Euler Solution Using Cartesian Grid with a Gridless Least-Squares Boundary Treatment

An approach that uses gridless or meshless methods to address the problem of boundary implementation associated with the use of Cartesian grid is discussed. This method applies the gridless concept only at the interface, whereas a standard structured grid method is used everywhere else. The Cartesian grid is used 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 of 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. The present method does not require the use of halo points. Subsonic, transonic, and supersonic flows are computed for the NACA 0012 and RAE 2822 airfoils, and the results compare well with solutions obtained by a standard Euler solver on body-fitted grids. The method is also used to calculate the flow over a three-element airfoil configuration, and the result is compared with the exact solution for this configuration obtained by conformal mapping.