Emergence of spatio-temporal dynamics from exact coherent solutions in pipe flow

Despite the ubiquity of turbulence, a theory explaining the emergence of its complexity has remained elusive even though the governing equations have been known for nearly two centuries. Landau [1] proposed in the 1940s that as the Reynolds number Re increases, turbulence arises through an infinite sequence of linear instabilities. In this picture each instability would add a new temporal frequency to the flow eventually resulting in the continuous spectrum that is characteristic of turbulence. Ruelle and Takens [2] refined this idea and established the current paradigm for the onset of turbulence: a few instabilities suffice to produce a continuous spectrum and render the flow chaotic. Such a scenario was first successfully demonstrated in laboratory experiments in a Taylor–Couette setup [3], and illustrations of other routes to chaos via period-doubling and intermittency followed shortly after [4]. This framework has been useful in explaining the origin of temporal chaos in many systems but falls short of describing turbulence because it fails to account for spatial features [5].

The importance of spatial interactions in fluid flows can be seen in pipes, channels and boundary layers already at low $\mathrm{Re}$. In these and other wall-bounded shear flows turbulence first appears in isolated patches surrounded by laminar flow [6]. This coexistence is possible because the laminar flow is linearly stable and finite amplitude disturbances are needed to seed turbulence. In pipe flow the Reynolds number (defined as $\mathrm{Re}\,:= \,{DU}/\nu$, with D the pipe diameter, U the constant mean driving speed and ν the kinematic viscosity of the fluid) is the sole governing parameter. Figure 1 illustrates the dynamics of a localised turbulent patch or puff in a 50-diameter long pipe at $\mathrm{Re}\,=\,1860$. While the mass-flux is kept constant, the driving pressure gradient, the streamwise length of the turbulent region and its propagation speed strongly fluctuate, and the internal arrangement of high and low velocity streaks within the puff evolves erratically (see figure 1(c)). These spatio-temporal fluctuations result in the eventual collapse of the puff back to laminar flow in a memoryless decay process [7]. Although the characteristic lifetime of puffs increases with Re, puffs remain transient and their dynamics is consistent with a chaotic repeller or saddle in the phase space of the Navier–Stokes equations [8, 9]. However, as Re is increased, puffs can also expand in length and split [10], thereby progressively filling the pipe with a spatially intermittent laminar-turbulent pattern [11]. In the limit of long pipes, a self-sustained turbulent state first arises once the puff-splitting rate exceeds the decay rate [12], which happens at Re ≈ 2040. Recently, this transition type has been shown to belong to the universality class of directed percolation for Couette flow [13].

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Despite recent progress in quantifying and modelling [14] the onset of turbulence in pipe flow as a spatio-temporal system whose fundamental units are puffs, surprisingly little is known about their origin. In fact, a framework providing a plausible explanation for their formation just begins to emerge. The key idea is that puffs stem from coherent structures (simple invariant solutions) of the Navier–Stokes equations that serve as elementary building blocks [15]. The simplest such structures in pipe flow are spatially periodic traveling waves [16], which are relative equilibria of the governing equations. More recently, streamwise localised solutions arising at a saddle-node bifurcation at Re = 1428 have been found [17] (figure 2). These solutions possess reflectional and π-rotational symmetry, and in this subspace the upper branch solution (UB in figure 2) is stable up to Re = 1532.5, which allows its computation directly by time-stepping of the Navier–Stokes equations. The lower branch solution (LB in figure 2) has a single unstable direction and is an edge state [18] locally separating phase–space trajectories that turn chaotic from those that go laminar. At Re ≈ 1533 the upper-branch solution undergoes a subcritical Neimark–Sacker bifurcation, with a narrow hysteresis band down to Re = 1531.3, followed by break-up into chaos at Re = 1540. At Re ≳ 1545 flow trajectories were observed to relaminarise, hinting at a global bifurcation turning the chaotic attractor into a chaotic saddle or repeller [17] (figure 2). The dynamics of a relaminarising run at Re = 1546 is shown in figure 3. The corresponding spatially localised spot exhibits chaotic yet mild modulations of structure, streamwise length and propagation speed. It is referred to as 'chaotic spot' in the rest of the paper in order to make clear that its behaviour is in contrast with the strong spatio-temporal fluctuations exhibited by turbulent puffs. This difference in spatio-temporal complexity, however, raises the question of what further qualitative changes occur in phase space as Re increases.

