Topics in Vortex Dynamics: Extreme Events, Optimal Closures and New Equilibrium Solutions

涡动力学主题：极端事件、最优闭合和新的平衡解

• 批准号: RGPIN-2020-05710
• 负责人: Protas, Bartosz
• 金额: $ 3.13万
• 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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759101
• 关键词: Topics; Vortex; Dynamics; Extreme; Events; Topics; Vortex; Dynamics; Extreme; Events

The long-term objective of my research program is to provide answers to classical and emerging open questions in the area of theoretical fluid dynamics. Even though the mathematical models on which fluid mechanics is based were introduced nearly two centuries ago, many of their fundamental properties are still to be fully understood. As a first objective, we will carry out a systematic search for singular behavior in flows of incompressible fluids governed by the Navier-Stokes equation. Recognized by the Clay Mathematical Institute as one of its "Millennium Problems", the question of the finite-time singularity formation is one of the central problems in mathematical fluid mechanics. While this is a problem in mathematical analysis, we will cast a new light on it by methodically finding solutions exhibiting the most extreme behavior in order to understand whether such behavior may lead to singularity formation. A key novelty of our approach is that such extreme flows will be sought systematically via numerical solution of suitable variational optimization problems. The second class of open questions is motivated by the development of simplified 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. While most of the work on this so-called "closure problem" to date has relied on purely empirical approaches, we will look for mathematically optimal solutions. In addition to offering an new way to design such closure strategies, our work will answer questions about their fundamental performance limitations. Finally, we will study problems in classical vortex dynamics where we will seek to discover new families of equilibrium solutions and explore their stability properties. This effort will fill important gaps in theoretical hydrodynamics and will also provide insights about the behavior of vortex rings ubiquitous in various technological and biological applications. While numerical optimization is widely used to solve practical problems, here it will be employed to shed light on the inner workings of the mathematical models of fluid flow. Many of our research questions are specifically framed to provide new insights beyond what can be established with rigorous analysis, such that these insights will point to the direction in which the mathematical analysis can be extended and sharpened. Given the interdisciplinary nature of this program, the proposed research and training will integrate methods from theoretical fluid mechanics, applied and numerical analysis, optimization and control, as well as large-scale scientific computing. In the long term our findings will have impact on various fields where fluid dynamics plays an important role, such as mechanical and chemical engineering, atmospheric and environmental science as well as numerical weather prediction.

我的研究计划的长期目标是为理论动力学领域的古典和新兴的开放问题提供答案。即使在将近两个世纪前引入了基于流体力学的数学模型，但其许多基本属性仍待完全了解。 作为第一个目标，我们将系统地搜索由Navier-Stokes方程控制的不可压缩流体的流动。被粘土数学研究所认为是其“千年问题”之一，有限的奇异性形成的问题是数学流体力学的核心问题之一。尽管这是数学分析中的一个问题，但我们将通过有条不紊地找到表现出最极端行为的解决方案来了解这种行为是否会导致奇异性形成，从而对其进行新的启示。我们方法的主要新颖性是，将通过合适的变分优化问题的数值解决方案系统地寻求这种极端流。 第二类开放问题是由开发简化湍流模型的动机。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。尽管迄今为止，大多数所谓的“封闭问题”的工作都依赖于纯粹的经验方法，但我们将寻找数学上最佳的解决方案。除了提供一种设计这种封闭策略的新方法外，我们的工作还将回答有关其基本绩效限制的问题。 最后，我们将研究经典涡流动力学中的问题，我们将寻求发现平衡解决方案的新家族并探索其稳定性。这项工作将填补理论流体动力学中的重要空白，还将提供有关在各种技术和生物学应用中无处不在的涡流行为的见解。虽然数值优化被广泛用于解决实际问题，但在这里，它将用于阐明流体流数学模型的内部工作。我们的许多研究问题都是专门构建的，可以提供超出严格分析可以建立的新见解，以便这些见解将指向可以扩展和锐化数学分析的方向。鉴于该计划的跨学科性质，拟议的研究和培训将整合理论流体力学，应用和数值分析，优化和控制以及大规模科学计算的方法。从长远来看，我们的发现将对流体动力学起着重要作用的各个领域产生影响，例如机械和化学工程，大气和环境科学以及数值天气预测。