On Richtmyer–Meshkov instability in dilute gas-particle mixtures

Abstract

Richtmyer–Meshkov instability (RMI) in gas-particle mixtures is investigated both numerically and analytically. The linear amplitude growth rate for a RMI in a two-phase mixture is derived by using a dusty gas formulation for small Stokes number (St⪡1.0), and it is shown that the problem can be characterized by mass loading and St. The model predictions are compared with numerical results under two conditions, i.e., a shock wave hitting (1) a perturbed species interface of air and SF6 surrounded by uniformly distributed particles, and (2) a perturbed shape particle cloud in uniform air. In the first case, the interaction between the instability of the species perturbation and the particles is investigated. The multiphase growth model accurately predicts the growth rates when St⪡1.0, and the amplitude growth normalized by the two-phase RMI velocity shows good agreement with the single-phase RMI growth rate as well. It is also shown that the two-phase model results are in accordance with the growth rates obtained from the simulations even for cases corresponding to St≈10. However, for St⪢10, particles do not follow the RMI motion, and the RMI growth rate agrees with the original Richtmyer’s model [R. D. Richtmyer, “Taylor instability in shock acceleration of compressible fluids,” Commun. Pure Appl. Math. 13, 297 (1960)]. Preferential concentration of particles are observed around the RMI roll-ups at late times when St is of order unity, whereas when St⪡1.0, the particles respond rapidly to the flow, causing them to distribute within the roll-ups. In the second problem, the two-phase RMI growth model is extended to study whether a perturbed dusty gas front shows RMI-like growth due to the impact of a shock wave. When St⪡1.0, good agreement with the multiphase model is again seen. Moreover, the normalized growth rates are very close to the single-phase RMI growth rates even at late times, which suggest that the two-phase growth model is applicable to this type of perturbed shape particle clouds as well. However, when St is close to unity or larger (St>1.0), the particles do not experience impulsive acceleration but rather a continuous one, which results in exponential growth rates as seen in a Rayleigh–Taylor instability.