A hybrid fluctuation splitting scheme for two-dimensional compressible steady flows

This paper provides a genuinely multidimensional upwind scheme which is robust and (second-order) accurate at all flow regimes of interest for aerodynamic and turbomachinery applications. Firstly, a review of the state of the art of multidimensional upwind schemes for scalar advection equations is provided; then, the case of the Euler system is considered. For such a case, it is shown that, within the framework of fluctuation splitting residual-distribution schemes (or, equivalently, of linear finite elements), one can obtain: i) a linear second-order-accurate matrix scheme which is fully adequate for solving subsonic flows; ii) a nonlinear scalar scheme which is optimal for supersonic shockless flows and also resolves monotonically shocks separating two supersonic-flow regions; iii) a linear first-order-accurate matrix scheme which resolves strong shocks without spurious oscillations. A fully satisfactory methodology is therefore proposed by combining such three methods within a hybrid approach, which selects the most suitable scheme for each single element in the discrete domain, depending on the local flow conditions. The methodology is finally extended to the case of the (Reynolds averaged) Navier–Stokes equations using a standard finite element discretization of the viscous terms. Results are provided which demonstrate the accuracy and robustness of the proposed approach for several test cases involving inviscid, laminar and turbulent flows.