Vapour bubble collapse problems lacking spherical symmetry are solved here using a numerical method designed especially for these problems. Viscosity and compressibility in the liquid are neglected. Two specific cases of initially spherical bubbles collapsing near a plane solid wall were simulated: a bubble initially in contact with the wall, and a bubble initially half its radius from the wall at the closest point. It is shown that the bubble develops a jet directed towards the wall rather early in the collapse history. Free surface shapes and velocities are presented at various stages in the collapse. Velocities are scaled like (Δp/ρ)^½ where ρ is the density of the liquid and Δp is the constant difference between the ambient liquid pressure and the pressure in the cavity. For Δp/ρ=10^6cm^2/sec^2 ≈ 1 atm/density of water

the jet had a speed of about 130m/sec in the first case and 170m/sec in the second when it struck the opposite side of the bubble. Such jet velocities are of a magnitude which can explain cavitation damage. The jet develops so early in the bubble collapse history that compressibility effects in the liquid and the vapour are not important

Chapman, Richard B.

Plesset, Milton S.

English

Caltech Authors - Main

J. Fluid Mech. (1971), vol. 47, part 2, pp. 283-290 
Printed in Great Britain 
283 
Collapse of an initially spherical vapour cavity in the 
neighbourhood of a solid boundary 
By MILTON S. PLESSET AND RICHARD B. CHAPMAN 
California Institute of Technology 
(Received 5 June 1970 and in revised form 30 October 1970) 
Vapour bubble collapse problems lacking spherical symmetry are solved here 
using a numerical method designed especially for these problems. Viscosity and 
compressibility in the liquid are neglected. Two specific cases of initially spherical 
bubbles collapsing near a plane solid wall were simulated: a bubble initially in 
contact with the wall, and a bubble initially half its radius from the wall at the 
closest point. It is shown that the bubble develops a jet directed towards the 
wall rather early in the collapse history. Free surface shapes and velocities are 
presented at various stages in the collapse. Velocities are scaled like (!)..p{p)i 
where p is the density of the liquid and !)..p is the constant difference between the 
ambient liquid pressure and the pressure in the cavity. For 
!)..pjp = l06 cm2fsec2 ~ 1 atmfdensity of water 
the jet had a speed of about 130 mfsec in the first case and 170 mfsec in the 
second when it struck the opposite side of the bubble. Such jet velocities are of 
a magnitude which can explain cavitation damage. The jet develops so early in 
the bubble collapse history that compressibility effects in the liquid and the 
vapour are not important. 
Introduction 
The study of the behaviour of a bubble in a liquid is greatly simplified by the 
assumption of spherical symmetry. Following Rayleigh's (1917) classical 
analysis of a problem first solved by Besant, the inviscid collapse of a spherical 
cavity in a homogeneous, incompressible liquid under a constant ambient 
pressure, numerous authors have studied the behavour of spherical bubbles under 
a wide range of conditions. Far less is known about the non-spherical behaviour 
of bubbles. Because problems lacking spherical symmetry have proved too com-
plex for direct analysis, they have been investigated primarily by qualitative 
reasoning, experiments, and perturbations from spherically symmetric solutions. 
A problem of primary importance is the interaction of a collapsing bubble 
with a solid surface. The earliest theory of cavitation damage was based on the 
high pressures developed near a spherical cavity which has collapsed to a small 
fraction of its initial size. A more recent theory includes the pressures developed 
during rebound caused by the compression of a small amount of permanent gas 
contained in the bubble. Calculations discussed by Plesset (1966) indicate that 
stresses produced by the collapse and subsequent rebound of a spherical bubble 
284 M.S. Plesset and R. B. Chapman 
fall off rapidly as the distance from the bubble is increased and are too small 
to damage a solid surface unless the surface is quite close to the bubble. Thus 
the presence of a solid boundary will have an important effect in destroying the 
spherical symmetry of any bubble capable of producing damage. 
Another explanation of cavitation damage is the theory, first suggested by 
Kornfeld & Suvorov (1944), that damage is caused by the action of liquid jets 
formed on bubbles near the solid surface. A perturbation study by Rattray 
(1951) suggested that the effect of a solid wall in disturbing the flow during the 
collapse of an initially spherical bubble could cause the formation of a liquid jet 
directed towards the wall. Experiments by Benjamin & Ellis (1966) later con-
firmed that jets form on bubbles collapsing near a solid wall. Large vapour bubbles, 
generally about one centimetre in radius, were grown from small nuclei by the 
application of a negative pressure. High-speed photographs were taken of these 
bubbles as they collapsed near a plane solid surface. The ambient pressure was 
maintained at about 0·04 atm during collapse so that collapse velocities would be 
reduced to facilitate the photography. These bubbles were nearly spherical as 
they started collapsing. First, they became elongated in the direction normal 
to the wall; then they tended to flatten and form an inward moving jet on the 
side of the bubble opposite the wall. 
