Franc J-P. Fundamentals of Cavitation

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FUNDAMENTALS OF CAVITATION62

Shape Time Velocity (m/s)

A 0.63 7.7

B 0.885 19

C 0.986 42

D 1.013 65

E 1.033 100

F 1.048 125

G 1.066 129

H 1.082 129

I 1.098 128

Wall

Initial sphere

h/R

= 1

J 1.119 128

4.2 - Numerical results of PLESSET and CHAPMAN (1971)

for the collapse of a bubble near a wall

The time is non-dimensionalized using the reference time

•

, which is

close to the R

AYLEIGH collapse time. The velocity given in the table is the velocity of

the upper point of the interface on the axis of symmetry. The initial distance h from

the bubble center to the wall is equal to the initial bubble radius R

4.3

Theoretical calculations

for the growth and collapse

of a vapor bubble near a wall

for

gg= =hR

(where h is the distance to the wall

and R

is the maximum radius)

The diagram shows particle paths

during growth and collapse.

[from B

LAKE & GIBSON, 1987]

Reprinted, with permission, from the

Annual Review of Fluid Mechanics,

(www.annualreviews.org).

Collapse of a bubble close to a free surface

If the bubble collapses close to a free surface, a re-entrant jet develops in the

direction opposite to the free surface. The behavior of the free surface itself

depends on the value of the ratio h, of the bubble's maximum radius to the initial

2.0

1.6

1.2

0.8

0.4

0.0

– 1.2 – 0.8 – 0.4 0.0 0.4 0.8

Rigid boundary

4 - BUBBLES IN A NON-SYMMETRICAL ENVIRONMENT 63

distance of its center from the free surface [CHAHINE 1982]. For

h<03.

, the free

surface is not greatly disturbed, whereas for larger values it is violently disturbed

by a counterjet (fig. 4.4).

4.4

Schematic view of the collapse

of a bubble near a free surface

with a counterjet

At the beginning of the collapse, the

liquid particles move preferentially

towards regions where they encounter

the weakest resistance. This is either the free surface or the face opposite the wall.

This movement generates a high pressure, which triggers the re-entrant jet. Hence,

it is possible to qualitatively anticipate the place where the re-entrant jet will form

by identifying the less constrained region.

Collapse of a bubble confined between two walls

4.5

Schematic view of the collapse

of a bubble between two parallel walls

In the case of a nucleus situated in the

middle of a very confined domain

between two parallel walls, growth is fastest in the directions parallel to the wall.

During the subsequent collapse phase, the reverse occurs: there is a rapid

equatorial contraction and the bubble takes the shape of an hourglass, before

splitting into two symmetric bubbles. Each one then develops a re-entrant jet

directed towards the nearest wall.

These results were obtained from a large number of studies, both numerical and

experimental. The most important of them include: N

AUDÉ & ELLIS (1961), ELLIS

(1965), BENJAMIN & ELLIS (1966), PLESSET & CHAPMAN (1971), LAUTERBORN & BOLLE

(1975), BLAKE & GIBSON (1981, 1987), CHAHINE (1982), TOMITA & SHIMA (1986),

HAHINE (1990a), ALLONCLE, DUFRESNE & TESTUD (1992), BLAKE (1994), ISSELIN et

al. (1996).

Other situations were also studied, such as the collapse of a bubble in a shear layer,

or on the axis of a rotating flow [C

HAHINE 1990b, YAN & MICHEL 1998]. In the latter

case, conservation of the moment of momentum prevents the liquid particles situated

on the interface from reaching the axis of rotation, so that a complete collapse of

the bubble is not possible. This point was experimentally checked by F

ILALI (1997).

FUNDAMENTALS OF CAVITATION64

4.3.3. BLAKE'S ANALYTICAL APPROACH

Principle

This approach is based on the concept of the KELVIN impulse, which was proposed

to get round a theoretical difficulty encountered in hydrodynamics when the

motion of a solid body in an infinite fluid domain is considered. Due to the non-

uniform convergence of integrals relative to the fluid momentum, it is not possible

to apply the momentum theorem directly to a fluid and body system. The impulse

concept was extended by B

LAKE and his co-authors (see BLAKE & GIBSON 1987, for

example) to the case of the evolution of a bubble in a non-symmetric flow field, in

order to give a rather simple analysis of the re-entrant jet and its direction. This

approach is developed within the framework of the potential flow theory.

