A critical point for bifurcation cascades and featureless turbulence

TL;DR

This paper investigates how bifurcation cascades and turbulence coexist in fluid flows, revealing the nature of transitions and attractors through analysis of toroidal pipe flow at different curvatures.

Contribution

It demonstrates that bifurcation cascades and turbulence can share phase space, identifying the supercritical transition as Ruelle--Takens type and analyzing competing attractors.

Findings

Supercritical transition is of Ruelle--Takens type.

Bifurcation cascades and turbulence can coexist in phase space.

Transition mechanisms depend on pipe curvature.

Abstract

In this Letter we show that a bifurcation cascade and fully sustained turbulence can share the phase space of a fluid flow system, resulting in the presence of competing stable attractors. We analyse the toroidal pipe flow, which undergoes subcritical transition to turbulence at low pipe curvatures and supercritical transition at high curvatures, as was previously documented. We provide decisive evidence that the nature of the supercritical transition is of Ruelle--Takens type and that, in a narrow range of intermediate curvatures, its dynamics competes with that of sustained turbulence emerging through subcritical transition mechanisms.

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Full text

The flow between two rotating cylinders—Taylor–Couette flow—is a classic example of transition initiated by linear instabilities (see Ref. Eckmann (1981) and references therein). When a governing parameter of the system—typically the Reynolds number $Re$ in fluid dynamics—is increased past a critical value, the system undergoes a supercritical Hopf bifurcation and the stable state of the flow changes from a steady fixed point to a periodic limit cycle. A further increase of the governing parameter causes a secondary Hopf (Neimark–Sacker) bifurcation 111A bifurcation occurring when the fixed point of a discrete-time map changes stability via a pair of complex-conjugate eigenvalues (multipliers) with unit modulus Neimark (1959); Sacker (1964); Kuznetsov (2004)., which renders the limit cycle unstable and introduces a new frequency in the flow, increasing its spatio–temporal complexity. The system can reach a chaotic state following a third bifurcation. Taylor–Couette flow, in certain configurations, actually follows a Ruelle–Takens scenario 222When a system undergoes three Hopf-type bifurcations, starting from a stationary solution as a parameter is varied, then it is likely that the system possesses a strange attractor with sensitivity to initial conditions after the third bifurcation. The power spectrum of such a system will exhibit one, then two, and possibly three independent basic frequencies. When the third frequency is about to appear, simultaneously some broad-band noise will appear if there is a strange attractor Ruelle and Takens (1971); Newhouse et al. (1978); Eckmann (1981). as was established experimentally in Ref. Gollub and Swinney (1975).

In linearly stable flows, on the other hand, subcritical bifurcations may occur and transition from a state to another is triggered by the introduction of finite-amplitude perturbations. This is a predominantly nonlinear process that relies on the existence of a non-trivial set other than the laminar state Waleffe (1995). Relaminarising and transitional trajectories are separated in phase space by a manifold known as the “edge of chaos” Skufca et al. (2006); Schneider et al. (2007); De Lozar et al. (2012). The trajectories that make their way to the turbulent attractor are generally organised around a set of unstable solutions, e.g., travelling waves or relative periodic orbits, at least at low Reynolds numbers Kawahara et al. (2012); Budanur et al. (2017). Flows in channels and pipes are examples of this scenario, and are dominated by large-scale spatial and temporal intermittency at the onset of turbulence. Only recently theoretical models have been shown to capture the complex physics of transition and match laboratory measurements Barkley et al. (2015); Shih et al. (2015); Barkley (2016). Notably, there is an increasing body of experimental and numerical evidence suggesting an analogy between subcritical transition and non-equilibrium phase transition of the directed percolation type Shih et al. (2015); Lemoult et al. (2016); Chantry et al. (2017). The accuracy of this analogy is still a matter of debate Barkley (2016); Mukund and Hof (2018), but it is clear that flow structures encountered in subcritical and supercritical transition scenarios are fundamentally different in nature, and their description is rooted in distinct theoretical grounds.

