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

Abstract

Polymers in a turbulent flow are stretched out by the fluctuating velocity gradient and exhibit a broad distribution of extensions $R$ ; the stationary probability density function (p.d.f.) of $R$ has a power-law tail with an exponent that increases with the Weissenberg number $\mathit {Wi}$ , a non-dimensional 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. We show that while the turbulent flow is more effective at stretching small- $\mathit {Wi}$ stiff polymers, the Gaussian flow is more effective for high- $\mathit {Wi}$ polymers. This suggests that high- $\mathit {Wi}$ polymers (with large relaxation times) are stretched primarily by the cumulative effect of moderate strain rate events, rather than by short-lived extreme-valued strain rates. 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 $\mathit {Wi}$ , the p.d.f. quickly develops a power-law tail with an exponent that evolves in time and approaches its stationary value exponentially. At high $\mathit {Wi}$ , 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 $\mathit {Wi}$ , 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.