Adaptive Gridless Type Solver for Unsteady Flows

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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.