Moment Methods for Multiphase Flow and Non-Equilibrium Gasdynamics

我的研究计划的总体目标是发展新的中尺度模型的情况下，由一个基本的动力学方程。通常，这涉及到一个由大量单个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对实际情况的直接处理，因为计算费用将是压倒性的。在上述制度中，有许多现象对工程和科学领域具有直接的重要性，对于这些现象，还没有可接受的模型。这项科学研究计划将增加有关这些现象背后的物理学的科学知识，同时为现实世界的应用提供分析这些情况的工程软件。这一建议建立在我过去在开发这些模型方面取得的成功之上。我使用的技巧被称为“瞬间闭合”。在这种技术中，相关情况的粒子性质不会被忽略。相反，偏微分方程（PDE）的粒子的统计演变。目前这项建议的一个主要研究领域是多相流的有效建模。特别地，感兴趣的流由悬浮在背景介质中的大量小颗粒或液滴组成。这项工作将建立在我以前为加拿大核实验室（CNL）完成的一个研究项目的基础上。在本项目中，我的研究小组开发了一种新的用于预测放射性核素大气扩散的矩模型族。我们用于CNL项目的模型对粒子的高阶统计进行了建模，这些粒子完全由它们的位置，速度和大小来描述。然而，我们开发的模型的层次结构自然延伸到由任何数量的变量区分的粒子。这个发现补助金申请的主要目的是进一步发展我的新层次的多相模型。通过扩展模型，包括颗粒温度和蒸发模型，我们将表明，喷雾也可以很好地预测我的模型。再次，使用矩封闭，我开发了一个非平衡气体流动模型。在局部平衡之外，流体应力和热通量不再采取传统的形式。因此，这些数量的演变需要单独的偏微分方程。作为这个项目的一部分，我以前开发的一个模型将被进一步完善和扩展。虽然我之前的模型似乎有一个令人愉快的数学形式（一阶双曲系统），但它并不是在所有情况下都是全局适定的。改进这个模型，消除这些地区的不适定性将开发。

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.