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In this paper we show a mechanism increasing the spatio-temporal complexity of chaotic motions in pipe flow. As the Reynolds number is increased several chaotic saddles appear in phase space. Each of them is associated with localised spots with distinct propagation speed and streamwise length. Initially these saddles are rather isolated in phase space, but their successive merging results in trajectories that explore increasingly larger regions. In physical space, this translates into spot length and pressure fluctuations closer to those of turbulent puffs.

In order to render the problem more tractable we simplify the dynamics by restricting them subject to reflectional symmetry about a diametrical plane and π-rotational periodicity with respect to the pipe axis, as in [17]. The only exception to this is figure 1, which was run in full space. Direct numerical simulations of the Navier–Stokes equations in cylindrical coordinates are carried out using the hybrid spectral finite-difference code of Willis [19], which is based on a velocity-potential formulation automatically satisfying the continuity equation and thereby avoiding the computation of the pressure. The coupled boundary conditions on the potentials are solved to machine precision with the influence-matrix method. We refer the interested reader to [19] for details.

We use periodic boundary conditions in the streamwise direction, with computational domains long enough (40–100D) such that no interaction occurs between the upstream and downstream fronts of the turbulent patches. For Re < 1650 we adopt a time-step $\delta t=0.0025D/U$, 768 axial Fourier modes ($K=\pm 384$) for a length of 50D, 32 azimuthal Fourier modes (M = ±16) for $\theta \in [0,\pi )$ and N = 48 points in the radial direction. This grid resolution and time-step allow us to determine the bifurcation sequence leading to transient chaos with a precision better than ${\rm{\Delta }}\mathrm{Re}\,=\,\pm 1$. Both time-step and spatial discretisation are slightly finer than for previous simulations at larger Re in full space [9] that managed to achieve quantitative agreement with turbulent puffs lifetimes measured in extremely accurate and precisely controlled experiments [8]. For $\mathrm{Re}\,\geqslant \,1650$ we use a domain length of 100D and K = ±1024 to accurately resolve the increasingly turbulent dynamics.

We start by analysing the transition from persistent to transient chaotic spots, which was previously estimated to occur at Re ≳ 1545 [17]. In order to shed light on the underlying bifurcation mechanism, we perform a statistical study of the mean lifetime (τ) of the transients. Initial conditions for this study were taken from a chaotic spot at Re = 1545. Figure 4(a) shows the survival probability of the transients as a function of elapsed time for several Re > 1545. As in low-dimensional systems relaminarisation follows a memoryless decay characterised by the escape rate $1/\tau$ from a chaotic saddle [20]. As Re increases, the mean lifetime rapidly decreases and is well described by

$$\begin{eqnarray}&&\tau =\displaystyle \frac{a}{\mathrm{Re}\,-\,{\mathrm{Re}}_{\mathrm{bc}}},\end{eqnarray} \tag{ 1 }$$

with critical Reynolds number ${\mathrm{Re}}_{\mathrm{bc}}\,=\,1545$ and $a=7293D/U$ obtained from an inverse linear fit (blue dashed line in figure 4(b) and inset). Note that in experiments the lifetimes of turbulent puffs increase super-exponentially for Re > 1700 [7], which suggests that further qualitative changes must occur in phase space as Re increases. The simple scaling of equation (1) is typical of low-dimensional chaos [20], and suggests that the onset of transients may take place via a generic mechanism called boundary crisis [21]. In a boundary crisis, the attractor collides with its own basin boundary and opens up to become a strange or chaotic saddle [20].

