Andres Castillo Castellanos

Anne Sergent

Maurice Rossi

Reversal cycle in square Rayleigh-Bénard cells in turbulent regime

Keywords: Turbulent convection, convection in cavities, Bénard convection Chapter

We consider long-term data from direct numerical simulations of turbulent Rayleigh-Bénard convection inside 2D square cells. For the range of Rayleigh numbers Ra = 10 7 -10 8 and Prandtl numbers Pr = 3.0 -4.3, two types of flow regimes are observed: a regime consisting of consecutive reversals, when the global rotation switches signs; and a regime consisting of an extended cessation, when global rotation is absent. A filtering method discriminates these two regimes and allows to identify two characteristic time-scales for the former regime. A time re-scaling is then used to tune our records to a common duration thus putting into evidence a generic reversal cycle. This cycle is composed of three successive phases: acceleration, accumulation, and release including a rebound event. We complement this view in terms of global angular impulse, available mechanical energy, global kinetic energy, and their corresponding transfer rates. For a particular realisation of a reversal, each phase is described in terms of the flow patterns (large diagonal roll, counter-rotating corner-flows and thermal plumes) and tied to the corresponding energy processes. We conclude by performing linear as well as nonlinear stability studies to account for the triggering mechanism of the release.

Convection turbulente dans les cellules de Rayleigh-Bénard avec des conditions limites modifiées

Résumé :

Le présent document est consacré à l'étude des écoulements à grande échelle dans la convection turbulente de Rayleigh-Bénard, et de l'influence exercée par différents facteurs sur leur structure et leur évolution temporelle. Ce travail se compose de trois parties.

Dans la première partie, nous étudions la structure et l'évolution temporelle des écoulements à grande échelle à l'intérieur d'une cellule carré (2D) de type Rayleigh-Bénard. La caractérisation proposée mélange une approche statistique avec une approche physique en s'appuyant sur l'évolution du moment cinétique angulaire ainsi que sur l'évolution des énergies cinétiques et potentiel disponibles, pour mettre en évidence les mécanismes physiques sous-jacents. Nous essayons par la suite de relier ces mécanismes aux structures persistantes observées et à leur évolution temporelle.

Dans la deuxième partie, nous nous concentrons sur les changements dans la circulation à grande échelle résultant de l'imposition d'une condition de surface libre dans une cellule carré (2D) de type Rayleigh-Bénard, ainsi brisant la symétrie haut/bas. Cette configuration présente une température moyenne plus faible, un flux de chaleur amélioré et des fluctuations accrues par rapport au cas classique.

Dans la dernière partie, nous nous concentrons sur les changements dans la circulation à grande échelle, cette fois résultant de l'imposition d'un fort degré de confinement latéral dans une cellular rectangulaire mince (3D) de type Rayleigh-Bénard. Cette configuration présente la formation des grandes structures cohérentes dans la direction verticale, connues sous le nom des canaux turbulents, au lieu d'une circulation à l'échelle du système. Le confinement présente également une forte influence sur différentes quantités globales, tels que les numéros de Nusselt et Reynolds. Afin d'aider à l'exploration du régime sévèrement confiné, nous développons et validons un modèle de dimensionnalité réduit avec corrections inertielles. En utilisant ce modèle, nous sommes en mesure d'explorer l'espace des paramètres et effectuer des observations dans la limite d'un fort confinement. The present document is dedicated to the study of the large-scale flow patterns in turbulent Rayleigh-Bénard convection, and of the influence exerted by different factors on the flow structures and on their temporal evolution. This work is composed of three parts.

In the first part, we study the structure and the temporal evolution of the large-scale flow inside a square (2D) Rayleigh-Bénard cell. The proposed characterization combines a statistical analysis with a physical approach relying on the angular momentum as well as the kinetic and potential energies to highlight the underlying physical mechanisms.

We subsequently attempt to tie these mechanisms together to each of the distinctive flow patterns observed and to their temporal evolution.

In the second part, we focus on the changes in the large-scale flow resulting from imposing a stress-free surface on a square (2D) Rayleigh-Bénard cell, thus breaking the top/bottom symmetry. This configuration exhibits a lower bulk temperature, enhanced heat-flux and increased fluctuations with respect to the classical case.

In the last part, we focus on the changes in the large-scale flow resulting from imposing a strong degree of lateral confinement on a slim rectangular (3D) Rayleigh-Bénard cell. This configuration exhibits the formation of large coherent structures in the vertical direction, or turbulent channels, instead of a system-wide circulation. Confinement also exhibits a strong influence on several global quantities such as the Nusselt and Reynolds numbers. In order to assist in the exploration of the severely confined regime, we develop and validate a reduced dimensionality model with inertial corrections. Using this model, we are able to explore the parameter space and provide further insight in the limit of strong confinement.

List of Notations

Introduction

The term natural convection indicates the motion of fluid due to density variations. Thermal convection, in which density variations are induced by local temperature differences, is present inside stars and planets, in the atmosphere and the ocean, but also in engineering and in day to day life. The universal nature of thermal convection and its relevance to different fields is reflected by the abundance of publications on the subject.

Context

Consider an object such as a hot air balloon (or "Montgolfiere") placidly hovering in midair as displayed on figure 1.1a. The balloon has a density such that, there is a balance between the weight of the balloon and the buoyancy force, i.e. the force exerted by the displaced fluid. If we dispose of some heat source aboard, the air contained inside the balloon will increase its temperature, decreasing its density with respect to surrounding air and driving the hot air balloon upwards until a new equilibrium is reached. In this sense, it is the density variation due to local temperature differences that drives motion. This represents the basic principle of thermal convection.

