Moment Methods for Multiphase Flow and Non-Equilibrium Gasdynamics

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The overarching goal of my research program is the development of new meso-scale models for situations governed by an underlying kinetic equation. Typically, this involves a system comprised of a huge number of individual elements or particles that each follow simple laws. The sheer number of particles precludes a direct treatment for practical situations, as the computational expense would be overwhelming. There are many phenomena that are of immediate importance to engineering and scientific fields in the above-described regime for which acceptable models are not available. This scientific research program will increase scientific knowledge regarding the physics underlying these phenomena while providing engineering software for the analysis of these situations for real-world applications. This proposal builds on my past successes in the development of these models. The technique that I use is known as "Moment Closure". In this technique, the particle nature of the relevant situation is not ignored. Rather, partial differential equations (PDEs) for the evolution of statistics of the particles are developed. One main area of research for this current proposal is the efficient modelling of multiphase flows. In particular, the flows of interest are made up of a large number of small particles or droplets suspended in a background medium. This work will build on a previous research project that I completed for the Canadian Nuclear Labs (CNL). In this project, my research group developed a new family of moment models for the prediction of the atmospheric dispersion of radioactive nuclides. The model that we used for the CNL project models high-order statistics of of particles that are described completely by their position, velocity, and size. The hierarchy of models that we developed, however, naturally extends to particles that are differentiated by any number of variables. A main thrust of this Discovery Grant application is the further development of my new hierarchy of multiphase models. By extending the model to include particle temperature and a model for evaporation, we will show that sprays can also be well predicted by my model. Again, using moment closures, I have developed a model for non-equilibrium gas flows. Outside of local equilibrium, the fluid stress and heat flux no longer take a traditional form. Therefore, separate PDEs for the evolution of these quantities are required. As part of this project, a previous model that I developed will be further refined and extended. Though my previous model appears to have a pleasant mathematical form (a first-order hyperbolic system), it is not globally well-posed in all cases. Refinements of this model that eliminate these regions of ill-posedness will be developed.