Revisiting 2D Viscoelastic Kolmogorov Flow: A Centre-mode-driven transition

TL;DR

This paper revisits viscoelastic Kolmogorov flow, revealing a centre-mode instability at zero Reynolds number and analyzing its role in elastic turbulence and energy spectra, extending previous findings to new flow regimes.

Contribution

It demonstrates that the centre-mode instability persists at zero Reynolds number and links elastic turbulence features to specific flow structures.

Findings

Centre-mode instability exists at Re=0 in viscoelastic Kolmogorov flow.

Preferred wavelength of instability is twice the forcing wavelength.

The k^{-4} energy spectrum is explained by flow structures.

Abstract

We revisit viscoelastic Kolmogorov flow to show that the elastic linear instability of an Oldroyd-B fluid at vanishing Reynolds numbers ($Re$) found by Boffetta et al. (J. Fluid Mech. 523, 161-170, 2005) is the same `centre-mode' instability found at much higher $Re$ by Garg et al. (Phys. Rev. Lett. 121, 024502, 2018) in a pipe and Khalid et al. (J. Fluid Mech. 915, A43, 2021) in a channel. In contrast to these wall-bounded flows, the centre-mode instability exists even when the solvent viscosity vanishes (e.g. it exists in the upper-convective Maxwell limit with $Re=0$). Floquet analysis reveals that the preferred centre-mode instability almost always has a wavelength twice that of the forcing. All elastic instabilities give rise to familiar `arrowheads' (Page et al. Phys. Rev. Lett. 125, 154501, 2020) which in sufficiently large domains and at sufficient Weissenberg number ($W$) interact chaotically in 2D to give elastic turbulence via a bursting scenario. Finally, it is found that the $k^{-4}$ scaling of the kinetic energy spectrum seen in this 2D elastic turbulence is already contained within the component arrowhead structures.