We report on recent progress on our three-dimensional computational fluid dynamics (3D CFD) modeling of supersonic diode pumped alkali lasers (DPALs), taking into account fluid dynamics and kinetic processes in the lasing medium. For a supersonic Cs DPAL with laser section geometry and resonator parameters similar to those of the 1-kW flowing-gas subsonic Cs DPAL [A.V. Bogachev et al., Quantum Electron. 42, 95 (2012)] the maximum achievable output power, similar to 7 kW, is 25% higher than that achievable in the subsonic case. Comparison between semi-analytical and 3D CFD models for Cs shows that the latter predicts much higher maximum achievable output power than the former. Optimization of the laser parameters using 3D CFD modeling shows that very high power and optical-to-optical efficiency, 35 kW and 82%, respectively, can be achieved in a Cs supersonic device pumped by a collimated cylindrical (0.5 cm diameter) beam. Application of end-or transverse-pumping by collimated rectangular (large cross section similar to 2 - 4 cm(2)) beam makes it possible to obtain even higher output power, > 250 kW, for similar to 350 kW pumping power. The main processes limiting the power of Cs supersonic DPAL are saturation of the D-2 transition and large similar to 40% losses of alkali atoms due to ionization, whereas the influence of gas heating is negligibly small. For supersonic K DPAL both gas heating and ionization effects are shown to be unimportant and the maximum achievable power, similar to 40 kW and 350 kW, for pumping by similar to 100 kW cylindrical and similar to 700 kW rectangular beam, respectively, are higher than those achievable in the Cs supersonic laser. The power achieved in the supersonic K DPAL is two times higher than for the subsonic version with the same resonator and K density at the gas inlet, the maximum optical-to-optical efficiency being 82%.