On The Collapse of a Gas Cavity by an Imploding MoltenLead Shell and Richtmyer-Meshkov Instability

Victoria Suponitsky, Sandra Barsky, and Aaron Froese

General Fusion Inc., 108-3680 Bonneville Place, Burnaby, BC V3N 4T5,Canada

Email: victoria.suponitsky@generalfusion.com

1 ABSTRACT

With an eye to investigating the collapse of a sphericalgas cavity, we examine the propagation of a pressurepulse in molten lead,Pb in 1, 2 and 3 dimensions, us-ing the open source CFD software OpenFOAM. Re-sults are compared to the theoretical predictions of thecollapse of a bubble. We investigate the Richtmyer-Meshkov instability[1, 2] in cylindrical geometry, atAtwood numberA = −1 for a number of different ini-tial perturbations and pressure pulses.

2 INTRODUCTION

In this paper we consider the collapse of a gas cavityby an imploding molten lead (Pb) shell. This processcan be accompanied by development of Richtmyer-Meshkov instability [1, 2](RMI). Our interest lies inMagnetized Target Fusion, and in particular, in thefactors relevant to the compression of a gas cavityby liquid lead. A scale model of this system is cur-rently under construction at General Fusion Inc. In ourprototype model a cylindrical or spherical implodingpressure wave, created by pneumatic pistons, travelsthrough the molten lead and strikes the interface be-tween thePb and a gas. The gases of interest includeair, argon and plasma. The interface undergoes a rapidacceleration and travels toward the center, compress-ing the gas. Disturbances existing at the interface priorto the passage of the pressure wave may seed devel-opment of hydrodynamic instabilities at later stages.Therefore, the estimation of the disturbances’ growthrates and an exploration of the parameters affectingthis growth are of great importance with regard tothe initially required smoothness of the interface forefficient compression. This work primarily aims todeepen our understanding of the entire collapse pro-cess including hydrodynamic instabilities which mayinfluence the efficiency of the target compression inMagnetized Target Fusion. A secondary goal is to un-

derstand the limitations of our current numerical capa-bilities and to pave the way for improvements in thefuture.

When a pressure wave hits an interface separating twofluids, there is both a transmitted and reflected wave.The ratio between the latter waves depends on the ratioof acoustic impedances,Z = ρc, of each fluid, whereρ andc are the density and sound speed respectively.For our experiment, the reflected wave is a rarefac-tion wave[3]. If the interface has perturbations, the de-velopment of the Richtmyer-Meshkov instability[1, 2](RMI) may occur.

The vast majority of work concerning RMI has beendone in a planar geometry and a number of modelsdescribing linear as well as nonlinear stages of the per-turbation growth have been developed [4, 5]. Therehas been recent interest focused on RMI in convergentgeometries [6, 7, 8, 9, 10]. Pressure pulses in conver-gent geometries are more complicated than in the pla-nar case. The convergence itself leads to accelerationand amplification of the pulse, regardless of any insta-bilities. Furthermore there are secondary effects suchas reshock, which further complicate the scenario. Afull understanding of RMI in cylindrical or sphericalgeometry has yet to be achieved.

This article is organized as follows: the numericalmethod and setup are described in §3. Results of thecylindrical and spherical collapse of unperturbed cav-ity are given §4.1 and the results concerning RMI incylindrical geometry are presented in §4.2.

3 NUMERICAL METHOD AND SETUP

Simulations were performed using the open sourceCFD code OpenFOAM [11]. The compressible mul-tiphase solver ‘compressibleInterFoam’ was used inthese simulations. This solver implements the Vol-ume Fraction (VoF) method for interface tracking and

is suitable for the simulation of two compressible im-miscible isothermal fluids. A barotropic equation ofstate is implemented in the solver to close the systemof continuity and momentum equations. For each ofthe phases it is given by:

ρi = ρ0 +ψiP (1)

where ψ = 1/c2 is the compressibility andc is thespeed of sound. For a compressible phase the nomi-nal densityρ0 is set to zero. This results in an idealgas equation of state for isothermal flow. For an in-compressible phase, i.e. liquid,ρ0 is set to a nomi-nal density of the liquid under the normal conditions;the fluid density then remains essentially constant un-less the liquid is subjected to the very high pressures.The majority of the simulations were carried out in a

Time [sec]

P(t

)

0 2E-05 4E-05 6E-05 8E-05 0.00010

5E+08

1E+09

1.5E+09

Pmax

(b) Pmax

1E05

Figure 1: (a) Numerical setup for 2D simulations in acylindrical geometry; (b) Shape of the pressure pulseimposed on the outer boundary.

