Turbulence of incompressible liquid flows near local equilibrium and the secondary entropy maximum principle

The variational principle of the maximal entropy is considered for the description of weakly non-equilibrium turbulence on the basis of the Reynolds stress theory for flows of a viscous incompressible liquid. Equations completing the Reynolds stress theory are obtained on the basis of this principle; equations describing the interaction of the average flow with turbulence are derived for 3D turbulent flows. Reductions of the theory to 2D flows and weak turbulence are considered. Thermodynamic analogies are used, and an example of the Couette flow is analyzed.