The Variational Multiscale Method in Computational Fluid Dynamics
John Evans
I will discuss the development of the variational multiscale (VMS)
method
for solving partial differential equations in fluid dynamics. The
motivation is as follows: any reasonable method utilizing functions
capable
of resolving the exact solution will obtain it. However, in numerical
analysis, we typically employ finite-dimensional spaces of functions
that
are unable to resolve many fluid dynamical problems of interest. In such
situations, we hope to obtain reasonable approximations of coarse-scale
features of the flow without directly resolving fine-scale features. VMS
is a paradigm for obtaining such approximations. The VMS framework
accounts for the effect of unresolved scales on resolved scales, and
thus
is a general framework for the derivation of numerical methods for
problems
with multiple scales.
In my lecture, I will begin by discussing the fundamental features of
VMS
in the context of a linear model problem. This discussion will motivate
certain approximations of fine-scale features in the context of complex
flows. Following, I will discuss the use of VMS as a theoretical means
for
developing turbulence models. In particular, I will present a
formulation
of Large Eddy Simulation (LES) that is derived entirely from the
Navier-Stokes equations without recourse to any ad hoc devices, such as
eddy viscosity models. Finally, I will finish by discussing how VMS can
be
used to enforce desirable characteristics such as positivity and
monotonicity.