Eulerian Monte Carlo method for the joint velocity and mass-fraction probability density function in turbulent reactive gas flows

A new Eulerian (field) Monte Carlo method for solving an equation that describes the one-time one-point probability density of species mass fractions in turbulent reactive gas flows has been proposed in a previous article. In the present paper, this method is extended to an equation for the joint velocity and mass-fraction probability density function. The method is based on passing from Lagrangian variables used in Lagrangian Monte Carlo methods to Eulerian variables. In this manner, stochastic ordinary differential equations for the Lagrangian trajectories of fluid particles are transformed to partial stochastic equations. As compared to the classical hydrodynamics, the stochastic velocity field satisfies only the mean continuity constraint and not the instantaneous one. As a consequence, one has to introduce a stochastic density, which differs from the physical density but has the same mean value. The case of the mass-fraction probability density is revised. The equations differ from those derived previously: they can be written in divergent form. Both formulations, however, are statistically equivalent.