Roe-type schemes for dense gas flow computations

Dense gas dynamics studies the dynamic behavior of gases in the thermodynamic region close to the liquid–vapor critical point, where the perfect gas law is no longer valid, and has to be replaced by more complex equations of state. In such a region, some fluids, known as the Bethe–Zel’dovich–Thompson fluids, can exhibit non-classical nonlinearities, such as expansion shocks, and mixed shock-fan waves. In the present work, the problem of choosing a suitable numerical scheme for dense gas flow computations is addressed. In particular, some extensions of classical Roe’s scheme to real gas flows are reviewed and their performances are evaluated for flow problems involving non-classical nonlinearities. A simplification to Roe’s linearization procedure is proposed, which does not satisfy the U-property exactly, but significantly reduces complexity and computational costs. Such simplification introduces an additional error O(δx²), with δx the mesh size, with respect to the first-order accurate Roe’s scheme, and O(δx⁶) with respect to its higher-order MUSCL extensions. Numerical experiments, concerning a one-dimensional dense gas shock tube, supersonic flow of a BZT gas past a forward-facing step, and transonic dense gas flow through a turbine cascade, show a negligible influence of the adopted linearization procedure on the solution accuracy, whereas it significantly affects computational efficiency.