2D cylindrical geometry. The outer radius was set toRouter = 1.5m, and the initial interface was located atRinter f ace= 0.2m. A very small central part with ra-dius r < 0.01m has not been solved to avoid unnec-essarily small time steps due to the very small size ofthe grid cells in this region. The effect of ‘taking out’the centre was found to be negligible for the purposesof this work. A cylindrical mesh withnr = 2800 andnθ = 800 grid points in the radial and azimuthal direc-tions, respectively, was used in the simulations. At the

outer boundary,Router, a pressure pulse was imposedas time-dependent boundary conditionP(t). In mostcases a pulse duration wasTpulse≈ 1×10−4s, whichis approximately what we expect in our scale model.At the inner boundaryRinner = 0.01m a zero gradientboundary condition for both pressure and velocity wasprescribed. Initially both fluids were in rest and at at-mospheric pressure.

Simulations were carried out for both smooth and si-nusoidally perturbed interfaces. The former is mainlyfor the validation purposes and the latter for studyingpotential development of instabilities due to the initialimperfection of the interface. A schematic of the setupand shape of the pressure pulse are shown in Figs. 1(a)and (b), respectively. A 3D spherical implosion wasalso simulated in the case of a smooth interface. Forsuch simulations a 20o degree spherical segment wasused as a computational domain with periodic bound-ary conditions applied on either side of the segment. Inthe radial (r) and azimuthal (θ) directions the numberof grid points was reduced by factor of two comparedto the 2D cylindrical mesh and in the periodic direction(φ) nφ = 23 points was used.

For a perturbed interface, the initial sinusoidal pertur-bation, defined by its initial amplitudeA and azimuthalwave numbern, was imposed in the beginning of thesimulation. The initial amplitude was defined relativeto the radial location of the unperturbed interface as:

Aperturbation=rmax−Rinter f ace

Rinter f ace×100%, (2)

and the azimuthal moden is given by n =2πRinter f ace/λ, whereλ is the perturbation wavelength.

All but one of the simulations were carried out foran implosion from molten lead,Pb, into air. Onesimulation for an implosion fromPb into water wasperformed for comparison. The densities and soundspeeds are:Pb ρ1 = 10000kg/m3, c1 = 2000m/s;air ρ2 = 1kg/m3, c2 = 316m/s; water ρwater =1000kg/m3, cwater = 1500m/s. The Atwood numberfor these simulations isA= (ρ2−ρ1)/(ρ2+ρ1) =−1for Pb-air, andA = −0.81 forPb-water.

4 RESULTS

4.1 Pulse propagation and gas cavity col-lapse: unperturbed interface

In this section we present results for the pressure pulsepropagation through thePb in convergent geometryand a subsequent collapse of the air cavity with aninitially unperturbed interface. The results are com-pared with existing theoretical solutions. A summary

of the relevant processes affecting the pressure pulse is(i) amplification of the pressure, due to the convergentgeometry and steepening of the pressure pulse duringits propagation through thePb; (ii) nearly complete re-flectance of the pressure pulse back into the lead. Thereflected wave is a rarefaction wave that subjects themolten lead to tension which in turn causes cavitation;(iii) when thePb/air interface is hit by the pulse, theinterface undergoes a rapid acceleration and continuesits movement toward the center, compressing the air.A Pb shell converges towards center, and is separatedfrom the rest of the lead by the cavitation region.

