3.1 Spanwise-localized wall-mode and free-stream coherent structures in the ASBL

We successfully continued both EQ7-1 and EQ7-2 to ASBL conditions with $[L_{x},H,L_{z}]$ at $[4{\rm\pi},20,24{\rm\pi}]$ and $[5.5{\rm\pi},20,24{\rm\pi}]$ respectively, and with a resolution of $[48,121,512]$ grid points. In the ASBL, these states are spanwise-localized travelling waves similar to the free-stream and wall-mode travelling waves of DH14 respectively. We henceforth refer to them as the free-stream coherent structure (FCS) and the wall mode (WM) of the ASBL. After continuation into the ASBL, the FCS retains the shift-reflect symmetry of EQ7-1,

and the WM retains the $z$ -mirror symmetry of EQ7-2,

These symmetries were enforced throughout the continuations.

Figure 1 shows the bifurcation diagram of the FCS and WM states in the ASBL with $\mathit{Re}$ as control parameter. The two states appear as lower branches in saddle-node bifurcations at $\mathit{Re}=968$ for the FCS and $\mathit{Re}=348$ for the WM. The structure of the velocity fields shows little variation with $\mathit{Re}$ ; i.e. the states do not change qualitatively along the continuation. We choose to discuss them somewhat above their respective bifurcation points, at $\mathit{Re}=1000$ and $\mathit{Re}=400$ respectively.

Figure 1.

Bifurcation diagram of the FCS and WM solutions showing volume-averaged cross-flow energy $\langle E_{cf}\rangle _{V}=(1/V)\int _{V}(v^{2}+w^{2})\,\text{d}x\,\text{d}y\,\text{d}z$ as a function of Reynolds number. Both solutions emerge as lower branches in saddle-node bifurcations. The two circles indicate the states depicted in figure 2(c–f).

Figure 2.

(a,b) Two localized equilibria of plane Couette flow, (a) EQ7-1 and (b) EQ7-2 from Gibson & Brand (Reference Gibson and Brand2014) at $U_{\infty }H/{\it\nu}=1600$ . The colours at the boundaries of the box show the streamwise-averaged deviation of the streamwise velocity from the laminar profile. Isocontours of the ${\it\lambda}_{2}$ vortex detection criterion (Jeong & Hussain Reference Jeong and Hussain1995) are shown in purple. EQ7-1 has one pair of counter-rotating vortices, while EQ7-2 has two pairs. High- and low-speed streaks are indicated by isocontours of the deviation of the streamwise velocity from laminar flow: red for high-speed streaks (faster than laminar) and blue for low-speed streaks. (c,d) The FCS and WM travelling-wave solutions of the ASBL obtained by continuation from EQ7-1 and EQ7-2, with the same colour coding as in (a,b). (e,f) The roll–streak structures of the FCS and WM are indicated with isocontours of the streamwise-averaged perturbation velocity $u$ (colours) and streamwise-averaged vorticity ${\it\omega}_{x}$ (solid black for positive ${\it\omega}_{x}$ , dotted for negative). The FCS shown in (c,e) at $Re=1000$ consists of two weak high-speed streaks close to the wall and one low-speed streak that extends far into the free stream. The vortices are localized at $y\simeq 12$ , far above the laminar boundary layer. The WM shown in (d,f) at $Re=400$ contains two pairs of vortices, two low- and three high-speed streaks, all within the laminar boundary layer, $y\leqslant {\it\delta}_{99\,\%}\simeq 5$ .

Figure 2 compares the ASBL states with their progenitor states in plane Couette flow. Figure 2(a,b) shows EQ7-1 and EQ7-2 of plane Couette flow and (c,d) the corresponding FCS and WM states obtained by continuation to the ASBL, with vortices indicated by isosurfaces of ${\it\lambda}_{2}$ (Jeong & Hussain Reference Jeong and Hussain1995) in purple and velocity fluctuations by isosurfaces of the streamwise velocity in blue and red (see the figure caption for details). Figure 2(e,f) shows the roll–streak structure of the FCS and WM in the ASBL by isocontours of the downstream velocity (colours) and downstream vorticity (black). The first important observation is that both states are well localized in the spanwise direction to within a range of $|z|<10$ , considerably smaller than the boundaries of the computational domain at $L_{z}/2=12{\rm\pi}\simeq 38$ (see also figure 3).

Figure 3.

Localization properties of the FCS (blue) and WM (green) in the (a) spanwise and (b) wall-normal directions. Localization is indicated by the cross-flow energy $E_{cf}=v^{2}+w^{2}$ averaged in $x,y$ and $x,z$ respectively. The exponential drop-off in cross-flow energy with (a) $z$ and (b) $y$ confirms that the computational domain is sufficiently large. (b) The peak of the cross-flow energy for the FCS is at $y\simeq 12$ , far above the laminar boundary layer $y\leqslant {\it\delta}_{99\,\%}\simeq 5$ , while for the WM the peak is within it.

