Molecular mixing in a turbulent flow - Some fundamental considerations

A transport or evolution equation for the joint PDF of a reacting scalar and the gradient of that scalar is developed and presented. The derivation clearly indicates the relationship between the transport equation for the PDF of a reacting scalar and the transport equation for the joint PDF. It also demonstrates why molecular mixing cannot be modeled exactly in the simpler equation governing the PDF. Transport in scalar space due to chemical reaction and molecular mixing and transport in scalar gradient space due to chemical reaction are represented exactly. The molecular mixing results in an inverse diffusion process in time. Transport in scalar gradient space due to molecular mixing and vortex dynamics is identified and modeled via physical arguments. Nondimensionalization of the equation yields the important nondimensional groupings and demonstrates the multi-time scale nature of the problem.