On the essential structure of exact traveling-wave solutions in viscoelastic flow

TL;DR

This study maps the exact travelling-wave arrowhead solutions in viscoelastic Kolmogorov flow by combining branch continuation with direct numerical simulation. By tracing solution branches in the $(Wi, L)$ parameter space, it uncovers a progression from an isolated isola to two-sided reconnections and eventually to branches that persist at large $Wi$ in longer domains, including the emergence of localized single arrowheads and trains of arrowheads. It highlights that a polymer strand attached to the arch is not essential for persistence and that strand attachment is tied to extensional flow, with a robust diagnostic based on flow extension and centerline velocity distinguishing attached from detached strands. The minimal sustaining length $L_{min}$ is non-monotonic in $Wi$ and, for $Wi\ge 20$, detached-strand states control the bound via $L_{min}\approx 0.125\,Wi+1.5$, approaching the long-domain limit $L\approx 6\pi$ as $Wi\to\infty$, revealing localisation phenomena and complex multiplicity with potential implications for elastic turbulence and flow control in polymeric fluids.

Abstract

We examine elastic travelling-wave (`arrowhead') solutions in a viscoelastic, unidirectionally body-forced flow, focusing on their existence and morphological changes as the Weissenberg number, $\mathrm{Wi}$, and streamwise duct length, $L$, are varied. We find that branch topology varies from an isola at low $L$ through a two-sided reconnection at intermediate $L$ to a branch which exists at asymptotically large $\mathrm{Wi}$ for larger $L$. At intermediate $L$ more than two arrowhead solutions can coexist at a given $(\mathrm{Wi}, L)$ choice due to extra saddle node bifurcations. Secondly, the canonical arrowhead consists of two legs joined by an arched head that blocks throughflow and traps a counter-rotating vortex pair, while a polymer strand can emerge as a by-product of a strong extensional region attached/detached to the arrowhead arch. Thirdly, a minimal domain length $L_{\min}$ required to sustain an arrowhead is found to vary non-monotonically with $\mathrm{Wi}$; for $\mathrm{Wi}\ge 20$, detached-strand states control $L_{\min}$ with a relation $L_{\min}\approx 0.125\mathrm{Wi}+1.5$. And fourthly, in sufficiently long domains, the upper branch becomes a localised single arrowhead whose streamwise extent depends on $\mathrm{Wi}$, whereas the lower branch can proliferate into a train of arrowheads at high $\mathrm{Wi}$, a phenomenon not previously reported.

Figures (6)

• Figure 1: A schematic of Kolmogorov flow (a) and a typical arrowhead travelling wave structure (with a weak attached spike) (b).
• Figure 2: Branches of arrowhead solution continue in $\mathrm{Wi}$: (a) spanwise velocity fluctuations $\langle v^2 \rangle$ ($\langle\cdot\rangle$ denotes spatial averaging), (b) distance between the stagnation points, $D_{\mathrm{sp}}$. Thick lines indicate relative centerline velocity $u_c-c\geq 0$ throughout, corresponding to at most 1 stagnation point ($n_\mathrm{sp}\leq 1$). Insets: $u_{c}-c$ versus $x$. Dashed lines indicate $u_{c}-c=0$, and dotted lines mark the arch location. Flow fields of the 6 marked cases are shown in \ref{['fig:flowfield']}
• Figure 3: Typical structures of arrowheads: (a) I, near-onset arrowhead solution resembles to eigenfunction of centre-mode instability, (b) II, lower branch with attached strand, (c) intermediate unstable solution with detached strand (d) III, upper branch arrowhead with detached weak strand, (e) IV, modulated solution at $L=6\pi$, (f) upper asymptote solution. Contours denote the normalized trace of polymer conformation $\mathrm{tr}\bm{\alpha}/\max(\mathrm{tr}\bm{\alpha})$, lines indicate the streamlines on the coordinate moving with the phase speed.
• Figure 4: Transition between detached and attached strands in DNS: (a-c) instantaneous fields of strand measure $-\partial^2 \alpha_{xx}/\partial y^2$ (colors) and $\mathrm{tr}\bm{\alpha}$ (lines) at $t=50$, $1200$ and $2000$ and (d-f) measure of flow extension $R=|\Omega|/|S|$ at the same times, (g,h) Space-time $x-t$ diagrams of (g) $-\partial^2 \alpha_{xx}/\partial y^2$ and (h) $u_c-c$ at $y=0$. Green dashed lines indicate the times shown in panels (a–f). (i) the $x$-averaged relative centerline velocity $\langle u_c-c\rangle_x$ versus the arch-strand separation $D_\mathrm{st}$, measured as the distance between the two zero crossings of $-\partial^2 \alpha_{xx}/\partial y^2$ shown in (g).
• Figure 5: Influence of streamwise length: (a) localisation (VI) and (b) train of arrowheads in the long channel (colors: $\mathrm{tr}\bm{\alpha}/\max(\mathrm{tr}\bm{\alpha})$, lines: streamlines on the coordinate moving with $c$). Branches of AHs projected onto (c) the $\langle v^2 \rangle$ vs streamise length $L$, and (d) the arrowhead length ($L_\mathrm{ah}$) vs $L$ space, respectively. The black dash line is $L_\mathrm{ah}=L$.