**5. Discussion**

The wave generation at the head, the tilt direction, the advection of the head, the streaks inside turbulent bands and the tilt angle of the band are discussed and investigated in this paper. The inflectionalinstability argument of Xiao and Song [24] for the wave generation at the head and its potential relationship with the localized periodic-orbit theory of Kanazawa [7] are discussed. Based on the discussion, we propose that the tilt direction should probably be determined by the local inflectional spanwise velocity profile generated/introduced by the initial perturbation. The opposite tilt directions are rooted in the mirror symmetry of the spanwise velocity component. Besides, we measured the propagation speed of the head and the advection speed of the low-speed streaks in the bulk of turbulent bands at low Reynolds numbers up to *Re* = 1050. We found that the head propagates at constant speeds of *cx* = 0.85 and *cz* = 0.1 (absolute value) at all Reynolds numbers investigated. The low-speed streaks are advected roughly at the speed of the bulk speed in the streamwise direction with a slight decreasing trend as the Reynolds number increases, and the spanwise advection speed is nearly constant at approximately 0.07. Prior studies measured the tilt angle by treating the band as a tilted object [6,9]; alternatively, we here propose that the tilt angle of turbulent bands should be determined by the kinematics of the head and the streaks generated at the head. Specifically, the tilt angle can be calculated using the relative speed between the streaks in the bulk and the head, and, at least for *Re* 900, we obtained a good agreement with the experimental measurements of Paranjape [9]. We also speculate that the tilt angle of a band may be indefinite and system-dependent if the head does not exist as in narrow tilted domains and relatively small normal domains.

A few problems remain poorly understood and should be investigated in order to further understand the transition in channel flow.


lower than the values shown in Figure 2, which is about 0.85. This discrepancy may be attributed to the over-simplification of the local mean flow at the head by temporal and spatial averaging in their linear stability analysis, as well as by the region selection for the averaging. A possibility to elucidate the mechanism underlying the advection speed is to investigate the speed of the periodic orbit of Kanazawa [7].

