  The kinetic theory of gases has suggested the idea of viscosity to model the
effect of thermal fluctuations on the resolved flow. Supported by the assumed
analogy between molecules and the eddies in a turbulent flows, the idea of an
eddy viscosity has been put forward in the pioneering work by Lord Kelvin and
Osborne Reynolds. In over hundred years of turbulence modeling, the numerical
schemes adopted to simulate turbulent flow - with the exception of the Lattice
Boltzmann methods - have never exploited this analogy in any other way. In this
work, a gas-kinetic scheme is modified to simulate turbulent flow; the
turbulent relaxation time is deduced from assumed turbulent quantities. The new
scheme does not adopt an eddy viscosity, yet it relies even more strongly on
the analogy between thermal and turbulent fluctuations, as turbulence dynamics
is mathematically modeled by the Boltzmann equation. In the gas-kinetic scheme,
a measure of the degree of rarefaction is introduced, as the ratio between
unresolved and resolved time scales of motion. At low rarefaction, the
turbulent gas-kinetic scheme deviates negligibly from a conventional
Navier-Stokes scheme. However, as the degree of rarefaction increases, the
kinetic effects become evident. This phenomenon is evident in the mathematical
description of the turbulent stress tensor and also in numerical experiments.
This study does not propose an innovative turbulence model or technique. It
addresses the fact that the traditional coupling numerical scheme and
turbulence modeling might improve the physical consistence of numerical
simulations.
