Localised Arrowheads: The building blocks of elastic turbulence in rectilinear, sheared polymer flows

Abstract

Pressure-driven flow of a dilute polymer solution has been numerically observed to possess a form of elastic turbulence which is organised around the interactions of localised versions of 2-dimensional `arrowhead' travelling waves (Page et al. Phys. Rev. Lett. 125, 154501, 2020). As a step to confirming this theoretically, we identify spanwise-localised arrowhead travelling waves by tracking a symmetry-breaking bifurcation of the known spanwise-invariant (2D) arrowhead to a spanwise-periodic state and then discovering a secondary modulational instability to a spanwise-localised travelling wave. Spanwise-symmetric and asymmetric localised arrowheads exist with the latter having a phase speed slightly inclined to the streamwise direction. Computations capture the flow randomly switching between spanwise local and global arrowhead states in a streamwise-restricted domain, suggesting they form the building blocks of the chaos. Splitting events are also seen, in which a single localised state spawns multiple arrowheads. However, both cross-shear and spanwise velocities are small suggesting that this elastic turbulence will not be a good mixer.

Figures (6)

• Figure 1: Arrowhead solutions at $W=20, L_x=3\pi$ and $L_z=4\pi$. (a) Spanwise-periodic solution with wavelength $\pi$. Blue 3D structure shows the $\Sigma-\Sigma_0 = 450$ isocontour (where $\Sigma:= tr({\bf C})$ and $\Sigma_0$ is the base state value), with surrounding coloured slices showing $p$ ($z<0$) and $\Sigma-\Sigma_0$ ($z>0$) at midplanes where $x=3\pi/2$ (back) and $z=0$ (left and right). The arrowhead structure is shown on the back panel via $\hat{\Sigma}_{L2}(z)$ (red) and $\hat{v}_{L2}(z)$ (blue). The windowing function $W_f(z;a,b)$ is shown in red at the front when $a=\pi/2$, $b=\pi/4$, with transparent black planes showing where the periodic solution is truncated by the window. (b) Spanwise-localised solution.
• Figure 2: (a) The solution of Fig. \ref{['spanwise-solutions']}b with $W=20, L_x=3\pi$ continued in $L_z$, with obtained states marked by symbols. Linear dependence at small $1/L_z$ demonstrates spanwise localisation, while the branch is lost at larger $1/L_z$. Inset shows $\Sigma_{L2}(z)$ for these localised solutions when $L_z=4\pi,8\pi,12\pi$ and $16\pi$ (marked by coloured squares in (a)). Only $z>0$ is shown as $\Sigma_{L2}(z)$ is even in $z$. (b) The spanwise-localised solution with $W=20, L_z=6\pi$ continued in $L_x$, with the insets showing $\Sigma_{L2}(z)$ and $\Sigma^x_{L2}(x) \coloneqq (\int(\Sigma - \Sigma_0)^2 dydz / L_yL_z)^{1/2}$ at points marked by squares. The base state is linearly unstable for $L_x>3.04\pi$. These show that the solution of Fig. \ref{['spanwise-solutions']}b is spanwise localised, but not streamwise-localised.
• Figure 3: (a) The bifurcation plot of $(\bar{\Sigma}-\bar{\Sigma}_0) /\bar{\Sigma}_0$ (where a bar denotes a volume average) as $L_z$ and $W$ vary for $Re=0.5, \varepsilon=10^{-3}, \beta=0.9, L_x=3\pi$. Solid (dashed) lines show stable (unstable) solutions. Green curves denotes solutions where $W=14$, blue when $W=20$, and intermediate $W$ are shown in orange. Solution branches are annotated, with '2D arrowhead' denoting spanwise-invariant arrowheads, '$m=(\cdot)$' denoting spanwise-periodic arrowheads with $m$ wavelengths in the domain, and 'localised' denoting the branch that localises at large $L_z$ (as in Fig. \ref{['localising']}a). The red (blue) diamond at $L_z=4\pi$ correspond to the solutions plotted in Fig. \ref{['spanwise-solutions']}a (Fig. \ref{['spanwise-solutions']}b). Open circles show where branches could not be continued by time-stepping, implying a bifurcation. We zoom-in on (b) the bifurcation of the spanwise-invariant arrowhead ($BP_1$), and (c) the modulational pitchfork of the $m=2$ branch ($BP_2$). (d) $\Sigma_{L2}(z)$ of solutions $(i)$-$(vi)$ and the localised state of Fig. \ref{['spanwise-solutions']}b, showing only $z>0$ as $\Sigma_{L2}$ is even in $z$.
• Figure 4: An asymmetric arrowhead at $W=20, L_z=4\pi$ and $L_x=3\pi$ generated from an intial perturbation with $A=0.1$ (see main text). Isocontours of $\hat{\Sigma}_{L2}$ and $\hat{v}_{L2}$ are as in Fig. \ref{['spanwise-solutions']}a. Coloured slices show $\Sigma -\Sigma_0$ midplanes where $x=3\pi/2$ (back), $y=0$ (bottom) and $z=0$ (left). This arrowhead has phase speeds $c_x=0.886$ and $c_z=0.0060$ in the flow and span directions respectively (measured in the frame with no net volume flux in any direction), demonstrating that arrowheads are able to drift in the span direction.
• Figure 5: Elastic turbulence at $W=30$, $\beta=0.95$ in a domain with $L_x=3\pi$, $L_z=8\pi$. The timeseries of $\bar{(}\Sigma - \bar{\Sigma}_0) / \bar{\Sigma}_0$ and the rms velocity components (top). The narrow grey shaded region corresponds to the time horizon considered in Fig. \ref{['mixing']}a. The states $(i)-(vi)$ are marked and $\Sigma-\Sigma_0$ on the $y=0$ midplane is plotted (below). The ET drifts between local states (e.g. $(i)$) and global states (e.g. $(iii)$) over time.