Numerical implementation of mixing and molecular transport in LES/PDF studies of turbulent reacting flows

Probability Density Function (PDF) methods in combination with Large Eddy Simulations (LES) are a powerful tool for studying turbulent reacting flow problems and we are interested in the implementation of mixing and molecular transport in LES/PDF methods. The numerical methodology used for solution procedure is the hybrid particle/mesh method and a fractional step scheme is used to solve for transport, reaction and mixing sequentially. Mixing is modeled using the Interaction by Exchange with the Mean (IEM) model and the effects of molecular transport are incorporated as a mean drift term in the mixing step. This methodology avoids spurious production of scalar variance and also allows direct incorporation of differential diffusion effects. In this study, various numerical implementations of mixing and molecular transport are presented and evaluated, using the Method of Manufactured Solutions (MMS), for (1) accuracy, (2) detailed conservation, (3) realizability, and (4) stability. Moreover, the methodology is shown to be successful in capturing the effects of differential diffusion accurately with the additional property of ensuring realizability of species mass fractions. Finally and most importantly, we describe a new variance reduction technique by way of an implicit smoothing methodology. This smoothing scheme is shown to satisfy conservation, boundedness and regularity criteria. Moreover, for an appropriate choice of the smoothing length scale, significant improvements in accuracy can be achieved for an incremental increase in computational cost. Also, it is shown that with smoothing on a length scale greater than the grid size, the bias and statistical errors due to there being a finite number of particles in the Lagrangian Monte Carlo simulations scale as N tot - 1 and N tot - 1 / 2 respectively, where N tot is the total number of particles in the computational domain, whereas without smoothing these errors scale as N pc - 1 and N pc - 1 / 2 , where N pc is the much smaller number of particles in a computational cell.