极坐标系下谱元方法求解不可压缩Navier-Stokes方程

本文利用有限元的思想并结合谱方法的精度提出求解偏微分方程的谱元方法,并将谱元方法应用到极坐标系中;详细推导了在极坐标系下的谱元方法的具体计算公式,求解了极坐标系下的简单椭圆型二阶偏微分方程;并结合时间分裂方法应用于同心旋转圆筒间流体的流动问题,具体求解了原始变量速度和压力的不可压缩NavierStokes方程,均取得了满意的结果.

Spectral element method in polar coordinate for incompressible Navier-Stokes equations

A spectral element method that combines the ideas of the finite element method with the accuracy of spectral method is presented and applied in polar coordinate. The formulas of the spectral element method in polar coordinate are induced and described in detail. Spectral element method is a high-order weighted residual technique. In this method, the computational domain is broken into a series of elements, and the variable such as the velocity and pressure is represented as a high-order interpolation polynomial through Chebyshev collocation points, and then the element matrix is formed. Lastly, the system matrix is constructed by element stiffness matrix summation, and the answer is obtained by solving the linear equation set. Then the method is utilized to simulate the simple two order partial differential equation in polar coordinate in this paper. The incompressible Navier-Stokes equations expressed in terms of primitive variables velocity and pressure equation are computed. The time-splitting method for temporal discretization combining with the spectral element method for spatial discretization, were used to simulate the flow in the concentric vertical cylinders. The obtained results show to be in excellent agreement with the accurate results.