Attaining the Asymptotic Range in RANS Simulations

In engineering simulations involving turbulent fluid flows, the Reynolds-averaged Navier-Stokes (RANS) equations are still the most common mathematical model. The RANS equations require the use of a turbulence model to calculate the Reynolds stresses generated by the averaging of the momentum equations. Nowadays, the most popular turbulence models require the solution of additional transport equations that can range from one to seven equations.

In this paper we illustrate the difficulties in attaining and identifying the so-called asymptotic range in grid refinement studies performed for the numerical solution of the RANS equations in the flow over a flat plate. Three turbulence models are tested: two-equation, eddy-viscosity, k—ω SST and k-√kL turbulence models and the seven-equation Reynolds stress model SSG/LRR—ω. The three turbulence models are tested with second and first-order upwind schemes applied to the convective terms of the turbulence models transport equations.

The results show that even in this simple flow, attaining the asymptotic order of grid convergence requires unreasonable levels of grid refinement. Furthermore, even in strictly geometrical similar grids, the observed order of grid refinement can be extremely sensitive to the discretization schemes used in the turbulence model and to any disturbances in the data. An alternative and more efficient way to address the quality of an error estimation based on a single term expansion is to determine the change of the estimate of the exact solution with grid refinement.