Examples of thermally-driven flows are abundant in nature. The grainy appearance of the solar surface (see figure 1.1b) is generated by convection currents inside the Sun -hot, bright rising 'granules' of order 1 Mm surrounded by cooler, darker intergranular lanes [START_REF] Leighton | The solar granulation[END_REF]. Convection of the Earth mantle drives the generation of plate tectonics, volcanism, earthquakes and mountain building [START_REF] Bercovici | The generation of plate tectonics from mantle convection[END_REF]. The general circulation of the atmosphere is composed of thermal cells created by uneven heating of the Earth's surface by solar radiation coupled with rotation. At "smaller-scales", the sea-breeze, and major weather events (hurricanes, flash-floods, electrical storms, and tornadoes) are also driven by thermal convection (B. [START_REF] Stevens | Atmospheric moist convection[END_REF]. In the ocean, the thermohaline circulation is powered by the effect of thermohaline forcing (thermo-referring to temperature and haline-to the concentration of salt) and turbulent mixing (winds and tides) [START_REF] Rahmstorf | Thermohaline circulation: the current climate[END_REF]. Engineering applications for thermal convection are equally abundant. It is found nearly everywhere, from heat exchangers, industrial cooling equipment, and natural ventilation systems like solar chimneys, to domestic appliances and consumer electronic devices and even in our day to day life. , 2015). (c) Convection in a "soap" bubble [START_REF] Seychelles | Thermal convection and emergence of isolated vortices in soap bubbles[END_REF].

Most of these examples involve thermal convection in complex geometries or combined with other physical processes, such as magnetic forcing, radiative transfer, phase changes, chemical reactions or biological processes. The study of such complex systems requires the use of large material resources which may not be necessarily available at the time.

Because of this, most fundamental studies are centered on simplified systems such as thermal convection between two horizontal plates, convection between concentric spheres [START_REF] Egbers | The geoflowexperiment on iss (part i): experimental preparation and design of laboratory testing hardware[END_REF][START_REF] Travnikov | The geoflow-experiment on iss (part ii): numerical simulation[END_REF], convection in 'soap' films (see figure 1.1c, Adami, 2014;[START_REF] Seychelles | Thermal convection and emergence of isolated vortices in soap bubbles[END_REF], and a differentially heated cavity [START_REF] Vahl | Natural convection of air in a square cavity: a benchmark solution[END_REF][START_REF] Quéré | From onset of unsteadiness to chaos in a differentially heated square cavity[END_REF].

For this work, we focus on the Rayleigh-Bénard convection which involves a horizontal fluid layer of height H, which is heated uniformly from below and cooled uniformly from above in the presence of gravity. By convention the gravity force g = {0, -g, 0} is opposed to the unit vector in the y-direction.

T = T bot at y = 0 (1.1a) T = T top = T bot -∆T at y = H (1.1b)
Ideally the fluid layer extends over an infinite plane, but in practice, one must also consider the fluid container, usually a cylinder or a square box. The latter type is defined by a width W (see figure 1.2a) and for the 3-D case, a depth d. In the following, we assume side-walls to be adiabatic.

Additionally, we assume the boundaries to be rigid no-slip boundaries, except for one case [START_REF] Shang | Measured local heat transport in turbulent rayleigh-bénard convection[END_REF].

where the top boundary is assumed to be a free surface. u = 0 on rigid boundaries (no-slip condition) (1.2a) v = ∂ y u = ∂ y w = 0 on a free surface (stress-free condition) (1.2b) Above a critical temperature difference ∆T c , the buoyancy force overcomes the viscous damping force driving the warm fluid upwards (resp. cold fluid downwards) thus creating a convective current. At the onset of convection flow appears as steady two-dimensional or three-dimensional rolls. For temperature differences larger than ∆T c , the flow exhibits a systematic transition to turbulence via periodic and chaotic solutions. For this work, we are concerned with the (weakly-)turbulent convection regime.

In order to introduce some of the distinctive features of turbulent Rayleigh-Bénard convection, let us consider the schematic representation displayed in figure 1.2. A kinetic boundary layer develops in the vicinity of the plates and side-walls, where viscous effects are dominant and most of the frictional drag is concentrated. Similarly, a thermal boundary layer develops in proximity to the top and bottom plates where most of the temperature gradient is concentrated. The fluid contained outside of the viscous and thermal boundary layers is referred to as the bulk.

One important feature of turbulent Rayleigh-Bénard convection is the spontaneous development of spatially ordered coherent structures. Thermal plumes are mushroom-like coherent structures that detach from the boundary layers (see figure 1.2b) and are responsible for the transport of most of the heat across the domain (hot plumes rise from the bottom boundary layer, cold plumes fall from the top boundary layer.) The interaction between plumes and walls is responsible for the formation of a large-scale circulation (LSC), also called 'the wind of turbulence' [START_REF] Niemela | The wind in confined thermal convection[END_REF]. The LSC in turn deforms the plumes and deflects their trajectory creating a complex dynamics (for instance, the roll-like structure on figure 1.2b). The characterization of the spatial structure and the long-term evolution of the LSC, and their sensitivity to different parameters has become a major research subject in recent years.

Key system parameters and response

Rayleigh-Bénard convection is characterized by competing physical mechanisms: heat conduction, momentum dissipation, and the buoyancy force. For each mechanism we may define a characteristic velocity scale

U cond = κ H U visc = ν H U buoy = |∆ρ| ρ gH (1.3)
where H is the characteristic length of the system, κ (resp. ν) the thermal (resp. momentum) diffusivity coefficient. The buoyancy velocity scale U buoy is that of a fluid parcel, lighter than its surroundings with a density difference ∆ρ = -ρβ∆T, with β = -1 ρ dρ dT p being the volumetric thermal expansion coefficient.

