Appearance of deterministic mixing behavior from ensembles of fluctuating hydrodynamics simulations of the Richtmyer-Meshkov instability

Abstract

We obtain numerical solutions of the two-fluid fluctuating compressible Navier-Stokes (FCNS) equations, which consistently account for thermal fluctuations from meso- to macroscales, in order to study the effect of such fluctuations on the mixing behavior in the Richtmyer-Meshkov instability (RMI). The numerical method used was successfully verified in two stages: for the deterministic fluxes by comparison against air-SF6 RMI experiment, and for the stochastic terms by comparison against the direct simulation Monte Carlo results for He-Ar RMI. We present results from fluctuating hydrodynamic RMI simulations for three He-Ar systems having length scales with decreasing order of magnitude that span from macroscopic to mesoscopic, with different levels of thermal fluctuations characterized by a nondimensional Boltzmann number (Bo). For a multidimensional FCNS system on a regular Cartesian grid, when using a discretization of a space-time stochastic flux Z(x,t) of the form Z(x,t)→1/-tN(ih,nΔt) for spatial interval h, time interval Δt, h, and Gaussian noise N should be greater than h0, with h0 corresponding to a cell volume that contains a sufficient number of molecules of the fluid such that the fluctuations are physically meaningful and produce the right equilibrium spectrum. For the mesoscale RMI systems simulated, it was desirable to use a cell size smaller than this limit in order to resolve the viscous shock. This was achieved by using a modified regularization of the noise term via Zx,t→1/-tmaxh3,h03Nih,nΔt, with h0=ξh<h0,ξZ+. Our simulations show that for systems with Bo1 deterministic mixing behavior emerges as the ensemble-averaged behavior of several fluctuating instances, whereas when Bo≈1, a deviation from deterministic behavior is observed. For all cases, the FCNS solution provides bounds on the growth rate of the amplitude of the mixing layer.