Early Time Modifications to the Buoyancy-Drag Model for Richtmyer–Meshkov Mixing

Abstract

The Buoyancy-Drag model is a simple model, based on ordinary differential equations, for estimating the growth in the width of a turbulent mixing zone at an interface between fluids of different densities due to Richtmyer–Meshkov and Rayleigh–Taylor instabilities. The model is calibrated to give the required self-similar behavior for mixing in simple situations. However, the early stages of the mixing process are very dependent on the initial conditions and modifications to the Buoyancy-Drag model are then needed to obtain correct results. In a recent paper, Thornber et al. (2017, “Late-Time Growth Rate, Mixing, and Anisotropy in the Multimode Narrowband Richtmyer–Meshkov Instability: The θ-Group Collaboration,” Phys. Fluids, 29, p. 105107), a range of three-dimensional simulation techniques was used to calculate the evolution of the mixing zone integral width due to single-shock Richtmyer–Meshkov mixing from narrowband initial random perturbations. Further analysis of the results of these simulations gives greater insight into the transition from the initial linear behavior to late-time self-similar mixing and provides a way of modifying the Buoyancy-Drag model to treat the initial conditions accurately. Higher-resolution simulations are used to calculate the early time behavior more accurately and compare with a multimode model based on the impulsive linear theory. The analysis of the iLES data also gives a new method for estimating the growth exponent, θ (mixing zone width ∼ tθ), which is suitable for simulations which do not fully reach the self-similar state. The estimates of θ are consistent with the theoretical model of Elbaz and Shvarts (2018, “Modal Model Mean Field Self-Similar Solutions to the Asymptotic Evolution of Rayleigh-Taylor and Richtmyer-Meshkov Instabilities and Its Dependence on the Initial Conditions,” Phys. Plasmas, 25, p. 062126).