An energy approach to cavitation bubbles near compliant surfaces W. K. Soh Department
of Mechanical
Engineering,
University of Wollongong,
NS W, Australia
A potential flow model for cavitation bubbles near compliant surfaces is presented. The compliant surface is considered to be a layer of linear elastic material, so that there is a twoway transfer of energy between the compliant surface and the fluid. The conservation of energy leads to a nonlinear differential equation; it also brings out two nondimensional parameters, which give relationships between the properties of the compliant surface and the configurations of the bubble. The solutions of the nonlinear differential equation provide illustrations for the behavior of the bubble near a compliant surface and show the effects that compliant surfaces have on the strength of the Kelvin impulse. Keywords: bubbles,
cavitation,
compliant
surfaces,
Introduction One of the significant developments in the study of cavitation is in the dynamics of a single bubble.’ For a bubble near a rigid wall the momentum generated by the bubble is directed toward the wall. Observations on a single bubble near a rigid wall show that it expands and collapses. At the final stage of its collapse the momentum of the bubble appears as a highvelocity microjet of water that impinges on the wall. It is believed that cavitation damages are caused by the impingement of this highspeed microjet. The situation reverses for a bubble near a free surface. The momentum is directed away from the free surface. An integrated property of the momentum, that is, the Kelvin impulse, gives effective estimation of the momentum. The Kelvin impulse is calculated from a simple flow model which represents a bubble by a point source in the growing state and a point sink at a collapse state. The predicted behavior of the bubble is in agreement with experimental observations. The behavior of a cavitation bubble near a flexible surface was first studied by Gibson and Blake.* A compliant surface or a flexible surface will, under certain circumstances, behave like a free surface, which causes the momentum of the bubble to be directed away from the surface. These types of surfaces could have the potential of reducing cavitation damages. A followup work by Gibson and Blake3 defined the relevant parameters for the compliant surface by a set of equiv
Address reprint requests to Dr. Soh at the Dept. of Mechanical Engineering, University of Wollongong, P.O. Box 1144, Wollongong, NSW 2500, Australia. Received 20 March 1991; revised 13 December December 1991
0 1992 ButterworthHeinemann
1991; accepted
18
Kelvin
impulse,
potential
flow
alent mass and stiffness. In this way the response of the bubble near various types of surfaces, such as a perspex plate, different types of rubber, and free surfaces, can be classified by the appropriate range of values of these proposed parameters. Shima et al.4 carried out an experimental investigation on bubbles generated near a composite surface. The composite surface consisted of layers of foam rubber and nitrile rubber (NBR) sheets. By changing the thickness of these layers a range of surface properties, spring constant, and partial mass, as defined by Gibson and Blake,3 could be obtained. Their results show that the migration of the bubble depends on these parameters, the maximum bubble size, and also the relative location of the bubble from the surface. This study reinforces the concept proposed by Gibson and Blake3 that the dynamics of a bubble near a compliant surface is dependent on the dynamic response of the surface. In their concluding remarks, Shima et al.4 stated that the behavior of a bubble near a compliant surface is “ . . . influenced by the difference in characteristic time scales between the response of the surface and the bubble motion.” It is possible to generate a cavitation bubble in a real liquid from a finite amount of energy without incurring infinite pressure and forces. Because of its vast volume, the almost incompressible liquid is capable of providing sufficient “give” to accommodate the displacement of a tiny bubble. For a large bubble, such as a bubble generated in an underwater explosion, there is always the presence of a free surface, which deforms such as to absorb the displaced volume of the bubble. For most experiments, which generate bubbles in containers, the presence of the free surface is essentialotherwise the walls of the container will be forced outward. The same can be said about the incompressible flow analysis for cavitation bubbles; the free surface
Appl. Math. Modelling,
1992, Vol. 16, May
263
Cavitation
bubbles
near compliant
surfaces:
W. K. Soh
has to be included in order to avoid the occurrence of infinite energy and forces. The compliant surface will deform as the pressure on the surface changes. Strictly, there is a small but finite velocity component normal to the surface. However, the kinetic energy associated with this velocity component would be of the same order as the kinetic energy in the compliant material and is assumed negligible. It will be demonstrated later that this additional term of strain energy in the energy equation affects the growth rate of the bubble and thus characterizes the presence of a compliant surface.
