Critical Reynolds number for global instability of channel flow in subcritical scenario

We perform direct numerical simulations for the transitional pressure-driven channel flow and investigate the critical Reynolds number for global instability (Re$_{\mathrm{G}})$. In the channel flow, the critical Reynolds number relevant to local (infinitesimal) instability (Re$_{\mathrm{L}})$ is known as 5772, which is based on the channel half width ($h$/2) and the channel centerline velocity in laminar flow, by the linear stability theory. However, the understanding of Re$_{\mathrm{G}}$ is still unresolved because it is inherently non-linear. In this study, temporal progress of a turbulent spot that grows from finite disturbance in the laminar flow is analyzed. We use the small and the large computational boxes for identifying the lower critical number, which are $L_{x}$*$L_{y}$*$L_{z}=$6.4$h$*$h$*3.2$h$ and 102.4$h$*$h$*51.2$h$, respectively. For the small box, we determine Re$_{\mathrm{G}}$\textbar $_{\mathrm{S}}$ is 1400. The flow regime observed in the small box is only either laminar flow or fully-developed turbulence. As for the large box, we obtain Re$_{\mathrm{G}}$\textbar $_{\mathrm{L}}$ of 875 because of the emergence of the transitional structure named ``turbulent stripe,'' or a coexistence of oblique turbulent and laminar region. Because the spatial size of turbulent stripe is much larger than the small box, Re$_{\mathrm{G}}$\textbar $_{\mathrm{S}}$ indicates the critical Reynolds number for the flow without large-scale intermittency. Therefore, we found that the existence of turbulent stripe caught in large box would decrease the Re$_{\mathrm{G}}$ value compared to those proposed by previous studies.