A numerical simulation of the collapse of a cloud of bubbles has been used to demonstrate the development of an inwardly propagating shock wave which grows rapidly in magnitude.  The fully non-linear nonbarotropic homogeneous flow equations are coupled with single bubble dynamics and solved by a stable numerical scheme.  The computational results demonstrate the structure of the shock wave as well as its strengthening effect due to the coupling of the single bubble dynamics with the global dynamics of the flow through the pressure and velocity fields.  This appears to confirm the speculation of Morch and his co-workers that such shock formation is an important part of cloud collapse

Brennen, Christopher E.

Wang, Y.-C.

English

Caltech Authors - Main

FED-Vol. 194, Cavitation and Multlphase flow 
ASME 1994 
SHOCK WAVE DEVELOPMENT IN 
THE COLLAPSE OF A CLOUD OF BUBBLES 
Y.4. Wang and Christopher E. Brennen 
California Institute of Technology 
Pasadena, California 
ABSTRACT 
A numerical simulation of the collapse of a cloud of 
bubbles has been used t o  demonstrate the develop- 
ment of an inwardly propagating shock wave which 
grows rapidly in magnitude. The fully non-linear non- 
barotropic homogeneous flow equations are coupled 
with single bubble dynamics aud solved by a stable 
numerical scheme. The computational results demon- 
strate khe structure of the shock wave as well as its 
strengthening effect due t o  the coupling of the single 
bubble dynamics with the global dynamics of the flow 
through the pressure and velocity fields. This appears 
to  confirm the speculation of Morch and his co-workers 
that such shock formation is an important part of cloud 
collapse. 
Nomenclature 
A Dimensionless radius of the bubble cloud 
A. Dimensionless radius of the bubble cloud at the 
undisturbed reference condition 
Cp Pressure coefficient, (p - y o ) / i p l i 2  
C p ,  Pressure coefficient a t  infinity, ( y ,  - p ~ ) / ~ p U ~  
Cp,,, Minimum pressure c.oeficient a t  infinity 
Reference body size 
Dimensionless bubble radius 
Initial radius of babble a t  undisturbed reference 
condition 
Reynolds number, I i R o / v  
Surface tension of the liquid 
Reference velocity of the flow 
Weber number, p U % ~ / S  
Polytropic index of behavior of the gas inside 
the bubble. 
Fluid pressure 
Fluid pressure at undisturbed referewe condition 
Vapor pressure inside the bubble 
Pressure a t  infinity 
Dimensionless Eulerian radial coordinate 
measured from the center of cloud 
Dimensionless Lagrangian radial coordinate 
measured from the center of cloud and equal 
to r a t  undisturbed reference condition 
Dimensionless time 
Dimensionless duration of the low pressure 
perturbation 
Dimensionless radial velocity of fluid 
Void fraction of the bubbly mixture 
Void fraction of the bubble mixture at the 
undisturbed reference 
Density of liquid 
Cavitation number, (po  - p , ) / i p ~ "  
Dimensionless bubble population per unit 
liquid volume 
Kinematic viscosity of the liquid 
Introduction 
Much recent interest has focused on the dynamics and 
acoustics of finite clouds of cavitation bubbles because 
of the very destructive effects which are observed t o  oc- 
cur when such clouds form and collapse in a flow (see, 
for example, Bark and Berlekom 1978, Soyama e l  al. 
1992). This paper addresses the issue of the modelling 
of the dynamics of cavitation clouds, a subject whose 
origins can be traced to the work of van Wijngaarden 
(1964) who first attempted to model the behavior of 
a collapsing layer of bubbly fluid next t o  a solid wall. 
Later investigators explored numerical methods which 
incorporate the individual bubbles (Chahine 1982) and 
c,ontinuum models which, for example, analyze the be- 
havior of shock waves in bubbly liquid (Noordzij and 
van Wijngaarden 1974) and identify the natural fre- 
quencies of spherical cloud of bubbles (d'Agostino and 
Brennen 1983). Indeed the literature on the linearized 
dynamics of clouds of bubbles is growing rapidly (see, 
for example, Omta 1987, d'Agostino el at. 1988 & 1989, 
PURE 
SPHERICAL CLOUD 
BOUNDARY 
0 0 '\\ 
1 0  
0 
\ O 0 BUBBLE RADIUS 0 
Rlr .0  
' 0 '\ 0 0 0  O O  / 
\\. - _ _ _  0 0-5 BuB, MIXTURE 
(wPuLATION = fl PER 
UNIT LIQUID VOLUME) 
Figure 1: Schematic of a spherical cloud o f  bubbles 