X
h d
Free Surface A f/s
A
Bubble
The energy equation Following the point source approach of Blake,5 the presence of a rigid wall can be realized by placing an image source so that the wall lies between the source and its image. It can be shown that this configuration will give rise to an infinite force on the wall; consequently, if the wall is a compliant surface, the corresponding elastic energy of this surface caused by the pulsation of the bubble will also be infinite. In this situation the free surface cannot be ignored. Images of the bubble source as reflected by the free surface as sinks (and vice versa) are introduced. The resulting model, in the cylindrical coordinates (R, X, O), consists of the bubble as a source of strength M(t,) (which is a function of time, ti), located at (0, h, 0); an image for the wall at (0, h, 0); and two sinks for the free surface at (0, 2d  h, 0) and (0, 2d + h, 0). Here, h is the distance between the centroid of the bubble and the wall; and d is the depth of water above the wall. This simple configuration is shown in Figure 1, and the same configuration repeats throughout the Xaxis to ensure that the rigid surface and free surface conditions are satisfied. It also turns out that the force on the wall is finite as well as the strain energy on the compliant surface. The velocity potential is given by
I
Test Surface(Image)
I
I 0
A C
M
Test Surface
d h
I
I

I
S
r
h
Image
0
M
d
Free Surface (Image) *
”

”
d
Test Surface,Md~e
1h
@
Figure 1. Schematic diagram showing one period of the series of bubble images
radius of the bubble is less than h. Consider a horizontal compliant surface submerged in water of depth d. A bubble in water has its centroid h above the compliant surface. Let C be the control surface, which consists of the surface of the bubble, the free surface, and the compliant surface. The kinetic energy of a region bounded by these boundaries is given by
(1) where 6 = 
5
[r* + (X  h + 4dn)*] I’*
“zm
cc
$9 = 
2 [r* + (X + h + 4&r)*]“* n= m
(2)
m
43 =
x [r2 +
(x  2d  h + 4dn)*] “2
In the above integration the shape of the bubble is assumed to be spherical; in other words, an equivalent sphere of the same volume is used to represent the bubble. The results of the integration are obtained by using the theorem due to Gauss for the mean value of the velocity potential.’ This results in expressing the kinetic energy as
n+ %icot
44 =
x [r* + (x
+ 2d  h + 4dn)*] I’*
In the formulation of the energy equation the centroid of the bubble is considered to be stationary. This assumption, as found in experiments,*x6 is reasonable only during the first pulsation of the bubble, and the
264
Appl.
Math.
Modelling,
1992,
Vol. 16, May
nh
(
d )I
(4)
Equation (4) can be used as an approximation for the kinetic energy of a bubble that deviates from spherical shape; in this case the radius, R, will be defined as the mean radius of the bubble. In the present point source model the discrepancy in the velocity potential between a spherical bubble and a nonspherical one re
Cavitation bubbles near compliant surfaces: duces with increased distance. For the moment this assumption is considered to be valid for h larger than R,, the maximum radius of the sphere. Consider the situation for d to be very large compared with h; equation (4) is reduced to
(5) The work done by the bubble on the fluid domain inside the control surface C is the product of its volume and the constant vapor pressure, P,, and is given by
displaced by the depression TP
(7)
The work done by the fluid on the compliant surface is transformed into strain energy on the compliant surface. Consider that the compliant surface is a linear elastic material of thickness 7 and attached above a rigid wall. If E is the Young’s modulus of the compliant surface and P is the pressure distribution exerted by the fluid on the surface, then the strain energy can be expressed as
Sw where the pressure is derived from the unsteady Bernoulli equation. p =
.