The characteristic velocities may be combined to form a non-dimensional parameter known as the Rayleigh number

Ra ≡ Buoyancy Diffusivity = U 2 buoy U cond U visc = β∆TgH 3 κν (1.4)
which quantifies the competition of buoyancy forces compared to thermal and viscous diffusion. The Rayleigh number is a control parameter, much like the Reynolds number which depends on the fluid and on the flow regime. Typical values of Ra range from Ra ∼ 10 3 -10 4 near the onset of convective motion, up to Ra ∼ 10 24 -10 27 (!) for deep oceanic convection [START_REF] Chillà | New perspectives in turbulent Rayleigh-Bénard convection[END_REF]. In contrast, direct numerical simulations are currently limited to Ra ∼ 10 12 -10 13 in simple geometries (R. Stevens, Lohse, et al., 2011;[START_REF] Verzicco | High rayleigh number simulations in a slender laterally periodic domain[END_REF].

A second important parameter is the Prandtl number, which measures the relative importance of momentum (viscous) diffusivity to heat conduction by molecular dissipation.

Pr ≡ Viscous diffusivity Thermal diffusivity = U visc U cond = ν κ (1.5)
The Prandtl number depends only of the fluid properties. Typical values of Pr are around 0.7 -1.0 for air and other gases, from 1.0 -10.0 for water, and around 10 23 for the mantle of the Earth.

The geometry of the container is represented by one or more dimensionless aspect ratios.

For a rectangular container we define the longitudinal and transversal aspect ratios

Γ x ≡ W H Γ z ≡ d H (1.6)
where W is the width and d is the depth of the container.

The central elements of the Rayleigh-Bénard problem are summarized by [START_REF] Ahlers | Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection[END_REF] as follows: for a given fluid in a closed container heated from below and cooled from above, how effective is the transfer of heat and momentum, and which are the main features of the flow inside the container. In this sense, one of the main responses of the system are the Nusselt and Reynolds numbers. The Nusselt number represents the Comparison between simulations, experiments, and the prediction of the Grossmann-Lohse theory (R. [START_REF] Stevens | The unifying theory of scaling in thermal convection: the updated prefactors[END_REF].

ratio between the overall heat transfer with respect to the heat transfer by conduction Nu ≡ vT xz -κ∂ y T xz κ∆T/H (1.7) where v is the vertical velocity and • xz indicates the average over time and over a plane xz at constant y. We also define a Reynolds number based on the height H and a characteristic velocity U Re ≡ U H ν (1.8) where U may be a characteristic velocity of the large-scale circulation, or a root mean squared velocity.

Scaling theories for the Nusselt number

It is easy to understand why so many theoretical, numerical, and experimental studies have focused over the years on explaining the evolution of the system responses, Nu and Re, in terms of the input parameters: Ra, Pr, and the aspect ratios. Most of the earlier theories predict general scaling laws Nu ∼ Ra β Nu Pr γ Nu Re ∼ Ra β Re Pr γ Re (1.9) where examples include the β Nu = 1 3 regime by [START_REF] Malkus | The heat transport and spectrum of thermal turbulence[END_REF] which assumes the heatflux to be independent of the height of the cavity, and the hard turbulence regime with β Nu = 2 7 and β Re = 1 2 by [START_REF] Castaing | Scaling of hard thermal turbulence in Rayleigh-Bénard convection[END_REF]. The latter theory assumes the existence of a mixing zone where plumes are detached from the boundary layers and accelerate to match the velocities of the central region. A model unifying the different scaling regimes was developed by [START_REF] Grossmann | Scaling in thermal convection: a unifying theory[END_REF] based on the exact relations between the Nusselt number and the viscous and thermal dissipation rates [START_REF] Shraiman | Heat transport in high-Rayleigh-number convection[END_REF]. The main idea is to decompose the viscous and thermal dissipation rates into contributions from the bulk flow and from the boundary layers, then use dimensional analysis to estimate [START_REF] Funfschilling | Plume motion and large-scale circulation in a cylindrical rayleigh-bénard cell[END_REF]; (b) Reversal of the LSC inside a cylindrical cell (Xi and Xia, 2008); (c) Torsional oscillations of the LSC plane [START_REF] Funfschilling | Torsional oscillations of the large-scale circulation in turbulent Rayleigh-Bénard convection[END_REF]; and (d) off-center oscillations of the LSC plane [START_REF] Zhou | Oscillations of the largescale circulation in turbulent rayleigh-bénard convection: the sloshing mode and its relationship with the torsional mode[END_REF]. these separate contributions in terms of Ra and Pr. This theory identifies different scaling regimes and makes use of crossover functions to combine scalings in a way which would not be possible with simpler power laws, see figure 1.3. A detailed discussion of the historical aspects of the scaling theories can be found in [START_REF] Ahlers | Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection[END_REF].

Re-orientations of the large-scale circulation

Despite its apparent simplicity, the Rayleigh-Bénard convection exhibits some incredibly rich and complex dynamics of the large-scale flow. Generally, the 'wind' breaks one or more of the natural symmetries of the system. For a cylindrical cell, the 'wind' takes the form of roll-like structures. Over time, the 'wind' may go through a series of transformations and re-orientations which help restore the (statistical) symmetry of the system. The first type of re-orientation is known as the azimuthal meandering is displayed on figure 1.4a. The plane of the roll-like structure exhibits spontaneous and erratic reorientations through azimuthal displacements. A complete rotation of the entire structure may be observed without major changes in the flow speed [START_REF] Funfschilling | Plume motion and large-scale circulation in a cylindrical rayleigh-bénard cell[END_REF]. The second type of re-orientation is through a cessation of the roll-like structure, before restarting in a different direction, see figure 1.4b. Both kinds of reversals are often superposed to other events, such as torsional oscillations of the LSC plane displayed in figure 1.4c [START_REF] Funfschilling | Torsional oscillations of the large-scale circulation in turbulent Rayleigh-Bénard convection[END_REF] and off-center oscillations of the LSC plane displayed in figure 1.4d [START_REF] Zhou | Oscillations of the largescale circulation in turbulent rayleigh-bénard convection: the sloshing mode and its relationship with the torsional mode[END_REF]. The precise nature of these transformations is not fully understood.

