Exact Traveling Wave Solutions in Viscoelastic Channel Flow

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Exact Traveling Wave Solutions in Viscoelastic Channel Flow

Jacob Page, Yves Dubief, and Rich R. Kerswell

Phys. Rev. Lett. 125, 154501 – Published 6 October 2020

Abstract

Elasto-inertial turbulence (EIT) is a new, two-dimensional chaotic flow state observed in polymer solutions with possible connections to inertialess elastic turbulence and drag-reduced Newtonian turbulence. In this Letter, we argue that the origins of EIT are fundamentally different from Newtonian turbulence by finding a dynamical connection between EIT and an elasto-inertial linear instability recently found at high Weissenberg numbers [Garg et al., Phys. Rev. Lett. 121, 024502 (2018)]. This link is established by isolating the first known exact coherent structures in viscoelastic parallel flows—nonlinear elasto-inertial traveling waves (TWs)—borne at the linear instability and tracking them down to substantially lower Weissenberg numbers where EIT exists. These TWs have a distinctive “arrowhead" structure in the polymer stretch field and can be clearly recognized albeit transiently in EIT as well as being attractors for EIT dynamics if the Weissenberg number is sufficiently large. Our findings suggest that the dynamical systems picture in which Newtonian turbulence is built around the coexistence of many (unstable) simple invariant solutions populating phase space carries over to EIT, though these solutions rely on elasticity to exist.

Received 15 June 2020
Revised 3 August 2020
Accepted 7 September 2020

DOI:https://doi.org/10.1103/PhysRevLett.125.154501

Physics Subject Headings (PhySH)

Non-Newtonian fluids Transition to turbulence

Flow instability

Viscoelasticity

Fluid Dynamics

Authors & Affiliations

Jacob Page ^1,*, Yves Dubief², and Rich R. Kerswell ¹

¹DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, United Kingdom
²School of Engineering, University of Vermont, Burlington, Vermont 05405, USA

^*jacob.page@ed.ac.uk Present address: School of Mathematics, University of Edinburgh, Edinburgh, EH9 3FD, United Kingdom.

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Vol. 125, Iss. 15 — 9 October 2020

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Images

Figure 1
Top: time evolution of volume-averaged trace of the polymer conformation relative to the laminar value, $A ≔ ⟨ tr C ⟩_{V} / ⟨ tr C_{lam} ⟩_{V}$ , in a computation of EIT in a long domain, $l_{x} = 4 π$ , at $Re = 1000$ , $Wi = 20$ . Middle: snapshot of the flow extracted at $t = 250$ (see marker in the top panel), contours show $tr C / L^{2}$ , lines are the perturbation stream function for ( $u - U_{lam}$ ). Bottom: snapshot of transient EIT at $Wi = 30$ , with all other parameters held fixed. The flow here eventually settles onto a stable traveling wave (an arrowhead).
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Figure 2
Left: contours of linear growth rate $σ$ for most unstable symmetric instability waves in a two-dimensional channel flow of a FENE-P fluid ( $L = 500$ ) for streamwise wave numbers $k \in N$ . The dashed white line indicates marginal stability $σ = 0$ . The overlayed grid and symbols identify DNS runs in a box of length $l_{x} = 2 π$ : (blue circle) laminar, (red square) EIT, (green triangle) arrowhead. The blue lines identify where we have performed arclength continuation of the nonlinear traveling wave born in this instability (at $k = 2$ ). Right: visualization of the instability wave in the $x - z$ plane at points indicated by white circles in the stability diagram. Contours show the trace of the (perturbation) conformation, lines the stream function. From top to bottom: $(Re, Wi, k) = (60, 30, 2)$ , (2000,90,7), and (130,90,1).
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Figure 3
Arclength continuation in Wi at $Re = 60$ (blue) and in Re at $Wi = 15$ (maroon) in a box of length $l_{x} = π$ . The amplitude $A ≔ ⟨ tr C ⟩_{V} / ⟨ tr C_{lam} ⟩_{V}$ . The solution curve was initialized at $(Re, Wi) = (60, 30)$ by perturbing a laminar flow with the linear instability wave at $k = 2$ , which was allowed to saturate onto the stable upper branch solution. The saddle node at $Re = 60$ is at $Wi = 8.77$ ; the saddle node at $Wi = 15$ is at $Re = 253.71$ . The bifurcation point at $(Re, Wi) = (60, 26.9)$ is identified with a small black circle. Dashed lines indicate unstable states and solid lines are stable. The upper branches of the Wi and Re continuation curves are unstable between $Wi \approx 10$ and $Wi \approx 20$ (boundaries marked with blue stars) and below $\approx 115$ (red star), respectively. The phase speed of travelling waves is reported in Fig. 4 and snapshots of the TW at points highlighted on the curve are shown in Fig. 5.
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Figure 4
Phase speed of the TWs reported in Fig. 3. Left: fixed $Re = 60$ , continuation in Wi. Right: fixed $Wi = 15$ , continuation in Re. The symbols match those in Fig. 3 and correspond to the states reported in Fig. 5.
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Figure 5
Traveling waves corresponding to points identified in Fig. 3. Contours show $tr C / L^{2}$ , lines are the perturbation ( $u - U_{lam}$ ) stream function. The aspect ratio matches the snapshot of the long domain ( $l_{x} = 4 π$ ) calculation reported in Fig. 1.
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