Linear and nonlinear optimal growth mechanisms for generating turbulent bands

TL;DR

This paper investigates how linear and nonlinear energy optimizations can generate turbulent bands in channel flow, highlighting the roles of linear growth mechanisms and spatial localization in transition to turbulence.

Contribution

It introduces a combined linear and nonlinear optimization approach to identify key mechanisms for turbulent band formation, emphasizing the importance of large-scale flow and spatial localization.

Findings

Linear optimal perturbations predict energy growth at specific wavenumbers.

Localized nonlinear effects induce the formation of turbulent bands.

Large-scale vortices and lift-up mechanisms are crucial for transition.

Abstract

Linear and nonlinear energy optimizations in a tilted domain are used to unveil the main mechanisms allowing the creation of a turbulent band in a channel flow. Linear optimization predicts an optimal growth for streamwse and spanwise wavenumbers $k_x = 1.2$, $k_z = - 1.75$, corresponding to the peak values of the premultiplied energy spectra of direct numerical simulations. At target time, the linear optimal perturbation is composed by oblique streaks, which, for a sufficiently large initial energy, induce turbulence in the whole domain, due to the lack of spatial localization. When localization is achieved by adding nonlinear effects to the optimization, or by artificially confining the linear optimal to a localized region in the spanwise direction, a large-scale flow is created, which leads to the generation of a localised turbulent band. These results suggest that inducing transition towards turbulent bands in a tilted domain, two main elements are needed: a linear energy growth mechanism such as the lift-up for generating large-amplitude flow structures which produce inflection points; large-scale vortices ensuring spatial localisation. Remarkably, these two elements alone are able to generate an isolated turbulent band also in a large, non-tilted domain.