谱元法和高阶时间分裂法求解方腔顶盖驱动流

详细推导了谱元方法的具体计算公式和时间分裂法的具体计算过程 ;对一般的时间分裂法进行了改进 ,即对非线性步分别用 3阶 Adams-Bashforth方法和 4阶显式 Runge-Kutta法 ,粘性步采用 3阶隐式 Adams-Moulton形式 ,提高了时间方向的离散精度 ,同时还改进了压力边界条件 ,采用 3阶的压力边界条件 ;利用改进的时间分裂方法分解不可压缩 Navier-Stokes方程 ,并结合谱元法计算了移动顶盖方腔驱动流 ,提高了方法可以计算的 Re数 ,缩短了达到收敛的时间 ,并将结果与基准解进行比较 ;分析了移动顶盖方腔驱动流中 Re数对流场分布的影响。

Spectral element method and high order time splitting method for navies stokes equation

The calculational formula of the spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is induced in detail in this paper. The spectral element method is a high order weighted residual technique. In this method, firstly, the computational domain is broken into a series of elements, and the variable such as the velocity and the pressure is represented as a high order interpolation polynomial through Chebyshev collocation points, and then the element matrix is formed by finite element method. Lastly, the system matrix is constructed by element stiffness matrix summation, and the answer is obtained by solving the linear equation set. A formulation for splitting methods is also developed that results in high order time accurate schemes for the solution of incompressible Navier Stokes equations. The normal time splitting method is improved by applying third order explicit Adams Bashforth method and fourth order Runge Kutta method to the nonlinear connective terms, and third order implicit Adams Molten method to the linear terms. And the third order pressure boundary conditions for Possion equation for the pressure is employed as well. Using the high order time splitting method and high order pressure boundary conditions, the precision of the time discretization is advanced, and we can calculate the flow by longer time step to improve the efficiency of calculation. The advance of the precision for the time discretization is also useful for high Re fluid calculation. Then the high order time splitting method to split the incompressible Navier Stokes Equation, combining with the spectral element method, is utilized to simulate one typical sample in computational fluid dynamics, lid driven flow in closed square cavity. In the case of the same calculational accuracy and the same grid, the paper increase the Reynolds number from Re=100 that can be obtained using the normal time splitting method and the spectral element method to Re=600, and the time of convergence is shortened by using longer time step. The results show desirable agreement with the accepted benchmark solutions. Lastly, the changes of the field of velocity are analyzed according to different Reynolds Numbers. The validity of methods for fluid flow in this paper is proved.