Spherical Richtmyer-Meshkov instability for axisymmetric flow

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1 Mathematics and Computers in Simulation 65 (2004) Spherical Richtmyer-Meshkov instability for axisymmetric flow Srabasti Dutta a, James Glimm a,b, John W. Grove c, David H. Sharp d, Yongmin Zhang a, a Department of Applied Mathematics and Statistics, University at Stony Brook, Stony Brook, NY , USA b Center for Data Intensive Computing, Brookhaven National Laboratory, Upton, NY , USA c Methods for Advanced Scientific Simulations Group, Computer and Computational Science Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA d Complex Systems Group, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Abstract Front tracking has proved to be an accurate and efficient algorithm in the sense that tracking the interface can reduce the error significantly [9,10]. By applying this algorithm, we conduct numerical simulations of Richtmyer Meshkov (RM) instabilities in spherical geometry for axisymmetric flow. We demonstrate scaling invariance with respect to shock Mach number for fluid mixing statistics, such as growth rate and volume fraction. Here the mixing is related to bulk transport rather than molecular mixing. Our results are validated by convergence under both mesh refinement and statistical ensemble averaging. We also show that the spherical geometry will converge to planar geometry when the number of modes of interface perturbation goes to infinity IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Spherical geometry; Richtmyer Meshkov instability; Front tracking 1. Introduction The instability of a material interface under an acceleration by an incident shock was predicted theoretically by Richtmyer in 1960 [33]. Ten years later, Meshkov [31] confirmed, experimentally, Richtmyer s prediction. Since then, this interfacial instability has been referred to the Richtmyer Meshkov (RM) instability. Such instabilities are observed in supernovae explosions [27] and inertial confinement fusion (ICF) [6], and are thus of great importance to science and technology. Extensive theoretical and experimental studies of the RM instability have been carried out in the last three decades. With the advent of computer technology and increasing computing power, numerical studies of the RM instability have become very common. Corresponding author. addresses: kuttush@ams.sunysb.edu (S. Dutta), glimm@ams.sunysb.edu (J. Glimm), dhs@t13.lanl.gov (J.W. Grove), jgrove@lanl.gov (D.H. Sharp), yzhang@ams.sunysb.edu (Y. Zhang) /$ IMACS. Published by Elsevier B.V. All rights reserved. doi: /j.matcom

2 418 S. Dutta et al. / Mathematics and Computers in Simulation 65 (2004) Past research in experimental, analytical and numerical validation of the Richtmyer Meshkov instability has primarily been carried out in planar geometry [1,2,5,6,24,26,32,35,36,39]. However, recently some advances have been made in curvilinear geometry [7,16,17,19,30,34,37,38]. In this paper, we present a numerical study of a three-dimensional axisymmetric RM instability in spherical geometry. The general features of an RM unstable interface in spherical geometry are the following. As a spherical incident shock travels in the radial direction and collides with the perturbed material interface, it bifurcates into a transmitted shock and reflected wave. This stage is known as the shock refraction stage. At the end of the refraction stage, both the transmitted shock and reflected wave detach from the material interface. One wave propagates toward the origin, and the other wave away. For an open geometry, this outgoing wave will not interact with the material interface again. Accelerated by the incident shock, the material interface becomes unstable and fingers grow to form bubbles of light fluid and spikes of heavy fluid. The wave which moves toward the origin generates a pressure singularity at the origin, and is then reflected outward. As this reflected wave propagates outward, it interacts with the material interface again, a process known as reshock. Shock refraction occurs again, and this cycle continues. Therefore, the material interface is reshocked many times, though each time the shock strength is weaker. The numerical method we use for computing axisymmetric flow in the paper is a front tracking method. Front tracking is an adaptive computational method in which a lower dimensional moving grid is fit to and follows distinguished waves in a flow. Tracked waves explicitly include jumps in the flow state across the waves and keep discontinuities sharp. A key feature is the avoidance of finite differencing across discontinuity fronts and thus the elimination of interfacial numerical diffusion including mass and vorticity diffusion. In addition, nonlinear instability and post-shock oscillations are reduced at the tracked fronts. Front tracking as implemented in the code FronTier includes the ability to handle multidimensional wave interactions in both two [15,20,25] and three [13,14] space dimensions and is based on a composite algorithm that combines shock capturing on a spatial grid with a specialized treatment of the flow near the tracked fronts. FronTier is implemented for distributed memory parallel computers and some of the fundamental algorithms used in this code are described in [4,12,18,21 23]. In this paper, we study the main physical parameter (shock strength) in RM system. We demonstrate an important scaling law by applying the front tracking algorithm. This scaling law extends the one obtained by Zhang and Graham [38] for single mode RM system in cylindrical geometry to multimode system in spherical geometry. We also give a brief summary of main results regarding the efficiency of the front tracking algorithm [9,10]. Finally, we show the spherical RM system will converge to planar system when the number of modes in the initial perturbation of interface becomes sufficiently large. 