Prosperetti 1988). However, apart from some weakly 
non-linear analyses (Kumar and Brennen 1991, 1992, 
1993) only a few papers have addressed the highly non- 
linear processes involved during the collapse of a cloud 
of bubbles. Chahine and Duraiswami (1992) have re- 
cently conducted numerical simulations using a number 
of discrete bubbles and demonstrated how the bubbles 
on the periphery of the cloud develop inwardly directed 
re-entrant jets. However, most clouds contain many 
thousands of bubbles and it therefore is advantageous 
to  examine the non-linear behavior of continuum mod- 
eh, wkch is the subject of this paper. 
Another perspective on the subject of collapsing 
clouds was that introduced by M ~ r c h  and his co-workers 
( M ~ r c h  1980 & 1981, Hanson e l  al. 1981). They spec- 
ulated that the collapse of a cloud of bubbles involves 
the formation and inward propagation of a shock wave 
and that the geometric focusing of this shock at the 
center of cloud creates the enhancement of the noise 
and damage potential associated with cloud collapse. 
One of the purposes of the present work is to examine 
whether or not continuum models of the cloud manifest 
such phenomena. 
2 Basic Equations 
The paper addresses the problem of the dynamics of a 
spherical cloud of bubbles in an unbounded liquid at 
rest a t  infinity, as shown in figure 1. To incorporate 
the interactive effects that the cavitating bubbles have 
on themselves and on the pressure and velocity of the 
liquid flow, the bubble dynamics must be included in 
the non-barotropic, homogeneous flow model. The ba- 
sic equations used are same as those of d'Agostino and 
Brennen (d'Agostino & Brennen 1983; d'Agostino et 
al. 1988, 1989) except that all the nonlinear convective 
terms are retained since these are important in the con- 
text of the highly non-linear growth and collapse of the 
cloud. 
The dimensionless forms of continuity and momen- 
tum equations for the one-dimensional spherical bubbly 
mixture can be written as 
where D/Dt indicates the Lagrangian derivative, R(r, t )  
is the individual bubble radius, Cp(r, t )  is the pressure 
coefficient in the mixture, and the bubble population 
per unit liquid volume, 7, is related to the void fraction, 
a, by ( $ 7 r ~ ~ ) v  = a / ( l  - a ) .  The variables and equa- 
tions are non-dimens~onalized us~ng the ini621 bnbb'~ 
radius, Ro, a reference flow velocity, U, and the time 
scale, Ro/U. The pressure coefficient, Cp, and the bub- 
ble radius, R, are related by the Rayleigh-Plesset equa- 
tion: -.. 
where u is the cavitation number, We = p U 2 ~ o / ~  is 
the Weber number, S is the surface tension of liquid, 
Re = U Ro/v is the Reynolds number, v is the kinematic 
viscosity of liquid, and 6 is the polytropic index of the 
gas inside the bubble. 
The above three equations (I) ,  (2) and (3) with 
the appropriate boundary conditions can, in theory, be 
solved to find the unknown Cp(r,  t ) ,  ~ ( r ,  t) and R(r, t )  
for any spherical bubbly cavitating flow. However the 
non-linearities in the Rayleigh-Plesset equation as well 
a a 
as those in the Lagrangian derivative, DIDt  = z + u ~ ,  
mean that only linearized and weakly non-linear soh- 
tions have been obtained previously. 
The boundary conditions for the present analysis are 
as follows. The incompressible liquid flow outside the 
cloud, r 2 A(t), has the standard solution of the form: 
where Cp,(t) corresponds to the known driving pres- 
sure at infinity. At the center of cloud the symmetry of 
the problem requires 
0  10 20 30 40 50 60 7 0 8 0  9 0 1 0 0 1 1 0  
Cloud radius, r 
Figure 2: The dimensionless bubble size distribution in the 
cloud as a function of the dimensionless cloud radius at 
various times for a=0.4. CpMl,=-0.5, rwo=0.08%, rlo= 
1 cm, R o = 1 0 0  p n  and the dimensionless time, IG= 5 0 0 .  
At time t = O ,  it is assumed that the whole flow field 
is in equilibrium. I t  is also assumed, for simplicity, 
that all the bubbles have the same initial radius Ro. 
Thus we have the following simple initial conditions: 
R ( r , O )  = R o ,  E(r, 0) = 0, u(r, 0) = 0, Cp(r ,  0) = 0. 
In the present solutions we have used a simple sinusoidal 
driving pressure as follows: 
where Cp,,, is the minimum pressure coefficient at in- 
finity and t G  is the dirne~isionless duration of the low 
