condensing much of the information available ten years ago is displayed in figure 3.1 for pedagogy. It suggests that only the laminar flow can be observed, sufficiently far down the pipe, provided $Re < 2050$. Sustained turbulent states are only observed above this value. Two specific turbulent states exist : the first is referred to as *spatiotemporal intermittency* (STI) and features only spatially sparse turbulence. The other corresponds to the high-Reynolds number regime where turbulence, if seeded locally, expands continuously. This regime is characterised by *expanding slugs*. At any location sufficiently far downstream, the flow will eventually become turbulent in a sustained manner, and the turbulent state is locally indistinguishable from its prediction in periodic computations [180].

<figure><img src="image_65.png" /><figcaption>FIGURE 3.1 – Qualitative bifurcation diagram for pipe flow as a function of the Reynolds number $Re$ (from [181]).</figcaption></figure>

### 3.2.1 Localised puffs

In its sparest incarnation, the STI regime consists of individual coherent structures called turbulent puffs. Puffs are localised in the streamwise direction and remain coherent over their cross-section [182]. They were already identified in Reynolds’ experiment [183] and described as "flashes", or also as "streaks" in Ref. [184]. The localisation of turbulent puffs makes them relatively easy to trigger one by one using localised perturbations, e.g. transverse impulsive jets as in figure 3.2. However, puffs can also form from other types of upstream perturbations, including permanent ones [185]. The apparent stability of puffs (seen as unsteady objects) over short times suggests that they are manifestations of a turbulent equilibrium [151, 186], naively thought to be a metastable one. In fact, and this is one of the main findings in transition during the years 2000s, individual puffs appear stable only on the timescales typical of short pipe experiments, but they are never in equilibrium : either they relaminarise after a finite time [62, 63, 187–189], or they proliferate [13, 180, 184, 186, 190]. Observing the instability of a puff is hence only a matter of time. Despite this property, the ratio between the lifetime of the puffs by the typical timescale of the turbulent fluctuations inside them is always reported to be very large. It hence makes sense to study puffs as equilibrium regimes over intermediatly long timescales, and to look for a self-sustaining process (SSP). The situation was similar in small periodic domains, where Waleffe’s self-sustaining process (SSP) did not prevent the turbulent state built on a chaotic saddle of relaminarising. However, the SSP of localised states turns out to differ structurally from that of Waleffe, precisely because of the spatial localisation.

<figure><img src="image_66.png" /><figcaption>FIGURE 3.2 – Puff in experimental pipe flow for $Re = 1800$ visualised by Mearlmaid Pearlessence illuminated by a vertical sheet of light [191]. Flow from left to right.</figcaption></figure>
<figure><img src="image_67.png" /><figcaption>FIGURE 3.3 – Puff in numerical pipe flow for $Re = 2000$ visualised by isolevels of absolute vorticity. Flow from left to right. From Ref. [192]</figcaption></figure>

The total vorticity field inside a meridian plane of a puff is shown in figure 3.3, with the flow from left to right. The less coherent part corresponding to the 'turbulent' zone (between $x=285$ and 320 in units of the pipe radius) has a well-reported arrowhead structure. The fluctuations closest to the pipe wall are those located most upstream and are relatively large-scale. Since they are in a zone of low velocity, their propagation velocity is slower than that of the mean structure. From the point of view of the puff, these wall structures are "left behind". They correspond to the tails of streaks generated within the turbulent zone itself. The strongest velocity gradients, also visible in figure 3.3 because of the strong levels of absolute vorticity, occur at the steep upstream interface between the fluctuations. Since they are located in a faster zone of the flow, these small-scale structures have a propagation velocity which is positive in the frame of the puff : these small-scales structures catch up with the puff, thereby contributing to its re-energization. This has led Shimizu & Kida [193] to suggest a simple cyclic SSP to explain the puff's long lifetime : the production of small-scale turbulence is the result of the shear instability of the vortex layer corresponding to the tails of the low-speed streaks (cf figure 3.4). These streaks are the signature of the lift-up effect due to the vortical activity close to the wall (due to the turbulence), and the loop is closed. As for periodic domains, it is intuitive that if any of these links fails to occur fast enough, the loop is broken, the puff fails at sustaining and relaminarises. Note that this SSP is not two-dimensional since it involves the azimuthal direction, without which there is no transverse modulation and no low-speed streak. This cyclic mechanism differs effectively from that of Waleffe [30] : the spatial localisation of the fluctuations is crucial to ensure enough shear upstream from the puff and to give rise to the shear instability (labelled for simplicity "KH" like the Kelvin-Helmholtz instability). This difference has not yet been much emphasized. Note that it is consistent with the PDE models suggested by Barkley [180, 194]. In the deterministic part of these models, localised states exist at values of the Reynolds number smaller than those where the homogeneous steady states bifurcate. Necessarily such localised states need a specific mechanism to sustain themselves against viscous dissipation as well.

## 3.2.2 Turbulence proliferation

### Puff splitting

Since the natural organisation of the turbulent regime at low $Re$ is precisely a series of localised puffs, it makes sense to investigate the proliferation phenomenon by starting from a single turbulent puff first. Unlike Fourier modes and other delocalised initial conditions, choosing a localised initial condition has the important advantage of being experimentally reproducible. There is a deeper justification to this choice. As shown in the previous chapter, edge states, which are invariant sets of the system, possess the localisation property as soon as they are computed in large enough domains (or, in other words, when they are computed in the absence of finite-size effects). Minimal states also share this property. These special perturbations are well defined mathematically; they are not rigorously unique but do not rely on a long list of parameters. In addition they are, by construction, robust to domain changes. Initialising the system at initial time with such perturbations is a consistent, non-parametric way to excite the flow with only little perturbation energy. For pipe flow the choice to start from the edge state was considered natural for instance in order to excite slugs in Refs [152] and [153], whereas localised minimal seeds were considered in Ref. [195].
<figure><img src="image_68.png" /><figcaption>FIGURE 3.4 – Sketch of the Self-Sustaining Process for localised states in pipe flow (taken from Ref. [193])</figcaption></figure>

The mechanisms of proliferation are relatively well described qualitatively in the early pipe experi-
ments of Lindgren [184]. Their quantification in the immediate neighbourhood of the critical point can be
found in Ref. [13]. In this regime turbulence proliferation corresponds qualitatively to the "splitting" of
individual puffs, or rather to their *duplication* or *self-replication* (in the sense that the turbulent fraction
is supposed to increase), if we are to borrow vocabulary from biology and pattern formation [196].

<figure><img src="image_69.png" /><figcaption>FIGURE 3.5 – Mean lifetimes and splitting times versus *Re* in pipe flow. Taken from [197].</figcaption></figure>

In this plot (reproduced in figure 3.5 in double log data following Ref. [197]) the mean lifetime and
the mean splitting time are plotted versus the Reynolds number. The mean lifetime increases super-
exponentially with *Re* while the mean splitting time decreases also in a super-exponential manner. Both
mean values cross at a given value of *Re* where the probability to decay balances exactly that of splitting.
Above this value of *Re*, turbulence spreads statistically. This single plot, beyond the impressive statistical database needed to achieve it, represents a real conceptual progress since it offers at last, after almost 130 years of investigation, a rigorous definition of the critical point for pipe flow.

The dynamics of the splitting process [192] is shown as a still movie in figure 3.6. The representation makes use of $r \approx 0.8$ surfaces, where fewer small-scale fluctuations are present and where the streaks dynamics is particularly clear. The initial puff fluctuates in length with some vorticity intermittently shed downstream at a fast pace (comparable to the turn-over time $O(D/U)$). The splitting process proceeds in several steps and is successful only if all steps have occurred in a fast enough sequence. First a low-speed streak undergoes a statistically atypical elongation in the downstream direction (1st to 5th frame) considered to be a rare event. Then the associated flow destabilises downstream. This gives rise to a new puff a distance of roughly 40 radii downstream of the initial puff (9th frame). Depending on the quantity chosen, it is possible to detect that the puff grows in length before effectively splitting into two turbulent parts. Whether the puff first elongates and then splits, or first splits and only later nucleates a new puff, is not entirely clear and deserves more investigation. The link with extreme value theory and Gumbel distributions [198], like for the relaminarisation process [199], deserves as well a deeper look.

What makes the description of the STI regime relatively simple in the approximate range $Re \in (1800: 2350)$ is, beyond the one-dimensional geometry inherent to the pipe set-up, the quantized nature of the proliferation rules themselves. Each puff yields, at the next instant, either the same puff, two puffs, or no puff at all. This set of probabilistic rules lends itself very well to a description in terms of cellular automata, known in the presence of an absorbing state to reproduce well STI dynamics [200]. For slightly larger values of *Re* however, i.e. from 2350 to approximately 2700, an emerging trend competes with the quantized dynamics : continuous expansion. For $Re \ge 2700$, most expansion events can be described as continuous.

## Continuous expansion

Although laminar-like holes and interrupted growth sequences can still be found as rare events [201], the flow tends to a uniformly turbulent flow as time increases. It is tempting to try to define a second critical value of *Re* around 2700 but this is not as effective as in the case of the first critical point. Moxey & Barkley [12] have for instance defined an intermittent factor $\gamma$ (otherwise called "turbulent fraction" in other flows) by thresholding the local perturbation kinetic energy. In principle one would wish to define the second critical point at the value of *Re* where $\gamma$ reaches 100%. The dependence on the threshold does not make this definition very accurate yet, although it is again consistent with a transition from STI to full-fledged turbulence at around $Re \approx 2600 - 2700$. Using the second moment of the distribution of laminar lengths is another option used by the same authors, with similar conclusions.

The hydrodynamic mechanisms responsible for the elongation of a puff (i.e. in this regime a *slug*) are not clear as of now. As pointed out in Ref. [192], most dynamical events of importance take place at the leading edge. They are related to the production and the fast propagation downstream of vortical disturbances : when the fluctuations born upstream (whatever their origin) overtake the puff before any decay, this results in a temporary elongation of the puff. The same mechanism can well occur for an already well elongated slug. This can be labelled *leading edge growth* since the presence of the trailing edge does not seem crucial as long as there is a turbulence production mechanism.

Two different variants of leading edge growth have been identified qualitatively in the space-time diagrams from Ref. [153] : for $Re = 3000$ for instance, there is no production of turbulent kinetic energy at the downstream front (only at the upstream one), the leading edge can be described as *diffusive* as for puffs at lower $Re$. For $Re = 6000$ in turn, there is strong production of turbulent kinetic energy at the downstream front, as visualised by the vigorous small-scale vorticity present near the wall. The existence of these two types of leading edge fronts have later been confirmed, first in the Barkley model [180] then in DNS [202]. They have been labelled respectively "weak" and "strong" fronts. The transition from weak
to strong fronts in terms of probabilities is gradual as *Re* increases, with only strong fronts selected at high *Re*.

Although this has been considerably less investigated, an additional mechanism for slug elongation at the trailing edge has also been reported in Ref. [153]. It can be labelled by analogy "trailing edge growth" since it requires a trailing edge but, apparently, no leading edge. It is illustrated in figure 3.7 for *Re* = 4500. This mechanism is a direct extension of the SSP of Shimizu & Kida to the higher-*Re* regime, outside the standard puff regime. Once the vortex layer upstream has shed Kelvin-Helmholtz vortices in what seems like a standard three-dimensional shear instability, these vortices propagate with a positive velocity downstream. The trailing edge growth mechanism relies on the idea that the value of the streamwise advection velocity obeys a non-trivial distribution. While the fastest vortical disturbances travel fast enough to enter the turbulent zone, the slowest ones do not manage to catch up with the puff and are left behind it as it propagates. As a consequence there are remains of turbulent fluctuations just upstream of the turbulent zone. This possibly nucleates new turbulence that forms a new trailing edge interface, in a mechanism of growth by aggregation.
<figure><img src="image_70.png" /><figcaption>FIGURE 3.6 – Sequence of events leading to puff splitting in numerical pipe flow for $Re = 2300$ visualised by isolevels of absolute vorticity at $r \approx 0.8$ at a frame velocity equal to the bulk speed. Flow from left to right. [192].</figcaption></figure>
<figure><img src="image_71.png" /><figcaption>FIGURE 3.7 – Sequence of events leading to trailing edge growth of slugs in numerical pipe flow for $Re = 4500$ visualised by isolevels of absolute vorticity. Flow from left to right. Taken from Ref. [153].</figcaption></figure>
## 3.3 Two-dimensional shear flows

### 3.3.1 Experimental approach

The structuration of the transitional regimes is notoriously more challenging when the effective dimension is two rather than one. This includes all flows with two neutral directions, be they planar or not. The first historical mention of laminar-turbulent patterning occured, as often, in an experimental investigation of Taylor-Couette flow. It appeared first in conference proceedings [203] and later as a part of a long article in which D. Coles [204] describes the many regimes of Taylor-Couette flow. A photograph from the experimental campaign is included here in figure 3.8, where a simple turbulent helix can be recognised amidst the flow. It was also mentioned before publication in the lecture notes of R. Feynman [205]. This special flow case was baptized with humour "barber-pole turbulence" or, for geometrical reasons not yet understood, "spiral turbulence". In this thesis, the wording 'oblique stripe pattern' will be preferred, with the convention that 'oblique' refers to a non-zero angle with respect to the mean (streamwise) flow direction. At the time it was not clear whether this coherent structure, obtained when the two cylinders are nearly exactly counter-rotating, was a consequence of the end walls or not, or whether several wavelengths could be captured at once. Further experiments with different aspect ratios [206–209] and numerics [210, 211] have eversince provided ample evidence that the helical-shaped turbulence is an intrinsic regime of the flow and exists also in the presence of periodic boundary conditions.

<figure><img src="image_72.png" /><figcaption>FIGURE 3.8 – Experiments by D. Coles [204].</figcaption></figure>

There are some clear advantages yet also difficulties associated with the Taylor-Couette configuration that explain the frequent choice for plane Couette flow as a decent alternative. One of the undisputed advantages is the closed geometry. Difficulties include the fact that exact counter-rotation leads to laminar-turbulent patterns only in the limit $\eta \to 1$, where $\eta$ is the radius ratio. Since a set-up with $\eta \approx 1$ is difficult to engineer accurately, this implies a specific tuning for the angular velocities of the cylinder and a complicated parametric dependence. Beyond the engineering issues, having non-equal angular velocities implies a non-zero flow rate and hence advection of the patterns. Also, for $\eta$ far from 1, the natural azimuthal periodicity strongly influences the wavelength selection. This is at odds with the initial motivation of investigating two-dimensional patterns in the absence of finite-size effects. All these reasons combined together lead to the choice of plane Couette flow as a simpler candidate for the study of subcritical transition. Experimentally speaking, plane Couette flow with zero mass flux is difficult to achieve. Since the experiments of Tillmark [212] and Daviaud [213] all pCf apparatuses are similar and follow the sketch in figure 3.9. For technical reasons, the working fluid must be a liquid and the two walls must consist of a single flexible belt guided by two co-rotating cylinders. Most of the theoretical progress
of the 1990s has been achieved using the apparatus at CEA Saclay, notably in the experimental studies of F. Daviaud [213], J. Hegseth [214], O. Dauchot [215, 216], S. Bottin [43, 65, 217], A. Prigent [209, 218, 219]), and later R. Monchaux at ENSTA [220–223].

<figure><img src="image_73.png" /><figcaption>FIGURE 3.9 – Sketch of experimental pCf apparatus, taken from Ref. [223].</figcaption></figure>

The transitional range of pCf, parametrised by the Reynolds number $Re = U_\infty h/\nu$ based on the half-gap $h$ and the velocity of the walls $\pm U_\infty$, does not appear in experiments to depend quantitatively on the dimensions of the apparatus or of the numerical domain. Its bounds lie within the interval $Re \in (280 : 410)$. Meanwhile turbulence seems sustained statistically above $Re = 325$ [43, 65, 66, 217, 224, 225]. However, there are differences. The domain used in S. Bottin's experiments [217] has dimensions $L_x \times L_z = 35 \times 190$, expressed in units of the half-gap $h$. The STI regime has been found and well described, but it features no order, only laminar and turbulent patches without a clear organisation. The set-up used by A. Prigent has dimensions $L_x \times L_z = 340 \times 770$ in units of $h$. The upper transitional range, for $Re \ge 340$ at least, is much more ordered than in the smaller set-up. It is shown in figure 3.10 and can be qualified as the patterning range. The range of values of $Re$ below it, that includes the critical point around $Re = 325$, is again less ordered. Understanding how the transitional range is structured for diverging domain sizes and observation times ("in the thermodynamic limit") has motivated the use of direct numerical simulation for pCf.

A conceptually simpler flow case is plane Poiseuille flow (pPf), especially for experimentalists, since in contrast with pCf it only features rigid static walls. Like pipe flow, it possesses a non-zero flow rate in the laboratory frame. This issue implies that long observation times require long channels, with a special emphasis on the technically demanding plate parallelism. Transitional plane Poiseuille flow had remained a curiosity in the 2000s and until very recently, only tackled numerically by T. Tsukahara and co-authors and later experimentally [226], and thought to be a simple generalisation of pCf. It turned out later that the two flows have subtle differences when it comes to pattern formation and the existence of a critical point. There was however a recent explosion in the number of experimental studies of the transitional range of pPf [165, 227–230], backed by associated numerical studies [14, 161, 162, 231–234]. Both pressure-driven and mass-driven pPf have been considered depending on the authors, implying either $Re_τ$ or $Re_{cl}$ as the genuine governing parameter. We note as well emerging studies in a companion flow, namely zero mass flux plane Couette-Poiseuille flow [235, 236].

### 3.3.2 Numerical approach

Theoretical efforts to understand the organisation of these shear flows in the transitional ranges are not many. The spatial structure of the turbulent "spirals" was qualitatively understood in the early
<figure><img src="image_74.png" /><figcaption>FIGURE 3.10 – Adiabatic descent in plane Couette flow, Experiments by A. Prigent [224]</figcaption></figure>

works of Coles [203], although it is not clear that these works were well recognised in their time. Pomeau and Hayot [237] attempted to explain the presence of laminar gaps in Taylor-Couette experiments from the feedback by the pressure forces, whose consequence is to halt the progression of the turbulent interface. Hegseth [214] has written about the importance of the spanwise vorticity in the formation of the intermittent state. The numerical work of Tuckerman & Barkley [159, 161, 238] has been carried out almost exclusively in oblique tilted domains. Laminar-turbulent patterns emerge in the turbulent regime without difficulty provided the dimensions of the domain are large enough. The range of values of *Re* in which they are found is compatible with experiments. These studies lead to a good understanding of the structuration of the mean flow. The main drawback is that the angle and the wavelength of the patterns are imposed input parameters.

In parallel a series of PDE models, where most of the modelling hypotheses concern the wall-normal structuration, have been developed over two decades by P. Manneville in the hope to reproduce laminar-turbulent patterns at a cheaper computational cost [239–241]. It was eventually confirmed that this approach is qualitatively valid provided the reduction in the wall-normal direction is not too severe [241]. The resulting computational model [242] does not lend itself easily to analytical predictions, but can be used as a low-order proxy of generic shear flows. The cheaper cost allows one, rather than to simply compute faster, to reach domain sizes unavailable by DNS and to eventually conclude about the continuous/discontinuous character of the transition [243]. This is detailed in Section 4.4.

In the context of the 2000s, although puffs in pipe flow had been properly reproduced numerically,
<figure><img src="image_75.png" /><figcaption>FIGURE 3.11 – Numerical simulation of laminar-turbulent patterns in pCf by L.S. Tuckerman and D. Barkley [159] perfomed in a tilted domain (2005). Several copies concatenated together displayed in the usual streamwise-spanwise coordinates. Instantaneous perturbation kinetic energy shown in the mid-plane. Domain angle : 24°, original dimensions 10h × 2h × 40h.</figcaption></figure>

a proper numerical simulation of the transitional laminar-turbulent patterns of pCf or TCf was still lacking. The simulations in a tilted slender domain of Tuckerman and Barkley [159] capture the long-wavelength modulation of turbulence flow with an imposed angle. The qualitative match with experiments is however not fully satisfying. For instance the signed orientation of the turbulent stripes observed in Prigent's experiments [224] can by construction not be reproduced.

