Simulation of Rayleigh-Taylor fluid mixing instability (left, early time; right, late time). The cut plane (right) shows a height level with 5% light fluid. The mixing rate a = h/Agt2 0.07 falls within the experimental range.

James Glimm and Xiaolin Li, State University of New York, Stony Brook

Research Objectives
We conduct 3D simulations of acceleration-driven fluid mixing, based on the front tracking code FronTier and the untracked TVD/level-set code for comparison. We continue studying two types of mixing: the steady acceleration-driven Rayleigh-Taylor (RT) instability and the shock-driven Richtmyer-Meshkov (RM) instability. In addition, we also use the FronTier code for the study of jet breakup, spray, and efficient diesel engine combustion.

Computational Approach
We use the front tracking method to study the RT and RM instabilities. Front tracking features high resolution of physical quantities at the material interface, thus giving a more accurate solution to the physical problem. It eliminates mass diffusion across the interface, reduces mesh orientation effects, and reduces diffusion of vorticity from the interface (where it is deposited by a shock wave in the RM instability) into the interior. Recently, we have implemented a robust grid-based method to handle the interface geometry. This method resolves the interface topology at the level of a single rectangular grid block. It provides a high degree of robustness to the numerical procedures. It reconstructs the interface at every time step and automatically corrects interfacial tangling. The traditional front tracking, known in contrast as grid-free tracking, also has advantages, in controlling the quality of the interface elements (triangles) and refined interface meshing.

A hybrid combination of the two methods will combine the best features of both. It is best suited for study of both the RT and RM instabilities. The use of grid-free tracking at the initial stage of both problems gives an accurate startup of the problem, since any grid-based interface description would require a very fine grid to resolve the fine-scale features and small amplitudes typically used to initialize the simulation studies. Grid-based tracking can handle the late-time chaotic stage of the fluid interface mixing without difficulty.

Accomplishments
Experimental values for the RT mixing coefficient a lie in the range of 0.05-0.07. FronTier simulation gives a value at the upper end of this interval, while most simulation codes report values of a outside of this experimental range, for example, a = 0.03.

Using diffusion-based renormalization of the diffusive TVD simulation, we obtained agreement for all values of a. We studied numerical mass diffusion by comparing the density distributions at horizontal slices drawn from similar penetration distances and heights within the mixing zone. The comparison shows the expected complete absence of mass diffusion for FronTier, and an approximate 50% reduction of density contrast for the TVD simulation. On this basis, we computed a time-dependent reduced effective, or mass diffused Atwood number, and used it to obtain an effective a_f for the TVD simulation. Remarkably, a_f is approximately equal to 0.07.

We have conducted a number of axisymmetrically perturbed spherical pure and multimode RM simulations. A careful validation and numerical study was reported, and the influence of axisymmetry as a statistical bias (due to the "north pole effect") from pure spherical symmetry was observed and studied. FronTier appears to have overcome the problem of mesh orientation dependence which afflicts Eulerian codes.

Significance
Acceleration-driven fluid mixing instabilities play important roles in inertially confined nuclear fusion and stockpile stewardship. Turbulent mixing is a difficult and centrally important issue for fluid dynamics, and impacts such questions as the rate of heat transfer by the Gulf Stream, resistence of pipes to fluid flow, combustion rates in automotive engines, and the late-time evolution of a supernova. Our computational study will provide a better understanding of the development of these instabilities.

Publications
J. Glimm, X. L. Li, and Y. J. Liu, "Conservative front tracking," SIAM J. Num. Ana. (submitted).

J. Glimm, J. Grove, and Y. Zhang, "Interface tracking for axisymmetric flows," SIAM J. Sci. Comp. (submitted).

J. Glimm, J. Grove, X. L. Li, W. Oh, and D. Sharp, "A critical analysis of Rayleigh-Taylor growth rates," J. Comp. Phys. 169, 652 (2001).

http://www.ams.sunysb.edu/~shock/FTdoc/FTmain.html