Amplitude growth rate of a Richtmyer–Meshkov unstable two-dimensional interface to intermediate times

Abstract

The Richtmyer–Meshkov instability in an incompressible and compressible stratified two-dimensional ideal flow is studied analytically and numerically. For the incompressible problem, we initialize a single small-amplitude sinusoidal perturbation of wavelength λ, we compute a series expansion for the amplitude a in powers of t up to t(11) with the MuPAD computer algebra environment. This involves harmonics up to eleven. The simulations are performed with two codes: incompressible, a vortex-in- cell numerical technique which tracks a single discontinuous density interface; and compressible, PPM for a shock-accelerated case with a finite interfacial transition layer (ITL). We identify properties of the interface at time t = tM at which it first becomes ‘multivalued’. Here, we find the normalized width of the ‘spike’ is related to the Atwood number by (wm/λ)−0.5 = −0.33A. A high-order Pad approximation is applied to the analytical series during early time and gives excellent results for the interface growth rate a˙. However, at intermediate times, t > tM, the agreement between numerical results and different-order Padé approximants depends on the Atwood number. During this phase, our numerical solutions give a˙∝O(t−1) for small A and a˙∝O(t−0.4) for A = 0.9. Experimental data of Prasad et al. (2000) for SF6 (post shock Atwood number = 0.74) shows an exponent between −0.68 and −0.72 and we obtain −0.683 for the compressible simulation. For this case, we illustrate the important growth of vortex-accelerated (secondary) circulation deposition of both signs of vorticity and the complex nature of the roll-up region.