Air flow in the wake region of a two-dimensional (plane) body with a blunt base has been studied using numerical simulations. The objective of this study is (1) to observe the behavior of large dynamic structures in the plane wake at several Mach numbers from low (almost incompressible) up to M = 2.46 and examine their effect on the base pressure, and (2) to address the nature of the instability in the shear layers bounding the wake flow at M = 2.46 and observe the structures that arise from this instability. A code was developed for this study which solves the compressible Navier-Stokes equations in two or three dimensions. This code may be used for either Direct Numerical Simulations (DNS) or Large Eddy Simulations (LES). A spatial model is used, with the computational domain arranged around the trailing edge of a two-dimensional flat plate with a blunt base. Two-dimensional simulations were carried out at Mach numbers of M = 0.25, M = 1.20, and M = 2.46. At all Mach numbers, the flow was found to be unstable with respect to sinuous (antisymmetric) disturbances, with the critical Reynolds number increasing with increasing Mach number. These disturbances grow to a periodic state, and a Karman vortex street is formed. Examination of the supersonic cases revealed that expansion fans in the flow at the corners are the primary cause of the low base pressure, and that disruptions in the expansions raise the base pressure. At M = 2.46 and Reynolds numbers starting at Re = 100, 000, an intermittent shear layer instability was also found, excited by sinuous disturbances. The two instability 2 modes interact to produce a chaotic behavior. Above Re = 200, 000, the shear layer instability appears close to the base without sinuous disturbances, forming rows of vortices in the shear layers. Preliminary three-dimensional simulations were carried out at M = 2.46, examining the variation in the growth rate of three-dimensional disturbances with spanwise wavelength.