One approach commonly used in the study of cessation-led reversals, is to constrain the [START_REF] Chandra | Dynamics and symmetries of flow reversals in turbulent convection[END_REF], and (b) experimental and numerical studies on a slim rectangular geometry (image from [START_REF] Vasiliev | Reversals of large-scale circulation at turbulent convection in rectangular boxes[END_REF]. large scale circulation to a plane by restricting the fluid domain to a pure 2-D geometry [START_REF] Sugiyama | Flow reversals in thermally driven turbulence[END_REF][START_REF] Chandra | Dynamics and symmetries of flow reversals in turbulent convection[END_REF][START_REF] Podvin | A large-scale investigation of wind reversal in a square Rayleigh-Bénard cell[END_REF] or to a slim rectangular geometry of small aspect ratio in the transversal direction [START_REF] Vasiliev | Reversals of large-scale circulation at turbulent convection in rectangular boxes[END_REF][START_REF] Ni | Reversals of the large-scale circulation in quasi-2D Rayleigh-Bénard convection[END_REF], see figure 1.5. Sugiyama, Ni, et al., 2010 identified a region in the (Ra, Pr) space in which reversals of the large-scale flow are observed numerically in a (2-D) square and experimentally inside a rectangular geometry with Γ x = 1 and Γ z = 0.3. In such configurations, the flow is composed of a large diagonal roll and two counter-rotating corner-rolls. Flow reversals has been observed experimentally by Ni et al., 2015 for 0.84 ≤ Γ x ≤ 1.10 and 0.33 ≤ Γ z ≤ 0.25, and by [START_REF] Vasiliev | Reversals of large-scale circulation at turbulent convection in rectangular boxes[END_REF] for Γ x = 1 and 0.06 ≤ Γ z ≤ 0.2 for different values of Ra and Pr. However, given some of the differences between pure 2-D and (quasi-)2-D turbulent convection, it is not entirely clear if the reversals observed in each case correspond to the same phenomenon. A more complete characterization of the reversal dynamics is required to help addressing this question.

Motivation and objectives

The present document is dedicated to the study of the large-scale flow patterns in turbulent Rayleigh-Bénard convection, and of the influence exerted by different factors on the flow structures and on their temporal evolution. The proposed characterization combines a statistical analysis with a physical approach relying on the angular momentum as well as the kinetic and potential energies to highlight the underlying physical mechanisms. We subsequently attempt to tie these mechanisms together to each of the distinctive flow patterns observed and to their evolution. Investigating the nature of these coherent structures may be useful in modeling and controlling turbulent flows, as well as to improve our understanding of the large-scale dynamics.

The contents of this work are organized as follows. Part I (chapters §2 to §3) is focused on the numerical computation and outlines the methodology used in the characterization of large-scale flow patterns in turbulent convection. Part II (chapters §4 and §5) is dedicated to the study of spontaneous re-organization of the large-scale flow inside square (2-D) cells. Part III (chapter §6) deals with the effects on the large-scale flow of imposing a free surface on a square cell. The last part of this work (chapters §7 and §8) deals with the influence of lateral confinement on a slim rectangular geometry.

Chapter 2

Governing equations and statistical characterization tools

In the first part of this chapter we define the classic statement of the Rayleigh-Bénard problem in the framework of the Boussinesq approximation. Additionally, we detail the two main configurations studied throughout this work: a square (2-D) geometry with a rigid or a stress-free surface, and a slim rectangular (3-D) geometry. In the second part we introduce some quantities useful in the subsequent analyses. Sections §2.2 and §2.3 deal with different global quantities used to characterize the flow from energetic and hydrodynamic viewpoints. Sections §2.4 to §2.6 focused on the spatial structure of the flow and the separation of large-and small-scale flow patterns.

Problem description

Governing equations

For thermal convection it is necessary to write governing equations for the temperature, velocity and pressure fields, couple with equations of state. In buoyancy driven flows, if the density field does not deviate too much from an average value, the Boussinesq approximation is used to obtain a simpler set of governing equations, called the Oberbeck-Boussinesq equations or Boussinesq equations.

Navier-Stokes equations for compressible fluids

The starting point are the compressible Navier-Stokes equations in the following form

∂ t ρ + ∂ j (ρu j ) = 0 (2.1) ∂ t (ρu i ) + ∂ j (ρu j u i ) = ρg i + ∂ j τ ij (2.2)
with ρ being the density, and τ ij the stress tensor. For a Newtonian fluid the stress tensor τ ij is modeled as (2.3) with p being the thermodynamic pressure, µ the dynamic viscosity, δ ij the Kronecker delta and S ij the rate of strain tensor

τ ij = -pδ ij + 2µ S ij - 1 3 ∂ k u k δ ij
S ij ≡ 1 2 (∂ j u i + ∂ i u j ) (2.4)

Equations for mechanical energy in compressible fluids

If we multiply equation (2.2) by u one obtains an expression for the kinetic energy,

D Dt (ρe k ) = ρu i g i + p∂ i u i + ∂ j (u i τ ij ) -φ (2.5)
where D Dt indicates the material derivative, e k ≡ 1 2 u i u i the specific, i.e. per unit mass, kinetic energy, and φ the rate of viscous dissipation

φ ≡ 2µ(S ij - 1 3 ∂ k u k δ ij )(S ij - 1 3 ∂ k u k δ ij ) (2.6)
The change in specific potential energy e p ≡ g i x i , reads as D Dt (ρe p ) = -ρu i g i (2.7) Combining the above expression with (2.5) yields the following expression for the total mechanical energy D Dt (ρ(e k + e p )) = p∂ i u i + ∂ j (u i τ ij )φ (2.8)

Equation for internal energy in compressible fluids

Now consider e int as the specific internal energy. Using the first law of thermodynamics on the material volume Ω gives D Dt Ω ρ(e int + e k + e p )dΩ = ∂Ω (τ ij u i n jq i n i )dS (2.9) where q is the heat-flux, dS the surface element and n the outward normal unit vector.

