The continuum mechanics of turbulence: a generalized
Navier–Stokes-α equation with complete boundary
conditions
Eliot Fried
Department of Mechanical and Aerospace Engineering
Washington University in St. Louis
St. Louis, MO 63130-4899, USA
Abstract
The direct numerical simulation of turbulence at Reynolds numbers in excess of a few thousand
provides a formidible 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 of which are large eddy simulation
and closure approximations based on Reynolds averaged equations. While they reduce computational
costs, the additional dissipation associated with these methods can lead to artifically sluggish flows. A
method that avoids this difficulty is provided by simulations based on an equation—known as the Navier–
Stokes-α (NS-α) equation—obtained by Lagrangian averaging. Aside from the density ρ and the shear
viscosity µ of the fluid, the NS-α equation involves an additional material parameter α > 0 carrying
dimensions of length. In the context of Lagrangian averaging, α is the statistical correlation length of
the excursions taken by a fluid particle away from its phase-averaged trajectory. More intuitively, α can
be viewed as the characteristic length 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 the NS-α equation. That generalization involves
two additional material length scales, one of energetic origin and the other of dissipative origin. The
NS-α equation arises on equating these length scales. We explore the impact of these length scales on
the energy spectrum and inertial range.
In contrast to Lagrangian averaging, our formulation also delivers boundary conditions and a complete
thermodynamic framework. The boundary conditions also involve one more material length scale.
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 by direct numerical
simulations. When the additional material parameter associated with the boundary conditions is signed
to ensure satisfaction of the second law (of thermodynamics) at the channel walls the theory delivers
solutions that agree neither quantitatively nor qualitatively with observed features of turbulent plane
channel flow. On the contrary, excellent agreement arises 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 NS-α equation
on bounded domains, their analysis is predicated upon on thermodynamically stable boundary conditions
and therefore cannot pertain to turbulent flows. The question of whether initial-boundary-value problems
for the NS-α 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 NS-α 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 these questions will require novel analytical approaches.
References
• Fried, E. & Gurtin, M. E. 2005. Tractions, balances, and boundary conditions for nonsimple materials with
application to liquid flow at small length scales. Archive for Rational Mechanics and Analysis, in press.
• Marsden, J. E. & Shkoller, S. 2001. Global well-posedness for the Lagrangian averaged Navier–Stokes-α
(LANS-α) equations on bounded domains. Philosophical Transactions of the Royal Society of London A, 359,
1449–1468.
Converted from eliot-fried-ab-NSalphaPSU.pdf (PDF).