Minimal energy thresholds for sustained turbulent bands in channel flow

In this work, nonlinear variational optimization is used for obtaining minimal seeds the formation of turbulent bands in channel flow. Using together with energy bisection, we have found that threshold spatially patterned turbulence scales $Re^{-8.5}$ $Re>1000$ . The seed, which different to a much smaller domain, characterized by spot-like structure surrounded low-amplitude large-scale quadrupolar flow filling whole domain. This minimal-energy perturbation laminar has dominant wavelengths close $4$ streamwise direction and $1$ spanwise direction, spatial localization increasing Reynolds number. At $Re \lesssim 1200$ , seed evolves time, creating an isolated oblique band, whereas $Re\gtrsim quasi-spanwise-symmetric evolution observed, giving rise two distinct bands. A similar also at low $Re$ non-minimal optimal perturbations. highlights mechanisms flow, depending on number initial perturbation. selection one these appears be dependent probability decay newly created stripe, increases but decreases