Complexity of localised coherent structures in a boundary-layer flow

Complexity of localised coherent structures in a

boundary-layer flow

Taras Khapko1,2, Yohann Duguet3, Tobias Kreilos4,5, Philipp Schlatter1,2, Bruno Eckhardt4,6, and Dan S. Henningson1,2

1Linne FLOW Centre, KTH Mechanics, Osquars Backe 18, SE-100 44 Stockholm, Sweden2Swedish e-Science Research Centre (SeRC)

3LIMSI-CNRS, UPR 3251, F-91403 Orsay Cedex, France4Fachbereich Physik, Philipps-Universitat Marburg, Renthof 6, D-35032 Marburg, Germany

5Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077 Gottingen, Germany6J.M. Burgerscentrum, Delft University of Technology, Mekelweg 2, NL-2628 CD Delft, The Netherlands

August 27, 2013

Abstract

We study numerically transitional coherent structures in a boundary-layer flow with homogeneous suction at the wall (the

so-called asymptotic suction boundary layer ASBL). The dynamics restricted to the laminar–turbulent separatrix is investigated

in a spanwisely extended domain that allows for robust localisation of all edge states. We work at fixed Reynolds number and

study the edge states as a function of the streamwise period. We demonstrate the complex spatio–temporal dynamics of these

localised states, which exhibits multistability and undergoes complex bifurcations leading from periodic to chaotic regimes. It

is argued that in all regimes the dynamics restricted to the edge is essentially low-dimensional and non-extensive.

1 Introduction

Understanding how boundary-layer flows become turbu-lent is of crucial importance for many applications, inparticular for all aeronautic purposes, because of the in-creased drag associated with turbulent fluctuations. Sincein most applications drag needs to be kept as low as possi-ble, control strategies aim at delaying the transition fromlaminar to turbulent. The problem of transition to turbu-lence has long been addressed using linear stability theory,where typically one looks for a critical value of a govern-ing parameter above which the laminar base flow loses itsstability with respect to infinitesimal disturbances. For in-stance, in the case of the incompressible flat plate Blasiusboundary-layer flow, a Reynolds number Re = U∞δ/νcan be constructed using the free-stream velocity U∞,the local displacement thickness δ at the given stream-wise location, and the kinematic viscosity ν of the fluid.The critical value of Re given by linear stability theory isRec ≈ 520, which corresponds to an exponential amplifi-cation of Tollmien–Schlichting (TS) waves [1]. TS wavesare “connected” to the base flow, meaning that there isa continuous path in parameter space linking them to thelaminar state. TS waves can actually be triggered at lowerRe when the bifurcation occurring at Rec is in fact subcrit-ical, resulting in a finite-instability of the base flow to suchwaves. The “classical” transition route, typical of weakly

disturbed environments, corresponds to the sequence ofsecondary bifurcations undergone by these waves [2]. Stillaccording to linear stability theory, a control strategy thatwould shift the linear stability threshold would be consid-ered efficient. We consider here the case of an incom-pressible boundary-layer flow stabilised by homogeneoussuction at the wall. Suction counteracts the spatial de-velopment of the boundary layer, leading asymptoticallyto a steady laminar base flow independent of the planarcoordinates, called Asymptotic Suction Boundary-Layerflow (ASBL) [3], see fig. 1. Since suction shifts the lin-ear stability threshold to 54, 370 [4] it would qualify asan efficient control. However, in this flow as well as inother shear flows, a second, nonlinear path to turbulence,so-called bypass transition, can interfere with this controlstrategy. In the presence of stronger background fluctua-tions (noise, incoming turbulence, localised forcing, etc.),a drastically different, fully nonlinear picture emerges fortransition, termed “bypass route to turbulence”. Schemat-ically, additional solutions of the governing equations thatare not connected to the base flow appear at a finitevalue of Re = ReSN through a saddle–node bifurcation[5, 6]. The upper branch corresponds to solutions withlarger drag, representative of a stable turbulent regime.Its counterpart, the lower branch, corresponds to an un-stable separatrix dividing the phase space into two basins

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Taras Khapko et al.: Complexity of localised coherent structures in a boundary-layer flow

x

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Figure 1: Sketch of the asymptotic suction boundary layer.

of attraction for the laminar and turbulent state, respec-tively. The value of ReSN cannot be predicted by linearstability theory, and generally ReSN < Rec. Perhaps moreimportantly, control strategies based on linear theory alsofail at predicting whether ReSN is affected at all by thecontrol, see fig. 2. There is recent evidence that ReSN forboth Blasius and ASBL flows is as low as 300 [7]. Thispoints out the need for a nonlinear description of transitionprocesses extended to controlled boundary-layer flows.

