3.1 Upstream vortex: growth and fluctuation

We begin our analysis by investigating the flow structures upstream of the cylinder. Figure 1(b,c) shows two snapshots of streaks taken at $z=10~\unicode[STIX]{x03BC}\text{m}$ (from bottom plane) of the channel for $Wi=23$ . These two-dimensional streak plots illustrate the highly unsteady vortices immediately upstream of the cylinder. We observe the presence of a large recirculation region in front of the cylinder which clearly separates the dominant bulk flow into two streams. To quantify the onset of the upstream vortex, we monitor its length $\unicode[STIX]{x1D712}$ normalized by the cylinder diameter $d$ as a function of Weissenberg number at a particular height ( $z=10~\unicode[STIX]{x03BC}\text{m}$ ). The length is defined as the furthest upstream point with zero or negative velocity to the edge of the cylinder. This stagnant vortex extends far upstream with maximum normalized vortex length of approximately $\unicode[STIX]{x1D712}/d\approx 6$ , as in figure 1(b). The feature of the vortex is marked by flow recirculation with relatively low or negative velocity compared to the bulk flow. At higher $Wi$ , the vortex is highly unsteady and frequently collapses (figure 1 c) and regenerates in time.

Figure 2. (a) Normalized vortex length $\unicode[STIX]{x1D712}/d$ (see text) as a function of $Wi$ . Black dots represent the mean of vortex length sampled over 200 s and open triangles represent the maximum vortex length measured using holographic microscopy. The shaded region is bounded by 5th and 95th percentiles. Four regimes of vortex dynamics can be identified. I: steady Stokesian flow with fore–aft symmetry. II: emergence of a steady vortex in front of the cylinder. III: onset of weak time-dependence. IV: pulsing vortex which collapses and regenerates. Inset: r.m.s. fluctuation of $\unicode[STIX]{x1D712}/d$ . (b) Streak plot for regime I. (c) Streak plot showing vortex symmetry around the channel centreline at $Wi\approx 4$ . (d) Streak plot showing subsequent symmetry breaking at $Wi\approx 8$ in regime III.

In figure 2(a), we plot the mean vortex length (black dots) for all sample observations (80 snapshots over a duration of 200 s) as a function of $Wi$ ; the vortex length $\unicode[STIX]{x1D712}$ is normalized by the cylinder diameter $d$ . We note that for the Newtonian solvent, no vortex or irregular flow is observed at any flow rate. For the viscoelastic fluid, however, we can clearly identify four regimes as a function of $Wi$ . For $Wi\lesssim 2$ (regime I in figure 2 a), the flow around the cylinder possesses fore–aft symmetry identical to the creeping flow shown in figure 2(b). For $Wi\gtrsim 2$ , however, we see the emergence of a recirculation zone comprised of two rotating vortex rolls that are symmetric relative to the line $y=0$ passing the centre of the cylinder, as in figure 2(c). The flow is steady and the pattern does not change over several minutes. As $Wi$ is further increased, the stagnant region becomes highly elongated and extends further upstream along the centreline (regime II in figure 2 a), before reaching another flow transition at $Wi\approx 4$ . For $Wi\gtrsim 4$ (regime III in figure 2 a), the extent of the recirculation zone becomes unsteady and fluctuates weakly in time. This can be seen in the 5th and 95th percentile of all observed vortex lengths, shown by the shaded area in figure 2(a). These percentile curves begin to deviate from the mean in this regime, indicating variation in the sampled vortex lengths. The unsteadiness of the vortex length is also reflected in the increase of the root-mean-square (r.m.s.) fluctuation of $\unicode[STIX]{x1D712}/d$ , shown in the inset of figure 2(a). The r.m.s. fluctuation, however, saturates before reaching $Wi\approx 9$ . In this regime (III), the vortex varies significantly in size with $Wi$ and the lateral symmetry of the two vortex rolls is lost, as shown in figure 2(d). Lastly, for $Wi\gtrsim 9$ , the flow enters a regime where the vortex frequently collapses suddenly to a length of approximately $2d$ and then regenerates, as shown by the constant 5th percentile curve despite increasing $Wi$ . The vortex lengths frequently ‘pulse’ in time with large r.m.s. fluctuations, as shown previously by the streak plots in figure 1(b,c). The mean vortex length continues to grow with $Wi$ and extends far upstream, reaching $6d$ ( $3W$ ) at $Wi=16$ .

