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 turbulent high-order 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 quite a long time to converge. A convergence study was performed and for fully converged statistics the sensitivity of the wall-normal kurtosis component value on 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 scaled 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 and are the probable cause of 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.