On the importance of wall transpiration in wall modeled LES of non-equilibrium flows

Numerical simulations of high Reynolds number flows present significant challenges, particularly in predicting separated flows induced by mild adverse pressure gradients. Recent work has demonstrated that equilibrium wall model formulations require resolution of viscous pressure gradient sublayer (approximately Δ∇p^1/3/ν^2/3 < 10) in order to accurately capture separation. This requirement leads to a grid point scaling locally in the vicinity of separation points of N ∼ Re^4/3. The fundamental issue is that wall stress-based wall models do not necessarily predict the outer layer flow (or the wall pressure distribution), particularly at coarse resolutions, even if the exact wall stress distribution is provided. In contrast, slip wall models have been shown to capture flow separation at coarse resolutions, although the results are sensitive to the choice of slip lengths. We present an alternative derivation of the LES wall boundary condition that shows the necessity of wall transpiration in the limit of coarse resolution. Specifically, it is shown that this formulation satisfies the no-slip limit (Δ⁺ << 1) and recovers Lighthill's viscous correction to the inviscid equations when Δ >> δ. This approach is validated in a posteriori slip wall modeled calculations of the Boeing speed bump. Results are presented for a range of resolutions and predictive modeling approaches are discussed.