A continuum mechanical theory for turbulence: a generalized Navier–Stokes-α equation with boundary conditions

Abstract

We develop a continuum-mechanical formulation and generalization of the Navier–Stokes-α equation based on a recently developed framework for fluid-dynamical theories involving higher-order gradient dependencies. Our flow equation involves two length scales α and β. The first of these enters the theory through the specific free-energy α ²|D|², where D is the symmetric part of the gradient of the filtered velocity, and contributes a dispersive term to the flow equation. The remaining scale is associated with a dissipative hyperstress which depends linearly on the gradient of the filtered vorticity and which contributes a viscous term, with coefficient proportional to β ², to the flow equation. In contrast to Lagrangian averaging, our formulation delivers boundary conditions and a complete structure based on thermodynamics applied to an isothermal system. For a fixed surface without slip, the standard no-slip condition is augmented by a wall-eddy condition involving another length scale ℓ characteristic of eddies shed at the boundary and referred to as the wall-eddy length. As an application, we consider the classical problem of turbulent flow in a plane, rectangular channel of gap 2h with fixed, impermeable, slip-free walls and make comparisons with results obtained from direct numerical simulations. We find that α/β ~ Re ^0.470 and ℓ/h ~ Re ^−0.772, where Re is the Reynolds number. The first result, which arises as a consequence of identifying the specific free-energy with the specific turbulent kinetic energy, indicates that the choice β = α required to reduce our flow equation to the Navier–Stokes-α equation is likely to be problematic. The second result evinces the classical scaling relation η/L ~ Re ^−3/4 for the ratio of the Kolmogorov microscale η to the integral length scale L.