The above implies D Dt [ρ(e int + e k + e p )] = ∂ j (u i τ ij ) -∂ i q i (2.10) Taking the difference between equations (2.8) and (2.10) yields an equation for the internal energy e int D Dt (ρe int ) = φ -∂ i q i -p∂ i u i (2.11) Equation (2.11) is implicitly coupled with (2.1) and (2.2) through a phenomenological equation for the heat-flux, the equation of state e int = e int (T, p) and the fluid properties ρ = ρ(T, p), µ = µ(T, p), and so on.

The Oberbeck-Boussinesq approximation

Under the Boussinesq approximation, variations of density are neglected everywhere except in the buoyancy term. Additionally, we assume the density to depend linearly on the temperature only,

ρ(T) = ρ ref (1 -β[T -T ref ]) (2.12)
where β is the coefficient of thermal expansion at the reference state (T ref , p ref ). In our case, the reference temperature is taken as T ref = (T bot + T top )/2. All the properties of the fluid are considered as constants and are evaluated at this reference state.

This hypothesis is valid for most fluids if we consider that: a) the compressibility effects are negligible, b) the vertical scale is small enough so that density variations due to the hydrostatic pressure variations are negligible, and c) temperature variations in the flow are small. For most liquid configurations these conditions are generally satisfied, whereas for gases the following conditions must be verified

gβH r [1 + β (T -T ref )] ≪ 1 (2.13a) β (T -T ref ) ≪ 1 (2.13b)
where r is the specific gas constant measured in SI units, i.e. [J kg -1 K -1 ] (see for instance [START_REF] Tritton | Physical fluid dynamics[END_REF][START_REF] Hinch | Hydrodynamique physique 3e édition[END_REF].

The internal energy equation is simplified under the Boussinesq approximation. [START_REF] Getling | Rayleigh-Bénard Convection: Structures and Dynamics, Advanced series in nonlinear dynamics[END_REF][START_REF] Kundu | Fifth Edition[END_REF]. In that approximation, we suppose the heat flux follows Fourier law q i = -k∂ i T, the heating due to viscous dissipation of energy is negligible φ ρCv ≈ 0, and the internal energy follows the Ideal gas's law e int = C v T, one obtains a partial differential equation for T .14) with C v being the specific heat at constant volume, and k being the thermal conductivity.

D Dt T = 1 ρC v [∂ j (k∂ j T) -p(∂ i u i )] ( 2
The volume expansion term -p(∂ i u i ) may be written as .15) with C p being the specific heat at constant pressure. This leads to the following expression for the temperature field D Dt T = ∂ j (κ∂ j T) (2.16) where the thermal diffusivity κ is defined as κ = k ρCp .

-p(∂ i u i ) = -ρ(C p -C v ) D Dt T ( 2
We introduce p * (x, t) as the difference between the thermodynamic pressure and the hydrostatic pressure, commonly referred as driven pressure, -∂ i p * = -∂ i p + g i ρ ref (2.17) to obtain the Oberbeck-Boussinesq equations (2.18) where ν ref = µ ref /ρ ref is the kinematic viscosity of the fluid. Since fluid properties are always evaluated at the reference state, in the following we drop the subscript everywhere except for the reference temperature T ref .

∂ i u i = 0 ∂ t u i + ∂ j (u j u i ) = - 1 ρ ref ∂ i p * + ν ref ∂ j ∂ j u i -β(T -T ref )g i ∂ t T + ∂ j (u j T) = κ ref ∂ j ∂ j T

Dimensionless parameters

Fluid flow problems without electro-magnetic forcing or chemical reactions involve only mechanical and thermal variables, can be expressed in terms of four basic quantities: a length, a time, a mass and a temperature. We define the characteristic mass [M ], length [L], temperature [Θ], and velocity [U ] scales as follows. The relevant length scale corresponds to the height of the fluid layer H, while the temperature scale is fixed by the temperature difference ∆T between the top and bottom plates. A characteristic velocity scale is obtained from dimensional analysis by a balance between the friction and buoyancy forces (Bejan, 2013, Sec. 4.3., Scale analysis) which corresponds to the free-fall velocity U buoy divided by Pr 0.5

[M ] = ρH 3 , [L] = H, [Θ] = ∆T, [U ] = κ H gβ∆TH 3 κν (2.19)
We proceed to define a set of non-dimensional variables The Oberbeck-Boussinesq equations may be written in dimensionless form as

x = x [L] ũ = u [U ] t = t [L]/[U ] p = p * ρ[U ] 2 (2.
∂ i u i = 0 ∂ t u i + ∂ j (u j u i ) = -∂ i p + PrRa -0.5 ∂ j ∂ j u i + Prθδ i2 ∂ t θ + ∂ j (u j θ) = Ra -0.5 ∂ j ∂ j θ (2.22)
In addition, we impose the thermal boundary conditions θ = -1/2 at the top plate (y = 1/2) (2.23a) θ = 1/2 at the bottom plate (y = -1/2) (2.23b)

∂ n θ = 0 on the sidewalls (2.23c) and the mechanical boundary conditions u = 0 on rigid boundaries (no-slip condition) (2.24a) v = ∂ y u = ∂ y w = 0 on a free surface (stress-free condition) (2.24b) and the geometry of the domain. Three main configurations are considered in this work. Chapters §4 to §5 are dedicated to the study of square (2-D) cell with rigid boundaries (figure 2.1a), chapter §6 deals with square (2-D) cells with top stress-free surface and bottom no-slip condition (figure 2.1b)

x ∈ - 1 2 , 1 2 × - 1 2 , 1 2 with x ∈ R 2
whereas chapters §7 and §8 deal with a slim rectangular (3-D) geometry with rigid adiabatic walls under various degrees of lateral confinement, with Γ z being the aspect ratio in the transversal direction (figure 2.1c).

