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

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##### Citation

Faranda, D., Lucarini, V., Manneville, P., & Wouters, J. (2014). On using extreme
values to detect global stability thresholds in multi-stable systems: The case of transitional plane Couette flow.*
Chaos, Solitons and Fractals,* *64*, 26-35. doi:10.1016/j.chaos.2014.01.008.

Cite as: http://hdl.handle.net/11858/00-001M-0000-0019-B785-5

##### 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.