The feasibility of operating diode pumped alkali lasers (DPALs) with supersonic expansion of the gaseous laser mixture, consisting of alkali atoms, He atoms and (frequently) hydrocarbon molecules, is explored. Taking into account fluid dynamics and kinetic processes, both semi-analytical and three-dimensional (3D) computational fluid dynamics (CFD) modeling of supersonic DPALs is reported. Using the semi-analytical model, the operation of supersonic DPALs is compared with that measured and modeled in subsonic lasers for both Cs and K. The maximum power of supersonic Cs and K lasers is found to be higher than that of subsonic lasers with the same resonator and alkali density at the laser inlet by 25% and 70%, respectively. Using the 3D CFD model, the flow pattern and spatial distributions of the pump and laser intensities in the resonator are calculated for Cs DPALs. Comparison between the semi-analytical and 3D CFD models for Cs shows that the latter predicts much larger maximum achievable laser power than the former. These results indicate that for scaling-up the power of DPALs, supersonic expansion should be considered.