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

大雷诺数非定常粘性不可压缩流动的高效数值方法刘建国项目概述：拟发展和分析大雷诺数边界下非定常粘性不可压缩流动的高效、准确、可靠的数值方法，希望能直接模拟壁面湍流。三维不可压缩流动的有效数值计算目前还远远落后于实际需要。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悬而未决的边界存在，如正确的数值边界条件的涡度和有效的三维公式。该研究者和他的合作者Weinan E.已经发现了不同类型涡度边界公式之间的一些联系，例如局部通过全局和MAC方案通过Thom公式，通过明确处理涡度项和高阶（本质上）紧致差分方法，他们最近在涡度流函数（矢量）公式中引入了一个非常有效的四阶格式，以及一个完整的收敛理论。研究者建议继续使用涡度流函数和涡度速度公式来开发有效和精确的三维计算数值方法。研究者还计划系统研究一些二维和三维壁面型湍流的统计行为和相干结构，如驱动腔流、后向台阶流等，并与均匀各向同性湍流模拟进行详细的数值比较。在无壁流动中各向同性湍流的直接模拟方面已经做了大量的工作，而涉及有壁湍流的工作就少得多。然而，大多数湍流是在壁上产生的。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）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对湍流实验和理论研究的有效补充。本课题主要研究两个问题：(1)研究固体边界区域，特别是大雷诺数区域内粘性不可压缩流动的高效、鲁棒和可靠的数值计算方法。(2)利用这些方法通过直接数值模拟研究了二维和三维壁面湍流的统计行为和相干结构。

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.