The advantages of a numerical technique for stimulating non-spherical bubble 
collapse are clear. Experiments are difficult and give only sketchy results. Per-
turbations from spherically symmetric solutions are not valid for large deforma-
tions. A numerical solution, however, can check results and supply detailed in-
formation. Numerical methods can also be applied to situations which might be 
very difficult to produce in the laboratory. Mitchell, Kling, Cheesewright & 
Hammitt (1967) have considered simulation of bubble collapse using the Marker-
and-Cell technique, a general method for simulating incompressible, viscid 
flows with an assortment of boundary conditions including free surfaces. Because 
non-spherical collapse is of such interest, it is worthwhile to develop a method of 
simulation especially suited to these problems. 
Definition of the problem 
The asymmetries caused by a solid wall should be separated from those due to 
initial asymmetries in shape or velocity of the type analyzed in the linearized 
theory ofPlesset & Mitchell (1956). The bubble is therefore taken to be spherical 
and at rest at the initiation of the collapse, and any other extraneous asym-
metric effects such as gravity are also omitted. 
The following assumptions will be made: (1) The liquid is incompressible. 
(2) The flow is non-viscous. (3) The vapour pressure is uniform throughout the 
bubble interior. (4) The ambient pressure and the vapour pressure are constant 
with time. (5) The bubble contains no permanent gas. (6) Surface tension effects 
are negligible. 
This set of assumptions defines the problem as the non-spherical version of the 
classical Rayleigh collapse calculation. Only the first three assumptions are 
essential to the method of simulation developed here. The last three assumptions 
Vapour cavity collapse 285 
are made to keep the essential features of the problem in the foreground. With the 
absence of shocks, compressibility will not become important until speeds in the 
liquid are comparable with the speed of sound. Thus the liquid can be assumed 
to be incompressible with the understanding that solutions are valid for small 
Mach numbers only. In most cases of collapse, viscosity can be neglected unless 
the bubble is initially very small. For example, viscosity is unimportant for a 
spherical bubble collapsing in water under atmospheric pressure if the initial 
radius is I0-3 cm or greater. As for the assumption of uniform pressure inside 
the bubble, this assumption will remain valid as long as speeds on the bubble 
surface are below the speed of sound in the vapour. 
The problem is specified by the following conditions: Poo =ambient pressure, 
Pv = vapour pressure inside the bubble, R0 = initial radius of the bubble, 
b = initial distance from the plane wall to the centre of the bubble. 
Because the flow is irrotational, the velocity vector v can be written in terms 
of a velocity potential ¢. Since incompressibility is assumed, ¢ must satisfy 
Laplace's equation throughout the liquid. 
The pressure boundary conditions can be restated in terms of¢ and v with the 
aid of Bernoulli's equation 
8¢18t+fv2 +(PIP) = c(t). (1) 
Infinitely far from the bubble the velocity is zero, and the pressure is the ambient 
pressure. The velocity potential there is an arbitrary function of time only, which 
can be taken to be zero limit ¢(x, t) = 0. (2) 
lxl-->-ro 
Then on the free surface, 
(3) 
The final boundary condition on the potential is that its normal derivative must 
vanish at the solid wall. Initially the potential is uniformly zero. 
As a result of the assumptions, the solutions are characterized by the single 
parameter b I R 0 • A solution for a particular value of b I R 0 can be scaled to bubbles 
of any initial size under any positive collapsing pressure !1p. Velocities are 
independent of the size of the bubble, and are scaled like (!1plp)l. 
The method of simulation 
The\ irrotationality of these problems is exploited by solving them in terms 
of the velocity potential. A number of small time steps are used. Before each step 
the potential problem is solved and the velocity is calculated at a large number 
of points representing the free surface of the cavity. The displacement of a free 
surface point with a velocity v calculated at the start of the time step is approxi-
matedby 
!1x = v!1t. 
