This work examines numerical simulation of supersonic, subsonic, and transonic flows over a bluff body using several numerical techniques. Comparisons between computational fluid dynamics predictions and wind-tunnel test data are shown. The wind-tunnel test used a 7.66%-scale model of the crew module for NASA's multipurpose crew vehicle. A variety of freestream conditions consisting of three Mach numbers and three angles of attack are considered. The wind-tunnel test data include time-averaged integrated forces and moments, static pressure port measurements, and unsteady frequency response. Results obtained using the Spalart-Allmaras Reynolds-averaged Navier-Stokes turbulence model are compared to solutions computed using detached-eddy simulation. Furthermore, low-dissipation numerical fluxes are applied in order to assess their effect on solution fidelity. Across the cases examined, the Reynolds-averaged Navier-Stokes model has difficulty matching the vehicle surface pressures. For supersonic and transonic flows, detached-eddy simulation agrees well with the experiment. Subsonic flows still prove difficult for the methods explored in this work, but detached-eddy simulation and low-dissipation numerical fluxes provide the most accurate prediction of the test data.