An adaptive primitive-conservative scheme for high speed transcritical flow with an arbitrary equation of state

When fully conservative methods are used to simulate transcritical flow, spurious pressure oscillations and numerical instability are generated. The strength and speed of propagation of shock waves cannot be represented correctly using a semi-conservative or primitive method. In this research, an adaptive primitive-conservative scheme is designed to overcome the aforesaid two difficulties. The underlying cause for pressure oscillation is analyzed within the framework of Finite Volume Method (FVM). We found that the nonlinearity of the thermodynamic properties of transcritical fluids renders standard conservative numerical methods ineffective. In smooth regions, schemes based on primitive variable are used to eliminate spurious pressure oscillations. For the purpose of correctly capturing shock waves, the modified Roe Riemann solver for real fluid is utilized in regions where shock waves induce discontinuity. The adaptive numerical approach relies only on the speed of sound, eliminating the requirement to calculate the derivatives of thermodynamic quantities. A large number of numerical test cases conducted in one- and two-dimensional spaces have shown the robustness and accuracy of the proposed adaptive scheme for the simulations of high speed transcritical flows.