Adaptive Gridless Type Solver for Unsteady Flows

本文发展了一种新的无网格方法，用于求解Navier-Stokes方程和浅水波方程。在该方法中，点，而不是网格首先分布在计算域。为每个点选择点云。方程的空间导数在每个点处用某些系数和点云中的流量的线性组合进行评估。系数可以用最小二乘近似获得。该方法是有效的二阶精度。无粘项的计算采用迎风差分和中心差分两种方法。粘性项采用中心差分法求解。采用龙格-库塔法或带内迭代的LU-SGS隐式方法对方程进行时间离散。该方法可以很好地处理任意分布的点以及任意类型网格上的点。本文首先对可压缩Euler方程和Navier-Stokes方程进行了数值模拟。虽然该方法的守恒相容性可能是不确定的，该方法可以预测正确的激波强度和速度的可压缩欧拉方程。然后将该方法应用于具有人工可压缩性的不可压Navier-Stokes方程以及涡量流函数列式。用无网格方法对几个流动问题进行了数值计算，所得结果与已有的实验数据和其它数值计算结果进行了比较。对浅水方程的威廉姆森标准检验问题的数值结果与高精度谱变换方法的结果符合得很好。无网格方法在求解非定常流动问题时具有很好的可靠性和灵活性。

A novel gridless method, an alternative to conventional finite difference methods, has been developed for the Navier-Stokes equations and the shallow water equations. In the method, points instead of grids are first distributed over computational domains. A cloud of points are selected for each point. The spatial derivatives of the equations are evaluated at each point with linear combinations of certain coefficients and the flow quantities in the cloud of points. The coefficients can be obtained with least-square approximations. The method is effectively second order accurate. The inviscid terms are evaluated with either upwind or central difference manners. The viscous terms are evaluated with central difference manner. Runge-Kutta methods or LU-SGS implicit method with inner iterations are used for the temporal discretization of the equations. The method can work fairly well on points arbitrarily distributed as well as points of any kind of grids. The validity of the method is first examined for the compressible Euler and Navier-Stokes equations. Although the conservation consistency of the method may be uncertain, the method can predict the correct shock strength and speed of the compressible Euler equations. Then the method has been applied to the incompressible Navier-Stokes equations with artificial compressibility as well as vorticity-stream function formulation. The numerical results obtained with the gridless method for several flow problems are well compared with available experimental data and other numerical results. The numerical results obtained for Williamson's standard test problems of the shallow water equations agree well with those of a highly accurate spectral transform method. Reliability and flexibility of the gridless method for unsteady flow problems is quite satisfactory.