Pressure pulse propagation through thePb for cylin-drical and spherical geometries are shown Figs. 2 (a)and (b), respectively. Results are shown for two am-plitudes of the pulse:P1 = Pmax = 1.5× 108Pa andP2 = Pmax = 1.5× 109Pa. These amplitudes corre-spond to the pressures attainable in our scale model.A relation between particle velocity,Vp, and the pres-sure of the pulse is given by:

Vp =Pmax−Po

ρ1c1≈

Pmax

ρ1c1. (3)

Eq. 3 leads to the initial particle velocities ofVp ≈7.5m/s andVp ≈ 75m/s for pulsesP1 andP2, respec-tively. Linear behaviour is expected forP1 whereasP2 might exhibit non-linearities. Fig. 2 demonstratesthe following: (i) the amplification of the pressurepulse during its propagation throughPb is in excellentagreement with theoretical predictions [12];P∼ 1/

√r

for cylindrical andP ∼ 1/r for spherical geometries;(ii) the steepening of the pulse due to the nonlin-ear effects, and (iii) the time taken for the pulse topropagate from the outer boundary to the interface istpropagation= 6.5×10−4s, which agrees with the pre-scribed speed of soundc1 = 2000m/s.

VoF contours at the time the pressure pulse hits theinterface and later during the collapse are shown inFigs. 3(a) and (b) for the pulseP2 in cylindrical geom-etry. This figure clearly demonstrates the formation ofthePb shell moving toward the center and the forma-tion of a cavitation region behind the shell separatingit from the rest of the molten lead. Fig. 4 shows radialprofiles of the pressure, velocity and VoF during thecollapse. Results are presented for cylindrical geome-try and pressure pulseP1. VoF profiles clearly show thelocation of the interface, the formation and growth ofthe cavitation region once the interface starts movingtowards the center and also the increase of thePbshellthickness during the collapse process. From the pres-sure profiles we can see the pressure build up inside thePbshell, the sudden drop of the pressure in the cavita-tion region and increase of the pressure inside the air

r [m]

Pre

ssur

e[P

a]

0.20.40.60.811.21.40

1E+08

2E+08

3E+08t=5e-4s

t=1e-4st=2e-4s

t=3e-4st=4e-4s

t=6e-4s

t=6.5e-4s Interface~ 1/r 0.5Cylindrical

(a)

r [m]

Pre

ssur

e[P

a]

00.20.40.60.811.21.40

2E+08

4E+08

6E+08

8E+08

1E+09

t=6.5e-4s

t=1e-4st=2e-4s

t=3e-4st=4e-4s

t=5e-4s

t=6e-4s

Interface

(b)

~ 1/r

Spherical

Figure 2: Pressure pulse propagation from the outerboundary atr = 1.5m towards the interface atr = 0.2min (a) cylindrical and (b) spherical geometry. Solidand broken lines correspond to the pulse amplitudesof Pmax= 1.5×108PaandPmax= 1.5×109Pa, respec-tively. The pulse propagates from the left to the right.

as it is getting compressed. Several important featuresof the flow field can be observed. Firstly, when thepulse reaches the interface (Fig. 4a), the particle ve-locity and pressure are in a good agreement with Eq. 3especially if the slight increase in lead density is takeninto account. Secondly, once the interface starts mov-ing, its velocity which is equal to the fluid velocity atthe position of the interface, roughly corresponds to

Vinter f ace≈ 2×Vp, (4)

whereVp is given by Eq. 3 withPmax taken as a max-imum pressure of the pulse just before it hits the in-terface. This is in agreement for expected result oftwo fluids with large impedance ratio (§4.8 and §4.9in [13]). Thirdly, there is a gradual increase in veloc-ity in the cavitation region followed by further increaseinside thePb shell. Lastly, a shock wave is generatedinside the air as a result of the sudden acceleration ofthe interface. This acceleration of the interface can beviewed as an analogy to a piston starting from the restand suddenly moving with constant velocity into a qui-escent gas. In this situation, a shock front immediatelyappears, moving away from the piston with a constantand supersonic speed. Ahead of the shock front the

Radial Position [m]

Vo

F

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loci

ty[m

/s]

