3.1 Streak amplification

Figure 2 reports the instantaneous streamwise velocity fluctuation of case S960 on increasing the forcing amplitude. Given the spanwise wavelength of the forcing ( $\unicode[STIX]{x1D706}_{z}/h=1.5$ ), two streaks are driven in our computational domain (see table 1). Here, for brevity, we only present half of the spanwise domain. As the forcing amplitude is gradually increased (from (a) to (c) in figure 2), a streaky structure covering the entire streamwise domain clearly emerges. It is also interesting to note that the highly amplified streak appears to meander along the streamwise direction (figure 2 c) and this is clearly seen in figure 3, where a top view of figure 2(c) is given. This meandering behaviour of the driven streak is reminiscent of that reported for VLSMs in a turbulent boundary layer (Hutchins & Marusic Reference Hutchins and Marusic2007). However, more quantitative discussion of this feature will be deferred to §§ 3.2 and 3.3.

Figure 3.

Top view of the isosurfaces of instantaneous streamwise velocity ( ${u^{\prime }}^{+}=-3.5$ ) in figure 2(c). Here, the flow direction is from left to right.

The streaky structure driven by the body forcing in the numerical simulations is detected statistically using the definition of the triple decomposition in (2.1). Since the applied body forcing is steady and uniform in the streamwise direction, it is evident that the ensemble average in (2.1) is replaced by the average in time and the streamwise direction. The organised wave in a simulation with the forcing is then obtained by

(3.2) $$\begin{eqnarray}\tilde{\boldsymbol{u}}_{s}=\langle \boldsymbol{u}-\boldsymbol{U}_{0}\rangle _{x,t},\end{eqnarray}$$

where the subscript ‘ $s$ ’ on the left-hand side denotes the organised wave component obtained by a numerical simulation and $\langle \cdot \rangle _{x,t}$ indicates the average in time and in the streamwise direction. We also recall that $\boldsymbol{U}_{0}=(U_{0}(y),0,0)$ is the mean velocity of the reference (unforced) simulation.

Figure 4.

Mean cross-stream flow field for S960: (a) $\Vert \boldsymbol{f}h/u_{\unicode[STIX]{x1D70F},\mathit{ref}}^{2}\Vert =0.31$ ( $A_{s}^{+}=1.44$ ), and (b)  $\Vert \boldsymbol{f}h/u_{\unicode[STIX]{x1D70F},\mathit{ref}}^{2}\Vert =1.90$ ( $A_{s}^{+}=4.04$ ). The contours indicate the streamwise velocity, and the arrows represent the wall-normal and spanwise velocities. The contours are normalised by the centreline velocity of each simulation.

The cross-streamwise view of the mean velocity ( $\langle \boldsymbol{u}\rangle _{x,t}=\boldsymbol{U}_{0}+\tilde{\boldsymbol{u}}_{s}$ ) is shown in figure 4 for two different forcing amplitudes. The appearance of a low-velocity region is evident around $z/h=0.75$ , where the streamwise momentum is expected to be pumped away from the wall by the counter-rotating vortical forcing (see also figure 1). The low-speed region extends along the entire wall-normal location, including the logarithmic layer ( $0.05<y/h<0.2$ at this Reynolds number). Two symmetric high-velocity regions arise both sides of the low-velocity region, where the streamwise momentum is supposed to be brought down to the wall by the forcing. An increase in the forcing amplitude results in the narrower spanwise extent of the low-speed region (compare figure 4 a with 4 b), as was also reported for the near-wall case (Cassinelli et al. Reference Cassinelli, de Giovanetti and Hwang2017). Finally, it should be mentioned that $\tilde{\boldsymbol{u}}_{s}$ is dominated by the streamwise velocity: for example, when $\Vert \boldsymbol{f}h/u_{\unicode[STIX]{x1D70F},\mathit{ref}}^{2}\Vert =1.91$ , the maximum streamwise component of $\tilde{\boldsymbol{u}}_{s}$ is ${\tilde{u} ^{+}}_{s,max}=4.76$ , while its wall-normal counterpart is only ${\tilde{v}^{+}}_{s,max}=1.01$ (the superscript ‘ $+$ ’ denotes scaling in the inner units). This indicates that the formation of the streaky mean flow is a consequence of the lift-up effect, as explained in detail in § 2.1.

