Carnegie Mellon
Department of Mathematical 

Eliot Fried, Mechanical and Aerospace Engineering, Washington University

"The continuum mechanics of turbulence: a generalized Navier--Stokes-$\alpha$ model with complete boundary conditions"


The direct numerical simulation of turbulence at Reynolds numbers in excess of a few thousand provides a formidable computational problem, even with access to state-of-the- art supercomputers. For this reason, there remains a strong interest in alternative methods that resolve only large-scale motions while modeling small-scale motions via filtering. The most well-known approaches are large eddy simulation and closure approximations based on Reynolds averaged equations. While these methods reduce computational costs, the additional dissipation associated with filtering can lead to artificially sluggish flows. A method that avoids this difficulty is provided by simulations based on an equation -- the Navier- Stokes-$ \alpha$ equation -- obtained by Lagrangian averaging. Known as the Navier- Stokes-$ \alpha$ equation, that equation has the form

$\displaystyle \rho\dot{\bf v}=-\hbox{\rm grad}\mskip2mu p+\mu\triangle{\bf v}-\...
...\overset{\raisebox{-0.25ex}{$\mskip-1.5mu\scriptscriptstyle\circ$}}{{\bf D}},

which $ {\bf v}$ is the fluid velocity, subject to the incompressibility constraint $ \hbox{\rm div}\mskip2mu {\bf v}=0$, $ p$ is the pressure, $ \dot{\bf v}$ is the material time derivative of $ {\bf v}$, $ \triangle$ is the Laplace operator, $ {\bf D}=\textstyle{\frac{1}{2}}(\hbox{\rm grad}\mskip2mu {\bf v}+(\hbox{\rm grad}\mskip2mu {\bf v})^{\mskip-2mu\scriptscriptstyle\top\mskip-2mu}) $ is the stretch-rate,

$\displaystyle \overset{\raisebox{-0.25ex}{$\mskip-1.5mu\scriptscriptstyle\circ$}}{{\bf D}}=\dot{{\bf D}}+{\bf D}{\bf W}-{\bf W}{\bf D}, $

with $ {\bf W}=\textstyle{\frac{1}{2}}(\hbox{\rm grad}\mskip2mu {\bf v}-(\hbox{\rm grad}\mskip2mu {\bf v})^{\mskip-2mu\scriptscriptstyle\top\mskip-2mu}) $ the spin, is the corotational rate of $ {\bf D}$. Aside from the density $ \rho$ and the shear viscosity $\mu$ of the fluid, the Navier-Stokes-$ \alpha$ equation involves an additional material parameter $ \alpha>0$ carrying dimensions of length. Within the framework of Lagrangian averaging, $ \alpha$ is the statistical correlation length of the excursions taken by a fluid particle away from its phase-averaged trajectory. More intuitively, $ \alpha$ can be interpreted as the characteristic linear dimension of the smallest eddies that the model is capable of resolving. In this talk, we use the framework of Fried & Gurtin (2005) to develop an alternative continuum-mechanical formulation leading to a generalization of Navier-Stokes-$ \alpha$ equation. That generalization involves not one but two additional material length scales, one being of energetic origin and the other being of dissipative origin. In contrast to Lagrangian averaging, our formulation deliver boundary conditions and a complete thermodynamic framework. The boundary condition also involve yet one more material length scale. As an application, we consider the problem of classical problem of turbulent flow in a plane, rectangular channel with fixed, impermeable, slip-free walls and make comparisons with results obtained from experiment and direct numerical simulations. An interesting feature of our results is that when the additional material parameter 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 qualitatively nor quantitavely with the experimentally and numerically 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. Although Marsden & Shkoller (2001) recently established well-posedness results for the Navier-Stokes-$ \alpha$ equation on bounded domains, their analysis is based on thermodynamically stable boundary conditions and therefore cannot pertain to turbulent flows. The question of whether initial-boundary-value problems for the Navier-Stokes-$ \alpha$ equations are well-posed when boundary conditions appropriate to turbulence are imposed therefore remains open. An additional question of central importance concerns whether solutions to initial-boundary-value problems for the Navier-Stokes-$ \alpha$ equations converge to solutions of initial-boundary-value problems for the Navier-Stokes equations. Because of the nonstandard thermodynamic structure of the theory, it seems very likely that answers to the se questions will require novel analytical approaches. This work is

TUESDAY, November 15, 2005
Time: 1:30 P.M.