pressure perturbation. Consequently, for a cloud flow- 
ing with velocity U past a body of size D, tG will be 
of the order of magnitude of D / U ,  CpM,, will be the 
minimum pressure coefficient of the flow and a is the 
conventional cavitation number. 
3 Numerical Method 
A stable numerical scheme was derived which is second 
order accurate in time and third order accurate in space. 
The scheme involves iteration of the governing equa- 
tions a t  each time step. The program automatically 
adjusts the interval of each time step to  ensure that 
the fractional change in bubble radius between any two 
consecutive times does not exceed some specific value, 
say 0.1. This is essential for time marching through a 
violent bubble collapse. Let a tilde represent the quan- 
tities which change during an iteration. Superscripts i 
0  10 20 3 0 4 0  5 0 6 0 7 0 8 0 9 0 1 0 0 1 1 0  
Cloud radius, r 
Figure 3: The pressure coefficient distribution in the cloud 
as a function of the dimensionless cloud radius a t  various 
times. Parameters as in figure 2. 
and i + 1 represent the quantities a t  time t i  and t i + l  
respectively. At the beginning of this iterative proce- 
dure, the solution of the last time step is used for the 
initial tilde quantities. A complete time step proceeds 
as follows: 
- i+ l  
1) R~+' and (E) are calculated using a Taylor's 
series and the solution of the previous time step, 
' D R  i D% i R', (=) and (rn). 
- i+l  
2) Equation (3) is used to  calculate (w) using - i+ 1 - i + l  
Ri+', (g) and C'p 
3) Integrated versions of equations (1) and (2) are - i+l .-,+I 
used with Ri+l ,  (g) , Cp and iii+l to ob- - i + l  
tain new values of Cp and i i i + l .  
- i+l  
4) Steps 2 and 3 are repeated until i'+ and C p  
converge. Under-relaxation must be used in iterat- 
- i + l  
ing C'p to  get convergence. 
- i+ l  
5) The new values of (9) are calculated using - iA.1 
D R  ' D2R i D2R - ' - 
( = ) I ,  (m)  , and (asr) and the iteration is - i+l 
repeated by going back to step 2 until (g) ' 
converges. 
The rate of convergence in the loop 2) - 4) depends 
strongly on the initial void fraction ao; the larger ao, 
the slower the convergence. 
Cloud radius, r 
0 10 20 30 40 X1 60 70 80 90 1W 110 
0 100 200 Jm 400 500 
Time, t 
Figure 4: T h e  t ime history o f  dimensionless bubble size 
a t  three different positions in the cloud, ro=O, ro=0.9 A0 
and ro= Ao. Parameters as in figure 2. 
4 Results and Discussion 
For the purpose of illustrative calculations, parameters 
were chosen as follows. A cloud, composed of air bub- 
bles (k=1.4) of initial radius Ro=lOO pm and water 
at 20°C (p=1000 kg/m3, p=0.001 Nsec/m2, S=0.0728 
Nlrn), flows with velocity U=10 mlsec and has a void 
fraction ao=0.08 %. The cavitation number of the flow 
is a=0.4. The pressure perturbation experienced by the 
cloud has a duration of fG=5 x 10-3sec with Cp,,,= 
-0.5. Under these conditions the lowest resonant fre- 
quency of the cloud (d'Agostino and Brennen 1983) is 
about one-third of the single bubble natural frequency 
and therefore the calculations should exhibit the effects 
of non-linear cloud dynamics. 
The calculations clearly demonstrated the formation 
of an inwardly propagating shock wave as can been seen 
in figures 2 and 3. The location of shock waves can 
be identified by the minimum bubble radius as well as 
the peak of the pressure coefficient in the cloud. The 
shielding of the interior bubbles by the outer shell of 
bubbles is readily apparent in these results. This shield- 
ing is a common feature of the linearized dynamics of 
clouds (see, for example, Omta 1987, d'Agostino and 
Brennen 1989, Chahine and Duraiswami 1992). Note 
also that  the structure of the shock wave in figure 2 is 
qualitatively similar to  that of the steady state shock 
propagation analysis of Noordij and van Wijngaarden 
(1974). However, quantitative comparison is difficult 
to make because of the rapid strengthening (figure 3) 
of the spherical shock wave which leads to its rapid 
inward acceleration. The present calculation had to be 
Figure 5: Typical dimensionless velocity distribution in the 
cloud as a function o f  the dimensionless cloud radius. T h e  
shock is passing the position o f  rz85.183 at  this moment. 
Parameters as in figure 2. 
terminated when the strength of the shock became very 
