This work presents an immersed boundary (IB) technique for compressible, turbulent flows and applies the technique to simulate the effects of micro vortex generators (micro VGs) in controlling oblique-shock / turbulent boundary layer interactions. Both Reynolds averaged Navier-Stokes (RANS) and hybrid large-eddy / Reynolds-averaged Navier-Stokes (LES/RANS) turbulence closures are used with the IB technique. The approach is validated by comparing RANS and hybrid LES/RANS solutions obtained using the IB technique with RANS solutions obtained using a body-fitted mesh and with experimental laser Doppler anemometry (LDA) data collected at Cambridge University for Mach 2.5 flow over a single micro VG. Simulations of an impinging oblique shock / boundary layer interaction at Mach 2.5 with and without micro VG flow control are also performed, considering a.) the development of the entire flow in the wind tunnel using a RANS model and b.) an idealized, nominally two-dimensional interaction using both the RANS and the LES/RANS models Comparisons are made with experimental LDA data and surface pressure measurements from Cambridge University, and an analysis of the flow structure is performed. The results show that three-dimensional effects initiated by the interaction of the oblique shock with the sidewall boundary layers significantly influence the flow patterns in the actual experiment. The general features of the interactions with and without the micro VG array are predicted to good accord by the RANS / IB model. Results for the idealized interaction show the LES/RANS model captures a faster recovery of the separated boundary layer and a broader influence of the vortices generated by the micro VG array, compared with the RANS model.