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This hypothesis is put to test here by examining the behaviour of extremely long phase–space trajectories ($\gt {10}^{5}D/U$) of the attractor in the range $\mathrm{Re}\,\in \,[1540,1545]$. Figure 5 shows that the chaotic trajectories sporadically visit the neighbourhood of the lower branch solution, which is known to be the attractor at the basin boundary (or edge state [18]) in the subspace [17]. The edge state is visited more closely with increasing Re and, as ${\mathrm{Re}}_{\mathrm{bc}}$ is approached, the minimum distance to the edge state vanishes (bottom-right inset of figure 5). This trend supports the boundary crisis hypothesis and a cubic fit to the data yields ${\mathrm{Re}}_{\mathrm{bc}}\,=\,1545$, exactly as for the lifetimes analysis. Crises leading to chaotic transients were recently observed in Couette [22], channel [23] and pipe flows [24], highlighting their ubiquity in shear flows. Note however that all these studies were done in short periodic domains unable to capture spatial localisation, a salient feature of shear flow turbulence.

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The mild fluctuations displayed by the transient chaotic spot together with the simple lifetime scaling suggest that the dynamics involved is low-dimensional. However, traces of more complex behaviour can be observed at larger Re. Figure 6(a) shows the time evolution of the driving pressure gradient corresponding to a run at Re = 1580, which was initialised from a chaotic spot at Re = 1545 and relaminarises at a later time (not shown). Initially the dynamics remains on a low friction plateau (labelled 'A'), and the spot has a streamwise length and propagation speed similar to that observed at lower Re. This is evidenced by comparison of the space–time diagram of figure 6(b) up to $t=150D/U$, to its counterpart at Re = 1546 (figure 3(b)). Subsequently the dynamics explores a region of higher friction (labelled 'B'), which corresponds to a spatial expansion of the spot (see the snapshots of figure 6(c)). Thereafter the spot rapidly recedes in length and the dynamics settles anew on the low friction plateau A.

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The phase portraits of figure 7 illustrate the dynamics of the flow for several Re. At Re = 1546 the dynamics revolves around a fairly small region of phase space (A), whereas a much larger region is explored at Re = 1580. This qualitative change suggests that two separate regions of phase space, (A) and (B), originally dynamically isolated, have merged at a global bifurcation. The chaotic spot that results from the merger exhibits strong fluctuations in its propagation speed and spatial extent (see figures 6(b) and (c)). To elucidate the nature of the underlying global bifurcation we perform an additional set of runs with initial conditions taken from a turbulent flow at larger Re = 2000, thus ensuring that the dynamics are not biased to start from too close to A.

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4.1. Dynamics prior to the boundary crisis

We first reduce the Reynolds number from Re = 2000 to Re = 1540, which precedes the boundary crisis (Re_bc = 1545). Here A is an attractor and phase–space trajectories evolving towards it remain there forever (see the red curve in figure 7(c)). Other trajectories approach region B instead before finally relaminarising (black curve). Of a total of 300 such runs, 22% converged onto A (either directly or after an interim visit to B) while the remaining 78% visited B before relaminarising. The survivor function of the latter is marked with black symbols in figure 8(a), and its exponential trend confirms the existence of a second saddle B.

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Figure 7(c) shows that B holds transient dynamics that eventually leave towards either laminar flow (black curve) or coexisting attractor A (red curve). Following the boundary crisis, A also becomes a saddle and all trajectories ultimately relaminarise regardless of whether they temporarily settled on either A or B (figure 7(d)).

4.2. Weak saddle interaction after the crisis

4.3. Progressive merger of saddles

As Re is further increased the ghosts of the two saddles progressively blend into one sole entity (figure 9). At Re ≈ 1580 the characteristic times for transitions between A and B become shorter than their relaminarisation times, so most trajectories visit both A and B several times before relaminarising (see figures 6 and 7(b)). Henceforth the two saddles effectively become one and most phase–space trajectories explore the total phase space associated beforehand with A and B (see figures 7(b), (d) and 9(b)). As a consequence, a unique combined slope emerges in the survival function (green symbols in figure 8(a)). As the transition rate among saddles increases, the clearly identifiable regimes of figure 6 gradually transform into the puff-like dynamics of figures 10(a) and (b). The merger in phase space is accompanied by a rise in lifetimes with increasing Re (figure 8(b)) as trajectories bounce from saddle to saddle more and more frequently. This reverses the initial decreasing trend followed after the boundary crisis and is thus consistent with experimental observations [7, 8].

