Cosserat fluids and the continuum mechanics of turbulence: A generalized Navier-Stokes-alpha equation with complete boundary conditions

摘要

We here develop a continuum-mechanical formulation and generalization of the Navier???Stokes-alpha equation based on a general framework for fluid-dynamical theories involving gradient dependencies (Fried & Gurtin 2005). That generalization entails two additional material length scales: one of energetic origin, the other of dissipative origin. In contrast to Lagrangian averaging, our formulation delivers boundary conditions???involving yet another material length scale???and a complete framework based on thermodynamics applied to an isothermal system. As an application, we consider the classical problem of turbulent flow in a plane, rectangular channel with fixed, impermeable, slip-free walls and make comparisons with results obtained from direct numerical simulations. For this problem, only one of the material length scales involved in the flow equation enters the final solution. When the additional material length scale associated with the boundary conditions is signed to ensure satisfaction of the second law at the channel walls the theory delivers solutions that agree neither quantitatively nor qualitatively with observed features of plane channel flow. On the contrary, we find excellent agreement when the sign of the additional material parameter associated with the boundary conditions violates the second law. We discuss the implication of this result.