Efficient Numerical Methods for Unsteady Viscous Incompressible Flows

9805621粘性不可压缩流动的高效数值方法本项目将继续发展高效、高精度、高精度的非定常粘性不可压缩N-S方程的有限差分和有限元方法，重点是寻找更适合于数值计算的高效的时间推进格式和新的方程格式。这些方法的一些实际方面，包括在一般2D和3D领域的流的扩展，以及在更具挑战性的物理问题中的应用，将被研究。在涡量公式中，对高雷诺数流动采用了一种新的时间步长方法。对于这种流动，对流项和粘性项进行了显式处理。然后，通过运动学方程，根据涡度计算流函数，从而得到速度。这种新的时间步长方法的关键是不需要任何迭代就能从蒸汽函数中显式地得到边界上的涡量值。这消除了一些与涡度公式相关的传统困难。与原始变量公式更相似的是，研究人员在脉冲密度变量中使用了一种新的NSE公式，该公式不同于通过规范变换的速度。规范自由度使人们能够为冲量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。这类新的高效数值方法已经被用于研究实际问题，如高速翼型减阻机理的研究，以及热带地区强风暴的发展。这些方法提供了一个重要的工具，使科学家和工程师能够研究制造业和工业中的相关流体问题，这些问题以前是现有的数值技术无法解决的。它们代表了在有效计算此类问题的解方面向前迈出的重要一步，并且自然适合在高性能大规模并行计算机体系结构上实现。

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.