Euler solution using cartesian grid with least squares technique

Abstract

This paper discusses an approach that uses "gridless" or "meshless" methods to address the boundary or interface while standard structured grid methods are used everywhere else. The present method uses the Cartesian grid to specify and distribute the computational points on the boundary surface but not to define the geometrical properties. Euler fluxes for the neighbors of cut cells are computed using the gridless method involving a local least-squares curve fit to a "cloud" of grid points. The boundary conditions implemented on the surface points are automatically satisfied in the process of evaluating the surface values in a similar least-squares fashion. No halo points are needed. The overall scheme is robust, stable and converges well for a range of Mach numbers tested. Solutions from the proposed approach are computed for the NACA 0012 and RAE 2822 airfoils and the results are compared with those obtained by a standard Euler solver using body-fitted grids. For grids with equal resolution the method is less accurate for capturing shocks but an improvement in resolution of 60% gives a sharper shock front. The approach offers a number of advantages and its extension to three dimensions is straightforward. © 2003 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.