2 Pipe experiments

Experiments are carried out in two different pipe set-ups. In the first set-up, the pipe is made of glass tubes with an inner diameter of $D=2~\text{mm}$ ( $\pm 0.007~\text{mm}$ ), and has a total length of $8000D$ . Pipe sections are joined with 4 cm long acrylic connectors (de Lozar & Hof Reference de Lozar and Hof2009; Samanta, De Lozar & Hof Reference Samanta, De Lozar and Hof2011) which allow accurate matching and alignment of the inner diameter of the tubes. Some of the acrylic connectors have 0.4 mm holes drilled into them (perpendicular to the pipe axis) which can be used to perturb the flow or to measure the pressure. The working fluid is water which is supplied from a reservoir (see figure 1) positioned more than 20 m above the pipe. The reservoir is continuously overflowing, ensuring a precise pressure head to drive the flow. The reservoir is mounted on motorized vertical guide rails, permitting fine adjustments to the height of the reservoir, and hence the flow rate. The fluid enters the pipe through a convergence which assures laminar flow up to $Re\approx 5000$ (natural transition point).

Figure 1.

A schematic of the gravity-driven pipe set-up.

The flow rate is measured in an automated fashion by collecting the water at the pipe exit over a fixed period (typically 50 s, corresponding to around $2.5\times 10^{4}$ advective time units for the Reynolds number range investigated) and weighing the amount of water with electronic scales, thus obtaining the flow rate at regular intervals with an accuracy of better than 0.2 %. The fluid temperature is monitored by platinum resistance thermometers at the inlet and outlet of the pipe. Throughout the measurements, the temperature differences between these probes remain below 0.1 K, although the absolute temperature can change by several degrees. The mean value of these two probes is taken to be the temperature of the fluid in the pipe. The $Re$ value is then determined from the flow rate and taking the temperature dependence of the fluid viscosity into account.

The Reynolds number can change due to variations in fluid temperature or turbulent fraction (due to splitting and decay of puffs). For instance, the difference in frictional drag due to addition or disappearance of a single puff changes the Reynolds number by around 1. Any such change in the Reynolds number results in an adjustment of the vertical position of the reservoir to bring $Re$ back to its original value. Using this feedback method, an overall stability of the Reynolds number to within $\pm 5$ is achieved.

The flow is monitored by two differential pressure sensors (Validyne DP45), one located $75D$ (point S1 in figure 1) and the second $7875D$ (point S2 in figure 1) downstream of the perturbation point. Each sensor measures the pressure difference across two holes separated by $8D$ in the streamwise direction. Turbulent puffs cause a local increase in skin friction, and hence their passage causes a distinct increase in the pressure loss across the two points. Thus, the passage of a puff shows up as a peak in an otherwise flat pressure signal (see, for example, figure 7).

In broad outline, the second pipe set-up is identical to the first one. A crucial difference is that the pipe is of larger diameter ( $D=4\pm 0.01~\text{mm}$ ), and is shorter, with a length of $2500D$ . The pressure sensors monitor the flow at $250D$ and $2500D$ . The natural transition for this pipe is lower, being $Re\approx 3000$ .

3 Characteristics of puff turbulence

Turbulence at moderate Reynolds numbers ( $1700\lesssim Re\lesssim 2300$ ) in pipes occurs in the form of localized puffs separated by laminar regions. Laminar here refers to vanishing fluctuation levels, although the profile may not have developed its parabolic shape yet. In the interior of a puff, vortices are predominantly produced close to the upstream (trailing) edge (Hof et al. Reference Hof, de Lozar, Avila, Tu and Schneider2010). Vortices are then advected downstream, forming an elongated arrow-shaped downstream (leading) edge, and their amplitude decreases exponentially with distance from the upstream edge. The energetic part of puffs can be taken to be approximately $5D$ to $10D$ in length, while the downstream distance over which the profile recovers its parabolic shape is substantially larger (Hof et al. Reference Hof, de Lozar, Avila, Tu and Schneider2010). Puffs keep their shape and characteristic size for long times before they suddenly decay. Puffs, however, also interact: if two puffs are separated by less than $25D$ , the downstream puff will decrease in amplitude and eventually decay. Equally, it has been shown Hof et al. (Reference Hof, de Lozar, Avila, Tu and Schneider2010) that flattening of the velocity profile directly upstream of a puff results in its decay. Energetically, puffs appear to depend on the upstream laminar flow with a sufficiently parabolic profile (Samanta et al. Reference Samanta, De Lozar and Hof2011). The region directly downstream of a puff, on the other hand, is characterized by a flat velocity profile with decaying fluctuation levels, and here the flow is ‘refractory’. Even when external perturbations are applied to this region, they will swiftly decay and no additional turbulence can be sustained here. With further downstream distance, the flow profile slowly recovers its parabolic shape, and once the profile is sufficiently developed (at least $25D$ downstream of the original puff), a second puff can be triggered (Samanta et al. Reference Samanta, De Lozar and Hof2011). Much of this qualitative behaviour can be captured by a reaction–diffusion model (Barkley Reference Barkley2011; Barkley et al. Reference Barkley, Song, Mukund, Lemoult, Avila and Hof2015), where localized excitations equally require a minimum time during which the system has to recover (refractory region).