large for then the adjusted time step became so small 
that the progress in time had essentially ceased. 
Figure 4 shows the time history of the growth and 
collapse of bubbles a t  several different positions in the 
cloud. All the bubbles grow initially in the same way, 
but are later affected by the cloud dynamics. 
A typical velocity distribution in the cloud is shown 
in figure 5. As anticipated, the magnitude of u as well 
as the values of du ld r  mean that the global convective 
effects are important in the non-linear dynamics of the 
cloud. 
5 Conclusions 
This paper presents some preliminary results of the 
non-linear growth and collapse of a cloud of cavitat- 
ing bubbles. We have developed an algorithm which 
allows simultaneous solution of the equations govern- 
ing the cloud and have utilized this to  investigate the 
bubble dynamics within a collapsing cloud. The ob- 
jective is to investigate the significant enhancement of 
the cavitation noise (and damage potential) associated 
with the collective effects in cloud cavitation. Several 
years ago, M ~ r c h  and his co-workers speculated that 
one of the reasons for this enhancement was that an 
inward propagating shock wave forms during cloud col- 
lapse. The present calculations confirm the formation 
of such a shock wave and indicate that it rapidly gains 
strength. The investigations into the resulting cloud 
dynamics and noise are continuing. 
Acknowledgments 
The authors are very grateful for the support for this 
research provided by the Office of Naval Research under 
Contract N00014-91-5-1295, 
References 
[l] Bark, G.  and Berlekom, W.B. (1978). Experimen- 
tal investigations of cavitation noise. Proc. 12th 
ONR Symp. on Naval Hydrodynamics, 470-493. 
[2] Chahine, G.L. (1982). Cloud cavitation: theory. 
Proc. 14th ONR Symp. on Naval Hydrodynamics, 
165-194. 
[3] Chahine, G.L. and Duraiswami, R. (1992). Dy- 
namical interactions in a multibubble cloud. 
ASME J. Fluids Eng., 114, 680-686. 
[4] dXgostino, L. and Brennen, C.E. (1983). On the 
acoustical dynamics of bubble clouds. ASME Cav- 
itation and Multiphase Flow Forum, 72-75. 
[5] d'Agostino, L. and Brennen, C.E., Acosta, A.J. 
(1988). Linearized dynamics of two-dimensional 
bubbly and cavitating flows over slender surfaces. 
J. Fluid Mech., 192,  485-509. 
[6] d'Agostino, L. and Brennen, C.E. (1988). Acous- 
tical absorption and scattering cross-sections of 
spherical bubble clouds. J. Acoust. Soc. Am., 
84, 2126-2134. 
[7] d'Agostino, L. and Brennen, C.E. (1989). Lin- 
earized dynamics of spherical bubble clouds. J .  
Fluid Mech., 199,  155-176. 
[8] Hanson, I., Kedrinskii, V.K. and Mgirch, K.A. 
(1981). On the dynamics of cavity clusters. J. 
Appl. Phys., 15 ,  1725-1734. 
[9] Kumar, S. and Brennen, C.E. (1991). Non-linear 
effects in the dynamics of clouds of bubbles. J.  
Acoust. Soc. Am., 89, 707-714. 
[lo] Kumar, S. and Brennen, C.E. (1993). Some nonlin- 
ear interactive effects in bubbly cavitating clouds. 
J. Fluid Mech., 253, 565-591. 
[12] M ~ r c h ,  K.A. (1980). On the collapse of cavity 
cluster in flow cavitation. Proc. First Int. Conf. 
on Cavitation and Inhomogenieties in Underwater 
Acoustics, Springer Series in Electrophysics, 4, 95- 
100. 
[13] M ~ r c h ,  K.A. (1981). Cavity cluster dynamics and 
cavitation erosion. Proc. ASME Cavitation and 
Polyphase Flow Forum, 1-10. 
[14] Noordij, L. and van Wijngaarden, L. (1974). Re- 
laxation effects, caused by relative motion, on 
shock waves in gas-bubblelliquid mixtures. J. 
Fluid Mech., 66, 115-143. 
[15] Omta, R. (1987). Oscillations of a cloud of bubbles 
of small and not so small amplitude. J. ACOUS~. 
Soc. Am., 82, 1018-1033. 
[16] Prosperetti, A. (1988). Bubble-related ambient 
noise in the ocean. J. Acoust. Soc. Am., 84 ,  1042- 
1054. 
[17] Soyama, H., Kato, H. and Oba, R. (1992). Cavi- 
tation observations of severely erosive vortex cavi- 
tation arising in a centrifugal pump. Proc. Third 
I. Mech.E. Int. Conf. on Cavitation, 103-1 10. 
[18] van Wijngaarden, L. (1964). On the collective col- 
lapse of a large number of gas bubbles in water. 
Proc. 11th Id. Conf. Appl. Mech., Springer- 
Verlag, Berlin, 854-861. 
[ l l ]  Kumar, S. and Brennen, C.E. (1992). Harmonic 
cascading in bubble clouds. Proc. Int. Symp. on 
Propulsors and Cavitation, Hamburg, 171-179. 