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Kreilos et al [26] have also reported a mechanism increasing the lifetimes of chaotic transients. In their simulations of minimal-flow-unit plane Couette flow an attractor arises within an existing saddle. The attracting bubble bursts in a crisis and the new saddle features its own lifetime statistics, which differ from those of the pre-existing saddle. As a result, the combined ensuing lifetimes increase in a non-smooth and non-monotonic fashion, which is in stark contrast to the monotonous increase of lifetimes that follows the saddle merger observed here.

4.4. Bifurcation mechanisms behind saddle merger

In section 3 we have demonstrated that saddle A emerges as the chaotic attractor grows and finally collides with its basin boundary at Re_bc = 1545. This raises the question of whether saddle B results from a similar global bifurcation. Note however that the dynamics of B is much more complex than that of A: in figure 6(b) two distinct friction plateaus (B₁ and B₂) are clearly discernible within B, and the phase portrait of figure 7(b) also shows that the dynamics of B₁ and B₂ unfolds in two distinct regions of phase space. The space–time diagram of figure 6(b) and snapshots of streamwise velocity isosurfaces in figures 6(b) and (c) further evidence, now in physical space, that B₁ and B₂ correspond to two spots with different streamwise length and propagation speed.

In an attempt to elucidate the origin of saddle B, we performed an extensive study of its lifetimes statistics. Interestingly, at Re ∼ 1470 lifetime distributions feature two clear slopes corresponding to spots B₁ and B₂. These are similar to the lifetime distribution shown as red in figure 8(a) and hence not shown here. For Re ≳ 1470, a single slope is observed in the lifetime statistics, which indicates that B₁ and B₂ have merged to form B. We further reduced the Reynolds number in an attempt to detect the initiation of the merger, but this resulted in the sudden disappearance of B at around Re ∼ 1450, suggesting that, unlike what happens with A and B at higher Re, the dynamics associated with the independent B₁ and B₂ saddles is highly unstable and only becomes detectable upon their merging to form B. Altogether, the dynamics in the range 1460 < Re < 1540 is extremely complex and beyond the scope of this work.

Recently, Chantry et al [34] showed that the localised relative periodic orbit underlying saddle A emerges from a spatially periodic travelling wave (named N2) at a subharmonic Hopf bifurcation. Figure 11 shows the streamwise velocity along an axial line for N2 and for the flow snapshots of figure 6(c). The internal wave structure of snapshots of B₁ and B₂, consisting of wave trains of different lengths, is also strikingly reminiscent of N2. Despite this similarity we could not determine whether exact solutions underlying B₁ and B₂ also stem from N2 or other traveling-wave solutions. A Newton–Krylov solver specially devised to compute relative periodic solutions invariably failed to converge from any of the many snapshots of B₁ and B₂ that we seeded as initial guesses.

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In planar shear flow spanwise localisation has been shown to follow homoclinic snaking [35], which generates multiple spanwise-localised solutions spanning different counts of vortex-pairs [36]. Although an analogous mechanism may generate localised relative periodic orbits that underlie A, B₁ and B₂, [37] have recently demonstrated in the case of channel flow that the snaking mechanism is decisively disrupted in the streamwise direction. It was found instead that streamwise-localised relative periodic orbits of quantised lengths, exactly as the chaotic spots found here, appear in saddle-node bifurcations and extend to very high Re. These solutions are related by intricate connections in parameter space and by heteroclinic connections in phase space. We speculate that this type of solutions might set the stage for bifurcation cascades into chaotic attractors, and for the basin boundary metamorphoses and crises that finally result in the complexity of wall-bounded turbulent flows.

Several routes to chaos were proposed in the seventies to account for the emergence of disordered dynamics in systems far from equilibrium [2, 4, 38]. These routes were first illustrated in simple dynamical systems and later observed in fluid experiments featuring linearly unstable laminar flows [3, 4]. More recently, numerical studies in small computational domains have shown that the same routes apply to linearly stable flows [22–24], albeit with decisive differences. First, the bifurcation cascade leading to chaos cannot start from laminar flow, and stems instead from disconnected invariant solutions originated at saddle-node bifurcations. These solutions are typically unstable in full space, so that all aforementioned numerical investigations rely on restricting the dynamics to symmetry subspaces that stabilise them. A second essential difference is that shear flow turbulence is transient, which requires a transition from permanent to transient chaos as Re increases. While many numerical studies of shear flows in small domains have demonstrated the emergence of chaotic saddles, the associated transients invariably produce decreasing lifetimes for increasing Re. Experimental turbulent lifetimes rapidly surge instead, such that additional qualitative changes must take place in phase space. Finally, transitional shear flow turbulence is spatially localised and it is only through spatial proliferation of spots that turbulence becomes self-sustained [12, 39]. None of the theoretical and numerical studies cited takes this important spatial aspect into consideration, the former using models based on low-dimensional ODEs and maps, the latter employing numerical domains too small to capture localisation.