As will be shown in the following, puffs are the only long lived turbulent structures that can be excited at moderate Reynolds numbers ( $1700\lesssim Re\lesssim 2300$ ). Fully turbulent flow is unstable in this Reynolds number regime. If triggered, it quickly collapses into a sequence of puffs. Puffs are statistically identical, and any structural differences resulting from the initial disturbance creating them are quickly lost. It is also shown that not all perturbation mechanisms are capable of triggering puffs and that the observed puff frequency (turbulent fraction) immediately downstream of the disturbance point strongly depends on the perturbation used.

The evolution of individual puffs and the dependence of their structure on the initial perturbation were investigated by comparing puffs triggered by two different perturbations in the 4 mm pipe set-up. The impulsive jet perturbation involved injecting a jet of water orthogonal to the flow for a short period of time, through a hole of diameter 0.4 mm. The timing was controlled by a solenoid valve. The injection time was approximately $2D/U$ and the relative mass flux of the perturbation to that in the pipe was around $10^{-2}$ . The other perturbation was of a push–pull type, where an actuator was used to simultaneously inject and withdraw water from the pipe through two diametrically opposite holes, each of diameter 0.4 mm. The duration of this perturbation was around $20D/U$ and the relative mass flux was around $10^{-3}$ .

The centreline velocity was monitored at different distances downstream of the perturbation using laser Doppler velocimetry (LDV). For each perturbation type, the velocity signals were phase-averaged over 50 puffs, and the results are shown in figure 2. As can be seen from the signals at $10D$ and $50D$ , the perturbations are still evolving, and their shape depends on the perturbation type. However, beyond around $200D$ from the perturbation point, the puffs have developed their characteristic structure and, regardless of the initial perturbation, they are statistically identical (here shown for the puff centreline velocity signature). This is in line with Wygnanski & Champagne (Reference Wygnanski and Champagne1973), who reported developed puffs to be identical and independent of the original disturbance. Equally, it has been reported by de Lozar & Hof (Reference de Lozar and Hof2009) that puff lifetimes are independent of the initial perturbation. That puffs are statistically indistinguishable and appear to have no memory of how they were created is rooted in their chaotic properties and the sensitive dependence on initial conditions on chaotic sets which erases memory exponentially fast.

Figure 2.

Ensemble-averaged centreline velocities of turbulent puffs at different times after the perturbation, for an impulsive jet and a push–pull perturbation. For times smaller than around 200 advective time units, the average velocity is still evolving, and the signals from the two perturbations are different. Beyond this, the signals agree very well, and there is no further evolution in time, emphasizing the memoryless nature of puffs.

Figure 3.

Evolution of strongly disturbed inlet flow at $Re=2060$ in the 4 mm pipe. (a–c) The ratio of the centreline velocity $u_{c}$ (obtained by LDV) to the bulk velocity $U$ at $10D$ , $75D$ and $250D$ respectively from the entrance. (d) The pressure signals at $250D$ and $2500D$ .