The liquid domain D is considered to extend between the bubble with boundary S,

a large sphere S

at infinity and a boundary S

representing a wall or a free surface

according to the case under examination (fig. 4.6). The total liquid momentum is:

M Vdv grad dv dS

SS S

== =

ÚÚÚ ÚÚÚ ÚÚ

rrj rjn

(4.13)

where j is the velocity potential and

the outward unit vector, normal to the

surface.

4.6

Domain of integration

for B

LAKE's analytical approach

The momentum equation takes the

following integral form:

(4.14)

where

stands for the total forces resulting from the pressure on the three

boundaries:

FpdS

SS S

ÚÚ

(4.15)

The contribution from the closed surface of the bubble is zero as the pressure on

the interface is assumed constant and equal to the vapor pressure.

Consider now the quantity

defined by:

MdS

ÚÚ

rjn

(4.16)

4 - BUBBLES IN A NON-SYMMETRICAL ENVIRONMENT 65

With reference to equation (4.13),

can be interpreted as a kind of momentum

attached to the bubble only. This can be termed "bubble momentum". Then, the

integral momentum equation (4.14) can be written:

(4.17)

with

FpdS

=- -

{}

ÚÚ ÚÚ

nrjn

01 01

(4.18)

The calculation is somewhat tedious and the details are given in the appendix.

Finally,

is given by the following equivalent expressions:

rrr

grad

grad dS

∂j

∂n

nr n

∂j

∂n

(4.19)

can be considered as a force due to the presence of the boundary S

, which

changes the bubble momentum

according to equation (4.17). Thus, the bubble

momentum is related to the nature of the boundary S

If the boundary S

is far from the bubble, the S

integral is zero and one obtains:

= 0

(4.20)

which extends the validity of equation (4.10) of conservation of the bubble

momentum to the case of a bubble of any shape.

4.7

Bubble near a plane boundary

In the case of a plane boundary S

with unit normal

vector

(fig. 4.7) and a spherical bubble, the flow

is axisymmetric. From equation (4.19), the force

has only an axial component given by:

Fruudr

px r x

()

•

(4.21)

where u

and u

are the axial and radial components of the velocity respectively.

– h

FUNDAMENTALS OF CAVITATION66

The case of a solid wall

In the case of a solid wall, the flow is simulated by a source located at the bubble

center and its image with respect to the wall, so that the slip condition

= 0

fulfilled on the wall. Hence, the velocity potential is given by:

[]

xh r xh r

()

() ()

(4.22)

where h is the distance of the bubble center from the wall and q(t) is the source

intensity. On the wall S

, the axial and radial components of the velocity are given

by:

qt r

∂j

∂

∂j

∂

2232

()

(4.23)

so that

qt rdr

•

223

()

(4.24)

The bubble momentum has only an axial component, which is given by:

MFdt

qtdt

Sx px

ÚÚ

()

(4.25)

Therefore, the bubble momentum is positive and directed toward the wall, which

indicates that the re-entrant jet is necessarily directed toward the wall.

The case of a free surface

If the boundary S

is a free surface, then the boundary is free to expand in the

axial direction x and the bubble collapse induces no radial velocity. Therefore, the

boundary condition is assumed to be

= 0

on S

. This condition is fulfilled simply

by changing the sign of the second term of the velocity potential (expression 4.22).

In other words, the image source is replaced by a sink. Then, the flow velocity on

the free surface is:

qt h

∂j

∂

∂j

∂

()

2232

(4.26)

so that the force is:

qth

rdr

•

223

()

(4.27)

4 - BUBBLES IN A NON-SYMMETRICAL ENVIRONMENT 67

and the bubble momentum is:

MFdt

qtdt

Sx px

==-

ÚÚ

()

(4.28)

Hence, the bubble momentum M

is negative, so that the direction of the re-

entrant jet is opposite to the free surface.

Other configurations

If the plane

x = 0

is an interface between two liquids of different densities, r

for

x < 0

and r

for

x > 0

, then in expression (4.25), M

is modified by changing r into

()/()rr rr

21 21

. Thus, this factor gives the sign of M

and is an indicator of the

direction of the re-entrant jet.

Other dissymmetric effects, such as gravity, the vicinity of a stagnation point, or

the effect of an elastic or visco-elastic wall, can also be analyzed by the present

method (see B

LAKE & GIBSON 1987).

