Threshold transient growth as a criterion for turbulent mean profiles

TL;DR

This paper models transient growth of streaks in turbulent flows to identify a criterion for turbulence sustainability, suggesting short-term growth ability is more fundamental than long-term stability.

Contribution

It extends previous work by modeling streak transient growth as a two-stage linear process applicable to larger domains and higher Reynolds numbers.

Findings

Threshold transient growth decreases with increasing Re.

Most energetic streaks in larger domains approach optimal streaks.

Model predicts short-term growth ability as key to turbulence sustainment.

Abstract

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 $Re_{\tau}=180$. 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 $U(y)\widehat{\mathbf{x}}$, 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 $Re_{\tau}=180$, 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.}, 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.