Moment Methods for Multiphase Flow and Non-Equilibrium Gasdynamics

我的研究计划的首要目标是开发新的中尺度模型，用于由潜在的动力学方程控制的情况。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。粒子的绝对数量阻碍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，但却没有可接受的模型。这项科学研究计划将增加关于这些现象背后的物理学的科学知识，同时为这些实际应用的分析提供工程软件。这个建议建立在我过去开发这些模型的成功基础上。我使用的技术被称为“瞬间封闭”。在这种技术中，相关情况的粒子性质并没有被忽略。相反，偏微分方程（PDEs）的发展统计粒子的演变。目前研究的一个主要领域是多相流的有效建模。特别地，感兴趣的流是由悬浮在背景介质中的大量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，我的课题组开发了一组新的矩模型，用于预测放射性核素的大气弥散。我们在CNL项目中使用的模型是粒子的高阶统计模型，完全由它们的位置、速度和大小来描述。然而，我们开发的模型层次结构自然地扩展到由任意数量的变量区分的粒子。这个发现基金申请的主要推动力是进一步发展我的新的多阶段模型层次结构。通过将模型扩展到包括颗粒温度和蒸发模型，我们将证明我的模型也可以很好地预测喷雾。再一次，使用力矩闭包，我开发了一个非平衡气体流动的模型。在局部平衡之外，流体应力和热流不再采用传统形式。因此，需要对这些量的演化进行单独的pde。作为这个项目的一部分，我之前开发的一个模型将被进一步细化和扩展。虽然我之前的模型看起来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。将对该模型进行改进，以消除这些不适区域。

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.