Next, we look at the evolution of a fully turbulent flow. Figure 3 shows the time series of the ratio of the centreline velocity (obtained by LDV) to the bulk velocity, as well as pressure traces at different locations downstream of a turbulent pipe inlet at a Reynolds number of 2060. These measurements are carried out in the 4 mm pipe, where a nozzle with a 2 mm diameter is connected to the pipe inlet, continuously perturbing the flow. At $10D$ downstream of the perturbation point (figure 3 a), the flow appears to be uniformly turbulent, with strong velocity fluctuations, while the mean value of the centreline velocity deviates strongly from the case of laminar flow. At $75D$ from the perturbation point, however, the turbulence has broken up into narrow active regions separated by more quiescent flow (figure 3 b). The centreline velocity shows distinct minima, whereas, in between, fluctuation levels are lower and the centreline velocity is closer to the laminar value. As seen in figure 3(c), at $250D$ , the flow has eventually settled to a sequence of distinct puffs separated by regions of laminar flow. At the location of each puff, the centreline velocity drops sharply well below the laminar value of $2U$ , while the pressure drop (figure 3 d) across a puff is larger (due to the increased drag) and hence shows a distinct peak compared with the surrounding laminar flow. For any perturbation tested in this Reynolds number regime (see more examples below), flows will either settle to a sequence of puffs or relaminarize.

Although puffs are internally highly chaotic, sequences of puffs appear to be rather stable. Comparing the pressure signal (figure 3 d) at $250D$ with that recorded at $2500D$ , the puff pattern is practically unchanged. This raises the question of whether the puff pattern has reached a steady state or evolves on longer time scales.

Figure 4.

Radial (a–c) and axial (d–f) cross-sections of different disturbance geometries used to perturb the flow; all dimensions are in mm. (a,d) Multiple jets; continuous injection through 20 holes in the pipe wall, each having a diameter of 0.4 mm; (b,e) orifice with an open diameter of 2 mm; (c,f) semicircular obstacle with a diameter of 4 mm.

Figure 5.

Effectiveness of different perturbations in exciting puffs at a Reynolds number of 2100. The panels on the left show the ratio of the centreline velocity $u_{c}$ (measured by LDV) to that of the bulk flow velocity $U$ , $10D$ downstream of the perturbation. The panels on the right show the corresponding pressure signals $250D$ and $2500D$ downstream of the perturbation. (a) Multiple jets; continuous injection through 20 holes in the pipe wall; (b) 2 mm orifice; (c) semicircular obstacle (4 mm diameter); (d) impulsive jet perturbation (injection time of ${\approx}2D/U$ ).

While puffs can be excited at Reynolds numbers as low as 1700, not all perturbation mechanisms are suitable for triggering puffs, especially at low $Re$ . This is illustrated by measurements carried out in the 4 mm pipe at $Re=2100$ using different perturbations. In addition to the single-jet perturbation already described earlier in this section, three other perturbation mechanisms are used, and a schematic of their disturbance geometries is shown in figure 4. In figure 5, the panels on the left show the results of LDV measurements of the centreline velocity (normalized by the bulk velocity $U$ ) $10D$ downstream of the perturbation point. The corresponding panels on the right indicate pressure traces far downstream of the perturbation ( $250D$ and $2500D$ ), showing typical puff sequences that the perturbation method generates. Figure 5(a) shows measurements corresponding to the case when fluid is injected continuously through 20 holes in the pipe wall (figure 4 a). The measurements show (figure 5 a) that although the flow profile is strongly distorted (the centreline velocity clearly deviates from  $2U$ ) and the velocity signal shows significant fluctuations, no puffs are triggered and the flow downstream is laminar. As already pointed out by Hof et al. (Reference Hof, de Lozar, Avila, Tu and Schneider2010), continuous disturbances are inefficient in triggering puffs. By strongly flattening the profile, the entire flow is turned refractory and turbulence collapses. Figure 5(b) shows the results of placing a 2 mm orifice in the pipe (figure 4 b), and here occasionally single puffs emerge at irregular intervals. At slightly lower $Re$ ( ${\approx}2000$ ), the orifice perturbation also stops creating puffs and only gives rise to laminar flow. The results of blocking the pipe with a semicircular obstacle having a diameter of 4 mm (figure 4 c) are plotted in figure 5(c), showing a larger density of puffs. The impulsive single-jet perturbation described earlier, on the other hand, readily triggers puffs each time it is actuated, provided that the perturbation amplitude is high enough and injections are spaced out by at least 25 advective time units to avoid strong interactions. Data from the impulsive jet (injection time of ${\sim}2D/U$ and mass flux $10^{-2}$ of the pipe flow) are shown in figure 5(d), where puffs are triggered every 2 s. The same holds for the push–pull mechanism, where fluid is injected and withdrawn simultaneously from diametrically opposite holes in the pipe. Here, a perturbation time of ${\sim}20D/U$ and a relative flux of $10^{-3}$ were used. The length of the perturbation and the mass flux were chosen to be experimentally convenient, while ensuring that each perturbation event resulted in the production of precisely one puff. Due to the reproducible nature of impulsive perturbations (single impulsive jet and push–pull), they are used (with the perturbation times and fluxes mentioned here) for the remainder of the measurements. For the experiment with the impulsive jet, the pressure signal as the puff sequences pass the downstream sensor (at $2500D$ ) (dotted line), reiterates the fact that, even at this Reynolds number (2100), puff sequences do not change much over the time available even in the longest laboratory pipes.

