Low-speed preconditioning for strongly coupled integration of Reynolds-averaged Navier–Stokes equations and two-equation turbulence models

Abstract

Computational fluid dynamics codes using the density-based compressible flow formulation of the Navier–Stokes equations have proven to be very successful for the analysis of high-speed flows. However, solution accuracy degradation and, for explicit solvers, reduction of the residual convergence rates occur as the local Mach number decreases below the threshold of 0.1. This performance impairment worsens remarkably in the presence of flow reversals at wall boundaries and unbounded high-vorticity flow regions. These issues can be resolved using low-speed preconditioning, but there exists an outstanding problem regarding the use of this technology in the strongly coupled integration of the Reynolds-averaged Navier–Stokes equations and two-equation turbulence models, such as the k−ω shear stress transport model. It is not possible to precondition only the RANS equations without altering parts of the governing equations, and there did not exist an approach for preconditioning both the RANS and the SST equations. This study solves this problem by introducing a turbulent low-speed preconditioner of the RANS and SST equations that does not require any alteration of the governing equations. The approach has recently been shown to significantly improve convergence rates in the case of a one-equation turbulence model. The study focuses on the explicit multigrid integration of the governing equations, but most algorithms are applicable also to implicit integration methods. The paper provides all algorithms required for implementing the presented turbulent preconditioner in other computational fluid dynamics codes. The new method is applicable to all low- and mixed-speed aeronautical and propulsion flow problems, and is demonstrated by analyzing the flow field of a Darrieus wind turbine rotor section at two operating conditions, one of which is characterized by significant blade/vortex interaction. Verification and further validation of the new method is also based on the comparison of the results obtained with the developed density-based code and those obtained with a commercial pressure-based code.