Using the computed $\tilde{\boldsymbol{u}}_{s}$ , two quantities are computed to measure the response to the forcing. The first one is the norm of $\tilde{\boldsymbol{u}}_{s}$ ,

(3.3) $$\begin{eqnarray}\Vert \tilde{\boldsymbol{u}}_{s}\Vert =\sqrt{\frac{2}{V}\int _{V}\tilde{\boldsymbol{u}}_{s}^{\text{H}}\tilde{\boldsymbol{u}}_{s}\,\text{d}V},\end{eqnarray}$$

which represents the kinetic energy of the induced motion averaged over the given computational domain. We note that, if the forcing amplitude is small and the eddy viscosity in (2.3) correctly models the surrounding turbulence, $\tilde{\boldsymbol{u}}_{s}$ from (3.2) would be identical to $\tilde{\boldsymbol{u}}$ from the linear model (2.2). Although, in practice, this is unlikely to happen due to the crude nature of the eddy-viscosity model, computation of $\Vert \tilde{\boldsymbol{u}}_{s}\Vert$ as in (3.3) would be useful to assess the linear model (2.2), enabling the quantitative comparison of the amplification by the forcing in a real turbulent flow with that in the linear model (2.2). The second measure of the response to the forcing is chosen to be the streak amplitude and is defined as

(3.4) $$\begin{eqnarray}A_{s}^{+}={\textstyle \frac{1}{2}}(\max _{y,z}[\tilde{u} _{s}^{+}(y,z)]-\min _{y,z}[\tilde{u} _{s}^{+}(y,z)]),\end{eqnarray}$$

where $\tilde{u} _{s}^{+}$ is the streamwise component of $\tilde{\boldsymbol{u}}_{s}$ . This definition follows that used in a laminar boundary layer and previous studies (Andersson et al. Reference Andersson, Brandt, Bottaro and Henningson2001; Park et al. Reference Park, Hwang and Cossu2011; Alizard Reference Alizard2015; Cassinelli et al. Reference Cassinelli, de Giovanetti and Hwang2017).

Figure 5.

Flow response to the forcing for S960: (a) $\Vert \tilde{\boldsymbol{u}}_{s}^{+}\Vert$ , and (b) $A_{s}^{+}$ . The definitions of $\Vert \tilde{\boldsymbol{u}}_{s}^{+}\Vert$ and $A_{s}^{+}$ are given in (3.3) and (3.4), respectively.

The evolution of the two quantities on increasing the forcing amplitude is shown in figure 5. As expected, both $\Vert \tilde{\boldsymbol{u}}_{s}^{+}\Vert$ and $A_{s}^{+}$ increase linearly with $\Vert \boldsymbol{f}\Vert$ at small forcing amplitudes (i.e.  $\Vert \boldsymbol{f}h/u_{\unicode[STIX]{x1D70F},\mathit{ref}}^{2}\Vert \lesssim 0.5$ ). The linear fit is found to follow $\Vert \tilde{\boldsymbol{u}}_{s}^{+}\Vert /\Vert \boldsymbol{f}h/u_{\unicode[STIX]{x1D70F},\mathit{ref}}^{2}\Vert \simeq 6.2$ . This value is fairly close to $\Vert \tilde{\boldsymbol{u}}^{+}\Vert /\Vert \tilde{\boldsymbol{f}}h/u_{\unicode[STIX]{x1D70F}}^{2}\Vert =4.0$ obtained with the linear theory based on (2.2) (note that $\boldsymbol{f}=\tilde{\boldsymbol{f}}$ ), suggesting that the eddy viscosity in (2.3) is a reasonably good approximation (see also § 4.4). When the forcing amplitude is large enough, neither $\Vert \tilde{\boldsymbol{u}}_{s}^{+}\Vert$ nor $A_{s}^{+}$ increase linearly any more, showing a nonlinear behaviour. The largest forcing amplitude considered for the S960 case is $\Vert \boldsymbol{f}h/u_{\unicode[STIX]{x1D70F},\mathit{ref}}^{2}\Vert =8.09$ . However, as we shall see in § 4.1, this forcing amplitude is too high to study the physical processes in a natural unforced flow. Therefore, forcing amplitude greater than this value is not considered.

3.2 Streak instability: spectra and root mean square

Figure 6.

