Hybrid low-diffusion approximate Riemann solvers for Reynolds-stress transport

The paper investigates the use of low-diffusion (contact-discontinuity-resolving [Liou M.S.: {\em J. Comp. Phys.} {\bf 160} (2000) 623--648]) approximate Riemann solvers for the convective part of the Reynolds-averaged Navier-Stokes (\tsn{RANS}) equations with Reynolds-stress model (\tsn{RSM}) closure. Different equivalent forms of the \tsn{RSM-RANS} system are discussed and classification of the complex terms introduced by advanced turbulence closures is attempted. Computational examples are presented, which indicate that the use of contact-discontinuity-resolving convective numerical fluxes, along with a passive-scalar approach for the Reynolds-stresses, may lead to unphysical oscillations of the solution. To determine the source of these instabilities, theoretical analysis of the Riemann problem for a simplified Reynolds-stress transport model-system, which incorporates the divergence of the Reynolds-stress tensor in the convective part of the mean-flow equations, and includes only those nonconservative products which are computable (do not require modelling), was undertaken, highlighting the differences in wave-structure compared to the passive-scalar case. A hybrid solution, allowing the combination of any low-diffusion approximate Riemann solver with the complex tensorial representations used in advanced models, is proposed, combining low-diffusion fluxes for the mean-flow equations with a more dissipative massflux for Reynolds-stress-transport. Several computational examples are presented to assess the performance of this approach.