## plane Couette flow

Owing to the computational power of the machines available at KTH Stockholm, numerical simulations in a domain with $L_x \times L_z = 800 \times 356$ (still in units of $h$) were performed over observation times ranging from $2 \times 10^3$ to $2 \times 10^4 h/U_w$. The simulation is based on the spectral code SIMSON which assumes Fourier modes in $x$ and $z$, i.e. periodic boundary conditions in $x$ and $z$. For $310 \le Re \le 410$, for any initial condition (spot, edge state, noise, turbulent field from another value of $Re$) the flow evolves towards a state of laminar-turbulent coexistence, with interfaces always oblique with respect to the mean flow. The instances where the interface is locally not oblique all result in rapid relaminarisation or in a rapid evolution of the flow, suggesting that non-oblique interfaces are simply unstable. The development of a turbulent pattern from a noisy initial condition is documented in figure 3.12 for $Re = 350$. The noisy flow field was selected because it allows for a faster convergence to the final turbulent flow field. The early *receptivity* stage is well visible in the animated plots : streaks emerge rapidly from the noise at every location due to the lift-up effect. After $\approx 100-200$ time units the streaks have decayed in amplitude everywhere, except at a few random locations where streak breakdown occurs : this is the stage of *spot nucleation*. The third phase is slower and corresponds to *front propagation* : the spots grow, usually in an oblique fashion, split, reconnect, merge... until a statistically steady pattern with weakly varying turbulent fraction settles. We note in passing that the high resolution typical of spectral methods is comparable to the spatial resolution achievable nowadays in experimental PIV. The comparison of the streamwise velocity field from DNS (figure 3.12c) vs. experiments (figure 3.13) is excellent and leaves no doubt that the governing Navier-Stokes equations are well adapted to the hydrodynamic phenomena of interest.
<figure><img src="image_76.png" /><figcaption>FIGURE 3.12 – Development of a turbulent pattern at $Re = 330$ starting from a noisy initial condition. Streamwise velocity field in the mid-plane $y = 0$ at time $t = 200, 1865$ and $20,000$.</figcaption></figure>

plane Poiseuille flow

A similar computational series of runs was carried out in plane Poiseuille (pPf) driven by a fixed pressure gradient [14]. This time, unlike the protocol followed in pCf, a genuine adiabatic descent was performed by lowering the governing parameter $Re_τ$ in small steps. A computational domain with $L_x \times L_z = 250 \times 125$ was considered initially (expressed in units of the half-gap $h$). As $Re_τ$ decreased, the mean spacing between stripes was observed to increase, until for $Re_τ = 55$ it was decided to restart the process from a noisy initial condition but in a larger domain with $L_x \times L_z = 500 \times 250$ in order to avoid accumulating finite-size effects. Instantaneous representations of the wall shear stress (averaged between the two walls) is shown in figure 3.14 for $Re_τ=100,80,60$ and finally 40 in figure 3.15. As in pCf an oblique and irregular stripe pattern is present for all values of $Re_τ \le 90$; it invariably forms whatever the initial condition. It is hence interpreted, at least in the upper transitional range, as a modulational instability of the homogeneous turbulent state [209]. The snapshot at $Re_τ=40$ stands out as all stripes point in the same direction. This phenomenon was pointed out in Ref. [244] and analysed as a macroscopic bifurcation. For $Re_τ \ge 50$ the network of oblique stripes is tight and rigid, both orientations are equally possible and are in competition at all times. As $Re_τ$ is lowered below 50, one of the two orientations gets beaten
<figure><img src="image_77.png" /><figcaption>FIGURE 3.13 – Experimental velocity field $(u_x, u_z)$ of turbulent stripes in pCf obtained using Particle Image Velocimetry (2020). From Ref. [223].</figcaption></figure>

in finite time by the other one. As a consequence past a size-dependent transient all stripes propagate downstream in the same direction and move freely like particles in a gas [165, 245–247]. The existence of this lower transitional range with a different dynamics [234] has as of yet no equivalent in pCf.
<figure><img src="image_78.png" /><figcaption>FIGURE 3.14 – Pressure-driven pPf for $Re_τ=100, 80$ and $60$. Isolevels of the instantaneous wall shear stress fluctuation $τ'(x, z, t)$. Domain length $L_x = 250h$. From Ref. [14].</figcaption></figure>

### 3.3.3 Bifurcation diagrams

#### Qualitative diagrams

Qualitative bifurcation diagrams can be deduced from the database obtained earlier. The protocol chosen to determine these bifurcation diagrams is always the same and is standard in the presence of hysteresis : the starting point is the featureless regime (no matter how it was found) and the exploration is
<figure><img src="image_79.png" /><figcaption>FIGURE 3.15 – Pressure-driven pPf for $Re_τ$ = 40. Isolevels of the instantaneous wall shear stress fluctuation $τ'(x, z, t)$. Domain length $L_x$ = 500h. From Ref. [14]</figcaption></figure>

carried out by quasi-adiabatically reducing the Reynolds number. The descent is in practice never strictly adiabatic, and the easiest way is to choose small enough steps in $Re$ with long enough plateaus between parameter changes. In analogy with figure 3.1 for pipe flow, the qualitative bifurcation diagrams for pCf and pPf are represented along a unidimensional axis in figure 3.16, respectively top and bottom.

Several comments can be made at this stage. These two flows have in common the presence of an STI/pattern regime lcoated between the laminar and the turbulent ones. As an order of magnitude the width of the STI zone occupies one quarter of the value of the critical value $Re_g$. But these two flows also display clear qualitative differences. The fact that the numbers do not match is secondary and can be assigned to the historically different definitions of $Re$ (as discussed in Refs [238, 248]). There are also clear qualitative differences. In pCf the STI zone seems to consist of one part only, whereas in pPf there is a change of behaviour for $Re_τ$ slightly above 50 as pointed out only recently by Shimizu & Manneville [244] and mentioned on the previous page.

## Wavelengths and angles

On a more quantitative basis, as far as pCf is concerned the pattern range lies approximately in the interval $Re \in (330 : 410)$. As far as pressure-driven pPf is concerned, it is restrained to $Re_τ \in (50 : 90)$. We see in figure 3.12c that several values of the unsigned angle as well as several values of the wavelengths coexist within the same snapshot. This should act as a warning : wavelengths and angles might not be uniquely determined by $Re$. This is only one of the reasons why angles and wavelengths have not been much documented so far. Another reason is simply that such statistics have been sufficiently dominated by finite-size effects.

More recent computational data has been gathered for the patterns of pressure-driven pPf [14]. Wavelengths and angles have been extracted using several standard methods from image processing. Streamwise/spanwise wavelengths are reported versus $Re_τ$ (occasionally denoted $Re_τ^G$) in figure 3.17a, whereas angles are reported in figure 3.17b and compared with recent available experimental/numerical data. Despite the good quantitative match, a solid trend emerges : the angle $θ$ gets steeper as $Re$ decreases, culminating with $θ \approx 45°$ close to the critical point $Re_τ$. Analyzing the wavelengths $λ_x$ and $λ_z$ shows that they both evolve similarly with $Re$ for $Re_τ \ge 60$. Consequently the ratio $λ_z/λ_x$ and the angle
<figure><img src="image_80.png" /><figcaption>FIGURE 3.16 – Qualitative bifurcation diagrams for pCf (top) and pPf (bottom) (from Refs. [248] and [244], respectively).</figcaption></figure>

$\theta = \arctan \frac{\lambda_z}{\lambda_x}$ stay approximately constant for $Re_{\tau} \ge 60$. The increase of the angle $\theta = \arctan \frac{\lambda_z}{\lambda_x}$ for $Re_{\tau} \le 55$ is linked to an increase in the wavelength $\lambda_z$ for $Re_{\tau} \le 55$ that is no longer compensated by an increase in $\lambda_x$. This suggests yet another possible bifurcation point, that lies strictly between the usual thresholds $Re_g$ and $Re_t$, and more precisely in the interval (55, 60).

Barkley and Tuckerman [238] have suggested, from the streamwise momentum balance expressed in the laminar zones, one of the rare quantitative formulas of this field as a link between angles and wavelengths. If $\theta$ refers to the angle between the stripes and the streamwise direction, $\Lambda$ is the wavelength of the patterns, then

$$ \frac{Re \sin \theta}{\lambda} \approx \pi, \qquad (3.1) $$

where $\lambda$, $\lambda_x$ and $\lambda_z$ (respectively the wavelength, the streamwise wavelength and the spanwise wavelength) are connected geometrically via

$$ \frac{1}{\lambda^2} = \frac{1}{\lambda_x^2} + \frac{1}{\lambda_z^2}. \qquad (3.2) $$

Eq. (3.1) suggests, consistently with e.g. figure 3.12c, that for fixed $Re$ steeper stripe angles are associated with sparser turbulence. Statistics gathered from many published papers (as of 2011) have been gathered in Ref. [249] in an effort to directly verify the validity of Eq. 3.1. They are displayed in figure 3.18. A clear visual conclusion is as follows : the larger the domain size considered (both experimentally or numerically speaking), the more valid the formula.

## Turbulent fraction

The instantaneous turbulent fraction is simply the percentage of the wall covered on average by turbulent motion. In line with experimental studies [217], it can be defined using standard thresholding of appropriate wall quantities. Using for example the wall-normal velocity at the centerplane $v = v(y = 0)$
<figure><img src="image_81.png" /><figcaption>FIGURE 3.17 – Statistics of (top) : streamwise and spanwise wavelengths denoted respectively $l_x$ and $l_z$, and (bottom) : angle $\theta$ with respect to the streamwise direction in pPf, taken from Ref. [14]. Comparison to angles reported in Refs [233] and [165].</figcaption></figure>

(which vanishes exactly for laminar flow) leads to the instantaneous definition of the turbulent fraction $F_T(t)$ as

$$F_T(t) = \frac{\int \delta(v^2(x, z) > a) dxdz}{\int dxdz}, \qquad (3.3)$$

while the time-averaged turbulent fraction, an order parameter, is classically defined by

$$\langle F_T \rangle = \lim_{T \to \infty} \frac{1}{T} \int_0^T F_T(t) dt. \qquad (3.4)$$

Equivalently other definitions can be used, without any clear influence on the results. We begin by describing the bifurcations of $\langle F_T \rangle$ for pCf. The mean value of $F_T$ versus $Re$, gathered in principle over
<figure><img src="image_82.png" /><figcaption>FIG. 16. (Color online) Re sin θ/(πλ) for turbulent-laminar patterns in experiment and computations in various wall-bounded shear flows. Experimental measurements by Prigent et al. (Ref. 3) of plane Couette flow (full black circles) and Taylor–Couette flow (red crosses). Experimental measurements of rotor-stator flow by Cros and Le Gal (Ref. 10) (full blue triangles). Simulations of plane Poiseuille flow by Tsukahara et al. (Refs. 20 and 21) (full blue square). Simulations of Taylor–Couette flow by Meseguer et al. (Ref. 24) (hollow magenta pentagons) and by Dong (Refs. 22 and 23) (hollow magenta squares). Simulations of plane Couette flow by Duguet (Ref. 16) (hollow green circles), by Philips and Manneville (Ref. 19) (hollow magenta squares), and by Barkley and Tuckerman (Ref. 13) (black crosses).</figcaption></figure>

FIGURE 3.18 – Statistics gathered from many numerical and experimental studies as of 2011, compiled by [249] together with original caption with references therein.

infinite times, are reported in figure 3.19, together with the values of $F_T$ at intermediate times (corresponding to the time of nucleation). We note that the turbulent fraction at intermediate times grows continuously with $Re$, indicating that more and more spots get nucleated as $Re$ increases. The values of $<F_T>$, on the other hand, display a clear discontinuity near $Re \approx 325$. This is perfectly consistent with the experimental results of Bottin et al. [43, 217] who also predicted a discontinuous transition from measuring $<F_T>$ ($Re$). The discontinuity at large times constrains with the continuity at intermediate times. It shows that for $Re < 325$, nucleated spots shrink and collapse, whereas for $Re > 325$ a growth process takes place and the array of initial spots needs to mutate into a proper network of mature turbulent stripes before statistical equilibrium is reached. The good match, both quantitative and qualitative, between simulation and experiments is only partial: although the diagrams of $<F_T(Re)>$ match well, the oblique stripes from figure 3.12 have no experimental counterpart in Ref. [217] whereas they compare qualitatively well with Prigent’s experimental visualisations. Perhaps more importantly, the match between computational and experimental results, although encouraging, does not imply that finite-size effects have been mastered. On the contrary, both studies [66] and [217] suffer from similar limitations due to finite-size effects.

The analysis of $<F_T>$ in pPf is slightly more difficult for several reasons: there is no solid experimental data for comparison, and there are additional bifurcations. It was chosen to compute $<F_T>$ based on the observable $E_v = \int v^2 dy/2$. It is illustrated versus $Re_τ$ in figure 3.20 for different values of the thresholding. As for pCf though, the trend is monotonic: $<F_T>$ decreases with decreasing Reynolds number. Consistently with the observations above and in Ref. [244], a bifurcation occurs near $Re_τ = 50$, visible in all curves via a change of convexity. As already pointed out in Ref. [12], the values of $<F_T>$ depend strongly on the choice of thresholding, despite the lack of strong qualitative difference. The amplitude of the fluctuations around the mean value of $F_T$ increases around $Re_τ = 50$ and decreases for $Re_τ > 50$.
lower Reynolds numbers.

<figure><img src="image_83.png" /><figcaption>FIGURE 3.19 – Turbulent fraction $F_T$ as a function of $Re$ in pCf: The dotted (blue) line corresponds to the value at equilibrium, whereas the full (red) line corresponds to the turbulent fraction at nucleation time. From [66].</figcaption></figure>

<figure><img src="image_84.png" /><figcaption>FIGURE 3.20 – Turbulent fraction $F_T$ as a function of $Re_τ$ in pPf, based on the observable $E_v = \int v^2 dy/2$, with error-bars based on the standard deviation $\sigma(E_v)$. Different cut-off values have been used, each of the form $\langle E_v \rangle + \alpha\sigma(E_v)$. Same database as Ref. [14]. Courtesy Pavan Kashyap.</figcaption></figure>
<figure><img src="image_85.png" /><figcaption>FIGURE 3.21 – Friction factor $C_f$ vs bulk Reynolds number $Re_m$ in plane channel flow [244].</figcaption></figure>

Friction factor

It is also possible to return to the original Moody diagram (also named Fanning for channel flow) for the mean friction factor $C_f = C_f(Re)$, where by convention $Re$ is understood as the bulk Reynolds number $Re_b$. We note that $C_f$ can be rewritten as $C_f = 2(Re_τ/Re_b)^2$ independently of the chosen protocol. For pressure-driven pPf, $C_f$ is shown in figure 3.21 taken from Ref. [244]. The laminar branch $C_f \sim Re^{-1}$ as well as the experimental Prandtl law $C_f = O(Re^{-1/4})$ have been indicated for clarity. Two regimes intermediate between the laminar and the turbulent one are found. The first regime, in order of decreasing $Re$, is the pattern regime with the two competing orientations, in the approximate interval $700 \le Re_m \le 1600$, i.e. $50 \le Re_τ \le 90$. It is characterised in a first approximation by constant $C_f$. Note that, almost 100 years after the determination of the first Moody diagram, this is a new result. Statistics in smaller domains had not yet revealed this property of constant $C_f$, which might then be a property of the thermodynamic limit only. The second regime occurs for $Re_τ \le 50$ only, and features a value of $C_f$ much closer to its laminar counterpart. This is the regime with one dominant orientation, characterised by finite-length stripes in weak interaction with another [14, 165, 234, 246].

The direct implications of a conserved quantity (here $C_f$) are not yet known. $C_f = cst$ implies mathematically that the variations of $Re_τ$ and those of $Re_b$ are linked in a linear way. The value of $C_f$ in the pattern regime seems in a first approximation to be the same for all shear flows and is found to be very close to 0.01. This number could well be a *universal constant* of the patterning regime with a weak case-dependence, analogous in that respect to the von Kármán constant for developed turbulent flows [250]. Another analogy, certainly naive and incomplete but closer to the realm of phase transitions, is the $(p, v)$ diagram for a van der Waals mixture [251]. The constitutive law of a van der Waals mixture is given by $(p + \frac{a}{v^2})(v - b) = RT$. As the volume available to the confined gas is progressively increased, the transition from liquid to gas proceeds through a plateau in the pressure. The value of the pressure at the plateau is determined by the equality of the chemical potentials of the two pure phases. Note that the van der Waals gas is a classical example of thermodynamically isolated system, hence of equilibrium statistics, in contrast with the present non-equilibrium example of a shear flow. In addition, the phase transition in a van der Waals mixture is the typical example of first-order phase transition, a concept which does not have a strict analog in the hydrodynamic case because free energies are not defined. Nevertheless there is a rich potential to explore from this analogy in terms of nucleation and front propagation [179]. For instance the notion of Gibbs plateau is intrinsically linked to the existence of a latent heat, and it would
<figure><img src="image_86.png" /><figcaption>FIGURE 3.22 – Thermodynamic (p,v) diagram for a van der Waals mixture.</figcaption></figure>

be interesting to test how far such cross-disciplinary analogies can be pushed. Another example is the
relevance of the concept of unstable edge state to the present picture : is there for instance a robust link
between the unstable branch in the thermodynamic diagram of figure 3.22 and the edge branch ?

**3.4 The directed percolation hypothesis**

**3.4.1 Concept and motivation**

As demonstrated earlier and summarised in figure 3.23, the turbulent fraction in pCf seems to un-
dergo a discontinuity at a critical Reynolds number close to 325. This conclusion relies on a comparison
between the experiments of S. Bottin [217] and direct numerical simulation [66]. Are there however any

<figure><img src="image_87.png" /><figcaption>FIGURE 3.23 – Turbulent fraction $F_T$ as a function of $Re$ : Comparison between the experiments of S. Bottin [217], direct numerical simulation [66], and theoretical expectation from directed percolation theory (continuous transitions). Taken from Ref. [252].</figcaption></figure>
analytical theories to rely on? Nucleation theory is intrisically a first-order transition theory, and it is a priori relevant to explain how laminar-turbulent fronts start to move, but its inability to support any sustained laminar-turbulent coexistence makes it poorly adapted. In contrast, the series of experiments and numerical simulations in the 1990-2000s [43, 63, 188] has highlighted the possibility to have on-site reaminarisation at least for localised turbulent structures. Moreover, a few computational works, e.g. Xiong et al. [233] have reported bifurcation diagrams based on other (intensive) observables, e.g. the product $ReC_f$ in channel flow. Their results suggest a continuous transition for this choice of order parameter rather than the discontinuous transition put forward until 2010. Looking back in the earlier Taylor-Couette literature, a few studies actually also suggest results for $F_T(Re)$ consistent with continuous rather than discontinuous transition [208, 253]. In principle, such results become robust in the so-called thermodynamic limit of infinitely large domains over infinite observation times. The inconsistency between these results suggest remnant finite-size effects, but how is that possible given the huge domains considered? As an illustration, the typical size of a minimal flow unit is $O(1)$ while the largest domains used so far [66, 224] included at least $10^4$ such "subdomains", a priori a large enough number even in the absence of any reference. The concept of "subdomain" is relatively fuzzy, nevertheless it proves useful when modelling complex phenomena using cellular automata. This is the ideal size for the correlated sub-systems such that a network of such connected subsystems behaves like the macroscopic system. B. Hof and co-authors questioned the relevance of the MFU concept when it comes to determining the ideal size of a sub-system over which there is a finite probability to relaminarise. They hypothesised correctly that the correct size of a "sub-unit" for transitional shear flows should correspond more to the natural wavelength of laminar-turbulent stripes rather than that of minimal flow units. In other words, STI should correspond to the interaction of neighbouring stripes rather than streaks. Using this point of view, if the turbulent flow can accomodate an arbitrarily low density, then by definition of "continuity" $F_T$ should be continuous near zero (i.e. in the low-density regime). This is bad news in practice both for experimentalists and numericists, because it makes the determination of the continuous/discontinuous nature of the transition even more demanding in terms of domain size. Moreover, at equivalent front propagation speed, larger domains imply longer transition times too.

More is known if fluctuations are taken into account. Grassberger [254] and Janssen [255] have considered the case of a system with two states, one with fluctuations (our turbulent phase) and the other unique and absorbing, i.e. linearly stable and without fluctuations (our laminar phase). Moreover they have assumed no long-range order, a positive scalar order parameter (the turbulent fraction or the kinetic energy). In such circumstances when the effective dimension is one (e.g. in pipe flow), the system must undergo a phase transition which not only is continuous, but belongs to the universality class of directed percolation. In two or more effective dimensions (as for planar flows) both continuous and discontinuous phase transitions are possible. Belonging to a universality class is, from a physicist's point of view, a very strong property : it brings phenomena from different area of physics together through a set of unifying scaling relations. Like most critical phenomena, directed percolation is defined by a set of three independent critical exponents that depend only on the effective dimension. Suppose that the critical point is located at the value of the control parameter $R = R_c$, that $\epsilon = R - R_c$ (possibly including a division by $R_c$), that the positive scalar order parameter is denoted $\rho$ and its ergodic average $\langle \rho \rangle$. We denote also by $P(l > L)$ and $P(t > T)$ respectively the cumulative probability distributions of finding a laminar subdomain of size $l$ larger than a given $L$, or of duration $t$ larger than a given $T$. Critical phenomena are characterised by the following scaling relations [256, 257]

$$ \rho(\epsilon = 0, t) \sim t^{-\alpha} \qquad (3.5) $$

$$ \langle \rho(\epsilon) \rangle \sim \epsilon^{\beta} \qquad (3.6) $$

$$ P(l > L) \sim L^{-\mu_{\perp}}, \qquad (3.7) $$

$$ P(t > T) \sim T^{-\mu_{\parallel}}. \qquad (3.8) $$
Equation (3.5) is valid at $\varepsilon = 0$ only, it actually allows one in practice to locate $R_c$ and hence to define $\varepsilon = R - R_c$. Equation 3.6 expresses the continuity of the transition. Equations 3.7 and 3.8 are related to the distribution of large laminar gaps both in space and in time. The values of the critical exponents $\alpha, \beta, \mu_{\perp}$ and $\mu_{\parallel}$ in one and two dimensions are respectively (within 3 digits of accuracy) 0.159, 0.276, 1.784 and 1.84 in 1D, 0.45, 0.583, 1.204 and 1.549 in 2D [258]. Genuine experimental evidence for systems behaving in this way is scarce, mainly due to the eternal limitations by finite-size effects. Physical systems closest to achieving such a critical scaling include for instance one-dimensional thermal convection [259], and in two dimensions electroconvection in nematics [258]. In transitional shear flows, although that idea had been around since Pomeau's 1986 article [179], convincing evidence for directed percolation (DP) at the onset of the turbulent regime was lacking until the work of Lemoult et al. [225] in one effective dimension (based on both experiments and numerics).

<figure><img src="image_88.png" /><figcaption>FIGURE 3.24 – Space-time diagram from [225]. From left to right : $\varepsilon < 0$ (transient), $\varepsilon = 0$ (critical) and $\varepsilon > 0$ (sustained STI).</figcaption></figure>

The experimental scaling relations in Ref. [225], corresponding to an axially short Taylor-Couette set-up with a radius ratio very close to unity, are displayed in figures 3.24 and 3.25. Convincing evidence for DP in two-dimensional fluid systems, with a complete set of exponents, is so far restricted to the simulations of Chantry in a sinusoidal shear flow [243, 260] (with reduced resolution). Figure 3.27 is taken from Ref. [243], and it illustrates the domain size necessary to accomodate the relevant STI structures close to $\varepsilon = 0$, in comparison with domains used formerly in the literature (including Refs. [43] and [66]). It illustrates pedagogically why the numerical domain used in Ref. [66], which could accomodate only a maximum of 5 to 6 stripes at onset, necessarily suffered from finite-size effects.