w
=
o
WtJ2 2~[&+&] ++3tf’ad’c~ 4 + 2~(p.d~~O,W
E
+m2 2E
yJl(h,
The total energy, W,, is unknown at this stage. However, if R, is assigned to be the maximum radius of a bubble generated near a rigid surface, i.e., E is infinitely large, so that the coefficients of dM/dt and [dMldt12 vanish, then equation (13) will lead to a situation in which the radius is largest when M vanishes. The total energy can then be expressed as wo=FAp~:,
(14)
where AP = P,,,  P,.. The above equation can be written into nondimensional form with respect to the maximum radius of the bubble, R,; the density of the fluid, p; and the pressure difference, AP = (P,,,  PJ. In this way the radius of the bubble r and the time t are nondimensional. It follows that the nondimensional form of equation (13) is given by
r(t)2~ 1
L +L +A,[rf~(t)]~+ C,ti(t)
+ pgd)fi(&)(d
 h)
(10)
= ln
(15)
[,,,,“Ih)]
Application
to a bubble near a rigid wall
The energy equation for a bubble near a rigid surface can be derived from equation (15) by setting A, and C, to zero. The result is
where J,(h, d) is given by
Jdhd)
 r(t)‘] = 0
This energy analysis has brought out two scaling parameters, A, and C,, which account for the properties of the compliant surface. With the addition of terms that contain (dmldt), the energy equation is a nonlinear differential equation for r(t) as compared with the case of rigid surfaces, which gives a firstorder differential equation.
4
+ 4Z(P,,,
 h) (13)
 y[l
w
(12)
p
rr~(t)~ [
(Pam+p&)+
 h)
In the absence of any dissipation of energy, conservation of energy requires that the sum of the kinetic energy and the network done by the fluid should remain constant and equal to W,. Thus by adding equations (5), (6), (7), (lo), and (12), conservation of energy gives
ii/r(t# 2E 271. JO, +7p2
4 WO”t= 5 rR’P,tm
of the compliant surface:
AW,,, = 2EP,,,M(t,)(d
w;,= 4 work done by on sphere, that is, through the raising of the free surface at the atmospheric pressure P,,,, is given by
W. K. Soh
(11)
As the elastic surface is being compressed, the depression it creates will reduce the volume of fluid rises above the calm free surface, causing a readjustment of the work done by the free surface. This work done is equal to the product of P,,, and the volume
(16) If m(t) is expressed
in terms of r, that is,
dr m(t) = 47rr2dt
Appl.
Math.
Modelling,
(17)
1992,
Vol. 16, May
265
Cavitation bubbles near compliant surfaces: equation (16) can be expressed
W. K. Soh
as
l/2 $1
dr z = e1

r3]
1 r4
i
1
+[
r
2~
I
1
A parameter el is introduced here as the result of the square root. It alternates as positive unity (for growing bubble) or negative unity (for collapsing bubble) so that the expression inside the square root of (18) can be maintained positive throughout the computation. For example, eI is initially a positive unity and becomes a negative unity when r reaches the value of 1. This equation replaces the expression for drldt given by Rayleigh. In fact, the Rayleigh equation can be recovered by setting y in (18) to infinity. The period for the first pulsation of the bubble, T,, can be obtained from the integration of (18). Equation (18) is also used in the integration for the Kelvin impulse of the bubble. Blake’ gives the expression for the Kelvin impulse and also defines the buoyancy parameter S as
JpgR,
SE
(19)
AP
The impulse over the duration, T,, for the first pulsation of the bubble can be expressed as follows: Z,(T,) =
FS2r3
1
 m2 dl 16~~~
Blake postulated that the direction of the momentum, and hence the microjet, from the bubble is directly related to the sign of the impulse over the time T,., as given in (20). If the buoyancy term (the term that contains S2) is dominant, the impulse will become positive and hence produce a gross upward momentum from the bubble. On the other hand, if the source term (the term that contains y2) is dominant, the impulse will become negative and hence produce a gross downward momentum from the bubble. Thus the combinations of y and S that make the above integral zero will yield the condition that determines the direction of the microjet. Blake5 invoked the Rayleigh expression for drldt and found that equation (20) vanishes for yS = 0.442.
dm = dt
d(r3> 3m dt =z
(23)
The choice of either of these possibilities
Appl. Math. Modelling,
* +
0
0.1
0.3
0.2
Best (1990) Blake (1988) Present work
0.4
0.5
Delta Figure 2.