The existence of a lower branch not connected to thebase flow is typical of a wider class of transition scenariosclassified as “subcritical transition”, common to many wallbounded flows such as circular pipe flow, square duct flow,plane Couette flow, plane channel flow, see for instance [6,8]. As mentioned before, the phase space is split into tworegions, one where trajectories immediately return to thelaminar fixed point, the other where the turbulent stateis reached. Detailed investigations of the dynamics alongthe associated separatrix have become popular in the lastdecade owing to the concept of “edge state” and “edgemanifold” [9]. The edge manifold (or simply “edge”) is notonly the geometrical separatrix in phase space, but alsoan invariant subspace for the flow. Within this invariantmanifold there are attractors, so-called edge states. In thesimplest case, the edge state is unique and correspondsto a simple unstable solution of the system, such as afixed point or a limit cycle, and its stable manifold (theedge) separates two attractors in state space: the laminarstate and the turbulent state. For initial conditions closeenough to the edge, trajectories will then approach theneighbourhood of the edge state and later leave towardsone of the two attractors depending on which “side” of theedge the initial condition lies. An efficient and intrinsicallynonlinear control strategy should then consist of targetingthe edge state in order to shift the dynamics from theturbulent side of the edge to the laminar one [10].

When the system is not geometrically restricted by pe-riodic boundary conditions, the edge state corresponds toa coherent structure of the flow that is always spatially lo-calised [11, 12, 13, 14, 15]. Families of localised structuresin plane Couette flow have been connected to the snaking

ReSN

edge of chaos

ReC ReReC

A

TS waves

turbulent regime

Figure 2: Schematic bifurcation diagram for the uncontrolled(grey) and controlled (black) boundary layer.

bifurcation scenarios [16, 17, 18].

Several difficulties arise, making this picture often in-complete. The first one, typical of low values of Re, occurswhen the turbulent state is a chaotic saddle rather than anattractor. The stable manifold of the edge state is in thatcase entangled with the turbulent dynamics in a complexway, a manifestation being a finite probability of relam-inarising for almost any turbulent trajectory. Such be-haviour is in general linked to a boundary crisis occurringat some value of Re slightly above ReSN [19, 20, 21] involv-ing the stable manifold of the edge state. At higher Re,sudden relaminarization events are no longer observed inpractice, either because turbulent lifetimes are too long onaverage, or because spatial proliferation of turbulent fluc-tuations makes the local probability for relaminarizationonly weakly relevant [22, 23]. Another difficulty, whichwe will address here, is related to simple general ques-tions such as: “what is the expected nature of the edgestate?”, “can it be chaotic?”, “is the edge state unique?”and “is it robust to changes in the parameters?”. Some ofthese questions have been addressed using a simple low-dimensional phenomenological model in ref. [24]. The con-cept of edge state as a relative attractor is an asymptoticconcept only. As edge-tracking algorithms are iterativeby nature, numerical evidence for chaotic edge states ishard to justify since any erratic edge trajectory is alwayspotentially a transient approach to a more simple invari-ant state. Chaotic edge states have been nevertheless re-ported several times in the literature [25, 12, 13] and arelikely to be typical if not generic in extended domains.Phase-space coexistence of different (i.e. not symmetry-related) edge states has been reported too [26, 27]. Thesequestions are important from a fundamental point of viewbecause the separatrix forms part of the skeleton of thephase space. They are also crucial from a practical pointof view, a multitude of edge states calling for a non-trivialgeneralisation of the nonlinear control strategy suggestedin ref. [10]. Apart from transition processes, studying co-herent structures on the edge can also teach us about thedynamics of near-wall turbulence [28].


Recent numerical studies have focused on edge statesin various models of boundary-layer flows [29, 15, 30], inparticular in the case of ASBL [31, 27]. We here continueto explore the rich dynamics of edge states in the ASBLas a prototype for controlled boundary-layer flows. Weperform numerical simulations in a periodic domain thatis wide enough to display spatial localisation in the span-wise direction. Let us denote by Lx the streamwise lengthof the domain, which in case of periodic boundary con-ditions defines the fundamental streamwise wavenumberin a given computational domain. Unlike more classicalprocedures where the governing parameter varied is Re,here we keep the value of Re constant and consider thestreamwise wavelength Lx as a varying parameter. Vari-ous types of dynamics on the edge emerge from this nu-merical exploration, among them period doubling, multi-stability and Pomeau–Manneville intermittency that aresurprisingly reminiscent of many simpler systems such ascoupled logistic maps [32].

The paper is structured as follows. In sect. 2 we intro-duce the essential features of the ASBL and recall the nu-merical methods combined together to identify edge states.Sect. 3 begins with examples of the low-Re dynamics ofthe flow and the evidence for a chaotic saddle related tothe stable manifold of the edge state. The various statesidentified in Khapko et al. 2013 [27] are then tracked inparameter space vs. changes in Lx. Sect. 4 is devoted toa discussion of the results and suggests a more generalpicture of the dynamics on the basin separatrix. Finallythe relevance of such localised states for the study of theturbulent regime will be discussed.

2 Flow case and numerical methods

The asymptotic suction boundary layer (ASBL) is a zero-pressure-gradient boundary-layer flow above a flat plate atwhich constant homogeneous suction is applied. The flowis governed by the incompressible Navier–Stokes equations

∂u

∂t+ (u · ∇)u = −1

ρ∇p+ ν∇2u , (1)

together with the continuity equation

∇ · u = 0 . (2)

Here u = (u, v, w) is the velocity field of the flow in thestreamwise x, wall-normal y and spanwise z directions,respectively, p stands for the pressure, ρ for the fluid den-sity and ν for the kinematic viscosity. The correspondingboundary conditions are

(u, v, w)y=0 = (0,−VS , 0) , (3a)

(u, v, w)y=∞ = (U∞,−VS , 0) , (3b)

where U∞ and VS are the fixed free-stream and suctionvelocities.