Note that an algebraic fit of the form $F_T \sim (R - R_c)^\beta$ alone is in general not sufficient to validate the critical scaling hypothesis. This hypothesis is inseparable from the $F_T \ll 1$ framework, in other words from a 'dilute' regime where turbulence is scarce. An example of such misleading statistics can be found in [244] for the study of pressure-driven channel flow. Their figure 10 alone could suggest that the scaling $F_T \sim (R - R_c)^\beta$ is well verified in the range $R \in (1000 : 2500)$, which would be a very wide range. However the cumulative distributions of laminar gap size were computed in the same numerical domain in
<figure><img src="image_89.png" /><figcaption>FIGURE 3.25 – Statistics in the experiment of [225]. From top to bottom, from left to right : equilibrium turbulent fraction $<F_T>$ vs. Re (inset : $<F_T>$ vs. $\epsilon$), instantaneous turbulent fraction $F_T(t)$ vs $t$, CDF of laminar gap size, and CDF of laminar interval duration, all in log-log scale.</figcaption></figure>

<figure><img src="image_90.png" /><figcaption>FIGURE 3.26 – Comparison of two-dimensional domains of shear flows [243], including references such as [65, 66, 225, 261].</figcaption></figure>
Ref. [14]. Such distributions are expected to be algebraic close to the critical point, however they appear exponential for all parameters investigated. This demonstrates that a credible algebraic scaling for the turbulent fraction alone has no validity if the scaling relations are not all verified simultaneously. We illustrate this subtle point using two flow cases, namely pPf from Ref. [14] and aCf from Ref. [262]. The latter has been artificially widened in order to reduce finite-size effects due to the confinement [263]. For both flows the investigated distribution is compatible with an algebraic scaling of the form $F_T \sim \epsilon^\beta$ with values of $\beta$ compatible with DP theory (see figure 3.20 for pPf, and figure 3.28(top) for aCf). Figure 3.27 shows clearly that the distribution of laminar gaps in the streamwise direction of pPf is exponential for all parameters considered, as in standard STI regimes. This is in contrast with the case of aCf where, as *Re* is varied, figure 3.28(bottom) suggests a cross-over from an exponential towards an algebraic distribution, in other words a transition from an STI to a laminar regime via a critical scaling. The illustration of this point shows more generally that the parameter range where the DP scaling is expected to be valid is always very narrow. Moreover, this DP range differs markedly from the range of *Re* characterised by laminar-turbulent patterns.

<figure><img src="image_91.png" /><figcaption>FIGURE 3.27 – Laminar gap statistics in computational pressure-driven pPf from [14]. The exponential statistics suggest that the critical point is still out of reach.</figcaption></figure>
<figure><img src="image_92.png" /><figcaption>FIGURE 3.28 – Turbulent fraction vs. *Re*<sub>w</sub> (top) and laminar gap statistics in the numerical simulations of a Cf from [262], both in the axial (bottom left) and azimuthal direction (bottom right). The azimuthal extent has been artificially extended up to 128π. Unlike figure 3.27 the present cross-over from exponential to algebraic statistics suggests the proximity to the critical point.</figcaption></figure>
## 3.5 Large-scale flows

### 3.5.1 Motivation

One of the most striking manifestations of the laminar-turbulent coexistence in the planar case is the obliqueness of the coherent structures. Such a property is intrisically large-scale, since at smallest scale the coherent structures are mainly streaks known to be aligned with the streamwise direction. We hence need to define a large-scale component for the velocity field. The mean flow, understood as the temporal average of the velocity field, is the first candidate for a large-scale flow, since the smallest scales corresponding to turbulent fluctuations are usually absorbed by averaging. In the case of banded turbulence, the mean velocity field has been computed and modelled by Barkley & Tuckerman [238]. The limitation inherent to the mean flow is its absence of time-dependence and hence of dynamics. The common alternative is to extract the large-scale component directly from the velocity by using spatial filtering. Spatial filtering approaches are routinely used in higher-$Re$ turbulent flows for scale-by-scale energy budgets (see e.g. [264, 265] for pPf), in the presence of a multitude of scales. It is also central in the computational concept of Large Eddy Simulation, when scales below a given cut-off value are modelled instead of being simulated [266]. The low values of $Re$ typical of the transitional regime are less complex. Owing to a spectral gap property, the more basic distinction between "large" and "small" velocity scales is sufficient for the present purpose of explaining the morphogenesis of stripes and spots in wall shear flows.

A relatively simple computational experiment demonstrates the role played by large-scale flows in the formation of oblique stripes. The main trick is to play with the periodicity of the computational domain in order to annihilate precisely the large-scale flow, in order to understand what is lost in its absence. Suppose, as in Refs [267] and [238], that the computational domain of plane Couette flow is extended in the spanwise direction ($L_z \gg 200$) but not in the streamwise direction ($L_x \approx 10$, comparable to MFUs). As a consequence of this confinement no large-scale flow can form with low wavenumber $k_x$. The spreading of turbulence in this system was analysed in Ref. [267] based on a statistical approach. Spatial proliferation occurs only above a critical Reynolds number $Re = Re_z \approx 310$. No marked intermittency is present for $Re > Re_z$. As an alternative, consider now the case (as in Ref. [238]) where the extension is in the streamwise direction only and not spanwise, i.e. $L_x \gg 200$ but $L_z = 10$. The expansion of turbulent patches occurs only above $Re = Re_x \approx 400$ in a slug-like fashion (fronts moving in the streamwise direction with well determined front speeds). In these two configurations, the dynamics of the turbulent fluctuations is heavily influenced by the choice of the numerical domain. These domains been considered because they are free of large-scale flow. When compared to the case ideally unaffected by finite-size effects (where both dimensions $L_x$ and $L_z$ are simultaneously large), the transitional range is found in an $Re$-interval of approximately 320–410 [66]. This range of values are numerically very compatible with the values of $Re_x$ and $Re_z$. This suggests that the sustainment properties of the stripe patterns, including their infamous obliqueness, require a two-dimensional large-scale flow, with both components in $x$ and $z$.

For the case of a simple steady parallel stripe pattern of plane Couette flow, the large-scale velocity integrated from wall-to-wall is depicted in figure 3.29 using white arrows. The steadiness of the pattern implies that the large-scale flow matches the mean flow considered in [238]. It features two-dimensional jets along each side of the stripes, flowing in opposite directions on either side. The large-scale flow is most prominent where a laminar-turbulent interface can be identified: it is weaker inside the turbulent zones and falls down to zero in laminar domains. The same properties of the large-scale flow around stripes are found in all other extended planar flows, such as pPf and cPf [14, 233, 269]. The case of pipe or duct flows will be mentioned later.

### 3.5.2 Large/small scale coupling

We base our analysis on the existence of two distinct characteristic scales. The small scales $O(h)$ correspond to the coherence of the turbulent fluctuations inside a turbulent patch (streaks). The large scales can be associated with the diffusive tails of the streaks and can be anticipated to scale like $O(hR)$.
<figure><img src="image_93.png" /><figcaption>FIGURE 3.29 – Unfiltered and y-integrated large-scale velocity field of pCf for Re = 350, from [268].</figcaption></figure>

The hypothesis $R \gg 1$ justifies the names "large" and "small" scales, however we make an additional hypothesis of scale separation, i.e. we assume the existence of a range of intermediate scales $\Lambda$ between $O(h)$ and $O(hR)$ in which there is (ideally) no energy. We separate the flow field $u$ into small scales $\tilde{u} = Hu$ and large scales $U = Lu$ such that

$$u = Lu + Hu = U + \tilde{u}. \qquad (3.9)$$

The large-scale flow $U = Lu = G \star u$ is frequently defined using the plane-isotropic Gaussian low-pass Gaussian kernel $G$:

$$G(x, z) = \frac{1}{\sqrt{2\pi}} e^{-\frac{(x^2+z^2)}{2\Lambda^2}}. \qquad (3.10)$$

The high-pass filter is defined simply by the convolution kernel $I - G$. Although this decomposition can be carried out a priori for any flow field, the scale separation hypothesis is crucial in making it robust with respect to the choice of $\Lambda$. In practice, in units of $h$, the value of $\Lambda$ can be chosen anywhere in the interval (20 : 60) for most shear flows. $U$ includes the laminar base flow as well as the large-scale modulations typical of STI. The residual $\tilde{u}$ contains the rest, including what intuitively corresponds to the turbulent fluctuations.

We denote y-averaging with a bar (\bar{\cdot}). The incompressibility condition leads to the two-dimensional divergence-free condition in the $xz$-plane:

$$\partial_x \bar{U}_x + \partial_z \bar{U}_z = 0. \qquad (3.11)$$

The momentum equation in the wall-normal ($y$) direction can be simplified using the scale separation hypothesis. It results in a coupled system of PDE for the large and small scales, respectively:

$$(\partial_t + \mathbf{U} \cdot \nabla) U_y = -\partial_y P + R^{-1} \nabla^2 U_y - L ((\tilde{\mathbf{u}} \cdot \nabla) \tilde{u}_y) \qquad (3.12)$$

$$(\partial_t + \mathbf{U} \cdot \nabla) \tilde{u}_y = -\partial_y \tilde{p} + R^{-1} \nabla^2 \tilde{u}_y - H ((\tilde{\mathbf{u}} \cdot \nabla) \tilde{u}_y) \qquad (3.13)$$

The second equation for the small scales can be simplified at the laminar-turbulent interface, which can be defined as a place where $\tilde{u}_y$ is small enough without being zero. This hypothesis leads to $H ((\tilde{\mathbf{u}} \cdot \nabla) \tilde{u}_y)$
being negligible, such that the second equation expresses now the simple advection of $\tilde{u}_y$ by the large-scale flow $\mathbf{U}$. We also note the presence of the small-scale pressure gradient $-\partial_y \tilde{p}$ which does not generate any large-scale flow.

### 3.5.3 Spot structure

We begin by assessing the structure of the large-scale velocity field around an incipient turbulent spot, at early times before it proliferates. The emphasis is on the far field, which implies particularly large computational domains of size $L_{x,z} > 10^3$ and hence large memory requirements. In the case of plane Couette flow, a robust large-scale quadrupolar structure has been observed in earlier spot simulations [270, 271]. Fluid is sucked inward into the spot in the streamwise direction and ejected out of the spot in the outward spanwise direction as in figure 3.30 (left). As for spots in Poiseuille flow, the large-scale flow is found to differ in topology. The velocity field shown in figure 3.30 (right) is dipolar, consistently with the fact that the spot moves in the streamwise direction with respect to the mean fluid velocity. To our knowledge this dipolar structure had not been inferred from earlier spot simulations [272].

<figure><img src="image_94.png" /><figcaption>FIGURE 3.30 – Incipient spots in shear flows. Wall-to-wall integrated velocity field in the $xz$ plane. Left : pCf for $Re = 400$, right : pPf for $Re = 3200$.</figcaption></figure>

The far field velocity field can be extracted directly from the above simulations [273]. In contrast to the exponential scaling found e.g. for the localised steady state of Brand and Gibson [274], the velocity field decays algebraically with the distance from the spot. The analytical study of Ref. [275] aims at deriving this algebraic law directly from properties of the linearised Navier-Stokes operator. It is now confirmed by simulations that, independently of time and of the Reynolds number, the quadrupolar velocity field of pCf has its two in-plane components decaying like $r^{-3}$, with $r = \sqrt{x^2 + z^2}$ the distance from the center of the spot (see figure 3.31). The robustness of this scaling has been probed by reproducing the same spot in different computational domains of increasing size and with varying numerical resolution. For pPf as well as cPf, the decay is also algebraic but with an exponent -2 rather than -3.

These exponents match exactly those predicted for the velocity field of multipoles [273] in two spatial dimensions merely by kinematic theory. In this naive planar model sketched in figure 3.33, the spot corresponds to a two-dimensional distribution of vorticity $\omega(r, \theta)$ (in polar coordinates), strictly localised in the plane, so that the two-dimensional flow is irrotational outside $r > a$ (where $a$ is larger than the size of the region where turbulent fluctuations are found). This is indeed verified approximately in our simulations when the $y$-integrated wall-normal vorticity $\omega_y$ is considered as visible in figure 3.32. The associated two-dimensional flow ($\bar{U}_x, \bar{U}_z$) has been shown to be divergence-free in the plane in Eq. 3.11. It thus derives from a two-dimensional stream function $\Psi$ verifying $\nabla^2\Psi = \omega$. The solutions to this equation
<figure><img src="image_95.png" /><figcaption>FIGURE 3.31 – Tails of y-integrated streamwise velocity $\bar{U}_x$ in pCf for $Re = 400$, measured along the diagonal $x = z$, shown in increasingly large computational domains (log-log coordinates) [273]. The solid line corresponds to an algebraic fit of exponent $-3$.</figcaption></figure>

<figure><img src="image_96.png" /><figcaption>FIGURE 3.32 – Tails of y-integrated wall-normal vorticity $\bar{\omega}_y$ in pCf for $Re = 400$, measured along the diagonal $x = z$, shown in lin-log scale.</figcaption></figure>

can be expanded in the far-field as Taylor series with respect to the small parameter $1/r$. This gives rise
<figure><img src="image_97.png" /><figcaption>FIGURE 3.33 – Sketch of the two-dimensional kinematic model for spots.</figcaption></figure>

classically to a multipolar expansion of the form

$$ \Psi(r, \theta) = \sum_{k=1}^{\infty} \frac{1}{r^k} f_k(\theta) \qquad (3.14) $$

$$ = \frac{f_1(\theta)}{r} + \frac{f_2(\theta)}{r^2} + \dots \qquad (3.15) $$

The term in Eq. 3.15 corresponding to monopoles has been discarded for being unphysical. The velocity field deriving from this streamfunction can hence be expanded in the far field as the sum of a dipole (decaying as $r^{-2}$), a quadrupole (decaying as $r^{-3}$), and higher-order terms. Unlike Ref. [275] this kinematic approach makes no use of the momentum balance. It suffices to justify the algebraic nature of the decay of the velocity field as well as the exponents found. The upstream-downstream symmetry of the two-dimensional flow field ($\bar{U}_x, \bar{U}_z$) in plane Couette flow has the effect of cancelling the dipolar component. This explains why, in this flow, the quadrupolar component dominates the flow with an algebraic decay rate steeper than in pressure-driven flows.

Generalising this kinematic theory to three dimensions turns out more tricky. In three dimensions, even when taking the confinement by the walls into account the exponents found do not match the measured exponents. This issue has been noted also in recent studies of the dynamics of pointwise swimmers in the Stokes regimes in active matter studies [276].

### 3.5.4 Stripe development from spots

From the point of view of *morphogenesis*, we can analyse the growth of a given spot, as it turns with increasing time into an oblique stripe of finite length [268]. The transient and time-dependent character of the velocity field is hence crucial in the present context. Moreover, whatever the shear flow type considered, spots at early times are symmetric, hence the emergence of oblique stripes requires a breaking of symmetry.

The growth of a turbulent patch is shown in Fig. 3.34 for $Re = 360$ along with the corresponding large scale flow ($\bar{U}_x, \bar{U}_z$). The streamwise ends of a such a turbulent patch are characterised by so-called overhang regions where locally turbulent flow on one wall faces nearly laminar flow near the other wall, visible in figure 3.35. These regions correspond to a mismatch in the flow rates $\bar{U}_x \neq 0$, whereas $\bar{U}_x = 0$ everywhere else. As a consequence $\partial_x \bar{U}_x \neq 0$ in the overhang regions. Hence $\bar{U}_z \neq 0$ by virtue of Eq. 3.11. The resulting total large-scale flow is thus locally oblique with respect to the streamwise direction. This suggests that newly nucleated streaks at the tips of the spots are simply *advected by the large-scale*
<figure><img src="image_98.png" /><figcaption>FIGURE 3.34 – Obliquely growing spot in plane Couette flow at $R = 360$ (grey : streamwise velocity in the midplane) and associated $y$-integrated large-scale flow $(\bar{U}_x, \bar{U}_z)(x, z)$ (arrows). From left to right : $t = 200, 300$ and $400$. Simulations in a periodic domain with $\Lambda = 500$ and $1536 \times 33 \times 2048$ spectral modes. Only the subdomain $[-60 : 60] \times [-60 : 60]$ is displayed here.</figcaption></figure>

<figure><img src="image_99.png" /><figcaption>FIGURE 3.35 – PCF from [268]. Cut in the $z$ direction.</figcaption></figure>

flow, which itself has a non-zero angle with respect to the streamwise direction. As a consequence of the non-zero angle between $U$ and the streamwise direction, the growth of the spots is distorted by the presence of the large-scale flow and proceeds obliquely as well [268].

We can now suggest a mechanism to explain the emerging obliqueness of stripes. A forming stripe pattern should be seen as an initial array of growing turbulent spots that consist of small-scale streaks of finite streamwise extent. The main mechanism for the expansion of turbulence in the spanwise direction proceeds via the stochastic nucleation of new streaks [267]. The small fluctuations of $\tilde{u}_y$ near the interface, the signature of newly nucleated streaks, are advected in the direction of $U$, which is oblique with respect to the $x$ direction. The local orientation of $U$ hence determines the direction in which the turbulent patch elongates every time a new streak is added in a nucleation event.

The angle $\theta$ measured for turbulent stripes hence appears as a large-scale quantity. Stochasticity of the streak generation process acts on the large-scale flow as intrinsic noise that can even break the symmetry $\theta \leftrightarrow -\theta$ after a sufficiently long time. The global labyrinthine aspect of spots observed at late times results from a random sequence of symmetry breakings occuring at the extremities of the developing spot. For larger values of $R \ge 380$, streaks start to elongate in the streamwise direction while their spanwise rate of nucleation increases rapidly [267]. Those combined effects accelerate the spreading of the turbulent spot while the obliqueness of the turbulent interfaces is qualitatively preserved. As the spot continues to grow, other such discrete extreme events (named "budding" in the study of Ref. [277]) can allow for the outgrowth of additional stripes. The resulting field can be described as a labyrinthine spot shown in figure 3.36 [66]. The success of this approach is unfortunately limited by its inability to predict accurately the angle of the resulting laminar-turbulent interfaces.

The mechanism above has been the subject of experimental verification including a series of experiments carried out in the Paris region. In the context of pCf, the experimental study by Couliou and
<figure><img src="image_100.png" /><figcaption>FIGURE 3.36 – Labyrinthine development of a turbulent spot in pCf for $Re = 350$. From top to bottom : $t=201$, 1043 and 2102 $h/U_w$.</figcaption></figure>

Monchaux [220, 221], based on Particle Image Velocimetry, as mimicked the numerical strategy of Ref. [268]. The spectral gap hypothesis is satisfied. The quadrupolar structure of the velocity field and its orientation shown in figure 3.37 are consistent with the numerical predictions, although y-averaging cannot be performed from planar PIV data.

Experimental investigations of other shear flows have been performed as well using the same methodology, in pPf [278] as well as more recently cPf [269]. The structure of the large-scale flow around the pPf spot was initially described as quadrupolar although the lack of upstream/downstream symmetry suggests the presence of a dipolar component, consistently with Ref. [273]. The far field predictions are however far from being verified, and the nature of the decay of the velocity field can not be assessed from this experimental data. Given the short spanwise extent used ($L_z = 7.5h$) this is not really a surprise, and much wider domains are required to verify these predictions.
<figure><img src="image_101.png" /><figcaption>FIGURE 3.37 – Large-scale around a turbulent spot in experimental pCf, from Ref. [221].</figcaption></figure>

<figure><img src="image_102.png" /><figcaption>FIGURE 3.38 – Large-scale around a turbulent spot in a narrow channel of width $L_z = 10$ from Ref. [227] : streamwise velocity component (top), spanwise velocity component (bottom).</figcaption></figure>
### 3.5.5 Helical spots in annular flows

In this subsection we wish to explore the limits of the previous theory by examining a few flow cases where the spanwise extent can be conveniently varied. We focus on shear flows inside an annular geometry consisting of two coaxial cylinders. The radius ratio $r_i/r_o$ is denoted as $\eta$. The geometry (and terminology) is common to the Taylor-Couette family but the flow is forced axially rather than azimuthally. Two flow cases are actually considered : annular Poiseuille flow (aPf), where an axial pressure gradient is applied, and annular Couette flow (aCf), where the inner cylinder moves axially with a steady positive velocity. In the $\eta \to 1$ limit, aPf coincides with pPf whereas aCf coincides with pCf, they hence display exactly the same phenomenology as far as the transitional range is concerned. For finite $\eta$, e.g. here $\eta = 0.8$, by varying Re and making sure that the domain is long enough it is not difficult to identify a transitional range characterised by helical stripes wrapped around the inner cylinder. These stripes are directly comparable with the oblique stripes of pPf once unwrapped and mapped onto a planar domain. In particular the large-scale flow can again be computed, and it displays the same property of being parallel to the laminar-turbulent interfaces. However the pitch angle of the stripes apparently decreases together with $\eta$. Lowering $\eta$ below 0.5 for aCf, or 0.3 for aPf, brings unexpected results : the helical patterns are no longer found. For $\eta = 0.1$ the flow is still characterised by localised structures, however their dynamics features mainly splitting and decay, in a way much more reminiscent of cylindrical pipe flow than of pPf.

