A numerical method for heat transfer and flow calculations with curved boundaries employing Cartesian grid.

Abstract

A simple and accurate numerical treatment of curved boundaries for the solution of typical transport problems and flow problems is described. The study is motivated by the importance of curved boundaries in applied and fundamental problems and the difficulties of performing computations at or near curved boundaries. Existing methods, including those employing unstructured or body-fitted meshes, are either clumsy, of low-order accuracy, or burdened with high computational overhead. Our method employs the finite difference method and places boundary nodes on the intersection of curved boundaries with the Cartesian mesh lines. Those boundary nodes permit exact description of the boundary location and precise satisfaction of the boundary conditions. Such an arrangement creates locally skewed meshes that are well known to be ill-behaved when boundary nodes are too close to the nearest interior nodes. Difference formulas and integration algorithms that are skew-tolerant and retain the same order of accuracy as in conventional nodes, are derived and used for all interior nodes adjacent to the boundary. The method is thoroughly tested by calculating several classes of problems with curved boundaries, including a heat conduction problem, advection of passive scalars in a given flow field, and three flow problems, with fixed and moving curved boundaries. Comparisons of numerical results to analytical solutions and published computations in the literature show that strict second- and third-order accuracy are achieved with the method.