This work consists of an overall effort to apply a detailed and accurate computational fluid dynamic technique to the solution of practical high energy laser flows. In particular, a third generation of super sonic diffusion chemical laser analysis is introduced, namely, the complete solution of the Navier-Stokes equations for the laminar, super sonic mixing flow fields fully coupled with chemical kinetics for both the hot and cold reactions for HF. Multicomponent diffusion is treated in a detailed fashion. Solutions are obtained, firstly, for "cold flows", where the effects of chemical reactions and vibrational relaxation are not included. Although such a situation is purely artificial, the results do isolate some of the fluid dynamic aspects of chemical laser flows, and provide a set of data to be compared later with hot flow calculations. A set of numerical experiments using four different time dependent finite difference schemes show that relatively minor changes in the differencing procedure can lead to major variations in the results. A modification of the well-known Maccormack approach appears to be the best suited for mixing flows associated with chemical lasers. A comparison is next made between cold flows (with fully coupled chemical kinetics). the results show that temperature distributions are affected the most and velocity distributions the least by chemical energy heat release. The results have an impact on the interpretation of cold flow aerodynamic experiments in the laboratory, and their proper extrapolation to the real chemical laser flows. also, comparisons between the present Navier Stokes results and other, more approximate, existing calculations are made. Gradients are calculated as a natural part of the Navier Stokes solutions. Results are given for steady flows with large pressure gradients where advantages of the Navier Stokes solutions are delineated. In addition, the effect of unsteady fluctuations intentionally introduced at the cavity inlet are studied. Specifically, sinusoidal fluctuations in one stream and then both streams (primary and secondary) in various quantities e.g. pressure, density, u velocity and v velocity were simulated. Of these, the oscillations in v velocity with approximate frequency and amplitude produced a remarkable improvement in mixing. Such unsteady fluctuations also yielded peak laser gain which were larger by almost a factor of two compared to the steady case. the flow at which the upstream boundary has so far, in the above mentioned cases been assumed to be uniform with real effects like Boundary Layer and Base Flow having been neglected. For comparison purposes these effects are next included. the boundary layer profile and velocity at the inlet is shown to feed production of gain substantially. Base flow calculations were attempted but were not successful