Numerical approximation of fluid equations are reviewed. We identify
numerical mass diffusion as a characteristic problem in most simulation codes.
This fact is illustrated by an analysis of fluid mixing flows. In these flows,
numerical mass diffusion has the effect of over regularizing the solution.
Simple mathematical theories explain this difficulty.
A number of startling conclusions have recently been observed,
related to numerical mass diffusion. For a flow accelerated by multiple
shock waves, we observe an interface between the two fluids proportional
to Delta x-1, that is occupying a constant fraction of the available
mesh degrees of freedom. This result suggests
- (a) nonconvergence for the unregularized mathematical problem or
- (b) nonuniqueness of the limit if it exists, or
- (c) limiting solutions only in the very weak form of a
space time dependent probability distribution.
The cure for the pathology (a), (b) is a regularized
solution, in other words inclusion of all physical regularizing effects,
such as viscosity and physical mass diffusion. We do not regard (c) as
a pathology, but an inherent feature of the equations.
In other words, the amount and type of regularization of an
unstable flow is of central importance. Too much regularization, with a
numerical origin, is bad, and too little, with respect to the physics,
is also bad. For systems of equations, the balance of regularization
between the distinct equations in the system is of central importance.
At the level of numerical modeling, the implication from this insight
is to compute solutions of the Navier-Stokes, not the Euler equations.
Resolution requirements for realistic problems make this solution
impractical in most cases. Thus subgrid transport processes must be modeled,
and for this we use dynamic models of the turbulence modeling community.
In the process we combine and extend ideas of the capturing community
(sharp interfaces or numerically steep gradients) with conventional
turbulence models, usually applied to problems relatively smooth at
a grid level.
The numerical strategy is verified with a careful study of a 2D
Richtmyer-Meshkov unstable turbulent mixing problem. We obtain converged
solutions for such molecular level mixing quantities as a chemical reaction
rate. The strategy is validated (comparison to laboratory experiments)
through the study of 3D Rayleigh-Taylor unstable flows.
