Nonlinear dynamics in transitional wall-bounded flows

Doctoral Thesis in Engineering Mechanics

Nonlinear dynamics in transitional wall-bounded flows

MIGUEL BENEITEZ

Stockholm, Sweden 2021

kth royal institute of technology

Nonlinear dynamics in transitional wall-bounded flows

MIGUEL BENEITEZ

Doctoral Thesis in Engineering Mechanics KTH Royal Institute of Technology Stockholm, Sweden 2021

Academic Dissertation which, with due permission of the KTH Royal Institute of Technology, is submitted for public defence for the Degree of Doctor of Philosophy on Friday the 4th June 2021, 10:00 a.m. in F1, Lindstedsvägen 26, Stockholm

© Miguel Beneitez Galan 978-91-7873-899-1 TRITA-SCI-FOU 2021:17

Printed by: Universitetsservice US-AB, Sweden 2021

Para mi madre, a quien recuerdo cada d´ıa.

“We shall not cease from exploration And the end of all our exploring Will be to arrive where we started And know the place for the first time.”

T. S. Eliot

Nonlinear dynamics in transitional wall-bounded shear flows

Miguel Beneitez

FLOW Centre, KTH Royal Institute of Technology, Department of Engineering Mechanics

SE–100 44 Stockholm, Sweden

Abstract

This thesis focuses on numerical studies of subcritical transition to turbulence in shear flows. The thesis employs a framework based on nonlinear dynamics in the subsequent studies. The geometrical approach to subcritical transition pivots the concepts of edge manifold and edge state. Such concepts are explored in detail in the Blasius boundary layer. The identified edge trajectory is chaotic and presents a couple of high- and low-speed streaks akin to those identified in other shear flows. For long enough times the linear instability of the Blasius boundary layer coexists with the bypass transition scenario. The edge is thus reinterpreted as a manifold separating both routes.

On the edge manifold of the Blasius boundary layer, the fully localised minimal seed is identified. The minimal seed experiences a sequence of linear mechanisms: the Orr mechanism followed by the lift-up. The resulting pertur- bation approaches the same region in state space as identified from arbitrary perturbations.

These insights from the edge trajectory identified in the Blasius boundary layer inspired a low-dimensional model. The model illustrates the e↵ect of the laminar attractor becoming linearly unstable and it agrees qualitatively with other recent studies in the literature.

The edge has been identified as a hyperbolic Lagrangian coherent structure of infinite dimension. We show how two Lagrangian diagnostics can be used to locate the edge directly in state space. This allows us to revisit edge tracking as a method optimising a Lagrangian diagnostic instead of a binary algorithm.

The two last studies of the thesis focus on the optimally time-dependent (OTD) modes as a basis for the linearised dynamics about a base flow with arbitrary time-dependence. The OTD modes are explored for a periodic flow in pulsating plane Poiseuille flow. The resulting OTD modes can be linked to the spectrum of the Orr-Sommerfeld operator. The results revealed perturbations which span more than one period of the base flow. Finally, the OTD framework is used on the edge trajectory starting from the minimal seed in the Blasius boundary layer.

Key words: transition to turbulence, boundary layer, direct numerical simula- tions, edge state, bypass transition, turbulence at the onset, laminar-turbulent coexistence, linear instability

Icke-linj¨ ar dynamik i v¨ agg-bunden str¨ omning

Miguel Beneitez

FLOW Centre, Kungliga Tekniska h¨ogskolan, Institutionen f¨or Mekanik SE-100 44 Stockholm, Sverige

Sammanfattning

Denna avhandling behandlar numeriska studier om subkristisk ¨overg˚ang till turbulens i skjuvfl¨oden. I avhandlingens p˚af¨oljande studier anv¨ands begrepp fr˚an dynamiska system. Avhandlingens f¨orsta studie behandlar the edge manifold i Blasius gr¨ansskikt. Edge trajectory identifieras genom anv¨andningen av sk.edge tracking.

Ber¨aknandet av the edge trajectory visar att den ¨ar kaotisk samtidigt som den visar p˚a ett par h¨og- och l˚aghastighetsstr˚ak som liknar de som identifi- erats i andra skjuvfl¨oden. Edge tracking under l¨angre perioder visade p˚a att den linj¨ara instabiliteten i Blasius gr¨ansskikt samexisterar med scenariot f¨or bypass¨overg˚ang.

Det helt lokaliserade minimal seed identifieras i Blasius gr¨ansskikt. Minimal seed genomg˚ar Orr-mekanismen, f¨oljt av lift-up mekanismen. Resultatet, i detta fall, st¨orningen, liknar en edge trajectory som k¨annetecknas fr˚an ett godtyckligt initialtillst˚and. Insikterna fr˚an the edge trajectory, identifierade i Blasius, inspirerade till en l˚agdimensionell modell som kvalitativt st¨ammer

¨overens med de senaste studierna d¨ar den lamin¨ara attraktorn blir linj¨art instabil.

Identifieringen av the edge manifold kan vara komplex. I denna avhand- ling identifierar vi the edge som en hyperbolic Lagrangian diagnostics, sam- manh¨angande struktur. Vi visar hur tv˚a Lagrangian diagnostics kan anv¨andas f¨or att lokalisera the edge direkt i fasrummet. Detta g¨or det m¨ojligt f¨or oss att se edge tracking som en metod f¨or att optimera en Lagrangian diagnostics, ist¨allet f¨or att betrakta det som en bin¨ar algoritm.

De tv˚a sista studierna i avhandlingen behandlar studier av de optimala tidsberoende moderna (OTD) som grund f¨or den linj¨ara dynamiken. OTD ¨ar utforskade f¨or periodiska fl¨oden i oscillerande plant Poiseuille fl¨ode. Resultatet, OTD moderna, kan kopplas ihop med Orr-Sommerfelds spektrat. Resultaten avsl¨ojade st¨orningar som str¨acker sig l¨angre ¨an en period i basfl¨odet.

Slutligen anv¨ands OTD modernas ramverk p˚a edge trajectory i gr¨ansskiktet i Blasius med utg˚angspunkt i the minimal seed. Avhandlingen lokaliserar och unders¨oker bakomliggande optimala tidsberoende moder samtidigt som meka- niskmen bakom instabiliten utforskas.

Nyckelord: ¨overg˚ang till turbulens, gr¨ansskikt, direkta numeriska simulering- ar, edge state, bypass transition, lamin¨ar och turbulent samexistens, linj¨ar instabilitet

vi

Preface

This thesis deals with an approach to subcritical transition based on nonlinear dynamics. A brief introduction on the basic concepts and methods is presented in the first part. The second part contains six papers. The papers are adjusted to comply with the present thesis format for consistency, but their contents have not been altered as compared with their original counterparts.

Paper 1. M. Beneitez, Y. Duguet, P. Schlatter & D. S. Henningson, 2019. Edge tracking in spatially developing boundary layer flows. J. Fluid Mech.

881, 164–181.

Paper 2. C. Vavaliaris, M. Beneitez & D. S. Henningson, 2020. Optimal perturbations and transition energy thresholds in boundary layer shear flows.

Phys. Rev. Fluids 5, 062401(R).

Paper 3. M. Beneitez, Y. Duguet, P.Schlatter & D. S. Henningson, 2020. Edge manifold as a Lagrangian coherent structure in a high-dimensional state space. Phys. Rev. Research 2, 033258.

