been known that turbulence in channel flow appears in form of turbulent bands at low Reynolds numbers, which are tilted with respect to the streamwise direction [@TsukaharaKawaguchi2014; @TsukaharaKawamura2014; @Tuckerman2014; @Xiong2015; @Tao2018; @Kanazawa2018; @Shimizu2019; @Xiao2020; @Paranjape2020]. Similar band patterns were also observed in other types of shear flows at transitional Reynolds numbers [@Coles1965; @Prigent2002; @Barkley2005; @Duguet2010; @Tuckerman2011; @Duguet2013; @Rolland2015; @Rolland2016; @Chantry2017; @Reetz2019; @Tuckerman2020]. [[The obliqueness of turbulent bands and the sustaining mechanism of bands in tilted periodic domains have been explained from the point of view of the large scale flow and dynamical system approach [@Duguet2013; @Reetz2019; @Paranjape2020]. However, the mechanism underlying the specific tilt angles and the dynamcics of fully localised turbulent bands remain poorly understood.]{}]{} Latest numerical studies showed that individual turbulent bands in large channels can be sustained at $Re\gtrsim 660$, and once triggered, a turbulent band can grow transversely or split when the band length is sufficiently large [@Xiong2015; @Tao2018; @Shimizu2019; @Kanazawa2018]. Throughout this paper, [[constant-flux driven flow is discussed about and]{}]{} the Reynolds number is defined as $Re=U_ch/\nu$, where $U_c$ is the centerline velocity of the unperturbed parabolic flow, $h$ the half gap-width and $\nu$ the kinematic viscosity of the fluid.

Regarding the sustaining mechanism of 