On the convergence and scaling of high-order statistical moments in turbulent pipe flow using direct numerical simulations

Abstract

Direct numerical simulations of turbulent pipe flow in a flow domain of length L = 42R, friction Reynolds number in the range of 180 ≤ Reτ ≤ 1500, and two different wall-normal grid refinements were carried out and investigated in terms of high-order turbulence statistics. The phenomenology of large local wall-normal velocity fluctuations (velocity spikes) was discussed by means of time series and instantaneous flow-field realisations. Due to their rare appearance both in space and time, statistical high-order moments take a long time to converge. A convergence study was performed and for fully converged statistics the sensitivity of the grid resolution on the wall-normal kurtosis component value at the wall as well as the scaling behaviour of high-order statistics was investigated. The streamwise Reynolds stress as well as the streamwise skewness and the wall-normal flatness exhibited logarithmic Reynolds number dependencies in the vicinity of the wall and scaling laws were derived accordingly. In the bulk flow region, a sudden increase in magnitude in both the streamwise Reynolds stress and skewness was determined for the largest Reynolds number Reτ = 1500, while the profiles collapsed well in wall units for Reτ ≤ 720. Both Reynolds number dependencies in the near-wall and the bulk region could be related to large-scale outer-flow motions penetrating the buffer layer. While wavelengths related to larger-scale motions (λz ≈ 3R) were computed for Reynolds numbers up to Reτ = 720 by means of two-dimensional two-point velocity correlations, even larger wavelengths related to very-large-scale motions appeared for Reτ = 1500. They are probably the reason for the sudden increase in magnitude of streamwise Reynolds stress and skewness, respectively. With the aid of instantaneous flow-field realisations and conditional averaged statistics, the Reynolds dependency of the wall-normal flatness value at the wall was related to the scaling failure of the streamwise Reynolds stress peak. For the lowest Reynolds number (Reτ = 180), discrepancies between plane channel and pipe flow were found and discussed.