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ssur

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a]

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108

t= 6.5e-4s

(a)

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ty[m

/s]

Pre

ssur

e[P

a]

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108

t= 0.0015s

(d)

Radial Position [m]

Vo

F

Ve

loci

ty[m

/s]

Pre

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e[P

a]

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108

t= 8e-4s

(b)

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Vo

F

Ve

loci

ty[m

/s]

Pre

ssur

e[P

a]

0.5 0.4 0.3 0.2 0.10

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t= 0.002s

(e)

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Vo

F

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ty[m

/s]

Pre

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e[P

a]

0.5 0.4 0.3 0.2 0.10

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t= 0.001s

(c)

Radial Position [m]

Vo

F

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loci

ty[m

/s]

Pre

ssur

e[P

a]

0.5 0.4 0.3 0.2 0.10

50

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200

0

0.2

0.4

0.6

0.8

1

104

105

106

107

t= 0.004s

( f )

cavitation region

shell

Figure 4: This figure illustrates the radial profiles of: the pressure, broken green line; velocity, solid red line; andVoF, dash-dot blue line at and after the pulse hits the interface for cylindrical geometry,Pmax= 1.5×108Pa.

Figure 3: VoF contours at (a)T = 6.5×10−4sec, pulsehits the interface, and (b)T = 9.5× 10−4sec, dur-ing the cavity collapse. Cylindrical geometry,Pmax =1.5×109Pa.

gas is at rest, while behind the shock it moves with thevelocity equal to that of the piston, i.e. the interfacevelocity in our case (see [14] §3). It is worth notingthat while the shock inside the air is an important fea-ture of the flow field, it has little effect on thePb shellevolution and therefore it is not the focus of our work.

A correct prediction of the gas cavity collapse is nec-essary for the reliable estimation of the compressionefficiency in our system and also for further studiesof potential instabilities. Fig. 5(a) shows the interface

position during the collapse for all geometries for thelow amplitude pulse,P1. We compare our results to asemi-analytical solution of Kedrinskii (§ 1.4 in [15]),Fig. 5. The agreement is excellent for a 1D cartesiancase and at early times for cylindrical and sphericalgeometries. The main reason for the disparity betweenour results and Kedrinskii’s [15] is the finite durationof the pressure pulse in our simulations. By increas-ing the duration of the pulse the differences betweenour results and Kedrinskii are considerably reduced. Itis also worth noting that our current simulations allowthe air to escape the computational domain resulting ina complete collapse of the cavity. Thus, in our case theinterface is accelerated during the entire collapse. Inreality, however, the interface will undergo decelera-tion during the latest stages of compression because ofthe high gas pressure. This deceleration might be im-portant for the potential development of the Rayleigh-Taylor instability, but it is not our current focus.

It is also worth noting that while the acoustic pres-sure pulse hits the interface at the same time ofT =6.5×10−4s regardless of its initial amplitude and ge-ometry, the initial interface velocity and, consequently,the collapse time, are proportional to the pulse ampli-tude at the time it hits the interface. Therefore, theresults for different geometries and initial pulse ampli-

Time [sec]

Inte

rfa

ceP

osi

tion

[m]

0.001 0.002 0.003 0.0040

0.05

0.1

0.15

0.2

Spherical

Cartesian

Cylindrical

(a)

Time [sec]

Inte

rfa

ceP

osi

tion

[m]

0.002 0.004 0.006 0.008 0.010

0.05

0.1

0.15

0.2

Spherical

Cartesian

Cylindrical

(b)

Figure 5: Interface position during the cavity collapsefor cartesian, cylindrical and spherical geometries forthe pulse withPmax = 1.5× 108Pa. (a) OpenFoam:red solid line; Kedrinskii (§1.4 in [15]) analyticalsolution: blue dash-dot line. (b) OpenFOAM resultsfor the case where the pulse amplitude is the samefor all geometries when the pulse hits the interface atT = 6.5×10−4sand equal toP = 1.5×108Pa.

