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- Title
A continuum mechanical theory for turbulence: a generalized Navier–Stokes- α equation with boundary conditions.
- Authors
Fried, Eliot; Gurtin, Morton E.
- 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 2 h 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.
- Subjects
TURBULENCE; MATHEMATICAL continuum; BOUNDARY value problems; HEAT; FLUID dynamics
- Publication
Theoretical & Computational Fluid Dynamics, 2008, Vol 22, Issue 6, p433
- ISSN
0935-4964
- Publication type
Article
- DOI
10.1007/s00162-008-0083-4