The system admits a steady laminar solution with con-stant boundary-layer thickness,

(U , V , W ) = (U∞(1− e−yVS/ν),−VS , 0) , (4a)

p = const . (4b)

The Reynolds number Re = U∞δ∗/ν is based on the

laminar displacement thickness

δ∗ =

∫ ∞0

(1− u(y)/U∞) dy , (5)

which in this case is given analytically by δ∗ = ν/VS .Accordingly, the Reynolds number is given by the ra-tio of the free-stream velocity and the cross-flow veloc-ity, Re = U∞/VS . The free-stream velocity U∞ and theboundary-layer thickness δ∗ are used as characteristicunits for the non-dimensionalisation, and we write non-dimensional quantities without tilde.

For the current study direct numerical simulations ofASBL are performed using a fully spectral code in a chan-nel geometry [33] of finite wall-normal extent [0, Ly]. Thevelocity field u is decomposed in Nx and Nz Fourier modesin x and z directions, respectively, and Ny Chebyshevpolynomials in the y direction. Periodicity in x and zcomes as a consequence of the choice of a Fourier ba-sis. Dealiasing with the 3/2 rule is performed in the xand z directions. The Dirichlet boundary condition 3ais still imposed at y = 0, while the free-stream condi-tion (u, v, w) = (1,−1/Re, 0) corresponding to eq. 3b isnow imposed at y = Ly. A third-order Runge–Kuttaand a second-order Crank–Nicolson method are used forthe time advancement of the nonlinear and linear terms,respectively. Whereas the streamwise extent Lx of thecomputational domain is varied between 4π and 6π (inunits of δ∗), the height Ly and the width Lz are heldconstant, as well as the Reynolds number which is set toRe = 500 except when indicated. As shown in ref. [27],Ly = 15 and Lz = 50 are sufficient to accurately catchthe localisation properties of the edge states. A resolu-tion of Nx × Ny × Nz = 48 × 129 × 192 spectral modeswas found suitable for the investigation of edge stateswithin this numerical domain, resulting in a system withN = 4NxNyNz ≈ 4.8 × 106 degrees of freedom for thevelocities and pressure. The main results of this studyhave been cross-checked using a different spectral code,Channelflow [34] developed by John F. Gibson.

In order to track the dynamics on the laminar–turbulentseparatrix, a bisection is performed along the line join-ing an arbitrary initial condition to the laminar state [9].Each individual trajectory is followed until it approacheseither the turbulent or the laminar one. Such approachesare detected using predetermined thresholds for the quan-tity v′rms, which represents root-mean-square of the wall-normal velocity fluctuations. This iterative procedure re-sults in a trajectory that shadows the laminar–turbulent


Figure 3: Three-dimensional visualisation of turbulent ASBLfor Re = 270 in a numerical domain with Lx = 6π. Blue (low-speed streak) and red (high-speed streak) are the isosurfacesof streamwise velocity fluctuations u′ = −0.4 and u′ = 0.2.Vortices are visualised using the λ2 criterion [35] with the iso-surface λ2 = −0.01. Flow from lower left to upper right.

0.02 0.04 0.06 0.08 0.10Perturbation amplitude A

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etim

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Figure 4: Lifetimes associated to trajectories starting fromthe initial condition uA, vs. the disturbance amplitude A, atRe = 270. A = 1 corresponds to the snapshot shown in fig. 3.The maximum observation time is here 10, 000 time units.

boundary for arbitrary long times. By iterating for a suf-ficiently long time we can determine the nature of therelative attractor on the edge, i.e. the edge state.

3 Results

3.1 Low-Re transient turbulence

We begin with a short description of the turbulent stateobtained by numerical simulation. Our investigations sofar show that transition to turbulence can be observed inthe ASBL at least for Re & 250 provided that the com-putational domain is large enough. Fig. 3 shows a typicalturbulent state at Re = 270 and for Lx = 6π. A slightlyhigher computational domain with Ly = 25 was used inorder to account for the thickening of the boundary layertypical for the turbulent ASBL [36]. The flow featuresclear coherent structures close to the wall in the form ofhigh-speed streaks (shown in red) very near the lower wall,and low-speed streaks (blue) further up into the flow. The

Ecf ×

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Figure 5: Time series for the left-shifting state (L) at Lx =6π: (a) cross-flow energy Ecf ; (b) mean spanwise velocity 〈w〉.

flow shows however no sign of spanwise localisation. Theredistribution of momentum in the vertical direction ismainly due to the advection by streamwise vortices.