Paper 4. M. Beneitez, Y. Duguet, P.Schlatter & D. S. Henningson, 2020. Modeling the collapse of the edge when two transition routes compete.

Phys. Rev. E 102, 053108.

Paper 5. J. S. Kern, M. Beneitez, A. Hanifi & D. S. Henningson.

Transient linear stability of pulsating Poiseuille flow using optimally time- dependent modes. Submitted to J. Fluid Mech.

Paper 6. M. Beneitez, Y. Duguet, P.Schlatter & D. S. Henning- son. Finite-time stability of an edge trajectory in the Blasius boundary layer.

Manuscript.

June 2021, Stockholm Miguel Beneitez

Division of work between authors

The main advisor for the project is Dan Henningson (DH). Philipp Schlatter (PS) and Yohann Duguet (YD) act as co-advisors.

Paper 1. The moving box code has been developed by Miguel Beneitez (MB) with input from PS and YD. The computations were performed by MB with input from PS and YD. The paper has been written by MB with input from YD, PS and DH. The research was designed by all authors.

Paper 2. The Nek5000 implemented of the Blasius code has been developed by Christos Vavaliaris (CV) and MB building on the nonlinear adjoint code by Jacopo Canton and Enrico Rinaldi. The paper has been written by CV with input from MB and DH. The research was designed by all authors.

Paper 3. The OTD/LD implementation code has been developed by MB with input from PS and YD. The computations were done by MB with input from YD. The paper has been written by MB with input from YD, PS and DH. The research was designed by all authors.

Paper 4. The model has been developed by MB, YD and DH. The analysis of the model has been done by MB with input from YD. The paper has been written by MB with input from YD and DH. The research was designed by all authors.

Paper 5. The Nek5000 implementation of the OTD modes has been developed by Simon Kern (SK) with input from Prabal Negi and MB. The Matlab code was originally developed by MB and the improvements for the oscillating flow were done by SK. The simulations were done by SK with input from MB, AH and DH. The paper has been written by SK with input from MB, AH and DH.

The research was designed by all authors.

Paper 6. The OTD computations in Blasius have been performed by MB with input from PS and YD. The paper has been written by MB with input from YD, PS and DH. The research was designed by all authors.

Conferences

Part of the work in this thesis has been presented at the following international conferences. The presenting author is underlined.

M. Beneitez, Y. Duguet, P. Schlatter & D. S. Henningson. The edge of chaos as a Lagrangian Coherent Structure. 73^rdAnnual Meeting of the APS Division of Fluid Dynamics. Chicago, USA (Online), December 2020.

M. Beneitez, Y. Duguet, P. Schlatter & D. S. Henningson. What can we learn from the Edge about bypass transition?. European Turbulence Conference (ETC 17). Torino, Italy, September 2019.

C. Vavaliaris, M. Beneitez & D. S. Henningson. Optimal initial perturba- tions and the minimal seed of Blasius boundary-layer flow. European Turbulence Conference (ETC 17). Torino, Italy, September 2019.

viii

M. Beneitez, Y. Duguet, P. Schlatter & D. S. Henningson. What can we learn from the Edge about bypass transition?. IUTAM Transition. London, U.K., September 2019.

M. Beneitez, Y. Duguet, P. Schlatter & D. S. Henningson. The edge for boundary layer flows. Euromech Colloquium 598: Coherent structures in wall-bounded turbulence: new directions in a classical problem. London, U.K., August 2018.

Contents

Abstract v

Sammanfattning vi

Preface vii

Part I - Overview and summary

Chapter 1. Introduction 1

Chapter 2. Basic concepts 3

2.1. The incompressible Navier-Stokes equations 3

2.2. Flow configurations 3

2.3. Dynamical systems basics 5

2.4. Hydrodynamic stability and transition 7

2.5. Numerical methods 11

Chapter 3. The edge manifold 13

3.1. Hypersurface in state space 13

3.2. Edge states and the minimal seed 14

3.3. The edge in boundary layer flows 16

3.4. Geometry of the edge manifold 21

3.5. The edge as a Lagrangian coherent structure 25

Chapter 4. Basis for the linearised dynamics 30

4.1. Optimally time-dependent modes 30

4.2. Pulsating channel flow 33

4.3. Stability of a finite-time edge trajectory in Blasius 36

Chapter 5. Conclusions and outlook 38

Outlooks and future work 39

Acknowledgements 41

Bibliography 42

xii

Part II - Papers

Summary of the papers 51

Paper 1. Edge tracking in spatially developing boundary

layer flows 53

Paper 2. Optimal perturbations and transition

energy thresholds in boundary layer shear flows 77 Paper 3. Edge manifold as a Lagrangian coherent structure

in a high-dimensional state space 91

Paper 4. Modeling the collapse of the edge

when two transition routes compete 123

Paper 5. Transient linear stability of pulsating

Poiseuille flow using optimally time-dependent modes 147 Paper 6. Finite-time stability of an edge trajectory

in the Blasius boundary layer 193

Part I

Overview and summary

Chapter 1

Introduction

Fluid mechanics is a field of classical physics that has puzzled mathematicians, physicists and engineers for centuries, ever since buoyancy was explained by Archimedes in ancient Greece. Fluids are ubiquitous in nature, they surround us –as air or water–, are present in a myriad of industrial applications –aerodynamics,

turbomachinery or paper production just to mention a few examples– and flow through our bodies. The amount of scales considered in fluid dynamics is astonishing: from microscopic fluid application to the size of accretion disks in astrophysics. The motion of fluids is fascinating and improving our understanding on the topic is fundamental. However, curiosity is not the only motivation to dive into the study of fluid mechanics. Now, more than ever, we should aim to achieve the Sustainable Development Goals and find relevant engineering solutions within fluid mechanics to the existing challenges.

Figure 1.1: Early times sketch of a turbulent flow from Leonardo da Vinci’s Codex Leicester (1507-1509).

The motion of fluids can be deemed as laminar, characterised by ordered motion which is relatively simple. In the most simple cases, we can even describe the motion of fluids with an analytical expression. Radically di↵erent are turbulent flows in which the motion is chaotic and several temporal and di↵erent spatial scales interact. This was reported as early as the time of

2 1. Introduction

Leonardo da Vinci, as shown in a sketch from his Codex Leicester in figure 1.1.

da Vinci’s seminal work related to fluid mechanics has been recently brought into the limelight by Marusic & Broomhall (2021).

The transition between the two regimes, where the flow is initially lami- nar and becomes unstable, is filled with captivating questions. Studying the transition regime has goals such as triggering transition to increase mixing, or delaying it to reduce the friction coefficient on an aircraft. The initial stages of the transition to turbulence can be studied by looking at infinitesimal perturbations to laminar flow states, which is the subject of linear stability analysis. However, experimental evidence has shown that certain flows which remain laminar to infinitesimal perturbations can instead become unstable to perturbations of finite-amplitude (Reynolds 1883). The framework considered in the present thesis for the study of finite-amplitude perturbations is that of nonlinear dynamics.

Nonlinear dynamics, and in particular chaos theory, has experienced a tremendous progress since the time of Poincar´e with the coming of the computer age. Applied to fluid mechanics, it provides a geometrical understanding of the di↵erent possible states of the flow. Recent progress since Nagata (1990) has identified several families of steady unstable flow states which are more complex than the trivial laminar states, without being turbulent. Among these states we focus on the ones which are on the verge of turbulence (Itano & Toh 2001).

