The Joint Pdf of a Scalar and Its Gradient at a Point in a Turbulent Fluid

Abstract

A closed model equation is derived which describes the evolution of the single-point, joint probability density function of a reacting scalar and its gradient in a turbulent advection field. To obtain a closed form a relaxation model is assumed for molecular diffusion and a white noise Kubo approximation for the advection. With this model, the roles of mean velocity, molecular mixing, chemistry and turbulent advection are examined for their effects on the statistics of a transported scalar and its gradient. It is shown that their joint statistical dependence becomes less significant as the Reynolds number increases. An analogy between transport in composition space due to chemical reaction and mass conservation in compressible gas dynamics leads to a methodology for determining the statistical dependence induced by chemical reaction

An application of the model to the turbulent advection of an isotropic scalar gradient field shows that it yields a transition probability for the logarithm of the magnitude of the gradient which is a simple random walk with outward drift; that is, with increasing magnitude of the gradient. The time scale of both the drift velocity and the variance is determined.