On the power-law exponent of multimode Richtmyer–Meshkov turbulent mixing width

Abstract

Turbulent mixing induced by the Richtmyer–Meshkov (RM) instability occurs extensively in natural phenomena and engineering applications. Among the physical quantities characterizing the RM turbulent mixing, the mixing width has prominent importance. The total mixing width h can be divided into the spike mixing zone width hs and the bubble mixing zone width hb. For multimode perturbed RM problems that commonly occur in engineering practice, early instability develops rapidly into the self-similar regime. In this regime, it is widely accepted that hs,bt∼tθs,b, where t is the time and θs,b is the power-law exponent. However, this scaling law is associated with two open questions. (1) How should a reasonable reference interface be selected to segment h into hs and hb? (2) Are the resulting θs and θb equal to each other or not? To answer these two questions, in this study, we propose a general definition of reference interface based on the position corresponding to any fixed value of either the mass fraction, volume fraction, or density. Under this definition, the invariance of fraction and density profiles by self-similar transformation leads to hs,bt∼tθs,b with θs=θb. The general definition covers those provided in linear electronic motor experiment [Dimonte and Schneider, “Density ratio dependence of Rayleigh–Taylor mixing for sustained and impulsive acceleration histories,” Phys. Fluids 12, 304–312 (2000)] and shock tube experiment [Krivets et al., “Turbulent mixing induced by Richtmyer-Meshkov instability,” AIP Conf. Proc. 1793, 150003 (2017)]. Moreover, these two definitions are proved to be, respectively, special cases of newly proposed general definition. Finally, it is deduced that θs≠θb observed in high-density ratio experiments is possibly because the turbulent mixing has not entered a self-similar regime. Compared to the low-density ratio cases, mixing of high-density ratio is much more difficult to enter the self-similar regime.