Wall-modeling in LES is crucial to allow scale-resolving simulations of turbulent flows in industrial-scale devices. Numerous models have been developed and validated for incompressible flows, including the simple quasi-analytical model based on the Reichardt formula to approximate the "law of the wall." This work introduces a new scaling of the Reichardt formula to account for strong compressibility effects in wall-modeled LES (wmLES). The new wall model is validated against wall-resolved LES for three adiabatic turbulent channel flow cases at moderate Reynolds numbers, with increasing Mach numbers, ranging from the quasi-incompressible regime to the supersonic regime. The wall model is then used to perform a high-fidelity wmLES of a planar supersonic air ejector, using periodic spanwise boundary conditions. Results are compared to 2D RANS simulations and experimental data. The structure of the mixing layers is analyzed and the postprocessing tools using total exergy fluxes are presented. Discrepancies between RANS and LES are explained through an analysis of turbulent fluctuating contributions in both frameworks. Finally, the design of a new cylindrical supersonic ejector experiment is presented. Experimental characteristic curves are compared to axisymmetric RANS simulations. While RANS simulations predict the global behavior quite well in on- and off-design operations, large deviations are observed between experimental and numerical wall-pressure profiles. To gain additional insight into the flow physics, and also help explain the observed differences, a high-fidelity wmLES of the full ejector is performed for an operating point at the end of the on-design regime, just before the critical point. Results are analyzed and used to explain the consistent wall-pressure profile underestimation in RANS simulations, using an extension of the compound-choking theory.