Closure in Turbulence from first principles

Abstract

It has been recently demonstrated, [3], that according to the principle of release of constraints, absence of shear stresses in the Euler equations must be compensated by additional degrees of freedom, and that led to a Reynolds-type enlarged Euler equations (EE equations) with a doublevalued velocity field that do not require any closures. In the first part of the paper, the theory is applies to turbulent mixing and illustrated by propagation of mixing zone triggered by a tangential jump of velocity. A comparison of the proposed solution with the Prandtl's solutions is performed and discussed. In the second part of the paper, a semi-viscose version of the Navier-Stokes equations is introduced. The model does not require any closures since the number of equations is equal to the number of unknowns. Special attention is paid to transition from laminar to turbulent state. The analytical solution for this transition demonstrates the turbulent mean velocity profile that qualitatively similar to the celebrated logarithmic law.