Although all puff sequences in figure 5 are recorded at the same $Re$ , the different perturbation schemes give rise to very different flow patterns, varying from high to sparse puff densities, and even fully laminar flow. From such measurements, it is therefore impossible to define a critical point, and the question arises whether flows intrinsically depend on initial conditions or whether flows on much longer time scales settle to a steady state, where the turbulent fraction (puff density) is uniquely defined by the Reynolds number.

4 Lifetime measurements

Dynamically, puffs are chaotic repellors (Hof et al. Reference Hof, de Lozar, Kuik and Westerweel2008), and the chaotic dynamics makes a precise prediction of the escape time from the chaotic set (as well as the occurrence of splitting events) impossible. Hence, puffs evolve seemingly unaltered (regarding average quantities such as mean speed and size) on the chaotic set until they suddenly disappear. However, as has been shown previously, puffs have a characteristic lifetime which only depends on $Re$ (Hof et al. Reference Hof, de Lozar, Kuik and Westerweel2008; de Lozar & Hof Reference de Lozar and Hof2009; Avila et al. Reference Avila, Willis and Hof2010, Reference Avila, Moxey, de Lozar, Avila, Barkley and Hof2011). To test the different pipes used in this study and to provide a connection to previous work, we will in the following determine puff lifetimes for the two different set-ups used in this study and compare them with previous experimental and numerical lifetime studies.

In the 2 mm pipe, puffs were generated one at a time, such that there was only one puff in the pipe at any instant. The pressure signal from the sensor at $7875D$ was used to determine whether puffs generated at the perturbation point survived to this location or had decayed upstream. For a given $Re$ , 1000 puffs were investigated to obtain sufficient statistics for mean lifetimes. The survival probability of a puff is expected to follow $P(t)=\exp (-(t-t_{0})/\unicode[STIX]{x1D70F})$ , where $\unicode[STIX]{x1D70F}(Re)$ is the characteristic puff lifetime. As shown in figure 6, the measured puff lifetimes are in excellent agreement with earlier lifetime studies (Hof et al. Reference Hof, de Lozar, Kuik and Westerweel2008; de Lozar & Hof Reference de Lozar and Hof2009; Avila et al. Reference Avila, Willis and Hof2010, Reference Avila, Moxey, de Lozar, Avila, Barkley and Hof2011; Kuik, Poelma & Westerweel Reference Kuik, Poelma and Westerweel2010), which had been carried out in different pipe set-ups as well as obtained from direct numerical simulations. Measurements were repeated in the 4 mm pipe, where instead of a single jet, push–pull injection (with zero net mass flux) was used to create puffs. Again, the lifetime statistics are in excellent agreement with those of the 2 mm pipe and the studies cited above.

Figure 6.

Mean lifetime before puff decay in the 2 mm and 4 mm pipes compared with the non-diverging superexponential scaling in Avila et al. (Reference Avila, Moxey, de Lozar, Avila, Barkley and Hof2011) (dashed line).

Our data hence support the superexponential lifetime scaling suggested previously and confirm that puff statistics only depend on $Re$ (not on specifics of a given pipe set-up or the perturbation used). In particular, they rule out a transition scenario based on diverging lifetimes of single puffs. It should be noted that the non-diverging superexponential lifetime scaling only applies to single isolated puffs. A spatio-temporally intermittent flow, i.e. a flow comprising multiple interacting puffs, is expected to show diverging lifetimes (of turbulence overall) instead. While individual puffs still decay, the creation of new puffs (puff splitting) has been suggested (Avila et al. Reference Avila, Moxey, de Lozar, Avila, Barkley and Hof2011) to eventually outweigh this process. Because of the extremely long time scales, the divergence of the overall lifetime could not be shown explicitly by Avila et al. (Reference Avila, Moxey, de Lozar, Avila, Barkley and Hof2011); instead, this could only be inferred from the decay and splitting rates of single puffs. Statistics of isolated puffs can also not give any insight into the eventual flow pattern, which is determined by puff interactions. The single-puff statistics and the simple balance between puff splitting and decay rates predict a continuously increasing number of puffs above the critical point. With increasing turbulent fraction, on the other hand, puff–puff interactions will become important and are expected to saturate the growth process.

