Moment Methods for Multiphase Flow and Non-Equilibrium Gasdynamics

我的研究计划的首要目标是为受潜在动力学方程支配的情况开发新的中尺度模型。通常，这涉及到一个由大量单独的元素或粒子组成的系统，每个元素或粒子都遵循简单的定律。粒子的绝对数量排除了直接处理实际情况的可能性，因为计算费用将是巨大的。在上述制度中，有许多对工程和科学领域具有直接重要性的现象，但无法获得可接受的模型。这一科学研究计划将增加关于这些现象背后的物理基础的科学知识，同时为现实世界的应用提供分析这些情况的工程软件。这项建议建立在我过去在开发这些模型方面取得的成功的基础上。我使用的技巧被称为“瞬间闭合”。在这种技术中，粒子性质的相关情况是不能忽视的。相反，发展了粒子统计演化的偏微分方程组(PDE)。 目前这一提议的一个主要研究领域是多相流的有效建模。特别是，感兴趣的流动是由大量悬浮在背景介质中的小颗粒或液滴组成的。这项工作将建立在我之前为加拿大核实验室(CNL)完成的一个研究项目的基础上。在这个项目中，我的研究小组开发了一系列新的矩模型，用于预测放射性核素的大气扩散。我们用于CNL项目的模型模拟了粒子的高阶统计量，这些粒子完全由它们的位置、速度和大小描述。然而，我们开发的模型层次结构自然扩展到由任意数量的变量区分的粒子。这个Discovery Grant应用程序的一个主要目的是进一步开发我的新的多阶段模型层次结构。通过将模型扩展到包括粒子温度和蒸发模型，我们将表明我的模型也可以很好地预测喷雾。 再一次，利用力矩闭合，我建立了一个非平衡气体流动模型。在局部平衡之外，流体应力和热流不再采用传统的形式。因此，这些量的演化需要单独的偏微分方程组。作为这个项目的一部分，我之前开发的一个模型将进一步完善和扩展。尽管我之前的模型似乎有一个令人愉快的数学形式(一阶双曲型系统)，但它并不是在所有情况下都是全局适定的。将对该模型进行改进，以消除这些病态区域。

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.