Time [sec]

Inte

rfa

ceP

osi

tion

[m]

0.001 0.002 0.003 0.004 0.0050

0.05

0.1

0.15

0.2

OF; P=1.5e8PaOF; P=5.0e8PaOF; P=1.5e9PaKedrinskii; P=1.5e8PaKedrinskii; P=1.5e9Pa

Figure 6: Effect of the pressure pulse amplitude onthe collapse of the gas cavity. Results are shown forcylindrical geometry and scaled for comparison withlow amplitude pulse withPmax= 1.5×108Pa.

tudes can be scaled such that the initial interface ve-locity is the same. This is presented in Fig. 5(b) wherethe same results shown in Fig. 5(a) are scaled toP1.

Finally the effect of the initial pulse amplitude on thecollapse time is shown in Fig. 6 for the cylindricalgeometry. The results are scaled so that the initialinterface velocity corresponds toP1. It can be seenthat OpenFOAM results are identical for pulses withPmax = 1.5× 108Pa andPmax = 5× 108Pa indicatingthat at these amplitudes non-linear effects are insignifi-cant. Results also show that for high amplitude pulses,P2, the nonlinearity begins to manifest by increasingthe time of collapse due to the energy losses. Devia-tions from the linear pulse are very similar for Open-FOAM and Kedrinskii [15] results.

4.2 RM-InstabilityWe now turn our attention to the development of RMIduring the gas cavity collapse as a result of an initialimperfection of the interface. Effects of initial pertur-bation amplitude, azimuthal wave-number and pres-sure pulse amplitude are addressed.

The effect of the initial perturbation amplitude on theevolution of the perturbation during the collapse isshown in Fig. 7 by VoF contours. Simulations wereperformed for four different amplitudes (Eq. 2): (i)A1 = 0.5%; (ii) A2 = 2.5%; (iii) A3 = 5% andA4 =15% for the perturbation with azimuthal wave-numbern = 6 and pressure pulseP2. One can see that af-ter the pressure pulse strikes the interface the charac-teristic pattern of RMI, namely a mushroom of lightfluid penetrating into heavy fluid and vice versa, startsto develop. The finger of light fluid poking into theheavy fluid is usually called a bubble, and that of theheavy into the light is called a spike. If the shock startsin the heavy fluid the sinusoidal perturbation initiallyflattens, and then inversion occurs [6]. During inver-sion, what had been a finger of light fluid penetratinginto the heavy becomes a finger of heavy fluid pen-etrating into the light fluid. In the present case of avery large ratio of fluid densities, the spikes are sig-nificantly sharper when compared with those at lowerAtwood numbers. This has been also observed by Tianet al. [8]. It can be also seen that shape of the moltenlead shell and cavitation region are affected by the ini-tial imperfection of the interface.

The time evolution of the amplitudes of the spikes andbubbles at different initial perturbation amplitudes isplotted in Fig. 8. The amplitude of the spikes andbubbles are calculated as their radial position minusthe position of undisturbed interface. We also showthe average perturbation amplitude which is calculated

as (rmax− rmin)/2. For a very small amplitude ofA1 = 0.5% (Fig. 8a) the perturbation evolution can bedescribed as linear during the entire collapse becausethe growth rates of spikes and bubbles are almost sym-metrical. Also the perturbation growth does not seemto slow down during the collapse. With increasing am-plitude, nonlinear effects become obvious, as can beseen in Fig. 8(b),(c) and (d). A dominant nonlinear ef-fect is the increasing disparity of growth rates betweenspikes and bubbles: spikes appear to accelerate oncethe inversion has occurred, i.e. they grow faster thanlinear scaling with amplitude would imply, while thebubble growth is slower than linear scaling. This canbe also observed by the increase in deviation of aver-age perturbation amplitude from that of spike with theincrease in initial perturbation amplitude. Fig. 9 showsan alternative representation of the results where thetime evolution of spikes and bubbles is shown by theirradial position. The position of the undisturbed inter-face for comparison.

