First-order accurate finite difference schemes for boundary vorticity approximations in curvilinear geometries

In this work we develop first-order accurate, forward finite difference schemes for the first derivative on both a uniform and a non-uniform grid. The schemes are applied to the calculation of vorticity on a solid wall of a curvilinear, two-dimensional channel. The von Mises coordinates are used to transform the governing equations, and map the irregular domain onto a rectangular computational domain. Vorticity on the solid boundary is expressed in terms of the first partial derivative of the square of the speed of the flow in the computational domain, and the derived finite difference schemes are used to calculate the vorticity at the computational boundary grid points using combinations of up to five computational domain grid points. This work extends previous work (Awartani et al., 2005) [3] in which higher-order schemes were obtained for the first derivative using up to four computational domain grid points. The aim here is to shed further light onto the use of first-order accurate non-uniform finite difference schemes that are essential when the von Mises transformation is used. Results show that the best schemes are those that use a natural sequence of non-uniform grid points. It is further shown that for non-uniform grid with clustering near the boundary, solution deteriorates with increasing number of grid points used. By contrast, when a uniform grid is used, solution improves with increasing number of grid points used.