Linear stability of optimal streaks in the log-layer of turbulent channel flows

The importance of secondary instability of streaks for the generation of vortical struc-tures attached to the wall in the logarithmic region of turbulent channels is studied. Thestreaks and their linear instability are computed by solving equations associated withthe organized motion that include an eddy-viscosity modeling the effect of incoherentfluctuations. Three friction Reynolds numbers,Reτ=2000,3000, and 5000, areinvestigated. For all flow cases, optimal streamwise vortices (i.e., having the highestpotential for linear transient energy amplification) are used as initial conditions. Dueto the lift-up mechanism, these optimal perturbations lead to the nonlinear growthof streaks. Based on a Floquet theory along the spanwise direction, we observe theonset of streak secondary instability for a wide range of spanwise wavelengths whenthe streak amplitude exceeds a critical value. Under neutral conditions, it is shown thatstreak instability modes have their energy mainly concentrated in the overlap layer andpropagate with a phase velocity equal to the mean streamwise velocity of the log-layer.These neutral log-layer modes exhibit a sinuous pattern and have characteristic sizesthat are proportional to the wall distance in both streamwise and spanwise directions, inagreement with the Townsend’s attached eddy hypothesis (A. Townsend, the structureof turbulent shear flow, Cambridge university press, 1976 2nd edition). In particular,for a distance from the wall varying fromy+≈100 (in wall units) toy≈0.3h, wherehis half the height of the channel, the neutral log-layer modes are self-similar with aspanwise width ofλz≈y/0.3 and a streamwise length ofλx≈3λz, independently ofthe Reynolds number. Based on this observation, it is suggested that compact vorticalstructures attached to the wall can be ascribed to streak secondary instabilities. Inaddition, spatial distributions of fluctuating vorticity components show that the onsetof secondary instability is associated with the roll-up of the shear layer at the edgeof the low-speed streak, similarly to a three-dimensional mixing layer