Populating the wall layer, one eddy at a time: Resolvent analysis for Wall-Modelled LES

Computational cost precludes direct numerical simulation or wall-resolved large-eddy simulations of non-equilibrium, wall-bounded turbulent flows in realistic conditions. Wall-modelled large-eddy simulations (WMLES) and hybrid RANS/LES methods can be used to analyse these flows at much decreased cost, but require modelling of the near-wall layer and, in particular, a means to address the deficit of turbulent activity, or eddies, in the vicinity of the interface between the outer flow and the wall model. We report a computational framework to populate the wall region with synthetic but realistic eddies and reflect their integrated effect on the flow in the inner layer. Two means of generating spatio-temporal representations for the synthetic eddies are investigated: low-order, resolvent-based representations of the wall layer and a coarse-grained, data-driven spectral proper orthogonal decomposition (SPOD) model, both generated in turbulent channel flow at a friction Reynolds number, $R{e}_{\tau }=1000$. The eddy-augmented WMLES models are then tested in $R{e}_{\tau }=5000$ and 20,000 channels and compared with experimental and numerical data. The inherent scaling of the resolvent operator can be used to scale the resolvent model to higher Reynolds numbers (and potentially populate new, self-similar eddies as the wall layer grows in inner units), while the SPOD model is energetically optimal for reconstruction of the flow at Reynolds numbers close to that where it is obtained, but degrades as the Reynolds number is increased. The results show that the effect of the introduction of synthetic eddies is twofold: first, a direct contribution to the stress due to the presence of the synthetic eddies and, second, an improved prediction of the normal Reynolds stresses in the inner layer due to an accompanying, coupled reduction in the time- and length-scales of the variation of the URANS-like velocity in the inner layer. Implications and extensions of the method for more complex flows, for example external boundary layers with pressure gradient and separation, are briefly discussed.