2.2 Analysis of global quantities related to the mechanical energy, the temperature variance, and Nusselt number

Definition of statistical quantities

In this section, we detail some of quantities used in the description of energy exchanges in convective turbulence. It is convenient to introduce the following notation convention regarding temporal and spatial averages, and other meaningful quantities.

Temporal fluctuations

Consider φ(x, t) to be a given field which may be decomposed as follows

φ ′ (x, t) ≡ φ(x, t) -φ(x) φ(x) ≡ t f to φ(x, t) dt (t f -t o ) (2.25)
where φ(x) is the mean field or time-averaged field evaluated in the interval [t o , t f ] and φ ′ (x, t) is the fluctuating field. One introduces as well the variance Var φ, the standard deviation σ(φ), and the skewness coefficient Skew(φ) as 3/2 (2.26)

Var φ(x) ≡ φ ′ φ ′ σ(φ)(x) ≡ (φ ′ φ ′ ) 1/2 Skew(φ)(x) ≡ φ ′ φ ′ φ ′ (Var φ)

Spatial fluctuations

A similar decomposition may be performed in space rather than in time. For instance φ xyz is the average over a volume V , φ xz the average value over a plane A xz , and φ x the average value over a line L x in the x-direction. This introduces the following spatial decomposition (2.28)

{φ} xyz (x, t) ≡ φ(x,
Note that, any of these definitions can be trivially extended to other directions.

Spatio-temporal fluctuations

Additionally, we may use a spatial and temporal average, i.e. φ and introduce the spatio-temporal root mean square φ rms

φ rms ≡ (φ -φ ) 2 = φφ -φ φ (2.29)
where the following property of the averaging operator φ φ = φ φ , is implied. Again this is valid to any spatial average, the direction of which is indicated by the subscript.

Instantaneous Nusselt numbers

One of the main responses of the system under convection is the Nusselt number which quantifies the dimensionless heat-flux across the fluid layer Nu y (y, t) ≡ Ra 0.

Mechanical energy: global kinetic and total potential energies

Consider the instantaneous global kinetic and total potential energies

E k (t) ≡ 1 2 u i u i xyz E p (t) ≡ -Pr yθ xyz
This latter quantifies the amount of energy that is required to bring all fluid parcels against gravity from their position at time t to the reference level y = 0. However, one can introduce a more pertinent instantaneous quantity, namely the available potential energy (see for instance [START_REF] Winters | Available potential energy and mixing in density-stratified fluids[END_REF]Sutherland, 2010) and the notion of a background state of minimal potential energy.

Available and background potential energies

Given a state at time t, the background state associated refers to the temperature field obtained by rearranging inside the fluid domain, the fluid particles in order to decrease as much as possible the global potential energy E p (t). It is easy to understand that θ (x, y,t) the background state has no horizontal dependency and that its density decreases (i.e. temperature increases) monotonically with height. For a given time t, a fluid particle located in (x, y, z) is rearranged into the background state at y r (x, y, z, t). It is safe to assume that if the temperature field is continuous and differentiable, there exist a corresponding field y r (x, t) that is also continuous and differentiable. Since rearranging occurs through virtual adiabatic processes, temperature distribution of the background state is identical to the one of the instantaneous state.

Using the above remarks, it is straightforward to get the temperature distribution θ r (y, t) of the background state from the statistical distribution of temperature at time t. If we consider P θ ( θ) as the probability density function for the instantaneous temperature field and θ ∈ [θ min , θ max ], then y r (θ) is evaluated as the cumulative density function.

y r (θ) -y bot = θ θ min P θ ( θ) d θ (2.32)
Its inverse function corresponds to the temperature distribution of the background state denoted by θ r (y r ). The quantity y r (x, y, z, t) is obtained using the identity θ r (y r , t) = θ(x, y, z, t) (see [START_REF] Tseng | Mixing and available potential energy in stratified flows[END_REF]). The reference height field y r (x, t) illustrates the spatial dependence of the temperature field (figure 2.

2).

The background potential energy E b is defined as the potential energy of the background reference state E b (t) ≡ -Pr y r θ xyz while the change in potential energy between the instantaneous state E p and its background companion E b is defined as the available potential energy E a

E a (t) ≡ E p (t) -E b (t) = -Pr (y -y r )θ xyz
which represents the part of the potential energy which can be effectively transformed into motion (Lorenz, 1955;[START_REF] Winters | Available potential energy and mixing in density-stratified fluids[END_REF].

Energy pathways or energy conversion rates

The energy exchange process may be better grasped by considering the evolution of the mechanical energy and the available potential energy through the following exact relations (see [START_REF] Winters | Available potential energy and mixing in density-stratified fluids[END_REF] and [START_REF] Hughes | Available potential energy in Rayleigh-Bénard convection[END_REF].

Conversion rates for the global kinetic energy

If we multiply by u the momentum equation and average over the fluid volume V , we obtain the following relation for the conversion rate of E k

d dt E k = PrRa -0.5 Ra 0.5 θu y xyz Φy -2 S ij S ij xyz ǫ + V -1 u i ∂ n u i dS Φτ (2.33)
where the first bulk term corresponds to the convective vertical heat-flux Φ y , the second bulk term corresponds to the energy conversion rate due to viscous dissipation ǫ, while the last term is a boundary term that quantifies the input of mechanical energy due to external forcing Φ τ . If one considers the mechanical boundary conditions, from (2.33) it is straightforward to show that Φ τ = 0 for both no-slip and stress-free boundary conditions. The viscous dissipation ǫ, is written in this general form to allow for the use of a stressfree condition in §8, expressed in terms of the velocity gradient and the vorticity field