The theory invoked for the growth of the patterns [268], based on the presence of a large-scale flow, can again be used here in its steady version. It is actually simpler since it is based solely on the two-dimensional divergence-free property of the large-scale flow. Written in cylindrical coordinates consistent with the annular geometry, incompressibility is written :

$$ \partial_x u_x + \frac{1}{r} \partial_r (r u_r) + \frac{1}{r} \partial_{\theta} u_{\theta} = 0. \qquad (3.16) $$

An angle $\alpha$ with the streamwise direction can, as before, be defined by

$$ \tan \alpha(x, \theta, t) = \left| \frac{\overline{U_{\theta}(x, \theta, t)}}{\overline{U_x(x, \theta, t)}} \right|. \qquad (3.17) $$

where

$$ \overline{(\cdot)} = \frac{\int_{R_i}^{R_o} r(\cdot) dr}{\int_{R_i}^{R_o} r dr}. \qquad (3.18) $$

The key point is that the azimuthal extent of $2\pi$ can be interpreted as a large number in units of the gap $d = r_o - r_i$, if $\eta$ is close to 1. Fig. 3.40 displays the inner and outer perimeters, resp. $L_{zi}(\eta)$ and $L_{zo}(\eta)$, in units of $d$ as the radius ratio is varied. When $L_{zi}/d \gg 1$, the flow in the gap behaves as a planar flow. Otherwise the situation is closer to pipe flow with an effective dimension of one and no large-scale azimuthal flow. The intersection of an arbitrary line $L_z = \lambda_z^c$ with the curves $L_{zo}(\eta)$ and $L_{zi}(\eta)$ in Fig. 3.40 defines two values of $\eta$, respectively $\eta_1$ and $\eta_2$. This leads to three distinct ranges of values for $\eta$:

- for $0 < \eta \le \eta_1$, there is no space for large scales in the azimuthal direction, neither at the inner nor at the outer wall. As a consequence $\overline{U_{\theta}} = 0$ for all $r$, and $\partial_x \overline{U_x} = 0$.

- for $\eta_2 \le \eta \le 1$, azimuthal large scales can form at both inner and outer walls : $\partial_{\theta} \overline{U_{\theta}} = -\partial_x \overline{U_x}$ with $\overline{U_{\theta}} \ne 0$. The situation is then analogous to the planar case where oblique laminar-interfaces can be sustained.

- for $\eta_1 \le \eta \le \eta_2$, the situation is mixed : azimuthal large scale flows cannot be accommodated at all locations in the cross-section. A probabilistic approach is required.

Since the exact values of $\eta_1$ and $\eta_2$ depend on the threshold $\lambda_z^c$, the qualitative conclusions above need to be verified numerically. Figure 3.41 shows, for $\eta = 0.2$, two values of Re for which the two types of inclinations ($\theta = 0^\circ$ and $\theta \ne 0^\circ$) coexist spatially. Two antagonist effects likely to deviate from the planar
<figure><img src="image_103.png" /><figcaption>FIGURE 3.39 – Localised coherent structures in Annular Pipe Flow (aPf), from top to bottom : η = 0.8 and Reτ = 72, η = 0.5 and Reτ = 56, η = 0.3 and Reτ = 56, η = 0.1 and Reτ = 52. Three-dimensional isolevels of streamwise velocity fluctuations.</figcaption></figure>
<figure><img src="image_104.png" /><figcaption>FIGURE 3.40 – Azimuthal length as a function of radius ratio.</figcaption></figure>

<figure><img src="image_105.png" /><figcaption>FIGURE 3.41 – Localised coherent structures in APF, $\eta = 0.2$. From top to bottom : $Re_{\tau} = 56$ (top) and $Re_{\tau} = 52$ (bottom). Two-dimensional isolevels of radial velocity close to mid-gap.</figcaption></figure>

case compete here : wall curvature and azimuthal confinement. They are a priori linked together by the geometry. The two related parameters can however be made independent if the outer perimeter $L_z$ is chosen such that $L_z > 2\pi$. This is achieved in practice by simulating the same regimes in an artificial domain with an azimuthal wavenumber $\beta = 2\pi/L_z$ input as a non-integer. This numerical trick has been considered in Refs. [263,262] in the context of aCf. We can focus for instance on the case $\eta = 0.5$. When $L_z = 2\pi$, no laminar-turbulence is found whatever the value of $Re$. However for $L_z \gg 2\pi$ the flow for $\eta = 0.5$ features a transitional range characterised by clear oblique/helical stripe patterns. This rules out wall curvature as a possible explanation why oblique interfaces do not form. However it validates the idea that regular oblique patterning in a given $Re$-interval is the rule once azimuthal confinement by periodic boundary conditions is weak.

### 3.5.6 Boundary layer flows

While the annular examples above were opportunities to test the robustness of the theory with respect to azimuthal confinement, its robustness with respect to wall-normal confinement needs to be also addressed. In the planar case with two plates (pPf or pCf), a two-dimensional divergence-free property was established in Eq. 3.11 for the velocity field integrated from wall to wall. The crucial mathematical step to Eq. 3.11 is the property $[u_y]_{y=-h}^{y=h} = 0$, in other words the fact that the wall-normal mass flux is zero (due to wall impermeability in pCf and pPf). This property can be violated in the case with one wall only, i.e. boundary layer flows. This is illustrated in the Asymptotic Suction Boundary Layer flow (ASBL). For this flow, in the absence of a top wall the integration in $y$ of Eq. 3.11 can be carried out between the lower wall (by convention at $y = 0$) and any wall-normal position $y^*$, yielding

$$ \partial_x \bar{u}_x + \partial_z \bar{u}_z = -u_y(y^*) + u_y(0), \qquad (3.19) $$

where $\bar{u}_i = \int_0^{y^*} u_i(x, y, z, t) dy$ is the equivalent of the wall-integrated flow in the original Eq. (3.11). As a consequence, the loss of streamwise flow rate experienced when entering a localised turbulence patch in the $x$ direction does no longer have to be balanced by a spanwise flux like in contained shear flows.
<figure><img src="image_106.png" /><figcaption>FIGURE 3.42 – Two-dimensional yz-plane visualisations of the streamwise velocity fluctuations $u'$ in ASBL for (a) $Re = 333$ and (b) $Re = 260$ computed using R1 resolution. Only part of the wall-normal extent of the domain is shown. The black dashed line marks the height $y^+ = 100$. The values of the time-averaged 99% boundary-layer thickness are $\delta_{99} \approx 155$ and 27 for $Re = 333$ and 260, respectively. Computations obtained using moderate resolution [1]</figcaption></figure>
<figure><img src="image_107.png" /><figcaption>FIGURE 3.43 – (a) Mean streamwise velocity profiles for $Re = 270$, $Re = 280$ and $Re = 290$ in ASBL, with the lighter colours corresponding to increasing $Re$. The green dot-dashed line corresponds to the modified linear law. The green dotted line represents the empirical log law $u^+ = \frac{1}{\kappa} \ln y^+ + B$ for $Re = 270$ with $\kappa = 1.1$ and $B = 10.7$. (c) Two-dimensional premultiplied energy spectrum $\phi_{uu}^+(\lambda_x^+, \lambda_z^+)$ at position $y^+ = 12$ for $Re = 270, 280$ and $290$</figcaption></figure>

This loss of streamwise flow rate can instead be balanced by a wall-normal gradient of momentum or by both wall-normal and spanwise gradients. In the absence of clear rule, we simply state that oblique laminar-turbulent interfaces are not generic in ASBL, whereas in channel and Couette flows they are. This conclusion is true independently of $Re$. Direct numerical simulation in computational domains both large and high (i.e. $L_y$ large too), with well-adapted numerical resolution, has shown that there is indeed no ordered laminar-turbulent coexistence in ASBL even arbitrarily close to the onset of its turbulence regime [1]. In other words ASBL does not possess any transitional range. Moreover, all elements point towards a clear violation of the directed percolation hypothesis : transition seems discontinuous in the sense that the turbulent fraction directly goes from 100% to 0% as $Re$ is decreased below the critical point $Re = 269 \pm 1$. The value of this critical Reynolds number has been verified experimentally in a recent study. It falls within only 1% of the numerical value [279, 280]. An element of justification is found in turbulent statistics : for $Re = 270$ (just above onset) the turbulent regime verifies the logarithmic law of the wall, whereas this is far from true for any of the classical contained flow such as pipe or channels near their critical point. Turbulent ASBL is hence multiscale for all Reynolds numbers from onset.

In order to verify that the mechanism suggested here to discard sustained laminar-turbulent coexistence in ASBL applies in practice, a numerical manipulation of the flow has been implemented. Similarly to Ref. [281], the non-zero fluctuations of the wall-normal and spanwise velocity are brought to zero at a given distance from the wall using a volume force. This mimics a "ceiling effect", whereas it is expected that truly confined flows would exhibit a transitional range, just as pCf or pPf do. If $V_S$ is the suction velocity (independent of $y$), the additional body force writes $F = (0, -\lambda(y)(V_S + u_y), -\lambda(y)u_z)$, with $\lambda(y) = S((y - y_s)/d)$. The function $S$ is chosen as

$$S(u) = \begin{cases} 0, & u \le 0, \\ 1/(1 + e^{(1/(u-1)+1/u)}), & 0 < u < 1, \\ 1, & u \ge 1, \end{cases} \qquad (3.20)$$

The parameters values tested correspond to $y_s^+ = 100$ and $d^+ \approx 80$. The distance of $100\delta_\nu$ corresponds to the mean streak spacing for all shear flows, or in other words to the Kolmogorov length of inhomogeneous shear flows. It is believed that choosing a similar wall-normal extent would leave out any scale larger than this cut-off value, make the flow monoscale and hence make it susceptible to develop large-scale modulations. The results are encouraging, and at least the main hypothesis turns out to be correct : although the flow ultimately relaminarises for $Re = 280$, it develops transient interfaces during the relaminarisation process. Importantly, these interfaces (visualised furtively in figure 3.44) appear unambiguously oblique with respect to the streamwise axes. For $Re \ge 290$ the flow does not relaminarise
<figure><img src="image_108.png" /><figcaption>FIGURE 3.44 – Two-dimensional xz-plane visualisations with wall-normal damping for $Re = 280$ in ASBL. Streamwise velocity at $y^+ \approx 12$.</figcaption></figure>

and other furtive oblique interfaces are easily identified [1]. However, with the choice of filtering function and the parameters used no sustained STI or pattern regime was identified so far.

## 3.6 Multiphysics extensions

The range of Reynolds numbers corresponding to the transitional range for planar shear flows is relatively narrow. Moreover, it does not correspond to a very wide range of applications. Most flows encountered in everyday applications, because of the typical lengthscales and velocity scales involved are characterised by Reynolds numbers of the order of $10^5$ or more, where full-fledged turbulence is the rule. It is however possible, and relevant in a large range of realistic applications, to consider shear flows in the presence of an additional force $F$ such that the transitional range is pushed to such high Reynolds numbers. In primitive variables formulation, the non-dimensional incompressible equations read

$$ \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} = -\nabla p + \frac{1}{R}\nabla^2\mathbf{u} + \mathbf{F}, \qquad (3.21) $$

$$ \nabla \cdot \mathbf{u} = 0, \qquad (3.22) $$

where $p$ is the pressure field, $\mathbf{u}$ is the velocity field and $\mathbf{F}$ is an external force (of module $F$) which can be added to damp turbulent fluctuations. For purely shear-driven flows, $\mathbf{F} = 0$. For patterns to be pushed towards higher $R$, the additional force of interest should have a tendency to damp turbulent fluctuations. If the Reynolds number is denoted $R$ for simplicity, the range of $R$ where patterns can be observed can be noted a priori $[R_g(F) : R_t(F)]$. Equivalently, for a given value of the governing parameter $R$, patterns are sustained within a range $[F_t(R) : F_g(R)]$. The force $F$ can alternatively be seen as a way to control turbulence in applications when strong unsteadiness is considered as undesirable.

A first example from the literature is found in plane Couette flow with global spanwise rotation (RPCF). Cyclonic spanwise rotation is known to damp turbulent fluctuations [283]. In this case we can define $F$ as a measure of the rotation rate, for instance the non-dimensional rotation number $Ro$ propo- tional to the rotation rate $\Omega < 0$ -not to be confused with the Rossby number inversely proportional to it-. It was verified experimentally [282] that RPCF has a linearly stable laminar regime. Under suitable excitation, a regime of localised turbulence exists in a well defined range of values of $R$ and $F$. For fixed $F$, when $R$ is increased, as documented in figure 3.45 one encounters first a regime of spots within a laminar environment, then a patterned regime, an intermittent one and eventually turbulence. That sequence is qualitatively similar to that encountered for $F = 0$ when increasing $R$. It was verified using large nume- rical domains that this phenomenology is well captured numerically [284] for $R$ up to 6000, as shown in
<figure><img src="image_109.png" /></figure>

<figure><img src="image_110.png" /></figure>

FIGURE 3.45 – Top : Experimental parameter space $(\Omega, Re)$ for negative rotation $\Omega$ [282] (COU : laminar Couette flow, SPT : spots, TRS : turbulent stripe patterns, INT : intermittent turbulence, TUR : turbulent flow). Bottom : snapshot of the spot regime
<figure><img src="image_111.png" /><figcaption>FIGURE 3.46 – Stripe pattern in DNS of RPCF.</figcaption></figure>

figure 3.46. Above $R = 6000$ the phenomenology changes. The value of $R$ in Ref. [284] was pushed as high as $R = 20,000$ with relevant numerical resolution. Laminar-turbulent patterns are still found but only at a small distance from the wall. Further away from the wall the flow is characterised by turbulent fluctuations without any dominant large-scale structure. We witness in figure 3.47 the emergence as $R$ increases of *near-wall stripes* of increasingly small thickness. Moreover, the stripes on either wall are no longer steady as in pCf, instead they travel in opposite directions with velocities $O(1)$ corresponding to the large-scale velocity if not to the mean flow velocity. This new regime is very expensive to simulate and equally difficult to achieve experimentally with large dimensions. As a consequence, the parameter space remains to be properly explored.

Note that another related example is the classical Taylor-Couette flow when both cylinders rotate in opposite directions. This is not a surprise since Taylor-Couette is viewed as a limiting case of spanwise rotating pCf when the radius ratio tends towards unity. When this aspect ratio is close to 1, it is known that helical laminar-turbulent patterns can be found [207, 218, 253, 261].

A second area of application of the above concept is found in the presence of stable density stratification. Such applications abound in geophysics both in the atmospheric and oceanic contexts. Former investigations in small enough periodic domains have revealed a trend for turbulent regimes to relaminarise for strong enough density gradient [285, 286]. In analogy with the observations for pure shear flows, it was tempting to speculate that laminar-turbulent coexistence would be possible, for neighbouring parameters, provided large enough domains are used. This has been verified again in Ref. [284] in a symmetric ("open") channel geometry that turbulent-laminar stripe patterns could be maintained at higher $R$, with a grossly constant turbulent fraction, provided the stratification parameter (the Richardson number) would be increased with the same rate. As for RPCF, at really high values of $R$ patterns do not occupy the full gap width, they are located at a close distance from the wall amidst turbulent fluctuations. The value of $R$ in Ref. [284] was pushed also as high as $R = 20,000$ with the relevant numerical resolution. The two systems are yet not equivalent. For instance, it is believed that internal waves, made possible, can be excited at the right frequency by the large-scale modulations or by turbulent fluctuations, and start to interfere with the turbulence itself. Despite more recent continuations of this work [287–289] this scenario lacks as of now a proper investigation.

The last application of interest is found within the context of electrically conducting fluids in the presence of an imposed magnetic field. Applications of low-magnetic Reynolds number magnetohydrodynamics are frequent in industry when liquid metals are involved, e.g. in electromagnetic brakes or in the growth of semiconductor crystals. Similarly to stratified flows, there was former evidence, from simulations in small periodic domains, that the application of a magnetic field could make turbulent flows relaminarise [290, 291]. A similar approach was undertaken in Ref. [284] using much larger computational domains, for the case of plane Poiseuille flow, again for $R$ up to 20,000. The force $F$ is parametrised by a Hartmann number. The orientation of the magnetic field was chosen aligned with either $x$, $y$ or $z$. The only clear case, from the point of view of the formation of laminar-turbulent patterns, corresponds to the
<figure><img src="image_112.png" /><figcaption>FIGURE 3.47 – RPCF in (x, y) section as Ro is increased. Isolevels of absolute value of wall-normal velocity. Values of Re and Ro and Lx from top to bottom : (350, 0, 250), (750, -0.02, 250), (6 × 10³, -0.116, 48) and (2 × 10⁴, -0.2, 25.1).</figcaption></figure>
spanwise magnetic field. There again, laminar-turbulent patterns appear robust to increasing *F* until the laminar-turbulent coexistence is found only by the wall. More recent investigations of these regimes have pointed out similarities and a few discrepancies with non-magnetic cases in duct geometries [292, 293].

These separate investigations have been carried out with the aim of identifying features in common. Beyond the immediate message that laminar-turbulent patterns are robust to increases in *F* (provided *F* has the property to dampen out turbulent fluctuations), it represents an opportunity to measure the wavelengths of the patterns over a much wider range of *R* than is possible for *F* = 0, namely almost two decades from *R* = 300 to 20,000. These wavelengths are represented for a large class of patterns from various sources in a two-dimensional cluster plot (Λₓ, Λ_z) in figure 3.48(left). Although *R* varies by almost two decades, wavelengths vary over hardly a factor of 4 (an exception is represented by the MHD case Mzh at *R* = 20,000 which displays a smaller wavelength, may be influenced by the choice of the numerical domain). Although a very slow dependence of Λₓ and Λ_z on *Re* is not strictly ruled out, the present results rather suggest a lack of direct dependence, i.e. Λₓ, Λ_z = *O*(1). This is in contrast with the small scales which, as is clear from e.g. figures 3.47 but also figure 3.46, shrink with increasing *Re* as in all turbulent flows. A reduction to inner units has been performed in the spectrum of figure 3.48b, for the cases with increasing density stratification. Such cases are seen in figure 3.48a to have similar large-scale wavelengths (as indicated by the labels PP, S80 and S183). According to the classical wall turbulence theory, for featureless turbulent flows the near-wall peaks of the spectrum should collapse around λ_z⁺ = 100. This is the case here, to acceptable accuracy, even in the patterning regime. The large-scale peaks, however, do not scale at all in inner units and the outer unit suggestion from figure 3.48a is much more adequate. We conclude that, whereas small-scale structures scale in inner units, large-scale patterns scale in outer units. Such a result would not have been exploitable from pure shear flow data only.

The last teaching from this very global comparison concerns the correlation between the imposed (resp. measured) Reynolds numbers depending on the choice of protocol. The friction Reynolds number $Re_τ$ is plotted in figure 3.49 versus the bulk Reynolds number $Re$ defined for each case, in log-log coordinates. Although no accurate power-law fit dominates this plot, a linear relation of the kind $Re_τ = O(Re_b)$ is compatible with the data. Recall that for pure shear flows such a relation was suggested from the largest-domain investigations, in correspondence with the relation $C_f = cst$. A simple extrapolation of this law to the data of figure 3.49 suggests again that, at least as a rule of thumb, laminar-turbulent patterns are characterised by a constant friction factor, also whenever additional forces are present in the budget.

## 3.7 Conclusion

The last decade has been important for the study of transition to turbulence from a spatiotemporal point of view, mainly because numerical facilities are now a full alternative to the more classical experimental approach. In particular, large computational domains are now feasible using numerical simulation. The global issue of whether transition obeys the equivalent of a phase transition is now essentially solved : as far as the turbulent fraction as a function of the Reynolds number is concerned, continuous transition is the rule, with critical exponents compatible with the scenario of directed percolation [225, 260]. Quantitative evidence for this universal scenario requires however large computational domains that remain extremely costly to simulate, even as of 2021. However, understanding that such requirements are in order represents already an important theoretical progress. For pipe flow, the spreading of turbulence results from puff splitting events, and for higher Reynolds number from the growth of slugs [153, 180, 202]. For planar flows such as plane Couette flow and plane Poiseuille flow, the spreading of turbulence proceeds from a sequence of events leading from receptivity to streak growth, to localised spot nucleation to spatial proliferation [14, 66]. The role played by large-scale flows in the proliferation process is now better understood : being generated at the level of the laminar-turbulent interfaces, the large-scale flow simply advects the weakest perturbations at the interface in a direction necessarily oblique to the streamwise direction. The resulting growth is hence dominated by the oblique elongation of the nucleated patches of turbulence [268]. This mechanism has been validated by analysing its actual limits. It breaks down in
<figure><img src="image_113.png" /><figcaption>FIGURE 3.48 – Top : location of modulation wavelengths of the patterns in $(\Lambda_x, \Lambda_z)$ space for various cases listed in Ref. [284]. Bottom : energy spectrum for the density stratified cases PP, S80 and S183, with lengths displayed in inner units.</figcaption></figure>
<figure><img src="image_114.png" /><figcaption>FIGURE 3.49 – ($Re_τ - Re_b$) relation for many different flow cases together in the transitional regime.</figcaption></figure>

two cases : in annular flows [263, 294] because of the azimuthal confinement that rules out large-scale flows [295], and in boundary layer flows because the flow is not confined in the wall-normal direction. The mechanistic reasons why large-scale modulations form in the transitional range are still not fully understood. Several competing theories exist, mainly motivated by low-order models : Turing instability [296] modified with noise, crystallisation of turbulent puffs [297] or simply excitability [180], but a deeper investigation is needed to distinguish between all these credible scenarios from real data. Interestingly, the notion of transitional range appears irrelevant in the parallel boundary layer flow (ASBL), which hosts no laminar-turbulent patterns and features a wholly discontinuous phase transition. The case of boundary layer flows is shown to differ strongly from the channel geometries because of the absence of wall confinement [1]. The next chapter is dedicated to a deeper theory of transition applicable to spatially developing boundary layer flows.
4

A transition theory for spatially
developing boundary layer flows
## 4.1 Phenomenology of Blasius boundary layer transition

Boundary layer flows are, simultaneously, among the most obvious hydrodynamic examples of solid/fluid interaction, and a complicated case of transition to turbulence. The simplest example is probably the Blasius boundary layer flow, because of its two-dimensional steady solution, and this chapter is concerned about the nonlinear stability of this base flow with respect to incoming disturbances. Unlike e.g. Couette flow, whose academic origin is hard to hide, the Blasius flow configuration is prevalent in many real applications including aeronautics, propulsion and aerodynamics. This explains why experimental studies abound, in particular those based on wind tunnels experiments. The present chapter aims at unifying recent results obtained by attempting to apply the concepts stemming from dynamical systems theories to the Blasius boundary layer flow.

### 4.1.1 Flow set-up

The Blasius flow corresponds to the steady flow developing above a semi-infinite plate that starts at some origin $x = 0$. Despite the steadiness hypothesis, the flow experiences spatial development, such that tracers advected by that flow are constantly slowing down. Besides there is always entrainment of fluid from the region outside. The spatial development can be interpreted as the combined advection and viscous diffusion of the spanwise vorticity. It emanates from the leading edge, which acts as a singular source of vorticity.