Like EQ7-1 in plane Couette flow, the FCS in the ASBL consists of a single pair of staggered and tilted counter-rotating vortices. These vortices are located at $y\simeq 12$ , far above the laminar boundary layer thickness at ${\it\delta}=1$ or the laminar $99\,\%$ boundary layer thickness at ${\it\delta}_{99\,\%}=4.6$ . Located below the vortices is a large low-speed streak, i.e. a region where the streamwise velocity is slower than in the surrounding laminar flow. The low-speed streak originates at the wall and ends below the vortices at $y\approx 12$ , creating a small shear from which the vortices draw energy. It should be noted that the streak contour lines are nearly as dense near the vortices as at the wall, indicating that the shear near the vortices is large, and that the large wall-normal extent of the low-speed streak in figure 2(e) is not an artefact of the choice of contour levels. Thus, a key property of the FCS is that its vortices are sustained not by the laminar shear near the wall, but by shear far from the wall that the solution creates itself. The vortices push fluid to the sides by linear advection, creating a weak but large-scale circulation. This circulation pushes fluid from the free stream towards the wall on the right and left sides of the vortices, creating two weak high-speed streaks, and pulls fluid up in the centre, creating the low-speed streak. The speed of the travelling wave is $c_{FCS}=0.905$ , notably fast compared with the speed of turbulent spots in boundary layers (which travel at roughly $c=2/3$ , Levin & Henningson (Reference Levin and Henningson2007)) but consistent with the $1-c\sim 100\times O(\mathit{Re}^{-1})$ scaling estimated by DH14. It should be noted that the laminar profile reaches this wavespeed ( $u_{L}(y)=c_{FCS}$ ) at $y=2.35$ .

There are several notable similarities between the spanwise-localized FCS (figure 2) and the spanwise-periodic free-stream coherent structure of DH14. Both are derived from the EQ7 plane Couette solution of Gibson, Halcrow & Cvitanović (Reference Gibson, Halcrow and Cvitanović2009). The shift-reflect symmetry (3.1) of the FCS is the symmetry that remains after breaking the $z$ -periodicity of the DH14 free-stream solution (their equation (3.4)). Comparison of figure 2(e) with figure 3(e) of DH14 reveals similar structure and wall-normal position of the counter-rotating vortices of the two solutions, despite the differences in streamwise wavelength and Reynolds number ( $L_{x}=4{\rm\pi},\mathit{Re}=968$ here versus $L_{x}=10{\rm\pi},\mathit{Re}=80\,000$ in DH14). In both solutions, the slow-speed streaks extend farther into the free stream than do the high-speed streaks, although for the DH14 free-stream solution, there is a clear concentration near the wall and decay away from it, whereas the low-speed streaks of the FCS have a more uniform core that extends farther into the free stream.

The WM in the ASBL is notably different from the FCS. The WM has two pairs of staggered counter-rotating vortices much closer to the wall, below $y={\it\delta}_{99\,\%}=4.65$ . By linear advection (lift-up effect), the vortices create and sustain two low-speed and three high-speed streaks which are also very near the wall. The vortices are slightly tilted and inclined in the downstream direction, a pattern that is commonly observed in wall-bounded shear flows (Adrian Reference Adrian2007). With all structures closely attached to the wall and the pattern of two vortices leaning over a low-speed streak, the WM is similar to the spanwise-localized edge states of Khapko et al. (Reference Khapko, Kreilos, Schlatter, Duguet, Eckhardt and Henningson2013), although in the latter, the crossing of the vortices over the streak destabilizes the streak, leading to breakup and reformation at a different position, and resulting in a time-dependent relative periodic orbit solution. The speed of the WM travelling wave is $c_{WM}=0.87$ , a speed reached by the laminar profile at $y=2.04$ . The slower wavespeed of the WM compared with the FCS is consistent with its closer proximity to the wall. The WM is clearly related to the TW2-2 near-wall travelling wave of plane Poiseuille flow of Gibson & Brand (Reference Gibson and Brand2014), both in their origin from EQ7-2 and in the structure of their vortices and streaks. How the WM relates to the wall mode of DH14 is less clear. The latter was continued from a different solution of sliding Couette flow and has vorticity concentrated much closer to the wall (comparing figure 2(d,e) with DH14 figure 3(b), albeit at different parameter values).

At a broad level, the FCS and the WM are composed of the same typical flow structures – streamwise streaks and downstream-oriented vortices – as almost all known invariant solutions in shear flows. The WM displays the familiar interactions known from the self-sustaining process (Hamilton, Kim & Waleffe Reference Hamilton, Kim and Waleffe1995; Waleffe Reference Waleffe1997; Schoppa & Hussain Reference Schoppa and Hussain2002) or the vortex–wave interaction of Hall & Sherwin (Reference Hall and Sherwin2010). In the FCS, however, the strong vortices are not supported by a linear instability of the underlying streak (DH14), as complex nonlinear interactions dominate.