2. Numerical method and simulation set up Front tracking is a numerical method in which selected waves are explicitly represented in the discrete form of the solution. Examples include shock waves, contact discontinuities, and material interfaces. Other waves which may be tracked, such as leading and trailing edges of rarefaction waves, have continuous states but jumps in their first derivatives. Tracked waves are propagated using the appropriate equations of motion for the given model. For example, if the system of equations consists of a set of hyperbolic conservation laws, u t + f = h, then the instantaneous velocity s of a discontinuity surface satisfies the Rankine Hugoniot equations, s[u] = [f ] n. Here, n is the unit normal to the discontinuity surface.

3 S. Dutta et al. / Mathematics and Computers in Simulation 65 (2004) Fig. 1. Density plots for a spherical simulation with an unperturbed interface. The left image shows a shocked contact in the tracked case. The right image shows the untracked case at the corresponding time. The grid size is During a time step propagation, the type of a wave, and the flow field in a neighborhood of the wave determine a local time integrated velocity for each point on the wave in the direction normal to the wave front. Wave propagation consists of moving each point a distance s t in the normal direction as well as computing the time updated states at the new position. Tracking preserves the mathematical structure of the discontinuous waves by maintaining the discrete jump at the wave front, thus eliminating numerical diffusion. It also allows for the direct inclusion of the appropriate flow equations for the wave front in the numerical solution. For a detailed description of the front tracking algorithm, we refer to Glimm, Grove and Zhang [18]. The validation in spherical geometry was carried out by comparison with experiment, see Drake et al. [8] where supernova experiments and simulations were reported. For other validations in planar geometry, we refer to [11,29,28]. Front tracking is a fast algorithm in the sense that it can reduces the level of mesh refinement needed to achieve a specified error tolerance by a significant factor compared to corresponding methods without tracking, thus substantially reducing the computational time as well as memory usage for simulations with contacts or material interfaces. In [9,10], we reported a detailed quantitative comparison of errors in spherical shock refraction simulations by tracked and untracked methods by solving the 2D axisymmetric Euler system. The comparison study was carried out for an unperturbed interface since this case admits an easily understood exact solution which can be obtained by solving a 1D system of spherical Euler equations on a very fine mesh. A major conclusion is that the error in any shock capturing finite difference scheme is mainly caused by numerical diffusion across the contact. The density plots for the spherical simulations with and without tracking are shown in Fig. 1, where we see the untracked interface was highly smeared by numerical diffusion. The main results in [9,10] can be summarized as follows. The error in the neighborhood of the untracked contact is the main contribution to the total error. Tracking the contact can effectively reduce both the contact error and the total error by a significant factor. For a given error tolerance, the mesh size needed to achieve a fixed error tolerance can be reduced by as much as a factor of eight per spatial dimension. A factor of 8 4 = 4096 fewer space time zones for a three space dimensional computation are then required for comparable accuracy. We apply the front tracking algorithm to simulate the RM problem where a randomly perturbed axisymmetric heavy fluid sphere is surrounded by light fluid and a spherical shock inside the heavy fluid is moving outward. The numerical simulations are carried out in a quarter domain of [0,r 1 ] [0,z 1 ], with

