Viscous flow simulations at high Reynolds numbers without wall functions - Is y+≃1 enough for the near-wall cells?

The Reynolds-Averaged Navier Stokes equations supplemented by eddy-viscosity models are still the most common mathematical model for the simulation of wall-bounded viscous flows at high Reynolds numbers. Viscous flow simulations around complex geometries are more accurate with the direct application of the no-slip condition at walls, i.e. avoiding the use of wall functions. Determination of the shear-stress at the wall from its definition requires the use of near-wall cells which are typically requested to present a non-dimensional height of y⁺≃1 . However, the effect of this widespread rule of thumb on the numerical accuracy of the solutions has not been established and it may be dependent on the turbulence model choice.

In this paper, we discuss the numerical accuracy of viscous flow simulations performed with a RANS solver supplemented by three eddy-viscosity models in the calculation of statistically steady, incompressible flows: the one-equation model of Spalart & Allmaras and the two-equation models SST k–ω and k-√kL. The selected test cases are: a flat plate at Reynolds numbers Re equal to 10⁷, 10⁸ and 10⁹; the NACA 0012 airfoil at angles of attack of 0,4 and 10 degrees with Re=6×10⁶ and the KLVCC2 tanker at model and full scale Reynolds numbers, Re=4.6×10⁶ and Re=2.03×10⁹. Calculations are performed in sets of geometrically similar grids with different sizes of the near-wall cells height to assess the influence of this choice on the numerical uncertainty of the solutions.

The results show that the numerical accuracy of the simulations is dependent on the selected turbulence model. For y⁺ < 1, there is no significant effect of the near-wall cell height on the numerical uncertainty of the selected flow quantities for the Spalart & Allmaras and k-√kL models and so y⁺≃1 is sufficient to obtain numerical uncertainties of friction resistance coefficients smaller than 1% . On the other hand, with the SST k–ω model, y⁺≃1 leads to numerical uncertainties of friction resistance coefficients larger than 5% for all the present test cases. To attain the same uncertainty of the other turbulence models with the SST k–ω, one must use y⁺≃0.1.