This paper describes a methodology for the simulation of high Reynolds number flow over rigid and moving bodies. The calculations are performed on a structured Cartesian grid regardless of the geometric complexity or motion of the body. The approach is based on a modified version of the Brinkman Penalization method. This paper describes the implementation of a simple modified penalization method for compressible flows. To avoid oscillations in the vicinity of the body and to simulate shock-containing flows, a Weighted Essentially Non-Oscillatory scheme is used to discretize the spatial flux derivatives. Two turbulence models are used: the two-equation Menter SST Unsteady Reynolds-averaged Navier-Stokes model and a two-equation Detached Eddy Simulation. The method is implemented for both stationary and moving grids in a three-dimensional parallel code. Results are given for a grid dependence study, flow over airfoils at subsonic, transonic and supersonic conditions, and unsteady transonic flow over a cylinder and an airfoil at high angle of attack. The effect of the choice of turbulence model is shown. The moving grid capability is demonstrated for airfoils at a high angles of attack and a cylinder rotating at high speed. Finally, as a larger scale demonstration, the flow around a rotor blade with an active flap is simulated.