Linear non-normal energy amplification of harmonic and stochastic forcing in the turbulent channel flow

Abstract

To cite this version:Yongyun Hwang, Carlo Cossu. Linear non-normal energy amplification of harmonic and stochastic forcing in the turbulent channel flow. Journal of Fluid Mechanics, Cambridge University Press (CUP), 2010, 664 (December), pp. 51-73. 10.1017/s0022112010003629. hal-01020668 J. Fluid Mech. (2010 The linear response to stochastic and optimal harmonic forcing of small coherent perturbations to the turbulent channel mean flow is computed for Reynolds numbers ranging from Re τ = 500 to 20 000. Even though the turbulent mean flow is linearly stable, it is nevertheless able to sustain large amplifications by the forcing. The most amplified structures consist of streamwise-elongated streaks that are optimally forced by streamwise-elongated vortices. For streamwise-elongated structures, the mean energy amplification of the stochastic forcing is found to be, to a first approximation, inversely proportional to the forced spanwise wavenumber while it is inversely proportional to its square for optimal harmonic forcing in an intermediate spanwise wavenumber range. This scaling can be explicitly derived from the linearized equations under the assumptions of geometric similarity of the coherent perturbations and of logarithmic base flow. Deviations from this approximate power-law regime are apparent in the pre-multiplied energy amplification curves that reveal a strong influence of two different peaks. The dominant peak scales in outer units with the most amplified spanwise wavelength of λ z ≈ 3.5h, while the secondary peak scales in wall units with the most amplified λ + z ≈ 80. The associated optimal perturbations are almost independent of the Reynolds number when, respectively, scaled in outer and inner units. In the intermediate wavenumber range, the optimal perturbations are approximatively geometrically similar. Furthermore, the shape of the optimal perturbations issued from the initial value, the harmonic forcing and the stochastic forcing analyses are almost indistinguishable. The optimal streaks corresponding to the large-scale peak strongly penetrate into the inner layer, where their amplitude is proportional to the mean-flow profile. At the wavenumbers corresponding to the large-scale peak, the optimal amplifications of harmonic forcing are at least two orders of magnitude larger than the amplifications of the variance of stochastic forcing and both increase with the Reynolds number. This confirms the potential of the artificial forcing of optimal large-scale streaks for the flow control of wall-bounded turbulent flows.

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

“…Probably the most important new finding is that large-scale [44,52] and moderate-scale [45] structures can self-sustain when smaller-scale motions are artificially quenched in over-damped large-eddy simulations of the turbulent flow. A continuum of self-sustaining motions therefore exists which is, furthermore, statistically and dynamically consistent with the features of Townsend's attached eddies, including an approximate self-similarity in the logarithmic region [35,45,50,64]. The existence of coherent self-sustained processes at large scale is also confirmed by the computation of invariant large-scale solutions of the filtered (fully nonlinear) Navier-Stokes equations [52,56,61,62,64].…”

“…Further inspection of the dynamics of each of the attached eddies revealed that the streaks and quasi-streamwise vortical structures form a self-sustaining process [41,55,65] exactly the same as the one shown in figure 3 with a turn-over time scale of Tu τ /λ z 2 [54]. The streaks are amplified from quasi-streamwise vortices via a coherent lift-up effect [26,30,35]; they then undergo rapid oscillation via secondary instability and/or transient growth on top of the streaks [39,40,42,55]; the quasi-streamwise vortical structures are nonlinearly regenerated [41,54,55]. The time scale of the self-sustaining process corresponds well to that of the 'bursting' in the logarithmic region, indicating that the bursting is naturally embedded in the self-sustaining process [54].…”

“…Optimal streaks associated with the secondary peak correspond well to the observed near-wall streaks. Structures with scales broadly lying between these two peaks correspond to log-layer streaks and are approximately self-similar [35]. This approach has then been extended by computing the optimal response to harmonic and stochastic forcing [33,35,36], confirming the existence of an inner and an outer amplification peak corresponding to the buffer-layer and large-scale streaky structures with intermediate (log-layer) scales also amplified.…”

“…Structures with scales broadly lying between these two peaks correspond to log-layer streaks and are approximately self-similar [35]. This approach has then been extended by computing the optimal response to harmonic and stochastic forcing [33,35,36], confirming the existence of an inner and an outer amplification peak corresponding to the buffer-layer and large-scale streaky structures with intermediate (log-layer) scales also amplified. Experiments, where large-scale coherent streaks were steadily forced in the turbulent boundary layer, confirmed that coherent (temporally averaged) large-scale streaks can be transiently amplified in space [37].…”

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).…”