Cumulant expansion technique for the pdf formulation of turbulent combustion

A systematic exposition of single-point pdf methods for reacting flows based on cumulant expansion techniques is carried out. A second order approximation for the pdf equation is established. It is shown that transport in velocity-scalar space is governed by drift and diffusion formally consistent with the Fokker-Planck equation. The main result is the explicit representation of the diffusivity as a time integral over Lagrangian two-time correlations. This diffusivity is not positive definite, hence the width of the pdf may be reduced or increased depending on the evolution of the diffusivity. The relation of the present approach to the standard method is discussed for the example of the single scalar pdf.