4 420 S. Dutta et al. / Mathematics and Computers in Simulation 65 (2004) r 1 = z 1 > 0. Let R denote the distance from any point in the domain to the origin. The initial contact surface is located at the perturbed circle n max R(φ) = R 0 + A n cos(4nφ), n=n min where the azimuthal angle φ [0,π/2], and n min = 4, n max = 8 are the minimum and maximum frequencies respectively. The Fourier mode amplitudes A n are generated by Gaussian sampling. The amplitude standard deviation is about 0.02λ, where λ is the average wavelength of the perturbation. The initial configuration of the system contains three regions: the region behind the incident shock, the region between the incident shock and the fluid interface, and the region outside the interface. The states ahead of the shock are initialized by the prescription of the densities inside and outside of the contact surface, and the pressure and the velocities of two fluids. The state behind the shock is determined by prescribing Mach number of the shock. The top and the right boundaries of the computational domain have flow-through boundary conditions, so that all outbound waves will exit the domain. Reflecting boundary conditions are used at the left and the bottom boundaries due to the symmetry. At the left boundary, we also impose the zero radial velocity due to the axisymmetry of the flow. The density ratio across the fluids is 13.7:1. An incident shock with Mach numbers M = 10, 20, 50, 100, 200 or 300 located in the heavy fluid moves towards the contact interface. The above simulations, as well as those discussed in other sections, are based on numerical solutions to the Euler equations that describe the conservation of mass, momentum, and energy for a compressible fluid: ρ t + (ρv) + α ρu r = 0, (ρv) t + (ρv v) + P + α ρuv = ρg, ( r (ρe) t + ρv E + P ) ρu(e + P/ρ) + α = ρv g, ρ r where ρ is the mass density, v the fluid velocity, P the thermodynamic pressure, E = e + (1/2)v v the specific total energy, and e the specific internal energy. The variables ρ, P, and e are related by a thermodynamic equation of state P = P(ρ, e). For simplicity we assume a perfect gas equation of state P = (γ 1)ρe, with γ > 1. The simulations reported in this paper use the value γ = 5/3. The body force per unit mass, g, is taken as zero for the present discussion. The geometry parameter α has the value zero for rectangular geometries. For axisymmetric flow α = 1, while α = 2 for one dimensional spherical symmetry. For spherical symmetry, u is the radial component of velocity and r is the distance from a point to the origin. For axi-symmetry, u is the radial component of the projection of the fluid velocity into the x y plane and r is the distance of a point from the z-axis. 3. Scaling law The scaling law with respect to shock strength was first observed by Zhang and Graham [38] in their study of RM instabilities in single mode cylindrical geometry. Their calculation solved Euler equations

5 S. Dutta et al. / Mathematics and Computers in Simulation 65 (2004) in rectangular coordinates. Here, we extend such a scaling law to multimode spherical geometry. Our computation is based on solving an axisymmetric Euler system in (r, z) cylindrical coordinates. How will the shock strength affect the RM instability? In general, the stronger the incident shock is, the faster the material interface and the faster the transmitted shock. Following Zhang and Graham [38], we study a scaling law with respect to shock Mach number. In order to reveal the scaling behavior of an RM unstable system, we introduce the following dimensionless scaled version of the quantities: time t, growth rate v, amplitude a, and interface radius R, as follows: t scaled = Mc 0t, λ 0 a scaled = a λ 0, v scaled = v Mc 0, R scaled = R λ 0. Here, λ 0 is the average wave length of perturbation in the initial material interface at t = 0, M is the Mach number of the incident shock, and c 0 is the sound speed of fluid ahead of shock at t = 0. The main parameters in studying RM instability are amplitude a and growth rate v which are defined as a = 0.5(R sp R bb ), v = 0.5(v sp v bb ), where R sp and R bb are the distances from the origin to the tips of the spike and the bubble, respectively; v sp and v bb are the velocities of the spike and the bubble, respectively in the radial direction. Fig. 2 shows front plots at the same t scaled for six different shock Mach numbers M = 10, 20, 50, 100, 200, 300. We see the front shape is almost unchanged for different shock strength. We notice that the perturbation near the axis of symmetry is more amplified than others. This asymmetry is due to a north pole effect of axisymmetric flow [19]. The results for growth rate and amplitude of the RM interface driven by a shock of Mach number M = 10, 20, 50, 100, 200, 300 are superimposed in Fig. 3 and shown in terms of the scaled velocity and perturbation amplitude as a function of the scaled time. Fig. 3 shows that the scaled quantities are almost independent of the shock strength. Another important quantity characterizing the mixing process is the mean concentration β k (t, x) of fluid k at spatial position x and t. In inertial confinement fusion, this function contains all of the information concerning the expected penetration of the instability. In astrophysics, β k is a first order moment of the material interface geometry, an important ingredient for the statistical description of remnant formation. Now we define an averaging procedure. We introduce the polar coordinate system (R, φ) in the r, z plane by r = R cos φ, z = R sin φ, where R = r 2 + z 2, and φ is the azimuthal angle. We express flow field quantities in terms of these variables. The azimuthal average of a flow quantity q is defined by π/2 q (R, t) = 2 q(r,φ,t)dφ. (1) π 0 The function X k is the phase indicator for material k (k = 1, 2), i.e. X k (R,φ,t)equals one if (R, φ) in fluid k at time t, zero otherwise. The azimuthal average of X k is the mean fluid k concentration or volume (layer, mixture) fraction, β k (R, t) X k.