The turbulent regime in the chosen computational do-main at such low values of Re is actually only metastableand the flow can rapidly relaminarise after a (sometimesextremely long) finite time. An investigation of lifetimeswas performed at Re = 270 by considering the perturba-tion u′ to the laminar state U displayed in fig. 3, rescalingits amplitude by a factor A, considering new initial con-ditions uA(t = 0) = U + Au′ and finally measuring thelifetime of the turbulent regime as a function of A. Be-low a given threshold in A, all trajectories decay rapidlyto the laminar state. Above this threshold A ≈ 0.06 life-times increase dramatically and show huge fluctuations,with isolated initial conditions persisting forever. How-ever, averaged over smooth sets of initial conditions, themean lifetimes remain finite, indicating that the edge hasbeen crossed [25]. In fig. 4 the lifetimes are displayed asa function of A showing a very fractal landscape. Thissituation, analogous to most subcritical shear flows at lowenough values of Re, suggests the existence of a chaoticsaddle [37]. As recently shown for the case of pipe flow andplane Couette flow, [19, 20, 21], the creation of this chaoticsaddle is due to a boundary crisis as Re is increased. Thisexample of global bifurcation is typical in dynamical sys-tems [38] and emerges when an already existing chaoticattractor collides with its basin boundary, which is thestable manifold of the edge state. In all studies of shearflows, the chaotic attractor preceding the boundary cri-sis emerges from a sequence of local bifurcations from an“upper branch” state originating from a saddle–node bi-furcation. The other (unstable) state originating from thatsaddle–node is precisely the edge state. In practice, thestudies in refs. [20, 21] all started with the identificationof the edge state on the basin boundary, followed usingcontinuation techniques down to the saddle–node bifurca-


Ecf ×

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t

⟨w⟩

× 1

03

0 1000 2000 3000 4000 5000 6000−1

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1

(a)

(b)

Figure 6: Time series for the left-right shifting state (LR) atLx = 6π: (a) cross-flow energy Ecf ; (b) mean spanwise velocity〈w〉.

tion, where both lower and upper branches are created.This points out the importance of the edge state as thebackbone of the turbulent dynamics, not only of the basinboundary. For larger values of Re, lifetimes tend to in-crease on average. Whether the turbulent regime remainstransient or not at larger values of Re is beyond the scopeof this study, however the concept of edge state remainsrobust. Note that the numerical continuation performedin former studies demands for technical reasons an edgestate with trivial time dependence. In the remainder of thepaper we investigate edge states in spanwisely extendedASBL and their dynamics.

3.2 Periodic edge states for Lx = 6π

From here on, the Reynolds number is held fixed atRe = 500. The edge state in the numerical domain ofsize(Lx, Ly, Lz) = (6π, 15, 50) has been discussed in detail inref. [27], and we briefly summarise the main findings here.In this set-up three distinct edge states were found (mod-ulo translational symmetries in x and z), two of which arerelated by the symmetry z → −z. They all are localisedin the spanwise direction and their dynamics is exactlyperiodic in time. The latter two states can be seen infigs. 5(a) and 6(a), where the time evolution of the cross-flow energy Ecf for the states is shown. This quantity canbe considered a measure for the amplitude of streamwisevortices and is defined as:

Ecf =1

LxLz

∫Ω

(v′2 + w′2) dxdy dz , (6)

where v′ and w′ are the wall-normal and spanwise veloc-ity perturbations to the laminar solution (U, V,W ) and Ωstands for the computational domain. For all three statesthe time signal alternates between calm regions with rel-atively low energy and bursts where significantly higher

t

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−0.3 −0.2 −0.1 0 0.1 0.2 0.3

(a)

(b)

Figure 7: Space–time diagrams of the streamwise velocityfluctuations u′ averaged in x at y = 1 for (a) the state shiftingleft (L) and (b) the state shifting repeatedly left and right (LR).For the online version: from blue to red, negative (low-speedstreaks) to positive (high-speed streaks) values of u′.

values in energy are realised. The temporal dynamics aswell as the spatial structure of the edge states is shownin fig. 7 using a space–time diagram for the streamwisevelocity perturbation u′ evaluated at y = 1 and averagedin the x direction. During the calm phase the states con-sist of a pair of active high- and low-speed streaks witha slowly decaying low-speed streak on the side. After theburst of the cross-flow energy the structure is destroyedbut rapidly reforms modulo a shift in the spanwise direc-tion. Based on the direction of the translation we dis-tinguish between the state alternatively shifting left andright (LR), and the ones shifting constantly in one direc-tion, left (L, z < 0) or right (R, z > 0). The informa-tion about the translation direction can be extracted fromthe time series of the spatially-averaged spanwise velocity

〈w〉 =∫ Ly

0w(0,0) dy displayed in figs. 5(b) and 6(b), where

w(0,0) is the (0, 0) mode of the Fourier discretisation in thehorizontal plane at a given y position. Here 〈w〉 serves as alinear observable. Positive and negative values of 〈w〉 nearthe cross-flow energy peaks indicate shifts to the right andleft, respectively.