We begin by presenting the 'ideal set-up' and then confront it with more realistic cases comparable to wind tunnel conditions. The 'ideal' Blasius set-up is illustrated in figure 4.17(top). The velocity at the plate is exactly zero when the plate is still, whereas at an infinite distance above the plate the velocity vectors are oriented in the streamwise direction, and have constant amplitude $U_\infty$. Owing to viscous diffusion the displacement thickness grows with the distance $x$ from the leading edge as $\sqrt{x}$. This flow does not depend on any physical control parameter [298]. Reynolds numbers are based either on $x$ or on the layer displacement thickness $\delta^*$, and they change with $x$. The displacement thickness is defined classically from the streamwise velocity base flow profile $u(x, y)$ as

$$ \delta^*(x) = \int_0^\infty \left(1 - \frac{u(x, y)}{U_\infty}\right) dy \qquad (4.1) $$

Such Reynolds numbers are interpreted as a coordinate system rather than as a governing parameter. Numerically however, it is important to specify the value of the Reynolds number at the upstream end of the computational domain $Re_{\delta_0^*} = U_\infty\delta_0^*/\nu$, with $\delta_0^*$ corresponding to $\delta^*$ evaluated at this point. The laminar solution is given, for $x > 0$ sufficiently large, by the solution of the Blasius O.D.E. [298]. This equation rules the dynamics of the streamfunction, considered as a function of the reduced variable $y/\sqrt{x}$ only, under additional hypotheses such as steadiness and lubrication ($x \gg y$). The laminar Blasius solution has no analytical expression but it is easily integrated numerically. It is not strictly valid immediately close to the leading edge, however it constitutes an excellent approximation of the real laminar solution as soon as $Re_x = xU_\infty/\nu \ge 200$. The linear stability (in time or in space, or both) of the steady Blasius solution is a difficult mathematical problem. Under the lubrication hypothesis it is however possible to "freeze" the base flow locally as if the spatial development was too slow to matter. Under such conditions the system can be linearised around the base flow solution $u_0$, and an ansatz of the form $e^{\alpha t}$ is admissible. The linear stability computed under these assumptions is displayed in figure 4.2. Linearly unstable modes correspond to the zone $\mathcal{R}(\alpha) > 0$. They exist for $Re > 519$ [18]. These modes have finite streamwise wavenumber $k_x \neq 0$ and $k_z = 0$, they are localised inside the boundary layer, travel uniquely downstream, consist of co-rotating spanwise vortices and are called Tollmien-Schlichting waves. In real experiments these waves appear via wavepackets that also grow in length with $x$ [299] A large series of experiments are dedicated to exciting these waves, to the way they lose their stability with respect to three-dimensional disturbances and finally transition to a turbulent flow far enough downstream. This is the so-called classical transition scenario usually present in most textbooks.
The realistic set-up is very similar to the ideal one, yet it incorporates sources of inhomogeneity that are responsible for the transition as observed in realistic wind tunnel experiments [300]. This is sketched in figure 4.17(bottom). The potential sources of flow disturbance can be of various physical origins : incompressible vortical disturbances, acoustic waves, incoming turbulence, tripping wire, wake of a grid [301], etc... They are located typically "upstream" but not necessarily upstream of the leading edge, nor upstream of the computational domain if the latter starts at a finite value of $x = x_0$. In a simplified picture the sources of disturbance are rather located upstream of the zone where transition happens : laminar flow is considered under the influence of permanent disturbances of finite amplitude, while the turbulent flow further downstream is not subject to disturbances. In particular, the permanent character of the disturbances takes the system a priori away from the realm of initial-value problems. The reason why such a distinction needs to be made between ideal and realistic set-up is the following : the transition scenario might differ from the classical one provided the finite-amplitude disturbances are strong enough [302–305]. We base the present investigations on the underlying idea that the flow is destabilised by incoming turbulence that might be caused by either an upstream grid, or by any other non-specified reason. In order to keep the flow conditions well defined, the in-coming turbulence is considered as temporally permanent, spatially decaying with $x$, isotropic and homogeneous, and it is characterised by two parameters : the turbulence intensity $Tu$ and the integral lengthscale of the decaying turbulence, $L$. Only the effect of $Tu$ will be considered in what follows, with $Tu > 0$ defined classically by

$$Tu = \frac{1}{\sqrt{3}} \sqrt{u_{x,rms}^2 + u_{y,rms}^2 + u_{z,rms}^2} \qquad (4.2)$$

The sketch in figure 4.17 illustrates the three main transition scenarios found as $Tu$ is increased from low levels ($Tu < 1\%$) to high levels, via intermediate levels (everything inbetween) :

- For low enough $Tu$, the classical transition scenario is at play : the initial receptivity phase converts the weak disturbances into nascent TS waves which grow and later destabilise into $\Lambda$-shaped vortices. These vortices eventually break down into sustained turbulence further downstream.

- For intermediate $Tu$, the scenario is called *bypass transition* [307] : the receptivity phase leads to the spatial growth of coherent streaks (also called Klebanoff modes), and the transition is provoked by the destabilisation of these streaks.

- For large enough $Tu$ ($Tu > 6\%$) free-stream turbulence breaks initially into smaller structures, which later destabilise into turbulent spots.[308])

The present aim is to generalise the knowledge on edge states to better understand their role in the bypass process. The presence of TS waves in one of the scenarios interferes with the concept of edge state in a puzzling way that has not been addressed before. The high amplitude regime is not investigated here, but it has also been investigated in a possible relation with minimal seeds [309, 310].

### 4.1.2 Bypass scenario

The bypass scenario was initially identified in experiments, and it has eversince been validated in several numerical simulations. A rapid but more detailed description of the process is useful to determine the "ingredients" needed for a nonlinear theory of transition. Pedagogic experimental visualisations from Ref. [302] are displayed in figure 4.4. They correspond to the same flow yet with different upstream grids and different values of $Re$ at the level of the grid. These visualisations show sudden bursts of turbulence nucleating randomly on a well-ordered array of straight streamwise streaks. The presence of streamwise streaks is consistent with computational observations. For instance in figure 4.5, the excitation by free-stream turbulence is achieved by mimicking a synthetic turbulent flow upstream [303].

The receptivity stage upstream, that turns decaying turbulence into active streaks, is not fully understood. It is essentially based on linear mechanisms but does not feature any instability. It does not differ from the lift-up provided there is streamwise vorticity upstream. The first manifestation of nonlinear behaviour lies probably at the level of the first solitary streak breaking down, i.e. when the first incipients spot gets nucleated. This is where the analogy with edge state scenarios from the previous chapter become tempting (despite the lack of an obvious initial-value problem context). In the case of plane Couette flow, it was demonstrated in Ref. [149] how the occurrence of a streaky edge state and its linear instability were the essential nonlinear steps justifying the search for exact solutions (although
<figure><img src="image_115.png" /><figcaption>FIGURE 4.1 – Sketches of Blasius boundary layer flow in the laminar (top) and transitional cases (bottom). From Ref. [306].</figcaption></figure>
<figure><img src="image_116.png" /><figcaption>FIGURE 4.2 – Stability of the Blasius flow in a frequency-Reynolds number diagram (F, Re). Solid lines : iso-real part of the related eigenvalues, Dashed lines : iso-imaginary part. From Ref. [306]</figcaption></figure>

<figure><img src="image_117.png" /><figcaption>FIGURE 4.3 – Sketch of transition in Blasius boundary layer depending on the incoming turbulence level $Tu$. From Ref. [311].</figcaption></figure>

the instability of the edge state can also be investigated in a linearised fashion). As in planar flows, the nonlinear approach consists first in identifying numerically a relevant and non-trivial base flow, then in analysing its stability and predicting the structures responsible for the breakdown of the streaks.

The edge state, being defined as an invariant set, is the ideal candidate for such a base flow. Scrutini-sing figure 4.5 leads to additional information on the edge state to be sought. It is clear from this figure that the early nucleation of spots is a local phenomenon. It make hence sense to consider the *growth of a*
<figure><img src="image_118.png" /><figcaption>FIGURE 4.4 – Visualisations of the bypass process for several entrance grids and flow speeds. Experiment from Ref. [302].</figcaption></figure>

<figure><img src="image_119.png" /><figcaption>FIGURE 4.5 – Visualisation of the bypass process from Ref. [303]. Isovalues of streamwise velocity fluctuations, together isolevels of $\lambda < 0$ (green).</figcaption></figure>
<figure><img src="image_120.png" /><figcaption>FIGURE 4.6 – Instabilities of streaks generated in an experimental boundary layer flow, of varicose (top) or sinuous (bottom) typeFrom Ref. [312].</figcaption></figure>

localised disturbance on a localised streak of finite length in order to free oneself from the dependency on the chosen wavenumbers. Computationally this has the advantage to restrain the study to single localised initial disturbances, closer to computational studies such as Ref. [313]. A further comment about the choice for an initial value problem formulation can be made at this point. Under a permanent disturbance it would sound more adequate to look for another framework. Nucleation of spots in the bypass picture proceeds however as a series of discrete localised events (see figure 4.4). It thus makes sense to reverse the arrow of time and to assign to an incipient spot at time T a predecessor, i.e. a velocity field corresponding to an "initial" time t = 0. The relevance of the initial value concept would be less credible if the generation of new turbulence was continuous in time. The problem becomes again the identification of the most unstable modes of a relevant streaky base flow. This problem was addressed experimentally by Asai and co-workers who generated a straight low-speed streak in a boundary layer flow in order to study its instability and breakdown [312]. Two different symmetries characterise the instability of straight streaks : either spanwise-symmetric (varicose) or anti-symmetric (sinuous). Both types are illustrated in figure 4.6. In Ref. [314], the modal instability of a spatially periodic array of straight streaks has been considered instead, except that the base flow consists of synthetic streaks whose spatial structure was determined from a numerical fit. The same qualitative conclusions as the localised case arise : both symmetric and anti-symmetric streamwise modulations of the streaks stand out as least stable. The threshold amplitude levels required for this instability to occur might still depend on the localisation features [315]. Later, non-modal instability was also predicted [316] to lead to very similar structures without the need for a proper exponential streak instability. Some confusion is still present, entertained by the choice of a synthetic base flow which is not solution to the governing equations. The suggested alternative here is to generate a new relevant base flow using an edge tracking method, as it ensures that the resulting velocity field will belong to an invariant set of the governing equations.
4.1.3 Difficulties associated with state space picture

A pedagogically clear state portrait of the state space of the Blasius system would be a formidable achievement, both from a fundamental and applicative point of view. There are however fundamental obstacles to such a task. The main theoretical difficulty compared to, say, the pipe flow problem, might be due to the spatial development of the boundary layer. Let us list the possible invariant sets of interest for such a challenge.

The laminar base flow is steady and should hence be represented as a fixed point of the state space. The issue of its stability is more subtle. The definition of (linear) stability indicates that a fixed point is unstable as soon as one unstable direction exists. This is here clearly the case since TS waves, although analytically predicted for frozen base flows only, have been found experimentally. The troubling detail is that the critical Reynolds number of 519 suggested has no relevance to the stability problem since *Re* is not an external control parameter any longer. As a consequence, the linear instability of the Blasius base flow is of an *unconditional* type : the right perturbation at the right initial location leads to an exponentially growing mode at large times, and this is all that matters to declare the base flow unstable.

The notion of turbulent attractor is more problematic. Consider an initial disturbance with a spatially localised support. In the right conditions this perturbation will advect downstream while growing spa- tially. It will increase in size and its kinetic energy, enstrophy etc... will all diverge. The system can hence not support a bounded attractor, at least not using the usual energy-based metrics. One might argue that the situation is similar for slugs in pipe flow, which is true, however there is a difference. Slugs operate at constant Reynolds number. An extensive quantity used as observable would diverge with time during transition, but an intensive quantity would probably converge to a constant value. A genuine attractor is hence possible if the coordinate system is revised. In spatially developing flows yet, every perturbation is advected downstream, while it experiences increasingly larger boundary layer thicknesses as time evolves. The turbulent state needs locally to adapt to its new environment, characterised by the continuous ap- parition of new scales. This rules out the possibility for a well-defined turbulent equilibrium. Even by renormalising at every time the outer size of the turbulence patch or the boundary layer thickness, smaller and smaller scales would appear in its interior as time increases, as if the viscosity was decreasing with time.

The third possible regime is, without surprise, the edge state intermediate between laminar and tur- bulent... although there is no well-defined asymptotic turbulent state, the "turbulent" (resp. "laminar") fate of a given perturbation is often unambiguous when the perturbation energy or enstrophy is monitored in time, since it corresponds to a rapid growth (resp. decay) of the corresponding observable. The same argument holds as for the turbulent state, namely that there is a risk to deal with an unbounded relative attractor as an edge state. As shown in Ref. [317] bisection is still possible because of the self-similarity property of the edge state. Eventually, it is shown in Ref. [318] that the issue of long-time asymptotic states is in practice irrelevant because of the instability of TS waves.

There are further difficulties associated with the boundary layer context, all related to the issue of finite times. Most perturbations travel in the positive downstream direction no matter their wall-normal position [319]. Once perturbations have exited the computation domain it does not make sense to re- cycle them using periodic boundary conditions. The situation might seem analog in channels and pipes ; however in these parallel flows the equivalence between incoming and outgoing disturbances makes it rea- sonable to recycle outputs into inputs, computationally and even experimentally (see Ref. [320]). There is a simple, disappointing perhaps, but at least rigorous to handle this problem in spatially developing flows : one needs to i) use computational domains that are as long as possible and ii) consider all the dynamics as a *finite-time* phenomenon. The former condition is solely a matter of computational cost and duration. The latter has deep mathematical implications, since even the notion of attractor does not make any sense anymore. In particular, although the term "edge state" (formally an asymptotic regime only) is still used, the right concept here is only that of "edge trajectory". Fortunately, a majority of other concepts such as stable and unstable manifolds, Lyapunov exponents etc.. are easily generalisable
to the finite-time framework [321].

Eventually, provided the edge manifold can be defined in finite time, so can the minimal seeds. Their definition does not differ from the one used in previous flow cases since it is intrisically a finite-time notion with an associated time horizon $T$. Nonlinear optimal perturbations in boundary layer flows have been computed first in Ref. [309, 310] (initially without energy minimisation), whereas minimal states have been computed recently in Ref. [322].

## 4.2 Competition between two transition scenarios

The application of the classical edge tracking requires, as always, the definition of a global observable which vanishes for the laminar flow and reaches high values in the turbulent regime. The quantity chosen in this study is the entrophy associated with the streamwise vorticity only :

$$a = \left( \frac{1}{\mathrm{vol}(V)} \int_V |\omega_x|^2 \mathrm{d}v \right)^{\frac{1}{2}}, \qquad (4.3)$$

(note that the denominator vol(V) is not crucial in the definition). Reporting on the results of bisection best supposes a "short" and a "long" time horizon.

<figure><img src="image_121.png" /><figcaption>FIGURE 4.7 – Three-dimensional rendition of streamwise velocity perturbation along the edge trajectory at $t = 1050$ (symmetric case). From Ref. [317].</figcaption></figure>

### 4.2.1 Impact on edge tracking

The "short" time horizon corresponds to a computational domain of length $L_x = 2000\delta_0^*$, where $\delta_0^*$ is chosen such that $Re_{\delta_0^*} = \frac{U_\infty\delta_0^*}{\nu} = 300$. This allows to carry out converged edge tracking over a time horizon of $T \approx 2000$ [317]. The "long" time horizon corresponds to a computational domain of length $L_x = 3000\delta_0^*$, still with $Re_{\delta_0^*} = 300$. This is not much longer except that a new trick was used, as detailed in the appendix of Ref. [318] : perturbations to the base flow are considered in frame moving with prescribed velocity $0.8U_\infty$. This numerical technique allows for much longer edge tracking with $T \approx 15,000$. The technique might be useful in many instances where localised disturbances travel through spatially varying environments.
For the short time bisection there is essentially no strong difference from the planar cases, except that the bisection time is limited. A localised initial condition similar to Ref. [270, 313] was first imposed with spanwise symmetry. The reason for this choice was initially a scientific interest in hairpin vortices suggested to impose their symmetry, coupled to the initial hope to find a simpler edge state in the presence of discrete symmetries as in Refs [106, 128]. Two bounds $a_L$ and $a_T$ have been chosen by trial and error. The dynamics of $a(t)$ and other observables shows no sign of convergence and is characterised by a few large-amplitude bursts. A snapshot of the velocity field along the edge trajectory after roughly $10^3$ time units is shown in figure 4.7. The striking features of this perturbation include its spatial localisation in three directions, the robust presence of long low-speed streaks (in blue). The length of the state fluctuates in time as the main structure upstream sheds short-lived perturbations further downstream, in a manner comparable to puffs in pipe flow. Despite these fluctuations, it was found that a simple rescaling of most spatial quantities by the "local" displacement thickness $\delta$ simplifies much the dynamics.

$$ \Omega_x = (\delta_0^*/\delta)^{\frac{1}{2}} \left( \frac{1}{\mathrm{vol}(V)} \int_V |\omega_x|^2 \mathrm{d}v \right)^{\frac{1}{2}}, \qquad (4.4) $$

$$ \Omega_y = (\delta_0^*/\delta)^{\frac{1}{2}} \left( \frac{1}{\mathrm{vol}(V)} \int_V |\omega_y|^2 \mathrm{d}v \right)^{\frac{1}{2}}, \qquad (4.5) $$

$$ W = (\delta_0^*/\delta)^{\frac{3}{2}} \left( \frac{1}{\mathrm{vol}(V)} \int_V |u_z|^2 \mathrm{d}v \right)^{\frac{1}{2}}. \qquad (4.6) $$

$\delta$ can be evaluated in several ways. In Ref. [317] the location of the edge state was monitored at every time by the maximum local wall shear stress, and $\delta$ was simply the value of $\delta^*$ at this location. In Ref. [318] a velocity barycenter was determined at every time, and the value of $\delta^*$ was determined at this location. The rescaling follows Eq. 4.6, and the dynamics expressed in these variables is included in figure 4.8. This state portrait suggests an approximate time periodicity. Note that the original time coordinate also needs to be rescaled by $\delta^*$. This suggests that, even if the edge state was time-periodic in the rescaled variables, the time between two successive bursts would diverge in the original units. Besides, since the wall-normal and spanwise extents of the edge state both increases as fast as $\delta^*(t)$, larger domains are required for longer time horizons. This sets a severe limit to the number of bursts that can be captured in a single bisection. As for non-symmetric initial conditions, on short timescales the situation is not very different. No convergence towards a simple state is found, instead bursts seems unavoidable.

<figure><img src="image_122.png" /><figcaption>FIGURE 4.8 – Left : Phase portraits using rescaled variables (symmetric case).</figcaption></figure>

Despite the relative shortness of the edge tracking, figure 4.8 suggests that a relative attractor with a
self-similarity property can exist. We note that, in analogy with the reduction of continuous symmetries that has emerged recently [40], there is also a possibility to reduce continuous symmetries together with self-similarity [323], but it has never been used in fluids. The present self-similarity property is surprisingly straightforward to justify : the factor ($\delta/\delta_0^*$) prefactor rules how the boundary layer thickness evolves with time, as well as the effective width of the localised state. Quantities such as enstrophies, edge state dimensions etc... simply adapt to the ever-thickening boundary layer. It is surprising that the size of this coherent structure manages to adapt to its changing environment on such a fast timescale, rather than adiabatically given that the edge state travels at speed $O(U_\infty)$. Nevertheless it is simple enough to warrant that a priori the concept of relative attractor makes sense in the Blasius boundary layer. As we shall see yet, longer time horizons blur this simple picture.

<figure><img src="image_123.png" /><figcaption>FIGURE 4.9 – Phase portraits using rescaled variables in the non-symmetric case for subliminal (left) and supraliminal (right) perturbations.</figcaption></figure>

The long-time case has been analysed only in the presence of non-symmetric initial condition. Three space-time diagrams (using only rescaled variables) illustrate the situation : figure 4.8 documents the liminal perturbation, while figure 4.9 illustrates the late-time dynamics of supra- and sub-liminal perturbations, respectively. These diagrams have been constructed by monitoring at every time the streamwise position $x_G$ and speed $\dot{x}_G$ of the velocity barycenter, then using $\dot{x}_G$ as a moving frame. Figure 4.8 illustrated the dynamics non-symmetric edge trajectory. Again it features a small number of bursts that correspond in space to streak switching events. Although no conclusion related to the long-time dynamics can be made at this point, the dynamics of the fully localised Blasius edge state on such time scales is comparable to that of the fully localised edge state in ASBL : long localised streaky flow where low-
<figure><img src="image_124.png" /><figcaption>FIGURE 4.10 – Space-time diagram of $u_x(x, y, z, t)$ for $z = -10$ and $y = y_p$ for the trajectory away from the edge manifold. Streaks decay while the Tollmien-Schlichting wavepacket grows in amplitude and form a turbulent spot.</figcaption></figure>

and high-speed streaks switch spanwise position quasi-periodically. For the supra-liminal perturbation, the differences with other localised edge states are not striking : the central low-speed streak undergoes an instability which very rapidly (less than 50 rescaled time units) induces a turbulent flow. It takes approximately the same timescale of 50 reduced time units for the ensuing turbulent spot to spread in the spanwise direction and fill the computational domain downstream. The difference concerns the sub-liminal case. From $t^* = (\delta_0^*/\delta^*)t=1200$ to 1400, the edge state appears to decay in amplitude like in a standard relaminarisation event where the largest scales decay last. For $1400<t^*<1500$, spanwise-independent waves start to propagate downstream while increasing in amplitude. At $t^* \approx 1400$ these waves undergo several destabilisations at various z-positions. Each destabilisation invariably leads to a turbulent spot, and the flow appears fully turbulent for $t^*>1600$. There is no doubt that these waves are Tollmien-Schlichting waves. This is confirmed by their phase speed, range of wavelengths and by the space-time diagram $(x, t)$ displayed in figure 4.10. An interesting feature in this diagram is how TS waves grow upstream of the streaky edge state, not at the same location (the mechanism pointed out in Ref. [324] actually prevents TS waves to grow on streaks). Since streaks and TS waves travel at unambiguously different velocities, the TS waves are quickly left behind. It is thus not strictly impossible to observe, out of one single localised initial condition, two patches transitioning through different mechanisms.