Shock Wave Development in the Collapse of a Cloud of Bubbles

FED-Vol. 194, Cavitation and Multlphase flow 
ASME 1994 
SHOCK WAVE DEVELOPMENT IN 
THE COLLAPSE OF A CLOUD OF BUBBLES 
Y.4. Wang and Christopher E. Brennen 
California Institute of Technology 
Pasadena, California 
ABSTRACT 
A numerical simulation of the collapse of a cloud of 
bubbles has been used t o  demonstrate the develop- 
ment of an inwardly propagating shock wave which 
grows rapidly in magnitude. The fully non-linear non- 
barotropic homogeneous flow equations are coupled 
with single bubble dynamics aud solved by a stable 
numerical scheme. The computational results demon- 
strate khe structure of the shock wave as well as its 
strengthening effect due t o  the coupling of the single 
bubble dynamics with the global dynamics of the flow 
through the pressure and velocity fields. This appears 
to  confirm the speculation of Morch and his co-workers 
that such shock formation is an important part of cloud 
collapse. 
Nomenclature 
A Dimensionless radius of the bubble cloud 
A. Dimensionless radius of the bubble cloud at the 
undisturbed reference condition 
Cp Pressure coefficient, (p - y o ) / i p l i 2  
C p ,  Pressure coefficient a t  infinity, ( y ,  - p ~ ) / ~ p U ~  
Cp,,, Minimum pressure c.oeficient a t  infinity 
Reference body size 
Dimensionless bubble radius 
Initial radius of babble a t  undisturbed reference 
condition 
Reynolds number, I i R o / v  
Surface tension of the liquid 
Reference velocity of the flow 
Weber number, p U % ~ / S  
Polytropic index of behavior of the gas inside 
the bubble. 
Fluid pressure 
Fluid pressure at undisturbed referewe condition 
Vapor pressure inside the bubble 
Pressure a t  infinity 
Dimensionless Eulerian radial coordinate 
measured from the center of cloud 
Dimensionless Lagrangian radial coordinate 
measured from the center of cloud and equal 
to r a t  undisturbed reference condition 
Dimensionless time 
Dimensionless duration of the low pressure 
perturbation 
Dimensionless radial velocity of fluid 
Void fraction of the bubbly mixture 
Void fraction of the bubble mixture at the 
undisturbed reference 
Density of liquid 
Cavitation number, (po  - p , ) / i p ~ "  
Dimensionless bubble population per unit 
liquid volume 
Kinematic viscosity of the liquid 
Introduction 
Much recent interest has focused on the dynamics and 
acoustics of finite clouds of cavitation bubbles because 
of the very destructive effects which are observed t o  oc- 
cur when such clouds form and collapse in a flow (see, 
for example, Bark and Berlekom 1978, Soyama e l  al. 
1992). This paper addresses the issue of the modelling 
of the dynamics of cavitation clouds, a subject whose 
origins can be traced to the work of van Wijngaarden 
(1964) who first attempted to model the behavior of 
a collapsing layer of bubbly fluid next t o  a solid wall. 
Later investigators explored numerical methods which 
incorporate the individual bubbles (Chahine 1982) and 
c,ontinuum models which, for example, analyze the be- 
havior of shock waves in bubbly liquid (Noordzij and 
van Wijngaarden 1974) and identify the natural fre- 
quencies of spherical cloud of bubbles (d'Agostino and 
Brennen 1983). Indeed the literature on the linearized 
dynamics of clouds of bubbles is growing rapidly (see, 
for example, Omta 1987, d'Agostino el at. 1988 & 1989, 
PURE 
SPHERICAL CLOUD 
BOUNDARY 
0 0 '\\ 
1 0  
0 \ O 0 BUBBLE RADIUS 0 
Rlr .0  
' 0 
'\ 0 0 0  O O  / 
\\. - _ _ _  0 0-5 BuB, MIXTURE 
(wPuLATION = fl PER 
UNIT LIQUID VOLUME) 
Figure 1: Schematic of a spherical cloud o f  bubbles 
Prosperetti 1988). However, apart from some weakly 
non-linear analyses (Kumar and Brennen 1991, 1992, 
1993) only a few papers have addressed the highly non- 
linear processes involved during the collapse of a cloud 
of bubbles. Chahine and Duraiswami (1992) have re- 
cently conducted numerical simulations using a number 
of discrete bubbles and demonstrated how the bubbles 
on the periphery of the cloud develop inwardly directed 
re-entrant jets. However, most clouds contain many 
thousands of bubbles and it therefore is advantageous 
to  examine the non-linear behavior of continuum mod- 
eh, wkch is the subject of this paper. 
Another perspective on the subject of collapsing 
clouds was that introduced by M ~ r c h  and his co-workers 
( M ~ r c h  1980 & 1981, Hanson e l  al. 1981). They spec- 
ulated that the collapse of a cloud of bubbles involves 
the formation and inward propagation of a shock wave 
and that the geometric focusing of this shock at the 
center of cloud creates the enhancement of the noise 
and damage potential associated with cloud collapse. 
One of the purposes of the present work is to examine 
whether or not continuum models of the cloud manifest 
such phenomena. 
2 Basic Equations 
The paper addresses the problem of the dynamics of a 
spherical cloud of bubbles in an unbounded liquid at 
rest a t  infinity, as shown in figure 1. To incorporate 
the interactive effects that the cavitating bubbles have 
on themselves and on the pressure and velocity of the 
liquid flow, the bubble dynamics must be included in 
the non-barotropic, homogeneous flow model. The ba- 
sic equations used are same as those of d'Agostino and 
Brennen (d'Agostino & Brennen 1983; d'Agostino et 
al. 1988, 1989) except that all the nonlinear convective 
terms are retained since these are important in the con- 
text of the highly non-linear growth and collapse of the 
cloud. 
The dimensionless forms of continuity and momen- 