In the following, we will show that by exploiting the memoryless nature of puffs, extremely long observation times (of the order of $10^{8}$ advective time units) can be accomplished, which in turn allows the study of the long-term evolution of the spatio-temporally intermittent flow pattern.

5 Steady state and critical point

The central idea of the present study is to exploit this memoryless nature of puffs and the large separation between time scales, i.e. between the fast chaotic internal puff dynamics and the exceedingly slow evolution times of flow patterns. On the latter time scales, puffs have well-defined average properties and are individually indistinguishable. We show that it is then possible to have periodic boundary conditions (in a sense that will be clarified shortly), which allows us to monitor puff sequences for longer than $10^{7}$ advective time units.

Figure 7.

Pressure traces of a puff sequence near the pipe exit (position S2 in figure 1) and of the reproduced sequence near the entrance (position S1 in figure 1).

To do this, the flow is perturbed $150D$ downstream of the inlet (point P in figure 1) to create a sequence of puffs that serves as the initial condition. This puff sequence travels downstream, and puffs can freely interact, split and decay. The resulting puff sequence is then monitored close to the pipe exit (position S2 in figure 1) by a pressure sensor. Each time a puff is detected at this point, a new puff is created at the perturbation point P. In this way, the sequence of puffs leaving the pipe is recreated at the pipe entrance and the puff spacing is precisely kept, as shown in figure 7. If puffs are indeed memoryless and identical in all of their mean properties, as discussed in §§ 3 and 4, the recreated puff sequence should in a statistical sense be equally representative and eventually lead to the same asymptotic state. In this sense, the pipe set-up provides periodic boundary conditions for the puff sequences studied and allows one to monitor the evolution of these sequences for arbitrarily long times. If successful, this method should show whether puff sequences eventually settle to a well-defined statistical steady state with a fixed turbulent fraction.

In the following, starting from some initial distribution of puffs, the exiting puff sequence will be reproduced at the inlet as described above. The evolution of the puff pattern is then monitored for long times to probe whether eventually an equilibrium state is reached.

Figure 8.

Evolution of the turbulent fraction in the pipe. (a) Results from the 2 mm pipe, with the impulsive jet used as a perturbation. The curves in red, blue and green correspond respectively to $Re=2020$ , 2060 and 2100. (b) Result from the 2 mm pipe with the push–pull perturbation (black curve) and the 4 mm pipe with the impulsive jet (red curve), for $Re=2060$ . In the shorter 4 mm pipe, the temporal fluctuations are larger, because the turbulent fraction is averaged over a smaller system size.

We first present the results of experiments in the $D=2~\text{mm}$ pipe, with the impulsive jet used as the perturbation. Figure 8(a) shows the time evolution of the turbulent fraction for different values of $Re$ and different initial conditions. It should be noted that the time axis is in a logarithmic scale. Each colour corresponds to a particular $Re$ ; the curves in red, blue and green correspond respectively to $Re=2020$ , 2060 and 2100. The different shades of a colour correspond to different initial (time $t=0$ ) turbulent fractions. The lighter and darker shades correspond respectively to lower and higher values of the initial turbulent fraction. First, we discuss the runs at $Re=2020$ , which is 1 % below the value of $Re_{c}=2040$ proposed by Avila et al. (Reference Avila, Moxey, de Lozar, Avila, Barkley and Hof2011). The initial condition chosen was a sequence of 32 equally spaced puffs. Taking the average length of the turbulent portion of a puff to be $10D$ , this corresponds to a turbulent fraction of 4 %. The length of the turbulent part of a puff depends to a certain extent on the criterion used to discriminate it from the surrounding laminar flow. However, a different choice only scales the turbulent fraction by a factor and does not change the conclusions obtained. On monitoring the flow at $Re=2020$ starting from this initial condition, the puffs indeed eventually all decayed, as seen by the light red curve in figure 8(a). Figure 9(a) is a space–time plot showing the eventual return to laminar flow. It took more than $3\times 10^{7}$ advective time units before the system settled to laminar flow, and the longest surviving puff passed the pipe approximately 4000 times before it eventually decayed. The experiment was repeated for the same $Re$ but with a different initial condition (around four times larger turbulent fraction) and again the flow eventually completely relaminarized (dark red curve in figure 8 a).

