Optimal transient growth and very large–scale structures in 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 delÁlamo and Jiménez (2006), or 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 delÁlamo and Jiménez (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 δ.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.

Invariant solutions of minimal large-scale structures in turbulent channel flow for up to 1000

“…The selfsustaining process was also found to be almost identical to that in the near-wall region: 1) the streaky structure (VLSM) is amplified by the vortical structures (LSMs) via the lift-up effect (e.g. Cossu et al 2009;Pujals et al 2009;Hwang & Cossu 2010a;Willis et al 2010); 2) the amplified streak undergoes rapid streak meandering motion via secondary instability or transient growth (Park et al 2011); 3) the following nonlinear regeneration of the streamwise vortical structures (Hwang & Bengana 2016).…”

A note on optimal transient growth in turbulent channel flows

“…Further investigation, related to the computation of the optimal growth supported by a turbulent boundary layer, 16 revealed that the linearized equations for the perturbations used in Ref. 15 were not consistent with the ones used in previous linear stability analysis of turbulent mean flows 17 and of laminar flows with variable viscosity.…”

Self-sustaining processes at all scales in wall-bounded turbulent shear flows

“…Contrary to canonical turbulent free shear flows, the turbulent mean flow profile of most wall-bounded shear flows is linearly stable [28][29][30][31][32][33], leaving unanswered the question of how turbulent motions can extract energy from the mean flow. The important progress realized in the understanding of non-normal energy amplifications in linearly stable laminar flows [25] has, however, motivated a re-examination of this question.…”