The thermodynamic and transport properties of fluids above their critical point is far from that of a perfect gas. Strong nonlinearities in the equation of state and transport properties may modify the structure of turbulence but also require additional terms in turbulence models. The present work addresses the influence of a real-gas equation of state (Peng & Robinson, 1976) on turbulence properties and turbulence models. The numerical simulation of two single-species flows was conducted with the intent of validating the Large-Eddy Simulation (LES) methodology for flows under supercritical thermodynamic conditions. First, the set of equations for LES is presented, with appropriate approximations and assumptions, in particular those relevant to the nonlinear thermodynamic properties of real gases. A homogeneous isotropic turbulence case considered to study the influence of equation-of-state effects on the properties of turbulence. It is shown that the peculiar thermodynamic properties of single-species supercritical fluids do not affect the spectral content of the flow, which is promising for LES. Then, the case of a dense jet at supercritical pressure was computed with the LES methodology. This jet corresponded to an actual experimental setup, which allows for comparison with Raman measurements of density profiles.