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Supercritical ŕuids have a number of thermodynamic and chemical properties which make them attractive for use in environmentally friendly technologies. However, though the thermodynamic properties of supercritical ŕuids have been studied comprehensively, the dynamics of supercritical and transcritical ŕuid ŕows are less well explored and understood. Studying the behavior of such ŕuid ŕows through high-quality computational investigations could provide crucial insights useful for designing and controlling ŕow systems operating in supercritical and transcritical regimes. An accurate and robust computational framework is a prerequisite to conducting high-quality computational investigations. This work extends a high-ődelity computational framework for ideal gas ŕows by including complex thermodynamic models and realistic transport models near the critical point of the ŕuid where abrupt changes in density and transport properties occur with small temperature or pressure ŕuctuations. The spatial discretization is based on compact őnite difference methods that achieve high-order grid convergence and the high spectral resolution needed to resolve small scale ŕow structures. The computational approach achieves robustness by reducing the aliasing error and improving the spectral resolution of the viscous ŕuxes at high wavenumbers. No non-conservative correction or őltering is needed to maintain robustness for shock-free ŕows if physical or physics-based model dissipation is included. The framework is also compatible with applications of shock capturing schemes and approximated Riemann solvers and supports simulations on curvilinear meshes. Two problems involving compressible free-shear ŕows (temporal mixing layer) and wall-bounded ŕows (zero-pressure gradient ŕat plate boundary layer with a cold isothermal wall) are studied for dense gases to demonstrate the robustness and versatility of the proposed numerical formulation.