6 422 S. Dutta et al. / Mathematics and Computers in Simulation 65 (2004) Fig. 2. Cross-sectional front-plots of the instability when the spherical interface is pushed outwards by a shock from the heavy fluid. The color represents the variation in density. The interface is initially perturbed randomly with a minimum frequency 4 and a maximum frequency 8. The three images of the top row show the interface when hit by a shock of Mach numbers 10, 20 and 50, respectively. The second row shows the interface for Mach numbers 100, 200 and 300. All plots are taken at the same scaled time. Fig. 3. The left image shows the scaled perturbation growth rate for various Mach numbers M = 10, 20, 50, 100, 200 and 300. The minimum frequency of the perturbed contact is 4 while the maximum frequency is 8. The right image shows the scaled perturbation amplitude for various Mach numbers M = 10, 20, 50, 100, 200 and 300. For each Mach number, simulations are averaged over 20 realizations. Fig. 4 shows the volume fractions β k for the light and heavy fluids as functions of the scaled radius R scaled for a fixed scaled time. Again we see there is little change for different Mach numbers. From Figs. 3 and 4, we notice that only the case M = 10 shows a slight difference from the other cases, where the statistics are almost identical. So, the loss of scaling seems to occur for smaller

7 S. Dutta et al. / Mathematics and Computers in Simulation 65 (2004) Fig. 4. The figure shows the volume fraction of the heavy (left image) and light (right image) fluid obtained by doing simulations using shock of Mach numbers M = 10, 20, 50, 100, 200 and 300. For each Mach number, simulations are averaged over 20 realizations. Mach numbers and the scaling law holds extremely well for strong shocks with Mach numbers M 20. All the above simulations were done using Fourier polynomial perturbed interfaces. Simulations were also carried out to check the validity of the scaling law for Legendre polynomial perturbed interfaces. All the statistical quantities are scaled in the same way as was done in the case of Fourier polynomial perturbed interfaces. The simulation set-up is the same as the one in the Fourier case except for the interface perturbation. Fig. 5 shows the front plots of a Legendre polynomial perturbed interface when driven outwards by a shock of Mach number 10, 20, 50, 100, 200 and 300. The minimum degree of the polynomial is 32 and the maximum degree is 64. Again we notice that the evolved interfaces look similar and thus are independent of the incident shock strength. Fig. 6 shows that both growth rate and amplitude become insensitive to the shock strength when the Mach number M 20 as we observed earlier in the Fourier simulations. Thus from all these observations, we conclude that the growth of an axisymmetric spherical RM unstable interface follows a scaling law. Let v m1 be the growth rate of an interface driven by a strong shock of Mach number m 1. Then the growth rate of an interface driven by a strong shock of Mach number m 2 can be obtained from the following equation: v m2 (t) = m ( ) 2 m2 v m1 t (2) m 1 m 1 Similarly, the amplitude of the unstable interface follows the scaling law a m2 (t) = a m1 ( m2 m 1 t ) where a m2 is the amplitude of the interface hit by a strong shock of Mach number m 2 and, a m1 is the amplitude of the interface hit by a strong shock of Mach number m 1. From Figs. 2 and 4, the following (3)

