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

TL;DR

This paper proposes an extended Euler's model that incorporates fluctuations to address turbulence without closure problems, applicable to various fluid types and stabilizing turbulent motions through multivalued velocity fields.

Contribution

It introduces an enlarged Euler's (EE) model that includes additional equations for fluctuations, eliminating the closure problem in turbulence modeling.

Findings

EE model stabilizes turbulent flows with elastic shear waves

Fluctuations grow until neutral stability is reached

Applicable to superfluids and atomized fluids

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 leads to 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 motions, fluctuations grow until the new state attains a neutral stability in the enlarged (multivalued) class of functions, and those fluctuations can be taken as boundary conditions for the EE model. The approach is illustrated by an example.