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Polymers in a turbulent flow are stretched out by the fluctuating velocitygradient; the stationary probability distribution function (p.d.f.) ofextensions $R$ has a power-law tail with an exponent that increases with theWeissenberg number $Wi$, a nondimensional measure of polymer elasticity. Thisstudy addresses the following questions: (i) What is the role of thenon-Gaussian statistics of the turbulent velocity gradient on polymerstretching? (ii) How does the p.d.f. of $R$ evolve to its asymptotic stationaryform? Our analysis is based on simulations of the dynamics offinitely-extensible bead-spring dumbbells and chains, in the extremely dilutelimit, that are transported in a homogeneous and isotropic turbulent flow, aswell as in a Gaussian random flow. First, we recall the large deviations theoryof polymer stretching, and illustrate its application. Then, we compare polymerstretching in turbulent and Gaussian random flows and show that whileextreme-valued strain rates aid in stretching small-$Wi$ stiff polymers, theyare unimportant for high-$Wi$ polymers, which instead are stretched by thecumulative action of moderate strain-rates. This result is supported by ananalysis of the persistence time of polymers in stretched states. Next,beginning from a distribution of coiled polymers, we find that the p.d.f. of$R$ has the form of an evolving power-law, for low to moderate $Wi$, thoughthis is not the case at high $Wi$. In either case, the p.d.f. relaxes to itsstationary form exponentially. The corresponding time scales of equilibration,measured as a function of $Wi$, point to a critical slowing down at thecoil-stretch transition. Importantly, these results show no qualitative changewhen chains in a turbulent flow are replaced by dumbbells in a Gaussian flow,thereby supporting the use of the latter for reduced-order modelling.

Polymers in turbulence: stretching statistics and the role of extreme strain-rate fluctuations