ω k = ε kji ∂ j u i (ε kji being the Levi-Civita symbol), ǫ(t) ≡ 2 S ij S ij xyz = 2 ∂ j u i ∂ j u i xyz -ω k ω k xyz (2.34) = ∂ j u i ∂ j u i xyz -∂ j u i ∂ i u j xyz = ∂ j u i ∂ j u i xyz -V -1 u i ∂ i u n dS
From (2.34) it is possible to show for no-slip boundary conditions

ǫ(t) = ∂ j u i ∂ j u i xyz (2.35)
From equation (2.31c) it is easily found that Nu vol = (Φ y + 1 Where the second term in the right hand side vanishes when integrated over the fluid domain (see corresponding demonstration on the appendix D.1). Introducing the dynamical equation for the temperature (2.22) leaves

d dt E b = Pr y r ∂ j (u j θ) xyz -PrRa -0.5 y r ∂ j ∂ j θ xyz (2.40)
Based on the spatial dependence of y r on θ(x, t), [START_REF] Winters | Available potential energy and mixing in density-stratified fluids[END_REF] defines an auxiliary function Ψ such that ∂ j Ψ = y r ∂ j θ. Introducing this expression and making use of Green's first identity on the second term in equation (2.40) gives

d dt E b = V -1 Pr S u n Ψ dS + PrRa -0.5 ∂ j y r ∂ j θ -∂ j (y r ∂ j θ) xyz (2.41)
For a closed system, i.e. u n = 0 on all boundaries, leaves the following expression

d dt E b = PrRa -0.5 ∂ j y r ∂ j θ xyz Φ d -V -1 (y r ∂ n θ) dS Φ b2 (2.42)
where the bulk term Φ d quantifies the energy conversion rate due to diapycnal mixing and Φ b2 measures the conversion rate from external sources to E b . Since iso-thermal conditions are imposed on the top and bottom walls, y r (x, -0.5, z, t) = 0.5 and y r (x, 0.5, z, t) = -0.5, and due to the adiabaticity of side-walls, one obtains that Φ b1 = Φ b2 = 1 2 (Nu bot + Nu top ).

The diapycnal mixing can also be written as

Φ d ≡ ∂y r ∂θ ∂ j θ∂ j θ xyz (2.43)
Since by definition ∂yr ∂θ > 0, the mixing term Φ d is bound to be positive.

Conversion rates for the available potential energy

Finally, by substracting equation (2.38)

Global Temperature fluctuations

Consider the dimensionless form of the global temperature fluctuations around the reduced temperature θ = 0,

E θ (t) ≡ 1 2 θ 2 xyz (2.46)
The temperature fluctuation E θ is not an energy per se. However, we have decided to show it here since we follow the same process and resulting terms are alike to those obtained for the potential energy.

If we multiply the temperature equation by θ and integrate over the fluid volume, the rate of change d dt E θ is shown to verify the following relation, 2.47) where the bulk term ǫ θ > 0 corresponds to the global thermal dissipation rate and a boundary term Φ b3 provides the conversion rate from external sources to E θ . Since isothermal conditions are imposed on the top and bottom walls, θ(x, 0.5, z, t) = -0.5 and θ(x, -0.5, z, t) = 0.5, and due to the adiabaticity of side-walls, it becomes clear that

d dt E θ = -Ra -0.5 ∂ j θ∂ j θ xyz ǫ θ -V -1 θ∂ n θ dS Φ b3 (
Φ b1 = Φ b2 = Φ b3 = 1 2 (Nu bot + Nu top ).

Exact relations to the Nusselt number

We evaluate the time-averaged Nusselt numbers from equation (2.30) and (2.31c)

Nu y (y) = Ra 0.5 vθ xz -∂ y θ xz (2.48) Nu vol = Ra 0.5 vθ xyz -∂ y θ xyz (2.49)
For a system with adiabatic side-walls, Nu y has no dependence on the vertical direction y. As such one expects Nu = Nu vol = Nu bot = Nu top .

Since temperature and velocity remain bounded, the time-averaged values of conversion rates for E k , E p , E b , and E a must be zero [START_REF] Hughes | Available potential energy in Rayleigh-Bénard convection[END_REF]. This is also true for the rate of change for E θ . Averaging equations All these quantities should be equal as shown by Shraiman andSiggia, 1990 and[START_REF] Hughes | Available potential energy in Rayleigh-Bénard convection[END_REF]. These relations are valid for all of the configurations considered in this work.

Analysis of global quantities related to vorticity in 2-D

For the study of two-dimensional flows, the flow structure is characterized in term of different global functions of the vorticity. For a two-dimensional configuration x ∈ R 2 , vorticity has only one component perpendicular to the plane (xy), ω = ωe z . The first global quantity is the mean vorticity

C ≡ ω xy (2.53)
which is always zero for no-slip boundary conditions, but is non vanishing for the freesurface.

A second quantity is the global angular impulse L 2D

L 2D ≡ - 1 2 |x| 2 ω xy (2.54)
This quantity provides a measure of the organized motion around the center of a cavity. The flow dynamics and the time evolution of L 2D , may be better understood through some exact relations developed in the study of self-organization in 2-D turbulent flows [START_REF] Molenaar | Angular momentum of forced 2D turbulence in a square no-slip domain[END_REF][START_REF] Van Heijst | The effects of solid boundaries on confined two-dimensional turbulence[END_REF]. The rate-of-change for L 2D may be written as follows

d dt L 2D = PrRa -0.5 I + 1 Γ x II + PrM (2.55a)
with I, II, and M given as

I = (r n ω - 1 2 r k r k ∂ n ω) ds -2C (2.55b) II = 1 2 (u i u i )(n x y -n y x) ds (2.55c) M = - 1 2 r k r k ∂ x θ vol (2.55d)
with ds being an infinitesimal element of the boundary and the loop integral is counterclockwise. The derivation process of (2.55a) is presented on the appendix E. Observe, that imposing no-slip boundary conditions gives in return the same expressions from [START_REF] Molenaar | Angular momentum of forced 2D turbulence in a square no-slip domain[END_REF]. The bulk term M (boundary term resp. II) corresponds to the contributions from the external thermal (resp. mechanical) forcing to L 2D , and can be seen as a form of external torque. However, it is difficult to provide an interpretation for I, other than being a viscous response of the system to the external torque.

