Model analysis of the turbulent flows in a convergent–divergent channel and around a sphere

Abstract

The turbulence closure problem where the objective is to find a model that captures the behavior of the Reynolds stress tensor has proved to be a huge challenge. The commonly employed Boussinesq hypothesis is known for some limitations in general problems, as discussed by Pope (Turbulent flows, Cambridge University Press, Cambridge, 2000). In the present work we employ a tensorial decomposition technique as a mean to evaluate the Boussinesq hypothesis. This methodology projects the anisotropic Reynolds stress tensor, obtained from a high-fidelity simulation onto the rate-of-strain tensor enabling an estimate for the turbulent viscosity and the error associated with the assumption. This approach is applied to two flow types corresponding to internal and external flows: a convergent–divergent channel flow and the flow around a sphere. The high-fidelity data are from a direct numerical simulation (DNS) for a convergent–divergent channel flow (available in the literature) and from a large eddy simulation (LES) performed in the present work for the flow around a sphere using the software OpenFOAM. An additional analysis is made regarding the detachment of the boundary layer in each flow using three Reynolds average Navier–Stokes (RANS) models: $$\kappa$$ – $$\epsilon$$ ; $$\kappa$$ – $$\omega$$ ; SST. Employing the software CFX for the numerical simulations employing RANS models, the behavior of the boundary layer was compared with the DNS/LES data in both flows. The results for the tensor decomposition theory have shown that the Boussinesq hypothesis is not sufficient to capture the Reynolds stress tensor in both flow types. It would be necessary to use an extended basis presented in this theory in order to have a better capture of the turbulence behavior in these flows. Regarding the boundary layer study, the results obtained showed to be reasonably accurate in regions where a viscometric flow is more likely to occur, in accordance with the literature.