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Applied Mathematics Colloquium
Friday, December 1, 2006, 11:30 am
Cullimore Lecture Hall II
New Jersey Institute of Technology
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Turbulent
Mixing in Real (Non-Ideal) Fluids
James Glimm
Department of Applied Mathematics and Statistics
Stony Brook University and Brookhaven National Laboratory
Stony Brook, NY
Abstract
Turbulent mixing is an important but difficult problem in fluid
dynamics. We consider the mixing zone generated by the acceleration of
a fluid interface between two fluids of different densities. The case of
steady acceleration is called the Rayleigh-Taylor instability, and it gives
rise to a mixing zone of the two fluids, growing in thckness as time
increases. The mixing rate, experimentally, shows a universal growth rate,
as multiple of t2, but simulations in most cases disagree with
experiments by a factor of 2 or more. We report on a new class of simulations
(based on (a) an improved front tracking algorithm and on (b) inclusion
of real fluid effects) which agree with experiment. Here the real fluid
effect is surface tension for immiscible fluids and mass diffusion, initial
(t = 0) mass diffusion, or viscosity for miscible fluids. We document significant
dependence of the mixing rate on both physical effects (e.g. surface tension)
and on numerical artifacts such as mass diffusion (for untracked simulations).
We conclude that modeling of turbulent mixing is sensitive to details
of transport, surface tension, and to their numerical analogues. The full
dependence of the mixing rates on these physical phenoma and their numerical
analogues is only partially understood.
When we consider the averaged equations, which remove all of the microphysical
complexity of the mixing process and replace it with mean or averaged
flow properties, serious difficulties emerge in the problem of closure,
or the proper definition of the new nonlinear terms arising from the
averaging process to define the closed equations. The averaged equations
are never in conservation form, and for this reason the proper meaning
of the nonlinear terms and their discretization and regularization
is still a research issue. The numerical data base resulting from experimentally
validated simulations becomes invaluable in providing a rational and
systematic way to evaluate proposed closures. We find excellent agreement
between the direct average of the simulated data and closures we have
proposed. We also find a fair degree of insensitivity in comparing
the data to other closures.
It is a pleasure to thank the many colleagues and collaborators who
have contributed to the work presented here.
Finally, we discuss briefly plans at Stony Brook and Brookhaven National
Laboratory for computational science.