The rate of change of the potential of a point moving on the free surface is 
D¢1Dt = o¢1ot+v2 , 
or by (3) D¢1Dt = !1plp+!v2• 
(4) 
(5) 
(6) 
286 M. S. Plesset and R. B. Chapman 
The change in potential of a free surface point can then be approximated by 
(7) 
A detailed description of this numerical method can be found in a recent report 
by Plesset & Chapman (1970). 
Results of the calculations 
The collapse of an initially spherical bubble near a plane solid wall was simu-
lated for two cases. In case 1 the parameter bfR0 was unity; that is, the bubble 
boundary was in contact with the solid wall and tangent to it. In case 2 b f R0 
was 1·5; the closest distance from the bubble boundary to the solid wall was in-
itially half the radius of the bubble. Ninety-four time steps were used for case 1 
and seventy-seven for case 2. Calculations were stopped when the liquid jet 
reached the opposite wall of the bubble since the assumption of incompressi-
bility is no longer valid. The bubble shapes for selected time steps for cases 1 and 
2 are shown superimposed in figures 1 and 2, respectively. Table 1lists the time 
intervals in units of R0(pjflp)t from the initiation of collapse for each shape and 
the downward velocity on the upper portion of the bubble at the axis of symmetry. 
The velocities, which are scaled like (D..pfp)t, are given in metresfsec for the special 
value 
D..p 106 dynesfcm2 1 atm 
p = 1·0 gfcm3 ~ densityofwater· (8) 
The solid wall influences the bubble early in the collapse chiefly by reducing 
the upward motion of the lower portion of the bubble. As a result the bubble 
becomes elongated in the direction normal to the wall as was predicted by 
Rattray (1951). The bottom of the bubble still moves upwards towards the 
bubble centre in case 2, but since this upward motion is reduced, the centroid 
of the bubble moves towards the wall displaying the well-known Bjerknes effect. 
As the bubble acquires kinetic energy, this energy is concentrated in the upper 
portion of the bubble which eventually flattens and forms a jet. Once the jet is 
formed, the speed of its tip remains fairly constant. 
The behaviour of the upper portion of the bubble in case 2 is not very different 
from case l. The overall shapes appear quite different, however, because the 
bottom of the bubble must remain in contact with the solid wall in case 1 but is 
allowed mobility in case 2. The jet speed in case 2 (about 170 mfsec under ~tmos­
pheric D..p) is somewhat larger than the speed in case 1 (about 130m/sec). This 
behaviour is as expected since a bubble which is farther from the wall collapses 
to a smaller size and can concentrate its energy over a smaller volume. 
The jet appears to be the result of the deformation caused by the presence of 
the wall during the early part of the collapse. It is known from the linearized 
theory ofPlesset & Mitchell (1956) that a small deformation can lead to jetting 
much later in the collapse, but the jet formation found here appears before the 
jetting which might develop from a small initial perturbation. 
Although the bubble is initially fairly close to the wall in case 2, the final 
jet must pass through the liquid for a distance of more than five times its 
Vapour cavity collapse 
Initial sphere 
Wall 
Figure 1. Bubble surfaces from case 1. 
Initial sphere 
77777777777777777777 
Wall 
Figure 2. Bubble surfaces from case 2. 
287 
288 M.S. Plesset and R. B. Chapman 
diameter before it reaches the solid wall. The jet in case 1, which strikes the wall 
directly, seems the more capable of damage even though the jet speed is lower. 
Apparently cavitation bubbles must almost touch the wall initially to be capable 
of damaging it. 
Figure 1 Figure 2 
Velocity Velocity 
Shape Time (mfsec) Time (mfsec) 
A 0·63 7·7 0.725 10 
B 0·885 19 0·875 17 
0 0·986 42 0·961 35 
D 1·013 65 0·991 53 
E 1·033 100 1·016 94 
F 1-048 125 1·028 142 
G 1-066 129 1·036 160 
H 1·082 129 1-044 165 
I 1-098 128 1·050 170 
J 1-119 128 
TABLE I. Time interval from initiation of collapse, and the velocity of the bubble botmdary 
at the axial point most distant from the wall, for the cases illustrated in figure 1 and figure 2 
A jet of speed v directly striking a solid boundary produces an initial pressure 
given by the water hammer equation, 
P - p C V ( PsCs ) WH- L L PLCL +pscs ' (9) 
where the L and s subscripts refer to the liquid and the solid, respectively. 
Usually p 8 c8 is large compared to PLCL producing the approximation 
(10) 
Experiments by Hancox & Brunton (1966) have shown that multiple impacts 
by water at a speed of 90 mfsec can erode even stainless steel. 