An example of the power of such a geometrical approach to study flows can be found in Gibson et al. (2008).

As stated, fluid systems can take a wide number of forms. Among the flow systems, the study of archetypal flow configurations gives us the chance to isolate the factors which might a↵ect the result of any experiment. The study of these canonical flows, such as the flow over a flat plate, is still relevant in many industrial applications and is often the first step towards industrial design.

This thesis consists on numerical simulations of transitional flows due to finite-amplitude perturbations on canonical configurations. The computational studies in this thesis are complemented with physics-inspired low-order modelling to shed light into the relevant phenomena. The flow cases treated in the thesis, albeit geometrically simple, present unique challenges.

Thesis structure. Part I of the thesis continues with chapter 2, which in- troduces the relevant basic concepts in fluid mechanics and links them with dynamical systems theory. Chapter 3 deepens the concept of edge manifold and the coherent states within. In chapter 4 the OTD modes are introduced as a basis for the linearised dynamics around a reference trajectory. The overview of the thesis concludes in chapter 5. Part II of the thesis collects the original results in the form of papers.

Chapter 2

Basic concepts

2.1. The incompressible Navier-Stokes equations

The present thesis focuses on solutions to the incompressible Navier–Stokes equations for Newtonian fluids with homogeneous and time-invariant properties.

Newtonian fluids are those in which the fluid particle’s stress tensor is linearly related to the local strain rate at all points at every time instant. The equations of motion read,

@U

@t + U· rU = 1

⇢rp + ⌫r²U + f , (2.1)

r · U = 0, (2.2)

where U is a three-dimensional velocity field, p is the pressure and f is a volume forcing acting on the fluid. ⇢ is the density of the fluid and ⌫ is the dynamic viscosity, ⌫ = µ/⇢, where µ is the kinematic viscosity, and t denotes time. A detailed derivation of the Navier-Stokes equations can be found, for example, in the book by Batchelor (2000). Equations (2.1) and (2.2) are prescribed on a physical domain with coordinates (x, y, z), corresponding to the streamwise, wall-normal and spanwise directions respectively. The velocity field U has components (U, V, W ) in the corresponding directions. The Navier-Stokes equations can be made non-dimensional using the appropriate characteristic scales: U^⇤, L^⇤, the velocity and length scales, and the reference fluid properties µ^⇤and ⇢^⇤. The Reynolds number is then defined as Re = U^⇤L^⇤⇢^⇤/µ^⇤and acts as the control parameter of the equations. It represents the relative importance of the inertial forces with respect to the viscous forces.

2.2. Flow configurations

The so-called canonical flows, have a very low geometrical complexity, and simple boundary conditions. They are steady (@t = 0) and most of them are analytical, two-dimensional (w = 0 and @z = 0) solutions to equation (2.1) and (2.2). Firstly, we can consider fully developed parallel flows (@x = 0) between two infinite walls. These are parallel flows where the velocity profile only depends on y. The equations can be made non-dimensional with respect to the characteristic velocity U^⇤and half the distance between the walls h. The

4 2. Basic concepts

Figure 2.1: Sketch depicting the Blasius boundary layer away from the leading edge.

domain considered is y2 [ 1, 1]. The equations motion then reduce to:

1 Re

@²U

@y² = V@U

@y + @p

@x, (2.3)

@p

@y = 0, (2.4)

v = const., (2.5)

together with the appropriate boundary conditions. The foremost canonical flows between two plates are: plane Couette flow, the flow between two plates moving at constant speed in opposite directions. U^⇤ in this case is half the velocity di↵erence between the walls. Equations (2.1) and (2.2) are completed with the boundary conditions

U (y = 1) = 1, U (y = 1) = 1, (2.6)

which results in a linear velocity profile of the shape

U = y, (2.7)

and V = 0. Plane Poiseuille flow is the flow driven by a constant pressure gradient @xp = const. The characteristic velocity U^⇤is the centerline velocity.

This yields a parabolic profile of the form

U = 1 y², , (2.8)

and V = 0. The asymptotic suction boundary layer, is an open flow on which suction is applied at the wall. U^⇤corresponds to the free stream velocity, U¹, far away from the wall. They domain considered is y 2 [0, 1). It accepts an

2.3. Dynamical systems basics 5

analytical solution of the form,

U = 1 exp( yReVs/U₁), (2.9)

V = Vs/U₁, (2.10)

where V_s > 0 is the suction velocity, and y 2 [0, 1). Other canonical flow may consist of simple variations on the previous ones, such as the oscillating Poiseuille flow, which is the subject of Paper 5.

We consider now the flow above a stationary semi-infinite flat plate as an archetype spatially developing flows bounded only by one wall. This corresponds to the Blasius boundary layer. Although it is a canonical flow it presents an additional degree of complexity since the flow is varying in the x direction. The equations of motion for the Blasius boundary layer read,

U@U

@x + V@U

@y = 1 Re

@²U

@y², (2.11)

@p

@y = 0, (2.12)

where @_xU6= 0. A solution to this flow was proposed by Blasius (1907). The solution sufficiently far away from the leading edge is self-similar and steady.

No analytical solution is available in the Blasius boundary layer, however a numerical solution can be easily computed. The Blasius boundary layer is the subject of Papers 1, 2 and 6.

Perturbations can be considered about these flows as q, a high-dimensional vector containing all the degrees of freedom for the velocity and pressure perturbations about the canonical solutions above. The concepts from stability and dynamical systems in the following sections can be written in term of the perturbations q, so that the trivial solution q = 0 corresponds to the canonical solutions given above.

2.3. Dynamical systems basics

This section aims to cover the basic ideas used in the rest of the thesis from the field of dynamical systems. A comprehensive introduction to the subject can be found, for example, in the book by Strogatz (2001).

A dynamical system consists of n ordinary di↵erential equations (ODEs) and can be written in compact form as

˙x = f (x, t) t2 (t0, t0+ ⌧ ) (2.13) where x2 Rⁿis the state of the n-dimensional system, f (x, t) :Rⁿ⇥(t⁰, t0+ ⌧ )! Rⁿis a dynamical rule that specifies the immediate future of the state variables.

Considering a dynamical system in (2.13). We denoteTt(x₀) a trajectory in state space, where x0corresponds to the initial condition. Let F_t^t₀ be the flow map

F_t^t₀ :Rⁿ! Rⁿ

x0! x(t; t0, x0), (2.14)

6 2. Basic concepts

(a) (b) (c)

Figure 2.2: (a) Sketch of a steady state (plane Couette flow) as a stable fixed point (blue dot) towards which trajectories approach (blue lines). (b) Representation of the a stable limit circle solution (thick blue line) around an unstable fixed point (red dot) in the bent pipe. (c) Representation of a turbulent trajectory in a turbulent wing as a chaotic attractor (red lines). Neighbouring trajectories (blue lines) get attracted into the chaotic set.

which maps an initial position x0 at time t0 to its position at time t. The equation describing the linearised dynamics around this trajectory reads

˙q = L(Tt, t)q, L(Tt, t) =rf(x, t), (2.15) where L(Tt, t) is the linearised operator about the trajectoryTt(x0). We consider a propagator which evolves the initial condition of a perturbation q0, around the trajectoryTt as the ordered product of infinitesimal propagators

q(t) = ^t_t0q(t₀), ^t_t0= lim

t!0e^{L(Tt,tj) t}, (2.16) We will consider only deterministic dynamical systems. A system is de- terministic if there is a unique consequent to every state. As an example, a simple pendulum can be considered a dynamical system inR²where the state variables x = (✓, !) are its angle from the vertical ✓ and its angular velocity

!. The evolution rule f (x, t) for the pendulum can be derived from Newton’s second law F = ma.

