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The kinetic theory of gases has suggested the idea of viscosity to model theeffect of thermal fluctuations on the resolved flow. Supported by the assumedanalogy between molecules and the eddies in a turbulent flows, the idea of aneddy viscosity has been put forward in the pioneering work by Lord Kelvin andOsborne Reynolds. In over hundred years of turbulence modeling, the numericalschemes adopted to simulate turbulent flow - with the exception of the LatticeBoltzmann methods - have never exploited this analogy in any other way. In thiswork, a gas-kinetic scheme is modified to simulate turbulent flow; theturbulent relaxation time is deduced from assumed turbulent quantities. The newscheme does not adopt an eddy viscosity, yet it relies even more strongly onthe analogy between thermal and turbulent fluctuations, as turbulence dynamicsis mathematically modeled by the Boltzmann equation. In the gas-kinetic scheme,a measure of the degree of rarefaction is introduced, as the ratio betweenunresolved and resolved time scales of motion. At low rarefaction, theturbulent gas-kinetic scheme deviates negligibly from a conventionalNavier-Stokes scheme. However, as the degree of rarefaction increases, thekinetic effects become evident. This phenomenon is evident in the mathematicaldescription of the turbulent stress tensor and also in numerical experiments.This study does not propose an innovative turbulence model or technique. Itaddresses the fact that the traditional coupling numerical scheme andturbulence modeling might improve the physical consistence of numericalsimulations.

A Gas-Kinetic Scheme For The Simulation Of Compressible Turbulent Flows