3.1. The transition from laminar to elastically turbulent flow

Flow passing through a porous medium at low $Wi$ and low $Re$ is expected to drive a pressure drop that varies approximately linearly with flow rate for the Boger fluids used in our experiments, as per Darcy's law. Thus to delineate our flow regimes, we consider the pressure drop data presented in figure 6. In figure 6(a), we show the time-averaged polymeric contribution to the pressure drop ${\rm \Delta} \bar {p}-{\rm \Delta} \bar {p}_{n}$ between the inlet and outlet of the microchannel. Here, ${\rm \Delta} p$ is the pressure drop (i.e. $p_1 - p_2$) measured for the flow of the polymer solution, and ${\rm \Delta} p_n$ is the pressure drop measured for the Newtonian solvent. Both were recorded over 180 s at each imposed volumetric flow rate $Q$. We see that for low $Wi$ ($0.5\leq Wi \leq 1.8$), the pressure drop obeys the laminar model ($r^2 = 0.99$), with a slight deviation at $Wi = 1.8$, but the polymeric contribution breaks this expectation at higher flow rates $Wi \geq 2.6$, resulting in the often discussed increased flow resistance driven by elastic effects (Marshall & Metzner Reference Marshall and Metzner1967; Durst et al. Reference Durst, Haas and Kaczmar1981; Browne & Datta Reference Browne and Datta2021). We use the departure point of the pressure drop data from the linear trend to estimate the critical $Wi$ as $Wi_{c} = 1.8$, i.e. the highest $Wi$ data point to fall near the trend.

Figure 6.

(a) The polymeric contribution to the pressure drop across the flow cell. (b) Power spectra $E_{pp}$ for selected $Wi$ highlighting the gradual strengthening of ET for increasing $Wi$. The dashed line in (a) indicates a linear regression fit below $Wi_{c}$.

To further describe the transition from laminar to unstable flow, we present power spectra $E_{pp}$ of the fluctuating pressure $p' = {\rm \Delta} p - \bar {{\rm \Delta} p}$ against dimensionless frequency $f^* = f\lambda$ in figure 6(b). Despite the pressure drop deviation in figure 6(a) for $1.8\leq Wi \leq 2.6$, when comparing the pressure power spectra of laminar flow at $Wi = 0.5$ to that of an apparently unstable flow at $Wi = 2.6$, one observes that they are quite similar, with slightly increased power in the lower $f^*$ for the unstable state. Evidently, the near-critical unstable flow state is weakly transient (we quantify the flow transience in the Appendix). For increasing $Wi$, the transient state grows in power, and by $Wi = 8.2$, we observe sustained increased power throughout the lower frequency range, with a steep power-law decay exponent $\beta \approx 3$ (which is typical of ET) up to $f^*=1$, i.e. the fluctuation time scale $1/f$ matches that of the polymer relaxation time $\lambda$. We hypothesize that the higher frequency roll-off towards $f^* = 2$ down to the noise floor is due to polydispersity of the HPAA, and generalize that $1/\lambda$ is a common upper bound for capturing the majority of the energy of ET (Browne & Datta Reference Browne and Datta2021). While the power spectra show that a smooth macro-scale transition towards stronger ET occurs with increasing $Wi$, we next consider the local micro-scale evolution.

3.2. Instability: a local curvature balance

In § 1 we introduced a common scaling parameter $M = \sqrt {2\,Wi\,\lambda U/R}$ (McKinley et al. Reference McKinley, Pakdel and Öztekin1996; Pakdel & McKinley Reference Pakdel and McKinley1996) as a generalized predictor for the onset of elastic instability by coupling streamline curvature with normal stresses. The reported literature for $M$ typically uses characteristic values for $Wi$, $U$ and $R$. Instead, we substitute in local parameters such that $M$ can describe the flow on instantaneous streamlines. We first assume that the vector field is quasi-static as sampled (1/250 Hz $\ll 1/f$), then seed streamlines randomly in the $x^*$–$y^*$ plane about $z^* = 0$, and use a Runge–Kutta 4–5 integrator to progress streamlines forwards and backwards in time until they exit the 3-D volume. The streamlines provide a local radius of curvature $R_{L}$ in three dimensions directly, but we linearly interpolate for $U$ and $\dot {\gamma }$ at each streamline from the background grid of $\boldsymbol {u}$ and ${{\boldsymbol{\mathsf{D}}}}$ to provide $\lvert \boldsymbol {u} \rvert$ and the local Weissenberg number $Wi_{L} = N_{1}(\dot {\gamma }_{L})/2\sigma (\dot {\gamma }_{L})$. Thus in the present work, we discuss the local form of the Pakdel–McKinley criterion, $M_{L} = \sqrt {2\,Wi_{L}\,\lambda \,\lvert \boldsymbol {u} \rvert /R_{L}}$.