Figure 9.

Space–time plots with the puffs shown in blue on a yellow laminar background. (a) Portion of a space–time plot for $Re=2020$ , where all of the puffs eventually decay. (b) Portion of a space–time plot under a statistical steady-state condition for $Re=2060$ . Splitting and decay events are marked with red and black circles respectively.

In contrast to this, at $Re=2060$ , the flow settles to a statistical steady state where the turbulent fraction is non-zero. The measurements for this $Re$ are repeated for three different initial conditions: turbulent fractions of 4 %, 10 % and 17 % (corresponding respectively to 32, 80 and 136 puffs), the time evolutions of which are indicated respectively by the light, medium and dark blue curves in figure 8(a). While the first two of these initial conditions had sequences of equally spaced puffs, the third one had a random puff sequence that was obtained downstream of a disturbed inlet (nozzle entrance) in a separate experiment. In all of the cases, the flow reached a statistical steady state in which the turbulent fraction fluctuated about a well-defined mean. The mean value of the turbulent fraction was the same for the different initial conditions, being around 14 % (108 puffs). A portion of the space–time plot in the statistical steady state for $Re=2060$ is shown in figure 9(b), with splitting and decay events indicated by red and black circles respectively. It should be emphasized that the existence of an equilibrium level of the turbulent fraction is a direct consequence of spatial interactions and cannot be predicted from single-puff statistics.

Similarly, for $Re=2100$ , the evolutions for initial conditions corresponding to turbulent fractions of 4 %, 10 % and 22 % are shown by the light, medium and dark green curves respectively. For this $Re$ , the steady state corresponds to a turbulent fraction of around 21 %. The time taken to reach steady state is smaller while the frequency of fluctuations is higher as compared with $Re=2060$ , as would be expected because puff splittings occur more frequently at this higher  $Re$ .

Hence, we first observe sustained ‘puff turbulence’ at $Re=2060$ , and the critical point therefore falls in the range $2020<Re_{c}<2060$ , showing that the estimate obtained by a simple balance of decay and splitting times (Avila et al. Reference Avila, Moxey, de Lozar, Avila, Barkley and Hof2011) works well for the case of pipe flow. To test whether the determined steady-state turbulent fraction depends on the pipe set-up or the type of perturbation used, the experiment was repeated for $Re=2060$ in the 2 mm pipe, using the push–pull perturbation, and subsequently the pipe was changed to the 4 mm pipe. In the latter, the impulsive jet perturbation was used. As discussed in § 2, apart from having a greater diameter, this pipe has a shorter length and a lower $Re$ for the natural transition. The results from these runs are shown in figure 8(b) and in both cases the turbulent fraction fluctuates around the same mean as in the original 2 mm pipe experiments figure 8(a). It hence appears that the average turbulent fraction is independent of the pipe and only a function of  $Re$ .

Figure 10.

Probability distributions of laminar gap sizes at $Re=2060$ and 2100.

It is known that certain spatio-temporal systems exhibit scale-invariant flow patterns close to the critical point (Chaté & Manneville Reference Chaté and Manneville1988; Lemoult et al. Reference Lemoult, Shi, Avila, Jalikop, Avila and Hof2016). In such cases, the size distribution of laminar gaps follows a power-law distribution close enough to the critical point, and exponential distributions farther away. We computed the distributions in the steady state for $Re=2060$ and 2100, and these are shown in figure 10. For both of these Reynolds numbers, we find that the laminar gap sizes are exponentially distributed. According to the measurements discussed above, the critical point lies between $Re=2060$ and $Re=2020$ . Hence, our measurements could be as much as 2 % away from the critical point, which could be insufficient to observe scale-invariant behaviour. In the present set-up, a more accurate determination of the critical points and measurements even closer to it are not possible due to limitations in the long-term stability of the Reynolds number (0.25 %). The local dynamics of puffs and their interactions also play a role in deciding how close to the critical point, if at all, scale-free behaviour is observed, and some aspects are discussed in the next section.

