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

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 [lambda], 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/[lambda])[minus sign]0.5 = [minus sign]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[dot above]. 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[dot above][is proportional to]O(t[minus sign]1) for small A and a[dot above][is proportional to]O(t[minus sign]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 [minus sign]0.68 and [minus sign]0.72 and we obtain [minus sign]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.