Estimation of Discretization Errors in Viscous Flow Simulations Based on the RANS Equations: Wall-Resolved Versus Wall-Functions

In the present paper, we focus on the simulation of viscous flows at high Reynolds numbers using the Reynolds-averaged Navier–Stokes (RANS) equations. Time-averaging is used to define the mean flow properties and only deterministic simulations are considered. Therefore, numerical errors are a consequence of round-off, iterative and discretization errors. In carefully performed simulations, round-off and iterative errors are reduced to negligible levels when compared to the discretization error and so the numerical error is dominated by the contribution of the discretization error. The use of grid refinement studies is one of the most flexible and popular techniques for the estimation of discretization errors for steady simulations. Several methods have been proposed in the open literature and most of them share common features. The discretization error of a quantity of interest is described as a function of the typical cell size by power series expansions. The estimation of the exact solution requires numerical solutions in more than one grid and so a family of (nearly) geometrical similar grids needs to be generated. The requirement of grid similarity is a consequence of the definition of the typical cell size. In the numerical solution of the RANS equations, the determination of the shear-stress at the wall $τw$ can be performed in two alternative ways: directly from its definition, or using wall functions. The grid refinement strategy required by each case is significantly different. In the first option, the near-wall cell must be systematically refined as all the remaining grid cells. When wall functions are used, the size of the near-wall cell size should remain fixed. In this paper, we present the consequences of using the wrong refinement strategy, i.e., by keeping the size of the near-wall cell fixed when $τw$ is calculated from its definition and by refining the near-wall cell when $τw$ is determined from wall functions. The selected test case is the flow over a flat plate at Reynolds numbers of $107$ and $109$⁠. The results show that using the wrong grid refinement strategy can lead to misleading results that exhibit reasonable orders of grid convergence.