Correction to Euler's equations and elimination of the closure problem in turbulence

Abstract

It has been demonstrated that the Euler equations of inviscid fluid are incomplete: according to the principle of release of constraints, absence of shear stresses must be compensated by additional degrees of freedom, and that leads to a Reynolds-type multivalued velocity field. However, unlike the Reynolds equations, the enlarged Euler's (EE) model provides additional equations for fluctuations, and that eliminates the closure problem. Therefore the EE equations are applicable to fully developed turbulent motions where the physical viscosity is vanishingly small compare to the turbulent viscosity, as well as to superfluids and atomized fluids. Analysis of coupled mean/fluctuation EE equations shows that fluctuations stabilize the whole system generating elastic shear waves and increasing speed of sound. Those turbulent motions that originated from instability of underlying laminar motions can be described by the modified Euler's equation with the closure provided by the stabilization principle: driven by instability of laminar motion, fluctuations grow until the new state attains a neutral stability in the enlarged (multivalued) class of functions, and these fluctuations can be taken as boundary conditions for the EE model. The approach is illustrated by an example. © 2012 Copyright 2012 Author(s). This article is distributed under a Creative Commons Attribution 3.0 Unported License.