Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES) are performed of a round fluid jet entering a high-pressure chamber. The chemical compositions and temperatures of the jet and that of the fluid in the chamber are initially prescribed. The governing equations consist of the conservation equations for mass, momentum, species and energy, and are complemented by a real-gas equation of state. The fluxes of species and heat are written in the framework of fluctuation-dissipation theory and include Soret and Dufour effects. For more than two species, the full mass diffusion and thermal diffusion matrices are computed using high-pressure mixing rules which utilize as building blocks the corresponding binary diffusion coefficients. The mixture viscosity and thermal conductivity are computed using standard mixing rules and corresponding states theory. To evaluate the physical model and numerical method, LES is employed first to simulate a supercritical N-2 jet injected into N-2. Time averaged results show reasonable agreement with the experimental data. Then, DNS is conducted to study the spatial evolution of a supercritical N-2 jet injected into CO2. Time averaged results are used to compute the length of the potential core and the species diffusion characteristics. Spectral analysis is then applied on a time series data obtained at several axial locations and a dominant frequency is observed inside the potential core.