# Mathematical and numerical principles for turbulent mixing

Wednesday, July 29, 2009 - 9:00am - 9:50am

EE/CS 3-180

James Glimm (The State University of New York)

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.

A number of startling conclusions have recently been observed.

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 this pathology is a regularized solution, in other words

inclusion of all physical regularizing effects, such as viscosity and

physical mass diffusion.

In other words, the amount 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.

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 three dimensional Rayleigh-Taylor unstable flows.

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.

A number of startling conclusions have recently been observed.

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 this pathology is a regularized solution, in other words

inclusion of all physical regularizing effects, such as viscosity and

physical mass diffusion.

In other words, the amount 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.

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 three dimensional Rayleigh-Taylor unstable flows.