Threshold transient growth as a criterion for turbulent mean profiles

Abstract

Lozano-Duran et al. (J. Fluid Mech., vol. 914, 2021, p. A8) have recently identified the ability of streamwise-averaged turbulent streak fields ${\mathcal {U}}(y,z,t)\hat {\boldsymbol {x}}$ in minimal channels to produce short-term transient growth as the key linear mechanism needed to sustain turbulence at $Re_{\tau }=180$ . Here, in an attempt to extend this result to larger domains and higher $Re_{\tau }$ , we model this streak transient growth as a two-stage linear process by first selecting the dominant streak structure expected to emerge over the eddy turnover time on the turbulent mean profile $U(y)\hat {\boldsymbol {x}}$ , and then examining the secondary growth on this (frozen) streak field ${\mathcal {U}}(y,z)\hat {\boldsymbol {x}}$ . Choosing the mean streak amplitude and eddy turnover time consistent with simulations captures the growth thresholds found by Lozano-Duran et al. (2021) for sustained turbulence. In a larger domain at $Re_{\tau }=180$ , the most energetic near-wall streaks observed in simulations are close to the predicted optimal streaks. This most energetic streak spacing, approaches the optimal streak at $Re_{\tau }=550$ where the secondary growth possible on each also comes together. A key prediction from the model is that the threshold transient growth required to sustain turbulence decreases with increasing $Re_{\tau }$ . More fundamentally, the work of Lozano-Duran et al. (2021) and our results suggest a subtle but significant revision of Malkus's (J. Fluid Mech., vol. 1, 1956, pp. 521–539) classic hypothesis concerning realisable turbulent mean profiles. The key property for a realisable turbulent mean profile could be the ability to generate sufficient short-term transient growth rather than dependence on its (long-term) linear stability characteristics, which was Malkus's original idea.