Abstract
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.
Highlights
IntroductionFluid flow through irregular geometries (that is, geometries that do not conform to known coordinate lines) is encountered in many natural and industrial phenomena
Fluid flow through irregular geometries is encountered in many natural and industrial phenomena
We present a discretization of the computational domain and derive third and fourth order accurate standard finite difference schemes for the first derivative and we test these schemes on a uniform grid and four clustered grids
Summary
Fluid flow through irregular geometries (that is, geometries that do not conform to known coordinate lines) is encountered in many natural and industrial phenomena. The above reported previous work, [5] [16], remained silent about the use of schemes of higherorder local accuracy, and the impact of using clustering in the physical domain on the computed vorticity on the boundary This motivates the current work in which we develop and validate finite difference expressions, for the first derivative, that can be used in the accurate evaluation of vorticity on the flow domain boundary. The intention is to develop a fourth-order scheme of local accuracy to approximate the vorticity on the boundary for both uniform and non-uniform grids using five grid-point stencils, and to address the following points when transforming a curvilinear physical domain into a rectangular computational domain: - The type of grid spacing used: Typically, clustered grid near the boundary is expected to produce more accurate results. For the first derivative and we test these schemes on a uniform grid and four clustered grids
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