PDF modeling for inhomogeneous turbulence with exact representation of rapid distortions

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Abstract
A model for inhomogeneous turbulence is constructed that provides an exact representation of rapidly distorted homogeneous turbulence (RDT). The fundamental quantity modeled is the joint PDF of the velocity and wave vector which is related to the unit wavenumber vector. This joint PDF provides a model equation for the evolution of the directional spectrum, the integral over the wavenumber magnitude of the velocity spectrum. At this level the rapid pressure–rate-of-strain correlation is closed yielding exact equations in RDT. For decaying turbulence, the return-to-isotropy terms are modeled by stochastic diffusion equations for the velocity and wave vector. A general model of this type is constructed along with four simplified versions. The decay models are combined with the RDT model to give complete models for homogeneous turbulence, which are tested for several flows. The homogeneous models are then extended in a general manner to inhomogeneous turbulence.