Dissipative dynamical systems, such as the Navier–Stokes equations, fea- ture attractors. The solutions to the dynamical system (2.13) can undergo bifurcations which is a qualitative change in the dynamics of the system as the Reynolds number changes. Bifurcations are defined by a non-invertible Jacobian L(Tt, t). Bifurcations can be either supercritical, in which the evolution of the solutions is continuous, i.e. a small change in the control parameter results in a steady solution becoming unstable. Another attracting solution grows then from an infinitesimal amplitude. Bifurcations may also be subcritical, where the evolution of the solutions is discontinuous, i.e. a small variation of the control parameter causes the state to go from a solution to di↵erent one in a non-continuous way. In supercritical bifurcations bringing the control parameter

2.4. Hydrodynamic stability and transition 7

(a) (b)

Figure 2.3: Schematic bifurcation diagram of the amplitude of a perturbation A with respect to the control parameter Re. (a) supercritical bifurcation. (b) subcritical bifurcation. Arrows indicate which solution trajectories are attracted to. Discontinuous lines indicate unstable solutions.

back to its previous value results in the original solution becoming attracting again. The same does not happen in subcritical bifurcations and hysteresis is present (see figure 2.3 for a schematic representation).

2.4. Hydrodynamic stability and transition

Hydrodynamic stability is the area of fluid mechanics dealing with the early stages of the mechanisms causing a laminar flow to transition to turbulence.

Reference work in the field can be found in the books by Drazin & Reid (2004) and Schmid & Henningson (2001).

2.4.1. Dynamical systems and hydrodynamic stability

The dynamical systems approach can be applied to fluid systems with the following considerations; In practice the solution of the Navier–Stokes equations can be obtained numerically by discretising equations (2.1) and (2.2). This can be done by a Galerkin projection of the Navier–Stokes equations into a set of [Nx, Ny, Nz] complete functions. The discretised Navier–Stokes equations in the velocity formulation form a system of n = N_x⇥ Ny⇥ Nz⇥ 3 ODEs, where the factor of 3 accounts for the three velocity components which describe the flow field at any instant. Note, that in the case of discretising the Navier–Stokes equations in the velocity-vorticity formulation, the state of the system has a dimension of n = Nx⇥ N^y⇥ N^z⇥ 2, since only 2 components are necessary to describe the system at any time instant. The dynamical rule f (t) is here the right hand side of the discretised Navier–Stokes equations.

The time-evolution of an initial condition U₀by equation (2.1) and (2.2) completed with boundary conditions corresponds to a trajectory in the state space. This suggests a geometrical view of the dynamics; a steady state, e.g.

the laminar plane Couette flow, would correspond to a fixed point in state space. A time-periodic solution, such as the pulsating Poiseuille flow, would correspond to a closed line in state space. A chaotic solution, such as the turbulent state would correspond to a complex non-periodic object in state

8 2. Basic concepts

space. All these objects are labelled as stable or unstable depending on whether infinitesimal perturbations around them decay or grow asymptotically, whereas a case which does not grow nor decay would be labelled neutrally stable (see figure 2.2). Recent reviews of the transition to turbulence from a nonlinear dynamics point of view in wall-bounded flows can be found in Eckhardt et al.

(2007) and Manneville (2016).

2.4.2. Supercritical transition

Supercritical transition is concerned with the stability of flow states to infinites- imal perturbations and nonlinear saturation. Restricting our description to the canonical flows mentioned above, we consider the Reynolds number as the main control parameter for the Navier–Stokes equations. The description of the transition process is as follows:

For low Re there exists only one steady and stable solution, corresponding to the analytical (or self-similar solution) above. In this case the solution corresponds to a stable fixed point which is a global attractor, meaning that trajectories starting from any point on the state space will approach this solution.

Physically it means that any perturbation to the steady state will, eventually, decay due to viscous di↵usion.

Increasing Re, the initially steady state may become unstable to infinitesimal perturbations, i.e. linearly unstable, and a new solution (or more than one) appears. A bifurcation has then taken place and this new solution can be either a fixed point or a periodic orbit. Typically fluid systems that become turbulent in a supercritical way follow a Ruelle-Takens-Newhouse route to turbulence (Ruelle

& Takens (1971); Newhouse et al. (1978)), experience Pomeau-Manneville intermittency (Pomeau & Manneville 1980), or undergo a period-doubling cascade (Feigenbaum (1980); Cvitanovic (1989)).

2.4.3. Subcritical transition

The supercritical route to turbulence, in which initially a state becomes linearly unstable to infinitesimal perturbations, is not the only possibility. Many flows undergo transition to turbulence at a value of Re much lower than the critical one predicted by linear stability theory (such as plane Poiseuille flow), or even when the flow is linearly stable for all Re (as straight pipe flow). Whenever transition takes place below Re_cit receives the name of subcritical transition.

Starting from the same scenario as above where for low enough Re only a stable steady state exists, as the Re is increased new branches of solutions may appear in a saddle node bifurcation and coexist with the stable steady state.

These branches are not accessible by perturbing the reference stable state with infinitesimal perturbations. The non-trivial solutions are instead accessed by finite amplitude perturbations.

In a very simplified description, the new upper branch will eventually give birth to the turbulent state, while the lower branch corresponds to an unstable state. A separatrix in state space between the trajectories which approach

2.4. Hydrodynamic stability and transition 9

Turbulent Flow

Laminar Flow Edge State

Edge M anif

old

(a) (b)

Figure 2.4: (a) Phase portrait the 2D Dauchot and Manneville model. Two trajectories approaching T and L are shown in orange respective blue. E is the edge state of the system (b) Schematic representation of the dynamical systems view of subcritical transition.

the upper branch solutions and those approaching the steady solution can be hypothesised. This separatrix receives the name of the edge manifold (Skufca et al. 2006; Schneider et al. 2007), and it is a codimension one hypersurface in state space. It is codimension one, since there is only one dimension pointing out of the hypersurface. The unstable solutions from the lower branch which live on this separatrix receive the name of edge states. A deeper discussion and overview of the relevance of the edge manifold can be found in chapter 3 of the thesis.

A very illustrative work on the fundamental di↵erence between subcritical and supercritical transition can be found in the paper by Dauchot & Manneville (1997). They present a two-dimensional model of perturbations over a steady state with a control parameter akin to the Reynolds number. The model reads,

dx₁

dt = s1x1+ x2+ x1x2 (2.17) dx2

dt = s₂x₂ x²₁, (2.18)

where x1and x2are the variables of the model, and s1and s2are two parameters of the model. The work by Trefethen et al. (1993) serves as inspiration for the linear terms. The control parameter of the system is = 1 4s1s2. For s₁ = 0.1875 and s₂= 1, so that = 1.75 the system is bistable, and it has 3 equilibrium solutions which correspond to fixed points. One of them is the trivial state L = (0, 0), and two fixed points appear from a saddle-node bifurcation, E and T given analytically,

E =

✓1

2( 1 +p ), 1

4s2

( 1 +p )²

◆

, (2.19)

10 2. Basic concepts

Figure 2.5: Sketch summarising the di↵erent routes to turbulence by Morkovin (1994)

and

T =

✓1

2( 1 p ), 1

4s2

( 1 p

)²

◆

. (2.20)

The states L and T can be associated with the laminar and turbulent states in a fluid system, while E is an edge state. A state portrait for the parameters s1= 0.1875 and s2= 1 can be found in figure 2.4. The stability properties of the di↵erent fixed points can be studied by means of their Jacobian matrix.