In figures 7(a–d), we present the outcome of this approach for $M_{L}$ at selected $Wi$ near $Wi_{c}$, in figure 7( f) the p.d.f. of $M_{L}$, and in figure 7(g) $M_{max}$ and $M_{avg}$. For increasing $Wi$ approaching $Wi_{c} = 1.8$, streamline curvature and $M_{L}$ increase, particularly at the throat but also at the midpoint of each pore. Furthermore, the streamlines are pushed outwards incrementally towards the necks between rows for increasing $Wi$. This behaviour is generally symmetric between crossflow pores for $Wi = 0.9, 1.3$. However at $Wi = 1.8$, we see that this symmetry is broken. For any given column of pores in $y^*$, streamlines will tend to flood the neck region asymmetrically from one of the pores, leading to straightened streamlines and reduced $M_{L}$ at the opposing pore.

Figure 7.

(a–d) Streamlines of local $M_{L}$ evaluation for selected $Wi$ near $Wi_{c}$, with arrows highlighting asymmetry (sketch in (e)). ( f) The p.d.f. of the streamline fields. (g) The maximum and average $M_{L}$ at each $Wi$. Streamline curvature does increase with $Wi$ above $Wi_{c}$, but in doing so, symmetry is broken as streamlines flood towards neighbouring pores, thereby enforcing stability.

For $Wi = 2.6$ (figure 7d), asymmetric flooding behaviour is shown more distinctly. Regions with this effect have been highlighted with arrows, and a sketch is shown in figure 7(e). Thus we infer that pores with elastically unstable flow will stabilize their peers as the instability manifests as a flooding incursion into neighbouring pores. Notably, this effect also correlates along the streamwise direction, such that rows of pores in $x^*$ show a similar flow response. Above $Wi_{c}$, we can see that local curvature and $M_{L}$ continue to increase with $Wi$, but at a reduced rate (figure 7g), owing to the overall distribution of $M_{L}$ tailing towards higher values but maintaining a similar mean (figure 7g). From $M_{max}$ in figure 7(g), we can estimate our critical $M_{c}$ for this geometry by taking $M_{max}$ at $Wi_c = 1.8$, yielding $M_{c} = 2.5$, with the caveat that although $M_{L}$ is self-consistent for different $Wi$, it will underestimate the theoretical $M$ due to discrete sampling limitations. Nonetheless, $M_{c}$ here is in good agreement with reported values of $M_{c}$ based on $M$ ranging from 1 to 7.

In the transient unstable state, spatial heterogeneity resolves the increased stresses via localized regions of increased flow instability. Although the pressure power spectra indicate that these pockets of instability must be transient, tomogram storage for masking remains computationally burdensome, thus instead we time-resolve the fluctuations of ET at high $Wi$ where stronger low-frequency modes make it feasible to capture multiple state changes in the VOI within a 10 s recording. A sample description of the low-frequency velocity fluctuations is available for $Wi = 1.8$ in the Appendix. Moreover, $Wi_{c} \approx 2$ aligns with $De = \lambda U_{p}/ w_{p} = 1$ by definition, thus transient effects can grow in the array in this regime as polymer chains are advected between pores faster than they can relax.

3.3. Inter-pore heterogeneity

We consider the evolving flow state at $Wi = 8.2$ as a representative case for ET. We support this assertion with selected power spectral densities extracted from a point in the velocity field ($x^* = z^* = 0$, $y^* = -0.75$) to expand upon the prior spectral analysis of the pressure field. In figure 8 we present the global $E_{pp}$ spectra discussed previously, together with local fluctuating crossflow velocity spectra $E_{vp}$, at $Wi = 3.7$ and $Wi = 8.2$. Here, $E_{vp}$ and $E_{pp}$ show the algebraic decays typical of ET: for $E_{pp}(f) \sim f^{-\beta }$ and $E_{vp}(f) \sim f^{-\alpha }$, $\beta \approx 3$ and $\alpha \approx 3.5$, consistent with a recent review (Steinberg Reference Steinberg2021). Furthermore, the analysis of Steinberg (Reference Steinberg2019) deduced a scaling relation $\beta = 2(\alpha -2)$ that supports our experimental data. The Taylor frozen flow hypothesis (Taylor Reference Taylor1938) has been applied reasonably well for parts of ET flow. Specifically, experiments and simulations have noted similarly steep decay exponents for the temporal and spatial power spectra (Burghelea et al. Reference Burghelea, Segre and Steinberg2005; Burghelea, Segre & Steinberg Reference Burghelea, Segre and Steinberg2007; van Buel & Stark Reference van Buel and Stark2022). Thus our scaling in the frequency domain will also track the wavenumber contribution in the spatial domain, yielding the common observation that ET is governed by the nonlinear interaction of only a few large-scale modes (Burghelea et al. Reference Burghelea, Segre and Steinberg2005; Steinberg Reference Steinberg2019).

