Abstract

Recently, many authors have investigated the origin and growth of turbulent bands in shear flows, highlighting the role of streaks and their inflectional instability in the process of band generation and sustainment. Recalling that streaks are created by an optimal transient growth mechanism, and motivated by the observation of a strong increase of the disturbance kinetic energy corresponding to the creation of turbulent bands, we use linear and nonlinear energy optimizations in a tilted domain to unveil the main mechanisms allowing the creation of a turbulent band in channel flow. While linear optimal perturbations induce turbulence in the whole domain, spatially-localized perturbations obtained by nonlinear optimization, or by artificially confining the linear optimal to a localized region in the transverse direction, lead to the generation of a localised turbulent band through the formation of large-scale vortices. These results suggest that two main elements are needed for inducing turbulent bands in a tilted domain: i) a linear energy growth mechanism, such as the lift-up, for generating large-amplitude flow structures, which produce inflection points; ii) spatial localisation, linked to the presence or generation of large-scale vortices. We show that these elements alone can generate isolated turbulent bands also in large non-tilted domains. In such framework, using nonlinear optimization together with energy bisection, we find a scaling law for the minimal energy threshold needed to trigger inclined bands. A similar scaling is found in a much smaller domain, where laminar-turbulent stripes are not observed, but the values of the minimal energy and the shape of the minimal seed strongly differ. Finally, based on the evolution of such minimal seeds, two different mechanisms of formation of turbulent bands in channel flow are discussed, depending on the Reynolds number and initial energy of the perturbation. The selection of one of these two mechanisms appears to be dependent on the probability of decay of the newly-created stripe, which increases with time, but decreases with the Reynolds number.