8 424 S. Dutta et al. / Mathematics and Computers in Simulation 65 (2004) Fig. 5. Cross-sectional front-plots of the instability when the spherical interface is pushed outwards by a shock from the heavy fluid. The color represents the variation in density. The interface is initially perturbed randomly with a Legendre polynomial which has a minimum degree of 32 and a maximum degree of 64. The three images of the top row show the interface when hit by a shock of Mach numbers 10, 20 and 50, respectively. The second row shows the interface for Mach numbers 100, 200 and 300. All plots are taken at the same scaled time. Fig. 6. Statistics for a Legendre perturbed interface with a minimum polynomial degree of 32 and a maximum degree of 64. The left image shows the scaled perturbation growth rate for various Mach numbers M = 10, 20, 50, 100, 200 and 300. The right image shows the scaled perturbation amplitude for various Mach numbers M = 10, 20, 50, 100, 200 and 300. For each Mach number, simulations are averaged over 20 realizations. scaling law holds for the volume fraction: ( ( m2 β m2 (R m2 (t), t) = β m1 R m1 t m 1 ), m 2 m 1 t ),

9 S. Dutta et al. / Mathematics and Computers in Simulation 65 (2004) Fig. 7. Convergence of numerical simulation under mesh refinement. The left image shows the growth rate of a Fourier polynomial perturbed interface, when driven radially outward by a shock of Mach number 100, located in a heavy fluid. The interface has a minimum frequency 4 and a maximum frequency 8. Amplitude for each mode is randomly generated. The right image shows the corresponding amplitudes. where β m2 is the volume fraction of either the light or the heavy fluid when the interface is hit by a strong shock of Mach number m 2, β m1 is the volume fraction of either the light or the heavy fluid hit by a strong shock of m 1, and R m1 and R m2 stand for radius. The above scaling laws are important because we can now do numerical simulations or laboratory experiments using only one strong shock. And, by studying the results of that one experiment or simulation, we can generalize the result for all other strong shock cases. However, we should mention that all calculations are performed with perfect gases at Mach numbers up to 300. In practice, the range of applicability of scaling laws could be limited for real gases. Fig. 7 shows the results of a convergence test of the numerical simulations under mesh refinement. We observe that there is a good convergence for grid sizes r = 1/200 and r = 1/400 for the growth rate and amplitude. In the scaling law simulations, we used a mesh size of and the statistical quantities were averaged over 20 realizations with different initial perturbation generated by random sampling. 4. Convergence to planar geometry In this section, we study the convergence of spherical geometry to planar geometry as the number of modes of the initial perturbation approaches infinity. There are three geometric parameters in the configuration of the initial perturbed interface: the average wave length λ 0, the initial interface radius R 0, the average amplitude a 0. If we treat the wave length λ 0 as the length scale, then we define R 0 = R 0, ã 0 = a 0 λ 0 λ 0 to be the dimensionless initial radius and the dimensionless amplitude. Let n 0 be the number of modes of the initial perturbation in the first r, z quadrant as simulated. Then we have

10 426 S. Dutta et al. / Mathematics and Computers in Simulation 65 (2004) λ 0 = πr 0, 2n 0 which gives R 0 = 2 π n 0. Therefore, we can interpret the number of modes as the dimensionless initial radius of the contact interface. The spherical simulation was performed in a quarter domain of [0,r 1 ] [0,z 1 ], with r 1 = z 1 > 0. The initial contact surface is located at the perturbed circle R(φ) = R 0 + n max n=n min A n cos(4nφ), Fig. 8. The left two images show the evolution a randomly perturbed interface under the RM instability with the minimum frequency 4 and the maximum frequency 8. The right two images show the evolution of the random interface when the minimum frequency is 16 and the maximum frequency is 32. The shock is located in a heavy fluid and moves toward a light fluid. The top images show the cross-sectional density plots of spherical simulation, while the bottom images show the density plots of planar simulation. The grid size is The color represents the variation in density. All images are taken at the same scaled time.