tum equations for the one-dimensional spherical bubbly 
mixture can be written as 
where D/Dt indicates the Lagrangian derivative, R(r, t )  
is the individual bubble radius, Cp(r, t )  is the pressure 
coefficient in the mixture, and the bubble population 
per unit liquid volume, 7, is related to the void fraction, 
a, by ( $ 7 r ~ ~ ) v  = a / ( l  - a ) .  The variables and equa- 
tions are non-dimens~onalized us~ng the ini621 bnbb'~ 
radius, Ro, a reference flow velocity, U, and the time 
scale, Ro/U. The pressure coefficient, Cp, and the bub- 
ble radius, R, are related by the Rayleigh-Plesset equa- 
tion: -.. 
where u is the cavitation number, We = p U 2 ~ o / ~  is 
the Weber number, S is the surface tension of liquid, 
Re = U Ro/v is the Reynolds number, v is the kinematic 
viscosity of liquid, and 6 is the polytropic index of the 
gas inside the bubble. 
The above three equations (I) ,  (2) and (3) with 
the appropriate boundary conditions can, in theory, be 
solved to find the unknown Cp(r,  t ) ,  ~ ( r ,  t) and R(r, t )  
for any spherical bubbly cavitating flow. However the 
non-linearities in the Rayleigh-Plesset equation as well 
a a 
as those in the Lagrangian derivative, DIDt  = z + u ~ ,  
mean that only linearized and weakly non-linear soh- 
tions have been obtained previously. 
The boundary conditions for the present analysis are 
as follows. The incompressible liquid flow outside the 
cloud, r 2 A(t), has the standard solution of the form: 
where Cp,(t) corresponds to the known driving pres- 
sure at infinity. At the center of cloud the symmetry of 
the problem requires 
0  10 20 30 40 50 60 7 0 8 0  9 0 1 0 0 1 1 0  
Cloud radius, r 
Figure 2: The dimensionless bubble size distribution in the 
cloud as a function of the dimensionless cloud radius at 
various times for a=0.4. CpMl,=-0.5, rwo=0.08%, rlo= 
1 cm, R o = 1 0 0  p n  and the dimensionless time, IG= 5 0 0 .  
At time t = O ,  it is assumed that the whole flow field 
is in equilibrium. I t  is also assumed, for simplicity, 
that all the bubbles have the same initial radius Ro. 
Thus we have the following simple initial conditions: 
R ( r , O )  = R o ,  E(r, 0) = 0, u(r, 0) = 0, Cp(r ,  0) = 0. 
In the present solutions we have used a simple sinusoidal 
driving pressure as follows: 
where Cp,,, is the minimum pressure coefficient at in- 
finity and t G  is the dirne~isionless duration of the low 
pressure perturbation. Consequently, for a cloud flow- 
ing with velocity U past a body of size D, tG will be 
of the order of magnitude of D / U ,  CpM,, will be the 
minimum pressure coefficient of the flow and a is the 
conventional cavitation number. 
3 Numerical Method 
A stable numerical scheme was derived which is second 
order accurate in time and third order accurate in space. 
The scheme involves iteration of the governing equa- 
tions a t  each time step. The program automatically 
adjusts the interval of each time step to  ensure that 
the fractional change in bubble radius between any two 
consecutive times does not exceed some specific value, 
say 0.1. This is essential for time marching through a 
violent bubble collapse. Let a tilde represent the quan- 
tities which change during an iteration. Superscripts i 
0  10 20 3 0 4 0  5 0 6 0 7 0 8 0 9 0 1 0 0 1 1 0  
Cloud radius, r 
Figure 3: The pressure coefficient distribution in the cloud 
as a function of the dimensionless cloud radius a t  various 
times. Parameters as in figure 2. 
and i + 1 represent the quantities a t  time t i  and t i + l  
respectively. At the beginning of this iterative proce- 
dure, the solution of the last time step is used for the 
initial tilde quantities. A complete time step proceeds 
as follows: 
- i+ l  
1) R~+' and (E) are calculated using a Taylor's 
series and the solution of the previous time step, 
' D R  i D% i R', (=) and (rn). 
- i+l  
2) Equation (3) is used to  calculate (w) using 
- i+ 1 
- i + l  
Ri+', (g) and C'p 
3) Integrated versions of equations (1) and (2) are 
- i+l .-,+I 
used with Ri+l ,  (g) , Cp and iii+l to ob- 
- i + l  
tain new values of Cp and i i i + l .  
- i+l  
4) Steps 2 and 3 are repeated until i'+ and C p  
converge. Under-relaxation must be used in iterat- 
- i + l  
ing C'p to  get convergence. 
- i+ l  
5) The new values of (9) are calculated using 
- iA.1 
D R  ' D2R i D2R - ' - 
( = ) I ,  (m)  , and (asr) and the iteration is 
- i+l 
repeated by going back to step 2 until (g) ' 
converges. 
The rate of convergence in the loop 2) - 4) depends 
strongly on the initial void fraction ao; the larger ao, 
the slower the convergence. 
Cloud radius, r 
0 10 20 30 40 X1 60 70 80 90 1W 110 
0 100 200 Jm 400 500 
Time, t 
Figure 4: T h e  t ime history o f  dimensionless bubble size 
a t  three different positions in the cloud, ro=O, ro=0.9 A0 
and ro= Ao. Parameters as in figure 2. 
4 Results and Discussion 
For the purpose of illustrative calculations, parameters 
were chosen as follows. A cloud, composed of air bub- 
bles (k=1.4) of initial radius Ro=lOO pm and water 
at 20°C (p=1000 kg/m3, p=0.001 Nsec/m2, S=0.0728 
Nlrn), flows with velocity U=10 mlsec and has a void 
fraction ao=0.08 %. The cavitation number of the flow 
is a=0.4. The pressure perturbation experienced by the 
cloud has a duration of fG=5 x 10-3sec with Cp,,,= 
-0.5. Under these conditions the lowest resonant fre- 
quency of the cloud (d'Agostino and Brennen 1983) is 
about one-third of the single bubble natural frequency 
and therefore the calculations should exhibit the effects 
