Linear and nonlinear optimal growth mechanisms for generating turbulent bands

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 optimisations in a tilted domain to unveil the main mechanisms allowing the creation of a turbulent band in a channel flow. Linear transient growth analysis shows an optimal growth for wavenumbers having an angle of approximately $35^\circ$ , close to the peak values of the premultiplied energy spectra of direct numerical simulations. This linear optimal perturbation generates oblique streaks, which, for a sufficiently large initial energy, induce turbulence in the whole domain, due to the lack of spatial localisation. However, spatially localised perturbations obtained by adding nonlinear effects to the optimisation or by artificially confining the linear optimal to a localised region in the transverse direction are characterised by a large-scale flow and lead to the generation of a localised turbulent band. 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 generate isolated turbulent bands also in large non-tilted domains.

A structured input-output approach to characterizing optimal perturbations in wall-bounded shear flows

“…The real parts of these autocorrelations also show the same sign reversal near the channel center as the NLOP [46]. In plane Poiseuille flow, the destabilizing velocity autocorrelation instead vanishes near the channel center, which is consistent with both the behavior of the NLOP [47] and results analyzing optimal secondary energy growth [48]. In contrast, linear optimal perturbations of plane Poiseuille flow peak near the channel center; see e.g.…”

“…In contrast, linear optimal perturbations of plane Poiseuille flow peak near the channel center; see e.g. the comparison in [47]. The agreement of these results with those from various nonlinear analysis approaches provides further evidence that behavior associated with nonlinear effects can be captured using SIOA-based approaches.…”

Minimal energy thresholds for sustained turbulent bands in channel flow

“…This asymmetric wavepacket evolves via nucleation of new streaky structures (see Parente et al. (2021) concerning the mechanism of creation of the streaks) in the direction of the inclined laminar–turbulent interface, clearly forming a singular turbulent band, as can be observed for . The newly formed turbulent band continues growing in an oblique direction with angle until reaching the periodic boundaries, where it interacts with itself ().…”

“…In order to reduce the computational cost, some numerical studies cleverly considered computational domains tilted in the direction of the bands, as done for plane Couette flow by Barkley & Tuckerman (2005) and for plane Poiseuille flow by Tuckerman et al (2014). Very recently, Parente et al (2021) carried out an optimal growth analysis in a tilted domain with given angle. They showed that although linear optimization is able to recover the main wavenumbers observed by direct numerical simulation, nonlinear effects are necessary for providing large-scale flow and spatial localization of the perturbation able to generate a turbulent band.…”

“…Very recently, Parente et al. (2021) carried out an optimal growth analysis in a tilted domain with given angle. They showed that although linear optimization is able to recover the main wavenumbers observed by direct numerical simulation, nonlinear effects are necessary for providing large-scale flow and spatial localization of the perturbation able to generate a turbulent band.…”