11 S. Dutta et al. / Mathematics and Computers in Simulation 65 (2004) Fig. 9. Convergence of spherical growth rate to planar growth rate. The left image shows the growth rate of a randomly perturbed spherical and planar interface when the minimum frequency is 4 and the maximum frequency is 8. The right image shows the growth rate when the interface has a minimum frequency 16 and maximum frequency 32. The ensemble size is 10. where the azimuthal angle is 0 φ π/2, and n min and n max are the minimum and the maximum frequencies respectively. The Fourier mode amplitudes A n are generated by Gaussian sampling. The inner heavy fluid is surrounded by light fluid. The shock is initially located in the heavy fluid and moving radially outward. The boundary conditions are the same as the ones in Section 3. In the planar simulation, we place light fluid on the top of the heavy fluid in which the shock wave is moving upward. A reflecting wall is placed on the bottom of the domain. At the top boundary, there is a flow through boundary condition. The two sides have periodic boundary conditions. The dimensionless amplitude and frequency are same in both spherical and planar runs. Here we choose to keep the perturbation wave length same in all simulations so we need to adjust the initial interface radius accordingly when we increase the frequency of perturbation. In planar runs, we choose the width of the domain to be the length of the corresponding circular interface and the height of the interface to be the radius of the sphere. The physical parameters for both runs are the density ratio across the fluids is 13.7:1 and the shock Mach number M = 10. Fig. 8 presents the comparison of the density plots for the evolution of a spherical and a planar interface. Fig. 9 shows the spherical growth rate, along with the planar growth rate, of a randomly perturbed interface, where the perturbation has n min = 4, and n max = 8. The image also shows the growth rate when n min = 16, and n max = 32. All the statistical quantities are averaged over 10 realizations and the ratio between the average amplitude and the average wavelength is kept constant with varying n min and n max. The growth rate and time are scaled as in Section 3. From Fig. 9, we observe the convergence of spherical growth rate to planar growth rate is obtained when n min = 16 and n max = Conclusions We have reported a spherical shock refraction simulation by the front tracking method. We have also presented a brief summary of the efficiency of this algorithm.

12 428 S. Dutta et al. / Mathematics and Computers in Simulation 65 (2004) By applying this algorithm, we have analyzed two important dimensionless parameters (shock strength and frequency of perturbation) in a spherical RM system for axisymmetric flow. We find scaling invariant with respect to shock strength for fluid mixing statistics. The effectiveness of scaling is better for stronger shock strength. We also show that the convergence of spherical geometry to the planar one when the frequency of perturbation in initial interface goes to infinity. This frequency (or number of modes) is proportional to the dimensionless radius of initial spherical interface. The significance of this result is that high mode number instabilities can be studied in a planar geometry if low mode number spherical perturbations are missing or can be neglected [3]. Ablation front instabilities are also of interest, and amenable to the methods used here. The only modification required in this case is the flow of mass across the interface. Also related are sharp flame combustion models, which we are currently developing for application to study of Type Ia supernova. The sharp