6 Spatio-temporal intermittency and puff clustering

Above the critical point, the flow consists of turbulent puffs separated by laminar regions of varying length, and the overall pattern continuously changes in time, as shown in figure 9. Spatio-temporal intermittency is hence an intrinsic feature of flows in this regime. Overall, the spatio-temporal intermittent nature of the flow pattern that is assumed in the long-time limit is characteristic for spreading processes in out-of-equilibrium (i.e. dissipative) systems where a fluctuating state (here turbulence) competes with an absorbing state (here laminar flow). In pipe flow, the laminar state qualifies as an absorbing state because of its linear stability, i.e. if a region relaminarizes, it remains in the laminar state for all times unless perturbed by a neighbouring turbulent site. If all turbulence has decayed, the flow subsequently remains laminar. The analogy between linearly stable shear flows and non-equilibrium phase transitions from fluctuating into absorbing states was first pointed out by Pomeau (Reference Pomeau1986). Subsequently, the physical processes that give rise to the competition between laminar and turbulent flow regions were identified for pipe flow (Avila et al. Reference Avila, Moxey, de Lozar, Avila, Barkley and Hof2011). Here, spreading of turbulence occurs in the form of puff splitting, and the transient nature (Hof et al. Reference Hof, Westerweel, Schneider and Eckhardt2006) of puffs and interactions (Hof et al. Reference Hof, de Lozar, Avila, Tu and Schneider2010) gives rise to decay and relaminarization. Generally, stochastic spreading phenomena have been studied in great detail in statistical physics. Examples of absorbing state transitions (Hinrichsen Reference Hinrichsen2000) are epidemics (here, healthy individuals can only be infected by contact with infected ones) or water ‘percolating’ through a porous medium (here, dry sites can only become wet if they are in direct contact with wet sites). With increasing control parameter, these systems display a continuous phase transition with universal features. They belong to the so called DP universality class and are characterized by three independent critical exponents. The first predicts the density of active sites (i.e. in our case the puff density or turbulent fraction) as the critical point is surpassed. The second and third exponents determine the spatial and temporal correlations of the statistical steady state close to the critical point; in our case, these would be the typical sizes of the laminar gaps along the space and time directions in a space–time plot such as figure 9(b). The DP framework hence predicts the characteristics of the spatio-temporal intermittent patterns that emerge at the phase transition, but it does not make any predictions about the ‘microscopic’ details, i.e., in the present case, the chaotic internal dynamics of puffs and their stochastic interactions. Or, in other words, the universal properties that emerge at the critical point are independent of the details of the microscopic dynamics (Hinrichsen Reference Hinrichsen2000). In recent experiments and numerical simulations, transition in Couette flow was shown to fall into the directed percolation universality class (Lemoult et al. Reference Lemoult, Shi, Avila, Jalikop, Avila and Hof2016). Other recent experiments in Poiseuille flow have also proposed that evidence was found for directed percolation (Sano & Tamai Reference Sano and Tamai2016). In the latter case however, the critical point does not agree with the values found in recent studies (Xiong et al. (Reference Xiong, Tao, Chen and Brandt2015), Tsukahara (private communications) and Mukund, Parnajape and Hof (in preparation)). Moreover, the equilibration times proposed by Sano and Tamai are approximately five orders of magnitude shorter than the splitting and decay times observed in pipes (Avila et al. Reference Avila, Moxey, de Lozar, Avila, Barkley and Hof2011). The finding of our present study that the equilibration times exceed the spreading and decay times suggests that the flows investigated by Sano & Tamai (Reference Sano and Tamai2016) were far from reaching a statistical steady state.

While a connection between transition in pipe flow and the simple percolation framework may appear to be far fetched, DP is conjectured to universally apply to seemingly unrelated systems if the following four conditions are met (Janssen Reference Janssen1981; Grassberger Reference Grassberger1982; Hinrichsen Reference Hinrichsen2000): (i) the system undergoes a continuous phase transition from a fluctuating (turbulence) into a unique absorbing state (laminar flow); (ii) the system is characterized by a positive one-component order parameter (in our case, the turbulent fraction); (iii) there are no long-range interactions; (iv) absence of additional symmetries or quenched randomness. While pipe flow appears to meet conditions (i), (ii) and (iv), we will in the following take a closer look at puff interactions.