4.4. THE PATH OF A SPHERICAL BUBBLE

It is supposed here that the sources of dissymmetry considered in section 4.3 are

negligible, so that the bubble remains spherical. We look for the trajectory of the

bubble center in a flowing liquid.

Let

be the liquid velocity in the absence of the bubble and

the velocity of the

bubble center (fig. 4.8). The slip velocity

is defined by

rr r

WV V

4.8

Moving bubble

in a flowing liquid

The total mass of vapor and

gas inside the bubble, i.e.

the mass of the bubble, is

assumed constant, but its

volume can vary due to the

pressure variations it undergoes during its movement. If the density of the

gas/vapor mixture inside the bubble is r

, the mass of the bubble is:

= 2

(4.29)

FUNDAMENTALS OF CAVITATION68

The quantity M is the virtual mass defined by equation (4.2) so that the

quantity

2M/r

is simply the bubble volume.

The equation of motion of the bubble center is:

mV m g P D

BB B

=++

(4.30)

The first term on the right-hand side is the weight of the bubble, the second term,

, stands for the pressure forces and the third,

, is the viscous drag. Such a

superposition is currently used in aeronautics for the estimate of the force on a

slender wing.

The viscous drag is given by the expression:

DC R

=- (Re ) p

(4.31)

in which

(Re )

stands for the drag coefficient of the bubble in its motion

relative to the liquid. Re

is the bubble REYNOLDS number based on the relative

velocity W.

According to the early experiments of H

ABERMAN and MORTON (1953), the drag of

spherical bubbles is identical to that of solid spheres with the same radius. Thus C

is taken from data relative to solid spheres. For small bubbles whose REYNOLDS

number in the relative motion is smaller than about one, the drag is given by the

TOKES formula:

DRW= 6pm

(4.32)

so that the drag coefficient is:

(4.33)

For slightly larger R

EYNOLDS numbers, the OSEEN formula can be used:

8Re

(4.34)

Other empirical expressions are available when the R

EYNOLDS number becomes

large.

The determination of the pressure force

is more complex. We adopt here a

simplified and rather intuitive approach. It is assumed that the pressure force

results from the superposition of two effects,

— a combined slip/rocket effect, and

— a flow effect, so that the resulting pressure force is written:

rr r

PP P=+

(4.35)

4 - BUBBLES IN A NON-SYMMETRICAL ENVIRONMENT 69

To estimate the slip/rocket effect, we consider the bubble to have the slip velocity

in the liquid which is itself assumed at rest at infinity. From section 4.2.2, the

pressure force acting on the bubble in this configuration is:

dMW

()

(4.36)

where M is the virtual mass defined by equation (4.2). This equation is the

generalization of equation (4.8) in vector form. It takes into account both the

volume variation of the bubble and its slip with respect to the liquid.

To estimate the flow effect, it is assumed that the pressure force on the bubble due

to the liquid movement is the same as the pressure force acting on a liquid globule

of the same radius that could exactly replace the bubble and move with the liquid,

without slipping. For such a configuration, the momentum balance of the liquid

globule can be written:

dP Mg

(4.37)

The left-hand term is the variation of momentum of the liquid globule whereas the

right-hand terms stand for the pressure and the gravity forces respectively. The

coefficient 2 in the last term results from the fact that the virtual mass is only half

the mass of the displaced fluid for a sphere.

From the previous equation, one obtains:

dMg

-rt

(4.38)

Considering the small size of the globule and assuming that the radius of

curvature R' of the streamline is large enough with respect to the globule radius R,

then the time derivative of the velocity can be considered constant over the entire

globule. One thus obtains the approximate relation:

PMVMg

22@-

(4.39)

This expression is valid to second order in

RR/'

SIEH gives another form to the pressure force

. Using the EULER equation:

ggradp

(4.40)

he obtained the equivalent form:

Pgradp

=-ᐂ

(4.41)

where ᐂ stands for the bubble volume.

FUNDAMENTALS OF CAVITATION70

Finally, the total pressure force acting on the bubble can be estimated as follows:

rr r r

dMW

MVMgMVVMVMVMg

LBLBL

()

+-=--

()

-+ -22 32

˙˙

(4.42)

Although this estimate was obtained in a rather intuitive way, it is identical to the

one found by V

OINOV (1973), using a more theoretical approach.

