Poiseuille-Bénard流的出口边界条件及其谱元法计算

研究二维矩形管道中底部加热的不可压缩Ｐｏｉｓｅｕｉｌｌｅ－Ｂｅｎａｒｄ流的谱元法数值计算问题．讨论各种不同的出口边界条件的处理及其对谱元法数值模拟的影响．通过干扰区、干扰幅度和计算时间的比较，确定比较理想的出口边界条件．

OPEN BOUNDARY CONDITIONS IN SIMULATION BY SPECTRAL ELEMENT METHODS OF POISEUILLE B(E)NARD CHANNEL FLOW

2D simulation of Poiseuille-Benard channel flow by a spectral element method isperformed. The main purpose is to compare the effect of different open boundary conditions(OBCs) upon simulation results. A new boundary condition is applied in the context of spectralresolution, for which a new treatment technique is used. The computation are carried out forRe=10, Ri=150 and Pr=2/3. Among selected OBCs, a so-called Orlanski-type OBCs isproven to have better behavior as compared with the other OBCs.The choices of OBCs depend on the numerical methods to be used in the computation. Givena boundary condition, different treatment is required by different numerical methods. Four typesof OBCs are considered in this paper: (1) periodic condition; (2) Dirichlet condition, =0; (3)Neumann condition, =0; (4) Orlanski-type condition, = 0, where arethe related variables (velocity or temperature in this paper), V is a constant to determine, n isthe outward normal. Orlajnski-type OBCs is a non standard boundary condition in the contextof spectral approximation. It can be viewed as an approach, based on the following hypothesis:diffusion or pressure gradient or the sum of both is small at the outlet, the velocity is thereforeapproximately equal to the one situated on the characteristics. In other words, Orlanski-type OBCsis a generalized Dirichlet boundary condition which is adjusted following the evolution in time ofthe flow. This motivates us to apply a characteristics method to treating the Orlanski OBCs.Classical characteristics methods, implemented in the spectral approximation, cause however somedifficulties: it depends on the exact localization of the characteristic foots and on the polynomialinterpolation. These two procedures are generally expensive and possibly instable in the case ofhigh order numerical methods. These considerations lead us to introduce a local interpolationtechnique, proved numerically stable and of high precision.The Uzawa algorithm has been used to solve the resulting discrete equations stemming fromthe spectral method. The Uzawa ajgorithm is more efficient in terms of computational complexityand memory requirement than a direct approach, but it is sensitive on the properties of the algebraicsystems. We not only analyze OBCs effects to the computational results, but also compare theCPU time needed by different. QBCs. Numerical.results show that Orlanski-type OBCs gives moreaccurate results than the other OBCs. Thanks to the local characteristics method, Orlanski-typeOBCs maintains the symmetry property of the decoupled pressure system, and hence well adaptedto the conjugate gradient iterative procedtire, which is not the case for the Neumann-type OBCs.