Revisiting two-dimensional viscoelastic Kolmogorov flow: a centre-mode-driven transition

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. , vol. 523, 2005, pp. 161–170) is the same ‘centre-mode’ instability found at much higher $Re$ by Garg et al. ( Phys. Rev. Lett. , vol. 121, 2018, 024502) in a pipe and by Khalid et al. ( J. Fluid Mech. , vol. 915, 2021, A43) 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. , vol. 125, 2020, 154501) which in sufficiently large domains and at sufficient Weissenberg number ( $W$ ) interact chaotically in two dimensions 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 two-dimensional elastic turbulence is already contained within the component arrowhead structures.