Threshold transient growth as a criterion for turbulent mean profiles
Lozano-Duran et al (J. Fluid Mech., 914, A8, 2021) have recently identified the ability of streamwise-averaged turbulent streak fields $U(y,z,t)\widehat{\mathbf{x}}$ in minimal channels to produce short-term transient growth as the key linear mechanism needed to sustain turbulence at $Re_τ=180$. Here, in an attempt to extend this result to larger domains and higher $Re_τ$, 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 $U(y)\widehat{\mathbf{x}}$, and then examining the secondary growth on this (frozen) streak field $U(y,z)\widehat{\mathbf{x}}$. 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 $Re_τ=180$, 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 $Re_τ=550$ 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 $Re_τ$. 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.}, 521, 1, 1956) 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.