The above results unfortunately has nasty consequences for the classical bisection process. On short times it is still possible to monitor without ambiguity, out of a single observable $a(t)$, on which "side" of the edge manifold the perturbation lies : the laminar side is characterised by trajectories where $a(t)$ decrease monotonically to zero. Over longer times one observable is no longer enough for such a task : an initial decay has little meaning given that ultimately, all trajectories will experience transition (be is via TS waves or streaks) and display increasing $a(t)$. It is hence no longer possible to know on which side of the edge manifold a given state lies. This has nothing to do with the choice of the observable $a(t)$. It is not even known whether the edge manifold is still orientable and whether its sides have a particular meaning. The situation in practice is illustrated in figure 4.11 using precisely the observable $a(t)$ used in Ref. [318]. For a given choice of the bound $a_L$, some trajectories might be interpreted on short times as going laminar, whereas in fact they turn turbulent at later times. Bracketing a given trajectory becomes thus impossible as long as one cannot monitor whether the ensuing transition is due to the destabilisation of TS waves or not. Long-time bisection leads hence to an 'uncertain area'.

Although this is the initial definition adopted in other contexts, one can no longer claim that the stable manifold of the edge state divides the state space into two distinct basins of attraction. All perturbations end up at later time turbulent, except perhaps the ones lying directly on that manifold to be found. There is however an edge which, to the accuracy permitted in the "uncertain area", marks the boundary between two different types of trajectories : those experiencing transition through streak breakdown and
<figure><img src="image_125.png" /><figcaption>FIGURE 4.11 – Observable $a(t)$ vs. $t$ during application of the bisection algorithm. The green vertical line marks the maximum time for the edge tracking algorithm using that observable, and the two dotted lines stand for the observable bounds $a = a_L$ and $a_T$ (see text).</figcaption></figure>

those undergoing destabilisation of TS waves. In other words, the edge marks the boundary between bypass and classical transition. The latter involves trajectories with, in principle, an *early and transient relaminarisation stage* as in figure 4.9 (right). The existence of an uncertain area suggests that some perturbations might transition twice at two different locations, in a mixed-type transition.

### 4.2.2 Application to wind tunnel experiments

The conclusion from the previous results is difficult to handle : bisection in spatially developing flows can only be trusted over finite time horizons, however the edge state is defined as an asymptotic state only. This contradiction is not necessarily important for the original purpose of explaining the main mechanisms occuring during bypass transition. Indeed, as long as the excitation parameters do not favour the rapid growth of TS waves over streaks breakdown, bypass remains the only important phenomenon. It is hence important to delimitate the parameter range of applicability of the edge-bypass picture. Figure 4.12 contains parts of the experimental database by S. Shahinfar and J. Fransson [325, 326] for wind tunnels excited by incoming grid turbulence, namely the transition values of $Re_x = xU_\infty/\nu$ depending on the turbulence level $Tu$ (in %). These x-locations are compared with the locations where transition occurs latest in the present initial value problem. Such locations are close to $Re_x \approx 10^6$, to be compared also to the theoretical onset for TS waves at $Re_x = 9.1 \times 10^4$. This gives a lower bound for the $Tu$ levels required in practice to observe transition earlier than suggested by edge tracking. The minimal corresponding $Tu$ levels are around 1.5%. This corresponds to the "weak bypass regime" reported originally by Narasimha [327]. As a reminder, classical transition dominates for $Tu < 1\%$ whereas bypass is classically observed even up to 6%. The above limitations still offer a comfortable zone in which to investigate streak breakdown with the help of the edge state as a base flow.

### 4.2.3 Low-order modelling

The results from Ref. [318] make it clear that the bistable picture in the introduction of Chapter 3 does not apply to the Blasius flow, because the laminar fixed point is itself unstable. This property is not
<figure><img src="image_126.png" /><figcaption>FIGURE 4.12 – *Tu(%) versus Re<sub>x</sub>* for experimental bypass transition data [325], showing intermittency of 10% and 90% (thick lines) and 50% (dashed line). The arrows mark the *Re<sub>x</sub>*-limitation of the bisection algorithm. For larger times and *Re<sub>x</sub>*, mixed transition is expected (grey area).</figcaption></figure>

a consequence of the spatial development of the boundary layer, and it can a *posteriori* manifest itself in
other flows including planar ones. The instability of the laminar base flow to Tollmien-Schlichting waves
has been also reported in plane Poiseuille flow [20], in ASBL [328], annular pipe flow and many others
(it is shorter to insist instead on the fact that such unstable waves are *not* found only in pipe and pCf).
However in these flow parametrised by a meaningful Reynolds number *Re*, this instability occurs only
above a given value *Re* = *Re*<sub>c</sub> which in general exceeds by several orders of magnitude the onset of bypass
transition. For all these flow cases one expects, in analogy with the results in the Blasius flow, that the
notion of edge manifold changes or loses its relevance beyond *Re*<sub>c</sub>. Two such cases have been documen-
ted recently. The simplest is bent pipe flow [329] for which the turbulent regime can be in competition
with a Hopf bifurcation of the laminar flow. From a dynamical point of view, this case is hardly more
complex than the standard case without Hopf bifurcation. Indeed, the Hopf bifurcation is supercritical :
the possible attractors are, on one hand, either the fixed point or the limit cycle and, on the other hand,
the turbulent state. Bisection and edge tracking are expected to proceed well and to identify the edge
state. A second example is plane Poiseuille flow in a small periodic box [330]. This case is more difficult :
the edge state exists before *Re*<sub>c</sub> as a TW state, which can be continued beyond *Re*<sub>c</sub>. The instability to
TS waves is known to be subcritical [331] and to include many bifurcations. As a consequence, does the
TW state remain an edge state beyond *Re*<sub>c</sub>? The amount of open questions regarding the edge manifold
in the presence of an additional instability of the base flow suggests the use of a low-dimensional model.
The reduced cost of the simulations would leave ample room for parametric studies or exploration, as
opposed to the very costly Blasius simulations.

Autonomous model

The two-dimensional DM2D model by Dauchot and Manneville [17] was used in Chapter 3 to introduce
the concept of edge in the bistable context. In the same way we wish to introduce a new low-order model
that would cope with the above constraints, yet with maximal simplicity. It was decided to generalise the
DM2D model into a three-dimensional "DM3D" model, with the same variables $x_1$, $x_2$ and an additional variable $x_3$. The model should satisfy the following properties :

- when restricted to the two first dimensions $x_1$ and $x_2$ (i.e. when $x_3 = 0$), the model coincides with DM2D

- the laminar fixed point $L$ is linearly unstable above some parameter

- the nonlinear terms conserve the new total energy $E = x_1^2 + x_2^2 + x_3^2$.

These constraints lead to the three-dimensional below :

$$ \frac{dx_1}{dt} = s_1x_1 + x_2 + x_1x_2 \qquad (4.7) $$

$$ \frac{dx_2}{dt} = s_2x_2 - x_1^2 + \sigma x_3^2 \qquad (4.8) $$

$$ \frac{dx_3}{dt} = s_3x_3 - \sigma x_2x_3. \qquad (4.9) $$

Eqs. 4.7, 4.8, 4.9 are characterised by the two same constants $s_1 < 0$, $s_2 < 0$ as the original DM2D model ($s_1 = -0.1875$, $s_2 = -1$) [17]. It involves also the additional parameter $s_3$, interpreted as the exponential growth rate of the new instability of $L$. Furthermore, $s_3$ is the control parameter for this new study. The instability is only present if $s_3 > 0$. The constant $\sigma = -1$ allows the instability to saturate, it occurs in the third as well as the second equation by virtue of the conservation of energy by the nonlinear terms.

The bifurcation diagram of the DM3D model, as $s_3$ increases, is reported in figure 4.13. The hyper-plane $\mathcal{P}: \{x_3 = 0\}$ is an invariant set of the dynamics for all $s_3$. For $s_3 < 0$, the invariant sets of the system are only those of the DM2D model, embedded in $\mathbb{R}^3$. The three fixed points $L, E, T \in \mathbb{R}^3$ are respectively the stable "laminar" attractor $L$, the hyperbolic saddle point $E$, which acts as the "edge state", and the stable "turbulent" fixed point $T$. They remain the same for all $s_3$ but their stability will depend on $s_3$. Figure 4.13 can be read as follows :

- For $s_3 < 0$, $\mathcal{P}$ is locally stable and the dynamics transverse to $\mathcal{P}$ is contracting. The edge manifold $\Sigma$ is a two-dimensional surface invariant along $x_3$.

- At $s_3 = 0$, $L$ loses its stability in a supercritical pitchfork bifurcation. A new stable attractor $S$ emerges for $0 < s_3 < s_{3H} \approx 0.0296$. This is the start of zone I in figure 4.13.

- At $s_3 = s_{3H}$, via a supercritical Hopf bifurcation $S$ becomes unstable in favour of a stable limit cycle $C$. For $s_{3H} < s_3 < s_{3c} \approx 0.03125$, the limit cycle $C$ increases in size.

- At $s_3 = s_{3c}$, the attracting limit cycle $C$ collides with $\mathcal{P}$ simultaneously at the points $E$ and $L$.

- For $s_{3c} < s_3 < s_{3d} \approx 0.063$, $E$ still has one unstable direction but all trajectories outside $\mathcal{P}$ converge now to the turbulent attractor $T$. This is zone II in figure 4.13.

- At $s_3 = s_{3d}$, $S$ merges with the saddle point $E$ in a pitchfork bifurcation.

- Beyond $s_3 = s_{3d}$, $E$ still exists but it has now two unstable directions, include now one unstable direction transverse to $\mathcal{P}$. This is zone III.

The edge manifold appears robust for all $s_3 \le s_{3c}$, in the sense that it separates two separate basins of attraction. Note that the attractor in the "laminar basin" is not necessarily the fixed point $L$, but can also be instead $S$ or $C$. The collapse of the edge manifold begins at the boundary crisis at $s_3 = s_{3c}$. According to the typology defined by N. Lebovitz [332], the edge turns from strong to weak : it separates the state space locally but not globally. All orbits outside $\Sigma$ are attracted by $T$ and they may "only be distinguished by features such as orbital complexity and time to reach the attractor" [332]. However for $s_{3c} < s_3 < s_{3d}$, $E$ remains a saddle point with one unstable direction. Classical bisection would output $E$ as an edge state. The "edge manifold" exists, yet it no longer separates the state space into two basins of attraction, rather it delimitates the fast routes from the slow ones. The slow routes comprise all trajectories that transiently visit the neighbourhood of $L$ before converging towards $T$. The last bifurcation at $s_3 = s_{3d}$ corresponds to the loss of accessibility of $E$. Beyond $s_3 = s_{3d}$ no trajectory outside $\mathcal{P}$ leads to $E$. Classical bisection would fail in that case as it would output $T$. There is neither any edge state nor
edge manifold any longer [333].

<figure><img src="image_127.png" /><figcaption>FIGURE 4.13 – Bifurcation diagram for the DM3D model as $s_3$ is increased. L : laminar fixed point, E : saddle point (edge state), T : turbulent fixed point, S : secondary laminar fixed point.</figcaption></figure>

A surprisingly simple correspondence between the model and the hydrodynamic cases mentioned earlier can be claimed. The case of bent pipe flow corresponds to zone I in figure 4.13. It is essentially the bistable picture with the classical edge notion. The main difference is that the attractor in the laminar basin is not the laminar solution but another state bifurcating from it. Zone II corresponds to the dynamics discussed in Ref. [330] : the edge state solution still exists and can be identified using bisection. However its stable manifold does not separate the state space into two basins anymore, but into zones with different transition properties. State portraits corresponding to Zones I and II are displayed in figure 4.14. Eventually zone III displays the dynamics closest to the Blasius case, where bisection reveals unable to identify any edge state. The edge state existing for lower $s_3$ is still a solution of the system but it has changed stability and dynamical role.

### Non-autonomous model

Eq. 4.9 is still relatively ill-adapted to the description of the edge dynamics in the Blasius boundary layer because it depends explicitely on an external parameter $s_3$. By contrast, for the Blasius flow (or any spatial developing flow) one would expect the control parameter to evolve with time, in analogy with the Reynolds number which evolves with the position of the localised state as time progresses. Assuming for simplicity a single propagation velocity, Eq. 4.9 can be turned easily into an *non-autonomous* O.D.E. system by decreeing $s_3 = k_1t + k_2$ (with $k_1 > 0$). The resulting system reads

$$ \frac{dx_1}{dt} = s_1x_1 + x_2 + x_1x_2 \quad (4.10) $$

$$ \frac{dx_2}{dt} = s_2x_2 - x_1^2 + \sigma x_3^2 \quad (4.11) $$

$$ \frac{dx_3}{dt} = (k_1t + k_2)x_3 - \sigma x_2x_3, \quad (4.12) $$

The time interval is restrained to $t \in [t_0, t_0 + T_F]$, with $k_1 = 0.73/T_F$ and $k_2 = -0.1$. Two trajectories are represented in a three-dimensional state portrait displayed in Figure 4.15. The two corresponding initial conditions lie at $t = 0$ on either side of the original edge manifold (i.e. the manifold existing for $s_3 < 0$) with a small non-zero value of $x_3$. The non-autonomous dynamics is easier to describe than its autonomous counterpart : either the trajectory reaches $T$ directly, or it does so by first approaching transiently the neighbourhood of $L$, leaving $P$ locally near $L$ and approaching it again near $T$.
<figure><img src="image_128.png" /></figure>

<figure><img src="image_129.png" /></figure>

<figure><img src="image_130.png" /><figcaption>FIGURE 4.14 – Autonomous DM3D. Comparison between region I and II. Phase portraits for different values of $s_3$ close to the global bifurcation : (a) $s_3 = 0.031$. The system presents a hard edge between $C$ and $T$, region I (b) $s_3 = 0.03125$. Heteroclinic cycle close to $s_{3c}$ (c) $s_3 = 0.032$, beyond $s_{3c}$ the heteroclinic connection collapses and the edge $W^s(E)$ becomes of the weak kind, region II. The blue and orange trajectories illustrate the two newly born different routes to transition.</figcaption></figure>
<figure><img src="image_131.png" /><figcaption>FIGURE 4.15 – DM3D nonautonomous model. state portrait for two trajectories starting close to the edge of DM2D and to the invariant plane $\mathcal{P}$.</figcaption></figure>
<figure><img src="image_132.png" /><figcaption>FIGURE 4.16 -</figcaption></figure>

## 4.3 Low-order modelling using cellular automata

We would eventually like to test the validity, in the Blasius boundary layer, of the nucleation scenario involving a localised edge state. The nucleation scenario is based on the simple idea that the nucleation and the spreading phenomena are two decoupled phenomena. The main idea is to analyse quantitatively a real simulation of the bypass transition process in the Blasius boundary layer, in order to *learn* the basic rules of turbulence spreading. Then in a second phase, the nucleation scenario, in the form of a space-dependent nucleation probability, can be tested inside a simple automaton model featuring the same probabilistic spreading rules. The transition statistics obtained that way can be compared with the original statistics from the large-scale simulation or even from experiments. The nucleation scenario is later labelled 'valid' in case the turbulence statistics match well.

For simplicity, this study is restricted to short-time edge state dynamics explained earlier, so that no Tollmien-Schichting wave is expected to interfere with the transition process. This implies that the equivalent turbulence level at the entrance (located at $Re_{\delta^*} = 300$) must lie strictly above 2%. The chosen range of values of *Tu* is therefore 3-4%, while the integral lengthscale of the incoming turbulence is set to $L = 10$.

The whole transition process was reproduced using costly Large-Eddy numerical simulation (LES) of the incompressible Navier-Stokes equations with suitable boundary conditions, in a computational domain of width $L_z = 500\delta_0^*$. The same code and methodology as Brandt *et al.* [303] were used, as illustrated in Fig. 4.16(top). The massive amount of space/time data collected in such simulations was post-processed and reduced to binarised data on a plane, with a focus on laminar-turbulent coexistence. A given location at the wall is either laminar or turbulent with a given probability $p(R)$, where $R$ is the Reynolds number coordinate, e.g. $Re_{\delta^*}$. In practice, because of the finite correlation lengths within the flow, probabilities need not be defined pointwise but rather *cell*-wise, where the dimensions of a typical cell will be defined later. This reduced dynamics becomes akin to a two-dimensional probabilistic cellular automaton (PCA), consisting of two phases: laminar or turbulent, as illustrated in Fig. 4.16(bottom). Within this approach, turbulent is identified using a Boolean variable. In a given cell at a given time the laminar/turbulent nature of the flow depends, *in a probabilistic way*, on the neighbouring variables at previous times.
<figure><img src="image_133.png" /><figcaption>FIGURE 4.17 – Update rules determined directly from the LES simulations.</figcaption></figure>

<figure><img src="image_134.png" /><figcaption>FIGURE 4.18 – Probabilities $p_p, p_f, p_b$ and $p_s$ for turbulence to respectively persist, move forward, backwards or sideways among the cells defined in figure 4.17. These probabilities are deduced from the LES data and plotted versus the streamwise position $x - x_0$ for different values of the free-stream turbulence level $Tu$ from 3 to 4 (expressed in %).</figcaption></figure>

As an indicator for turbulent motion the local spanwise shear stress at the wall $Q = \partial w / \partial y \bigg|_{y=0}$ is used with a threshold value of 0.3. The size of a cell, as well as the timestep in the automaton, are deduced from the autocorrelation functions of $Q$ in space and time. The data is finally regularised using Gaussian filtering in order to remove spurious events. This yields a two-dimensional array with discretisation on $N_x \times N_z = 204 \times 76$ gridpoints.

The probability rules of the PCA are schematically shown in 4.17. Each temporal update follows two steps. The first step is deterministic and models the advection, translating all cells by one unit in the downstream direction. In the second step, a turbulent cell can persist with probability $p_p$ and infect neighbouring cells, spreading turbulence forward, backward or sideways (left and right) with probabilities $p_f, p_b$ and $p_s$, respectively. In addition, spontaneous nucleation of turbulence is allowed for with probability $p_c$ as in Ref. [259]. The four probabilities $p_p, p_f, p_b$ and $p_s$ are directly identified as functions of $R$ from the simulation data. Their small variations in the transitional region suggest to model these functions as constants, represented as solid lines in figure 4.18.

The exact choice for the nucleation rate $p_c(R)$ is however more delicate and has a crucial influence on the dynamics of the automaton. It is clear that $p_c = \text{cst}$ is not a physically relevant hypothesis. The experimental literature often assumes, instead, $p_c$ to be a Dirac function, consistently with the old "concentrated breakdown hypothesis" [334, 335]. Determining the exact function form for $p_c$ from data turned out to be too demanding in terms of simulations to be deduced and was given up. The nucleation rate has instead been modelled using a minimal one-dimensional approach to the edge state concept. This requires the knowledge of a scalar quantity $A(x(t))$ to be defined. Let $A_E$ be the amplitude of the edge state, assumed constant at leading order. For turbulence to develop downstream from a nucleation event, it is required that the amplitude $A = A_0$ at the entrance of the domain satisfies $A_0 \ge A_E$. The
<figure><img src="image_135.png" /><figcaption>FIGURE 4.19 – Top : Sketch to explain the nucleation model based on the edge state concept and yielding the nucleation rate $p_c(x)$. Bottom : Nucleation rate $p_c(x)$ estimated for various values of the free-stream turbulence level $Tu$ from 3 to 4 (expressed in %).</figcaption></figure>

instability of the edge state implies that, after some time corresponding to a distance $x$, the amplitude $A$ grows exponentially from $A_0$, until it reaches the threshold value $A_T$. The way how $A_0$ is distributed statistically at the entrance of the domain $x = 0$ determines indirectly the nucleation probability, using the differential relation $p_c(x)dx = p(A_0(x))dA_0$ illustrated in figure 4.19(top). A standard Gaussian distribution has been assumed here, which leads to an analytical expression for $p_c(R)$ documented in Ref. [336] and shown in Figure 4.19(bottom) for different values of $Tu$. The higher the value of $Tu$, the earlier the nucleation of a spot, and the narrower the distribution of $p_c(x)$. Direct measurements of $u_{r.m.s.}$, considered as a relevant choice for $A$, have yielded different distributions closer to Gamma distributions, yet the philosophy behind this modelling does not change. Note that no variant of the model ever predicts concentrated breakdown.

Eventually, by seeding the flow statistically using the nucleation rate $p_c$ displayed in figure 4.19 and using the constant fits from figure 4.18, the automaton reaches a steady regime with fluctuations. The main quantity of interest is the intermittency factor $\gamma(R)$, identical to the turbulent fraction introduced in the previous chapter. The qualitative as well as quantitative match with the experimental measurements, as well as with the computational results from the simulation, are very convincing. This suggests
<figure><img src="image_136.png" /><figcaption>FIGURE 4.20 – Intermittency factor $\gamma$ versus streamwise position. Top : experimental measurements from Ref. [304] (using a rescaled streamwise variable). Bottom : model versus LES measurements.</figcaption></figure>
that the decoupled dynamics "nucleation + spreading" is a relevant picture for modelling. The match is equally good when it comes to other statistical quantities [336]. Besides, we note that independently of the details of the model, the distribution for $\gamma$ is invariably sigmoid-shaped. This distribution differs from the distribution suggested by the Directed Percolation hypothesis, which exhibits a non-derivable singularity at $\gamma = 0$. Leaving aside the issue of spatial development, which probably has no influence on such local issues, the sigmoid-shaped scenario seems particularly adapted to the development of turbulence downstream of a grid, in the presence of a non-homogeneous nucleation rate. A similarly-looking curve was also found in recent experiments of channel flow [228, 337], although it was not necessarily recognised as such by the authors. By contrast, the nucleation rates in models compatible with the DP scaling are implicitely homogeneous. This important example suggests that, depending on the way turbulence can be nucleated spatially, different universal scenarios can be envisioned.