The dynamics during one period of the cycle is similarfor all three states, being equivalent to the one in the smalldomain [31], and bears strong resemblance with the self-sustaining cycle of wall turbulence described in ref. [39].In the beginning of the calm phase the state consists ofalmost streamwise-independent streamwise streaks. Oneof the low-speed streaks is accompanied by a pair of quasi-


Figure 8: Snapshot of the L state for Lx = 6π shortly beforea burst. Isocontours are u′ = −0.2 in blue, u′ = 0.1 in red andλ2 = −0.001 in grey. A pair of counter-rotating vortices canbe identified leaning over the low-speed streak. On the right,a high-speed streak can be seen, the remnant of the precedingburst. Flow from lower left to upper right.

|⟨w⟩| (n) × 104

|⟨w

⟩| (n

+1

) ×

10

4

7 8 9 10 11 127

8

9

10

11

12

Figure 9: First return map of the absolute value of meanspanwise velocity |〈w〉| at the peaks of the cross-flow energy Ecf

for the L state at Lx = 6π. The slope β of the approach to thediagonal, indicated by the dashed and solid lines, respectively,is close to 0.5, meaning that the state is stable.

streamwise vortices which sustain the streak through thelift-up effect. The vortices increase in strength and size,further bending the streak (see fig. 8). Eventually thevortices cross each other and the streak, thereby creatingregions where the fluid is pushed down instead of beinglifted up. This inevitably leads to the break-up of the low-speed streak, which corresponds to the burst in the cross-flow energy. During the breakdown process streamwisevortices are re-created in the vicinity of the destroyed low-speed streak. They lead to the creation of a high-speedstreak at the same spanwise position with two low-speedstreaks on the sides, one of which has stronger streamwisevortices than the other one, and the loop is closed (see alsoref. [27]). Thus, compared to the cycle from [39] there isan additional spatial aspect, manifesting itself in shifts ofthe structure in the spanwise direction.

Lx/π

Ecf_

max ×

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Lx/π

T

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600

800

1000

1200

1400

1600

1800

2000

(a)

(b)

Figure 10: Bifurcation diagram in terms of (a) cross-flow en-ergy peaks Ecf max and (b) inter-burst periods T . The branchesfor the L and LR states are represented with squares (red on-line) and circles (blue online), respectively. Larger symbolsrepresent stable (filled) and unstable (empty) states, whereaserratic behaviour is denoted with smaller symbols.

The linear stability of the edge states with respect toperturbations within the basin boundary can be checkedby considering return maps of |〈w〉| sampled at times cor-responding to the maxima of Ecf . In such a representationa period-k orbit corresponds to k different points on thediagonal of the k-th return map. The slope β near thesepoints in the return map should be equal and indicatesthe linear stability properties of the orbit: |β| > 1 indi-cates instability whereas |β| < 1 indicates stability. Fromfig. 9, which includes the approach to the L state in thefirst return map, we can for instance deduce that β ≈ 0.5and that the periodic orbit is hence a stable attractor onthe edge.

3.3 Bifurcation diagram

When the length Lx of the computational domain is re-duced to 4π, only few edge trajectories converge to a pe-riodic LR state while the others stay erratic even for verylong edge tracking times. In particular, we never got con-vergence to a periodic state when initiating the bisection


with random initial conditions. Despite being erratic thestate remains strongly localised in z for all times, withthe active part consisting of a pair of low- and high-speedstreaks. As in the periodic case, high-Ecf bursts are fol-lowed by a spanwise translation of the whole structure.However, the time between the bursts is not constant andthe shifts vary unpredictably in direction and distance.Understanding and characterising the bifurcations con-necting the periodic to the erratic regime is the focus ofthis study.

The long temporal period of the obtained states to-gether with the large number of degrees of freedom in-volved, makes numerical continuation not feasible with thetools at hand. Therefore, we chose to vary the parame-ter Lx in discrete steps, starting from Lx = 6π. SinceL and R states are the same under z → −z transforma-tion, it is sufficient to focus on one of them only. Startingfrom Lx = 6π the instantaneous flow fields of both L andLR states were shrunk to shorter domains. The resultingfields were used as initial conditions for the edge tracking,which was performed until the edge state in the consid-ered domain was found. Initially, Lx was decreased from6π to 4π in steps of 0.2π. As soon as qualitative changeswere observed at some length, the same procedure wasperformed from the last periodic state with smaller stepsize. Decreases in Lx were regularly complemented withincreases in Lx in order to check for hysteresis. Finallythe important bifurcation regions were re-sampled using afiner resolution of 0.05π in Lx. The simulations altogetherrepresent a total of nearly 106 CPU hours.

The results obtained using this procedure are displayedin fig. 10 via two bifurcation diagrams. In fig. 10(a), themaxima of Ecf are plotted for each value of Lx. We ob-tain two independent branches associated with L (squares)and LR (circles) states. Stable periodic states are repre-sented by large filled symbols. If the state is n-periodicit corresponds to n different points in the diagram. Er-ratic states are denoted with smaller symbols. Unequalamount of points in different cases is due to varying lengthof the trajectories on the edge and irregular periods be-tween two consecutive Ecf peaks. Nonetheless, each of theerratic simulations is an order of magnitude longer thanthe average time it takes to reach a periodic state, withup to 100, 000 time units in the longest chaotic simula-tions. If some range of values are frequently visited bythe chaotic trajectory the corresponding state in the dia-gram is marked with a large empty symbol. In fig. 10(b)an alternative representation is shown, where the time in-tervals T between consecutive peaks of Ecf are consideredinstead.

