An implicit low-diffusive HLL scheme with complete time linearization: Application to cavitating barotropic flows

Abstract

A numerical method for generic barotropic flows is presented, together with its application to the simulation of cavitating flows. A homogeneous-flow cavitation model is indeed considered, which leads to a barotropic state equation. The continuity and momentum equations for compressible flows are discretized through a mixed finite-element/finite-volume approach, applicable to unstructured grids. P1 finite elements are used for the viscous terms, while finite volumes for the convective ones. The numerical fluxes are computed by shock-capturing schemes and ad-hoc preconditioning is used to avoid accuracy problems in the low-Mach regime. A HLL flux function for barotropic flows is proposed, in which an anti-diffusive term is introduced to counteract accuracy problems for contact discontinuities and viscous flows typical of this class of schemes, while maintaining its simplicity. Second-order accuracy in space is obtained through MUSCL reconstruction. Time advancing is carried out by an implicit linearized scheme. For this HLL-like flux function two different time linearizations are considered; in the first one the upwind part of the flux function is frozen in time, while in the second one its time variation is taken into account. The proposed numerical ingredients are validated through the simulations of different flow configurations, viz. the Blasius boundary layer, a Riemann problem, the quasi-1D cavitating flow in a nozzle and the flow around a hydrofoil mounted in a tunnel, both in cavitating and non-cavitating conditions. The Roe flux function is also considered for comparison. It is shown that the anti-diffusive term introduced in the HLL scheme is actually effective to obtain good accuracy (similar to the one of the Roe scheme) for viscous flows and contact discontinuities. Moreover, the more complete time linearization is a key ingredient to largely improve numerical stability and efficiency in cavitating conditions.