Figure 11.

A space–time plot of pressure showing the relative motion of two puffs that start with an initial separation of $30D$ . The pressure is measured across $8D$ and normalized by the laminar pressure drop ( $P_{HG}$ ) across the same distance.

A first consequence of puff interactions is that the puff decay rates differ strongly from those for individual puffs measured by Avila et al. (Reference Avila, Moxey, de Lozar, Avila, Barkley and Hof2011). For the approximately 40 puffs shown in figure 9, single-puff statistics would predict at most one decay to occur over the $2\times 10^{6}$ time units displayed. However, six decays are observed. In general, the relatively dense puff pattern considerably limits the expansion rates and increases the puff decay rates (puffs in too close proximity can annihilate each other (Samanta et al. Reference Samanta, De Lozar and Hof2011)). Overall, these puff–puff interactions lead to a balance between splitting and decay events and hence are responsible for the flow assuming a steady turbulent fraction. Moreover, the time taken to reach equilibrium exceeds the time scales of splitting and decay at the critical point (Avila et al. Reference Avila, Moxey, de Lozar, Avila, Barkley and Hof2011), although the critical point has been approached to only within 1 %. Puff interactions are also directly visible in the space–time plot (figure 9) where puff clusters are observed, which propagate in the streamwise direction (diagonal lines going from top right to bottom left). To better understand the cause of puff clustering, we carried out a separate experiment where two puffs were generated $30D$ apart and monitored with pressure sensors every $500D$ . Five independent runs were carried out and the pressure signal as well as the arrival time at each of the sensor locations was averaged over these five independent puff pairs. Pressure values and arrival times at locations in between two adjacent sensors were linearly interpolated. The results are shown in a space–time plot in figure 11, in a frame that is moving with the mean speed of a single isolated puff. The mean speed of an isolated puff was measured in a separate experiment and its $Re$ dependence was found to be well described by $U_{p}=U(6.22\times 10^{-7}Re^{2}-2.83\times 10^{-3}Re+4.16)$ in the range $1700<Re<2250$ , where $U_{p}$ is the speed of the puff and $U$ is the bulk flow speed. This agrees well with previous measurements (de Lozar & Hof Reference de Lozar and Hof2009; Barkley et al. Reference Barkley, Song, Mukund, Lemoult, Avila and Hof2015). Figure 11 shows that the mean velocity of the upstream puff is the same as that of an isolated puff. At separations of less than around $100D$ , the downstream puff is affected by the upstream one and energetically weakened (Hof et al. Reference Hof, de Lozar, Avila, Tu and Schneider2010), as seen in a lower pressure drop across the puff. The weaker downstream puff then travels faster and the two puffs separate until they are around $100D$ apart; see also figure 27 in Barkley (Reference Barkley2016). Returning to figure 9, puff clustering is generally caused by puffs being in close range to their upstream neighbours. The energetically weakened puff travels faster while the speed of the upstream puff is not affected (also see figure 11). The faster puff then separates from its upstream neighbour so that it eventually slows down again, exactly as shown for the two interacting puffs in figure 11. However, at the same time, it gets into range of its next downstream neighbour, which in turn accelerates. The resulting cluster of puffs with increased speed then propagates down the pipe in a wave-like fashion. This clustering breaks the left–right symmetry in the space–time plot, whose visual appearance differs from the space–time plot for directed percolation. This could potentially have consequences for the nature of the transition.

Puff clustering clearly increases the interaction length beyond the nearest-neighbour limit. It typically involves a group of around five puffs. Whether the interaction stops at this range (i.e. falls off exponentially) or whether a weak interaction persists to much greater length (in the extreme case the entire pipe) is unclear. In the former case, the transition is likely to still fall into the DP universality class but would potentially require larger system sizes to determine the nature of the transition. Such interactions also have an influence on the range of $Re$ near the critical point where the universal scaling applies. The lack of scale invariance at $Re=2060$ suggests that the scaling range is considerably smaller than 2 % in $Re$ . The possible requirement of even larger system sizes and tighter control on the long-term stability of $Re$ poses considerable challenges for future experiments. In any case, pipe flow appears to be more complicated than Couette flow, where such interactions due to the upstream–downstream symmetry are absent.