Stony Brook AMS - Downloadable Preprints - 2009

Submitted to ASTRONUM proceedings
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SUNYSB-AMS-09-02	Nearly Discontinuous Chaotic Mixing	H. Lim, Y. Yu, J. Glimm, and D. H. Sharp

A new scientific approach is presented for a broad class of chaotic problems involving a high degree of mixing over rapid time scales. Rayleigh-Taylor and Richtmyer-Meshkov unstable flows are typical of such problems. Microscopic mixing properties such as chemical reaction rates for turbulent mixtures can be obtained with feasible grid resolution. The essential dependence of (some) fluid mixing observables on transport phenomena is observed. This dependence includes numerical as well as physical transport and it includes laminar as well as turbulent transport. A new approach to the mathematical theory for the underlying equations is suggested.

Submitted to High Energy Density Physics
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SUNYSB-AMS-09-03	Pore-Level Examination of Gel Destruction During Oil Flow	R.S. Seright, W.B. Lindquist and R. Cai

Pore-scale X-ray computed microtomography (XMT) images were obtained at a variety of oil (hexadecane) throughput values after gel placement in cores [involving a pore-filling Cr(III)-acetate-hydrolyzed polyacrylamide (HPAM) gel]. For each pore in our image volume, we followed oil and water saturations as a function of oil throughput. These studies were performed both in water-wet Berea sandstone and in hydrophobic porous polyethylene cores. In hydrophobic porous polyethylene, oil saturations increased and gel was destroyed (presumably dehydrated) quite quickly in the smallest pores. Also, oil saturations increased and gel was destroyed quickly in the largest pores. In contrast, oil saturations rose much more gradually for the most common or intermediate-size pores (around 10?4 mm3). The minimum in oil saturation vs. pore size may result from a balance between gel dehydration by oil film growth vs. gel extrusion.

In contrast, in water-wet Berea sandstone, increases in oil saturation occurred evenly over all pore sizes (10?6 to 0.02 mm3) for all oil throughput values. Consistent with imbibition and drainage studies performed before gel placement, oil apparently had equal access to Berea pores of all sizes and, thus, uniformly dehydrated gel in pores of all sizes. Gel extrusion did not appear to be significant in the Berea pores.

SPE Journal, to appear
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SUNYSB-AMS-09-04	Reflections and Prospectives	J. Glimm

Intellectual challenges and opportunities for mathematics are greater than ever. The role of mathematics in society continues to grow; with this growth comes new opportunities and some growing pains; each will be analyzed here.

AMS Presidential Address
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SUNYSB-AMS-09-05	Nonideal Rayleigh-Taylor Mixing	H. Lim, J. Iwerks, J. Glimm, and D. H. Sharp

Rayleigh-Taylor mixing is a classical hydrodynamic instability, which occurs when a light fluid pushes against a heavy fluid. The two main sources of nonideal behavior in Rayleigh-Taylor (RT) mixing are regularizations (physical and numerical) which produce deviations from a pure Euler equation, scale invariant formulation, and nonideal (i.e. experimental) initial conditions. The Kolmogorov theory of turbulence predicts stirring at all length scales for the Euler fluid equations without regularization. We interpret mathematical theories of existence and non-uniqueness in this context, and we provide numerical evidence for dependence of the RT mixing rate on nonideal regularizations, in other words indeterminacy when modeled by Euler equations. Operationally, indeterminacy shows up as non unique solutions for RT mixing, parametrized by Schmidt and Prandtl numbers, in the large Reynolds number (Euler equation) limit. Verification and validation evidence is presented for the large eddy simulation algorithm used here. Mesh convergence depends on breaking the non-uniqueness with explicit use of the laminar Schmidt and Prandtl numbers and their turbulent counterparts, defined in terms of subgrid scale models. The dependence of the mixing rate on the Schmidt and Prandtl numbers and other physical parameters will be illustrated. We demonstrate numerically the influence of initial conditions on the mixing rate. Both the dominant short wavelength initial conditions and long wavelength perturbations are observed to play a role. By examination of two classes of experiments, we observe the absence of a single universal explanation, with long and short wavelength initial conditions, and the various physical and numerical regularizations contributing in different proportions in these two different contexts.

