Keywords: large-eddy simulation, wall modeling, high Reynolds number flow, separated flow.

In this paper, a new simple idea is presented to remove a major source of error in large-eddy simulation with wall-modeling (i.e., when the wall shear stress is modelled and the viscous near-wall layer is not resolved): the error in estimating the wall shear stress from a given outer-layer velocity field by solving auxiliary near-wall RANS equations in the inner-layer. The proposed wall model includes the non-equilibruim effects of convection and pressure-gradient, e.g., adverse pressure gradient leading to separation, and hence this non-equilibrium model can be expected to be applicable to a broader range of flow conditions than the equilibrium log-law model.

By considering the behaviour of turbulence length scales near a wall, the cause of the errors is diagnosed and solutions that remove the impact of these errors on the computed turbulence are proposed based solidly on physical reasoning. The proposed non-equilibrium wall-model dynamically matches the total stresses and heat fluxes at the matching location (where the top of the RANS solution set equal to the instantaneous LES solution at the corresponding location), and also accounts for the physics that the unresolved stresses and heat fluxes are increased in the wall-normal direction toward the wall in the inner-layer RANS. It is noted that the wall-model and the arguments leading to the proposed method are presented for compressible flows, but everything extends trivially to incompressible flows.

The resulting dynamic non-equilibrium wall-model is validated against the corresponding experiments of shock-wave or turbulent boundary layer interaction by Souverein et al. [1]. The results are first validated on the undisturbed zero-pressure-gradient supersonic turbulent boundary layer at the high Reynolds number: freestream Mach number of 1.69 and momentum thickness based Reynolds number of 5x10⁴, and then the wall-model is applied to the shock-wave or turbulent boundary layer interacting separated flow at the same Mach number and Reynolds number. It is important to note that this is a much higher Reynolds number than that the wall-resolved LES or DNS can reach.

The resulting method is shown to accurately predict undisturbed equilibrium boundary layers at high Reynolds number, with both realistic instantaneous fields without overly elongated unphysical near-wall structures as often seen in DES-type approach and accurate statistics in both turbulence quantities and skin friction without showing the typical logarithmic layer mismatch.

1

L.J. Souverein, P. Dupont, J.F. Debieve, J.P. Dussauge, B.W. van Oudheusden, F. Scarano, "Effect of Interaction Strength on Unsteadiness in Turbulent Shock-Wave-Induced Separations", AIAA Journal, 48(7), 1480-1493, July 2010.

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