Mathematical Sciences: Efficient Numerical Methods for Large Reynolds Number Unsteady Viscous Incompressible Flows

数学科学：大雷诺数不稳定粘性不可压缩流的有效数值方法

• 批准号: 9505275
• 负责人: Jian-Guo Liu
• 金额: $ 5.81万
• 依托单位: Temple University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-15 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9505275&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Efficient; Numerical; Methods

EFFICIENT NUMERICAL METHODS FOR LARGE REYNOLDS NUMBER UNSTEADY VISCOUS INCOMPRESSIBLE FLOWS Jian-Guo Liu Project Summary: The investigator proposes to develop and analyze efficient, accurate, and reliable numerical methods for unsteady viscous incompressible flows in the presence of boundaries with large Reynolds number, in hopes of simulating directly wall-generated turbulence. Efficient numerical computations of 3-D incompressible flow are currently far behind the practical needs. Indeed, even some very basic issues in formulating a numerical method are unsettled whenever boundaries are present, such as the correct numerical boundary conditions for the vorticity and efficient 3-D formulation. Some connections between different types of vorticity boundary formulations, such as local via global and the MAC scheme via Thom's formula, have already been found by this investigator and his collaborator Weinan E. With explicit treatment of the vorticity term and high-order (essentially) compact difference approach, they recently introduced a very efficient fourth order scheme in vorticity-stream function (vector) formulation, together with a complete convergence theory. The investigator proposes to continue developing efficient and accurate numerical methods for the three dimensional computation using both the vorticity-stream function and vorticity-velocity formulations. The investigator also plans to conduct a systematic study of statistical behavior and coherent structure of some 2-D and 3-D wall-bounded turbulence flows such as the driven cavity flow, backward-facing step flow, etc. and carry out a detailed numerical comparison with the homogeneous isotropic turbulence simulations. Extensive work has been done on direct simulation of isotropic turbulence in wall-free flows, much less work has involved wall-bounded turbulence. However, most turbulence is generated at the walls. Understanding turbulent flows is a grand challenge comp arable to other prominent scientific problems such as the large-scale structure of the universe and the nature of subatomic particles. In contrast to many of the other grand challenges, progress on the basic theory of turbulence translates nearly immediately into a wide range of engineering applications and technological advances that affect many aspects of everyday life. Direct numerical simulation for Navier-Stokes equation is an effective tool that complements experimental and theoretical investigation of turbulence. The investigator proposes to study two problems in the field: (1) Developing efficient, robust and reliable numerical methods for viscous incompressible flows in domains with solid boundaries, especially in the cases with large Reynolds number. (2) Using these methods to study statistical behavior and coherent structure of two and three dimensional wall-bounded turbulence via direct numerical simulations.

大雷诺数的有效数值方法 非定常粘性不可压缩流 刘建国 项目概要： 研究人员建议开发和分析有效的，准确的，可靠的数值方法，非定常粘性不可压缩流的存在下，具有大雷诺数的边界，希望直接模拟壁产生的湍流。三维不可压缩流动的有效数值计算目前远远不能满足实际需要。事实上，即使是一些非常基本的问题，在制定一个数值方法是未解决的边界，如正确的数值边界条件的涡量和有效的3-D制定。 本研究者和他的合作者Weinan E. 随着涡量项的显式处理和高阶（本质上）紧致差分方法，他们最近在涡流函数（矢量）公式中引入了一个非常有效的四阶格式，以及一个完整的收敛理论。研究人员建议继续开发有效和准确的数值方法，用于使用涡量流函数和涡量速度公式进行三维计算。 研究者还计划对一些二维和三维壁面湍流（如驱动空腔流、后台阶流等）的统计行为和相干结构进行系统的研究，并与均匀各向同性湍流模拟进行详细的数值比较。对于无壁流动中各向同性湍流的直接模拟已经做了大量的工作，而对于有壁湍流的模拟工作则少得多。 然而，大多数湍流产生在壁处。 理解湍流是一个巨大的挑战，与其他突出的科学问题，如宇宙的大尺度结构和亚原子粒子的性质。与许多其他重大挑战相比，湍流基本理论的进展几乎立即转化为广泛的工程应用和技术进步，影响日常生活的许多方面。Navier-Stokes方程的直接数值模拟是湍流实验和理论研究的有效补充。 本文主要研究两个方面的问题：（1）发展高效、鲁棒、可靠的数值方法，求解具有固体边界的粘性不可压缩流场，特别是大雷诺数情况下的粘性不可压缩流场。(2)利用这些方法对二维和三维有壁湍流的统计行为和相干结构进行了直接数值模拟。