Figure 8.

Power spectral densities of the fluctuation pressure $E_{pp}(f)$ and the fluctuating crossflow velocity $E_{vp}(f)$ at moderate and high $Wi$. The power law decays for $E_{pp}(f) \sim f^{-\beta }$ and $E_{vp}(f) \sim f^{-\alpha }$ follow a scaling $\beta = 2(\alpha -2)$ (Steinberg Reference Steinberg2019).

For a local description of ET flow, in figures 9(a,b) we show streamlines coloured by $\lvert \boldsymbol {u} \rvert ^*$ and $M_{L}$, respectively, to inspect local flow properties between the central rows of pores (labelled as $r_a$ and $r_b$, respectively). Generally, flow through row $r_a$ remains stable from pore columns 0–2, with a flow rate $\lvert \boldsymbol {u} \rvert ^*$ above the volume average, straightened streamlines, and low $M_{L}$. Flow through row $r_b$, however, has an opposite reaction for the same quantities: reduced $\lvert \boldsymbol {u} \rvert ^*$, strongly curving streamlines, and increased local $M_{L}$ across pore columns 1–3, but a reduction in pore 4. Remarkably, we can also see flow through $r_a$ transition to a state similar to flow through $r_b$ by pore 4, in this case driven by out-of-plane streamline curvature towards $+z^*$. Preceding the unstable flow in pore 4, we observe a reduction in $\lvert \boldsymbol {u} \rvert ^*$ in pore 3 as either the elastic instability or other rheological parameters drive increased pore resistance. Finally, we also see the same correlation behaviours noted in the prior section: streamwise correlation and crossflow inverse correlation of pores with unstable flow. The inversely correlated state is particularly compelling, with no $y^*$-adjacent pair of pores responding with mirrored unstable flow. This further supports our hypothesis that pore flooding can promote stability in adjacent pores. In pores farther from the unstable region, however, we do observe adjacent stable flow states, such as at $x^* = y^* = 0$ in figure 9. Therefore, the adjacent peer stabilization effect may be alternatively classed as a crossflow exclusion of instability. Streamwise correlation generally holds true, apart from the transition states noted in pores 3 and 4 as $r_a$ goes unstable and $r_b$ reduces curvature, respectively. A likely explanation for the streamwise consistency is flow history, which we will investigate by continuing this analysis throughout time in § 3.4.

Figure 9.

Instantaneous streamlines for the HPAA flow under a state of ET at $Wi = 8.2$, coloured by (a) the dimensionless velocity magnitude $\lvert \boldsymbol {u} \rvert ^*$, and (b) the Pakdel–McKinley criterion $M_{L}$. We have numbered columns of pores along $x^*$.

To describe the laminar and unstable flow states, we take a snapshot also at $Wi = 8.2$, but at an instant when $r_a$ and $r_b$ are in consistent states along $x^*$. Figure 10 presents both the local $M_{L}$ streamlines previously described together with planes of $\lvert \boldsymbol {u} \rvert ^*$, $v^*$, $\xi$ and $\dot {\gamma }$. Here, $r_a$ and $r_b$ have almost fully reversed their states from the figure 9 snapshot, with $r_a$ uniformly unstable between pores along $x^*$ and in $r_b$ repeating pores with laminar flow. Hence we can relate the instability to local rheological effects. As discussed previously, pores with unstable flow lead consistently to increased flow resistance (locally decelerated) compared to the laminar channel, but importantly, we see here that the laminar channel $r_b$ is also shear-dominated with strong rate-of-strain. The shear-thinning behaviour of the fluid is likely in effect, leading to a locally accelerated flow reaching $\lvert \boldsymbol {u} \rvert ^* = 3.6$. Along the unstable zone, we observe competing influences: while the fluid is strongly extensionally strain hardening, and these pores manifest with extensionally dominant flow, the rate-of-strain is locally reduced. Thus while strain hardening may be an influence in the evolution of the local instability, we cannot confirm this hypothesis in the present work.

Figure 10.

(a,b) Instantaneous streamlines for the HPAA flow under a state of ET at $Wi = 8.2$, together with (c–f) $x^*$–$y^*$ slices showing $\lvert \boldsymbol {u} \rvert ^*$, $\xi$, $v^*$ and $\dot {\gamma }$, respectively.

3.4. Feedback between streamline curvature and flow resistance

To describe the temporal evolution of the unstable flow state, we first refer back to the fluctuating pressure power spectra from figure 6(b). As we mentioned previously, $f^* \approx 1$ appears to be a natural upper limit for the frequency of the spatiotemporal modes of ET as fluctuations approach $1/\lambda$ but decay rapidly at higher frequencies (figure 8). Therefore, we compare snapshots with a temporal stride of approximately $\lambda$, but with some variation permitted to capture disturbances evolving away from $\lambda$.