of non-linear cloud dynamics. 
The calculations clearly demonstrated the formation 
of an inwardly propagating shock wave as can been seen 
in figures 2 and 3. The location of shock waves can 
be identified by the minimum bubble radius as well as 
the peak of the pressure coefficient in the cloud. The 
shielding of the interior bubbles by the outer shell of 
bubbles is readily apparent in these results. This shield- 
ing is a common feature of the linearized dynamics of 
clouds (see, for example, Omta 1987, d'Agostino and 
Brennen 1989, Chahine and Duraiswami 1992). Note 
also that  the structure of the shock wave in figure 2 is 
qualitatively similar to  that of the steady state shock 
propagation analysis of Noordij and van Wijngaarden 
(1974). However, quantitative comparison is difficult 
to make because of the rapid strengthening (figure 3) 
of the spherical shock wave which leads to its rapid 
inward acceleration. The present calculation had to be 
Figure 5: Typical dimensionless velocity distribution in the 
cloud as a function o f  the dimensionless cloud radius. T h e  
shock is passing the position o f  rz85.183 at  this moment. 
Parameters as in figure 2. 
terminated when the strength of the shock became very 
large for then the adjusted time step became so small 
that the progress in time had essentially ceased. 
Figure 4 shows the time history of the growth and 
collapse of bubbles a t  several different positions in the 
cloud. All the bubbles grow initially in the same way, 
but are later affected by the cloud dynamics. 
A typical velocity distribution in the cloud is shown 
in figure 5. As anticipated, the magnitude of u as well 
as the values of du ld r  mean that the global convective 
effects are important in the non-linear dynamics of the 
cloud. 
5 Conclusions 
This paper presents some preliminary results of the 
non-linear growth and collapse of a cloud of cavitat- 
ing bubbles. We have developed an algorithm which 
allows simultaneous solution of the equations govern- 
ing the cloud and have utilized this to  investigate the 
bubble dynamics within a collapsing cloud. The ob- 
jective is to investigate the significant enhancement of 
the cavitation noise (and damage potential) associated 
with the collective effects in cloud cavitation. Several 
years ago, M ~ r c h  and his co-workers speculated that 
one of the reasons for this enhancement was that an 
inward propagating shock wave forms during cloud col- 
lapse. The present calculations confirm the formation 
of such a shock wave and indicate that it rapidly gains 
strength. The investigations into the resulting cloud 
dynamics and noise are continuing. 
Acknowledgments 
The authors are very grateful for the support for this 
research provided by the Office of Naval Research under 
Contract N00014-91-5-1295, 
References 
[l] Bark, G.  and Berlekom, W.B. (1978). Experimen- 
tal investigations of cavitation noise. Proc. 12th 
ONR Symp. on Naval Hydrodynamics, 470-493. 
[2] Chahine, G.L. (1982). Cloud cavitation: theory. 
Proc. 14th ONR Symp. on Naval Hydrodynamics, 
165-194. 
[3] Chahine, G.L. and Duraiswami, R. (1992). Dy- 
namical interactions in a multibubble cloud. 
ASME J. Fluids Eng., 114, 680-686. 
[4] dXgostino, L. and Brennen, C.E. (1983). On the 
acoustical dynamics of bubble clouds. ASME Cav- 
itation and Multiphase Flow Forum, 72-75. 
[5] d'Agostino, L. and Brennen, C.E., Acosta, A.J. 
(1988). Linearized dynamics of two-dimensional 
bubbly and cavitating flows over slender surfaces. 
J. Fluid Mech., 192,  485-509. 
[6] d'Agostino, L. and Brennen, C.E. (1988). Acous- 
tical absorption and scattering cross-sections of 
spherical bubble clouds. J. Acoust. Soc. Am., 
84, 2126-2134. 
[7] d'Agostino, L. and Brennen, C.E. (1989). Lin- 
earized dynamics of spherical bubble clouds. J .  
Fluid Mech., 199,  155-176. 
[8] Hanson, I., Kedrinskii, V.K. and Mgirch, K.A. 
(1981). On the dynamics of cavity clusters. J. 
Appl. Phys., 15 ,  1725-1734. 
[9] Kumar, S. and Brennen, C.E. (1991). Non-linear 
effects in the dynamics of clouds of bubbles. J.  
Acoust. Soc. Am., 89, 707-714. 
[lo] Kumar, S. and Brennen, C.E. (1993). Some nonlin- 
ear interactive effects in bubbly cavitating clouds. 
J. Fluid Mech., 253, 565-591. 
[12] M ~ r c h ,  K.A. (1980). On the collapse of cavity 
cluster in flow cavitation. Proc. First Int. Conf. 
on Cavitation and Inhomogenieties in Underwater 
Acoustics, Springer Series in Electrophysics, 4, 95- 
100. 
[13] M ~ r c h ,  K.A. (1981). Cavity cluster dynamics and 
cavitation erosion. Proc. ASME Cavitation and 
Polyphase Flow Forum, 1-10. 
[14] Noordij, L. and van Wijngaarden, L. (1974). Re- 
laxation effects, caused by relative motion, on 
shock waves in gas-bubblelliquid mixtures. J. 
Fluid Mech., 66, 115-143. 
[15] Omta, R. (1987). Oscillations of a cloud of bubbles 
of small and not so small amplitude. J. ACOUS~. 
Soc. Am., 82, 1018-1033. 
[16] Prosperetti, A. (1988). Bubble-related ambient 
noise in the ocean. J. Acoust. Soc. Am., 84 ,  1042- 
1054. 
[17] Soyama, H., Kato, H. and Oba, R. (1992). Cavi- 
tation observations of severely erosive vortex cavi- 
tation arising in a centrifugal pump. Proc. Third 
I. Mech.E. Int. Conf. on Cavitation, 103-1 10. 
[18] van Wijngaarden, L. (1964). On the collective col- 
lapse of a large number of gas bubbles in water. 
Proc. 11th Id. Conf. Appl. Mech., Springer- 
Verlag, Berlin, 854-861. 
[ l l ]  Kumar, S. and Brennen, C.E. (1992). Harmonic 
cascading in bubble clouds. Proc. Int. Symp. on 
Propulsors and Cavitation, Hamburg, 171-179. 