flame models are characterized by externally given (laminar) flame speed and heat release. The simulations will be reported in future papers. Acknowledgements This work has been supported by the following research grants. S. Dutta was supported by the US Department of Energy under grant DE-FG02-90ER25084, the National Science Foundation grant DMS , and by the Los Alamos National Laboratory under contract number L. J. Glimm was supported by the Los Alamos National Laboratory under contract number L, the US Department of Energy under grants DE-FG03-98DP00206, DE-AC02-98CH10886, and W ENG-36, the National Science Foundation grant DMS , the Army Research Office grant DAAG J.W. Grove and D.H. Sharp were supported by the US Department of Energy. References [1] U. Alon, J. Hecht, D. Mukamel, D. Shvarts, Scale invariant mixing rates of hydrodynamically unstable interfaces, Phys. Rev. Lett. 72 (1994) [2] U. Alon, J. Hecht, D. Ofer, D. Shvarts, Power laws and similarity of Rayleigh Taylor and Richtmyer Meshkov mixing fronts at all density ratios, Phys. Rev. Lett. 74 (1995) [3] B. Cheng, J. Glimm, D.H. Sharp, Dynamical evolution of the Rayleigh Taylor and Richtmyer Meshkov mixing fronts, Phys. Rev. E 66 (2002) 1 7. Paper No [4] I.-L. Chern, J. Glimm, O. McBryan, B. Plohr, S. Yaniv, Front tracking for gas dynamics, J. Comp. Phys. 62 (1986) [5] L.D. Cloutman, M.F. Wehner, Numerical simulation of Richtmyer Meshkov instabilities, Phys. Fluids A 4 (1992) [6] G. Dimonte, Nonlinear evolution of the Rayleigh Taylor and Richtmyer Meshkov instabilities, Phys. Plasmas 6 (5) (1999) [7] G. Dimonte, C.E. Frerking, M. Schneider, B. Remington, Richtmyer Meshkov instability with strong radiatively driven shocks, Phys. Plasma 3 (2) (1996) [8] R.P. Drake, H.F. Robey, O.A. Hurricane, B.A. Remington, J. Knauer, J. Glimm, Y. Zhang, D. Arnett, D.D. Ryutov, J.O. Kane, K.S. Budil, J.W. Grove, Experiments to produce a hydrodynamically unstable spherically diverging system of relevance to instabilities in supernovae, Astrophys. J. 564 (2002) [9] S. Dutta, J. Glimm, J.W. Grove, D.H. Sharp, Y. Zhang, Error comparison in tracked and untracked spherical simulations, University at Stony Brook preprint no. AMS and LANL report no. LA-UR , 2003.

13 S. Dutta et al. / Mathematics and Computers in Simulation 65 (2004) [10] S. Dutta, J. Glimm, J.W. Grove, D.H. Sharp, Y. Zhang, A fast algorithm for moving interface problems, in: V. Kumar, et al. (Eds.), Computational Science and its Applications, ICCSA 2003, LNCS 2668, LANL report no. LA-UR , Springer-Verlag, Berlin, Heidelberg, 2003, pp [11] E. George, J. Glimm, X.L. Li, A. Marchese, Z.L. Xu, A comparison of experimental, theoretical, and numerical simulation Rayleigh Taylor mixing rates, Proc. Natl. Acad. Sci. 99 (2002) [12] J. Glimm, J.W. Grove, X.-L. Li, K.-M. Shyue, Q. Zhang, Y. Zeng, Three dimensional front tracking, SIAM J. Sci. Comp. 19 (1998) [13] J. Glimm, J.W. Grove, X.-L. Li, D.C. Tan, Robust computational algorithms for dynamic interface tracking in three dimensions, SIAM J. Sci. Comp. 21 (2000) [14] J. Glimm, J.W. Grove, X.-L. Li, N. Zhao, Simple front tracking, in: G.-Q. Chen, E. DiBenedetto (Eds.), Contemporary Mathematics, vol. 238, American Mathematical Society, Providence, RI, 1999, pp [15] J. Glimm, J.W. Grove, W.B. Lindquist, O. McBryan, G. Tryggvason, The bifurcation of tracked scalar waves, SIAM J. Comput. 9 (1988) [16] J. Glimm, J.W. Grove, Y. Zhang, Numerical calculation of Rayleigh Taylor and Richtmyer Meshkov instabilities for three dimensional axi-symmetric flows in cylindrical and spherical geometries, SUNY Preprint SUNYSB-AMS-99-22, LANL Preprint LA-UR , [17] J. Glimm, J.W. Grove, Y. Zhang, Three-dimensional axisymmetric simulations of fluid instabilities in curved geometry, in: M. Rahman, C.A. Brebbia (Eds.), Advances in Fluid Mechanics III, WIT Press, Southampton, UK, 2000, pp [18] J. Glimm, J.W. Grove, Y. Zhang, Interface tracking for axisymmetric flows, SIAM J. Sci. Comput. 