Structural Wall-modeled LES Using a High-order Spectral Difference Scheme for Unstructured Meshes

The combination of the high-order unstructured Spectral Difference (SD) spatial discretization scheme with Sub-Grid Scale (SGS) modeling for Wall-Modeled Large-Eddy Simulation (WMLES) is investigated. Particular focus is given to the use of wall-function approaches and to the relevant optimal coupling with the numerical scheme and the SGS model, a similarity mixed type model featuring newly designed discrete filters with specified cutoff length scale. To take full advantage of the discontinuous Finite Element (FE) structure which characterizes the SD scheme, wall-modeling is accomplished within the first wall element by using the information from the farthest solution points from the wall. Compared to the customary used first off-wall node, this point provides more accurate information to the wall-function, thus improving the quality of the solution. Two different law-of-the-wall are tested, a classical three-layers wall-function based on the equilibrium assumption and a more general formulation to account for the pressure gradient in more complex configurations. Moreover, the mixed scale-similarity SGS model is used in the entire computational domain without any particular adjustment inside the wall-modeled region. Numerical tests on the classical test case of the turbulent channel flow at different Reynolds numbers and on the channel with periodic constrictions at Re (h) = 10,595 give evidence that the results are extremely sensitive to the choice of the solution points used to provide the informations to the law-of-the-wall. In particular, it is shown that significant improvements in the results can be attained by solving the wall-function away from the wall, rather than at the first off-wall solution point as customary done. The combination of the selected wall-modeling strategies and the similarity mixed formulation proves to be remarkably accurate, even in the presence of boundary layer separation, thus opening the path to further exploit the high-order SD platform, as well as a broad range of other similar methodologies, for WMLES. Extensions of the methodology are envisaged to include more sophisticated wall-modeling approaches incorporating turbulent sensors to switch to no-slip conditions in laminar regions.