Benjamin & Ellis (1966) present two series of photographs ofbubbles collapsing 
near a solid wall in figures 3 and 4 of their paper. The collapse illustrated in their 
figure 4 is very similar to case 2. The collapse illustrated in their figure 3 falls 
between case 1 and case 2. Benjamin & Ellis estimated the jet speed in their 
figure 3 to be about 10m/sec under an ambient pressure of about 0·04atm. The 
vapour pressure of the water is very important at this reduced pressure. Since 
Benjamin & Ellis did not mention the temperature of the water, this pressure 
cannot be determined directly. However, D.p can be deduced from the total col-
lapse time which they gave as 10 msec. The total collapse time for a spherical 
bubble is, according to Rayleigh, 
T = 0·915R0(pfD.p)i. (11) 
The total collapse times for cases 1 and 2 are only slightly greater since most 
Vapour cavity collapse 289 
of the time is consumed early in the collapse while the bubble is nearly spherical. 
For collapse near a solid wall, then, the total collapse time is roughlyt 
r ~ R0(pj:lp)!. (12) 
Since R0 ~ l·Ocm and r = lOmsec, the pressure:difference for the collapse in 
figure 3 of Benjamin & Ellis is approximately 
:lp = Poo -Pv ~ l04 dynesjcm2 ~ 0·01 atm. (13) 
A vapour pressure of 0·03 atm corresponds to a temperature of about 76 °F. 
Speeds for one atmosphere pressure difference should be increased by a factor of 
ten giving an estimated jet speed of roughly lOOm/sec so that the experimental 
observations of Benjamin & Ellis are compatible with the calculations performed 
here. 
As general conclusions we may say that it appears very likely that cavitation 
damage with collapsing vapour bubbles is caused by the impact of the jet pro-
duced by the presence of the adjacent solid wall. Further, it appears reasonable 
to say that only those cavitation bubbles quite near the solid boundary can pro-
duce damage whether by a jet or by any radiated shock. From the calculations 
presented here, we see that for a bubble near the wall the jet is formed early in 
the collapse history so that the many complications of the late stages of cavity 
collapse do not enter. These familiar complications include the instability of the 
spherical shape toward the end of collapse, the effects of high bubble wall velo-
cities on the behaviour of the vapour in the bubble, and the effects of compres-
sibility, not only in the vapour phase, but in the liquid as well. It is also very 
evident that the jet appears before there is any possibility of radiating a shock. 
It is not clear that the impact, or 'water-hammer' stress of (10) is the 
mechanism of damage to the solid. For the case of the spherical bubble initially 
in contact with the wall and for :lp = 1 atm in water, we have v- 130 mjsec and 
cL ,..., 1500 mfsec so that PwH ,..., 2000 atm. 
While this is a most impressive impact stress, it is not obvious that it is the im-
portant damaging mechanism since the duration of this stress is so short. We 
may estimate this duration as being no longer than the time for the impact 
signal to traverse the radius of the jet. For a bubble with an initial radius 
R0 = 0·1 em, this time is rwH,..., I0-7 sec. On the other hand the stagnation 
pressure is approximately Ps ,..., !pv2 ,..., 800 atm, 
which will have a duration of the order of the length of the jet divided by its 
velocity v. This pressure pulse may be the source of the damage because its 
duration is an order of magnitude greater. 
Finally, we may say that cavitation damage should have a close relationship 
with liquid impact damage and inferences from studies of the latter should be 
useful for cavitation damage. We may also use calculations of the kind presented 
here to get reasonably accurate estimates of cavitation stress pulses. 
t Rattray derived the formula 
T (!J.p)!- ( R0) (R0) 2 - - = 0·915 1+0·41- +0 -
~ p ~ ~ 
from his perturbation analysis. 
19 FLM 47 
290 M.S. Plesset and R. B. Chapman 
REFERENCES 
BENJAMIN, T. B. & ELLIS, A. T. 1966 Phil. Trans. Roy. Soc. A 260, 221. 
HANCOX, N. L. & BRUNTON, J. H. 1966 Phil. Trans. Roy. Soc. A 260, 121. 
KoRNFELD, M. & SuvoRov, L. 1944 J. Appl. Phys. 15, 495. 
MITCHELL, T. M., KLING, C. L., CHEESEWRIGHT, R. & HAMMITT, F. G. 1967 U. of Michi-
gan, College of Eng., Rep. 07738-5-T. 
PLESSET, M.S. 1966 Phil. Trans. Roy. Soc. A 260, 241. 
