Investigating the Mixing-Hydrodynamic Cascade Timescale Ratio in Transient Non-Equilibrium Turbulence Using a Fokker-Planck Equation Approach

We investigate the mixing– hydrodynamic cascade timescale ratio (C_b2) in transient, non-equilibrium binary mixing variable density turbulence (VDT). We represent the turbulent flow as a stochastic β-process using a Fokker-Planck equation that evolves a probability density function in time and in bounded mass fraction space. The FPE is integrated to obtain the evolution equation for the density – specific volume covariance (b). This evolution equation for b is compared to the equation obtained from ensemble averaging the Navier-Stokes equations, where the mixing cascade or destruction of b is given by ε_b. We use C_b2 to represent ε_b as a linear drag model with a hydrodynamic timescale and a proportionality parameter C_b2. The evolution equation for b derived from the FPE is then used to find an expression for timescale ratio C_b2 as a function of flow statistics, which is then modeled, using a new transient timescale, as an algebraic function of mean flow statistics. Simulations of homogeneous VDT using a Monte Carlo model based on the FPE, and solutions of the BHR equation for b using the new model for C_b2 are compared with direct numerical simulations at several Atwood numbers, and for different initial conditions. We discuss extensions to in-homogeneous flows.