Large-eddy simulation study of the logarithmic law for second and higher-order moments in turbulent wall-bounded flow

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.