Premultiplied streamwise wavenumber spectra of wall-normal velocity (S960): (a)  $A_{s}^{+}=0$ (unforced); (b) $A_{s}^{+}=0.54$ ; (c) $A_{s}^{+}=1.44$ ; (d) $A_{s}^{+}=2.96$ ; (e) $A_{s}^{+}=3.42$ ; (f)  $A_{s}^{+}=4.04$ ; (g) $A_{s}^{+}=4.76$ ; and (h)  $A_{s}^{+}=5.30$ . The thick white dashed line indicates $y=0.35\unicode[STIX]{x1D706}_{x}$ .

Now, we investigate the emergence of other flow structures from the driven streamwise-uniform streaks. The focus of this section is given to the detection of wall-normal and spanwise velocity fluctuations at $1<\unicode[STIX]{x1D706}_{x}/h<5$ , as this is the main statistical feature of LSMs (Hwang Reference Hwang2015). All velocity spectra analysed in this section have been normalised using the friction velocity of the reference (unforced) simulation. Figure 6 shows the evolution of the streamwise wavenumber spectra of the wall-normal velocity, as the streak (and forcing) amplitude is increased. In the absence of any forcing (figure 6 a), the spectra are well aligned along the linear ridge $y=0.35\unicode[STIX]{x1D706}_{x}$ , as pointed out by Hwang (Reference Hwang2015). At the top end of the linear edge (i.e.  $y/h\geqslant 0.4{-}0.6$ ), the spectra show their peak at $\unicode[STIX]{x1D706}_{x}/h\simeq 1{-}2$ , as in other direct numerical simulations (e.g. Hoyas & Jiménez Reference Hoyas and Jiménez2006). Although this streamwise wavelength is a little shorter than $\unicode[STIX]{x1D706}_{x}/h\simeq 3{-}4$ observed in the streamwise velocity spectra (Monty et al. Reference Monty, Hutchins, Ng, Marusic and Chong2009), it is evident that this part would correspond to LSMs, given its wall-normal location and streamwise wavelength. Application of the forcing is found to significantly change the wall-normal velocity spectra. Even at fairly small forcing amplitudes (figure 6 b,c), the spectral intensity around the channel centre for $\unicode[STIX]{x1D706}_{x}/h\simeq 1{-}2$ is slightly elevated compared to the unforced case (figure 6 a). As the forcing amplitude is further increased, the spectra reveal a well-manifested peak at $\unicode[STIX]{x1D706}_{x}/h\simeq (1{-}2)h$ and $y/h\geqslant 0.4{-}0.6$ , consistent with the location of LSMs (figure 6 d,e). At excessively large forcing amplitudes (figure 6 $f$ – $h$ ), excitation at larger streamwise length scales is observed. However, the excited streamwise wavelength does not exceed $\unicode[STIX]{x1D706}_{x}/h\simeq 10$ at least for $A_{s}^{+}<5$ (figure 6 g).

Through the elevation of the forcing amplitude, it has been consistently observed that the forcing significantly affects the structures at smaller length scales in the near-wall and logarithmic regions. Indeed, the increase in the forcing amplitude gradually reduces the activity in the logarithmic region (figure 6 b–e), significantly weakening even the near-wall processes eventually (figure 6 f–h). It is not clear yet why the application of such a body forcing significantly inhibits the processes at smaller scales in the near-wall and logarithmic regions. However, this feature reminds us of the suppression of the near-wall process, previously observed when a large-scale counter-rotating vortical body forcing is applied at low Reynolds numbers (Schoppa & Hussain Reference Schoppa and Hussain1998; Willis et al. Reference Willis, Hwang and Cossu2010; Canton et al. Reference Canton, Örlü, Chin and Schlatter2016). Indeed, the inhibition of the structures at smaller scales in figure 6 is consistent with previous observations. However, in the present study, the suppression of the near-wall and logarithmic regions did not yield any skin-friction reduction, unlike previous studies at low Reynolds numbers. This is because the forcing itself has elevated the skin-friction generation process at the scale of the applied forcing via the enhanced lift-up effect (see also the discussion in § 3.1). Finally, we note that the observed suppression of the processes at smaller scales is clearly due to an interaction of the forcing with other scales. Therefore, further discussion on this issue is beyond the scope of the present study.

