Stony Brook AMS - Downloadable Preprints - 2007

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SUNYSB-AMS-07-02	Compressible Multi Species Multi Phase Flow Models	W. Bo, H. Jin, D. Kim, X. Liu, H. Lee, N. Pestieau, Y. Yu, J. Glimm and J. W. Grove

Multi-phase flow equations are defined through an ensemble average of microphysical equations characterized by distinct phases. A central problem for multi-phase flow models is closure, or the proper definition of averages of nonlinear terms. We consider compressible multi species multiphase flow with surface tension and transport. We propose closures which satisfy boundary conditions and conservation constraints. These closures and related closures of Abgrall and Saurel are compared in a validation study to spatial averages of DNS (direct numerical simulation), { i.e.}, simulation solutions of the microphysical equations. The DNS data are themselves validated by agreement with laboratory experiment. We find excellent validation agreement for the closures proposed here. The Abgrall-Saurel closures, in comparison to the same validation data, are good, but less satisfactory.

Submitted to Phys. Rev. E
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SUNYSB-AMS-07-03	On validation of turbulent mixing simulations for Rayleigh-Taylor instability	Hyunsun Lee, Hyeonseong Jin, Yan Yu and James Glimm

The purpose of this paper is to analyze the validation achieved in recent simulations of Rayleigh-Taylor unstable mixing. The simulations are already in agreement with experiment; mesh refinement or insertion of a calibrated subgrid model for mass diffusion will serve to refine this validation and possibly shed light on the role of unobserved long wave length perturbations in the initial data. In this paper we present evidence to suggest that a subgrid model will have a barely noticeable effect on the simulation. The analysis is of independent interest, as it connects a validated simulation to common studies of mixing properties. The average molecular mixing parameter $\theta$ for the ideal and immiscible simulations is zero at a grid block level, as is required by the exact microphysics of these simulations. Averaging of data over volumes of $(4\Delta x)^3$ to $(8\Delta x)^3$ yields a conventional value $\theta \sim 0.8$, suggesting that fluid entrainment in front tracked simulations produces a result similar to numerical mass diffusion in untracked simulations. The miscible simulations yield a nonzero $\theta \sim 0.8$ in agreement with experimental values. We find spectra in possible approximate agreement with the Kolmogorov theory. A characteristic upturn especially in the density fluctuation spectrum at high wave numbers suggests the need for a subgrid mass diffusion model, while the small size of the upturn and the analysis of $\theta$ suggest that the magnitude of the model will not be large. We study directly the appropriate settings for a subgrid diffusion coefficient, to be inserted into simulations modeling miscible experiments. This is our most definitive assessment of the role for a subgrid model. We find that a Smagorinsky type subgrid mass diffusion model would have a diffusion coefficient at most about 0.15\% of the value of the physical mass diffusion for the (mass diffusive) experiment studied.

Submitted to Phys. Fluids
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SUNYSB-AMS-07-04	Verification and Validation for Turbulent Mixing	Hyeonseong Jin and James Glimm

Verification and validation constitute a relatively new requirement for numerical analysis. It is one which results from the growing practical use of computation as a basis for engineering decisions. While a number of procedures have been proposed for these tasks, their application to the more difficult cases remains a research issue. Here we explain methods developed by ourselves and collaborators for verification and validation of simulations for chaotic, multiscale flows, specifically for turbulent mixing simulations.

Submitted to Nonlinear Analysis
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SUNYSB-AMS-07-05	Multi Scale Models for Fluid Mixing	H. Lim, Y. Yu, H. Jin, D. Kim, H. Lee, J.Glimm, X.-L. Li and D. H. Sharp

Recent work of the authors and colleagues on the turbulent mixing of compressible fluids is developed and extended with an emphasis on the multiscale aspects of this work. Specifically, we study an interplay between micro and macro aspects of mixing.

