Minimal seeds for turbulent bands

In this work, nonlinear variational optimization is used for obtaining minimal seeds for the formation of turbulent bands in channel flow. Using nonlinear optimization together with energy bisection, we have found that the minimal energy threshold for obtaining spatially-patterned turbulence scales with $Re^{-8.5}$ for $Re>1000$. The minimal seed is constituted by a localized spot-like structure surrounded by a low-amplitude large-scale quadrupolar structure filling the whole domain. This minimal-energy perturbation of the laminar flow has dominant wavenumbers equal to $0.15$ and $4$ in the streamwise and spanwise directions, respectively, and is characterized by a more marked spatial localization when the Reynolds number increases. At $Re \lesssim 1200$, the minimal seed evolves in time creating an isolated oblique band. Whereas, for $Re\gtrsim 1200$, an almost spanwise-symmetric evolution is observed, giving rise to two distinct bands. A similar evolution is found also at low $Re$ for non-minimal optimal perturbations. This highlights two different mechanisms of formation of turbulent bands in channel flow, depending on the Reynolds number and initial energy of the perturbation. The selection of one of these two mechanisms appears to be affected by the probability of decay of the newly-created stripe, which increases with time, but decreases with the Reynolds number.