In this paper, the local grid refinement is focused by using a multi-block technique. The Cartesian grid numerical method is developed for simulating unsteady, viscous, incompressible flows with complex immersed boundaries. A finite volume method is used in conjunction with a two-step fractional-step procedure. The key aspects that need to be considered in developing such a multi-block solver are imposition of interface conditions on the inter-block boundaries and accurate discretization of the governing equation in cells that are with block-interface as a control surface. A new interpolation procedure is presented which allows systematic development of a spatial discretization scheme that preserves the spatial accuracy of the underlying solver. The present multi-block method has been tested by two numerical examples to examine its performance in the two dimensional problems. The numerical examples include flow past a circular cylinder symmetrically installed in a Channel and flow past two circular cylinders with different diameters. From the numerical experiments, the ability of the solver to simulate flows with complicated immersed boundaries is demonstrated and the multi-block approach can efficiently speed up the numerical solutions.
In the numerical simulation of complex physical phenomena, the crucial requirement is predictability, i.e., that the simulation results remain faithful to the actual physical processes. Errors resulting from a lack of spatial resolution are particularly deleterious. However, over-resolving is computationally expensive. As a result, how to efficiently and effectively solve the partial differential equations which represent the mathematical model of physical problems concerned becomes a subject of active research in numerical analysis (Chacon and Lapenta, 2006, Ding and Shu, 2006). In general, there are two approaches to obtain accurate solution of PDEs. One approach is to employ high-order numerical method, and the other is to improve the resolution through the computational grid. Mesh refinement is desirable to improve spatial resolution by using uniform or non-uniform grids.