The behaviour of the spanwise velocity spectra on increasing the forcing amplitude has also been found to be qualitatively the same as that of the wall-normal velocity spectra. Figure 7 shows the streamwise wavenumber spectra of the spanwise velocity for the unforced and forced cases. Similar to the wall-normal velocity spectra, the forcing resulting in $A_{s}^{+}\simeq 3{-}4$ excites the spectral intensity at $\unicode[STIX]{x1D706}_{x}/h\simeq 1{-}2$ and $y/h\geqslant 0.4{-}0.6$ , while weakening the activity of the structures in the near-wall and logarithmic regions.

Figure 7.

Premultiplied streamwise wavenumber spectra of spanwise velocity (S960): (a)  $A_{s}^{+}=0$ (unforced), and (b) $A_{s}^{+}=3.42$ . The thick white dashed line indicates $y=0.35\unicode[STIX]{x1D706}_{x}$ .

The observation made with figures 6 and 7 suggests that an intense cross-streamwise velocity structure at $\unicode[STIX]{x1D706}_{x}/h\simeq 1{-}2$ is excited at low amplitudes of the body forcing. It is important to remember that the applied forcing is uniform in the streamwise direction ( $\unicode[STIX]{x1D706}_{x}=\infty$ ); thus this cross-streamwise velocity structure at a finite streamwise length scale should not be the direct outcome of the forcing. Instead, it suggests that another physical mechanism is at play in the generation of this structure. In this respect, it is encouraging to compare the observed streamwise wavelength of the cross-streamwise velocity structure with that of the streak instability in the previous theoretical analysis. The stability analysis using an eddy-viscosity model (Park et al. Reference Park, Hwang and Cossu2011) predicted that a streaky base flow with $\unicode[STIX]{x1D706}_{z}/h=4$ becomes unstable at $\unicode[STIX]{x1D706}_{x}/h\simeq 3{-}6$ . At first glance, this prediction does not seem to match well with $\unicode[STIX]{x1D706}_{x}/h\simeq 1{-}2$ in the present numerical experiment. However, the spanwise spacing of the streak in the present study is $\unicode[STIX]{x1D706}_{x}/h\simeq 1.5$ , considerably smaller than $\unicode[STIX]{x1D706}_{z}/h=4$ in Park et al. (Reference Park, Hwang and Cossu2011). Indeed, a recent extension of the theoretical analysis to smaller spanwise wavelengths showed that there exists a self-similar relation between the spanwise wavelength of the imposed streak and the streamwise wavelength of its instability (Alizard Reference Alizard2015). If this is adopted, a streak at $\unicode[STIX]{x1D706}_{z}/h\simeq 1.5$ would become unstable at $\unicode[STIX]{x1D706}_{x}/h\simeq 1.13{-}2.25$ , showing a fairly good agreement with $\unicode[STIX]{x1D706}_{x}/h\simeq 1{-}2$ observed in the present study.

Figure 8.

Cross-streamwise view of the root mean square of the velocities for (a,c,e)  $A_{s}^{+}=0$ and (b,d,f)  $A_{s}^{+}=3.42$ (S960): (a,b) streamwise velocity; (c,d) wall-normal velocity; and (e,f) spanwise velocity. The thick white line indicates the location where the cross-streamwise mean velocity is $0.9U_{c}$ ( $U_{c}$ is the centreline velocity of the simulation).

Further evidence of the streak instability mechanism is found in the cross-streamwise view of the velocity fluctuations in figure 8, where the left and right columns visualise the unforced and forced cases, respectively. We remind the reader that the streaky mean flow shows an even symmetry about $z/h=0.75$ (figure 4) and that the eigenstructure of a sinuous-mode streak instability is mathematically defined by odd symmetry of the streamwise and wall-normal velocities and by even symmetry of the spanwise velocity about this axis (see e.g. Andersson et al. Reference Andersson, Brandt, Bottaro and Henningson2001), i.e. 

(3.5) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}{\check{u}}(y,z^{\ast })=-{\check{u}}(y,-z^{\ast }),\\ \check{v}(y,z^{\ast })=-\check{v}(y,-z^{\ast }),\\ \check{w}(y,z^{\ast })=\check{w}(y,-z^{\ast }),\end{array}\right\}\end{eqnarray}$$

