Adaptive Gridless Type Solver for Unsteady Flows

针对Navier-Stokes方程和浅水方程，提出了一种新的无网格方法，替代了传统的有限差分方法。在该方法中，首先将点而不是网格分布在计算域上。为每个点选择一个点云。方程的空间导数在每个点上用一定的系数和点云中流量的线性组合来计算。系数可以用最小二乘近似求得。该方法具有有效的二阶精度。不粘项用逆风或中心差法计算。粘性项用中心差分法求值。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化。该方法可以很好地处理任意分布的点以及任何类型的网格点。首先对可压缩Euler方程和Navier-Stokes方程验证了该方法的有效性。虽然该方法的守恒一致性可能存在不确定性，但该方法可以准确预测可压缩欧拉方程的冲击强度和速度。然后将该方法应用于具有人工可压缩性的不可压缩Navier-Stokes方程以及涡流函数的表述。对几个流动问题，用无网格法得到的数值结果与已有的实验数据和其他数值结果进行了比较。浅水方程的Williamson标准测试问题的数值结果与高精度谱变换方法的结果吻合较好。无网格法求解非定常流问题的可靠性和灵活性令人满意。

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.