Designing large-eddy simulation of the turbulent boundary layer to capture law-of-the-wall scaling

Abstract

Law-of-the-wall (LOTW) scaling implies that at sufficiently high Reynolds numbers the mean velocity gradient ∂U/∂z in the turbulent boundary layer should scale on u∗/z in the inertia-dominated surface layer, where u∗ is the friction velocity and z is the distance from the surface. In 1992, Mason and Thomson pointed out that large-eddy simulation (LES) of the atmospheric boundary layer (ABL) creates a systematic peak in ϕ(z)≡(∂U/∂z)/(u∗/z) in the surface layer. This “overshoot” is particularly evident when the first grid level is within the inertial surface layer and in hybrid LES/Reynolds-averaged Navier–Stokes methods such as “detached-eddy simulation,” where the overshoot is identified as a “logarithmic layer mismatch.” Negative consequences of the overshoot—spurious streamwise coherence, large-eddy structure, and vertical transport—are enhanced by buoyancy. Several studies have shown that adjustments to the modeling of the subfilter scale (SFS) stress tensor can alter the degree of the overshoot. A comparison among simulations indicates a lack of grid independence in the prediction of mean velocity that originates in surface layer deviations from LOTW. Here we analyze the broader framework of LES prediction of LOTW scaling—including, but extending beyond, “the overshoot.” Our theory includes a criterion that is necessary to remove the overshoot but is insufficient for LES to produce constant ϕ(z)≡1/κ through the surface layer, and fully satisfy the LOTW. For mean shear to scale on u∗/z in the surface layer, we show that two additional criteria must be satisfied. These criteria can be framed in terms of three nondimensional variables that define a parameter space in which systematic adjustments can be made to the simulation to achieve LOTW scaling. This occurs when the three parameters exceed critical values that we estimate from basic scaling arguments. The essential difficulty can be traced to a spurious numerical LES viscous length scale that interferes with the dimensional analysis underlying LOTW. When this spurious scale is suppressed sufficiently to retrieve LOTW scaling, the LES has been moved into the supercritical “high accuracy zone” (HAZ) of our parameter space. Using eddy viscosity closures for SFS stress, we show that to move the simulation into the HAZ, the model constant must be adjusted together with grid aspect ratio in coordinated fashion while guaranteeing that the surface layer is sufficiently well resolved in the vertical by the grid. We argue that, in principle, both the critical values that define the HAZ and the surface layer constant κ when LOTW scaling is achieved can depend on details of the SFS (and surface stress) models applied in the LES. We carried out over 110 simulations of the neutral rough-surface ABL to cover a wide portion of the parameter space using a low dissipation spectral code, the Smagorinsky SFS stress model and a standard model for fluctuating surface stress. The overshoot was found to systematically reduce and ϕ(z) was found to systematically approach a constant value in the surface layer as the three parameters exceeded critical values and the LES moved into the HAZ, consistent with the theory. However, whereas constant ϕ(z) was achieved over nearly the entire surface layer as the LES is moved into the HAZ, the model for surface shear stress continues to disrupt LOTW scaling at the first couple grid levels, and the predicted von Kármán constant κ is lower than traditional values. In a comprehensive discussion, we summarize the primary results of subsequent studies where we minimize the spurious influence of the surface stress model and show that the surface stress model and the SFS stress model constant influence the predicted value of the von Kármán constant for LES in the HAZ.