The origin of the vortex regions may be understood in terms of minimization of the flow extension due to high fluid extensional viscosity, similar to other contraction geometries (Boger Reference Boger1987; Rodd et al. Reference Rodd, Cooper-White, Boger and McKinley2007). In such geometries, the flow field develops recirculating vortices in order to produce an effectively longer entrance region for the flow to increase in velocity gradually, which reduces the extension rate $\dot{\unicode[STIX]{x1D716}}\sim \unicode[STIX]{x2202}U/\unicode[STIX]{x2202}x$ around the cylinder.

We note that in the high Trouton ratio limit (high extension), the length of a vortex extending upstream from the lip of a planar contraction has been predicted to increase linearly with $Wi$ and is independent of the Trouton ratio (Lubansky et al. Reference Lubansky, Boger, Servais, Burbidge and Cooper-White2007). We find similar behaviour here in that the (normalized) vortex length increases linearly with $Wi$ (figure 2 a) although quantitative comparison is not applicable between the two geometries.

Although the observed vortex is measured in an $x{-}y$ plane, the structure and dynamics are far from two-dimensional. In fact, the structure of the flow switches from two bistable modes in the $z$ direction, as we explore next.

3.2 Upstream vortex: 3D structures

In order to visualize the full flow structure upstream of the cylinder, we use holographic particle tracking to reconstruct the 3D flow field. To identify the stagnant region, characterized by negative stream-wise velocity, we plot the iso-surface where the velocity magnitude is zero. Figure 3(a,b) shows two snapshots of the spatial structure of the stagnant region at different times. It is clear that the flow is in fact made up of a pair of two separate recirculation zones originating near the corner of the cylinder with the walls. The vortex regions extend upstream along the top and bottom surfaces. Moreover, the vortex growth along one wall is accompanied by the suppression of the vortex on the other wall, which is confined to the space immediately in front of the cylinder. The switching between the two states occurs irregularly in time: as the stagnant zone collapses on one side, the other vortex reforms.

The flow symmetry breaking in the $z$ -direction is evident in figure 3(c,d), where the corresponding velocity field in a $x{-}z$ cross-section along the channel centreline is plotted. The presence of the adverse flow (blue) clearly alters the surrounding bulk flow, as shown by the streamwise flow (red) in figure 3(c,d). The generation of adverse flow originates along the upstream side of the cylinder that drives flow along top or bottom surface. In the pulsing regime, the extended vortex region can separate from the cylinder and move upstream along the top or bottom surface. Meanwhile, the other vortex expands in the $z$ -direction and fills the space directly in front of the cylinder.

Figure 3. Three-dimensional structure of the stagnant vortex upstream of the cylinder in the pulsing regime ( $Wi=23$ ). (a) Snapshot of the stagnant vortex defined by the iso-surface for zero speed $U=0$ , showing the dominance of the vortex near the top wall. (b) The complementary case where the vortex near the bottom wall dominates. (c,d) Velocity map along a cross-section passing through the channel centreline for the cases respectively shown in (a,b).

Figure 4. (a,b) Surfaces of equal velocity in front of the cylinder at two different time instances. Regions of zero $x$ -velocity form in front of the cylinder along the top or bottom walls. High-velocity regions ( $U=6.9~\text{mm}~\text{s}^{-1}$ isosurface) separate along the left and right sides of the channel. (c) The volume of the high-speed flow region ( $U\geqslant 6.9~\text{mm}~\text{s}^{-1}$ ) compared to the volume of the back flow $U\leqslant 0$ over a length of $1W$ upstream of the cylinder. The strong correlation indicates that the vortex region acts to constrict the bulk flow into a smaller region.