The effect of the azimuthal wave-number on the per-turbation growth is shown in Fig. 10 for the perturba-tion with initial amplitude ofA3 = 5% and pressurepulseP2. This figure clearly demonstrates that the dis-parity in the growth of spikes and bubbles increaseswith an increase in azimuthal wave-number. It alsoseems that even at initial stages, before inversion, thegrowth rates do not really scale with wave-mode num-ber. A close-up of the perturbation structure at latestages of the collapse for the different azimuthal modesis shown in Fig. 11 by the VoF contours. It can beobserved that for higher azimuthal modes a secondaryKelvin-Helmholtz instability starts to develop on thesides of the spike because of the strong velocity shear.

For the sake of completeness the perturbation evolu-tion for the pulses of different strength,P1 and P2

are shown in Figs. 12(a) and (b), respectively, forthe perturbation with amplitudeAmax = 5% and az-imuthal moden = 6. Fig. 12(a) shows radial positionof the spike, bubbles and the unperturbed interface,and Fig. 12(b) shows the amplitude of spike and bub-bles relative to the undisturbed interface. The resultsfor a weaker pulse are scaled with the amplitude ratioof the pressure pulses for the comparison; the curvesdo not collapse, indicating the lack of a linear relation.

Finally, for comparison, we show some results of im-plosion of molten lead into water. The equation ofstate used for water is Eq. 1 with appropriate valuesfor ρo andc, i.e. both fluids are basically incompress-ible. The Atwood number isA = −0.81 and ratio ofacoustic impedances isZ1/Z2 = 13.3. In this case thedisturbance evolution is considerably different from

that discussed earlier. Since the water can hardly becompressed, the interface moves only bit toward thecenter before it springs back. Phase inversion occurs,however, the spike penetration is not as dramatic asbefore. Also no cavitation has been observed in thecase. Fig. 13 illustrates the perturbation evolution inthis case.

REFERENCES

[1] R.D. Richtmyer. Taylor instability in shock ac-celeration of compressible fluids.Commun. PureAppl. Math., 13:297-319, 1960.

[2] E.E. Meshkov. Instability of the interface of twogases accelerated by a shock wave.Fluid Dyn.,4(5):101-104, 1969.

[3] Y. Yang and Q. Zhang and D.H. Sharp. Small am-plitude theory of Richtmyer-Meshkov Instability.Phys. Fluids, 6:1856-873, 1993.

[4] M. Latini and O. Schilling and W.S. Don.High-resolution simulations and modeling ofreshocked single-mode Rychtmyer-Meshkov in-stability: Comparison to experimental data and toamplitude growth model predictions.Phys. Fluids,19:024104, 2007.

[5] M. Brouillette The Richtmyer-Meshkov Instabil-ity. Annu. Rev. Fluid Mech, 34:445-68, 2002.

[6] Q. Zhang and M.J. Graham A numerical study ofRichtmyer-Meshkov instability driven by cylindri-cal shocks.Phys. Fluids, 10:974–992, 1998.

[7] J.R. Fincke, N.E. Lanier, S.H.Batha, R.M. Hueck-staedt, G.R. Magelssen, S.D. Rothman, K.W.Parker and C.J.Horsfield. Postponement of Sat-uration of the Richtmyer-Meshkov Instability ina Convergent Geometry. Phys. Rev. Letters,93(11):115003, 2004.

[8] B. Tian and D. Fu and Y. Ma. Numerical inves-tigation of Richtmyer-Meshkov instability drivenby cylindrical shocks.Acta Mech Sinica, 22:9–16,2006.

[9] M. Lombardini and D.I. Pullin. Small-amplitudeperturbations in the three-dimensional cylindri-cal Richtmyer-Meshkov instability.Phys. Fluids,21:114103, 2009.

[10] R. Krechetnikov. Rayleigh-Taylor andRichtmyer-Meshkov instabilities of flat andcurved interfaces.J. Fluid Mech., 625:387, 2009.

[11] OpenFOAM. www.openfoam.org.[12] Landau L. D. and Lifshitz E. M. Fluid Mechanics

(2nd Ed.).Pergamon Press, 1987.[13] Thompson, P. A. Compressible-Fluid Dynamics.