While the modulation in cross-flow energy of the LRstate changes with Lx, the orbit stays exactly periodicuntil Lx = 4.25π. At Lx = 4.2π it becomes weakly un-stable, and the trajectory spends considerable time in itsvicinity before leaving and exploring different part of the

phase space. Either it starts drifting to the left (whichwill be described in the next paragraph), or a new branchof periodic LR states is obtained, which is characterisedby a shorter period T and slightly lower values of Ecf max.This new LR-branch has been tracked down to as low asLx = 3.6π, were the periodic behaviour is finally lost.The latter state was not identified in ref. [27] for Lx = 4π,though edge tracking was performed from many variousrandom initial conditions. It was obtained here by follow-ing the LR-branch in small steps of Lx, suggesting that ithas a small basin of attraction in this parameter range.

The stability range for the L state is narrower comparedto the LR state, as it does not extend below Lx = 4.95π.For the next value we probed, Lx = 4.9π, a longer butstable 3-period state emerges as relative attractor on theedge, the full period of which consists of three consecu-tive shifts in each direction. We denote it by L3R3 (notethat in fig. 10 two of the three symbols are very close toeach other). Between 4.85π and 4.75π the dynamics iserratic in terms of the time evolution of the cross-flow en-ergy, however repeating the same pattern of translationsin the spanwise direction, with four consecutive shifts ineach direction for Lx = 4.85π, three consecutive shiftsin each direction for Lx = 4.8π and irregular combina-tions between two and three shifts in each direction forLx = 4.75π. The corresponding trajectories show no in-dication of convergence. Thus, even if the underlying or-bits are stable, considerably longer simulation times maybe required for convergence in this case. For Lx = 4.7πPomeau–Manneville intermittency is observed in the formof chaotic dynamics alternating with visits to an unstableL2R2 orbit. Below this value the dynamics is erratic, ex-cept in a stable period-1 and a period-2 window between4.2π and 4.4π, where an L state is recovered again. Addi-tionally, an intermittently chaotic L state is obtained fromthe LR-branch at 4.2π and on the L-branch at 4.15π. It ismarked with empty squares for the corresponding valuesof Lx in the bifurcation diagram. Due to the interestingdynamics on this branch, some parameter values will bediscussed in more detail below.

3.4 Investigation of the L-branch

3.4.1 Period-3 state (Lx = 4.9π)

For Lx = 4.9π a longer periodic state is obtained. As isevident from fig. 11, the state is 3-periodic in terms of thetime evolution of Ecf . Part of the space–time diagram forthis state is shown in fig. 12(a), where the state is seen toalternate between three shifts in each direction. We thusdenote this state as L3R3.

3.4.2 Intermittent state (Lx = 4.7π)

Lowering Lx down to 4.7π on the L-branch, the peri-odic behaviour is lost. Part of the time evolution of Ecf

and 〈w〉 in this case is shown in fig. 13. The dynamics


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16000

(a) (b) (c)

Figure 12: Space–time diagrams of the streamwise velocity fluctuations u′ averaged in x at y = 1 for some states on theL-branch: (a) Period-3 state at Lx = 4.9π (L3R3); (b) Intermittent state at Lx = 4.7π; (c) Chaotic state at Lx = 4.5π. Thecolourmap is the same as in fig. 7.

Ecf ×

10

4

0 0.5 1 1.5 2 2.5

x 104

0

2

4

t

⟨w⟩

× 1

03

0 0.5 1 1.5 2 2.5

x 104

−1

0

1

(a)

(b)

Figure 11: Time series for the 3-periodic L3R3 state atLx = 4.9π: (a) cross-flow energy Ecf ; (b) mean spanwise ve-locity 〈w〉.

seems to repeatedly spend considerable amount of timein the vicinity of the state with two shifts in each direc-tion (L2R2), before suddenly leaving and being quicklyre-injected again. These transient approaches to the L2R2

state can be clearly seen in fig. 14, where the maxima ofthe cross-flow energy Ecf max are plotted against time forthe full trajectory. Again, in order to assess the stabil-ity of this 2-period state the second return map of |〈w〉| isconsidered (see fig. 15). Clearly, both slopes near the diag-onal are larger than 1, meaning that the underlying L2R2

state exists at least for neighbouring parameters and ishere not stable. A part of the space–time diagram captur-ing this phenomena is shown in fig. 12(b), where regularL2R2 shifts are interrupted by short aperiodic motion.

This type of behaviour was first described by Pomeauand Manneville in 1980 as a key element of transition tochaos through intermittency [40]. Depending on the bi-furcation leading from periodic to intermittent dynamicsdifferent types of intermittency are defined. In practice, inorder to identify the relevant intermittency scenario, con-verged statistics of the inter-burst times would be neededfor a continuous range of values Lx close to the bifurcationpoint, which is too costly.

