复杂边界旋转Navier-Stokes方程的几何方法及二度并行算法  

A Geometry Method for Rotating Navier-Stokes Equations with Complex Boundary and Bi-Parallel Algorithm

作　　者：李开泰[1] 于佳平[1] 刘德民[2] 

机构地区：[1]西安交通大学理学院,西安710049 [2]新疆大学数学与系统科学学院,乌鲁木齐830046

出　　处：《应用数学学报》2012年第1期1-41,共41页Acta Mathematicae Applicatae Sinica

基　　金：国家自然科学基金(10971165;10771167;10871156;11001216);国家高科技研究发展计划(2009AA01A135);成都飞机设计研究院资助项目

摘　　要：本文提出了一种求解复杂边界旋转Navier-Stokes方程的微分几何方法及其二度并行算法.此方法可用于求解透平机械内部叶片间流动和飞行器外部绕流等复杂流动问题.假设流动区域可以用一系列光滑曲面■_k,k=1,2,…,K分割为一系列子区域(称作流层),通过应用微分几何的方法,三维N-S算子可以分解为两类算子之和:建立在曲面■_k切空间上"膜算子"和曲面■_k法线方向的"挠曲算子",将挠曲算子应用欧拉中心差商来逼近,由此得到建立在■_k上的"2D-3C"N-S方程.求解2D-3C N-S方程并且反复迭代直到收敛.我们得到"二度并行算法",它是2D-3C N-S方程并行算法与k方向的同时并行.这个算法的优点在于,(1)可以改进由于复杂边界造成的不规则三维网格引起的逼近解的精度;(2)为克服边界层的数值效应,在边界层内可以构造很密的流层,形成三维多尺度的网格,是一个很好的边界层算法;(3)这个方法不同于经典的区域分解算法,这里的每个子区域只需要求解一个"2D-3C"N-S方程,而经典区域分解方法要在每个子区域上求解三维问题.In this paper, a new algorithm based on the differential geometry to solve the 3D rotating Navier-Stokes equations with the complex Boundary is proposed. This algo- rithm possesses two parallel functions along two directions. Hence it is called the bi-parallel algorithm. This method can be applied to the channel flow between two blades in the turbomachinery, the circulation flow through the aircrafts with the complex geometric shape of the boundary. Assume that a domain in E^3 can be decomposed into a series of the subdomains, which is called ＂the flow layer＂, by a series of the smooth surfaces k. Applying the differential geometry method, the 3D Navier-Stokes operator can be split into two kind of the operators： the ＂membrane＂ operator based on tangent space at the surface and ＂bending＂ operator along the transverse direction. The bending operators are approximated by the finite difference quotients. When restricting 3D Naver-Stokes equations on the interface surface k, a bi-parallel algorithm can be constructed along two directions： the ＂bending＂ transverse and the ＂membrane＂ direction. The advantages of the method are that （1） it can improved to approximate the solution of the boundary layer; （2） It is easy to generate two-level meshes; （3） This method is different from the classical domain decomposition method, it is sufficiency to solve the two dimensional sub-problems without solving the 3D sub-problems.

关 键 词：旋转Navier—Stokes方程 微分几何方法 二度并行算法 

分 类 号：O212.7[理学—概率论与数理统计]