On the Simulation of Statistically Unsteady Flows with the RANS Equations

Many engineering applications are still based on the solution of the Reynolds-Averaged Navier-Stokes (RANS) equations that require the definition of mean flow quantities and the averaging of the continuity and momentum equations. Most RANS turbulence models available in the open literature have been developed for statistically steady flows, i.e. time-averaging is applied to the flow variables and to the continuity and momentum equations. In external flows around bluff bodies or large angles of incidence in streamlines bodies, wide wakes are generated due to massive flow separation and vortex shedding will occur. In such conditions, time-averaging is not a reasonable option for the definition of the mean flow, because time variations generated by the vortex shedding phenomena will be considered as turbulence fluctuations. As for the statistically steady flows, the role of the Reynolds stresses is to damp the turbulence fluctuations and allow the determination of the mean flow. However, there is no guarantee that turbulence models developed for time-averaged RANS will also be appropriate for statistically unsteady flows. In this paper, we present simulations for the flow around a circular cylinder at Reynolds numbers ranging from sub-critical (transition in the near-wake) to super-critical (transition on the cylinder upstream of separation) performed with three turbulence models: the Shear-Stress Transport (SST) k−ω two-equation eddy-viscosity model, an explicit algebraic Reynolds Stress model (EARSM) based on a SST k−ω and a Reynolds Stress model (RSM). Two-dimensional and three-dimensional approaches are compared and the contribution of statistical, iterative and discretization errors to the numerical uncertainty are estimated for all flow conditions. With this simple flow, we assess if the turbulence model is able to provide the required diffusion to damp turbulence fluctuations and allow the determination of the mean flow solution.