Mathematical Sciences: Hydrodynamics and Magnetohydrodynamics

数学科学：流体动力学和磁流体动力学

• 批准号: 9307644
• 负责人: Bruce Turkington
• 金额: $ 6万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-07-15 至 1996-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9307644&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Hydrodynamics; Magnetohydrodynamics

The proposed research is directed toward some models of turbulence in fluids and magnetofluids. Deterministic and statistical equilibrium theories of coherent structures in turbulence will be explored using both analytical and numerical methods. These theories, which complement the cascade-oriented concept of a driven and dissipative system, provide models for the long-time, large-scale behavior of flows and fields governed by conservative evolution equations. Their equilibrium solutions are characterized by variational principles, such as a maximum entropy principle with constraints dictated by the conservation laws for the governing dynamics. Theoretical tools and computational algorithms will be developed to solve the constrained optimization problems that arise from these general principles. A variety of fundamental, prototype problems derived from specific physical applications will be treated with these methods; they include, for instance, the organized vortex structures in a shear layer, and the most probable state of a magnetically-confined plasma. Most of the tractable problems of this kind are essentially two-dimensional; however, some aspects of the analogous three-dimensional problems will be also tackled using extensions of the same methods. The principal aim of this research is to develop a basic understanding of the physics of turbulence. The turbulent medium can be an ordinary fluid, such as water flowing in a pipe or air streaming past a wing; or, it can be a magnetized plasma, such as ionized gas in a thermonuclear fusion reactor. Simplified mathematical models of these complex physical problems are formulated in order to investigate some fundamental properties of the phenomena they exhibit. Particular attention is focussed on the coherent or organized structures that emerge and persist in the presence of turbulent disorder. These structures are studied quantitatively by deriving and then solving equations that captu re the coarse-grained behavior of the system while partially ignoring its fine-grained fluctuations. In this way, useful predictions can be made of physical systems having a very large number of degrees of freedom, which otherwise would be beyond the grasp of current methods of mathematical analysis or computer simulation. An interdisciplinary approach will be adopted in which abstract analytical tools, modern computing techniques, and solid physical reasoning are employed in a unified manner.

提出的研究是针对流体和磁流体中的湍流的一些模型。我们将使用解析和数值方法来探索湍流中相干结构的确定性和统计平衡理论。这些理论补充了以级联为导向的驱动和耗散系统的概念，为守恒演化方程所支配的流动和场的长期、大尺度行为提供了模型。它们的平衡解由变分原理来表征，例如具有由支配动力学的守恒定律所规定的约束的最大熵原理。将开发理论工具和计算算法来解决由这些一般原则引起的约束最优化问题。从具体的物理应用中衍生出的各种基本的、原型的问题将用这些方法来处理；例如，它们包括剪切层中的有组织的涡旋结构，以及磁约束等离子体的最可能的状态。大多数这类容易处理的问题本质上是二维的；然而，类似的三维问题的某些方面也将使用相同方法的扩展来处理。这项研究的主要目的是对湍流物理学有一个基本的了解。湍流介质可以是普通流体，如管道中流动的水或流经机翼的空气；也可以是磁化等离子体，如热核聚变反应堆中的电离气体。建立了这些复杂物理问题的简化数学模型，以研究它们所表现出的现象的一些基本性质。特别关注的是在混乱状态下出现并持续存在的连贯的或有组织的结构。这些结构的定量研究是通过推导和求解描述系统粗粒度行为而部分忽略细粒度涨落的方程来实现的。通过这种方式，可以对具有非常多自由度的物理系统做出有用的预测，否则这将超出当前的数学分析或计算机模拟方法的掌握。将采用跨学科方法，其中抽象分析工具、现代计算技术和固体物理推理将以统一的方式使用。