High speed jets in cross flows are central to fuel injection in supersonic combustion scramjet engines. In supersonic combustion scramjet engines, the sonic under expanded transverse jet of fuel is injected into a supersonic cross flow of air, where efficient mixing of fuel and air is one of the major critical issues. Due to the limited flow residence time inside the combustion chamber, the enhancement of supersonic turbulent mixing of jet fuel and cross-flow air is a critical issue in developing supersonic air-breathing engines. The accurate estimation and detailed physical understanding of the turbulent mixing mechanisms plays an important role in combustor design of scramjet engines. This numerical study aims at understanding the complex physical phenomenon involved in mixing of fuel jet and air and the associated turbulence characteristics, boundary layer capture and flow separation. In the current study the flow field resulting from the transverse injection of fuel jet into cross-flow of air is simulated numerically by solving the appropriate governing equations for a 2-dimensional flow. Numerical simulations are used to study an under-expanded jet injected into a supersonic cross flow. This study examines the flow structure, separation topology and performance characteristics of an under expanded transverse jet issuing normally into supersonic free stream. The influence of the compressibility effect on the shock wave structure and on the vortex system ahead and behind of the jet are studied by solving Favre averaged Navier Stokes (FANS) equations with SST k-ω turbulence model. The influence of the jet Mach number and jet-to-cross-flow pressure ratio on shock wave structure of the flow and jet penetration depth are studied. The simulated numerical results are compared with the experimental data available in the literature. Grid independence study is carried out for all the simulations carried out in the work to have good accurate results. It was found out that wall pressure profile of transverse jet injection for the high jet-to-cross-flow pressure ratio is predicted more accurately by the SST k-ω turbulence model. The jet penetration depth found out to be increasing with the increase in jet-to-cross-flow pressure ratio and fuel jet slot width.