Selected streamline fields are presented in figure 11 to describe generally instability propagation forward in time. We start the sequence in figure 11(a), where an individual pore in the VOI is unstable (3, the last pore). All of the pores upstream of this point are stable in this row. Forward in time at $t^* = 0.7$ (figure 11b), we see that pores 2 and 3 are now unstable; the unstable region has evidently propagated upstream. This trend continues for $t^* = 1.8$ and $2.8$ (figures 11c,d), with the full row now unstable at $t^* = 2.8$. Upstream propagation continues at $t^* = 3.9$ (figure 11e), with relaminarization occurring in pore 3. Clearly, flow history is important in this context, for both the onset of instability as well as the termination thereof. We support this notion by considering the streamline curvature of pores upstream of the local maximum $M_{L}$. The transition from a laminar state to a strongly unstable flow is spatially smooth, with streamline curvature and elastic stresses building for pores progressing along increasing $x^*$ towards a pore with a locally maximum state of instability (protruding streamlines). Positive feedback between streamline curvature and flow resistance is a likely explanation: in a given pore, local streamlines respond with increased curvature due to raised resistance in neighbouring pores downstream. In a given pore with unstable flow, streamlines protrude from the pore centre (figure 7e), and this excursion becomes more pronounced over time while being accompanied by locally increased flow resistance. Thereafter, flow in an adjacent pore upstream responds to the raised resistance with streamline protrusion, hence the instability propagates upstream in the array.

Figure 11.

Snapshot streamline fields of $M_{L}$ over selected $t^*$ at $Wi = 8.2$. Note that the black line in (a) indicates $y^* = -0.75$, which is used in figure 12.

To further explain the instability propagation, we present space–time diagrams in figure 12 extracted along $y^* = \pm 0.75$ from separate recordings at the same $Wi = 8.2$. For reference, we also show the advection length scale $U_{p}\lambda /w_{p}$ overlaid on the figure. Here, we see that for a given point $x^*$, flow maintains a consistent laminar or unstable state over long $t^*$. Whereas for a time $t^*$, the fluctuating state persists over $x^*$ with a scale on a par with the advection length scale. Polymer chains in the flow would thus be in a state where strain is accumulated throughout a cohesive pocket of instability. Crossflow fluctuation may propagate rapidly upstream, as it does near $t^{*}_{0} = 2$ for $x^* = 2\unicode{x2013}3$, or the laminar state could linger as it does along $t^{*}_{0} = 2\unicode{x2013}4$ for $x^* = 1$. Similar behaviour is noted in the termination of instability shown in figure 12(b), where, for example, fluctuations decay along low $x^* = 0\unicode{x2013}2$ over a long time scale surpassing the recording limit. These time scales are not surprising when we compare the space–time diagrams to the fluctuating pressure power spectra from figure 6, where the lowest frequencies continued to gain power for decreasing $f^*$ towards the lower limit $0.06 f\lambda$ (period $17\lambda$). Thus although an upstream propagation speed of instability can be estimated roughly as $\approx$$-0.3U_p$, the progression from laminar to unstable flow is spatially smooth but not temporally, in support of the spectral characteristics of ET.

Figure 12.

Space–time diagrams along $y^* = \pm 0.75$ of the crossflow velocity magnitude ratio $|v|/|\boldsymbol {u}|$ from separate recordings at $Wi = 8.2$. We also include the advection length scale $U_{p}\lambda /w_{p}$ as the pink bar.

From our simplified ordered porous medium, we have elucidated viscoelastic flow effects relevant to porous media flow in particular, and to the evolution of vessel-scale modes for ET in general. Namely, we have highlighted the importance of spatial correlation via flow history effects as well as via communication between pore flow states. An area for further research, however, would be probing longer time scales; such are not feasible for a volumetric time-resolved study given current processing times, but a 2-D study of ET in our ordered porous medium would be a promising direction for future work, to better quantify the propagation nature of unstable flow near $Wi_{c}$. Furthermore, while ET has been quantified in 3-D randomly packed media (Browne & Datta Reference Browne and Datta2021), packing structure and disorder are known to be significant for the onset of instability for viscoelastic flow (Walkama et al. Reference Walkama, Waisbord and Guasto2020; Haward et al. Reference Haward, Hopkins and Shen2021) and for chaotic mixing for creeping porous media flow (Turuban et al. Reference Turuban, Lester, Le Borgne and Méheust2018, Reference Turuban, Lester, Heyman, Le Borgne and Méheust2019), and these are thus topics of ongoing research.