Optimal Transient Growth and Very Large-Scale Structures in Zero-Pressure Gradient Turbulent Boundary Layers

Abstract : We are interested in the optimal energy growth of perturbations sustained by a zero pressure gradient turbulent boundary layer. We use the mean flow proposed by Monkewitz et al. (2007), the turbulence dynamics being modeled by an eddy viscosity added in the disturbance equations following the approach of del Alamo and Jimenez (2006), or Pujals et al. (2009) in the turbulent channel flow case. Although all the considered turbulent mean profiles are linearly stable, they support transient energy growths due to the non-normality of the operator. We find that the most amplified perturbations are streamwise uniform and correspond to streamwise vortices evolving into streamwise streaks. Consistently with the study of del Alamo and Jimenez (2006), we find that two distinct peaks of the optimal growth exist for sufficiently large Reynolds numbers: a primary one scaling in outer units and a secondary one scaling in wall units. The optimal structures associated with the peak scaling in wall units correspond well to the most probable streaks observed in the buffer layer and their moderate energy growth is independent of the Reynolds number. The energy growth associated with the peak scaling in outer units is larger than that of the inner peak. The optimal perturbations associated with this primary peak consist in very large-scale structures with a spanwise wavelength of the order of 8. Since such very large-scale structures have not been observed yet in turbulent shear flows, preliminary experiments aiming at forcing such structures and studying their growth have been conducted. We find that large-scale turbulent streaks can be forced using well-shaped roughness elements embedded in the boundary layer. Their amplitude can reach about 13.5% of the free-stream velocity before decaying.