ObjectiveThe Richtmyer-Meshkov instability is similar to the Rayleigh-Taylor instability. They both develop when an interface between fluids of two different densities is accelerated. For Rayleigh-Taylor instabilities, the interface is accelerated gradually by a force, such as gravity. For the Richtmyer-Meshkov instability, the interface is accelerated impulsively, here by the passage of a shock wave from one fluid to the other. Both types of instabilities show the characteristic bubble and spike morphology, in which the spike is terminated by a mushroom cap structure. In addition, Kelvin-Helmholtz instabilities develop along the interface due to the shear flow between the two fluids. The Richtmyer-Meshkov instability grows more slowly that the Rayleigh-Taylor case, showing a linear rather than an exponential time behavior. The objective of this study is to calculate how the growth rate of the instability depends on the Mach number of the incident shock, the size of the density jump, the compressibility of the two fluids, and the size and wavelength of the initial perturbation of the interface.
The simulations were carried out using the PROMETHEUS computer code. This code solves Euler's equations for compressible gas dynamics on a two-dimensional Cartesian grid using the Piecewise-Parabolic Method (PPM). In order to obtain much greater numerical resolution than previously possible, the code was modified to run on massively parallel computers.
The code was restructured using a domain decomposition technique, in which the two-dimensional computational grid was divided up into a number of smaller tiles. Communication between adjacent tiles involves only exchanging the values at the boundary points. Because of the large number of operations to update each grid point in the PPM algorithm and the small amount of communication required, parallelization of the code is extremely efficient on all current architectures. The results shown here were obtained on a 16,384-processor MasPar MP-2, which achieves a performance of 4 gigaFLOPS. This is approximately 2/3 of the peak speed of the computer. The calculation shown here required approximately 2 days of CPU time and would have required almost three weeks of dedicated time on a single processor of a CRAY C90. The code also achieves impressive performance and excellent scalability on the CRAY T3D, the SGI Power Challenge, the Convex Exemplar, the IBM SP2, and the CRAY C90. The instability in the figure was calculated using a 1920 x 480 grid. This large resolution is required to correctly simulate the shear instability between the two fluids, the structure of the spikes, and the complex shock interactions that take place to the right of the interface.
Understanding the rate of mixing caused by fluid instabilities is important to a wide range of applications in science and engineering. This has been more difficult for Richtmyer-Meshkov instabilities than for Rayleigh-Taylor, since constructing analytic theories has been more difficult. In fact, a number of analytic theories have been developed that disagree with each other significantly in certain parameter regimes. By comparing the results of these calculations and those produced by other codes with laboratory experiments obtained using high-energy lasers and linear electric motors, it is hoped that we can reach an understanding of which, if any, of the analytic theories is correct and why. A collaboration has already been established with scientists at Lawrence Livermore National Laboratory, Los Alamos National Laboratory, Stony Brook, and the Naval Research Laboratory in an effort to address these issues.
Current plans involve performing an exhaustive parameter study to investigate regimes where the analytic theories differ. This should enable us to determine if any of the theories can predict the correct behavior in all areas of parameter space, and if not, perhaps why they fail and how to fix them. In the future, as more powerful computers become available, the study will be extended to three-dimensional calculations, experiments and theories.
Bruce Fryxell
George Mason University
fryxell@neutrino.gsfc.nasa.gov
301-286-8567
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