3.4.3 Chaotic state (Lx = 4.5π)

Reducing Lx further, no periodic states are identified onthe L-branch between Lx = 4.6π and Lx = 4.45π. Thetime signal of the cross-flow energy, shown in fig. 16 forLx = 4.5π, is erratic, while still experiencing calm and


Ecf ×

10

4

0 0.5 1 1.5 2 2.5

x 104

0

2

4

t

⟨w⟩

× 1

03

0 0.5 1 1.5 2 2.5

x 104

−1

0

1

(a)

(b)

Figure 13: Time series for the intermittent state at Lx =4.7π: (a) cross-flow energy Ecf ; (b) mean spanwise velocity〈w〉.

→

t

Ecf_

max ×

10

4

0 2 4 6 8 10

x 104

2.5

3

3.5

4

4.5

5

Figure 14: Cross-flow energy maxima Ecf max for the inter-mittent state at Lx = 4.7π. The starting time for fig. 12(b)and fig. 13 corresponds to t = 54, 800 in this representation,marked with the vertical arrow in the figure.

|⟨w⟩| (n) × 104

|⟨w

⟩| (n

+2

) ×

10

4

1

2

2 4 6 82

3

4

5

6

7

8

Figure 15: Second return map of |〈w〉| at the peaks of Ecf

for the intermittent L2R2 state at 4.7π. Both slopes, indicatedby the dashed lines, are approximately the same and are largerthan 1, with β1 ≈ β2 ≈ 1.3, proving that the L2R2 orbit isunstable on the edge.

Ecf ×

10

4

0 0.5 1 1.5 2 2.5

x 104

0

2

4

t

⟨w⟩

× 1

03

0 0.5 1 1.5 2 2.5

x 104

−1

0

1

(a)

(b)

Figure 16: Time series for the erratic state at Lx = 4.5π:(a) cross-flow energy Ecf ; (b) mean spanwise velocity 〈w〉.

bursting phases. As can be seen from the space–timediagram in fig. 12(c), the state does not change struc-turally, keeping the same characteristic lengthscales andtimescales. However its spatio-temporal dynamics is nolonger regular but features a random walk, with unpre-dictable spanwise shifts occurring at non-regular times.The various return maps did not reveal any clear struc-ture and we refer to this state as ”chaotic” or ”erratic”.

3.4.4 Period doubling (Lx = 4.3π)

Below Lx = 4.45π the state which repeatedly shifts onlyin one direction becomes attracting again. However un-like the previously identified L state, two consecutivetranslations are no longer equivalent (see fig. 17). Thespatio-temporal dynamics consists of two shifts of differ-ent lengths in the same direction. Their duration is differ-ent by less than 20% and the total period of the state isapproximately twice the period found for larger Lx. Wethus denote this edge state by L2 and call somewhat abu-sively ”period-doubled”. Somewhere between Lx = 4.3πand Lx = 4.25 the state undergoes a reverse period dou-bling bifurcation, and the original L state is recovered. Itis stable down to Lx = 4.2π, below which the dynamicsbecomes chaotic again. Notable intermittent approachesto another periodic state were identified for Lx = 4.15π,as well as for Lx = 4.2π for other non-converging edgetrajectories.

4 Discussion, perspectives

When confined to the laminar–turbulent separatrix,the present system exhibits complex dynamics, withcoexistence of more than one attractor. The complexityraises questions about how those attracting states areconnected together. They can either belong to differentfamilies of solutions or be connected through local


Ecf ×

10

4

0 2000 4000 6000 80000

2

4

t

⟨w⟩

× 1

03

0 2000 4000 6000 8000−1

−0.5

0

(a)

(b)

Figure 17: Time series for the 2-periodic L2 state at Lx =4.3π: (a) cross-flow energy Ecf ; (b) mean spanwise velocity〈w〉.

bifurcations. Multistability is typical in low-dimensionalsystems (see ref. [41] and related references) and can bealso found in hydrodynamical settings (see e.g. ref. [42]).For some values of the parameter Lx we identified herean unusual competition between a periodic and anerratic state. Performing edge state tracking in thissituation, very long simulation times can be neededto decide whether the algorithm has converged to achaotic state or is still converging to a simpler state.This demonstrates the limitations of the edge trackingprocedure for understanding and characterising the fullstructure of the laminar–turbulent boundary using afinite (small) number of simulations only. Conversely,the linear stability properties of an edge state give noindication about the size of its basin of attraction, thelatter being a fully nonlinear concept. In some situationsthis can lead to a misinterpretation of the dynamicalrole of some stable solutions, as pointed out in ref. [43]using a counterexample from a low-dimensional toy model.

Still, we were able to obtain a finer characterisation ofthe edge structure than, for instance, in ref. [26]. Wesuggest here a qualitative explanation of some exotic be-haviours expected on the edge, such as multistability andintermittency. Our suggestion is based on compiling re-sults on exact coherent structures from other subcriticalshear flows, notably the influence of varying either thestreamwise or spanwise wavelength of the numerical do-main. The use of arc-length continuation robustly showedthat most fundamental solutions such as travelling wavesor steady states are born and destroyed in saddle–node bi-furcations (see for instance fig. 11(b) in [44] and fig. 14 in[45]). Among these solutions, those with only one unsta-ble eigendirection should in principle correspond to edgestates. The coexistence of several loops of solutions inparameter space is typical and thus multistability should