24 (2002) (LANL report no. LA-UR ). [19] J. Glimm, J.W. Grove, Y. Zhang, S. Dutta, Numerical study of axisymmetric Richtmyer Meshkov instability and azimuthal effect on spherical mixing, J. Stat. Phys. 107 (2002) [20] J.W. Grove, Anomalous waves in shock wave fluid interface collisions, in: B. Lindquist (Ed.), Current Progress in Hyperbolic Systems: Riemann Problems and Computations, vol. 100 of Contemporary Mathematics, American Mathematics Society, Providence, RI, 1989, pp [21] J.W. Grove, The interaction of shock waves with fluid interfaces, Adv. Appl. Math. 10 (1989) [22] J.W. Grove, Applications of front tracking to the simulation of shock refractions and unstable mixing, J. Appl. Numer. Math. 14 (1994) [23] J.W. Grove, FronTier: a compressible hydrodynamics front tracking code; A short course in front tracking, LANL report no. LA-UR , Los Alamos National Laboratory, [24] J.W. Grove, R. Holmes, D.H. Sharp, Y. Yang, Q. Zhang, Quantitative theory of Richtmyer Meshkov instability, Phys. Rev. Lett. 71 (21) (1993) [25] J.W. Grove, R. Menikoff, The anomalous reflection of a shock wave at a material interface, J. Fluid Mech. 219 (1990) [26] J. Hecht, U. Alon, D. Shvarts, Potential flow models of Rayleigh Taylor and Richtmyer Meshkov bubble fronts, Phys. Fluids 6 (1994) [27] M. Herant, S.E. Woosley, Postexplosion hydrodynamics of supernovae in red supergiants, ApJ 425 (1994) [28] R.L. Holmes, B. Fryxell, M. Gittings, J.W. Grove, G. Dimonte, M. Schneider, D.H. Sharp, A. Velikovich, R.P. Weaver, Q. Zhang, Richtmyer Meshkov instability growth: experiment, simulation, and theory, J. Fluid Mech. 389 (1999) LA-UR [29] R.L. Holmes, J.W. Grove, D.H. Sharp, Numerical investigation of Richtmyer Meshkov instability using front tracking, J. Fluid Mech. 301 (1995) [30] A.L. Kuhl, Spherical mixing layers in explosions, in: J.R. Bowen (Ed.), Dynamics of Exothermicity: Honor of Antoni Kazimierz Oppenheim, Gordon and Breach, 1996, pp [31] E.E. Meshkov, Instability of a shock wave accelerated interface between two gases, NASA Tech. Trans. F-13 (1970) 074. [32] J.D. Ramshaw, Simple model for linear and nonlinear mixing at unstable fluid interfaces with variable acceleration, Phys. Rev. E 58 (5) (1998) [33] R.D. Richtmyer, Taylor instability in shock acceleration of compressible fluids, Commun. Pure Appl. Math. 13 (1960) [34] H.F. Robey, J.O. Kane, R.P. Drake, B.A. Remington, O.E. Hurricane, H. Louis, R.J. Wallace, J. Knauer, P. Keiter, D. Arnett, D.D. Ryutov, An experimental testbed for the study of hydrodynamic issues in supernovae, Phys. Plasmas 8 (5) (2001)

14 430 S. Dutta et al. / Mathematics and Computers in Simulation 65 (2004) [35] Y. Yang, Q. Zhang, D.H. Sharp, Small amplitude theory of Richtmyer Meshkov instability, Phys. Fluids 6 (5) (1994) [36] D.L. Youngs, Numerical simulation of mixing by Rayleigh Taylor and Richtmyer Meshkov instabilities, Laser Particle Beams 12 (1994) [37] Q. Zhang, M.J. Graham, A numerical study of the Richtmyer Meshkov instability in cylindrical geometry, in: G. Jourdan, L. Houas (Eds.), Proceedings of the 6th International Workshop on the Physics of Compressible Turbulent Mixing, Imprimerie Caractère, Marseille, France, 1997, pp [38] Q. Zhang, M.J. Graham, Scaling laws for unstable interfaces driven by strong shocks in cylindrical geometry, Phys. Rev. Lett. 79 (1997) [39] Q. Zhang, S. Sohn, An analytical nonlinear theory of Richtmyer Meshkov instability, Phys. Lett. A 212 (1996)