Extreme Events and Optimal Closures in Fluid Mechanics

流体力学中的极端事件和最佳闭合

• 批准号: RGPIN-2014-04400
• 负责人: Protas, Bartosz
• 金额: $ 2.48万
• 依托单位: McMaster University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=566496
• 关键词: Extreme; Events; Optimal; Closures; Fluid

This application seeks funding for Dr. Protas' ongoing research program which will offer a new perspective on a number of classical and emerging fundamental problems in the area of theoretical fluid dynamics. By performing carefully designed and executed numerical computations we will shed light on how well the rigorous mathematical analysis can describe actual flow behaviours. We will also develop optimal ways of simplifying the mathematical description of complex fluid flows. The first class of open questions we will investigate addresses forms of extreme behavior possible in flows of incompressible fluids. Such "extreme behavior" may concern the spontaneous growth of energy-like quantities characterizing the flow structure and might potentially manifest itself in the formation of singularities in finite time. Related questions arise, for instance, in the characterization of the maximum possible mixing of passive scalars or convective heat transfer. In addition to assessing the fundamental performance limitations of various flow processes in science and engineering, these issues are in the first place relevant for our basic understanding of how the mathematical models of fluid flow behave. An example of such a question is one of the Clay Institute's "Millennium Problems" for the mathematical community concerning the well-posedness of the Navier-Stokes system in 3D which remains notoriously unresolved. The ultimate goal of the proposed research program is to determine systematically, via a suitable process of numerical optimization, smooth Navier-Stokes flows in 3D exhibiting a worst-case behavior in order to understand whether such behavior may be compatible with singularity formation in finite time. As intermediate objectives, we will address analogous questions for a range of related simplified flow models in which such singular behavior is known, or suspected, to exist. These results will provide key new insights about the sharpness of the mathematical analysis and the properties of mathematical fluid models with ubiquitous applications, bridging in this way mathematical analysis with large-scale scientific computations. The second class of open problems is motivated by the development of reduced-order models for turbulent flows. Given the multiscale complexity exhibited by such flows, their complete resolution in numerical computations will always be a challenge and some form of approximation is needed to represent unresolved dynamics. Although such problems have been studied for a long time, almost all of the work has been based on purely empirical approaches. Mathematically optimal solutions to this so-called "closure problem" are the second major open question we will address. In addition to offering an entirely new way to design such closure strategies, our work will also answer questions about the fundamental performance limitations of these approaches. A common theme in this proposed research effort is that these different questions will be framed in terms of variational optimization problems amenable to solutions using modern methods of numerical optimization. During this project we will offer uniquely broad interdisciplinary training to a large number of trainees at different levels. It will span applied analysis and large-scale scientific computation, and will be combined with applied fields such as fluid mechanics and model reduction.

该申请为Protas博士正在进行的研究计划寻求资金，该计划将为理论流体动力学领域的许多经典和新兴的基本问题提供新的观点。通过执行精心设计和执行的数值计算，我们将阐明严格的数学分析能够描述实际流动行为。我们还将开发最佳方法来简化复杂流体流的数学描述。我们将调查的第一类开放问题解决了不可压缩流体流中可能的极端行为形式。这种“极端行为”可能涉及表征流动结构的能量样量的自发生长，并可能在有限的时间内形成奇异性。例如，在表征被动标量或对流传热的最大可能混合时会出现相关问题。除了评估科学和工程中各种流动过程的基本绩效限制外，这些问题首先与我们对流体流量的数学模型的基本了解有关。这个问题的一个例子是粘土研究所的“千年问题”，用于数学社区，涉及3D中Navier-Stokes System的良好性，这仍然众所周知未解决。拟议的研究计划的最终目标是通过合适的数值优化过程系统地确定3D中平滑的Navier-Stokes流动，表现出最坏情况的行为，以了解在有限时间内是否与奇异性形成兼容。作为中间目标，我们将针对一系列相关的简化流程模型解决类似的问题，其中这种奇异行为是已知或怀疑存在的。这些结果将为数学分析的清晰度以及使用无处不在的应用程序的数学流体模型的特性提供关键的新见解，并通过大规模的科学计算以这种方式进行数学分析进行桥接。开放问题的第二类是由为湍流降低的模型的开发而激发的。鉴于此类流量表现出的多尺度复杂性，它们在数值计算中的完整分辨率将永远是一个挑战，并且需要某种形式的近似值来表示未解决的动力学。尽管已经研究了这些问题已有很长时间了，但几乎所有工作都是基于纯粹的经验方法。在数学上为此所谓的“封闭问题”的最佳解决方案是我们将要解决的第二个主要开放问题。除了提供一种设计这种封闭策略的全新方法外，我们的工作还将回答有关这些方法基本绩效限制的问题。这项提出的研究工作中的一个共同主题是，这些不同的问题将以现代数值优化方法适应解决方案的变异优化问题来构成。在此项目中，我们将在不同级别的大量受训人员为大量学员提供独特的跨学科培训。它将涵盖应用分析和大规模的科学计算，并将与诸如流体力学和模型还原等应用领域结合使用。