Azimuthal organization of large-scale motions in a turbulent minimal pipe flow

Direct numerical simulation data for turbulent minimal pipe flows with Re-t = 927, 1990, and 2916 are examined to explore the azimuthal (or spanwise) organization of their large-scale structures. We chose a streamwise-minimal unit with a streamwise domain length of L-x(+) approximate to 1000, which is the characteristic streamwise length of near-wall streaks. The spanwise scales of most of the energetic motions and their contributions to the total energy are comparable with those of the streamwise long-domain simulation. In the azimuthal energy spectra of the streamwise velocity fluctuations (u), the large-scale energy increases with Re, and three outer peaks (lambda(theta) = 0.7-0.8, pi/2 and pi) become evident when Re-t = 2916. The presence of the outer peaks at lambda(theta) = 0.7-0.8 and pi/2 is consistent with the results of the long domain simulation. The peak at lambda(theta) = 0.7-0.8 is associated with large-scale motions and the other two peaks are associated with very-large-scale motions (VLSMs). The maximum spanwise wavelength increases linearly with the wall normal distance from the wall. A k(g)(-1) region is evident in the range 0.3R < lambda(z) (=r lambda(theta)) < R, which indicates the presence of self-similar motions. The conditional two-point correlation with a cut-off wavelength of lambda(z) = 0.9R shows that there is a strong correlation between the enhanced energy in the outer region and the wall attached structures, which were extracted from the time evolution of the streamwise-averaged u field (u(2D)). The spanwise sizes (l(z)) of the attached u(2D) structures scale with their height (l(y)) in the log region and their time scales (l(t)) follow l(t)u(t)/l(z) = 2, which is consistent with the bursting time scale. Their span wise sizes lie in the range R < l(z) < 3R, for which l(t) increases significantly, which indicates that these structures are associated with VLSMs and make the dominant contributions to the enhanced energy in the outer region. These structures penetrate to the wall region as a manifestation of the footprint and modulate the small-scale energy. The negative-u(2D) structures induce congregative motions in the near wail region. Published under license by AIP Publishing