Acoustical absorption and scattering cross-sections of spherical bubble clouds.

Bark, G. and Berlekom, W.B. (1978). Experimental investigations of cavitation noise. Proc. 12th ONR Symp. on Naval Hydrodynamics, 470-493.

Bubble-related ambient noise in the ocean.

Cavitation observations of severely erosive vortex cavitation arising in a centrifugal pump.

Cavity cluster dynamics and cavitation erosion.

Chahine, G.L. (1982). Cloud cavitation: theory. Proc. 14th ONR Symp. on Naval Hydrodynamics, 165-194.

Chahine, G.L. and Duraiswami, R. (1992). Dynamical interactions in a multibubble cloud. ASME J. Fluids Eng., 114, 680-686.

Cloud cavitation: theory.

d'Agostino, L. and Brennen, C.E. (1988). Acoustical absorption and scattering cross-sections of spherical bubble clouds. J. Acoust. Soc. Am., 84, 2126-2134.

d'Agostino, L. and Brennen, C.E. (1989). Linearized dynamics of spherical bubble clouds. J . Fluid Mech., 199, 155-176.

d'Agostino, L. and Brennen, C.E., Acosta, A.J. (1988). Linearized dynamics of two-dimensional bubbly and cavitating flows over slender surfaces. J. Fluid Mech., 192, 485-509.

dXgostino, L. and Brennen, C.E. (1983). On the acoustical dynamics of bubble clouds. ASME Cavitation and Multiphase Flow Forum, 72-75.

Dynamical interactions in a multibubble cloud.

Experimental investigations of cavitation noise.

Hanson, I., Kedrinskii, V.K. and Mgirch, K.A. (1981). On the dynamics of cavity clusters. J. Appl. Phys., 15 , 1725-1734.

Harmonic cascading in bubble clouds.

Kumar, S. and Brennen, C.E. (1991). Non-linear effects in the dynamics of clouds of bubbles. J. Acoust. Soc. Am., 89, 707-714.

Kumar, S. and Brennen, C.E. (1992). Harmonic cascading in bubble clouds. Proc. Int. Symp. on Propulsors and Cavitation, Hamburg, 171-179.

Kumar, S. and Brennen, C.E. (1993). Some nonlinear interactive effects in bubbly cavitating clouds. J. Fluid Mech., 253, 565-591.

Linearized dynamics of spherical bubble clouds.

Linearized dynamics of two-dimensional bubbly and cavitating flows over slender surfaces.

M ~ r c h , K.A. (1980). On the collapse of cavity cluster in flow cavitation. Proc. First Int. Conf. on Cavitation and Inhomogenieties in Underwater Acoustics, Springer Series in Electrophysics, 4, 95-100.

M ~ r c h , K.A. (1981). Cavity cluster dynamics and cavitation erosion. Proc. ASME Cavitation and Polyphase Flow Forum, 1-10.

Non-linear effects in the dynamics of clouds of bubbles.

Noordij, L. and van Wijngaarden, L. (1974). Relaxation effects, caused by relative motion, on shock waves in gas-bubblelliquid mixtures. J. Fluid Mech., 66, 115-143.

Omta, R. (1987). Oscillations of a cloud of bubbles of small and not so small amplitude. J. ACOUS~. Soc. Am., 82, 1018-1033.

On the acoustical dynamics of bubble clouds.

On the collapse of cavity cluster in flow cavitation.

On the collective collapse of a large number of gas bubbles in water.

On the dynamics of cavity clusters.

Oscillations of a cloud of bubbles of small and not so small amplitude.

Prosperetti, A. (1988). Bubble-related ambient noise in the ocean. J. Acoust. Soc. Am., 84 , 1042-1054.

Relaxation effects, caused by relative motion, on shock waves in gas-bubblelliquid mixtures.

Some nonlinear interactive effects in bubbly cavitating clouds.

Soyama, H., Kato, H. and Oba, R. (1992). Cavitation observations of severely erosive vortex cavitation arising in a centrifugal pump. Proc. Third I. Mech.E. Int. Conf. on Cavitation, 103-1 10.

van Wijngaarden, L. (1964). On the collective collapse of a large number of gas bubbles in water. Proc. 11th Id. Conf. Appl. Mech., SpringerVerlag, Berlin, 854-861.

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Shock Wave Development in the Collapse of a Cloud of Bubbles

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