In figure 4(a,b), we distinguish the bulk flow from the vortex region by comparing the high-velocity iso-surface $U=6.9~\text{mm}~\text{s}^{-1}$ with the low-velocity iso-surface $U=0~\text{mm}~\text{s}^{-1}$ , measured simultaneously. We find that when the extended stagnant vortices are present, the bulk flow separates into two high-velocity regions surrounding the stagnant region. Similarly, periods of weakened vortex development correspond to transitional states where the bulk flow occupies a larger cross-section at a lower velocity. In figure 4(c), we compare the volume of the stagnant vortex (low-velocity region) and the bulk flow (high-velocity region) within $1W$ upstream of the cylinder, which shows that the two signals are indeed highly correlated.

The flow symmetry also breaks down in the span-wise $y$ -direction, resulting in a shift of the primary flow to either the left or right of the channel. For instance, figure 4(a) shows the region has shifted the predominant flow towards the left side and figure 4(b) shows the dominant flow on the right. This results in a greater flow along the right side of the cylinder. In fact, the flow rates around either side of the cylinder are found to be anti-correlated due to the constant volume flow rate, but can vary by approximately 20 % from the mean, suggesting flow switching and symmetry breaking in the span-wise $y$ direction. The lateral symmetry breaking and fluctuations may be intimately linked to the onset of finite flow perturbations observed in Pan et al. (Reference Pan, Morozov, Wagner and Arratia2013) and Qin & Arratia (Reference Qin and Arratia2017).

Lastly, although the variation in flow to either side of the cylinder may produce further flow fluctuation downstream, the flow instability associated with the vortex formation is only weakly communicated downstream. In fact, we find that the flow disturbance actually propagates far upstream.

3.3 Disturbance propagation and elastic waves

The unsteady vortex is accompanied by bulk flow instabilities far upstream. To quantify the fluctuation in the unstable flow, we conduct particle tracking velocimetry focused on a small window in the channel centreline ( $z=30~\unicode[STIX]{x03BC}\text{m}$ ) and monitor the instantaneous stream-wise velocity $u_{c}$ at various channel $x$ locations. The fluctuation is then obtained by subtracting the mean velocity from the instantaneous velocity, $u_{c}^{\prime }=u_{c}-\overline{u}_{c}$ . To facilitate comparison between various flow rates, we normalize the velocity fluctuation with the mean centreline velocity $\overline{U}_{c}$ far from the cylinder. In figure 5(a,b), we plot the time series of the normalized velocity fluctuations at $-5W$ and $-3W$ upstream of the cylinder. We observe large fluctuations (20 % of the mean flow) for the viscoelastic flow compared to the Newtonian solvent (grey line). The flow downstream of the cylinder at a comparable location, however, sees a different type of fluctuation. The amplitude of the fluctuation is smaller and the signal shows frequent jumps to high velocity amidst periods of dwelling at low velocities. This suggests two different instabilities up- and downstream of the cylinder.

Figure 5. (a) Fluctuation of the centreline velocity, $u_{c}^{\prime }$ , normalized by the mean velocity far from the cylinder at an upstream location $x=-5W$ for $Wi=23$ , where upstream vortices are pulsing, (b) at $x=-3W$ and (c) downstream at $x=3W$ . The grey line represents the flow of the Newtonian fluid at similar flow rates in each of these plots. (d) The normalized root mean square of the centreline velocity fluctuation at various channel locations and Weissenberg numbers.

Figure 6. (a) Cross-correlation between pairs of stream-wise velocity signals measured simultaneously at different locations for $Wi=23$ . The pairwise locations are indicated in the legend. (b) The elastic wave speed computed via the peak shift time as a function of $Wi$ , obtained from the pair $-2W\otimes -W$ , where the disturbance is strongest. Inset: lag time and various time scales to compute the elastic wave. Error bars represent the uncertainty estimated from statistical bootstrapping using time series sub-intervals.

The impact of the vortex instability is not confined to the vicinity of the cylinder but propagates upstream. In figure 5(d), we plot the root mean square velocity fluctuation $\unicode[STIX]{x1D70E}$ normalized by $\overline{U}_{c}$ at various $x$ locations. For low Weissenberg number ( $Wi\lesssim 1$ ), we see very little velocity fluctuation at any channel location. As $Wi$ increases, we see that the flow downstream of the cylinder becomes unstable, as expected from the well-known wake instability. The flow upstream, however, also shows a significant increase in velocity fluctuations. As $Wi$ further increases, the upstream velocity fluctuation grows rapidly in strength and propagates increasingly further upstream. At $Wi=23$ , it can be felt over $10W$ upstream of the cylinder and the largest fluctuation magnitude is in front of the cylinder at $-3W$ . Very far upstream ( $20W$ above), the flow is found to be steady, with fluctuations close to the instrument noise.