PLESSET, M.S. & CHAPMAN, R. B. 1970 Div. of Eng. and Appl. Science, Calif. Inst. of 
Tech. Rep. no. 85-49. 
PLESSET, M.S. & MITCHELL, T. P. 1956 Quart. Appl. Math. 13, 419. 
RATTRAY, M. 1951 Ph.D. thesis, California Institute of Technology. 
RAYLEIGH, LORD 1917 Phil. Mag. 34, 94. 


Collapse of an initially spherical vapour cavity in the

neighbourhood of a solid boundary

J. Fluid Mech. (1971), vol. 47, part 2, pp. 283-290 
Printed in Great Britain 
283 
Collapse of an initially spherical vapour cavity in the 
neighbourhood of a solid boundary 
By MILTON S. PLESSET AND RICHARD B. CHAPMAN 
California Institute of Technology 
(Received 5 June 1970 and in revised form 30 October 1970) 
Vapour bubble collapse problems lacking spherical symmetry are solved here 
using a numerical method designed especially for these problems. Viscosity and 
compressibility in the liquid are neglected. Two specific cases of initially spherical 
bubbles collapsing near a plane solid wall were simulated: a bubble initially in 
contact with the wall, and a bubble initially half its radius from the wall at the 
closest point. It is shown that the bubble develops a jet directed towards the 
wall rather early in the collapse history. Free surface shapes and velocities are 
presented at various stages in the collapse. Velocities are scaled like (!)..p{p)i 
where p is the density of the liquid and !)..p is the constant difference between the 
ambient liquid pressure and the pressure in the cavity. For 
!)..pjp = l06 cm2fsec2 ~ 1 atmfdensity of water 
the jet had a speed of about 130 mfsec in the first case and 170 mfsec in the 
second when it struck the opposite side of the bubble. Such jet velocities are of 
a magnitude which can explain cavitation damage. The jet develops so early in 
the bubble collapse history that compressibility effects in the liquid and the 
vapour are not important. 
Introduction 
The study of the behaviour of a bubble in a liquid is greatly simplified by the 
assumption of spherical symmetry. Following Rayleigh's (1917) classical 
analysis of a problem first solved by Besant, the inviscid collapse of a spherical 
cavity in a homogeneous, incompressible liquid under a constant ambient 
pressure, numerous authors have studied the behavour of spherical bubbles under 
a wide range of conditions. Far less is known about the non-spherical behaviour 
of bubbles. Because problems lacking spherical symmetry have proved too com-
plex for direct analysis, they have been investigated primarily by qualitative 
reasoning, experiments, and perturbations from spherically symmetric solutions. 
A problem of primary importance is the interaction of a collapsing bubble 
with a solid surface. The earliest theory of cavitation damage was based on the 
high pressures developed near a spherical cavity which has collapsed to a small 
fraction of its initial size. A more recent theory includes the pressures developed 
during rebound caused by the compression of a small amount of permanent gas 
contained in the bubble. Calculations discussed by Plesset (1966) indicate that 
stresses produced by the collapse and subsequent rebound of a spherical bubble 
284 M.S. Plesset and R. B. Chapman 
fall off rapidly as the distance from the bubble is increased and are too small 
to damage a solid surface unless the surface is quite close to the bubble. Thus 
the presence of a solid boundary will have an important effect in destroying the 
spherical symmetry of any bubble capable of producing damage. 
Another explanation of cavitation damage is the theory, first suggested by 
Kornfeld & Suvorov (1944), that damage is caused by the action of liquid jets 
formed on bubbles near the solid surface. A perturbation study by Rattray 
(1951) suggested that the effect of a solid wall in disturbing the flow during the 
collapse of an initially spherical bubble could cause the formation of a liquid jet 
directed towards the wall. Experiments by Benjamin & Ellis (1966) later con-
firmed that jets form on bubbles collapsing near a solid wall. Large vapour bubbles, 
generally about one centimetre in radius, were grown from small nuclei by the 
application of a negative pressure. High-speed photographs were taken of these 
bubbles as they collapsed near a plane solid surface. The ambient pressure was 
maintained at about 0·04 atm during collapse so that collapse velocities would be 
reduced to facilitate the photography. These bubbles were nearly spherical as 
they started collapsing. First, they became elongated in the direction normal 
to the wall; then they tended to flatten and form an inward moving jet on the 
side of the bubble opposite the wall. 