A

LxA

ch

aos

ch

aos

ch

aos

Lx

ch

aos

(a)

(b)

Figure 18: Conceptual sketch of (a) exactly periodic solu-tions as a function of Lx and (b) the corresponding bifurcationdiagram for the dynamics on the edge. Solid lines representsolutions with one unstable direction, whereas dashed lines –with more than one. The thin vertical dashed lines connectthe saddle–node bifurcation points between the upper and thelower part of the figure. Hatched areas indicate possible inter-mittency zones.

be found on the edge when several solutions with only oneunstable multiplier overlap in a given range of parameters.Such a situation has already been reported in ref. [26] forthe travelling waves in pipe flow with m = 2 symmetry. Inthe vicinity of these saddle–node bifurcations, type-I in-termittency [40] should be found. In the present study wedeal with spanwisely localised solutions, hence the widthof the domain Lz is irrelevant once large enough. How-ever we do report multistability in some windows of theparameter Lx and intermittency for other values of Lx.

The above scenario is summarised graphically in fig. 18.We suppose (by direct analogy with former studies) thatperiodic solutions such as those identified for Lx = 6πemerge as loops in fig. 18(a). The y-axis refers to a suitablescalar observable, for instance Ecf max (used in fig. 10(a))or v′rms, the quantity used to iteratively find edge states.Knowledge of the stability properties of each branch (in-cluding possible local bifurcations along these branches)allows to list the solutions potentially behaving locally asrelative attractors on the edge. Locating the saddle–nodebifurcations associated precisely with these solutions alsoallows for predicting regions in Lx where intermittencycan be expected on the edge. This is shown in fig. 18(b),


which is graphically deduced from fig. 18(a) and comparesconceptually well with the numerical bifurcation diagramof fig. 10(a).

It is noteworthy that multistability and intermittencyare not observed if the width of the domain Lz is small.In ref. [31] edge states which are periodic in both wall-parallel directions are studied and the range in Lx thatwe investigated here is also covered. However, only oneedge state is found; it is only the spanwise localisationthat allows multiple states to coexist in the same domain.

We note, as many authors before us, that the total num-ber of unstable directions of solutions lying on the edge isnecessarily low, being equal to 1 for edge states and finiteotherwise. For the parameter values where edge trajec-tories stay erratic, consisting of transient approaches tofinite-amplitude solutions with at least 2 unstable eigen-values, the number of unstable Lyapunov exponents mustalso remain small. Moreover, the robust spatial locali-sation of these edge states, verified in ref. [27], impliesthat increasing the domain size in the direction of locali-sation does not modify the dynamics, and hence does notchange the number of unstable Lyapunov exponents evenin chaotic regimes. The dynamics on the edge, though gen-uinely spatio–temporal, can hence safely be interpreted asan example of non-extensive and low-dimensional dynam-ics. This has to be contrasted with the associated turbu-lent regime existing for the same parameters. The turbu-lent regime is not spatially localised and thus representsextensive chaos. Quantitative estimation of the number ofunstable Lyapunov exponents of typical turbulent trajec-tories is a hard task. A numerical estimate of this numberwas given in ref. [46] in the case of a “minimal” turbulentchannel flow at Reτ = 180, and was found to be 352, i.e.very large. Though analogies between all these differentflows are to be taken cautiously, the different orders ofmagnitude are unambiguous. The dynamics on the edgefeatures a very small number of unstable dimensions whilethe dynamics in the turbulent regime is highly unstable.This strongly supports the investigation of edge states asa laboratory for investigating the dynamics and the self-sustenance mechanisms of coherent structures arising inturbulent flows.

We have here identified and characterised several edgestates in spanwisely extended ASBL for different stream-wise wavelengths. All these states are structurally thesame, consisting of an active localised pair of low- andhigh-speed streaks with streamwise vortices. The dynam-ics of the streaks is built on the same elements as in thesimplest L/LR cycles, i.e. bursts and spanwise shifts, re-gardless of the nature of the regime. Very similar coher-ent structures and dynamics were also found to be theminimal self-sustaining elements of the near-wall turbu-lence [47, 39, 48, 28]. Moreover, comparing the three-dimensional visualisations of both turbulence and the edgestate in figs. 3 and 8, similarities are obvious, let alone

the localisation properties, and closer comparison deservesmore intensive investigation.

In summary we have studied the dynamics on the lam-inar–turbulent separatrix for a boundary-layer flow in aspanwisely extended set-up with varying streamwise pe-riodicity. We were able to find a multitude of periodicand erratic edge states, which all share the same struc-ture. The bifurcation diagram obtained by compiling allthese results contains a variety of interesting phenomenawhose study is usually restricted to the framework of low-dimensional ODEs or iterated maps. Those include multi-stability, intermittency and period doubling. Besides theirproperties as examples of simple dynamics embedded in ahigh-dimensional space, the identified states also consti-tute an interesting prototype flow for understanding theself-sustaining mechanism of near-wall turbulence.

Acknowledgements

T. Kh. would like to thank Paul Manneville and PredragCvitanovic for discussions about the bifurcation diagram.Computer time provided by SNIC (Swedish National In-frastructure for Computing) is gratefully acknowledged.

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