Applied Analysis of the Navier-Stokes and Related Equations

纳维-斯托克斯及相关方程的应用分析

• 批准号: 9900635
• 负责人: Charles Doering
• 金额: $ 18.5万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-06-01 至 2003-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9900635&HistoricalAwards=false
• 关键词: Applied; Analysis; Navier; Stokes; Related

This fundamental research in mathematical physics and applied mathematics focuses on the challenges presented by the incompressible Navier-Stokes and related equations of fluid dynamics. The Navier-Stokes equations constitute the basic mathematical model for fluid flow, and are believed to contain turbulent dynamics among their solutions. Turbulence in fluid mechanics remains one of the outstanding challenges for theoretical physics and applied mathematics with important applications in many fields of science and engineering. The work is carried out via modern applied and numerical analysis by the principal investigator and a mathematics graduate student doing doctoral dissertation work.The project has three specific objectives. The first is to extend a rigorous technique for deriving theoretical bounds on turbulent flow quantities to new applications for heat transport in Marangoni convection, and more generally to turbulence in fluid systems with imposed stress boundary conditions. Second, we aim to extend the background field method for applications to unbounded flow domains to derive theoretical limits on turbulent drag coefficients for flows past a compact body. It remains an open problem, for example, to establish a limit on the drag experienced by a sphere moving at high speed through a viscous fluid which is both mathematically rigorous and physically relevant. The third objective is to investigate small length scales appearing in turbulent flows by means of a set of dynamical equations derived for an analytic extension of solutions of the Navier-Stokes equations. Mathematical results in this area will produce strict lower bounds on the small length scales associated with high wavenumber exponential decay of the Fourier power spectrum in turbulent velocity fields.

这项在数学物理和应用数学方面的基本研究集中在不可压缩的Navier-Stokes和流体动力学相关方程式所面临的挑战上。 Navier-Stokes方程构成流体流的基本数学模型，并且被认为在其溶液中包含湍流动力学。 流体力学的湍流仍然是理论物理学和应用数学的重要挑战之一，这些数学在许多科学和工程领域都有重要应用。 这项工作是通过主要研究者和数学研究生进行博士学位论文工作的现代应用和数值分析进行的。该项目具有三个特定的目标。 首先是扩展一种严格的技术，用于将湍流数量的理论界限推导到Marangoni对流中的热传输的新应用，更通常是在具有应力边界条件的流体系统中进行湍流。 其次，我们旨在将应用程序的背景字段方法扩展到无界流量域，以在湍流阻力系数上的理论限制，用于流过紧凑的身体。 例如，要通过高速移动的球体通过粘性流体移动的球体所经历的阻力限制，这仍然是一个空旷的问题，这既是数学上严格又有物理上相关的。 第三个目标是通过一组得出的动力学方程来研究在湍流中出现的小长度尺度，这些方程是用于纳入Navier-Stokes方程解决方案的分析扩展的。 该区域中的数学结果将在与湍流速度场中傅立叶功率谱的高度波数指数衰减相关的小长度尺度上产生严格的下限。