The advantages of a numerical technique for stimulating non-spherical bubble 
collapse are clear. Experiments are difficult and give only sketchy results. Per-
turbations from spherically symmetric solutions are not valid for large deforma-
tions. A numerical solution, however, can check results and supply detailed in-
formation. Numerical methods can also be applied to situations which might be 
very difficult to produce in the laboratory. Mitchell, Kling, Cheesewright & 
Hammitt (1967) have considered simulation of bubble collapse using the Marker-
and-Cell technique, a general method for simulating incompressible, viscid 
flows with an assortment of boundary conditions including free surfaces. Because 
non-spherical collapse is of such interest, it is worthwhile to develop a method of 
simulation especially suited to these problems. 
Definition of the problem 
The asymmetries caused by a solid wall should be separated from those due to 
initial asymmetries in shape or velocity of the type analyzed in the linearized 
theory ofPlesset & Mitchell (1956). The bubble is therefore taken to be spherical 
and at rest at the initiation of the collapse, and any other extraneous asym-
metric effects such as gravity are also omitted. 
The following assumptions will be made: (1) The liquid is incompressible. 
(2) The flow is non-viscous. (3) The vapour pressure is uniform throughout the 
bubble interior. (4) The ambient pressure and the vapour pressure are constant 
with time. (5) The bubble contains no permanent gas. (6) Surface tension effects 
are negligible. 
This set of assumptions defines the problem as the non-spherical version of the 
classical Rayleigh collapse calculation. Only the first three assumptions are 
essential to the method of simulation developed here. The last three assumptions 
Vapour cavity collapse 285 
are made to keep the essential features of the problem in the foreground. With the 
absence of shocks, compressibility will not become important until speeds in the 
liquid are comparable with the speed of sound. Thus the liquid can be assumed 
to be incompressible with the understanding that solutions are valid for small 
Mach numbers only. In most cases of collapse, viscosity can be neglected unless 
the bubble is initially very small. For example, viscosity is unimportant for a 
spherical bubble collapsing in water under atmospheric pressure if the initial 
radius is I0-3 cm or greater. As for the assumption of uniform pressure inside 
the bubble, this assumption will remain valid as long as speeds on the bubble 
surface are below the speed of sound in the vapour. 
The problem is specified by the following conditions: Poo =ambient pressure, 
Pv = vapour pressure inside the bubble, R0 = initial radius of the bubble, 
b = initial distance from the plane wall to the centre of the bubble. 
Because the flow is irrotational, the velocity vector v can be written in terms 
of a velocity potential ¢. Since incompressibility is assumed, ¢ must satisfy 
Laplace's equation throughout the liquid. 
The pressure boundary conditions can be restated in terms of¢ and v with the 
aid of Bernoulli's equation 
8¢18t+fv2 +(PIP) = c(t). (1) 
Infinitely far from the bubble the velocity is zero, and the pressure is the ambient 
pressure. The velocity potential there is an arbitrary function of time only, which 
can be taken to be zero limit ¢(x, t) = 0. (2) 
lxl-->-ro 
Then on the free surface, 
(3) 
The final boundary condition on the potential is that its normal derivative must 
vanish at the solid wall. Initially the potential is uniformly zero. 
As a result of the assumptions, the solutions are characterized by the single 
parameter b I R 0 • A solution for a particular value of b I R 0 can be scaled to bubbles 
of any initial size under any positive collapsing pressure !1p. Velocities are 
independent of the size of the bubble, and are scaled like (!1plp)l. 
The method of simulation 
The\ irrotationality of these problems is exploited by solving them in terms 
of the velocity potential. A number of small time steps are used. Before each step 
the potential problem is solved and the velocity is calculated at a large number 
of points representing the free surface of the cavity. The displacement of a free 
surface point with a velocity v calculated at the start of the time step is approxi-
matedby !1x = v!1t. 
The rate of change of the potential of a point moving on the free surface is 
D¢1Dt = o¢1ot+v2 , 
or by (3) D¢1Dt = !1plp+!v2• 
(4) 
(5) 
(6) 
286 M. S. Plesset and R. B. Chapman 
The change in potential of a free surface point can then be approximated by 
(7) 
A detailed description of this numerical method can be found in a recent report 
by Plesset & Chapman (1970). 
