  Polymers in a turbulent flow are stretched out by the fluctuating velocity
gradient; the stationary probability distribution function (p.d.f.) of
extensions $R$ has a power-law tail with an exponent that increases with the
Weissenberg number $Wi$, a nondimensional measure of polymer elasticity. This
study addresses the following questions: (i) What is the role of the
non-Gaussian statistics of the turbulent velocity gradient on polymer
stretching? (ii) How does the p.d.f. of $R$ evolve to its asymptotic stationary
form? Our analysis is based on simulations of the dynamics of
finitely-extensible bead-spring dumbbells and chains, in the extremely dilute
limit, that are transported in a homogeneous and isotropic turbulent flow, as
well as in a Gaussian random flow. First, we recall the large deviations theory
of polymer stretching, and illustrate its application. Then, we compare polymer
stretching in turbulent and Gaussian random flows and show that while
extreme-valued strain rates aid in stretching small-$Wi$ stiff polymers, they
are unimportant for high-$Wi$ polymers, which instead are stretched by the
cumulative action of moderate strain-rates. This result is supported by an
analysis 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 $Wi$, though
this is not the case at high $Wi$. In either case, the p.d.f. relaxes to its
stationary form exponentially. The corresponding time scales of equilibration,
measured as a function of $Wi$, point to a critical slowing down at the
coil-stretch transition. Importantly, these results show no qualitative change
when chains in a turbulent flow are replaced by dumbbells in a Gaussian flow,
thereby supporting the use of the latter for reduced-order modelling.
