*2.2. Wave Generation at the Head and the Tilt Direction of the Band*

In this section, firstly we discuss about some recent studies on the dynamics of the head. Therefore, a part of the results shown below is not original. It has been noticed that the head drives the growth of turbulent bands by continually generating waves, in the form of alternating high- and low-speed streaks and arrays of vortices, while moving into the adjacent laminar region [7–9,24]. Figure 3 shows the wave-like structure of the head. Contours of streamwise velocity fluctuation are plotted in the *x*-*z* plane at *y* = −0.8 (close to the wall, see Figure 3a), at *y* = −0.5 (Figure 3b) and in the mid-plane *y* = 0 (see Figure 3c). It can be seen that the flow is characterized by high-speed streaks close to the wall. In the mid-plane, the flow is characterized by low-speed streaks in the bulk, which almost merge and form a connected low-speed region, and is characterized by a high speed region at the head (see the yellow spot in Figure 3c). At *y* = −0.5, the flow exhibits wave-like alternating low and high-speed streaks. The large-scale (compared with the wave-like streaky structures) flow in the neighborhood of the head manifests a circulation (see [6,24]), which is counter-clockwise for a right-going band as shown. Duguet and Schlatter [27] proposed a mechanism for the formation of large-scale flow around turbulent bands in plane-shear flows. Their theory applies to the large-scale flow associated with the bulk region of the band and describes the band as the advection of small-scale structures (streaks) by the large-scale flow. However, they did not explicitly study the large-scale flow at the head.

**Figure 3.** Large-scale circulation flow in the neighborhood of the head. Streamwise velocity is plotted as the colormap in the *x*-*z* cut plane at: *y* = −0.8 (**a**); *y* = −0.5 (**b**); and *y* = 0 (**c**). In each panel, the in-plane velocities are plotted as vectors. The dotted rectangle (size 6 *h* × 10 *h*) marks approximately the area where the first visible wave that is continually generated at the head in the frame of reference co-moving with the head.

The dotted rectangle in Figure 3 marks the approximate region in which the first visible high speed streak is periodically generated. The vector plot of the in-plane velocities shows that, at *y* = −0.5 (see Figure 3b), the vectors in the rectangle overall point to the positive *z* direction, and, at *y* = 0 (see Figure 3c), the vectors overall point to the negative *z* direction. This hints that there should be an inflection in the spanwise velocity profile in this region, which may be inflectionally unstable. Based on this observation, Xiao and Song [24] investigated the local mean flow at the head and attributed the wave generation at the head to an inflectional instability associated with the modified local mean flow. For the ease of discussion, we measured the averaged velocity profiles at the head again in a different region and for a different turbulent band compared to those reported in [24] (see Figure 4). Both streamwise and spanwise velocity profile (the parabolic base flow is not included) show inflection. These profiles are measured at a right-going band similar to the upper one in Figure 1 and the one shown in Figure 3. Figure 4b shows the unstable region in the wavenumer plane (the region enclosed by the bold line) and Figure 4c shows the streaky flow pattern of the most unstable disturbance (see also [24,25]). It can be seen that these streaks are tilted about the streamwise direction and the tilt direction is the same as the waves that can be seen at the head of right-going bands in Figures 1 and 3. Besides, the most unstable wave move downward, i.e., in the negative spanwise direction (see the arrow), just as the head of the right-going band. By the symmetry of channel flow about the *x*-*y* plane, it can be inferred that the velocity profiles at the head of a left-going band will be similar to those shown in Figure 4d, with the sign of the spanwise velocity changed. We performed a similar linear analysis here and show the unstable region in the wavenumber plane in Figure 4e and the most unstable disturbance in Figure 4f. Clearly, we can see a spanwise symmetry in the distribution of eigenvalues and in the flow pattern by comparing to Figure 4b,c. The waves shown in Figure 4f are tilted in the opposite direction compared with the waves in Figure 4c and move in the positive spanwise direction, which is consistent with the structure and kinematics of the head of a left-going band. In a word, linear stability analysis gives qualitatively similar flow structures and kinematics as that of the head. The nonlinear development of disturbances was shown to give similar flow structures as those at the head [24]. Therefore, Xiao and Song [24] proposed that the growth of turbulent bands is driven by the inflectional instability locally at the head. Further, Song and Xiao [25] performed a non-modal analysis of the inflectional velocity profiles and showed an Orr-mechanism via which disturbances can achieve a fast growth in energy at the early stage (by a factor of 100 within about 15 time units for *Re* = 750). Subsequently, the modal instability takes part and starts to dominate the growth at later points of time. The linear instability together with the fast non-normal growth at the early stage are able to result in a fast growth of the unstable waves at the head. Reaching a certain amplitude, the waves become turbulent when nonlinearity sets in and subsequently evolve inside the bulk of the band in the form of streaks and vortices.