Only a few flow cases present both transition scenarios. Taylor–Couette flow undergoes a sequence of supercritical bifurcations when the cylinders are co-rotating, while subcritical transition is observed when the cylinders are counter-rotating Coles (1965); Eckmann (1981); Andereck et al. (1986). A similar behaviour is observed in rotating plane Couette flow Tsukahara et al. (2010). In both cases, however, the two transition scenarios were not known to interact with each other until the present study and one contemporary to it Christopher et al. (2019). In spatially developing boundary layers recent work suggests that at high Reynolds numbers the edge of chaos can effectively be interpreted as a manifold dividing classical supercritical and subcritical bypass-type transition Beneitez et al. (2019).

The flow in a bent pipe is a rare case in which the nature of transition is altered without such a clear separation. The change takes place as the pipe-to-torus diameter ratio $\delta=D/D_{T}$ is gradually increased Kühnen et al. (2015) and will be the focus of this Letter. Figure 1 sets the stage by presenting an overview of this flow case in the ($\delta$, $Re$) parameter space. Transition to turbulence is subcritical at low curvatures and qualitatively similar to the one in straight pipes Rinaldi et al. (2019); Sreenivasan and Strykowski (1983); Kühnen et al. (2015). For larger $\delta$, instead, transition is initiated by a supercritical Hopf bifurcation Sreenivasan and Strykowski (1983); Kühnen et al. (2014, 2015); Canton et al. (2016) and all elements point towards a bifurcation cascade Kühnen et al. (2014). The dynamics of the flow at intermediate curvatures, however, remains largely unexplored, leaving unanswered the question of whether the two transition scenarios interact, and if so how. In this Letter we address this question and show that characteristic structures of a bifurcation cascade and a subcritical transition scenario can coexist at a fixed combination of all governing flow parameters.

For the same combination of $\delta$ and $Re$, we find two competing attractors with complementary basins, namely sustained turbulence and a stable travelling wave (relative equilibrium) originated by the supercritical Hopf bifurcation predicted by linear theory. This is in contrast with what can be observed, e.g., in a linearly unstable channel flow, where only one asymptotic state exists regardless of whether transition is caused by Tollmien–Schlichting waves or by by-pass mechanisms 333Above the critical Reynolds number $Re\approx 5772$ channel flow becomes linearly unstable and the critical mode of instability is called Tollmien–Schlichting wave. Transition to turbulence may occur through the breakdown of such waves for carefully chosen initial conditions. However, in the presence of other disturbances of finite amplitude, the critical mode is by-passed by the nucleation and expansion of localised turbulent patches Schmid and Henningson (2001). Bent pipes are not the first flow case for which two competing stable—but not necessarily steady—solutions are documented, see, e.g. Taylor–Couette flow Coles (1965) or the flow through a sinuous stenosis Samuelsson et al. (2015). However, the competing solutions in Taylor–Couette appear to be either all of supercritical type or reached through a hysteresis cycle Christopher et al. (2019), while those in a stenotic flow belong to two branches originated by the same saddle–node bifurcation, and do not bear the imprint of the two fundamentally different transition scenarios. We perform direct numerical simulations (DNS) of the flow in a toroidal pipe 444The incompressible Navier–Stokes equations are solved using the open-source code Nek5000 Fischer et al. (2008). The numerical setup was previously validated for straight and bent pipes El Khoury et al. (2013); Noorani et al. (2013). and articulate our analysis in two steps.

First, we shed light on the bifurcation cascade at large curvatures, and provide evidence for a secondary Hopf bifurcation. The occurrence of a second bifurcation was “conjectured” in Ref. Kühnen et al. (2015) but “could not be pinpointed” experimentally (quotations from Refs. Kühnen et al. (2014, 2015)). Second, we move our attention to the region of the parameter space where the neutral curve of the flow intersects the threshold for subcritical transition 555Defined as 50% intermittency in Ref. Kühnen et al. (2015), hereafter referred to as critical point, see Fig. 1.

