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