Cartesian grid method for the compressible Euler equations using simplified ghost point treatments at embedded boundaries

•Two new ghost point approaches are introduced.•These approaches use fluid points as mirror points without interpolation.•Supersonic flow over simple to complex embedded boundaries is presented.•A good agreement is found when compare to the results in the literature. We introduce two new approaches called the simplified and the modified simplified ghost point treatments for solving the 2D compressible Euler equations near embedded boundaries for the Cartesian grid method. These approaches are second order accurate for second order schemes near the embedded boundaries, if the wall boundary is in the middle between fluid and ghost points. We assign values to the ghost points near embedded solid boundaries from mirror points in the fluid to reflect the presence of the solid boundaries. In the simplified ghost point treatment, we consider the closest grid points on the grid lines through the ghost points in the x- and y-directions as the mirror points of the ghost points depending on which directions are closest to the directions normal to the embedded boundaries. In the modified simplified ghost point treatment, we choose mirror points not only on the grid lines through the ghost points in the x- or y-directions, but also on the diagonals through the ghost points. The primitive variables at the mirror points are mirrored to the ghost points using local symmetry boundary conditions. The simplified ghost point treatments at embedded boundaries are tested for supersonic flow over a circular arc airfoil and a circular cylinder. Applications to supersonic flow over multiple circular cylinders and a 2D model of the F-22 fighter aircraft demonstrate the flexibility of the ghost point treatments. Another advantage of these new approaches is that they are easily extendable to higher order methods and to 3D. The Cartesian grid method requires a larger number of grid points than the standard body-fitted grid method. We found a good agreement between the results obtained with the simplified and the modified simplified ghost point treatments and the reference solutions in the literature.