with $z^{\ast }=z-0.75h$ . Here, $\check{\boldsymbol{u}}(y,z^{\ast })=({\check{u}},\check{v},\check{w})$ is the eigenfunction of the streak instability, and the instability mode in physical space is constructed as $\check{\boldsymbol{u}}(y,z^{\ast })\text{e}^{\text{i}k_{x,m}x}$ ( $k_{x,m}$ is the streamwise wavenumber of the instability). Given the mathematical definition, the streamwise and wall-normal velocity fluctuations solely from the sinuous-mode instability should be zero along $z/h=0.75$ and the spanwise velocity fluctuation is supposed to have a local maximum/minimum along this axis. Therefore, in the presence of background turbulence, the streamwise and wall-normal velocity fluctuations of the sinuous-mode instability would always show a local minimum along $z/h=0.75$ , while the spanwise one should exhibit a local maximum/minimum along this axis. These features indeed appear in the computed cross-streamwise view of turbulent velocity fluctuations in figure 8: both the streamwise and wall-normal velocity fluctuations exhibit two symmetric peaks, with a local minimum given along $z/h=0.75$ (figure 8 b,d), while the spanwise one shows a local maximum along $z/h=0.75$ (figure 8 f). We also note that the cross-streamwise view of the velocity fluctuations is qualitatively the same as the one in a laminar boundary layer (Andersson et al. Reference Andersson, Brandt, Bottaro and Henningson2001), indicating the presence of a sinuous-mode instability. In § 3.3, we will provide further evidence of the sinuous-mode instability by visualising the most energetic DMD mode (see figure 11).

3.3 Dynamic mode decomposition

A DMD is performed to examine the flow structure in detail. For the snapshots of the DMD, a time series of a specific Fourier component of the velocity field in the lower half of the wall-normal domain is taken from the simulation with a sufficiently large streak amplitude. The spanwise wavenumber is chosen to be the same as that of the forcing ( $k_{z}=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D706}_{z}$ with $\unicode[STIX]{x1D706}_{z}/h=1.5$ ), while a range of different streamwise wavenumbers ( $k_{x}=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D706}_{x}$ ) are considered ( $2<\unicode[STIX]{x1D706}_{x}/h<5$ ). For given $k_{z}$ and $k_{x}$ , each snapshot is constructed such that

(3.6) $$\begin{eqnarray}\bar{\boldsymbol{u}}(y,z;k_{x})=\hat{\boldsymbol{u}}(y;k_{x},k_{z})\text{e}^{\text{i}k_{z}z}+\hat{\boldsymbol{u}}(y;k_{x},-k_{z})\text{e}^{-\text{i}k_{z}z}.\end{eqnarray}$$

We note that $\bar{\boldsymbol{u}}(y,z;k_{x})$ is the streamwise Fourier component given only with $\pm k_{z}$ spanwise Fourier components. The DMD modes and the corresponding eigenvalues $\unicode[STIX]{x1D707}_{j}$ are then computed following the procedure in § 2.2. The time interval, the number of POD modes and the number of snapshots for the DMD are carefully chosen to ensure good temporal resolution of large-scale dynamics: $\unicode[STIX]{x0394}tu_{\unicode[STIX]{x1D70F}}/h=0.009$ , $r=30$ and $N=1000$ (see appendix A for a detailed discussion on the choice of parameters). Once the DMD is performed, the associated angular frequency is computed by taking the real part of $\unicode[STIX]{x1D714}_{j}=\text{i}\ln \unicode[STIX]{x1D707}_{j}/\unicode[STIX]{x0394}t$ . Since the snapshot in (3.6) represents a streamwise Fourier component, the downstream propagating speed of the DMD mode is computed by

(3.7) $$\begin{eqnarray}c_{j}=\frac{\unicode[STIX]{x1D714}_{j}}{k_{x}},\end{eqnarray}$$

where $c_{j}$ is the phase speed of the $j$ th DMD mode. The relative amplitude $\unicode[STIX]{x1D6FC}_{j}$ of all of the DMD modes is finally determined (see also § 2.2).

Figure 9.

DMD analysis with $\unicode[STIX]{x1D706}_{x}=3.57h$ for $A_{s}^{+}=3.42$ (S960): (a) eigenspectra in the $\unicode[STIX]{x1D707}_{r}$ – $\unicode[STIX]{x1D707}_{i}$ plane; and (b) mode amplitude $|\unicode[STIX]{x1D6FC}_{j}|$ with phase speed $c_{j}$ (the mode amplitude is normalised by the largest one).

Figure 10.

Variation of phase speed ( $c/U_{c}$ ) with streamwise wavelength: ▫, $A_{s}^{+}=4.11$ (S490), $\ast$ , $A_{s}^{+}=4.04$ (S960); ○, $A_{s}^{+}=3.77$ (S2050). The horizontal line indicates $c=0.9U_{c}$ .

