative in the fixed frame of

reference.

After some calculations, one obtains:

pr t p

(,,)

˙˙ ˙ ˙ ˙

cos

( sin )

sin

=+ -

++ -

+---

•

(4.6)

At the surface of the sphere, the pressure distribution is given by:

pR t p

RR R WR WR W

(,,)

˙˙ ˙ ˙ ˙

cos sin

=+ + +

()

•

222

(4.7)

The lack of symmetry is due to the term cosq. Integration over the surface of the

sphere gives the resulting force:

R R WR WR

dMW

=- +

()

˙˙

()

(4.8)

where M is the virtual mass already defined in equation (4.2).

The equation of motion of the sphere is then:

dMW

F+=

()

(4.9)

In the case of a sphere of constant radius, equation (4.9) reduces to equation (4.3).

In the case of a constant translation velocity W, if the sphere is shrinking (

R < 0

the resulting force R

is positive, so that the sphere is pushed by the liquid. This is

due to higher pressure values at the back of the sphere i.e. for

qp> /2

. Meanwhile,

F is negative which means that the motion of a sphere at constant velocity requires

an adverse force if its volume decreases.

4.2.3. APPLICATION TO BUBBLES

It is assumed here that the bubble remains spherical. This assumption is particularly

well fulfilled for small bubbles, since surface tension, which tends to give a spherical

shape to the interface, becomes more efficient for small radii of curvature.

FUNDAMENTALS OF CAVITATION60

Equation (4.9), with

F = 0

since the bubble motion is free, and

m @ 0

due to the

negligible mass of gas within it, becomes:

MW MW MW()

˙˙

=+=0

(4.10)

Hence, the total virtual impulse MW of the bubble remains constant.

If the bubble collapses

(

)M < 0

, then

W > 0

, i.e. the bubble accelerates. This effect

is called the rocket effect. Inversely, an exploding bubble will decelerate. Hence, a

bubble of varying volume tends to move relative to the ambient liquid. The present

effect is purely inertial, since all other effects, such as gravity, have been neglected.

One can conjecture that a bubble will depart from its spherical shape if the

maximum pressure difference between two points on the interface is larger

than the characteristic pressure difference

2S R/

due to surface tension. From

equation (4.7), the pressure difference between the back and the front of the

bubble is

˙˙

WR WR+

()

, so that the condition of deviation from sphericity can be

written:

˙˙

WR WR

()

>>3

(4.11)

As an example, a bubble of constant radius accelerated at

/Wms= 10

in water

will loose its spherical shape if its radius is greater than about

238SW mm/

.r@

The problem is then to describe the subsequent evolution of the shape of the

interface. This is done in the following section. However, in the present case of a

collapsing, accelerated spherical bubble, it is obvious that the back of the interface,

which is exposed to higher pressures, will undergo a deformation such that the

liquid will tend to penetrate the bubble and give rise to a re-entrant jet.

4.3. NON-SPHERICAL BUBBLE EVOLUTION

4.3.1. PRINCIPLE OF PLESSET-CHAPMAN NUMERICAL MODELING

Consider an initially spherical bubble, close to a solid wall, which collapses under

a constant pressure p

•

. The proximity of the wall will alter the sphericity of the inter-

face. P

LESSET and CHAPMAN (1971) computed the time evolution of the bubble for

various distances from the wall. The principle of the simulation is presented below.

1. To be rigorous, the derivation of equation (4.11) should refer to a solid sphere which is

removed instantaneously, generating an initially spherical cavity in the liquid. At this

initial time, the velocities and acceleration in the liquid are unchanged and the initial

trend for the further deformation of the cavity is given by this pressure difference.

4 - BUBBLES IN A NON-SYMMETRICAL ENVIRONMENT 61

At a given instant t, the velocity potential j is calculated from the LAPLACE

equation

Dj = 0

using a classical numerical method (e.g. boundary elements),

together with the appropriate boundary conditions on the wall, the bubble

interface and at infinity. The velocity on the interface is deduced from the velocity

potential using the equation

Vgrad=j

and the lagrangian derivative of the

potential

ddtj /

at the bubble surface is computed from the generalized BERNOULLI

equation, giving:

dt t

Vgrad

interface

∂

•

(4.12)

Here, surface tension is ignored and the bubble is assumed to contain only vapor at

vapor pressure p

As the velocity of the liquid particles on the interface is known at time t, the new

position of the bubble interface at

tdt+

can be readily deduced assuming that the

interface nodes are moving with fluid velocity

. In addition, the new value of the

velocity potential on the interface can be computed from equation (4.12). Then, the

APLACE equation is solved at

tdt+

using the updated boundary conditions, and

so on. Therefore, computation of the time evolution of the bubble appears as a series

of boundary value problems each one separated by a time step.

4.3.2. SOME GENERAL RESULTS

Collapse of a bubble close to a wall

Figure 4.2 presents the computational results of PLESSET and CHAPMAN (1971) for a

bubble of 1 mm initial radius whose center is initially 1 mm from the wall.

During collapse, the interface close to the wall tends to flatten while a hollow

develops on the opposite face with a change in curvature under very high

pressure. Then, a re-entrant jet, directed towards the solid wall, develops, with a

rapidly increasing velocity. Finally, the jet pierces the bubble and strikes the solid

wall.

Because of the high velocities involved, the re-entrant jet is often considered as a

possible hydrodynamic mechanism of cavitation erosion. In general, the lifetime of

the bubble increases due to the presence of the wall. The particle paths during the

growth and subsequent collapse are shown in figure 4.3 reproduced from B

LAKE

and GIBSON (1987).