Results of the calculations 
The collapse of an initially spherical bubble near a plane solid wall was simu-
lated for two cases. In case 1 the parameter bfR0 was unity; that is, the bubble 
boundary was in contact with the solid wall and tangent to it. In case 2 b f R0 
was 1·5; the closest distance from the bubble boundary to the solid wall was in-
itially half the radius of the bubble. Ninety-four time steps were used for case 1 
and seventy-seven for case 2. Calculations were stopped when the liquid jet 
reached the opposite wall of the bubble since the assumption of incompressi-
bility is no longer valid. The bubble shapes for selected time steps for cases 1 and 
2 are shown superimposed in figures 1 and 2, respectively. Table 1lists the time 
intervals in units of R0(pjflp)t from the initiation of collapse for each shape and 
the downward velocity on the upper portion of the bubble at the axis of symmetry. 
The velocities, which are scaled like (D..pfp)t, are given in metresfsec for the special 
value D..p 106 dynesfcm2 1 atm 
p = 1·0 gfcm3 ~ densityofwater· (8) 
The solid wall influences the bubble early in the collapse chiefly by reducing 
the upward motion of the lower portion of the bubble. As a result the bubble 
becomes elongated in the direction normal to the wall as was predicted by 
Rattray (1951). The bottom of the bubble still moves upwards towards the 
bubble centre in case 2, but since this upward motion is reduced, the centroid 
of the bubble moves towards the wall displaying the well-known Bjerknes effect. 
As the bubble acquires kinetic energy, this energy is concentrated in the upper 
portion of the bubble which eventually flattens and forms a jet. Once the jet is 
formed, the speed of its tip remains fairly constant. 
The behaviour of the upper portion of the bubble in case 2 is not very different 
from case l. The overall shapes appear quite different, however, because the 
bottom of the bubble must remain in contact with the solid wall in case 1 but is 
allowed mobility in case 2. The jet speed in case 2 (about 170 mfsec under ~tmos­
pheric D..p) is somewhat larger than the speed in case 1 (about 130m/sec). This 
behaviour is as expected since a bubble which is farther from the wall collapses 
to a smaller size and can concentrate its energy over a smaller volume. 
The jet appears to be the result of the deformation caused by the presence of 
the wall during the early part of the collapse. It is known from the linearized 
theory ofPlesset & Mitchell (1956) that a small deformation can lead to jetting 
much later in the collapse, but the jet formation found here appears before the 
jetting which might develop from a small initial perturbation. 
Although the bubble is initially fairly close to the wall in case 2, the final 
jet must pass through the liquid for a distance of more than five times its 
Vapour cavity collapse 
Initial sphere 
Wall 
Figure 1. Bubble surfaces from case 1. 
Initial sphere 
77777777777777777777 
Wall 
Figure 2. Bubble surfaces from case 2. 
287 
288 M.S. Plesset and R. B. Chapman 
diameter before it reaches the solid wall. The jet in case 1, which strikes the wall 
directly, seems the more capable of damage even though the jet speed is lower. 
Apparently cavitation bubbles must almost touch the wall initially to be capable 
of damaging it. 
Figure 1 Figure 2 
Velocity Velocity 
Shape Time (mfsec) Time (mfsec) 
A 0·63 7·7 0.725 10 
B 0·885 19 0·875 17 
0 0·986 42 0·961 35 
D 1·013 65 0·991 53 
E 1·033 100 1·016 94 
F 1-048 125 1·028 142 
G 1-066 129 1·036 160 
H 1·082 129 1-044 165 
I 1-098 128 1·050 170 
J 1-119 128 
TABLE I. Time interval from initiation of collapse, and the velocity of the bubble botmdary 
at the axial point most distant from the wall, for the cases illustrated in figure 1 and figure 2 
A jet of speed v directly striking a solid boundary produces an initial pressure 
given by the water hammer equation, 
P - p C V ( PsCs ) WH- L L PLCL +pscs ' (9) 
where the L and s subscripts refer to the liquid and the solid, respectively. 
Usually p 8 c8 is large compared to PLCL producing the approximation 
(10) 
Experiments by Hancox & Brunton (1966) have shown that multiple impacts 
by water at a speed of 90 mfsec can erode even stainless steel. 
Benjamin & Ellis (1966) present two series of photographs ofbubbles collapsing 
near a solid wall in figures 3 and 4 of their paper. The collapse illustrated in their 
figure 4 is very similar to case 2. The collapse illustrated in their figure 3 falls 
between case 1 and case 2. Benjamin & Ellis estimated the jet speed in their 
figure 3 to be about 10m/sec under an ambient pressure of about 0·04atm. The 
vapour pressure of the water is very important at this reduced pressure. Since 
Benjamin & Ellis did not mention the temperature of the water, this pressure 
cannot be determined directly. However, D.p can be deduced from the total col-
lapse time which they gave as 10 msec. The total collapse time for a spherical 
bubble is, according to Rayleigh, 
T = 0·915R0(pfD.p)i. (11) 
The total collapse times for cases 1 and 2 are only slightly greater since most 
Vapour cavity collapse 289 
of the time is consumed early in the collapse while the bubble is nearly spherical. 
