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.