Computational fluid dynamics was used to study the flow through a scaled, mixed-compression, high-speed inlet with a rotating cowl at Mach 4.0 conditions. First, steady Reynolds-averaged Navier-Stokes computations were undertaken with a range of popular turbulence models including the Spalart-Allmaras model, the realizable k-ε model, the cubic k-ε model, and the Menter shear stress transport (SST) model to assess the impact on the inlet operating state. It was found that two models, the Spalart-Allmaras model and the Menter SST model, predicted the inlet to become unstarted at a cowl angle where the experimental data indicated the inlet remained started. Next, steady-state flow structures were studied at three discrete cowl positions, identifying highly three-dimensional flow features including regions of separated flow and spanwise gradients that became stronger as the cowl opened. Finally, the study culminated with the development of the new transient model, which allowed for time-accurate investigation into the unstart, restart, and hysteresis. The evolution of the separation bubbles was shown to be a major factor in the hysteresis, causing the inlet to restart at an angle different from where it unstarted. The utility of unsteady Reynolds-averaged Navier-Stokes computations to capture the complex time-dependent details of such flows was demonstrated.