Efficient Numerical Methods for Unsteady Viscous Incompressible Flows

DMS-9805621 粘性不可压缩流动的有效数值方法 刘建国 项目摘要 本项目的重点是继续发展非定常粘性不可压缩Navier-Stokes方程（NSE）的高效、精确、高阶有限差分和有限元方法，重点是寻找更适合数值计算的有效时间步长程序和方程的新公式。 一些实际方面的方法，包括扩展到一般的2D和3D域的流，和应用程序更具挑战性的物理问题，将进行调查。 在涡量公式中，一个新的时间步进程序用于高雷诺数流动。 对于这样的流动，对流和粘性项显式处理。流函数，因此速度，然后通过运动学方程从涡量评估。 新的时间步进方法的效率的关键是边界上的涡量的值是显式地获得从蒸汽函数没有任何迭代。 这消除了与涡量公式有关的一些传统困难。 更类似于原始变量公式，研究人员在脉冲密度变量中使用了新的NSE公式，该公式通过规范变换与速度不同。规范自由度使人们能够为脉冲场和规范场指定简单而具体的边界条件，从而消除了一些传统的困难，如压力边界条件。 这类新的有效的数值方法已经被用来研究诸如高速翼型减阻机理的研究以及热带地区强风暴的发展等真实的世界问题。 这些方法提供了一个重要的工具，使科学家和工程师研究相关的流体问题，在制造业和工业，以前无法解决目前可用的数值技术。 他们代表了一个重要的一步，在有效的计算解决这些问题，自然是适合高性能大规模并行计算机架构上的实施。

DMS-9805621 EFFICIENT NUMERICAL METHODS FOR VISCOUS INCOMPRESSIBLE FLOWS Jian-Guo Liu Project Summary This project will focus on continuing the development of efficient, accurate, high order finite difference and finite element methods for the unsteady viscous incompressible Navier-Stokes equations (NSE), with emphasis on finding efficient time stepping procedures and new formulations of the equations better suited to numerical computation. Some practical aspects of the methods, including extensions to flows in general 2D and 3D domains, and applications to more challenging physical problems, will be investigated. In the vorticity formulation, a new time-stepping procedure is used for high Reynolds number flows. For such flows the convection and viscous terms are treated explicitly. The stream function, and hence the velocity, is then evaluated from the vorticity via the kinematic equation. The key to the efficiency of the new time-stepping procedure is that the value of the vorticity on the boundary is obtained explicitly from the steam function without any iteration. This eliminates some traditional difficulties associated with the vorticity formulation. More akin to the primitive variable formulation, the investigator is using a new formulation of the NSE in the impulse density variable which differs from the velocity by a gauge transformation. The gauge freedom enables one to assign simple and specific boundary conditions for both the impulse and gauge fields, thus eliminating some traditional difficulties such as the pressure boundary condition. This new class of efficient numerical methods has already been used to study such real world problems as the investigation of the mechanism of drag reduction on airfoils at high velocities, as well as the development of severe storms in tropical latitudes. These methods provide an important tool that allow scientists and engineers to study related fluid problems in manufacturing and industry that were previous unsolvable with currently available numerical techniques. They represent a significant step forward in the efficient computation of solutions to such problems, and are naturally suited for implementation on high performance massively parallel computer architectures.