Finite-amplitude elastic waves in viscoelastic channel flow from large to zero Reynolds number

Abstract

Using branch continuation in the FENE-P model, we show that finite-amplitude travelling waves borne out of the recently discovered linear instability of viscoelastic channel flow (Khalid et al., J. Fluid Mech., vol. 915, 2021, A43) are substantially subcritical reaching much lower Weissenberg ( $Wi$ ) numbers than on the neutral curve at a given Reynolds ( $Re$ ) number over $Re \in [0,3000]$ . The travelling waves on the lower branch are surprisingly weak indicating that viscoelastic channel flow is susceptible to (nonlinear) instability triggered by small finite-amplitude disturbances for $Wi$ and $Re$ well below the neutral curve. The critical $Wi$ for these waves to appear in a saddle node bifurcation decreases monotonically from, for example, $\approx 37$ at $Re=3000$ down to $\approx 7.5$ at $Re=0$ at the solvent-to-total-viscosity ratio $\beta =0.9$ . In this latter creeping flow limit, we also show that these waves exist at $Wi \lesssim 50$ for higher polymer concentrations, $\beta \in [0.5,0.97)$ , where there is no known linear instability. Our results therefore indicate that these travelling waves, found in simulations and named ‘arrowheads’ by Dubief et al. (Phys. Rev. Fluids, vol. 7, 2022, 073301), exist much more generally in $(Wi,Re, \beta )$ parameter space than their spawning neutral curve and, hence, can either directly, or indirectly through their instabilities, influence the dynamics seen far away from where the flow is linearly unstable. Possible connections to elastic and elasto-inertial turbulence are discussed.

Transition to turbulence in viscoelastic channel flow of dilute polymer solutions

“…The study of EIT has also provided an alternative explanation to the upper limit of turbulent DR, also known as the maximum drag reduction state (Samanta et al 2013;Choueiri, Lopez & Hof 2018;Lopez, Choueiri & Hof 2019). Interestingly, there are recent studies that have found the nonlinear elasto-inertial exact coherent structures in the EIT regime, named arrowhead structures (Page, Dubief & Kerswell 2020;Buza et al 2022;Dubief et al 2022), which link the EIT and elasto-inertial linear instability. An extensive review of these instabilities can be found on Castillo-Sánchez et al (2022) and Datta et al (2022).…”

“…This boundary indeed represents the critical perturbation amplitude A c in figure 7. Since there are theoretical arguments that the so-called exact coherent states (ECSs) form a part of the basin boundary (Kawahara 2005;Wang, Gibson & Waleffe 2007), the dynamics on or near this boundary could play an important role in finding new ECSs in viscoelastic flows or EIT (Page et al 2020;Buza et al 2022;Dubief et al 2022), which will be a subject of interesting future work.…”

“…Interestingly, there are recent studies that have found the nonlinear elasto-inertial exact coherent structures in the EIT regime, named arrowhead structures (Page, Dubief & Kerswell 2020; Buza et al. 2022; Dubief et al. 2022), which link the EIT and elasto-inertial linear instability.…”

Multistability of elasto-inertial two-dimensional channel flow

“…The nonlinear evolution of the viscoelastic centre mode leads to a saturated 'arrowhead' travelling wave (Page, Dubief & Kerswell 2020) that is strongly subcritical (Wan, Sun & Zhang 2021;Buza et al 2022b). The arrowhead can be continued down to the inertialess limit where it is found to exist at experimentally realisable values of the Weissenberg number (Buza et al 2022a;Morozov 2022). Finite-amplitude structures that are similar in appearance to the exact arrowhead travelling waves have been observed in experiments at low Re (Choueiri et al 2021) and have also been seen intermittently in numerical simulations of EIT at high Re (Page et al 2020;Dubief et al 2022).…”

Inertial enhancement of the polymer diffusive instability

“…The finite-amplitude state resulting from this instability is an ‘arrowhead’ travelling wave (Page, Dubief & Kerswell 2020; Buza et al. 2022 a ; Morozov 2022) which has been observed in channel flow EIT (Dubief et al. 2022) and, in retrospect, ET in two-dimensional Kolmogorov flow (Berti & Boffetta 2010).…”