Taking into account equations (4.30) and (4.42), the following equation of motion

for the bubble center is obtained:

MVMgMVMVVD

LBL

21 3+

=- -

+--

()

rrrr

˙˙

(4.43)

The first term on the right-hand side is the sum of the bubble weight and the

buoyancy force. The second term is due to the pressure gradient in the liquid, the

third to the effect of bubble volume variations, the so-called rocket effect, and the

last term is the viscous drag on the bubble. Note that equation (4.43) has the trivial

solution

if it is assumed that

Usually, the terms in

are negligible, so that equation (4.43) reduces to the

SIEH equation (see the reference in chapter 3 and also JOHNSON & HSIEH 1966):

MV Mg MV M V V D

BLBL

rrrr

˙˙

=- + - -

()

+23

(4.44)

The velocity

, which characterizes the carrier flow, is assumed to be known,

while the virtual mass M, which principally depends upon the bubble radius R, is

deduced from the R

AYLEIGH-PLESSET equation. The solution of equation (4.43) or

(4.44) coupled with the R

AYLEIGH-PLESSET equation allows us to compute the

bubble velocity

and hence, the bubble path.

Other approaches have been used to address this problem. V

ILLAT (1943) and

OISSEY (1972) give the following expression for the total force on a sphere of

constant radius, using a no-slip condition at the boundary, at small R

EYNOLDS

numbers:

rrrrr

TRVVMVVMgR

Vu Vu

LB LB

()

6326

pm pmr

˙˙

()

(4.45)

The first term is S

TOKES drag. The last term is called BASSET drag. It represents the

effect of the evolving viscous wake on the motion of the sphere itself. If we adopt

the model of a sphere with negligible mass, as is the case for a bubble, we have

T = 0

, which gives:

M V M g M V R V V Basset drag

BLBL

rrr

˙˙

=- + - -

()

+23 6pm

(4.46)

4 - BUBBLES IN A NON-SYMMETRICAL ENVIRONMENT 71

The term

()

MV V

must be added to the right-hand side of equation (4.46) in

the case of a bubble of variable radius in order to take the rocket effect into

account. Then, equation (4.46) reduces to the H

SIEH equation (4.44) considering

that the last two terms represent the total drag

Evaluation of the B

ASSET and other terms in equation (4.46) can be found in an

experimental study by S

RIDHAR and KATZ (1995). These authors have added a term

to take into account the lift caused by nuclei passing through a region of high

vorticity.

REFERENCES

ALLONCLE A.P., DUFRESNE D. & TESTUD P. –1992– Etude expérimentale des bulles

de vapeur générées par laser. La Houille Blanche 7/8, 539-544.

BATCHELOR G.K. –1967– An introduction to fluid dynamics.

Cambridge University Press.

LAKE J.R. –1994– Transient inviscid bubble dynamics. Proc. 2

Int. Symp.

on Cavitation, Tokyo (Japan), 9-18.

LAKE J.R. & GIBSON D.C. –1981– Growth and collapse of a vapor cavity near

a free surface. J. Fluid Mech. 111, 123-140.

LAKE J.R. & GIBSON D.C. –1987– Cavitation bubbles near boundaries.

Ann. Rev. Fluid Mech. 19, 99-128.

ENJAMIN T.B. & ELLIS A.T. –1966– The collapse of cavitation bubbles

and the pressures thereby produced against solid boundaries.

Phil. Trans. Roy. Soc. London A 260, 221-240.

HAHINE G.L. –1982– Experimental and asymptotic study of non-spherical bubble

collapse. Appl. Sci. Res. 38, 187-197.

HAHINE G.L. –1990a– Numerical modeling of the dynamic behavior of bubbles

in non-uniform flow fields. Proc. ASME Symp. on Numerical Methods

for Multiphase Flows, FED 91, Toronto (Canada), 57-65.

HAHINE G.L. –1990b– Non-spherical bubble dynamics in a line vortex.

Proc. ASME Cavitation and Multiphase Flow Forum, FED 98, Toronto (Canada),

121-127.

LLIS A.T. –1965– Parameters affecting cavitation and some new methods for

their study. Cal. Inst. Techn. Hydro. Labo, Rpt E115-1.

FILALI E.G. –1997– Étude physique de l'implosion axiale de tourbillons cavitants

érosifs formés dans une chambre tournante. Thesis, Grenoble University (France).

FOISSEY C. –1973– Théorie asymptotique de la cavitation naissante. Ensta, Rpt 014,

Paris (France).