Regarding the bifurcation cascade, we focus on curvature $\delta=0.05$, which is far enough from the critical point to guarantee no influence of the subcritical transition scenario. By means of a modal stability analysis we verified that the steady flow becomes linearly unstable at $Re=3713$ and nonlinear simulations up to $Re=4000$ confirmed the nature of the supercritical Hopf bifurcation Canton et al. (2016). Figure 2(a) depicts the state of the system at this Reynolds number: the flow is constituted by a travelling wave generated by the first supercritical Hopf bifurcation and all trajectories in the phase space eventually converge to the corresponding relative equilibrium. The power spectral density (PSD) of the velocity signal at a fixed point presents an isolated peak at frequency $f_{1}\approx 0.48$, followed by its harmonics (made dimensioneless with bulk velocity and pipe diameter, i.e. $f=f^{*}D/U$). On the other hand, a PSD analysis of integral quantities, such as the kinetic energy $E_{k}$ and friction factor $f$, reveals no frequencies (except zero frequency). This is expected for systems with continuous symmetries, as the true dimension of the system is revealed in a symmetry-reduced phase space only—such as that constituted by $E_{k}$ and $f$—see e.g. Ref. Willis et al. (2013). Upon increasing the Reynolds number well beyond the critical one, the system undergoes a secondary Hopf bifurcation to what appears as a quasi-periodic state in the velocity state space, but is actually a relative periodic orbit in a symmetry-reduced state space. The attractor appears as a $T^{2}$ torus in the velocity space, depicted in figure 2(b) for $Re=4500$, and the PSD of the point velocity signal presents two incommensurable frequencies, $f_{2}\approx 0.22$ and $f_{3}\approx 0.37$, with their linear combinations and higher harmonics. Once again, the PSD of integral quantities shows one less frequency, indicating that the flow is on a relative periodic orbit. As the Reynolds number is increased further the flow eventually becomes turbulent in the whole pipe. The trajectory in the corresponding phase space is chaotic and the PSD of both point- and integral quantities is characterised by a broadband spectrum, see Fig. 2(c). Further bifurcations could not be accurately pinpointed with additional simulations between $Re=4500$ and $6000$, but only the appearance of broadband noise and eventually chaos. If this system were to follow a Ruelle–Takens route, either one of two scenarios could be observed between $Re=4500$ and $6000$: a Neimark–Sacker bifurcation would lead the system to a $T^{2}$ torus—in the $E_{k}-f$ phase space—followed either by a direct transition to chaos, or by a fourth bifurcation to a $T^{3}$ torus before the appearance of chaos. However, three-frequency tori are rare in hydrodynamical systems, albeit not impossible Oteski et al. (2015). We conclude that, for curvatures sufficiently larger than the ones at which subcritical transition occurs, from $\delta\approx 0.025$ up to unity, the flow indeed undergoes a bifurcation cascade that leads it to chaos, similarly to what is observed for Taylor–Couette Coles (1965).

