Variational and Multiscale Methods in Turbulence with Particular
Emphasis on Large Eddy Simulation 

T.J.R. Hughes

(The University of Texas at Austin)

Abstract

We present a review of progress in applying the Variational Multiscale
Formulation to turbulent flows.  We begin with a description of the
type of engineering applications we are interested in and follow with
a critical evaluation of the conventional approach to Large Eddy
Simulation based upon the filtered equations.  We then discuss
modeling and consider the classic Smagorinsky model (inspired by Von
Neumann) and the Spectral Eddy Viscosity concept of Heisenberg.  With
this background, we argue that numerical analysis considerations
suggest a multiscale framework for modeling.  We then describe the
Variational Multiscale Formulation which obviates many of the
shortcomings of the filtered equations and possesses enhanced
potential for modeling.  We then demonstrate numerically that the
simplest instantiations of the idea lead to improved performance on
homogeneous flows, and equilibrium and non-equilibrium turbulent
channel flows.  We m! ention in closing the results of some other
investigators who have also obtained very good results with various
implementations of the ideas and finally we describe current research
activities.