L and T have both stable eigenvalues, and they are the attractors of the system.

E has a positive and a negative eigenvalue, it is a saddle point and the edge state of the system. Figure 2.4 (a) illustrates two trajectories which start on di↵erent sides of the edge manifold and approach the edge state during a finite time before going towards the L or T attractors. Parallels can be drawn in fluid systems as illustrated by figure 2.4 (b). The next chapter provides one with a deeper discussion of the concepts of edge manifold and edge states.

2.4.4. Natural and bypass transition in boundary layers

Boundary layers can become turbulent in many ways. We consider the scenario based on incoming perturbations from outside the boundary layer, as illustrated in figure 2.5. Two main transition scenarios are possible. For low levels of incoming perturbations, which correspond e.g. to flight conditions, natural transition occurs. Linear stability theory predicts a linear instability of the flow to two-dimensional spanwise-invariant Tollmien-Schlichting (TS) waves followed by their destabilization (Boiko et al. (1994); Schmid & Henningson (2001); Schlichting & Gersten (2016)).

2.5. Numerical methods 11

x z

Figure 2.6: Top view of DNS simulations of transitional Blasius boundary layer.

Contours of the perturbation velocity (red and blue) and the 2criterion for vortex identification (green) (a) bypass transition and (b) classical transition.

Flow from left to right.

In contrast, higher incoming disturbance levels (typically above 2% of the free stream velocity) trigger bypass transition. This transition scenario, characteristic in turbomachinery, features streaks (streamwise velocity defects) sustained via streamwise vorticity and the lift-up mechanisms (Klebano↵ 1955;

Morkovin 1969). The streaks then experiment secondary instabilities (Henning- son et al. 1993; Brandt et al. 2004; Schlatter et al. 2008; Vaughan & Zaki 2011).

The secondary instability of the streaks will cause their breakdown, originating turbulent spots that invade the flow (Matsubara & Alfredsson (2001); Kreilos et al. (2016b)). The two transition scenarios in a Blasius boundary layer can be seen in figure 2.6 above for bypass transition and below for natural TS wave transition.

2.5. Numerical methods

The simulations of the Navier–Stokes equations in Papers 1, 3, 4 and 6, and partly in paper 2, were performed with the in-house code SIMSON (Chevalier et al. 2007). It is a pseudo-spectral code, in which the solution is expanded into Nx and Nz Fourier modes in the stream- and spanwise directions respectively.

Ny Chebyshev modes are considered in the wall-normal direction, where the Chebyshev tau method is used. The Navier–Stokes equations are solved in the velocity-vorticity formulation. The solution is advanced in time using a second-order Crank–Nicolson method for the linear terms and a low-storage fourth-order Runge–Kutta method for the nonlinear terms. The computation of the advection term is performed in physical space following the 3/2-rule for dealiasing.

In order to ensure the periodicity in the streamwise direction when simulat- ing spatially developing flows a fringe region is added in the physical domain, where fluctuations are removed and the flow is forced to adapt to the initial condition. The fringe is implemented in the form of a forcing in the right-hand side of the Navier–Stokes equations. The forcing reads F = (x)(uref u), where u_ref is the desired reference solution. (x) is the fringe function as described in Chevalier et al. (2007). The code used in this thesis is parallelised using Message Passing Interface (MPI) in the x and z directions (Li et al. 2009).

12 2. Basic concepts

The other large scale numerical code used in the present thesis is Nek5000 (Fischer et al. 2008). It is a high-performance open-source CFD code which solves the Navier–Stokes equations using the spectral element method developed by Patera (1984). The particularity of spectral element methods is that the computational domain is discretised using quadrilateral or hexahedral elements.

In each element the velocity is represented through polynomials of order N constructed through the Gauss–Lobato–Legendre (GLL) points, while the pres- sure uses a polynomial of order N 2. The nonlinear term is treated explicitly by third order extrapolation while the linear terms are treated implicitly. The solution is advanced in time using a third-order backward-di↵erentiation algo- rithm. The divergence-free condition needs to be enforced, which is done via a third order pressure correction scheme. Nek5000 is partially used to obtain the results in paper 2 and it is the only numerical code in paper 5.

Chapter 3

The edge manifold

3.1. Hypersurface in state space

As briefly introduced in chapter 2.3, the edge manifold is a codimension one hypersurface separating trajectories which approach either the laminar or the turbulent attractor. The edge has been a central concept in the dynamical systems approach to subcritical transition in the last decade.

The concept of edge was formally introduced by Skufca et al. (2006), however the first study on the topic was performed by Itano & Toh (2001) where the laminar-turbulent separatrix was identified albeit it was not yet named edge.

Skufca et al. (2006) studied a Galerkin reduced-order model of a parallel flow, where a laminar stable state and a transient turbulent state coexist. Based on the lifetime of the trajectories, namely the time it takes for a trajectory to reach the laminar attractor within a threshold, Skufca et al. (2006) were able to identify the structure in state space separating the trajectories which visited the transient turbulent state from those which quickly became laminar. Trajectories within the edge were shown to have infinite lifetimes without relaminarising.

The laminar-turbulent separatrix in shear flows was studied even before the concept of edge was established. Itano & Toh (2001) identified for the first time a travelling wave solution in plane channel flow lying on the boundary between the turbulent and laminar basins of attraction. The state identified by Itano &

Toh (2001) was part of an e↵ort to understand the bursting events in near-wall turbulence. The characteristic bursting of streaks has been identified as part of a nonlinear self-sustained process (SSP) (Hamilton et al. 1995) in near-wall turbulence. A generic self-sustained process for wall-bounded shear flows has been conjectured by Wale↵e (1997). The process consists of streaks that wiggle and burst creating streamwise rolls which redistribute the mean shear, creating new streaks.

Itano & Toh (2001) first identified the bursting of streaks as travelling waves experiencing an instability. This result was later revised in Toh & Itano (2003) and interpreted as a periodic-like solution on the hypersurface separating the laminar and turbulent basins of attraction. The identified solution undergoes calm and active phases in agreement with the SSP described by Hamilton et al.

(1995). The seminal work of Itano & Toh (2001) gave origin to a series of studies about edge states discussed in detail in section 3.2.

14 3. The edge manifold

3.2. Edge states and the minimal seed

The work identifying structures on the laminar-turbulent separatrix of Itano &

Toh (2001) was followed by Schneider et al. (2007, 2008), who settled the ideas of edge and edge state. A well established algorithm to identify edge states receives the name of edge tracking, we will also refer to it as the ”classical”

edge tracking. The method was first introduced by Itano & Toh (2001) and Skufca et al. (2006). However, Itano & Toh (2001) refer to it as a shooting method, where the di↵erences with the method used by Skufca et al. (2006) will be specified later in the section.

The edge tracking algorithm is based on an observable a(t) which is able to distinguish trajectories approaching the laminar or the turbulent attractor.