Loci of zeroimpulse
for bubble near a rigid surface
In other words, yS < 0.442 indicates that the jet is directed toward the rigid surface, and yS > 0.442 indicates that it is directed away from the rigid surface. The work by Best and Blake8 has refined the above condition in an energy analysis that includes the motion of the bubble and has obtained better agreement with experimental results. In the present energy approach, equations (17) and (18) are substituted into (20) for r and m, and a result very close to that of Best and Blake’ is obtained, except where y is close to but smaller than unity. This is expected, since the assumption of the bubbles being spherical in this analysis is no longer valid for y close to but less than unity. The present result is curve fitted by the equation yS = 0.438  0.1786 The comparison ure 2.
(21)
of these results is plotted in Fig
Solution of the energy equation Since equation (15) is a nonlinear differential equation, it gives two possible expressions for dmldt as shown below, with the two solutions corresponding to a positive or a negative square root:
(22)
where
266
+
(18)
depends
1992, Vol. 16, May
on the initial condition. It can be assumed that initially, the terms related to strain energy vanish, so that the bubble behaves as if it is influenced by a rigid surface. This implies that the positive square root of equation (22) has to be used and that the initial growth rates for
Cavitation bubbles near compliant Table 1. Rm (mm)
7 (mm)
(FPI,
ikEa)
h (mm)
d (mm)
A,
C?
A,IC:
15 15 21 20
2.68 1.34 1.39 1.25
5 5 5 5
10000 10000 10000 20000
30 30 40 40
200 200 200 200
0.00023 0.00011 6.8E05 3.2E05
20.0 10.0 5.0 2.5
5.7E07 l.lE06 2.7E06 5.3E06
4.0
25 ‘I5l2’5 0 6 &r
m=5
(24)
25 ‘
[email protected] 06
which is the growth rate for a bubble near a rigid surface. A situation may result in experimentations, for example, as shown in Table I, in which A,IC: becomes very small. Under this condition the terms inside the square root of equation (22) can be expanded to give the following approximation for dmldt: dm =dt
’ C,
!??(I 3
_
l
+ * * * +
3.5 B
’
‘5
3.0 
I
If
a__
l
2.0:
r3)
ce12.5 Ce5.0 Cello.0 Ce20.0 Ce=O.O
l
I
I
2
3
4
Gamma m2(~+$)]
+0(s)
(25)
It is evident that (25) is virtually independent of A,. Equations (25) and (23) were solved for C, equal to 2.5, 5.0, 10.0, and 20.0. The calculated values of r and m with respect to time were used for the evaluation of the Kelvin’s impulse, as given in (20). The loci where the Kelvin’s impulse is zero are shown in Figure 3. It is evident that as C, increases, the zeroimpulse line (line of zero Kelvin’s impulse) is shifted toward the
41 * * * * +
Ce12.5 Ce15.0 Ce=lO.O Ce120.0 Ce=O.O
2 3:
W. K. Soh
Some examples for the values of A,IC$
r and m are given by r=
surfaces:
Figure 4.
Plots of period, T,, versus y for various values of C,
origin. This indicates that for a given configuration (i.e., given values of y and 6), as C, is reduced in magnitude, the strength of the Kelvin’s impulse of a bubble will be smaller; one can thus expect that the momentum of the impinging microjet will be correspondingly reduced. Figure 4 shows the plots of the period against y for various value of C,. It is interesting to note that these periods are generally longer than the period for a bubble near a right surface (C, = 0). This longer time scale means slower development of the bubble and may indicate a lower velocity for the microjet. When this longer period is considered together with the lower value of impulse, as given in Figure 3, a lower force (hence lower pressure) being exerted on the surface is expected.
