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: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635163
• 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项目寻求资助，该项目将为理论流体动力学领域的一些经典和新兴基础问题提供新的视角。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如何很好地描述实际的流动行为。我们还将开发简化复杂流体流动的数学描述的最佳方法。我们将研究的第一类开放问题涉及不可压缩流体流动中可能出现的极端行为形式。这种“极端行为”可能涉及表征流动结构的类能量的自发增长，并可能在有限时间内奇点的形成中潜在地表现出来。相关的问题出现了，例如，在表征最大可能混合的被动标量或对流传热。除了评估科学和工程中各种流动过程的基本性能限制外，这些问题首先与我们对流体流动的数学模型如何表现的基本理解相关。这类问题的一个例子是克莱研究所为数学界提出的“千年问题”之一，涉及三维Navier-Stokes系统的适定性，这一问题至今仍未得到解决。所提出的研究计划的最终目标是系统地确定，通过适当的数值优化过程，三维光滑的Navier-Stokes流表现出最坏情况，以便了解这种行为是否可能与有限时间内的奇点形成相容。作为中间目标，我们将解决一系列相关的简化流动模型的类似问题，其中已知或怀疑存在这种奇异行为。这些结果将为数学分析的尖锐性和数学流体模型的特性提供关键的新见解，并以这种方式将数学分析与大规模科学计算联系起来。第二类开放问题是由湍流降阶模型的发展引起的。考虑到这些流动所表现出的多尺度复杂性，它们在数值计算中的完全解决将始终是一个挑战，需要某种形式的近似来表示未解决的动力学。虽然这些问题已经研究了很长时间，但几乎所有的工作都是基于纯粹的经验方法。这个所谓的“封闭问题”的数学最佳解决方案是我们将要解决的第二个主要开放问题。除了提供一种全新的方法来设计这样的闭包策略外，我们的工作还将回答有关这些方法的基本性能限制的问题。在这项研究工作中，一个共同的主题是，这些不同的问题将在变分优化问题方面进行框架，这些问题可以使用现代数值优化方法来解决。在这个项目中，我们将为大量不同层次的学员提供独特的广泛的跨学科培训。它将跨越应用分析和大规模科学计算，并将与流体力学和模型简化等应用领域相结合。