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.

大雷诺数非稳态粘性不可压缩流的高效数值方法刘建国项目摘要：研究者提出开发和分析存在大雷诺数边界的非稳态粘性不可压缩流的高效、准确和可靠的数值方法，以期直接模拟壁面生成的湍流。三维不可压缩流的高效数值计算目前远远落后于实际需求。事实上，只要存在边界，即使是制定数值方法时的一些非常基本的问题也无法解决，例如涡度的正确数值边界条件和有效的 3-D 公式。 本研究人员和他的合作者Weinan E已经发现了不同类型的涡度边界公式之间的一些联系，例如局部通过全局和MAC格式通过Thom公式。通过对涡度项的明确处理和高阶（本质上）紧致差分方法，他们最近在涡度流函数（矢量）公式中引入了一种非常有效的四阶格式以及完整的收敛理论。研究人员建议继续开发使用涡量流函数和涡量速度公式进行三维计算的高效且准确的数值方法。 研究者还计划对一些2D和3D壁面湍流的统计行为和相干结构进行系统研究，例如驱动空腔流、向后台阶流等，并与均匀各向同性湍流模拟进行详细的数值比较。在直接模拟无壁流动中的各向同性湍流方面已经进行了大量工作，而涉及壁界湍流的工作则少之又少。 然而，大多数湍流是在壁处产生的。 与宇宙的大尺度结构和亚原子粒子的性质等其他突出的科学问题相比，理解湍流是一个巨大的挑战。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影响日常生活许多方面的广泛工程应用和技术进步。纳维-斯托克斯方程的直接数值模拟是补充湍流实验和理论研究的有效工具。 研究者建议研究该领域的两个问题：（1）开发有效、稳健和可靠的数值方法，用于具有固体边界的域中的粘性不可压缩流，特别是在雷诺数较大的情况下。 (2) 利用这些方法通过直接数值模拟研究二维和三维壁面湍流的统计行为和相干结构。