## 4.4 Conclusion

Establishing a state space picture for turbulent spots in the Blasius boundary layer is not a trivial generalisation of the approach followed for parallel shear flows. However, although the concept of turbulent attractor is not relevant due to the spatial development, the concept of edge manifold is tractable and can be investigated using the classical bisection algorithm. Unsteady edge trajectories become hence the new object of study. Their internal dynamics reveals localised streaks with bursts and streak switching, which is comparable to parallel flows such as channel flow or the boundary layer with suction [115]. An additional complication blurs the picture : the edge trajectory is well defined only over moderate times [317]. Over long timescales the linear instability of the laminar base flow to Tollmien-Schlichting waves can not be ignored. These waves form localised wavepackets that lead invariably to turbulent spots later downstream. Their very existence makes the usual concept of edge irrelevant as such, however trajectories on the edge manifold remain interpretable as mediators between two types of transition : the bypass and classical scenarios [318]. This distinction appears relevant in wind tunnel experiments for levels of incoming turbulence $Tu$ above 2%. The structure of state space close to the edge manifold has been understood and reproduced using a three-dimensional O.D.E. model system, obtained as a generalisation of the two-dimensional Dauchot-Manneville [333]. Eventually, realistic LES simulations of the bypass process have been analysed directly. The short-time edge model can be used to relate theoretically the nucleation rate of turbulent spots to the distribution of incoming turbulence. The relevance of this model has been tested within the context of a data-driven cellular automaton, where the spreading probabilities have been directly inferred from the large-scale simulations [336]. The model reproduces without any difficulty the mean intermittency curve, a sigmoid-shaped curve different from the critical scalings from directed percolation, explained by the non-homogeneous nucleation rate. This project in its entirety suggests that the realistic bypass transition process in spatially boundary layers can be regarded, on moderate timescales, as an *activation process* where the edge state plays the role of the unstable saddle.
# Conclusions and outlooks

## 5.1 Conclusions

### 5.1.1 State space organisation

A large part of the research illustrated in this thesis deals with a temptative sketch of the state space of a fluid system where transition to turbulence is possible from some (but not all) initial conditions. In most cases a robust picture needs to include an *edge* hypersurface that separates the state space in two different basins of attraction [81, 97, 100]. That picture gets challenged in a few cases : when the turbulent state is not a formal attractor, or when the laminar state itself is unstable. The edge manifold has been shown to persist in such cases although its nature as a separatrix is no longer guaranteed in the latter case [333]. Two locations on this edge manifold are worth deeper investigations : the local attractors within the edge, namely *edge states* [177], and the lowest-energy perturbations namely *minimal seed* [178]. The former forms an invariant set often displaying chaotic dynamics, while the latter is a well-defined optimal way of triggering transition in the frame of initial value problems (IVPs). Generalising such a picture to a framework different from IVPs is a welcome challenge. Resolvent analysis has been an increasingly popular tool in this regard [16, 25] although the problem of initial conditions is not fully absent of that formulation either. Another case where the edge picture needs an update is that of subcritical boundary layer flow. While parallel boundary layer flows (e.g. ASBL) do not introduce any additional difficulty with respect to the other planar flows [338], an additional complication arises in spatially developing boundary layers due to the absence of governing parameter. Achieving a description of transitioning boundary layers in terms of dynamical systems was not a trivial generalisation of the concepts developed by former authors : the mathematical notion of turbulent attractor itself is formerly fragile, and the notion of bounded edge trajectories does not emerge unless a rescaling by the boundary layer thickness is considered [318]. For spatially developing boundary layer flows, it turns out that the notion of edge trajectory is more useful that the (asymptotic) notion of edge state. Consequently a finite-time description and formalism are better suited to the description of the bypass transition process, as adopted in the recent PhD manuscript of Miguel Beneitez [339].

### 5.1.2 Phenomenology depending on the scale of observation

The increase in computing power in the last decades has accompanied the recent progress in understanding transition to turbulence. In particular, locally well-resolved simulations of the governing equations are now routinely achieved. However the spatial extent of the flows under consideration is in computations always finite. This has lead to difficulties related to the interpretation of the computation results. Some confusion has inevitably taken place when considering minimal flow units (MFUs), in other words spatially periodic domains of smallest size able to accommodate non-trivial flow dynamics. Designed historically [28] as a numerical means to simulating the flow at a cheaper price and in isolation from the largest turbulent scales, the MFU concept has evolved into an autonomous paradigm of computational turbulence, independently of the experimental reality. In such small domains, the velocity field is fully cor-
related with itself at least in the spatial sense. This artificial constraint turns turbulence into mainstream temporal chaos, and the clean application of dynamical systems concepts proceeds as expected, despite the technical difficulties associated with continuous symmetries and high dimension. Although the limitations of this deterministic view have been frequently mentioned in the last decade, notably by Paul Manneville, a few concepts developed in that framework have proved robust to generalisation. This is the case in particular for edge states and minimal seeds: as the computation domain size is increased, these structures become progressively spatially localised. In the limit of infinitely large domains the localisation property becomes robust. Edge states and minimal seeds remain hence relevant to spatially unbounded experimental contexts, although their natural instability makes their experimental investigation arduous. Since edge states are equilibrium regimes, they are governed by a nonlinear process usually referred to as *self-sustaining process* (SSP). The first SSP was originally suggested by F. Waleffe and co-authors in the context of a periodic cell of plane Couette flow [9]. The present contribution demonstrates the multiplicity of SSPs, and the crucial differences between different SSPs, notably in the presence of localisation [130, 153, 193].

The possibility to change from one scale of observation to another one, simply by considering different spatial extents among the computational parameters, completes this description. For a given periodic domain (with a size fixed in outer units), reducing the driving Reynolds number makes the turbulent state feel increasingly the boundary conditions. Whatever the domain size, below a size-dependent value of *Re* the flow dynamics will manifest self-correlation effects due to the spatial periodicity. The consequence is usually a boundary crisis turning a sustained (although possibly intermittent) turbulent state into a chaotic regime with finite lifetimes, i.e. a chaotic saddle. This phenomenon, rather than a computational nuisance, should be seen as a handwavy explanation for the occurrence of local relaminarisation in shear flows. Similarly, increasing the domain size makes the apparition/disappearance of the turbulent state qualitatively closer to the concept of absorbing phase transitions. Important efforts have been devoted to the question of whether or not subcritical transition to turbulence obeys the universal class of directed percolation independently of finite-size effects. Although the emerging answer is predominantly positive for parallel flows, again this thesis has highlighted different scenarios specific to boundary layer cases. For parallel boundary layer flow cases a discontinuous transition seems to be the rule [1], because wall-localised structures happen to be ruled out by kinematic effects. For spatially developing boundary layer flows, the situation is more subtle. The non-homogeneity of the base flow in the streamwise direction implies a non-homogeneous spot nucleation in space (but homogenous in time). By contrast in parallel flows, usually described as IVPs, spots have to be nucleated at early times but with equal probability through the spatial domain. Although parallel flows seem to fall, in the thermodynamic limit, into the class of directed percolation, for spatially boundary layer flows this description of the nucleation rate leads to a dramatically different dynamics: the turbulent fraction increases like a sigmoid with the distance downstream, not like a power law. This new distinction allows one to shed light on formerly ambiguous results. In recent experiments of channel flow [228] for instance, turbulence was generated upstream using a honeycomb and left to decay by itself, and the departure from power-law in intermittency statistics was rapidly attributed to finite-size effects. The present point of view (based on an analysis of the nucleation rate both as a function of space and time) suggests a situation closer to the bypass case as in developing boundary layer flows, with intermittency statistics fully consistent with a sigmoid curve.

## 5.2 Outlooks

### 5.2.1 Patterning mechanism

Although a number of new elements has contributed to a better understanding of the transitional regime of wall-bounded shear flows, much remains to be understood. In particular the missing piece in the puzzle seems to concern the origin of the streamwise localisation of turbulent patches, from which much follows. Suggesting self-sustaining processes compatible with streamwise localisation is a necessary first step. However it does not necessarily indicate why streamwise localisation is the only stable option
for the turbulent flow below some value of the Reynolds number. Another related aspect is the mechanism present at the onset of laminar-turbulent patterning, before defects start to appear inside the patterns themselves. Recent simulations suggest that the laminar-turbulent structures are the most pronounced instances of a simpler amplitude modulation [244]. Does this modulation arise as an instability of the homogeneous turbulent flow? Can it even be compared qualitatively to a Turing instability mechanism as found in standard reaction-diffusion systems [296]? Or is a fully different picture needed, like the crystallisation picture recently suggested in Ref. [297]. Beyond the onset, the finite-amplitude patterns themselves keep some of their best secret unanswered. The data in chapter 3 has shown that the angle as well as the wavelength selection, and their dependence on the Reynolds number, are not trivial. The values determined from numerics correspond to a given numerical or experimental protocol, which suggests multistability, a feature already noted in Ref. [66]. It is yet not understood whether the range of possible wavelengths and angles obeys well-defined rules, and whether such rules can be deduced from first principles, either the Navier-Stokes equations or even model equations. The thesis of Pavan Kashyap, currently PhD student at LISN, will aim at answering some of these open questions.

### 5.2.2 Rough walls

One of the reference engineering plots in hydrodynamics is the Moody diagram, which features the energy loss in the flow due to mostly viscous effects. It is a well-known fact that at sufficiently high Reynolds numbers the "smooth" part of the $C_f$ curves is a poor description of the reality, and that the mean roughness of the walls needs to be taken into account. This observation naturally suggests that no study of transition to turbulence is complete without the eventual inclusion of wall roughness. The high-Reynolds number range where such effects manifest themselves calls for a modelling strategy. It is not fully understood how much roughness plays a role in the intermittent range of values of $Re$ where spatio-temporal intermittency is the dominant dynamics. To start with, there has not been any convincing dynamical systems approach to the rough case so far. Although the linear stability of the corresponding laminar flow can be envisioned, the recurrent issue of selecting the right computation domain size emerges again. Although recently a minimal flow unit framework was suggested for rough flows [340], the intrinsically disordered nature of the roughness landscape appears rather incompatible with the notion of 'minimal' domain.

A preliminary study of the patterning regime of plane Couette flow under the hypothesis of rough walls has appeared recently [341, 342]. It is based on a simple statistical model where the influence of the roughness elements amounts in average to a damping force, applied in the streamwise direction and dependent only on the distance from the wall. This study suggests that the transition phenomenon is hardly affected by weak roughness amplitudes of less than 10%, consistently with the fate of the high-$Re$ picture when considered at sufficient low $Re_τ$ [343]. For larger amplitudes exotic structures such as non-oblique stripes were predicted. These results call for a wider range of values of $Re$ of investigation, and possibly other flows, and compared with experimental and model-free numerical investigations. Another point of view on turbulence can be adopted by considering, rather than developed turbulent regimes, transients such as the growth of turbulent spots. This would be closer to the modelling approach adopted by Y. Pomeau [179] and later D. Barkley [180] based on front motion. Finally, "rough" wall geometries do not reduce to rigid solid walls that are not flat. A number of natural situations exists where knowledge of the fluid-structure interaction is crucial. They include non exhaustively vegetated canopies [344, 345], soft and even hairy surfaces [346].

### 5.2.3 Control of subcritical transition

In some circumstances (most trivially for drag reduction) one wishes the flow to remain laminar at all times. Since the laminar state is usually already linearly stable, the aim of control in such a case would be, rather than stabilising the laminar state even more, to either destabilise the turbulent state, or to reduce the size of its attraction basin. The robust presence of a saddle state at the root of the
laminar-turbulent distinction makes it ideal for nonlinear control strategies. The goal becomes simply to play on the parameters of the system such that a given initial condition finds itself on the other side of the boundary. This simple idea has been surprisingly under-exploited. A notable exception is the study by G. Kawahara [347] who managed to stabilise numerically a minimal flow unit of turbulent plane Couette flow by adding suitably quantified spanwise rotation. It is my personal opinion that simple instances of subcritical flow transition can open a new avenue of control strategies with an emphasis on nonlinear rather than linear properties. However, as often size matters : a successful algorithm in a minimal flow unit can fail in an arbitrary large domain. As seen in Chapter 3 the dynamical systems perspective becomes of a questionable interest when dealing with extended flow systems or experimental condition. Therefore an effective control strategy should be developed by taking this limitation into account. Recently, a simple methodology was suggested by Hof and co-workers [348] based on a mean flow perspective. A blunted mean flow profile is less efficient at supporting Waleffe's self-sustaining process [30], therefore any way to sufficiently blunt that mean flow should make relaminarisation effective. This method is heavily reminiscent of Large Eddy Break-up devices developed much earlier in (higher-$Re$) wind tunnels investigations (see e.g. Refs [349, 350]).

### 5.2.4 Non-Newtonian rheologies

The results presented in this thesis are limited to the context of Newtonian fluids. Basic evidence suggests that one of the simplest yet effective passive control strategies to reduce the drag is to change the fluid. This can be achieved efficiently by, in the case of liquids, adding polymers [351] whose concentration is a new parameter of the problem. There are well-known analytical closures in the literature for such rheologies such as the Fene-P model. The addition of an additional parameter (usually in the form of a Weissenberg number $W_i$) opens up a fully different zoology of states including the celebrated Maximum Drag Reduction state (see e.g. [352]) and the recently discovered regime of elasto-inertial turbulence [353]. Interestingly for the present purpose, the classification between edge and turbulent regimes put forward in Newtonian fluids also has its relevance in the presence of polymers at least for moderate values of $W_i$, mainly because the structure of the edge states make them poorly sensible to the polymers [354]. The situation for other rheologies remains almost entirely open as of now. The inclusion of surfactants (e.g. washing-up product into water), either added directly into the bulk of the fluid or inside gas bubbles fed into the flow, is also potentially interesting as another way of reducing drag (see e.g. Ref. [355]).

### 5.2.5 Towards more turbulent shear flows

Ultimately, like in the conclusions of most hydrodynamical studies I am expected to discuss the extension of the present theories to higher Reynolds number. Two questions arise for the present thesis. Firstly, how does the transition process evolve for higher values of $Re$? Secondly, now that we can pretend to understand the transient transition process, does it teach us anything new about the nature of the turbulent flow itself?

As for the first question, it was partially treated here in Chapter 2, where the dynamical systems picture can naturally be pushed to higher values of $Re$. Surprisingly, the transition process becomes simpler for larger $Re$ at least when the computational domain scales in outer units : finite lifetimes of the turbulent flow are too large to differ from the infinite lifetime phenomenology, edge states feature critical layers and their eigenspectrum simplifies too [107]. However it is not obvious that continuation in outer units is the most relevant option from a physical point of view [136]. It was also discussed how this picture evolves in arbitrarily long computational domains. For localised states only continuation in outer units makes full sense because inner units are not appropriate for localised flows. For outer unit continuation it was confirmed that the usual saddle point picture holds, however without any critical layer in physical space. The phenomenology of a rapid spot nucleation for high enough initial disturbance amplitude, based on the existence of a high-$Re$ localised edge state, is hence expected to be a robust picture.
As for the second question, the answer is less clear. The general trend in shear flows is that the edge regime is increasingly intermingled with the turbulent state as *Re* is lowered. Conversely, as *Re* increases, several features are worth remarking. The edge and the turbulent regime are increasingly separated both in state space and in energy levels. Although, loosely speaking they were born together in a saddle-node bifurcation, the edge and turbulent regimes are also increasingly different from the point of view of the temporal and spatial scales involved. As often speculated, a full dynamical systems description of the turbulent state at high *Re*, in terms of invariant sets and periodic solutions, seems vain: the state space gets increasingly complicated and high-dimensional. Besides the multi-scale and multi-frequency nature of the flow lends itself poorly to the identification of one-frequency solutions. If such solutions exist they will be both hard to find, and difficult to justify as templates for the turbulent dynamics [356]. The high-*Re* fate of edge states is, again, different. Edge states remain essentially large-scale structures (at odds with the initial framework of the 'near-wall turbulence' for which they were originally introduced). The only small scales supported by such solutions are found within critical layers, should they exist. However the self-sustaning process at the core of the edge state's existence stays the same as put forward in Ref. [108]. On the other hand, the self-sustained process associated with the turbulent flow, currently re-visited by several teams at higher *Re* [357-359], becomes increasingly complicated. That leads to an auxiliary question of, perhaps, an even deeper interest: are critical layers an important concept or a mathematical oddity? Can they be related successfully to the occurrence of uniform momentum zones in developed turbulent flows [360]? Or, more philosophically, to the occurrence of blow-up singularities and of a finite dissipation in infinite-*Re* turbulent flows [361]? It is not excluded that the state space concepts developed in simple shear flows, all revolving around the notion of unstable states, sheds some light on this mysterious and singular infinite-*Re* limit.
Acknowledgements

I would like to warmly thank all the members of the committee for accepting to take the time to read this thesis. More generally I would like to thank warmly all people who have at some point believed in my ability to contribute to science. If you recognise yourself in this description you are automatically thanked!

There are a few people who deserve personal acknowledgements because they played a special role in my evolution since the last thesis defended. Rich Kerswell supported my application for a post-doc with him in Bristol, and allowed me into the small circle of experts working on subcritical transition. It was a pleasure to work with Rich together with Ashley Willis and Chris Pringle. Once in the business I have to acknowledge the rare opportunity I had to spend time with recognised scientists and get personal tutoring from them. Looking back in retrospect I feel very lucky that I could spend so much time with Bruno Eckhardt, Predrag Cvitanović, Laurette Tuckerman, Dwight Barkley, Paul Manneville, Genta Kawahara and Olivier Dauchot, and learn so much formally and informally from them even without working on a common project. The story continued at KTH Mechanics in Stockholm for a second post-doctoral position. I owe truly a lot to Dan Henningson, Philipp Schlatter for the collaboration that started there and is still active and fruitful today. In general I would like all the nice people that I met at KTH Mechanics who (without knowing it) taught me to how to take pleasure in accomplishing productive work in team, among them Luca, Geert, Shervin, Espen, Antonios and many more.

Time came to apply for further positions and this is perhaps the time when you realise most who helped you for real. Thanks again to Alvaró Meseguer, to Bruno Eckhardt and to Lennaert van Veen, and to all anonymous committee members who have supported my various applications. Sorry for the positions that I did not take. In the end I chose to go back to my hometown Paris by joining CNRS, and I am still thankful to Patrick Le Quéré for having supported my CNRS application at LIMSI in 2009. I thank those at LIMSI who have shown some interest in my work and personality, and among them Claudine Delcarte, Caroline Nore, Olivier Le Maître, Luc Pastur, François Lusseyran, Laurent Martin Witkowski, Michel Pons, Lionel Mathelin, Onofrio Semeraro and Alessandro Bucci.

Other scientists deserve genuine acknowledgements. That includes in no particular order former colleagues from my office at LMFA in Ecully, Wouter Bos, Denis Martinand and Hatem Touil, Franciliens such as Eduardo Wesfreid, with whom I took pleasure in organising workshops, Romain Monchaux, Marie Couliou, Bérengère Dubrulle, François Daviaud, Olivier Dauchot, most people at ENSAM/Dynfluid, Joran Rolland, foreign colleagues such as Björn Hof, Marc Avila, Burak Budanur, Stefan Zammert, Divakar Viswanath, Peter Schmid, Grégoire Lemoult, Michael Schatz, Roman Grigoriev, Nigel Goldenfeld, Collecille Caulfied, Carlo Cossu, Cédric Beaume, Yongyun Hwang, Florence Marcotte, Thomas Boeck, all the French and Japanese participants to the France-Japan workshops that we organised (Genta Kawahara, Takahiro Tsukahara, Masaki Shimizu, Koji Fukudome, and their many students). I thank the whole community of applied mathematicians working on chimera states, notably Yuri Maistrenko, Oleh and Iryna Omel'chenko, Matthias Wolfrum, Eckehard Schöll, Arkady Pikovsky, Anna Zakharova, Laurent Larger and Bogdan Penkovskyi. Thank you for the invitations. I liked the people I met in this community and I
learned so much from this excursion outside the usual territories. And thank you to all the attendants to
all colloquia that I organised, in particular those who spread the word that Euromech 565 held in Cargèse
in 2014 was a success.

Funding was more than necessary to achieve all these projects. In almost chronological order I would
like to thank the European Commission for funding my first post-doc, Dan Henningson for funding not
only my second post-doc but also almost all subsequent projects done at KTH, Bruno Eckhardt and
the DFG for funding my stays in Marburg, Genta Kawahara, The French Ministry of Foreign affairs as
well as CNRS for funding in total 6 visits to Japan, The Airbus Group (formerly Fondation EADS) for
funding my first project at LIMSI, Euromech for funding the colloquium EC565 held in Cargèse, the
Agence National de la Recherche for funding the ETAE project led by Laurent Martin Witkowski, and
the French Ministry of Research and Higher Education for funding three theses under my supervision as
well as conferences and a few laptops.

Numerical simulations relies on numerical codes, and a large part of my research after the PhD was
based on original DNS codes developed by more skilled colleagues rather than by me. So this is another
opportunity to thank Ashley Willis, Philipp Schlatter, Shihe Xin, John F. Gibson, the developers of
Channelflow and Dedalus, as well as Yann Fraigneau for allowing me to use the codes developed by them.
Consistently with the need to provide more open-source codes to scientists, I did not try to claim that
these works are mine since I simply used them.

When it comes to the writing of this thesis, all the people who gave me constructive advices are
thanked. Those who tried to scare me or to tell me that I was too late are not thanked. Special thanks
go to Frédéric Alizard for giving me his LaTeX template.

Eventually, I have to thank individually all the students that contribute to this thesis since some of
the work contained here is in priority their work. Chronologically, these thanks go to Robin Larsson,
Ludomir Oteski, Taras Khapko, Takahiro Ishida, Shrayas Acharya Neelavara, Wang Zhe, Chaitanya Pa-
ranjape, Antoine Faugaret, Miguel Beneitez, Pavan Kashyap, as well as Kento Higashitsutsumi, Kohei
Kunii, Kazuki Takeda, Elyes Sleimi. Thank you and good luck for what follows. The other students that
I have supervised are also implicitly acknowledged.

