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

Michail Zak

doi:10.1063/1.4765064

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.

Highlights

IntroductionIn Newtonian mechanics, the models of continua are derived from the Newton’s laws. in addition to that, several mathematical restrictions, (such as space-time differentiability, the Lipchitz conditions etc.) which are not required by the Newton’s laws, are exploited
In Newtonian mechanics, the models of continua are derived from the Newton’s laws. in addition to that, several mathematical restrictions, which are not required by the Newton’s laws, are exploited
A computational strategy that may be applicable for the EE formulation of the problem of turbulence has been proposed in our previous publications, [4-6], and it can complement the results of this paper

Summary

Introduction

In Newtonian mechanics, the models of continua are derived from the Newton’s laws. in addition to that, several mathematical restrictions, (such as space-time differentiability, the Lipchitz conditions etc.) which are not required by the Newton’s laws, are exploited. As follows from Eq (1), non-differential velocity field does not effect stresses It means that requirement of differentiability of the velocity field in the model of inviscid fluid is introduced only for mathematical convenience, and it can be removed. The EE systems do not include the continuity equation for fluctuations These differences follow from the fact that the Reynolds velocity field, strictly speaking, is single-valued since the stress tensor of the Navier-Stokes equations does not have zero components. The large-scale fluctuations ( Re → ∞ ) are sizable with mean velocities, and they significantly contribute to the motions These properties suggest that the general picture of turbulence is better captured by the Euler rather than Navier-Stokes equation, and in particular, by EE equations, regardless of whether the underlying pre-instability laminar flow is viscous or nonviscose.

In the classical case when

Discussion and conclusion