A computational study of sonic light-gas jet injection into a supersonic cross flow was conducted. The scope of the numerical analysis encompassed many studies that affect how the flow-field is numerically modeled and the behavior, specifically mixing, of the flow-field itself. A single, round injector was used for the Baseline design. Simulated conditions involved sonic injection of helium heated to 313 K into a Mach 4 air cross-stream with average Reynolds number 5.77 e+7 per meter and a freestream momentum flux ratio of 2.1. Experiments at these conditions were available for comparison. The primary numerical flow solver employed was GASP v. 4.2. The Menter Shear Stress Transport (SST) turbulence model was used, since the algorithm has good capability of solving both wall-bounded and free-shear flows. The SST model was able to capture the mixing behavior of the complex flow-field. Important numerical parameters that affect the capabilities of the numerical solver were studied for the Baseline injector. These sensitivity studies varied the choice of turbulent Prandtl number, Schmidt number, freestream turbulence intensity, boundary layer size, steady and unsteady approaches and computational software packages. A decrease in the turbulent Prandtl number resulted in better mixing behavior of the prediction and better agreement with the experiment. An increase in the turbulent Schmidt number had a small adverse effect on the predictions. The mixing characteristics remained constant with an increase in freestream turbulence intensity. The best Baseline prediction was then compared to three different injector configurations: an aerodynamic ramp consisting of four injectors in an array, a diamond injector both aligned and yawed 15° to the oncoming flow. The Computational Fluid Dynamics (CFD) tools were more accurate compared to experiment in the prediction of the aeroramp injector than the diamond-shaped injectors. The aeroramp injector slightly improved mixing efficiency over the Baseline injector at these conditions. Both of the diamond-shaped injectors had similar mixing as the Baseline injector but did not predict significant improvement in penetration for the analyzed conditions. Additional studies involving the interaction of transverse injection with impinging oblique shock waves were performed. The impingement of a shock upon light gas jet injection increased mixing. The closer the shock is to the injection point, the larger the effect on mixing and vorticity. The last analyses involved a numerical comparison of a non-reacting model to a reacting hydrogen-air model. The reacting analysis prediction had an improved spreading rate and larger counter-rotating vortex pair with downstream distance over the non-reacting analysis. The mixing was not significantly altered by the addition of hydrogen-air reactions to the numerical equations. The numerical tools used are capable of reasonable accuracy in predicting the complex flow-field of jet injection into a supersonic freestream with proper choice of models and parameters. Numerical modeling offers a way to study the entire flow-field thoroughly in a cost and time efficient manner. / Ph. D.