is in the positive $x$ direction. Red color represents higher velocity and blue represents lower velocity compared to the base flow. (c-f) Contours of the non-modal energy growth in the $\alpha$-$\beta$ wavenumber plane at $t=10$, 20, 30 and 40.](Fig3b.pdf "fig:"){width="90.00000%"}

The non-modal energy growth is shown as contours in the wavenumber plane at $t=10$, 20, 30 and 40; see Figure \[fig:unstable\_region\_and\_nonmodal\_growth\](c-f). It can be seen that at the very early stage, the most amplified mode is outside the linearly unstable region. Nevertheless, linearly unstable ones quickly become the most amplified modes after approximately $t=20$. Already at $t=30$, the linearly unstable modes are amplified by a few hundred times in energy.

![\[fig:non\_modal\_most\_unstable\] (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 