Hidden symmetries to a Hanjalic–Launder semiempirical model of turbulence

The article is devoted to examining algebraic closure relationships which are used in the Theory of Semiempirical Models of Turbulence. As an example, the dynamics of a far plan turbulent wake is investigated and the so-called locally equilibrium approximation of second-order moments (tangential Reynolds stresses) is considered. Applicability of this algebraic approximation for tangential Reynolds stresses is analyzed by the method of differential constraints in the context of investigation of compatibility of the original mathematical model (the classical ($e$, $\varepsilon$, $ < u' v' >$) — model of turbulence) with an added differential constraint (i.e. with an algebraic closure relationship for tangential Reynolds stresses). We show that the compatibility condition obtained coincide with the condition that a Hamiltonian vector field generated by the velocity field of the turbulent flow under consideration admits a symplectic symmetry of the canonical transformations.