On using Extreme Values to detect global stability thresholds in multi-stable systems: The case of transitional plane Couette flow

TL;DR

This paper uses Extreme Value Theory to identify the global stability threshold in multi-stable plane Couette flow by analyzing the extremes of perturbation energy fluctuations, providing a novel method for detecting critical transitions.

Contribution

It introduces a new approach employing EVT to determine the global stability threshold in fluid dynamics, linking extreme value distribution changes to stability transitions.

Findings

The shape parameter of the GEV for minima changes sign at the stability threshold.

As R approaches Rg, the probability of very low minima increases.

Maxima distribution remains bounded near the threshold.

Abstract

Extreme Value Theory (EVT) is exploited to determine the global stability threshold $R_g$ of plane Couette flow --the flow of a viscous fluid in the space between two parallel plates-- whose laminar or turbulent behavior depends on the Reynolds number R. Even if the existence of a global stability threshold has been detected in simulations and experiments, its numerical value has not been unequivocally defined. $R_g$ is the value such that for $R > R_g$ turbulence is sustained, whereas for $R < R_g$ it is transient and eventually decays. We address the problem of determining $R_g$ by using the extremes - maxima and minima - of the perturbation energy fluctuations. When $R_g$, both the positive and negative extremes are bounded. As the critical Reynolds number is approached from above, the probability of observing a very low minimum increases causing asymmetries in the distributions of maxima and minima. On the other hand, the maxima distribution is unaffected as the fluctuations towards higher values of the perturbation energy remain bounded. This tipping point can be detected by fitting the data to the Generalized Extreme Value (GEV) distribution and by identifying $R_g$ as the value of $R$ such that the shape parameter of the GEV for the minima changes sign from negative to positive. The results are supported by the analysis of theoretical models which feature a bistable behavior.