Optimal transient growth and very large–scale structures in turbulent boundary layers

Abstract

The optimal energy growth of perturbations sustained by a zero pressure gradient turbulent boundary is computed using the eddy viscosity associated with the turbulent mean flow. It is found that even if all the considered turbulent mean profiles are linearly stable, they support transient energy growths. The most amplified perturbations are streamwise uniform and correspond to streamwise streaks originated by streamwise vortices. For sufficiently large Reynolds numbers two distinct peaks of the optimal growth exist, respectively scaling in inner and outer units. The optimal structures associated with the peak scaling in inner units correspond well with the most probable streaks and vortices 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 and scales linearly with an effective turbulent Reynolds number formed with the maximum eddy viscosity and a modified Rotta–Clauser length based on the momentum thickness. The corresponding optimal perturbations consist of very large–scale structures with a spanwise wavelength of the order of 8δ. The associated optimal streaks scale in outer variables in the outer region and in wall units in the inner region of the boundary layer, in which they are proportional to the mean flow velocity. These outer streaks protrude far into the near wall region, having still 50% of their maximum amplitude at y+ = 20. The amplification of very large–scale structures appears to be a robust feature of the turbulent boundary layer: optimal perturbations with spanwise wavelengths ranging from 4δ to 15δ can all reach 80% of the overall optimal peak growth.