Submitted to PNAS
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SUNYSB-AMS-09-06	Weakly compressible two-pressure two-phase flow	H. Jin, and J. Glimm

We analyze the limiting behavior of a compressible two-pressure two-phase flow model as the Mach number tends to zero. Formal asymptotic expansions are derived for the solutions of compressible two-phase equations. Expansion coe±cients through second order are evaluated in closed form. Underdetermination of incompressible pressures is resolved by information supplied from the weakly compressible theory. The incompressible pressures are uniquely speciŻed by certain details of the compressible °uids from which they are derived as a limit. This aspect of two phase °ow in the incompressible limit appears to be new, and results basically from closures which satisfy single phase boundary conditions at the edges of the mixing zone.

Submitted to Acta Mathematica Scientia
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SUNYSB-AMS-09-07	Verification and Validation for Turbulent Mixing Simulations	H. Lim, J. Iwerks, Y. Yu, J. Glimm, and D. H. Sharp

We present highlights from and supplementary material related to two recent studies giving verification and validation of a new approach to the simulation of turbulent mixing. The verification is based on (i) a mesh refinement study of a circular Richtmyer-Meshkov unstable flow, (ii) code comparison to a well documented code and (iii) comparison to a simple analytic model. The validation is based on simulation agreement with Rayleigh-Taylor unstable experiments of Smeeton-Youngs and of Mueschke-Andrews. The mesh refinement verification gives convergence for such molecular level variables as the probability density functions for the concentrations, temperatures and a chemical reaction rate. The validation study, beyond obtaining near perfect agreement with experiment, explores the various factors in the simulations that result in this agreement and in the differences between the two experiments. The significant variables are: fluid transport parameters, dimensionless groups (not widely recognized to be significant) to characterize the dominant short wavelength initial perturbations and experimentally measured (long wavelength) initial perturbations.

Submitted to Physica Scripta
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SUNYSB-AMS-09-08	Mathematical, Physical and Numerical Principles Essential for Models of Turbulent Mixing	H. Lim, Y. Yu, J. Glimm, and D. H. Sharp

We propose mathematical, physical and numerical principles which are important for the modeling of turbulent mixing, especially the classical and well studied Rayleigh-Taylor and Richtmyer-Meshkov instabilities which involve acceleration driven mixing of a fluid discontinuity layer, by (respectively) a continuous acceleration or an impulsive delta function force.
The fundamental mathematical issue is the nonuniqueness and thus indeterminancy of solutions of the 3D compressible Euler equation. Verification (demonstration that the numerical solution of the equations is mathematically correct) is meaningless for such a model of turbulent mixing. Uniqueness requires physical fluid transport, i.e. the compressible (multifluid) Navier-Stokes equation.
The same fundamental issue, formulated in terms of physics, is that the properties of the mixing depend on dimensionless ratios of the transport coefficients, namely the Schmidt number (viscosity/mass diffusion) and Prandtl number (viscosity/thermal conductivity). Validation (meaning that the simulation equations correctly model the problem to be solved) is impossible without specification of fluid transport. It is in effect an effort to validate an answer for 0/0.
The fundamental issue, formulated in numerical terms is that the physical transport terms are so small that they cannot be resolved at feasible grid levels. Large eddy simulations (LES) are needed. For these, subgrid scale (SGS) terms must be added to the equations, to correctly reflect the influence of the unresolved transport on the grid scales that are resolved. In the absence of such an approach, numerical artifacts intrude, leading to apparently converged solutions, with answers that depend on the computer code.
Plainly, this issue, in its three guises, poses a challenge for verification and validation (V&V), and since V&V is a major scientific enterprise, it is of great importance.

Submitted to IMA Proceedings
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