A particle formulation for treating differential diffusion in filtered density function methods

We present a new approach for treating the molecular diffusion term in filtered density function (FDF) methods for modeling turbulent reacting flows. The diffusion term accounts for molecular `transport' in physical space and molecular `mixing' in scalar space. Conventionally, the FDF is represented by an ensemble of fluid particles and transport is modeled by a random walk in physical space. There are two significant shortcomings in this transport model: (1) the resulting composition variance equation contains a spurious production term, and (2) because the random walk is governed by a single diffusion coefficient, the formulation cannot account for differential diffusion, which can have a first-order effect in reacting flows. In our new approach transport is simply modeled by a mean drift in the particle composition equation. The resulting variance equation contains no spurious production term and differential diffusion is treated easily. Hence, the new formulation reduces to a DNS in the limit of vanishing filter width, a desirable property of any LES approach. We use the IEM model for mixing. It is shown that there is a lower bound on the specified mixing rate such that the boundedness of the compositions is ensured. We also present a numerical method for solving the particle equations which is second-order accurate in space and time, obeys detailed conservation, enforces the boundedness constraints, and is unconditionally stable.