In the PDF transport equation, molecular mixing appears in terms of the conditional scalar dissipation rate (CSDR), a square, symmetric matrix of functions depending on the composition variables. For a statistically homogeneous system (i.e. the mixing statistics are independent of the spatial location), a Fokker-Planck (FP) equation describes the evolution of the PDF in composition space with the CSDR as the diffusion matrix. In principle, for a given CSDR, one can solve for the composition PDF given its initial condition (e.g. completely segregated feed streams). In practice, this must be done with a Monte-Carlo (MC) code, leading to statistical noise; or by making use of simplifications such as conditioning on the mixture fraction whose univariate PDF has an assumed form (e.g. a beta PDF).
Recently we have developed an alternative solution method for the FP equation using quadrature-based moment methods (QBMM). Given a mathematically consistent form for the CSDR, the basic idea is to solve the moment equations corresponding to the FP equation, and then to use QBMM to find quadrature weights and abscissas for evaluation of the chemical source terms. Conceptually, the deterministic equations for the moments (and, hence, the abscissas) replace the stochastic differential equations for the MC notional particles. Remarkably, the moment equations for molecular mixing are closed when the CSDR is chosen to relax towards a beta PDF. Thus, the MC algorithm is replaced by a high-order ODE solver for the set of coupled moments, thereby eliminating the statistical noise.
In this presentation, a general overview of QBMM for turbulent mixing is provided, followed by a specific application using moments up to first order for reacting species (i.e. equivalent to the conditional moment closure). The implementation of QBMM for turbulent reacting flows in CFD codes at the level of RANS and LES will be discussed. Specific issues related to solving for moments, such as realizability, will also be covered.