Trajectories take infinitely long times to reach the attractor, and thus e↵ective bounds for a trajectory approaching the laminar or turbulent attractor are defined. A certain trajectory is labelled as turbulent if at any point on the trajectory a(t) > a_T and as laminar if a(t) < a_L. The algorithm starts from an initial condition u^(L) known to relaminarise and an initial condition u^{(T )}, known to trigger subcritical transition. We consider the average perturbation u¹ = (u^(L)+ u^{(T )})/2. This perturbation is then evolved in time until the observable a(t) crosses the laminar or turbulent bound. The corresponding initial conditions are then updated as u^(L) u¹or u^{(T )} u¹, depending on which bound was crossed. Repeating this process a number of times would generate a discrete sequence of initial conditions u¹, u², . . . , uⁿ so that uⁿ lies on the edge manifold W^s(E) as n! 1. Note that in the literature the edge manifold has also been denoted ⌃.

Due to the numerical accuracy of the simulations and the unstable nature of the edge state, the classical edge tracking can only be done for a finite times before it becomes necessary to apply a refinement. The classical edge tracking is illustrated in figure 3.1. The di↵erence between the shooting method in Itano &

Toh (2001) and other commonly used ones is the choice of u^(L)and u^{(T )}. Itano

& Toh (2001) used the full velocity field as u^{(T )} and the quasi-two-dimensional component of the velocity as u^(L). Skufca et al. (2006) chose instead points in the previous trajectories which left the edge manifold in di↵erent directions.

Schneider & Eckhardt (2008) and Duguet et al. (2008) used the laminar state and a state on the last turbulent trajectory as u^(L) and u^{(T )}.

Classical edge tracking has been successfully used in many parallel flow configurations. After Itano & Toh (2001), Kawahara & Kida (2001) found an edge state corresponding to a periodic orbit in plane Couette flow. More steady and chaotic edge states were identified in plane Couette flow (Wang et al. 2007; Schneider & Eckhardt 2008). These first studies of edge states in shear flows were done in periodic domains and consisted of non-localised states.

The jump to extended domains with a laminar-turbulent interface was done in the work of Willis & Kerswell (2009); Duguet et al. (2009); Mellibovsky et al. (2009); Duguet et al. (2010). The common physical manifestation to most of the edge states studies is the presence of streaks and rolls, fitting the

3.2. Edge states and the minimal seed 15

Figure 3.1: Sketch describing the classical bisection algorithm. The sketch presents the laminar attractor L, the turbulent attractor T , the edge state E.

The unstable manifold of the edgeW^u(E) and the edge manifold, or stable manifold of the edge state W^s(E). The diamond shape corresponds to the minimal seed and the circle around L corresponds to its energy. Top: Two initial conditions u^{(T )} and u^(L) are defined for the edge tracking algorithm.

Bottom: u^{(T )} and u^(L)have been updated based on the algorithm described in the text yielding and now follow remain on the edge for a finite time.

SSP description by Hamilton et al. (1995). Worth mentioning is the work of Shimizu & Kida (2009), which describes a self-sustained process for pu↵s in pipe flow and a recent study by Paranjape et al. (2020) identifying another SSP based on oblique turbulent structures. Some of the spanwise-localised states were linked to homoclinic snaking (Burke & Knobloch 2007; Schneider et al.

2010). There are other families of exact solutions in state space of parallel shear

16 3. The edge manifold

flows, as first reported by Nagata (1990), however not all of them lie within the laminar-turbulent separatrix.

A particularly relevant point on the state manifold is the point closest in energy to the laminar attractor. This point receives the name of minimal seed as first introduced by Cossu (2005) for low-dimensional models. The first application in fluid systems was reported by Pringle & Kerswell (2010); Pringle et al. (2012). The minimal seed is the most dangerous disturbance in the sense that an infinitesimal perturbation towards the turbulent attractor would enable transition to turbulence with minimal energy. The minimal seed will thus be the initial condition of a trajectory on the edge manifold that starts closest to the laminar attractor and evolves towards the edge state. The term has been used slightly di↵erently in the literature, Cherubini et al. (2011a) refer to the minimal seed as “basic building block” for subcritical transition to turbulence.

However, we will follow the definition initially given by Pringle et al. (2012).

Initial work on optimal perturbations relied on brute force studies surveying a selection of initial conditions to identify optima. Viswanath & Cvitanovi´c (2009) showed that, by adding up to three flow fields only an energy gain up O(10⁴) could be achieved in pipe flow. Duguet et al. (2010a) looked for optimal perturbations in plane Couette flow by using a linear combination of linear optimal modes. The work by Pringle & Kerswell (2010) instead identified finite-time perturbations considering the optimisation of a key functional, i. e.

the perturbation energy, for a fixed time horizon. This optimisation procedure is based on an iterative nonlinear direct adjoint method Rabin et al. (2012);

Eaves & Caulfield (2015). Such optimisation methods have successfully been used to compute the minimal seed in plane Couette flow by Monokrousos et al.

(2011) among other shear flows (Rabin et al. 2012; Duguet et al. 2013; Cherubini et al. 2015; Rinaldi et al. 2019). A recent review on the topic can be found in Kerswell (2018).

3.3. The edge in boundary layer flows

The edge studies on boundary layer flows can be classified into studies on parallel and spatially developing flows. In the former category, Kreilos et al. (2013) computed the first nonlocalised edge state in a small computational domain in the ASBL. Khapko et al. (2013) identified for the first time spanwise localised edge states in the ASBL. The analysis of the structure on the edge trajectories is addressed in Khapko et al. (2014). Khapko et al. (2016a) computed fully localised edge states. The edge states computed in the ASBL supported streak and rolls. A streak-switching phenomena was observed, which causes the edge state to drift sideways. The results from Khapko et al. (2016a) confirm the role of edge states as mediators of bypass transition by comparing the bursting and calm phases along the edge trajectory with nucleation evens from noise.

The minimal seed in the ASBL was computed by Cherubini et al. (2015). Edge states have also been computed in a parallel approximations of the boundary layer (Biau 2012).

3.3. The edge in boundary layer flows 17

Figure 3.2: Three-dimensional visualization view from above of the edge trajectory in the Blasius boundary layer in Beneitez et al. (2019) at times t = 1700, 2750 and 3550 (from top to bottom), isosurfaces of streamwise per- turbation velocity with respect to spanwise mean with values 0.06 and 0.08 (red and blue, respectively), and ₂= 1.5· 10 ⁵ (green). Flow from left to right. The black lines are separated by a distance of 200 in units of ₀^⇤. The last snapshot includes the cross sections presented in figure 3.3

The edge in spatially developing boundary layers has been studied in Cherubini et al. (2011). Cherubini et al. (2011) identified an edge trajectory in the Blasius boundary layer computed in a narrow computational domain, giving rise to a non-localised structure. The geometrical restrictions in spatially developing flows were overcome in the worked by Duguet et al. (2012). They identified a fully localised edge trajectory starting from a symmetric initial condition in the xy-plane. The edge trajectory lost memory of any initial conditions. The authors identified a robust core with little time dependence and secondary structures located further downstream. The secondary structures were much more time dependent than the robust core. Longitudinal vortices were identified staggered on the flanks of the streak. Hairpin vortices formed in the region above where the streaks pinch. The hairpin vortices present were found to be a robust feature of the edge trajectory. The identified structure contained the characteristic streaks and rolls of the SSP reported by Wale↵e (1997), however sinusoidal streak instabilities did not give rise to hairpin vortices.

The perturbation characteristic of the edge trajectory grows in size with the boundary layer thickness ^⇤. Dynamically rescaling the observables for edge tracking with the local ^⇤ suggested a quasicyclic solution.

