The logarithmic law for the mean velocity in turbulent boundary layers haslong provided a valuable and robust reference for comparison with theories,models, and large-eddy simulations (LES) of wall-bounded turbulence. Morerecently, analysis of high-Reynolds number experimental boundary layer data hasshown that also the variance and higher-order moments of the streamwisevelocity fluctuations $u'^{+}$ display logarithmic laws. Such experimentalobservations motivate the question whether LES can accurately reproduce thevariance and the higher-order moments, in particular their logarithmicdependency on distance to the wall. In this study we perform LES of very highReynolds number wall-modeled channel flow and focus on profiles of variance andhigher-order moments of the streamwise velocity fluctuations. In agreement withthe experimental data, we observe an approximately logarithmic law for thevariance in the LES, with a `Townsend-Perry' constant of $A_1\approx 1.25$. TheLES also yields approximate logarithmic laws for the higher-order moments ofthe streamwise velocity. Good agreement is found between $A_p$, the generalized`Townsend-Perry' constants for moments of order $2p$, from experiments andsimulations. Both are indicative of sub-Gaussian behavior of the streamwisevelocity fluctuations. The near-wall behavior of the variance, the ranges ofvalidity of the logarithmic law and in particular possible dependencies oncharacteristic length scales such as the roughness scale $z_0$, the LES gridscale $\Delta$, and sub-grid scale (SGS) mixing length $C_s\Delta$ areexamined. We also present LES results on moments of spanwise and wall-normalfluctuations of velocity.
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