It seems like, as a flow becomes more turbulent, there should be more possible states in the fluid. Shouldn't that correspond to an increase in entropy? Steam tables seem to only depend on temperature and pressure and not 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.