18 3. The edge manifold

(a) (b) (c)

Figure 3.3: Illustration of a streak switching event along an edge trajectory in the Blasius boundary layer at t = 3550 (Beneitez et al. 2019). (y, z) cross- sections of ux indicated in figure 3.2. Locations of the cross-sections left to right x = 2140, 2420 and 2980.

Figure 3.4: Space-time diagram of ux(x, yf, zf, t) for z = 10 and y = 1.5 for the trajectory slightly below the edge manifold in the Blasius boundary layer (Beneitez et al. 2019). The streaks decay while the TS wavepacket grows in amplitude, forming a turbulent spot.

The most recent work to date in about edge states in spatially developing flows is that of Beneitez et al. (2019). This work consisted in a longer time edge tracking. The goal was to explore the dynamics further in time than in Duguet et al. (2012) and without constraints on the geometry of the initial conditions. To do so, Beneitez et al. (2019) perform an edge tracking for the Blasius boundary layer in a domain twice as long as the one in Duguet et al.

(2012). Moreover, since the edge tracking is to be performed for a fully localised perturbation travelling downstream, a moving box technique is implemented.

The moving box allows to track the perturbation within the boundary layer while retaining a relatively small computational domain. The method consists of Galilean changes of the reference frame of the form u u c_mve_x, where the domain moves in the streamwise direction with a piecewise constant speed cmv. The moving box technique in Beneitez et al. (2019) allowed for simulations

3.3. The edge in boundary layer flows 19 in the box up to t = 1.2⇥ 10⁴. These simulations allowed for edge tracking up to t⇡ 4700, three times longer than the previous work by Duguet et al. (2012).

Beneitez et al. (2019) showed that, starting from a non-symmetric initial condition, the classical edge tracking yields an edge trajectory experiencing the characteristic streak switching present in other flows (Khapko et al. 2016a) for moderate times. The trajectory does not settle to a quasicyclic behaviour, as hypothesised in Duguet et al. (2012), but looks instead chaotic. Three snapshots illustrating the edge trajectory can be seen in figure 3.2. The streak switching is a spatiotemporal phenomenon in the Blasius boundary layer, it is illustrated in the cuts from the edge at t = 3550 in figure 3.2 (bottom) which are shown in figure 3.3. The streak-switching appears to be general to all boundary layers, including the edge states in plane Poiseuille flow reported in Zammert &

Eckhardt (2014); Neelavara et al. (2017).

The streak-like perturbation from the moderate-time edge trajectory causes a modification of the base flow. This modification gives rise to TS waves characteristic of the linear instability of the boundary layer at later times. Initial conditions with an energy content below the energy of a nearby trajectory on the edge manifold, will follow a trajectory approaching the laminar attractor.

For long enough times these trajectories will experience the linear instability of the base flow and become turbulent via the “classical” route. Trajectories with an energy content above the energy of a nearby trajectory on the edge will instead go to turbulence via the bypass route. Figure 3.4 shows for a fixed yf = 1.5 and zf = 10 the velocity profile of u(x, yf, zf, t), which illustrates the coexistence of both transition scenarios.

The choice of observable a(t) in the edge tracking algorithm is the streamwise vorticity !xin Beneitez et al. (2019). It is shown in figure 3.5 for two trajectories becoming turbulent in di↵erent ways. The vertical line indicates the point where the two bracketing trajectories separate 2% in terms of the observable a(t).

State portraits can help us to understand the global structure of the state space, in the work by Beneitez et al. (2019) a state portrait is generated with the variables

⌦_x = ( ^⇤₀/ )¹²

✓ 1

vol(V ) Z

V|!x|²dv

◆¹₂

, (3.1)

⌦_y= ( ₀^⇤/ )¹²

✓ 1

vol(V ) Z

V |!y|²dv

◆¹₂

, (3.2)

W = ( ₀^⇤/ )³²

✓ 1

vol(V ) Z

V|uz|²dv

◆¹₂

. (3.3)

The prefactor ( ^⇤₀/ ) accounts for the boundary layer growth as indicated in Duguet et al. (2012). A state portrait illustrating the two di↵erent routes to turbulence can be seen in figure 3.6.

The work in Beneitez et al. (2019) considered an edge tracking from a pair of tilted counter rotating vortices showing that the trajectory reaches a

20 3. The edge manifold

2000 4000 6000 8000 10000 12000 14000 t

10^-4 10^-3

a(t)

Bypass transition Classical transition

Figure 3.5: Two bracketing trajectories in the Blasius boundary layer which become turbulent following the bypass route (red) and the classical route (blue).

Vertical line (green) indicates the point where the trajectories separate 2%, horizontal lines correspond to the bounds aL and aT.

Figure 3.6: State portrait of the Blasius boundary layer using variables (⌦x, ⌦y, W ), the laminar fixed point (blue) is located at (0,0,0). A trajec- tory (red) goes to turbulence following bypass transition, while the other (blue) approaches the laminar attractor and becomes turbulent via TS waves. Arrows indicate the direction of the trajectory.

3.4. Geometry of the edge manifold 21 relative attractor. Vavaliaris et al. (2020) have computed the minimal seed in the Blasius boundary layer reaching the same relative attractor in state space.

The computation of the minimal seed in Vavaliaris et al. (2020) is done using an adjoint based optimisation code, first used in Rinaldi et al. (2019). It optimised the perturbation energy at a prescribed time in the way introduced by Pringle &

Kerswell (2010); Pringle et al. (2012) using the rotated gradient update method described in Foures et al. (2013).

In contrast with parallel flows, the minimal seed in the Blasius boundary layer is not only defined by one Reynolds number (Rabin et al. 2012). In their work Vavaliaris et al. (2020) show that the minimal seed depends on the length of the box L_x and the optimisation time T_obj. The parameters L_x and T_obj will define the range of local Reynolds number spanned in the evolution of the localised perturbation. Vavaliaris et al. (2020) identified the minimal seed for Tobj= 400 located at Re = 425.1 based on the centre of mass of u, with an energy of Ec= 3.644⇥ 10 ². To ensure that the trajectory lies within the edge manifold the perturbation shape is optimised for the two bracketing trajectories with energies EL= 3.6⇥ 10 ²and ET = 3.7⇥ 10 ², and then an edge tracking is performed using the trajectory with ET = 3.7⇥ 10 ²and u^{(T )}.

Vavaliaris et al. (2020) observed that the minimal seed in the Blasius boundary layer relies on a series of linear mechanisms, identifying first the Orr mechanism, which is inherently two-dimensional. It is characterised by the initial conditions tilting about its spanwise axis against the shear and detilting to align with the shear. This mechanism was observed in the perturbation between t 2 [0, 60]. The Orr is followed by the lift-up mechanism, which is three-dimensional. In the lift-up mechanism streamwise vortices transport fluid particles with high and low streamwise momentum in the wall-normal direction giving rise to streaks. The lift-up mechanism was identified in the time interval t2 [70, 200]. The presence of these mechanisms is common to other shear flows and has been identified in plane Couette flow (Duguet et al. 2013), ASBL Cherubini et al. (2015) and small computational domains in the Blasius boundary layer Cherubini et al. (2011a). Once the lift-up mechanism has taken place, the perturbation resembles the edge trajectory reported in Beneitez et al.