The DMD result of the S960 simulation is shown in figure 9 for $\unicode[STIX]{x1D706}_{x}/h=3.57$ and $A_{s}^{+}=3.42$ . The DMD modes are deemed to be neutral, given that the eigenvalues $\unicode[STIX]{x1D707}_{j}$ sit well on the unit circle (figure 9 a). This also indicates that the snapshots are indeed collected from a statistically stationary flow with a good resolution. Their asymmetry is due to the processing of complex-valued snapshots instead of real-valued ones. Using (3.7), the computed eigenvalues are subsequently transformed to the phase speed of the DMD modes. The contribution of each DMD mode is then assessed by computing its amplitude $\unicode[STIX]{x1D6FC}_{j}$ , as shown in figure 9(b). The phase speed of the DMD modes appears mainly in the range of $0.7\lesssim c/U_{c}\lesssim 1.1$ ( $U_{c}$ is the centreline velocity of each simulation), and, in particular, the most energetic DMD mode exhibits $c/U_{c}\simeq 0.9$ .

It is important to mention that the computed phase speed of the most energetic DMD mode is found not to depend significantly on the streamwise wavelength and the Reynolds number, as long as the streak amplitude is chosen to be $A_{s}^{+}>3$ . Figure 10 shows the phase speed of the most energetic DMD mode with respect to the streamwise wavelength for all the considered Reynolds numbers. It appears that the most energetic DMD mode, for different streamwise lengths and Reynolds numbers, robustly exhibits $c/U_{c}\simeq 0.9$ . This suggests that the propagation speed of the flow structure generated by the streak instability scales well with the centreline velocity, consistent with del Álamo & Jiménez (Reference del Álamo and Jiménez2009), who showed the outer scaling of the propagation speed of large-scale structures.

Figure 11.

Visualisation of the streak instability using the most energetic DMD mode from the S960 simulation ( $\unicode[STIX]{x1D706}_{x}/h=3.73$ and $A_{s}^{+}=4.73$ ). (a) The low-speed streak with $u_{s}^{\prime }=-0.03U_{c}$ ( $u_{s}^{\prime }\equiv u_{s}(y,z)-\langle u_{s}(y,z)\rangle _{z}$ , where $\langle \cdot \rangle _{z}$ denotes average in the spanwise direction). (b) The DMD mode $\boldsymbol{u}_{dmd}$ , where the green isosurfaces are a positive streamwise velocity $(0.8\max [u_{dmd}])$ , the blue ones a negative streamwise velocity $(u=0.8\min [u_{dmd}])$ and the red ones a negative wall-normal velocity $(v=0.8\min [v_{dmd}])$ . (c) The low-speed streak with the DMD mode $\boldsymbol{u}_{s}-\langle \boldsymbol{u}_{s}\rangle _{z}+\unicode[STIX]{x1D6FE}\boldsymbol{u}_{dmd}$ , where $\unicode[STIX]{x1D6FE}$ is an appropriate tuning constant for visualisation, and where the blue isosurface indicates a negative streamwise velocity, while the red ones indicate a negative wall-normal velocity.

Finally, the most energetic DMD mode is examined. Figure 11(a,b) shows the streaky uniform mean flow and the most energetic DMD mode for $\unicode[STIX]{x1D706}_{x}/h=3.73$ , respectively, and their combination is given in figure 11(c). Figure 11(c) clearly reveals that the structure is composed of a streamwise-meandering low-speed streak (blue isosurface in figure 11 c) and a streamwise-alternating pattern of wall-normal velocity structures (red isosurface in figure 11 c). We note that this is a robust feature of the computed DMD mode and it does not significantly depend upon the choice of the DMD parameters, such as the streamwise wavelength $\unicode[STIX]{x1D706}_{x}$ , the number of snapshots $N$ and the number of POD modes $r$ . Therefore, the spatial structure of the DMD mode in figure 11 clearly indicates that the flow structure with an intense cross-streamwise turbulent kinetic energy in § 3.2 is directly linked to the sinuous-mode streak instability. In this respect, it is finally worth mentioning that the structure of the most energetic DMD mode with the streaky mean flow highly resembles the general form of the traveling-wave solutions in Hwang, Willis & Cossu (Reference Hwang, Willis and Cossu2016), which is a mathematical representation of the self-sustaining process.