Macro/Micro Perspectives for Turbulent Mixing: Large Scale and Atomic Scale Mixing Properties
James Glimm
Department of Applied Mathematics and Statistics
State University of New York at Stony Brook
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. A main problem for such flows is to sort out the distinct effects of small and large scale mixing.
We study both large scale and atomic scale mixing properties for classical hydrodynamic instabilities and mixing flows. The instability is driven by acceleration directed across a density discontinuity in the fluid. Assuming small scale initial perturbations of the interface, a highly complex mixing zone develops when acceleration is applied to the fluids. This simple sounding mixing flow has been notoriously difficult to predict. Standard simulations may give results differing from experiments by factors of two or more. We ascribe these differences to numerical artifacts in the simulations, specifically numerical mass diffusion.
A number of additional startling conclusions have recently emerged. 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 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 this pathology is a regularized solution, in other words inclusion of all physical regularizing effects, such as viscosity and physical mass diffusion. Once this is done, the solution appears to depend on the ratio of the coefficients in these terms, such as the Schmidt number, or if the solution is under resolved, on a numerical and code dependent Schmidt number.