A weak-shear correction to the Smagorinsky model for high-Reynolds-number wall-modelled simulations

The Lilly-Smagorinsky model is known to work well in high-Reynolds HIT, but is too dissipative for wall-bounded flows. In wall-modelled LES, this leads to the log-layer mismatch and over prediction of streamwise velocity fluctuations. It has been shown that the mismatch is in part caused by the wall-stress model, but excluding this does not fully resolve the issue for the Smagorinsky model. Various heuristics exist to improve the model, e.g., based on wall damping, or the dynamic procedure, but theoretical insights into how the model coefficient should behave near the wall remain elusive. We show that, with increasing resolution, predictions of the Lilly-Smagorinsky model become very accurate away from the wall. However, when approaching the wall, an important aspect is the prediction of the subgrid mean shear. While the classical Lilly analysis links the coefficient to the Kolmogorov constant in inertial range turbulence, we show that this is not consistent with correct levels of subgrid mean shear in weak-shear turbulence. Based on generalizations of Lumley’s co-spectrum, we formulate a correction to the Smagorinsky model, that incorporates weak-shear effects. This solves the log-layer mismatch and leads to accurate predictions of the von Karman measure when compared to DNS.