Does entropy change with Reynolds number?

In Thermodynamics, entropy is introduced as an equilibrium state function, i.e. as a quantity depending on the thermodynamic state only. The basic request is that a macroscopic system, whose microscopic state depends on a huge number of degrees of freedom, can be fully characterized by only a few physical quantities like temperature and pressure. Such a huge reduction of the number of effective parameters required to describe the system can only be achieved under special conditions: the presence of the so-called thermodynamic equilibrium states.

At thermodynamic equilibrium, no macroscopic motion is present. Therefore, no velocity field, no Navier-Stokes equation and no Reynolds number. This is the reason one cannot find any reference to Reynolds number in steam tables.

In many cases, it is possible to introduce a local thermodynamic description of a macroscopically moving fluid, by introducing a field of thermodynamic variables describing the thermodynamic behavior of a fluid element small enough to be uniform, but large enough to contain still a large number of molecules. Within such local thermodynamic equilibrium (LTE) approach, turbulent fluxes can be characterized by a local entropy production term.

What about turbulent flux and the increase of the number of possible states? This has to do with a statistical mechanics description. However most of what we learn is connected with statistical mechanics at equilibrium. Only at thermodynamic equilibrium, we can interpret entropy as a measurement of the number of microstates compatible with a fixed volume and energy. However, all the possible mechanical states of the same energy are counted and no dependence on the kind of flux is allowed.