Formation of spanwise vorticity in oblique turbulent bands of transitional plane Couette flow, part 2: Modelling and stability analysis

This article presents a modelling of the formation of spanwise vorticity in the turbulent streaks of the oblique bands and spots of transitional plane Couette flow. A functional model is designed to mimic the coherent flow in the streaks. The control parameters of the model are extracted from Direct Numerical Simulations (DNS) statistical data. A Reynolds stress is proposed to study the effect on the instability of this additional force maintaining the baseflow. Local (quasi-parallel) temporal stability analysis is performed on that model to investigate the linear development of the spanwise vorticity. Results show that average profiles, even if they have an inflection, are stable: the shear layers inside the velocity streaks are responsible for the vorticity formation. Emphasis is put on the convective or absolute nature of the instability, depending on the location in the band. This shows that a transition from a convective to an absolute instability occurs in the zone in between fully turbulent and laminar flow. The group velocity of the most unstable modes compares well to the advection velocity of spanwise vorticity measured in DNS. This investigation is completed by the global (non-parallel) stability analysis of a typical band case. Eventually, the possible cycles of sustainment of localised low Reynolds number turbulence of shear flows are discussed in the light of these results.