3.2 Dynamical properties of the free-stream coherent structure

In this section, we show that the dynamical evolution of the FCS is detached from the wall. Figure 4 shows that the FCS has $O(10)$ unstable modes (eigenvalues with positive real part). The edge states of Khapko et al. (Reference Khapko, Kreilos, Schlatter, Duguet, Eckhardt and Henningson2013, Reference Khapko, Duguet, Kreilos, Schlatter, Eckhardt and Henningson2014), in comparison, have a single unstable eigenvalue by construction. Figure 5 shows the roll–streak structure of the four leading unstable eigenmodes. These are the most dynamically relevant eigenmodes, since they will dominate the evolution of trajectories that approach the FCS transiently. The leading eigenmodes are concentrated around the vortical structures of the FCS at $y\simeq 12$ , far from the wall. The linear dynamics near the FCS is thus concentrated near the free-stream part of the solution, and the streaks that extend close to the wall do not play a significant role.

Figure 4.

The leading eigenvalues of the FCS at $Re=1000$ with restriction to the shift-reflect symmetry subspace (3.1). With an increased resolution of $[64,121,576]$ grid points, the leading 10 eigenvalues differ by $10^{-4}$ , confirming that the solution is well converged.

Figure 5.

The four leading eigenmodes of the FCS. For complex eigenmodes, the real part is shown. The left part of each plot shows isocontours of the streamwise-averaged deviation of the streamwise velocity from laminar flow, the right part the isocontours of the streamwise-averaged streamwise vorticity. For all eigenmodes, the streaks and roll structures are localized in the free stream, close to the vortices of the FCS.

Figure 6.

Space–time plots of the transition from FCS to turbulence at $Re=1000$ . The averaged cross-flow energy $E_{cf}$ for the growth of a small perturbation of the FCS is shown as a function of time and the (a) wall-normal and (b) spanwise directions. The perturbation remains small for approximately $350$ advective time units. After $t\simeq 350$ , the perturbations grow slowly in the spanwise direction and from the free stream towards the wall. When perturbations reach the laminar boundary layer $y\leqslant {\it\delta}_{99\,\%}$ at approximately $t\simeq 800$ , they strengthen and later spread rapidly along the wall, as shown by the increased slope of the red area in (b).

To study the dynamics of transients near the FCS, we perturb the FCS slightly and follow the time evolution of the perturbed initial condition. Here, we present results obtained by rescaling the FCS by a factor of $1.01$ ; the qualitative results are unchanged if a small random perturbation is added instead. A movie in the supplementary material available online at http://dx.doi.org/10.1017/jfm.2016.242 shows the evolution in time. Figure 6 shows space–time plots of the cross-flow energy $E_{cf}=v^{2}+w^{2}$ (Kreilos et al. Reference Kreilos, Veble, Schneider and Eckhardt2013) as a function of $y$ and as a function of $z$ , in each case averaged over the remaining two directions. No changes are visible for the first 350 time units as the exponential amplifications of the linearly unstable modes grow to large amplitudes. (This time of course depends on the magnitude of the initial perturbation.) At $t\simeq 350$ , velocity perturbations near the vortical structures at $y\simeq 12$ and $z=0$ grow to amplitudes comparable to the FCS velocity field. These perturbations spread in the $y$ direction mostly towards the wall and in the spanwise direction at roughly equal but slow propagation speeds. This slow spreading continues until the fluctuations reach the laminar boundary layer at $t\simeq 900$ . After this, the fluctuations strengthen and energetic turbulence begins to spread along the wall and also back into the free stream. The rate of spanwise spreading is notably faster after the perturbations for $t\geqslant 1000$ . This indicates that there are two different mechanisms by which turbulence spreads from the FCS: a slow growth of weak perturbations in the free stream followed by a faster more energetic growth fuelled by the shear of the laminar boundary layer.

Figure 7.

Time evolution of the FCS, illustrating the spreading of weak perturbations towards the wall, followed by growth in energy and faster spreading in the spanwise direction ( $t=700$ (a), 900 (b), 1100 (c), 1300 (d), 1500 (e), 1700 (f)). The plots show isocontours of the downstream velocity fluctuations in colours and isocontours of the downstream vorticity in black.

The observation that turbulence spreads rapidly once it reaches the wall is confirmed by the plots of the roll–streak structure in figure 7, with six snapshots of the trajectory taken at equidistant time intervals ${\rm\Delta}t=200$ . In (a), the state has not changed much from the FCS and the vortical structures (black isocontours) are located high in the free stream. In (b), the perturbations have grown to the laminar boundary layer, where they are strongest. In (c–f), the stronger perturbations spread in the spanwise direction and start to grow in the wall-normal direction, creating a larger turbulent boundary layer, in which disturbances would ultimately grow to reach the upper boundary of the computational domain (Schlatter & Örlü Reference Schlatter and Örlü2011).