Self-similar motions for the logarithmic variation of spanwise turbulence intensity in turbulent pipe flow*

According to Townsend’s attached eddy hypothesis, the streamwise and spanwise turbulence intensities exhibit a logarithmic variation at extremely high Reynolds numbers (Re). However, the spanwise component shows a logarithmic variation even at very low Re, absent in the streamwise component. We investigate this discrepancy by focusing on self-similar motions responsible for the logarithmic variation of the wall-parallel component. We examine the direct numerical simulation dataset of turbulent pipe flows at Re_τ = 1000–6000. The two-dimensional (2D) energy spectra of the streamwise and spanwise velocities (Ф_uu and Ф_ww) are decomposed using linear stochastic estimation based on the cross-spectrum across the near-wall and logarithmic regions. The 2D spectra are decomposed into two components: Ф_ss and Ф_vl. Ф_vl represents the cumulative energy contribution from tall wall-attached motions. Ф_ss, obtained by subtracting Ф_vl from Ф, scales well with the friction velocity and wall-normal distance, indicating self-similar motions. The energy in Ф_vl, associated with very-large-scale motions, can distort the self-similar scaling in the logarithmic region. For w, Ф_ss aligns along the linear relationship between the wall-parallel wavelengths over a broad range, and the energy contained in Ф_vl is weak compared to the streamwise component. These differences in spectral contributions explain the presence of the logarithmic variation of the spanwise turbulence intensity at lower Re.