For collapse near a solid wall, then, the total collapse time is roughlyt 
r ~ R0(pj:lp)!. (12) 
Since R0 ~ l·Ocm and r = lOmsec, the pressure:difference for the collapse in 
figure 3 of Benjamin & Ellis is approximately 
:lp = Poo -Pv ~ l04 dynesjcm2 ~ 0·01 atm. (13) 
A vapour pressure of 0·03 atm corresponds to a temperature of about 76 °F. 
Speeds for one atmosphere pressure difference should be increased by a factor of 
ten giving an estimated jet speed of roughly lOOm/sec so that the experimental 
observations of Benjamin & Ellis are compatible with the calculations performed 
here. 
As general conclusions we may say that it appears very likely that cavitation 
damage with collapsing vapour bubbles is caused by the impact of the jet pro-
duced by the presence of the adjacent solid wall. Further, it appears reasonable 
to say that only those cavitation bubbles quite near the solid boundary can pro-
duce damage whether by a jet or by any radiated shock. From the calculations 
presented here, we see that for a bubble near the wall the jet is formed early in 
the collapse history so that the many complications of the late stages of cavity 
collapse do not enter. These familiar complications include the instability of the 
spherical shape toward the end of collapse, the effects of high bubble wall velo-
cities on the behaviour of the vapour in the bubble, and the effects of compres-
sibility, not only in the vapour phase, but in the liquid as well. It is also very 
evident that the jet appears before there is any possibility of radiating a shock. 
It is not clear that the impact, or 'water-hammer' stress of (10) is the 
mechanism of damage to the solid. For the case of the spherical bubble initially 
in contact with the wall and for :lp = 1 atm in water, we have v- 130 mjsec and 
cL ,..., 1500 mfsec so that PwH ,..., 2000 atm. 
While this is a most impressive impact stress, it is not obvious that it is the im-
portant damaging mechanism since the duration of this stress is so short. We 
may estimate this duration as being no longer than the time for the impact 
signal to traverse the radius of the jet. For a bubble with an initial radius 
R0 = 0·1 em, this time is rwH,..., I0-7 sec. On the other hand the stagnation 
pressure is approximately Ps ,..., !pv2 ,..., 800 atm, 
which will have a duration of the order of the length of the jet divided by its 
velocity v. This pressure pulse may be the source of the damage because its 
duration is an order of magnitude greater. 
Finally, we may say that cavitation damage should have a close relationship 
with liquid impact damage and inferences from studies of the latter should be 
useful for cavitation damage. We may also use calculations of the kind presented 
here to get reasonably accurate estimates of cavitation stress pulses. 
t Rattray derived the formula 
T (!J.p)!- ( R0) (R0) 2 
- - = 0·915 1+0·41- +0 -~ p ~ ~ 
from his perturbation analysis. 
19 FLM 47 
290 M.S. Plesset and R. B. Chapman 
REFERENCES 
BENJAMIN, T. B. & ELLIS, A. T. 1966 Phil. Trans. Roy. Soc. A 260, 221. 
HANCOX, N. L. & BRUNTON, J. H. 1966 Phil. Trans. Roy. Soc. A 260, 121. 
KoRNFELD, M. & SuvoRov, L. 1944 J. Appl. Phys. 15, 495. 
MITCHELL, T. M., KLING, C. L., CHEESEWRIGHT, R. & HAMMITT, F. G. 1967 U. of Michi-
gan, College of Eng., Rep. 07738-5-T. 
PLESSET, M.S. 1966 Phil. Trans. Roy. Soc. A 260, 241. 
PLESSET, M.S. & CHAPMAN, R. B. 1970 Div. of Eng. and Appl. Science, Calif. Inst. of 
Tech. Rep. no. 85-49. 
PLESSET, M.S. & MITCHELL, T. P. 1956 Quart. Appl. Math. 13, 419. 
RATTRAY, M. 1951 Ph.D. thesis, California Institute of Technology. 
RAYLEIGH, LORD 1917 Phil. Mag. 34, 94. 


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Collapse of an initially spherical vapour cavity in the neighbourhood of a solid boundary

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