Buoyancy–Drag modelling of bubble and spike distances for single-shock Richtmyer–Meshkov mixing

The Buoyancy–Drag model is a simple model, based on ordinary differential equations, for estimating the growth of a turbulent mixing zone at an interface between fluids of different density due to Richtmyer–Meshkov and Rayleigh–Taylor instabilities. The early stages of the mixing process are very dependent on the initial conditions and modifications to the Buoyancy–Drag model are needed to obtain correct results. In a recent paper, Youngs & Thornber (2019), the results of three-dimensional turbulent mixing simulations were used to construct the modifications required to represent the evolution of the overall width of the mixing zone due to single-shock Richtmyer–Meshkov mixing evolving from narrowband initial random perturbations. The present paper extends this analysis to give separate equations for the bubble and spike distances (the depths to which the mixing zone penetrates the dense and light fluids). The data analysis depends on novel integral definitions of the bubble and spike distances which vary smoothly with time. Results are presented for two pre-shock density ratios, ρ₁∕ρ₂=3and20. New insights are given for the variation of asymmetry of the mixing zone with time. At early time, values of the spike-to-bubble distance are very high. The asymmetry greatly reduces as mixing proceeds towards a self-similar state. For the overall (integral) mixing width, W, the Buoyancy–Drag model gives satisfactory results at both density ratios using the same parameters. However, for the bubble and spike distances the behaviour is very different at the two density ratios. The method used to analyse the data provides a new way of estimating the self-similar growth exponent θ (W∼t^θ). The values obtained are approximate because of the difficulty in running the three dimensional simulations far enough into the self-similar regime. Estimates of θ are consistent with the theoretical value of 1/3 given by the model of Elbaz & Shvarts (2018). The corrected form of the Buoyancy–Drag model gives accurate fits to the data for W,h_bandh_sover the whole time range with θ =1/3 for ρ₁∕ρ₂=3 and θ=0.35 for ρ₁∕ρ₂=20.