Grid Refinement Techniques for the 𝜸-Re𝜽𝒕 Transition Model in FUN3D

There has been an increased focus on the overall accuracy and grid convergence of Reynolds-averaged Navier-Stokes (RANS)-based transition models from the recent AIAA and NATO-AVT workshops. Even though satisfactory grid convergence could be achieved for simple two-dimensional flow configurations, it required mesh counts that are substantially larger than those used in typical applications. In this paper, we focus our efforts on understanding how the grid resolution and topology influences the accuracy and convergence of results by studying the Schubauer and Skramstad flat-plate configuration and the NLF-0416 airfoil at an angle of attack equal to five degrees using FUN3D, a second-order finite-volume code. By focusing on these cases, we can analyze both natural and separation-induced transition scenarios. Multiple grid refinement strategies are investigated. First, we determine the relative effectiveness of zonal streamwise refinement in the transition region within structured grids as an alternative to the costly option of globally uniform refinement of a baseline grid. We also complement this zonal technique by globally varying the wall-normal resolution keeping the streamwise resolution fixed. The zonal streamwise refinement can accurately model natural transition in a flat-plate boundary layer and separation-induced transition, but struggles to accurately model natural transition in airfoil flows. A series of unstructured prismatic grids that have similar node counts and viscous wall spacings as the structured hexahedral grids are also tested, and they do not achieve grid convergence until an extremely fine resolution. Last, we employ adjoint-based unstructured grid adaptation in FUN3D to natural and separation-induced transition on the NLF-0416 airfoil. The adjoint-based refinement process converges to the same solution as the baseline family of structured grids, but leads to smaller errors on coarser grids.