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边界拟合坐标系下的差分有限元破开算子法 被引量:3

AN EFFICIENT OPERATOR-SPLITTING METHOD COUPLINGFDM AND FEM FOR THE NAVIER-STOKES EQUATION UNDERBOUNDARY-FITTED COORDINATE SYSTEM
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摘要 在边界拟合坐标系下给出耦合了有限差分法和有限单元法的新型的破开算子法.利用该方法,Navier-Stokes方程被分解成对流方程和扩散方程,对流方程采用稳定性好的有限差分法求解,而扩散方程则采用有限单元法求解.由于计算是基于非均匀网格,采用边界加密的51×51网格就达到了前人在计算雷诺数为5000的方腔环流时采用的257×257均匀网格的效果,对于瑞利数Rt=107的竖腔自然对流的计算进一步表明,提出的方法是有效的.对于Re=105的高雷诺数方腔受驱环流,得到了非稳态、非周期的和带有随机特征的流场结构. An operator splitting numerical method combining finite difference method (FDM) and finite element method(FEM) is proposed in this paper by using boundary-fitted coordinate system. The governing equation is split into convection and diffusion equations and solved by finite difference method and finite element method, respectively. The Navier-Stokes equations are expressed in terms of stream function and vorticity. Using this method, a cavity flow was simulated at Re = 5000, with 51×51 grids, and the numerical results agree very well with the computed result by Chia et al. using 257×257 grid. To further test the proposed method, a nature convection problems with horizontal temperature gradients was used. The numerical result agrees very well with the previous results. The high Reynolds numbers flows were investigated by the author at Re = 30000,60000 and 105, unsteady and non-periodic flows have been obtained.The basic idea of the operator-splitting method with FDM and FEM using boundary-fitted coordinate system have been proposed by Zhan and Zhang (1988) for the convection and diffusion problems of heated discharge acted on by tide and current. When solving a convection-diffusion equation, if one uses only the finite difference method, he will have the difficulty in treating boundary conditions, especially when the computational region is not irregular. If he uses only the finite element method, he will have the difficulty in treating the convection term because of the numerical instability. Using the proposed method here, the difficulties can be overcome. First, using time-splitting method, the convection-diffusion equation can be split to two equations: the convection equation and the diffusion equation. Then under boundary-fitted coordinate system, the convection equation and the diffusion equation can be solved by using the up-wind scheme and nine-node Lagrangian quadratic isoparametric elements of FEM, respectively. In this paper, the convection-diffusion equation for the vorticity has been solved using the operator-splitting method and the Poisson equation for the stream function using FEM. It can be seen that the coupled FDM-FEM method is stable and efficient method and contains the advantages both FDM and FEM. It is an ideal method in both theory and practice.In the last two decades, a large number of operator-splitting algorithms have been reported for the solution of the convection-diffusion equation. For example, an operator splitting algorithm for the numerical solution of a coupled system of convection diffusion-reaction equations, governing the transport of non-pollutants in porous media is presented by Khan and Liu (1995) using a backward method of characteristic, a finite-element method and explicit Runge-Kutta method.
作者 詹杰民
出处 《力学学报》 EI CSCD 北大核心 2002年第4期616-621,共6页 Chinese Journal of Theoretical and Applied Mechanics
基金 国家自然科学基金(19802023)资助项目
关键词 边界拟合坐标系 有限差分法 有限单元法 破开算子法 非稳态流 boundary-fitted coordinate system, FDM, FEM, operator-splitting method, unsteady flow
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