Based on these discussions, here we propose that the moving direction and the tilt direction of a band are probably determined by what type of local flow is formed when a localized perturbation is introduced: One similar to that shown in Figure 4a generates a right-going band and one similar to that shown in Figure 4d generates a left-going band. In fact, the technique proposed by Song and Xiao [25], with which we generated the bands in Figure 1, is based on this mechanism. The key of the technique is to impose a localized body force that moves with the speed of the head and induces a locally inflectional flow. It can offer a control on the tilt direction of the generated bands because it offers a control on the spanwise velocity profile (to be similar to either the one in Figure 4a or the one in Figure 4c) and on the moving direction of the force. The efficacy of the technique in turn supports that some key characteristics of a band are determined by the local inflectional mean flow at the head.

**Figure 4.** Linear instability of the modified velocity profile at the head that is spatially and temporally averaged in the rectangle shown in Figure 3b: left-going bands (**a**–**c**); and right-going bands (**d**,**e**). (**a**,**d**) Velocity profiles. Wall-normal component is very small and neglected; (**b**,**d**) The maximum eigenvalue in the wave number plane, in which *α* is the streamwise wave number and *β* the spanwise wavenumber. The bold line marks the neutral stability curve; (**c**,**f**) Contours of streamwise velocity of the most unstable disturbances at the cut plane *y* = −0.5. Red and blue colors represent high speed and low speed regions, respectively. The arrows show the direction of the spanwise wave speed. Similar analysis for the modified velocity profiles averaged in different regions were reported in [24,25].

Although the linear instability, as well as the non-normality, associated with the local mean flow seem to be the mechanism underlying the wave generation and growth of turbulent bands, how this inflectional local mean flow is formed and sustained is still not sufficiently understood. Tao et al. [6] observed that, when the computational domain is too small, a band may interact with its periodic image and decay. Based on this observation, they proposed that the sustainment of a turbulent band relies on the secondary large-scale flow surrounding the band, and a close neighbor may affect this large-scale flow and eliminate the band. Given that a turbulent band is driven by the head, this observation seems to imply that the head of a band is sustained by the large-scale flow, see Figure 3. However, Kanazawa [7] proposed a completely different scenario. They added a damping term to the Navier–Stokes equations, using which they suppressed the formation of the body of a band and isolated the head, and observed that the head can be self-sustained as a nonlinear periodic orbit. This periodic orbit is characterized by an array of streaks and vortex tubes that resemble the flow structure at the head. Because the band does not form under the damping, the large-scale flow is also absent, although there is still a local circulation flow associated with the localized periodic orbit itself. This seems to contradict the conclusion of Tao et al. [6] that a band relies on the large-scale flow surrounding the band. Further, Kanazawa [7] studied the bifurcation of the periodic orbit in the damped system and reported a saddle-node bifurcation that gives rise to the periodic orbit. Below the saddle-node bifurcation point, no such exact coherent structures exist. Therefore, the authors proposed that this self-sustained periodic orbit and the subsequent bifurcations to torus and chaos is responsible for the formation and sustaining mechanism of turbulent bands. However, they failed to obtain a periodic orbit and reproduce the bifurcations as the damping parameter vanishes, i.e., in the

Navier–Stokes equations without an artificial damping. Obtaining such a periodic solution may finally elucidate the appearance and self-sustaining mechanism of fully localized turbulent bands [7].

Kanazawa [7] did not show why and how exactly this periodic orbit generates wave-like streaks or vortices, rather, only described them as the characteristics of the periodic orbit. In fact, the inflectional instability proposed by Xiao and Song [24] may be related to this periodic orbit. The possible connection is that the circulation associated with the periodic orbit may be locally inflectional and responsible for the wave generation. The inflectional profiles of Xiao and Song [24] are just temporal-spatial averages at the head and only depend on *y*. The averaging leaves out the streamwise and spanwise dependence of the real local flow at the head; therefore, Xiao and Song [24] pointed out that this may be why their stability analysis cannot quantitatively capture some characteristics of the waves at the head, such as the value of the tilt angle of the waves with respect to the streamwise direction. The analysis of this three-dimensional periodic orbit may be needed to more quantitatively understand the dynamics of the head.

## **3. The Bulk**

The bulk of a turbulent band is defined as the elongated part that is sufficiently far from the head and tail, which does not significantly vary on large-scale and can be considered to be at an 'equilibrium state'.
