Relaxation of initial probability density functions in the turbulent convection of scalar fields

The evolution of an initially binary (zero unity) scalar field undergoing turbulent and molecular mixing is studied in terms of conservation equations for the probability density function of the scalar property. Attention is focused on the relaxation of the dynamic system to a state independent of the intial conditions. A few existing methods are discussed and evaluated and a new mechanistic model is proposed. Classical iteration techniques are used to obtain an equation for the single point probability density and the unperturbed Green’s function. It is suggested that use of the true Green’s function or perturbed propagator of the system might be necessary in order to obtain the correct evolution of the probability density function.