Last but not least, I would like to dedicate these writings to my dear wife Elena, to my son Ilya and to
my grandfather Viorel Sergiescu, who inspired me as a theoretical physicist and is just about to celebrate
his 100th birthday. Merci encore une fois.
Bibliographie

[1] T. KHAPKO, P. SCHLATTER, Y. DUGUET et D. HENNINGSON, "Turbulence collapse in a suction boundary layer", *J. Fluid Mech.*, t. 795, p. 356-379, 2016.

[2] Y. HASEGAWA, M. QUADRIO et B. FROHNAPFEL, "Numerical simulation of turbulent duct flows with constant power input", *J. Fluid Mech.*, t. 750, p. 191-209, 2014.

[3] A. FAUGARET, Y. DUGUET, Y. FRAIGNEAU et L. M. WITKOWSKI, "Influence of interface pollution on the linear stability of a rotating flow", *J. Fluid Mech.*, t. 900, A42-1, 2020.

[4] A. LOZANO-DURÁN et J. JIMÉNEZ, "Effect of the computational domain on direct simulations of turbulent channels up to $re_τ=4200$", *Physics of Fluids*, t. 26, n° 1, p. 011702, 2014.

[5] Y. HWANG, "Statistical structure of self-sustaining attached eddies in turbulent channel flow", *Journal of Fluid Mechanics*, t. 767, p. 254-289, 2015.

[6] A. TOWNSEND, *The structure of turbulent shear flow*. Cambridge university press, 1980.

[7] Y. HWANG et Y. BENGANA, "Self-sustaining process of minimal attached eddies in turbulent channel flow", *J. Fluid Mech.*, t. 795, p. 708-738, 2016.

[8] A. J. SMITS, B. J. MCKEON et I. MARUSIC, "High-Reynolds number wall turbulence", *Annual Review of Fluid Mechanics*, t. 43, 2011.

[9] J. HAMILTON, J. KIM et F. WALEFFE, "Regeneration mechanisms of near-wall turbulence structures", *J. Fluid Mech.*, t. 287, p. 317-348, 1995.

[10] P ALFREDSSON et M. MATSUBARA, "Free-stream turbulence, streaky structures and transition in boundary layer flows", in *Fluids 2000 Conference and Exhibit*, 2000, p. 2534.

[11] N. NIKITIN et V. PIMANOV, "Numerical study of localized turbulent structures in a pipe", *Fluid Dynamics*, t. 50, n° 5, p. 655-664, 2015.

[12] D. MOXEY et D. BARKLEY, "Distinct large-scale turbulent-laminar states in transitional pipe flow", *Proceedings of the National Academy of Sciences*, t. 107, n° 18, p. 8091-8096, 2010.

[13] K. AVILA, D. MOXEY, A. de LOZAR, M. AVILA, D. BARKLEY et B. HOF, "The onset of turbulence in pipe flow", *Science*, t. 333, n° 6039, p. 192-196, 2011.

[14] P. V. KASHYAP, Y. DUGUET et O. DAUCHOT, "Flow statistics in the transitional regime of plane channel flow", *Entropy*, t. 22, n° 9, p. 1001, 2020.

[15] W. S. SARIC, H. L. REED et E. J. KERSCHEN, "Boundary-layer receptivity to freestream disturbances", *Annual Review of Fluid Mechanics*, t. 34, n° 1, p. 291-319, 2002.

[16] A. TOWNE, O. T. SCHMIDT et T. COLONIUS, "Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis", *J. Fluid Mech.*, t. 847, p. 821-867, 2018.

[17] O. DAUCHOT et P. MANNEVILLE, "Local versus global concepts in hydrodynamic stability theory", *J. Phys. II*, t. 7, n° 2, p. 371-389, 1997.

[18] P. SCHMID et D. HENNINGSON, *Stability and Transition in Shear Flows*. New York : Springer-Verlag, 2001.
[19] S. C. REDDY, P. J. SCHMID et D. S. HENNINGSON, "Pseudospectra of the Orr–Sommerfeld operator", *SIAM Journal on Applied Mathematics*, t. 53, n° 1, p. 15–47, 1993.

[20] S. A. ORSZAG, "Accurate solution of the Orr–Sommerfeld stability equation", *J. Fluid Mech.*, t. 50, n° 4, p. 689–703, 1971.

[21] L. N. TREFETHEN et M. EMBREE, *Spectra and pseudospectra : the behavior of nonnormal matrices and operators*. Princeton University Press, 2005.

[22] F. CHARRU, *Instabilités hydrodynamiques*. EDP Sciences, 2012.

[23] J. MOEHLIS, H. FAISST et B. ECKHARDT, "A low-dimensional model for turbulent shear flows", *New J. Phys.*, n° 56, p. 56, 2004.

[24] J. F. GIBSON, J. HALCROW et P. CVITANOVIĆ, "Visualizing the geometry of state space in plane Couette flow", *J. Fluid Mech.*, t. 611, p. 107–130, 2008.

[25] P. J. SCHMID, "Nonmodal stability theory", *Annu. Rev. Fluid Mech.*, t. 39, p. 129–162, 2007.

[26] D. HENNINGSON, "Comment on 'Transition in shear flows. Nonlinear normality versus non-normal linearity' [Phys. Fluids 7, 3060 (1995)]", *Physics of Fluids*, t. 8, n° 8, p. 2257–2258, 1996.

[27] F. WALEFFE, *Physics of Fluids*, n° 9, p. 883, 1997.

[28] J JIMÉNEZ et P MOIN, "The minimal flow unit in near-wall turbulence", *J. Fluid Mech.*, t. 225, p. 213–240, 1991.

[29] W SCHOPPA et F. HUSSAIN, "Coherent structure generation in near-wall turbulence", *J. Fluid Mech.*, t. 453, p. 57–108, 2002.

[30] F. WALEFFE, "On a self-sustaining process in shear flows", *Phys. Fluids*, t. 9, n° 4, p. 883–900, 1997.

[31] F. WALEFFE, *private communication*, 2008.

[32] G. KAWAHARA et S. KIDA, *J. Fluid Mech.*, n° 449, p. 291, 2001.

[33] W. WALEFFE, *J. Fluid Mech.*, n° 435, p. 93, 2001.

[34] H. FAISST et B. ECKHARDT, *Phys. Rev. Letters*, n° 91, p. 224502, 2003.

[35] H. WEDIN et R. KERSWELL, *J. Fluid Mech.*, n° 508, p. 333, 2004.

[36] M. NAGATA, *J. Fluid Mech.*, n° 217, p. 517, 1990.

[37] J. GIBSON, J. HALCROW et P. CVITANOVIĆ, "Visualizing the geometry of state space in plane Couette flow", *J. Fluid Mech.*, t. 611, p. 107, 2008.

[38] J. HALCROW, J. GIBSON, P. CVITANOVIĆ et D. VISWANATH, "Heteroclinic connections in plane Couette flow", *J. Fluid Mech.*, t. 621, p. 365, 2009.

[39] P. CVITANOVIĆ et J. GIBSON, "Geometry of the turbulence in wall-bounded shear flows : periodic orbits", *Physica Scripta*, t. 2010, n° T142, p. 014007, 2010.

[40] N. B. BUDANUR, P. CVITANOVIĆ, R. L. DAVIDCHACK et E. SIMINOS, "Reduction of SO(2) symmetry for spatially extended dynamical systems", *Phys. Rev. Lett.*, t. 114, n° 8, p. 084102, 2015.

[41] A. P. WILLIS, P. CVITANOVIĆ et M. AVILA, "Revealing the state space of turbulent pipe flow by symmetry reduction", *J. Fluid Mech.*, t. 721, p. 514–540, 2013.

[42] B. BUDANUR, K. SHORT, M. FARAZMAND, A. WILLIS et P. CVITANOVIĆ, "Relative periodic orbits form the backbone of turbulent pipe flow", *J. Fluid Mech.*, t. 833, n° 1, p. 274–301, 2017.

[43] S. BOTTIN, "Streamwise vortices as finite-amplitude solutions of the Navier–Stokes equations", *Physics of Fluids*, p. 143–155, 1998.

[44] B. HOF, C. W. VAN DOORNE, J. WESTERWEEL, F. T. NIEUWSTADT, H. FAISST, B. ECKHARDT, H. WEDIN, R. R. KERSWELL et F. WALEFFE, "Experimental observation of nonlinear traveling waves in turbulent pipe flow", *Science*, t. 305, n° 5690, p. 1594–1598, 2004.

[45] C. C. PRINGLE et R. R. KERSWELL, "Asymmetric, helical, and mirror-symmetric traveling waves in pipe flow", *Phys. Rev. Lett.*, t. 99, n° 7, p. 074502, 2007.
[46] A. DE LOZAR, F. MELLIBOVSKY, M. AVILA et B. HOF, "Edge state in pipe flow experiments", *Phys. Rev. Lett.*, t. 108, n° 21, p. 214502, 2012.

[47] T. DURIEZ, J.-L. AIDER et J. E. WESFREID, "Self-sustaining process through streak generation in a flat-plate boundary layer", *Phys. Rev. Lett.*, t. 103, n° 14, p. 144502, 2009.

[48] B. SURI, J. TITHOF, R. O. GRIGORIEV et M. F. SCHATZ, "Forecasting fluid flows using the geometry of turbulence", *Phys. Rev. Lett.*, t. 118, n° 11, p. 114501, 2017.

[49] B. SURI, L. KAGEORGE, R. O. GRIGORIEV et M. F. SCHATZ, "Capturing turbulent dynamics and statistics in experiments with unstable periodic orbits", *Phys. Rev. Lett.*, t. 125, n° 6, p. 064501, 2020.

[50] S. K. ROBINSON, "Coherent motions in the turbulent boundary layer", *Annual Review of Fluid Mechanics*, t. 23, n° 1, p. 601-639, 1991.

[51] P. CVITANOVIĆ, "Recurrent flows : the clockwork behind turbulence", *J. Fluid Mech.*, t. 726, p. 1-4, 2013.

[52] T. KREILOS, "Turbulence transition in shear flows and dynamical systems theory", thèse de doct., Univ. Philipps Marburg, 2014.

[53] P. CVITANOVIC, R. ARTUSO, R. MAINIERI, G. TANNER, G. VATTAY, N. WHELAN et A. WIRZBA, "Chaos : classical and quantum", *ChaosBook.org* (Niels Bohr Institute, Copenhagen 2005), t. 69, 2005.

[54] P. GASPARD, *Chaos, scattering and statistical mechanics*. Cambridge University Press, 2005, t. 9.

[55] P. CVITANOVIĆ, "Dynamical averaging in terms of periodic orbits", *Physica D : Nonlinear Phenomena*, t. 83, n° 1-3, p. 109-123, 1995.

[56] P. CVITANOVIC et B. ECKHARDT, "Periodic orbit expansions for classical smooth flows", *Journal of Physics A : Mathematical and General*, t. 24, n° 5, p. L237, 1991.

[57] B. ECKHARDT et G. OTT, "Periodic orbit analysis of the Lorenz attractor", *Zeitschrift für Physik B Condensed Matter*, t. 93, n° 2, p. 259-266, 1994.

[58] E. OTT, *Chaos in dynamical systems*. Cambridge university press, 2002.

[59] R. DEVANEY, *An introduction to chaotic dynamical systems*. CRC Press, 2018.

[60] B. R. HUNT et E. OTT, "Defining chaos", *Chaos : An Interdisciplinary Journal of Nonlinear Science*, t. 25, n° 9, p. 097618, 2015.

[61] G. KAWAHARA et S. KIDA, "Periodic motion embedded in plane Couette turbulence : regeneration cycle and burst", *J. Fluid Mech.*, t. 449, p. 291-300, 2001.

[62] A. DARBYSHIRE et T. MULLIN, "Transition to turbulence in constant-mass-flux pipe flow", *J. Fluid Mech.*, t. 289, p. 83-114, 1995.

[63] B. HOF, J. WESTERWEEL, T. M. SCHNEIDER et B. ECKHARDT, "Finite lifetime of turbulence in shear flows", *Nature*, t. 443, n° 7107, p. 59, 2006.

[64] A. P. WILLIS et R. R. KERSWELL, "Critical behavior in the relaminarization of localized turbulence in pipe flow", *Phys. Rev. Lett.*, t. 98, n° 1, p. 014501, 2007.

[65] S. BOTTIN et H. CHATÉ, "Statistical analysis of the transition to turbulence in plane Couette flow", *European Phys. J. B*, t. 6, n° 1, p. 143-155, 1998.

[66] Y. DUGUET, P. SCHLATTER et D. S. HENNINGSON, "Formation of turbulent patterns near the onset of transition in plane Couette flow", *J. Fluid Mech.*, t. 650, p. 119, 2010.

[67] T. TÉL et Y.-C. LAI, "Chaotic transients in spatially extended systems", *Physics Reports*, t. 460, n° 6, p. 245-275, 2008.

[68] P. GASPARD et S. A. RICE, "Scattering from a classically chaotic repellor", *The Journal of chemical physics*, t. 90, n° 4, p. 2225-2241, 1989.
[69] C JUNG, T TÉL et E ZIEMNIAK, "Application of scattering chaos to particle transport in a hydro-dynamical flow", *Chaos : An Interdisciplinary Journal of Nonlinear Science*, t. 3, n° 4, p. 555-568, 1993.

[70] L OTESKI, Y DUGUET et L. PASTUR, "Lagrangian chaos in confined two-dimensional oscillatory convection", *J. Fluid Mech.*, t. 759, p. 489, 2014.

[71] T. M. SCHNEIDER, B. ECKHARDT et J. VOLLMER, "Statistical analysis of coherent structures in transitional pipe flow", *Phys. Rev. E*, t. 75, n° 6, p. 066313, 2007.

[72] G. YALNIZ, B. HOF et N. B. BUDANUR, "Coarse graining the state space of a turbulent flow using periodic orbits", *Phys. Rev. Lett.*, t. 126, p. 244502, 2021.

[73] J. HALCROW, J. F. GIBSON, P. CVITANOVIC et D. VISWANATH, "Heteroclinic connections in plane Couette flow", t. 621, p. 365-376, 2009.

[74] M. FARANO, S. CHERUBINI, J.-C. ROBINET, P. DE PALMA et T. SCHNEIDER, "Computing heteroclinic orbits using adjoint-based methods", *J. Fluid Mech.*, t. 858, 2019.

[75] N. B. BUDANUR et B. HOF, "Heteroclinic path to spatially localized chaos in pipe flow", *J. Fluid Mech.*, t. 827, 2017.

[76] S. SMALE et al., "Differentiable dynamical systems", *Bulletin of the American mathematical Society*, t. 73, n° 6, p. 747-817, 1967.

[77] L. van VEEN et G. KAWAHARA, "Homoclinic tangle on the edge of shear turbulence", *Phys. Rev. Lett.*, t. 107, n° 11, p. 114501, 2011.

[78] J. R. T. LUSTRO, G. KAWAHARA, L. van VEEN, M. SHIMIZU et H. KOKUBU, "The onset of transient turbulence in minimal plane Couette flow", *J. Fluid Mech.*, t. 862, 2019.

[79] P. DOOHAN, A. P. WILLIS et Y. HWANG, "Two-scale interaction in near-wall turbulence", *Bulletin of the American Physical Society*, t. 64, 2019.

[80] L. N. TREFETHEN, A. E. TREFETHEN, S. C. REDDY et T. A. DRISCOLL, "Hydrodynamic stability without eigenvalues", *Science*, t. 261, n° 5121, p. 578-584, 1993.

[81] M. BENEITEZ, Y. DUGUET, P. SCHLATTER et D. S. HENNINGSON, "Edge manifold as a Lagrangian coherent structure in a high-dimensional state space", *Phys. Rev. Research*, t. 2, n° 3, p. 033258, 2020.

[82] H.-D. CHIANG et L. F. ALBERTO, *Stability regions of nonlinear dynamical systems : theory, estimation, and applications*. Cambridge University Press, 2015.

[83] G. HALLER, "Lagrangian coherent structures", *Annu. Rev. Fluid Mech.*, t. 47, n° 1, p. 137-162, 2015.

[84] G. HALLER et G. YUAN, "Lagrangian coherent structures and mixing in two-dimensional turbulence", *Physica D*, t. 147, n° 3, p. 352-370, 2000.

[85] T. PEACOCK et G. HALLER, "Lagrangian coherent structures : the hidden skeleton of fluid flows", *Phys. Today*, t. 66, n° 2, p. 41-47, 2013.

[86] A. JUNGINGER et R. HERNANDEZ, "Lagrangian descriptors in dissipative systems", *Phys. Chem. Chem. Phys.*, t. 18, n° 44, p. 30282-30287, 2016.

[87] G. HALLER, "Lagrangian coherent structures from approximate velocity data", *Phys. Fluids*, t. 14, n° 6, p. 1851-1861, 2002.

[88] A. HADJIGHASEM, M. FARAZMAND, D. BLAZEVSKI, G. FROYLAND et G. HALLER, "A critical comparison of Lagrangian methods for coherent structure detection", *Chaos*, t. 27, n° 5, p. 053104, 2017.

[89] G. HALLER et T. SAPSIS, "Lagrangian coherent structures and the smallest finite-time lyapunov exponent", *Chaos*, t. 21, n° 2, p. 023115, 2011.

[90] H. BABAEE et T. P. SAPSIS, "A minimization principle for the description of modes associated with finite-time instabilities", *Proceedings of the Royal Society A : Mathematical, Physical and Engineering Sciences*, t. 472, n° 2186, p. 20150779, 2016.
[91] H. BABAEE, M. FARAZMAND, G. HALLER et T. P. SAPSIS, "Reduced-order description of transient instabilities and computation of finite-time lyapunov exponents", *Chaos*, t. 27, n° 6, p. 063 103, 2017.

[92] C. MENDOZA et A. M. MANCHO, "Hidden geometry of ocean flows", *Phys. Rev. Lett.*, t. 105, n° 3, p. 038 501, 2010.

[93] A. M. MANCHO, S. WIGGINS, J. CURBELO et C. MENDOZA, "Lagrangian descriptors: a method for revealing phase space structures of general time dependent dynamical systems", *Commun. Nonlinear Sci.*, t. 18, n° 12, p. 3530–3557, 2013.

[94] C. LOPESINO, F. BALIBREA-INIESTA, V. J. GARCÍA-GARRIDO, S. WIGGINS et A. M. MANCHO, "A theoretical framework for Lagrangian descriptors", *Int. J. Bifurcat. Chaos*, t. 27, n° 01, p. 1 730 001, 2017.

[95] S. NAIK et S. WIGGINS, "Finding normally hyperbolic invariant manifolds in two and three degrees of freedom with hénon-heiles-type potential", *Phys. Rev. E*, t. 100, n° 2, p. 022 204, 2019.

[96] M. R. ALLSHOUSE et T. PEACOCK, "Lagrangian based methods for coherent structure detection", *Chaos*, t. 25, n° 9, p. 097 617, 2015.

[97] T. ITANO et S. TOH, "The dynamics of bursting process in wall turbulence", *Journal of the Physical Society of Japan*, t. 70, n° 3, p. 703–716, 2001.

[98] S. TOH et T. ITANO, "A periodic-like solution in channel flow", *J. Fluid Mech.*, t. 481, p. 67, 2003.

[99] S. ZAMMERT et B. ECKHARDT, "Periodically bursting edge states in plane Blasius flow", *Fluid Dynamics Research*, t. 46, n° 4, p. 041 419, 2014.

[100] B. ECKHARDT, T. M. SCHNEIDER, B. HOF et J. WESTERWEEL, "Turbulence transition in pipe flow", *Annu. Rev. Fluid Mech.*, t. 39, p. 447–468, 2007.

[101] H. E. NUSSE et J. A. YORKE, "A procedure for finding numerical trajectories on chaotic saddles", *Physica D*, t. 36, n° 1-2, p. 137–156, 1989.

[102] D. SWEET, H. E. NUSSE et J. A. YORKE, "Stagger-and-step method: detecting and computing chaotic saddles in higher dimensions", *Phys. Rev. Lett.*, t. 86, n° 11, p. 2261, 2001.

[103] E. M. BOLLT, "The path towards a longer life: on invariant sets and the escape time landscape", *Int. J. Bifurcat. Chaos*, t. 15, n° 05, p. 1615–1624, 2005.

[104] T. M. SCHNEIDER et B. ECKHARDT, "Edge of chaos in pipe flow", *Chaos*, t. 16, n° 4, p. 041 103, 2006.

[105] T. M. SCHNEIDER, B. ECKHARDT et J. A. YORKE, "Turbulence transition and the edge of chaos in pipe flow", *Phys. Rev. Lett.*, t. 99, n° 3, p. 034 502, 2007.

[106] Y. DUGUET, A. P. WILLIS et R. R. KERSWELL, "Transition in pipe flow: the saddle structure on the boundary of turbulence", *J. Fluid Mech.*, t. 613, p. 255–274, 2008.

[107] J. WANG, J. GIBSON et F. WALEFFE, "Lower branch coherent states in shear flows: transition and control", *Phys. Rev. Lett.*, t. 98, n° 20, p. 204 501, 2007.

[108] P. HALL et S. SHERWIN, "Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures", *J. Fluid Mech.*, t. 661, p. 178, 2010.

[109] P. HALL et F. T. SMITH, "On strongly nonlinear vortex/wave interactions in boundary-layer transition", 1989.

[110] F. SMITH et R. BODONYI, "Amplitude-dependent neutral modes in the hagen-Poiseuille flow through a circular pipe", *Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences*, t. 384, n° 1787, p. 463–489, 1982.

[111] M. NAGATA, "Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity", *J. Fluid Mech.*, t. 217, p. 519–527, 1990.

[112] D. OLVERA et R. R. KERSWELL, "Optimizing energy growth as a tool for finding exact coherent structures", *Phys. Rev. Fluids*, t. 2, n° 8, p. 083 902, 2017.