Three-dimensional simulation of a Richtmyer–Meshkov instability with a two-scale initial perturbation

Three-dimensional high-resolution simulations (up to 8 billion zones) have been performed for a Richtmyer–Meshkov instability produced by passing a shock through a contact discontinuity with a two-scale initial perturbation. The setup approximates shock-tube experiments with a membrane pushed through a wire mesh. The simulation produces mixing-layer widths similar to those observed experimentally. Comparison of runs at various resolutions suggests a mixing transition from unstable to turbulent flow as the numerical Reynolds number is increased. At the highest resolutions, the spectrum exhibits a region of power-law decay, in which the spectral flux is approximately constant, suggestive of an inertial range, but with weaker wave number dependence than Kolmogorov scaling, about k−6/5. Analysis of structure functions at the end of the simulation indicates the persistence of structures with velocities largest in the stream-wise direction. Comparison of three-dimensional and two-dimensional runs illustrates the tendency toward forward cascade in three dimensions, versus inverse cascade in two dimensions. Comparison of the full simulation with a simulation of a single-scale perturbation indicates that the coupling of the disparate scales leads to destruction of the small-scale bubbles and spikes except near the spike growing from the large-scale perturbation. Finally, an analysis of the sub-grid-scale stresses in filtered data indicates significant correlation of the resultant forward and back transfer of energy with the determinant of the rate-of-strain tensor of the resolved scale flow. A possible relation between this trend and alignment of vorticity on small scales with the principal directions of strain on large scales is discussed. The observed correlation lends support to the use of sub-grid-scale models proportional to the determinant of the rate-of-strain tensor for large-eddy simulation.