(2019). The perturbation then retains its streaky structure while growing in size with increasing ^⇤. Using the rescaled variables (3.1)-(3.3) Vavaliaris et al.

(2020) show that the trajectory starting from the minimal seed reaches the same relative attractor in state space as in Beneitez et al. (2019) in a faster and more efficient way.

3.4. Geometry of the edge manifold

The generally accepted picture of the state space is illustrated in Duguet et al.

(2008). It consists of a turbulent set, which supports turbulence on one side of the edge and a stable laminar state in the other. This picture is displayed in the model introduced in equations. (2.17) and (2.18), (Dauchot & Manneville 1997) and illustrated in figure 2.4. Three objects can thus be identified in the state

22 3. The edge manifold

X Y

X Z

X Y

X Z

X Y

X Z

(a) (b) (c)

X Y

X Z

(d)

Figure 3.7: Contours of streamwise perturbation velocity (blue and red for negative respective positive) and 2 vortex criterion (green) contours of the minimal seed’s evolution in the Blasius boundary layer. xy and xz views; flow from left to right right. The double arrow gives the length of the minimal seed for comparison (22 units). Contour for (a)-(c): u = 8· 10 ³, 6· 10 ³; (d): u = 6· 10 ², 5· 10 ², 2 = 1· 10 ⁵. (a) t = 0; the minimal seed is non-symmetric and backwards-leaning. (b) t = 60; the perturbation is leaning forward after the Orr mechanism’s e↵ect. (c) t = 150; the perturbation is getting elongated and consists of high- and low-speed streaks created by the lift-up mechanism. (d) t = 2500; the perturbation is considerably longer, while maintaining its streaky structure; the 2contours help identify its active core (Duguet et al. 2012), characteristic of an edge trajectory.

space: the chaotic set, the edge state and the laminar fixed point. This global geometry, although compatible with the observations of subcritical transition is challenged in certain scenarios.

The spontaneous decay of turbulent fluctuations implies that there is a trajectory connecting the turbulent saddle with the laminar attractor. This scenario was explored by Chantry & Schneider (2014) in plane Couette flow.

They provide evidence that the edge is instead wrapped around the turbulent structure, drawing parallels with the low-order models of subcritical transition in Lebovitz (2012). The edge manifold can thus be labelled as strong if it separates two basins of attraction or weak if it does not (Lebovitz 2012).

Not only does sudden relaminarisation of turbulent trajectories challenge the typical picture of the state space. Recent studies where the laminar fixed point becomes linear unstable in plane channel flow (Zammert & Eckhardt 2019), the bent pipe (Canton et al. 2020) and the Blasius boundary layer (Beneitez et al. 2019) call into question the global picture of the state space.

3.4. Geometry of the edge manifold 23 Zammert & Eckhardt (2019) explored the arrangement of the di↵erent routes to transition in plane channel flow above the critical Reynolds number. Their findings show that both the supercritical and subcritical transition scenarios have the same turbulent attractor as destination, which is characteristic of a weak edge. Zammert & Eckhardt (2019) studied the transition times of both transition scenarios identifying a sharp transition between them. The study of Canton et al. (2020) identified a local bifurcation of the laminar state in the bent pipe, where the laminar fixed point becomes linearly unstable in favour of a limit cycle. Their results show that the global structure of the state space remains unaltered by the local bifurcation of the laminar state and that an edge manifold separating two basins of attraction is still in place, the edge is thus of the strong type.

The Blasius boundary layer becomes linearly unstable at Re ⇡ 519, and the local Reynolds number of a localised disturbance is intrinsically linked to the streamwise coordinate. Beneitez et al. (2019) identified an edge trajectory in the Blasius boundary layer and found that the edge e↵ectively separates the trajectories approaching the laminar attractor and the turbulent attractor for moderate times. It was shown that a linear instability in the form of two- dimensional Tollmien–Schlichting (TS) waves manifests itself in the Blasius boundary layer. The origin of the TS waves was linked to the modification of the mean flow caused by the localised disturbance characteristic of bypass transition.

This “classical” transition scenario consisting of TS waves competes with the bypass transition route. The work of Beneitez et al. (2019) and Zammert

& Eckhardt (2019) suggest the reinterpretation of the edge as the boundary between two transition routes.

Beneitez et al. (2020a) give a very simplified three-dimensional model to study the edge geometry as the laminar fixed point becomes linearly unstable.

This three dimensional model reads dx₁

dt = s1x1+ x2+ x1x2 (3.4) dx2

dt = s₂x₂ x²₁+ x²₃ (3.5)

dx3

dt = s₃x₃ x₂x₃, (3.6)

where s3and are two parameters of the model. Note that s1 and s2are kept as in the example in section 2.4.3. It is an extension of the model proposed in equations (2.17) and (2.18). The model contains a new variable x₃, along which the linear instability of the state L will manifest itself. The extension of the model is done such that: (i) the nonlinear terms have no contribution to the energy, defined as H = x²₁+ x²₂+ x²₃and (ii) the dynamics on the 2D plane P defined by x3= 0 remains unchanged.

The bifurcation analysis of interest in the model is for s₃ 0. The model in equations (3.4)-(3.6) showed three distinct dynamical regions. The first region between s3= 0 and s3= 0.032 shows how a new stable fixed point S, additional

24 3. The edge manifold

C

(a) (b) (c)

Figure 3.8: Phase portrait the 3D model in Beneitez et al. (2020a). (a) the state space at s3= 0.01 (b) The state space at s3= 0.001 and (c) The state space at s3= 0.031. The edge manifold is computed based on the technique described in Beneitez et al. (2020) section II.

to the already mentioned L, T and E, is created in a pitchfork bifurcation at s3= 0. In this region a limit cycle is created from the fixed point at s3= 0.03, a result qualitatively akin to the results reported on the bent pipe (Canton et al. 2020). The edge manifold starts to bend around the new stable solution while two well separated basins of attraction still exist. Figure 3.8 (a)-(b) shows how the edge manifoldW^s(E) bends around the new fixed point S, and figure 3.8 (c) shows the new limit cycle solution.

The limit cycle generated at s3 = 0.03 collides with the edge manifold at s3= 0.03125 causing a global bifurcation which changes the structure of the state space. Consequently only the basin of attraction of T remains and the edge goes from being strong to weak. The edge state E still has only one unstable direction, which lies withinP. This behaviour is consistent with the observations in Zammert & Eckhardt (2019), where the edge state is still accessible even when base flow instability competes with bypass transition.

At s3 = 0.063 the newly created solution S merges with E, so that E becomes unstable also in the x₃ direction. In this case the edge manifold ceases to exist, and the edge state, is no longer accessible from outside of P. No sharp transition is then observed between the two existing routes to turbulence, where a parallel to the results in Beneitez et al. (2019) can be drawn. The transition time ⌧trwas introduced by Zammert & Eckhardt (2019), mathematically defined as arg min_t>0kTt(x) Tk < ✏T. Tt(x₀) refers to a trajectory initialised at certain x₀= (x₁, x₂, x₃) position in state space and ✏_T defines a threshold for a trajectory to be identified as close to the attractor T . It is a useful quantity to understand the edge geometry in the cases mentioned above. Figure 3.9 shows ⌧trfor the qualitative behaviours explained above: (i) a strong edge (ii) a weak edge separating two routes to transition and (iii) a collapsed edge.

The work in Beneitez et al. (2019) opens new questions about the nonlinear dynamics approach in the Blasius boundary layer. This flow falls into the