Polymers in a turbulent flow are stretched out by the fluctuating velocity gradient and exhibit a broad distribution of extensions 𝑅; the stationary probability distribution function (p.d.f.) of 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 show that while the turbulent flow is more effective at stretching small-Wi stiff polymers, the Gaussian flow is more effective for high-Wi polymers. This suggests that high-Wi polymers (with large relaxation times) are primarily stretched by the cumulative effect of moderate strain-rate events, rather than by short-lived extreme-valued strain rates; we confirm this behaviour by analysing the persistence time of polymers in stretched states. Next, we show that, beginning from a distribution of coiled polymers, the p.d.f. of 𝑅 exhibits two distinct regimes of evolution. At low to moderate Wi, the p.d.f. quickly develops a powerlaw tail with an exponent that evolves in time and approaches its stationary value exponentially. This result is supported by an asymptotic analysis of a stochastic model. At high Wi, the rapid stretching of polymers first produces a peak in the p.d.f. near their maximum extension; a power law with a constant exponent then emerges and expands its range towards smaller 𝑅. The 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.

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