The propagation of upstream disturbances suggests a mechanism for transmitting perturbations against the primary flow direction, even in the presence of strong advection. Specifically, we investigate relaying of disturbances by computing the two-point cross-correlation between the two observation points separated by a distance  $\ell$ . In figure 6(a), we compute the cross-correlation coefficient for two streamwise velocity magnitudes measured simultaneously at $-3W$ and $-2W$ upstream of the cylinder. First, we see that the two signals are highly correlated, with a maximum $\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D70F})$ reaching almost 0.8. Note that perfect correlation has $\unicode[STIX]{x1D70C}=1$ , perfect anti-correlation has $\unicode[STIX]{x1D70C}=-1$ , while uncorrelated signal has $\unicode[STIX]{x1D70C}=0$ . However, the peak shift time $\unicode[STIX]{x1D70F}_{p}$ is approximately 0.14 s, which indicates that the velocity signal at $-3W$ leads $-2W$ by 0.14 s. This lead time turns out to be much longer than the advection timescale, $\ell /\overline{U}_{c}$ , which is approximately 0.018 s.

The propagation of the disturbance due to the elastic instability has a smaller effect on the flow in the cylinder wake. When we compare the velocity signal measured simultaneously $1W$ upstream and downstream of the cylinder we find that the signals are weakly correlated compared to those upstream. However, notably, the peak shift time is similar to the correlations made at two points upstream of the cylinder. This weaker correlation is expected from the very different velocity time series shown in figure 5(b,c), where the characteristics of the fluctuations are markedly different. Although some of the flow fluctuations are communicated downstream of the cylinder due to the finite relaxation time, the instability mechanisms of the flow upstream and downstream the cylinder are clearly distinct; the first relies on the propagation of elastic wave while the latter relies on the elastic hoop stresses.

The increase in peak shift time occurs if there is a wave travelling upstream with wave speed $c_{e}$ that goes against the bulk advection in the lab frame. The net result is a reduction in speed and increase in the time needed to travel, $\unicode[STIX]{x1D70F}_{p}=\ell /(\overline{U}_{c}-c_{e})$ . We note that the minus sign implies the wave is going in the direction opposite to the bulk flow and the shift time reflects the competition between the wave speed going upstream and the advection of fluid downstream. All quantities other than $c_{e}$ can be directly measured – we can then compute $c_{e}$ . Since the flow is steady without fluctuation below $Wi=4$ , we only report results for $Wi\gtrsim 4$ . Additionally, the accuracy of the method relies on a large velocity fluctuation from which a time shift can be measured. We therefore focus on positions with large fluctuations and note that only the flow in the pulsing regime has a clear wave speed. The peak shift times are measured to extract the corresponding wave speed between two locations $-2W\otimes -W$ , shown in figure 6(b) for various Weissenberg number.

As shown in figure 6(b), the elastic wave speed increases with Weissenberg number for $Wi\gtrsim 10$ , rather than maintaining a constant value. This implies that as $Wi$ increases, the disturbance increases in strength and propagates upstream faster. A comparison of the measured mean advection time, $l/\overline{U}_{c}$ , and peak shift times (inset of figure 6 b) show that both timescales decrease in a similar power-law relationship with $Wi$ . This is in contrast to the fluid relaxation time $\unicode[STIX]{x1D706}(\dot{\unicode[STIX]{x1D6FE}})$ , which decreases only slightly over this range of $Wi$ , showing that the wave speed does not simply scale with the polymer relaxation time. An increase in the wave speed upstream of the cylinder with $Wi$ gives an indication of an absolute instability, rather than one that is convected with the flow (Monkewitz Reference Monkewitz1988; Van Saarloos Reference Van Saarloos2003). Nevertheless, the finite extent of the upstream perturbations suggests there is a mechanism for dissipation or slow down of the elastic wave.