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43872-AC9
Equation-Free, Coarse-Grained Computation for Multiphase Flow Modeling
In our research we use an “equation-free'', coarse-grained computational approach to accelerate molecular dynamics-based computations of granular flows. Work during this year involved the use of this computational framework to accelerate the computations of demixing (or segregation) occurring in dissimilar particles subject to an upward gas flow (gas-fluidized beds). We explored the coarse-grained dynamics of these phenomena in gently fluidized beds of solid mixtures of different densities, typically a slow process for which reasonable continuum models are currently unavailable. We considered well-known phenomena for which the derivation of continuum models is still in flux; mixing and demixing (segregation) can occur when dissimilar particle mixtures of different sizes and/or densities are subject to a strong enough upward fluid flow. A few different continuum models, more phenomenological or more rigorous have been proposed, which often reproduce the phenomena in a qualitatively correct manner; however, quantitative agreement is still generally elusive. Furthermore, kinetic theory-based continuum models for binary mixtures are much more complicated than those for uniform particles, and numerical simulation becomes more time-consuming.
Accelerating the computation using (quantitative) microscopic models would therefore be invaluable for such problems. The objective of our work was to enable accelerated integration of MD-based microscopic simulations of dense particulate flows. We deliberately choose demixing occurring in narrow beds (cross sectional area of 15x15 particle diameters with periodic boundary conditions for both lateral directions, as a test problem, so that the exact results can readily be computed and used to critically test the coarse-grained computations. These are quasi-1D flows, where the coarse-grained gas flow is effectively 1D, while particle simulation is fully 3D; demixing becomes more pronounced in such narrow beds.
We sought coarse-grained variables (or ``observables'') that could be used in continuum descriptions. As in the two-fluid modeling approach it is natural to think of hydrodynamic variables as candidate coarse observables. From direct simulations, we observed that in the course of demixing in quasi-1D beds, the process strongly depends on the local density, which makes the 1D volume fraction profiles themselves sufficient coarse observables; when we suddenly randomize (only) the individual particle velocities (hence the granular temperature as well) snd continue the simulation, the demixing progresses essentially undisturbed. We further recognized that cumulative particle distribution functions (CDFs, and more precisely, their inverses which are bounded by 0 and 1) are more convenient coarse observables: CDFs are smoother than volume fraction profiles, suffer from less noise, and facilitate the “lifting” procedure. When the time series of coarse observables (obtained by direct integration of the microscopic simulator) are smooth and slowly varying, one can estimate their local time derivatives and then project the values at a future time (e.g. using forward Euler or more sophisticated schemes). We recognized that if one can initialize the microscopic simulator consistent with the future (projected) values of the coarse observables, one can actually accelerate the overall computation. This simple idea underpins coarse projective integration.
In equation-free computations, traditional continuum numerical techniques are directly applied to the outcome of appropriately initialized short bursts of microscopic simulation, and the macroscopic equations are "integrated'' or "solved'' without ever being written
down. We identified four coarse observables based on the ICDF that helped perform ensemble-averaged coarse projective integration over a number of realizations of the MD simulation. These observables vary slowly and smoothly in time (occasional oscillations disappear for larger ensembles); in a sense, we performed a pseudospectral solution of the unknown governing equations for ICDF evolution.
Our equation-free coarse-grained approach was capable of accelerating (by a factor of two to ten) computations of dense particulate flows and, in particular, of demixing occurring in gas-fluidized beds of dissimilar particles.. This approach holds promise for the prediction of coarse-grained behavior at practically relevant spatial and temporal scales. We deliberately considered a quasi-1D illustrative problem in our study, in order to demonstrate the viability of the approach. As a consequence of the problem considered in this study, the coarse observables were one-dimensional discretized ICDFs. For systems involving higher dimensional flows, candidates for coarse observables may include marginal and conditional ICDFs. More work for such systems has to be done to identify proper coarse observables an an efficient lifting operator, a vital components of this approach. Ensemble averaging reduces fluctuations among the realizations, giving better quantitative representations; the computation of each realization readily parallelizes across computational nodes.
More sophisticated equation-free algorithms (e.g. coarse fixed point algorithms can be used to find stable as well as unstable steady states; quantify their stability and perform numerical bifurcation analysis; exploiting such tools to investigate the coarse-grained dynamics of mixing and segregation (and other particulate flow problems) is the subject of current research, during the second year of the grant.
