Scaling-up flowing-gas diode pumped alkali lasers (DPALs) to megawatt class power is studied using accurate threedimensional computational fluid dynamics model, taking into account the effects of temperature rise and losses of alkali atoms due to ionization. Both the maximum achievable power and laser beam quality are estimated for Cs and K lasers. We examined the influence of the flow velocity and Mach number M on the maximum achievable power of subsonic and supersonic lasers. For Cs DPAL devices with M = 0.2 - 3 the output power increases with increasing M by only ∼20%, implying that supersonic operation mode has only small advantage over subsonic. In contrast, the power achievable in K DPALs strongly depends on M. The output power increases by ∼100% when M increases from 0.2 to 4, showing a considerable advantage of supersonic device over subsonic. The reason for the increase of the power with M in both Cs and K DPALs is the decrease of the temperature due to the gas expansion in the flow system. However, the power increase for K lasers is much larger than for the Cs devices mainly due to the much smaller fine-structure splitting of the 2P states (∼58 cm-1 for K and ∼554 cm-1 for Cs), which results in a much stronger effect of the temperature decrease in K DPALs. For pumping by beams of the same rectangular cross section, comparison between end-pumping and transverse-pumping shows that the output power is not affected by the pump geometry. However, the intensity of the output laser beam in the case of transverse-pumped DPALs is strongly non-uniform in the laser beam cross section resulting in higher brightness and better beam quality in the far field for the end-pumping geometry where the intensity of the output beam is uniform.