Efficient Numerical Methods for Viscous Incompressible Flows

粘性不可压缩流的高效数值方法

• 批准号: 0512176
• 负责人: Jian-Guo Liu
• 金额: $ 28.34万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0512176&HistoricalAwards=false
• 关键词: Efficient; Numerical; Methods; Viscous; Incompressible

The investigator develops a class of efficient and accurate numericalmethods for the unsteady viscous incompressible Navier-Stokesequations (NSE) based a new unconstrained formulation of NSE withfully dissipation in contrast to the traditional formulation where theStokes operator is dissipative only in the divergence freefields. This class of NSE solver is unconditionally stable withexplicit treatment of both pressure and convection terms. Moreover, inthis class of finite element methods, there is no requirement of theso called inf-sup condition. The cost of solving NSE is greatlyreduced to solving a standard heat equation and a standard Poissonequation at each time step for general three dimensional fluid problems.The simplicity of the method also enable the PI to develop a class ofnumerical methods for complex fluids such as magneto-hydrodynamics,liquid crystal polymers, geodynamo, climate modeling, and large eddyturbulence simulations;Computational Fluid Dynamics has grown from a mathematical curiosityto become an essential tool in almost every branch of fluid dynamics,from aerospace propulsion to weather prediction and has receivedextensive attention throughout the international community since theadvent of the digital computer. The accuracy and efficiency of theproposed schemes will allow us to simulate general three-dimensionaltime-dependent flows with a reasonable turn-over time. It is expectedthat the proposed fast algorithms will become an important tool formany scientists and engineers in numerous scientific and industrialapplications of current interest.

研究人员发展了一类求解非定常粘性不可压缩Navier-Stokes方程(NSE)的高效而精确的数值方法，该方法基于一种新的完全耗散的无约束形式，而不是传统的Stokes算符只在散度自由场中耗散的形式。这类NSE求解器是无条件稳定的，同时显式处理了压力项和对流项。此外，在这类有限元方法中，没有所谓的inf-sup条件。对于一般的三维流体问题，求解NSE的成本大大降低为在每个时间步求解一个标准的热方程和一个标准的Poisson方程。该方法的简单性还使PI能够开发一类用于复杂流体的数值方法，如磁流体动力学、液晶聚合物、地质发电机、气候模拟和大涡湍流模拟；计算流体动力学已经从一个数学奇闻成长为几乎每个流体动力学分支的基本工具，从航空航天推进到天气预报，自从数字计算机出现以来，它在国际上受到了广泛的关注。所提出的格式的精度和效率将使我们能够以合理的翻转时间模拟一般的三维时变流动。预计所提出的快速算法将成为科学家和工程师在当前感兴趣的众多科学和工业应用中的重要工具。