Submitted to special issue of CMAME on stochastic modeling of multiscale/multiphysics problems
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SUNYSB-AMS-07-06	Data-Based Analysis of Winner-Loser Models of Hierarchy Formation in Animals	W.B. Lindquist and I.D. Chase

We review winner-loser models, the currently popular explanation for the occurrence of linear dominance hierarchies, via a three-part approach. 1) We isolate the two most significant components of the mathematical formulation of three of the most widely-cited models and rigorously evaluate the components' predictions against data collected on hierarchy formation in groups of hens. 2) We evaluate the experimental support in the literature for the basic assumptions contained in winner-loser models. 3) We apply new techniques to the hen data to uncover several behavioral dynamics of hierarchy formation not previously described. The mathematical formulations of these models do not show satisfactory agreement with the hen data; key model assumptions have either little, or no conclusive, support from experimental findings in the literature. In agreement with the latest experimental results concerning social cognition, the new behavioral dynamics of hierarchy formation discovered in the hen data suggest that members of groups are intensely aware both of their own interactions as well as interactions occurring among other members of their group. We suggest that more adequate models of hierarchy formation should be based upon behavioral dynamics that reflect more sophisticated levels of social cognition.

Submitted to Bulletin of Mathematical Biology
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SUNYSB-AMS-07-07	Network Flow Modeling via Lattice-Boltzmann Based Channel Conductance	Y. Sholokhova, D. Kim and W.B. Lindquist

We introduce a network flow model in which the single phase conductance between neighboring sites (pore bodies) is computed via lattice-Boltzmann (LB) computation of actual channel configurations derived from X-ray computed tomographic (XCT) images of Fontainebleau sandstone samples. The LB computations for individual channel conductance are compared to models in which channel conductance is computed via a pore body-throat-pore body series resistance, with the conductance of each individual element (pore body, throat) based on shape factor measurements. The shape factor measurements provide conductances assuming cross sectional shape is predominantly triangular. The LB computations based upon actual channel geometry (as captured by the XCT images) show that the variation in conductance possible for channels having similar cross sectional shape factor is much larger than is suggested by triangular models. Fits to the dependence of median value of conductance versus shape factor from the LB-based computations reveal a power law dependence, in contrast to the linear dependence observed for triangular, square and circular cross section approximations. The observed power law dependence has a larger exponent than predicted by rectangular or elliptic cross section models.

Bulk absolute permeabilities for each of the sandstone images is modeled using both the LB-based and the shape factor-based network models. The LB-based network models produce bulk absolute permeability values that fit published data more accurately than the shape factor-based models.

Submitted to Journal of Colloid and Interface Science
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SUNYSB-AMS-07-08	Transonic Shock Formation in a Rarefaction Riemann Problem for the 2-D Compressible Euler Equations	J. Glimm, X. Ji, J. Li, X. Li, P. Zhang, T. Zhang and Y. Zheng

It is perhaps surprising for a shock wave to exist in the solution of a rarefaction Riemann problem for the compressible Euler equations in two space dimensions. We present numerical evidence and generalized characteristic analysis to establish the existence of a shock wave in such a 2D Riemann problem, defined by the interaction of four rarefaction waves. We consider both the customary configuration of waves at the right angle and oblique angles as well. We discover two forms of shock formation in those configurations as the angle between the waves is varied.

Submitted to SIAM Journal on Applied Mathematics
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SUNYSB-AMS-07-09	Chaos, Transport, and Mesh Convergence for Fluid Mixing	H. Lim, Y. Yu, J.Glimm, X.-L. Li and D. H. Sharp

Chaotic mixing of distinct fluids produces a convoluted structure to the interface separating these fluids. For miscible fluids (as considered here), this interface is defined as a $50\%$ mass concentration isosurface. For shock wave induced (Richtmyer-Meshkov) instabilities, we find the interface to be increasingly complex as the computational mesh is refined. This interfacial chaos is cut off by viscosity, or by the computational mesh if the Kolmogorov scale is small relative to the mesh. In a regime of converged interface statistics, we then examine mixing, i.e. concentration statistics, regularized by mass diffusion. For Schmidt numbers significantly larger than unity, typical of a liquid or dense plasma, additional mesh refinement is normally needed to overcome numerical mass diffusion and to achieve a converged solution of the mixing problem. However, with the benefit of front tracking and with an algorithm that allows limited interface diffusion, we can assure convergence uniformly in the Schmidt number. We show that different solutions result from variation of the Schmidt number. We propose subgrid viscosity and mass diffusion parameterizations which might allow converged solutions at realistic grid levels.

Submitted to Acta Mathematicae Applicatae Sinica
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