;
Concluding
1 t
t
I 00.00
I
I
0.10
0.20
I 0.30
Delta Figure 3. Loci of zeroimpulse face for various values of C,
for bubble near a compliant sur
remarks
It must be noted that Benjamin and Ellis9 had discussed the problems with infinite energy and forces that lead to the use of the concept of Kelvin impulse in the study of cavitation bubbles. The present analysis makes use of a pattern of images to obtain finite force and energy. It can be considered as an artificial means to model an experimental situation in which the free surface is always present.
Appl.
Math. Modelling,
1992, Vol. 16, May
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Cavitation bubbles near compliant surfaces:
W, K. Soh
By considering the exchange of energy between an elastic compliant surface and a cavitation bubble a nonlinear differential equation is derived. This has brought out two parameters, A, and C,, which are relevant to the study of cavitation bubbles near compliant surfaces. The relative magnitude of A, and C, will determine the characteristics of the bubble. An example for very small value of A,ICz is illustrated here. It is evident that the zeroimpulse line, which moves toward the origin of the y&axes as the value of C, increases, follows the expectation of Blake and Gibson’ that the bubble generated near a compliant surface has a greater tendency to be repelled by the surface.
Nomenclature ‘4
= 23n
[,,,;‘I
C,
= 167~s
(d  h)
d E h
depth of water Young’s modulus of compliant surface distance between initial center of bubble and a boundary (rigid surface or compliant surface) Kelvin impulse r d* i = In 1 h(2d  h) 1 nondimensional source (sink) strength
:;(h, 4
4)
Wf,)
m(t) = 
Mt1)
P P, P atm r R &
268
h)]

P
R2, J AP
Math. Modelling,
time
interval of time between the generation a bubble and its first collapse axial coordinate
of
Greek characters buoyancy parameter a=
J
~&nax 
AP
P atm  P,. geometrical parameter; y = h/R,,, components of velocity potential as defined inequations (1) and(i) velocity potential thickness of compliant surface
References
source (sink) strength; volumetric flow rate pressure vapor pressure in a bubble pressure on free surface radial coordinate nondimensional radius; r = R/R, radius of bubble maximum radius of a bubble
Appl.
nondimensional
1992, Vol. 16, May
Blake, J. R. and Gibson, D. C. Cavitation bubbles near boundaries. Annu. Rev. Fluid Mech. 1983, 19, 99123 Gibson, D. C. and Blake, J. R. Growth and collapse ofcavitation bubble near flexible boundaries. Seventh Ausrrulasiun Hydraulics and Fluid Mechunics Conference. Inst. of Eng. (Aust.), Brisbane, August 1980. pp. 283286 Gibson, D. C. and Blake, J. R. The growth and collapse of bubbles near deformable surfaces. Appl. Sci. Res. 1982, 38, 215224 Shima, A., Tomita, Y., Gibson, D. C., and Blake, J. R. The growth and collapse of cavitation bubbles near composite surfaces. J. Fluid Mech. 1989, 203, 199214 Blake, J. R. The Kelvin impulse application to cavitation bubble dynamics. J. Ausf. Math. Sot. Ser. B 1988, 30, 127146 Wong, K. C., Soh, W. K., and Blake, J. R. High speed visualization of vapour cavity near boundaries. Tenth Australasian FIuid Mechanics Conference, Inst. of Eng. (Aust.), Melbourne, December 1989, pp. 227230 MilneThomson, L. M. Theoretical Hydrodynamics, 5th ed. Macmillan, New York, 1968, p. 96 Best, J. P. and Blake, J. R. Underwater explosion bubble dynamics, 1, Kelvin impulse and spherical bubbles. Private communication, 1990 Benjamin, T. B. and Ellis, A. T. The collapse of cavitation bubbles and the pressures thereby produced against solid boundaries. Philos. Trans. R. Sot. London 1966, A260,221240