not respect the incompressibility constraint, which is however fulfilled by the projection method when the Navier-Stokes equations are solved. [[We want to point out that other velocity profiles with similar inflectional properties may be used to design the force.]{}]{}

Stability property of the modified base flow
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Here, we briefly show the linear stability property of the modified base flow [[ $(U_x+1-y^2, 0, U_z)$]{}]{}. [[The modal stability of this profile has been studied in @Xiao2020. For the ease of discussion, we reproduced the data in Figure \[fig:unstable\_region\_and\_nonmodal\_growth\](a, b).]{}]{} The profile bears linear instabilities in a wide region in the wavenumber plane (the region enclosed by a bold line in panel (a)). The most unstable mode $(\alpha,\beta)=(0.32, -1.96)$ shows wave-like streaks tilted with respect to the streamwise direction, see the visualisation of the streamwise velocity in the $x$-$z$ cut plane at $y=-0.5$ (top) and in the $z$-$y$ cut plane (bottom) in panel (b). However, @Xiao2020 mentioned that the modal growth rate of the most unstable mode seems smaller than the observed growth rate of the streaks at the head of turbulent bands. [[In this paper, we performed a non-modal analysis and will show that the unstable modes can actually achieve a 