flow.](Fig4a.pdf "fig:"){width="55.00000%"} ![\[fig:non\_modal\_most\_unstable\] (a) The energy amplification $E(t)/E(0)$ of the optimal perturbation optimised at $t=30$ for the mode $(\alpha,\beta)=(0.32,-1.96)$, which is the most unstable mode of the modal instability analysis. $E(t)$ denotes the kinetic energy of the disturbances integrated in the whole domain at time $t$. (b-d) Change of the flow field of the flow described in panel (a). Contours of the streamwise velocity in the $z$-$y$ cut plane at $x=0$ is plotted at $t=0$, 15 and 30. Red color represents higher velocity and blue represents lower velocity compared to the base flow.](Fig4b.pdf "fig:"){width="95.00000%"}

Figure \[fig:non\_modal\_most\_unstable\](a) shows the energy amplification of the mode $(\alpha,\beta)$ = $(0.32,-1.96)$. The flow is initialised with the optimal perturbation that is optimised at $t=30$ in the non-modal analysis. The flow field is visualised in the $z$-$y$ cut plane at $t=0$, 15 and 30 in panel (b-d). As shown in panel (b), the initial flow field manifests flow structures that mainly tilt against the mean shear of the spanwise flow component (see Figure \[fig:target\_profiles\](b)) in the region $-0.6\lesssim y\lesssim 0.6$. Quickly, only after about 15 time units, the tilt direction of the structures is inverted by the underlying mean shear, see panel (c), during which the 