Maple Press Company, 1984.[14] R. Courant and K. O. Friedrichs. Supersonic

Flow and Shock Waves.Springer-Verlag, 1976.

[15] V. K. Kedrinskii. Hydrodynamics of Explosion.Springer-Verlag, 2005.

Figure 7: Effect of the initial perturbation amplitude shown by the VoF contours. Red is thePb, blue is the gas,and the mixed colours indicate the cavitation regime. Perturbation isn = 6, and the pressure pulse amplitude isPmax= 1.5×109Pa. Black circles correspond to the interface position for theunperturbed case.

Time [sec]

Per

turb

atio

nA

mplit

ude

[m]

0.0007 0.0008 0.0009 0.001-0.006

-0.004

-0.002

0

0.002

0.004

0.006

A1= 0.5%

(a)

SpikeBubbleAverage

Time [sec]

Pert

urb

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nA

mplit

ude

[m]

0.0007 0.0008 0.0009 0.001-0.03

-0.02

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0.01

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0.03

A2 = 2.5%

(b)

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mp

litu

de

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0.0007 0.0008 0.0009 0.001-0.06

-0.04

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A3 = 5%

(c)

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nA

mp

litu

de

[m]

0.0007 0.0008 0.0009 0.001

-0.1

-0.05

0

0.05

0.1

A4 = 15%

(d)

Figure 8: Effect of the perturbation amplitude: green and blues lines - amplitude of the spikes and bubblesrelative to the unperturbed interface; red lines - averagedamplitude calculated as(rmax− rmin)/2. Perturbation isat azimuthal mode ofn = 6, and the pressure pulse amplitude isPmax= 1.5×109Pa.

Time [sec]

Inte

rface

Posi

tion

[m]

0.0007 0.0008 0.00090

0.05

0.1

0.15

0.2

A4 = 15%A3 = 5%A2 = 2.5%A1 = 0.5%

SpikeBubbleUnperturbed

Figure 9: Radial location of spikes and bubbles(green and blue lines) for different amplitudes ofthe initial perturbation. Interface location for theunperturbed case (red line) is also shown. Resultsare for a pulse withPmax= 1.5×109 Pa and a per-turbation with azimuthal mode ofn = 6.

Time (sec)

Per

turb

atio

nA

mplit

ude

[m]

0.0007 0.0008 0.0009 0.001-0.06

-0.04

-0.02

0

0.02

0.04

n=3n=6n=9

Figure 10: Effect of azimuthal mode number: Am-plitude of spikes and bubbles relative to the posi-tion of the unperturbed interface for perturbationswith n= 3 (red lines),n= 6 (green lines) andn= 9(blue lines). Results are shown for the perturbationwith initial amplitude ofA3 = 5% and pulse withPmax= 1.5×109Pa.

Figure 11: Close-up look on the perturbation structure during the collapse (T = 9.25× 10−4sec) at differentazimuthal modes. Perturbation amplitude isA3 = 5% and pulse withPmax= 1.5×109Pa.

Time [sec]

Inte

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osi

tion

[m]

0.0007 0.0008 0.0009 0.0010

0.05

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0.2 SpikeBubbleUnperturbed

Pmax =1.5e9 Pa

Pmax =1.5e8 Pa

(a)

Time [sec]

Per

turb

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mp

litud

e[m

]

0.0007 0.0008 0.0009 0.001

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Pmax =1.5e9 Pa

Pmax =1.5e8 Pa

(b)

Figure 12: Effect of the pressure pulse amplitude: (a) Radial location of spikes and bubbles (green and blue lines)along with interface location for the unperturbed case (redline). (b) Perturbation amplitude (spikes and bubbles)relative to the unperturbed case. Perturbation amplitude isA3 = 5% and azimuthal wave-number isn = 6.

Figure 13: Perturbation evolution for implosion of a moltenlead (blue) into water (red). Results are shown bythe VoF contours for the perturbation withA3 = 5%, azimuthal mode number ofn = 6 and pressure pulse withPmax= 1.5×109Pa.