The natural viscosity of turbulence

Abstract

In this work, we extend the notion of natural viscosity introduced by Truesdell in 1964 for the simple fluid in a viscometric flow to the study of homogeneous turbulent shear flow of a Navier–Stokes fluid. Within the framework of Navier–Stokes equations, first, we prove the existence of the natural viscosity of turbulence of a Navier–Stokes fluid in the limit of zero shear rate in homogeneous turbulent shear flow and derive its exact expression. Secondly, we show that, in contrast with the simple fluid in which case its natural viscosity is a material constant, the natural viscosity of turbulence is a function of time and always non-negative, different from the eddy viscosity introduced by Boussinesq in 1877, which could take on negative values as shown in experiments and differs from one model to another in turbulence modelling. In addition, we analyse Prandtl's mixing length model and a few linear and non-linear K–ε models related to the concept of eddy viscosity and compare their performance in approximating the natural viscosity of turbulence introduced herein.