We now turn our attention to the critical point, i.e. the area within $0.02\lesssim\delta\lesssim 0.03$ and $4000\lesssim Re\lesssim 5000$. These boundaries are approximate, as is the grey shaded area in Fig. 1. In this region of the parameter space we perform DNS in domains of length $L_{s}=4$–$8\,\lambda_{s}$—corresponding to approximately $10$–$20D$—where $\lambda_{s}$ is the wavelength of the travelling wave predicted by the linear theory at each curvature 666These domains are long enough for accommodating sustained turbulence as well as over 4 periods of the fundamental travelling wave. Note that turbulence is not intermittent in this region of the parameter space Sreenivasan and Strykowski (1983). When not simulating the whole torus, the domain length is restricted to integer multiples of the fundamental wavelength such that any dynamics present in these domains would also appear in the full toroidal geometry. This restriction does not introduce additional feedback effects, nor does it alter the nature of transition as was previously observed in Refs. Noorani et al. (2013); Noorani and Schlatter (2015); Canton et al. (2016); Rinaldi et al. (2019).. As a first step we verify that both sub- and supercritical behaviours can still be isolated. In order to check for subcritical behaviour we perform DNS for $\delta=0.022$ and $Re=4500$ (point A in Fig. 1). Each simulation is initialised with a single turbulent puff, selected from the ones computed in Ref. Rinaldi et al. (2019), which expands—becoming a slug at this $Re$—and fills the whole computational domain with turbulent flow, in agreement with the findings in Refs. Sreenivasan and Strykowski (1983); Kühnen et al. (2015); Rinaldi et al. (2019). At this Reynolds number, in fact, we are above the intermittency range where turbulent puffs are observed Sreenivasan and Strykowski (1983); Kühnen et al. (2015); Rinaldi et al. (2019). An appropriate perturbation is sufficient to initiate the by-pass mechanism that renders the flow homogeneously turbulent. The second test, to check for supercritical behaviour, is at $\delta=0.028$ and $Re=4600$ (point B in Fig. 1). This point is located below the subcritical transition threshold and just above the neutral curve—which for this curvature marks the first Hopf bifurcation at $Re=4570$. The flow is initialised with a parabolic streamwise velocity profile perturbed with either random noise or the unstable eigenmode, as was done in Ref. Canton et al. (2016)—in both cases increasing the energy of the flow field by approximately 1%. These simulations always converge to the nonlinear travelling wave created by the supercritical Hopf bifurcation, confirming the supercritical transition route previously discussed for $\delta=0.05$.

Having established both behaviours—subcritical in point A and supercritical in B—close to criticality, we explore the region above the neutral curve between points B and C. We investigate three pairs of $(\delta,Re)$, i.e. $(0.026,4750)$, $(0.024,4900)$, and $(0.022,5050)$, the latter corresponding to point C in Fig. 1, which is analysed in detail in the following and in Figs. 3 and 4.

Unlike in other regions of the parameter space, for $(\delta,Re)=(0.022,5050)$ the flow shows sensitivity to initial conditions, which dictate its asymptotic state. For these critical values of $\delta$ and $Re$ an initial condition consisting of a randomly perturbed parabolic velocity profile slowly converges to a stable nonlinear travelling wave, as predicted by the modal analysis and as observed for higher curvatures. The flow trajectory projected on the energy–friction phase space collapses to a single point, illustrated in blue in Fig. 4. On the other hand, if the laminar flow is perturbed with a sufficiently energetic localised disturbance—a puff in our case—the disturbance grows and invades the whole length of the pipe, turning the flow into a persistently turbulent state 777Experiments in Refs. Sreenivasan and Strykowski (1983); Kühnen et al. (2015) indicate persistent turbulence for these values of $\delta$ and $Re$. through a subcritical-like transition process (red line in Fig. 4). For this particular combination of $\delta$ and $Re$, sufficiently energetic means that the kinetic energy of the puff accounts for more than 4.8% of the kinetic energy of the flow field. If the puff has lower initial energy, it will only transiently grow in size before eventually disappearing (orange line in Fig. 4). In this case the flow state visits the neighbourhood of the turbulent attractor in phase space, but eventually converges to the travelling wave. Snapshots of the flow taken from these three different trajectories are reported in Fig. 3 which visually illustrates the temporary coexistence of travelling wave and puff.

The flow behaviour discussed in the present work clearly shows that a narrow region of the parameter space $(\delta,Re)$ presents two attractors, evidence of two completely different system dynamics, which compete and have complementary and finite basins of attraction. The two attractors represent sustained turbulence and the travelling wave originated by a supercritical Hopf bifurcation. In analogy with the generally accepted picture of the phase space for subcritical flows, we sketch a modified phase space in Fig. 5 where the laminar, steady-state attractor is replaced by the travelling wave and a saddle state acts as a mediator between flow trajectories. The two attractors documented in this Letter embody two entirely different transition scenarios, which are shown to coexist in a fluid system, and lead the flow to two diametrically opposed unsteady asymptotic states.