A compensation-defect model for the joint probability density function of the scalar difference and the length scale of dissipation elements

Abstract

Dissipation element analysis is a new approach to study turbulent scalar fields. Gradient trajectories starting from each material point in a fluctuating scalar field φ' (x,t) in ascending and descending directions will inevitably reach a maximal and a minimal point. The ensemble of material points sharing the same pair ending points is named a dissipation element. Dissipation elements can be parametrized by the length scale l and the scalar difference Δφ', which are defined as the straight line connecting the two extremal points and the scalar difference at these points, respectively. The decomposition of a turbulent field into dissipation elements is space filling. This allows us to reconstruct certain statistical quantities of fine scale turbulence which cannot be obtained otherwise. The marginal probability density function (PDF) of the length scale distribution had been modeled in the previous work based on a Poisson random cutting-reconnection process and had been compared to data from direct numerical simulation (DNS). The joint PDF of l and Δφ' contains the important information that is needed for the modeling of scalar mixing in turbulence, such as the marginal PDF of the length of elements and conditional moments, as well as their scaling exponents. In order to be able to predict these quantities, there is a need to model the joint PDF. A compensation-defect model is put forward in this work and the agreement between the model prediction and DNS results is satisfactory. © 2008 American Institute of Physics.