An Efficient Method for Flow Computations for Bodies With Curved Boundaries

Abstract

The difficulties of performing flow computations for bodies with arbitrary curved boundaries are well known. Many approaches have been advocated, but few have proven to be efficient, accurate, and simple to program. The earliest approach of using staircase boundaries to approximate curves is simple but crude, and would be prohibitively expensive if accurate, fine mesh results are desired. The use of meshes based on body-fitted curvilinear coordinates typically involve much more complex arithmetic and hence incur a high computational overhead. The inefficiency of curvilinear coordinates have spurred the current popularity of unstructured mesh formulations, which appear less restrictive but seem to lead to comparable overheads. The present paper describes a method based on returning to the intuitive concept of defining the boundary by placing boundary nodes on points of intersection of the boundary with the basic mesh lines. Such an arrangement creates locally acute non-uniform meshes which are well known to be ill-behaved. This is shown to be the consequence of singular behaviors in the limiting case of large “acuteness”, or the ratio of adjacent mesh increments. Differencing formulas and integration algorithms which are tolerant of such singularities, but retain the same order of accuracy as conventional nodes, are derived and used for all nodes adjacent to the boundary. In addition, algorithms for computing the normal and tangential gradients along the surface are also derived, based on vector combination of gradient components which can be easily evaluated from functions values at the nodes. Such gradients are needed for stress and pressure boundary conditions in flow computations. The method is demonstration by second order computations for the impersively started concentric cylinder Couette flow. It is shown that the method is efficient and produces accurate results.