Stochastic partial differential equations as a tool for solving the joint velocity-scalar probability density function transport equation

Abstract

International audienceIn this chapter, we outline the Eulerian (Field) Monte Carlo Method (EMC) for solving the joint velocity-scalar PDF transport equation in turbulent reactive flows. The EMC method is based on stochastic Eulerian fields, which evolve according to stochastic partial differential equations (SPDEs). These SPDEs belong to the class of quasi-linear equations. Characteristic curves of the SPDEs can cross. As a result, multi-valued solutions for the velocity and scalar fields can appear. We give an example illustrating that the entropy-dissipative interpretation of these SPDEs is inadequate and introduce their entropy conservative interpretation. We show that with this interpretation, the derived SPDEs are indeed equivalent to the PDF we want to solve. Capturing multi-valued solutions of the SPDEs by efficient algorithms is an important issue. Numerical schemes which satisfy entropy increase condition are not appropriate for the multivalued solutions. A numerical scheme is therefore proposed to solve these SPDEs and is evaluated on a simplified configuration