High-Order Finite Difference Schemes for the First Derivative in Von Mises Coordinates

Third- and fourth-order accurate finite difference schemes for the first derivative of the square of the speed are developed, for both uniform and non-uniform grids, and applied in the study of a two-dimensional viscous fluid flow through an irregular domain. The von Mises transformation is 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 schemes are used to calculate the vorticity at the computational boundary grid points using up to five computational domain grid points. In all schemes developed, we study the effect of coordinate clustering on the computed results.