The first reported bifurcation cascade in a spatially extended shear flow, Hagen–Poiseuille flow in a long cylindrical pipe [17], laid out a two-step transition of a streamwise localised relative periodic orbit into a localised chaotic spot. Here we have shown that a boundary crisis in which the attractor collides with its own basin boundary is responsible for the transition from permanent to transient chaos. After the boundary crisis, lifetimes decrease rapidly with increasing Re following the simplest possible scaling law in low dimensional dynamical systems [40]. The dynamics of the resulting transient spot is only mildly chaotic, involving but weak fluctuations of pressure drop, propagation speed and length, typical of merely temporal chaos. We have shown that strong spatio-temporal chaotic fluctuations arise from a progressive merger of temporally chaotic saddles in phase space. Each saddle is associated with a chaotic spot of distinct characteristic streamwise length, propagation speed and pressure drop, such that their merger results in a spot whose length oscillates erratically. Completion of the saddles merger is followed at higher Re by phenomena typically observed in experiments [7] such as rapid lifetime growth with increasing Reynolds number (see figure 8(b)) or the enhancement of spatio-temporal fluctuations through an increase in the transition rate between the phase space regions originally occupied by isolated saddles. Overall, the dynamics after the merger is consistent with that of the turbulent puff. Furthermore, a spatial mechanism of occasional violent growth appears to be a precursor to splitting, which we have observed to start occurring at larger Re ≳ 1700 (see figure 12). Increasing lifetimes and spatial expansion of spots set the basis for an analogy between turbulent transition and directed percolation [13, 41], thus rendering our work relevant to future development of a rational foundation for this phenomenology.

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A multitude of streamwise periodic travelling waves with different symmetries have been found in the last decade. These waves only require subharmonic bifurcations that have been shown to be generic [34] to produce streamwise localised solution branches that could, in turn, give rise to temporally chaotic saddles. Hence, analogous scenarios will most probably be at play within symmetry subspaces other than that considered here. Piecing together the progress made in different subspaces in order to explain the complexity observed in full space remains however an outstanding challenge. It is unclear as of today how to tackle the problem in the abscence of symmetry restrictions given that the localised edge state in full space is chaotic at all Reynolds numbers hitherto investigated [42, 43]. A quantitative characterisation of the progressive complexification of the dynamics, namely through the analysis of the evolution of the Lyapunov spectrum of spatio-temporal transients (see [44] for an application to coupled-map lattices), is beyond the scope of this work and therefore left for future study.

Beyond the example of pipe flow, we expect our findings to be relevant also for flows with two spatially extended directions, such as channel and Couette flows. In these cases turbulence appears in the form of patches that are localised in both the streamwise and spanwise directions [6] and that feature yet richer spatial self-organisation processes [45]. Despite the challenge, doubly localised invariant solutions have recently been discovered in these geometries [46] and are promising candidates for building blocks of turbulent dynamics in doubly extended systems. More generally, the mechanisms shown here may be relevant to other extended systems driven far from equilibrium. In particular, we expect our route to complexity to be generic in excitable media, for which theories of temporal chaos also fail to properly characterise the spatio-temporal dynamics observed.

Support from the Deutsche Forschungsgemeinschaft (DFG) through grant FOR 1182 and computing time from the Jülich Supercomputing Centre (grant number HER22) and Regionales Rechenzentrum Erlangen (RRZE) are acknowledged. FM (Serra Húnter fellow) also acknowledges financial support from grants FIS2013-40880-P and 2014SGR1515 from the Spanish and Catalan governments, respectively.