Analysis of spatial structure of the flow: thermal and kinetic boundary layers

The thermal and kinetic boundary layers have a primordial role on the efficiency of heat transport and on different statistical properties. Near the top and bottom plates we define thermal (resp. kinetic) boundary layers of thickness δ θ (resp. δ p ). A kinetic boundary layer of thickness δ w may be also defined along the side-walls. In this section we present different estimates of the boundary layer thicknesses.

Thickness of the thermal boundary layers

i.) Based on the Nusselt number. We suppose that the temperature gradient is concentrated at the top and bottom boundary layers of constant thickness δ θ , and the center region, called the bulk of almost constant temperature 1 2 (θ top + θ bot ). As a consequence

δ θ ≈ 1/(2Nu) (2.56)
provides an estimate of the boundary layer thickness by supposing a top-bottom symmetry.

ii.) Based on the root mean squared temperature profile. One alternative considers the fluctuations of the temperature field to characterize the average thickness of the thermal boundary layer. A local thickness may be derived from the temperature temporal fluctuations σ(θ(x, t)), as done for instance in [START_REF] Wang | Spatial variations of the mean and statistical quantities in the thermal boundary layers of turbulent convection[END_REF]. Another estimate of the average thickness is obtained from the distance of the closest peak in the spatio-temporal fluctuations θ rms xz to the top and bottom plates, as done for instance in [START_REF] Kerr | Rayleigh number scaling in numerical convection[END_REF]S. Wagner, and Shishkina, 2013). We use the latter estimate except when specified otherwise.

iii.) Based on the thermal dissipation profile. The use of thermal dissipation layer based on the spatial distribution of the thermal dissipation rate has been proposed [START_REF] Petschel | Dissipation Layers in Rayleigh-Bénard Convection: A Unifying View[END_REF]. The boundary layer thickness is estimated by the closest distance from the top and bottom plates to the vertical position where (∂ j θ)(∂ j θ) xz (y) = Nu.

Thickness of the kinetic boundary layers at bottom and top plates

i.) Based on the root mean squared horizontal velocity profile. One alternative considers the spatio-temporal fluctuations of the horizontal velocity component to characterize the kinetic boundary layer at the plates, for instance see [START_REF] Kerr | Rayleigh number scaling in numerical convection[END_REF]. The average thickness is estimated from the closest distance from the top and bottom plates to the peak in u rms xz . However, this definition supposes to the presence of a rigid boundary surface. In the following, we use this estimate except when specified otherwise.

ii.) Based on the viscous dissipation profile. The spatial distribution of the viscous dissipation rate has been proposed [START_REF] Petschel | Dissipation Layers in Rayleigh-Bénard Convection: A Unifying View[END_REF] as an alternative to the fluctuation profiles. In this case, the boundary layer thickness is estimated by the closest distance from the top and bottom plates to the vertical position where

∂ j u i ∂ j u i xz (y) = Nu -1.
In the 3-D cases, we prefer to use a slightly different definition: instead of the transversal average, we use a definition based on the median plane z = 0, i.e. the thickness is based on the closest distance between the position where ∂ j u i ∂ j u i x (z = 0) = (Nu -1) to the top and bottom plates. The median plane z = 0 is chosen in order to avoid as much as possible the influence of the side-walls.

In the 2-D cases, in chapters §4 to §6 we use

∂ j u i ∂ j u i x (y) = Nu -1.

Analysis of spatial structure of the flow: Half-range Fourier decomposition

A simplified representation of the large-scale flow by projecting into Fourier modes has been used extensively in configurations such as 2-D cells [START_REF] Chandra | Dynamics and symmetries of flow reversals in turbulent convection[END_REF] and cylindrical cells (Xi and Xia, 2008). Ideally, the basis for such projection should verify the boundary conditions and respect the natural symmetries of the targeted system. Furthermore, selecting the relevant modes requires a great deal of knowledge of system. For simplicity, let us consider a square (2-D) cell with isothermal top and bottom walls.

If we consider the cell to be centered around (x = 0, y = 0), the following symmetries are identified [START_REF] Podvin | A large-scale investigation of wind reversal in a square Rayleigh-Bénard cell[END_REF]:

i. A reflexion symmetry with respect to the vertical axis,

S x    u v θ    (x, y) →    -u v θ    (-x, y) (2.57)
ii. A reflexion symmetry with respect to the horizontal axis, S y

   u v θ    (x, y) →    u -v -θ    (x, -y) (2.58)
iii. A rotation of origin of the cell center, R π = S x ⊙ S y

In order to observe the spatial dependence of the temperature and velocity field, [START_REF] Das | Asymmetric squares as standing waves in Rayleigh-Bénard convection[END_REF] proposed fields to be projected into modes describing rolls. This decomposition [START_REF] Das | Asymmetric squares as standing waves in Rayleigh-Bénard convection[END_REF].

was initially proposed in the context of stress-free isothermal boundaries and periodic sidewalls. This projection has been used extensively by [START_REF] Chandra | Dynamics and symmetries of flow reversals in turbulent convection[END_REF][START_REF] Chandra | Flow reversals in turbulent convection via vortex reconnections[END_REF] in the study reversals in a 2-D Rayleigh-Bénard cell. The primitive fields are expressed as a half-Fourier series in the plane (x, y)

u(x, y, t) = p,q≥1 u pq (t) sin(pπx) cos(qπy) (2.59a) v(x, y, t) = p,q≥1 v pq (t) cos(pπx) sin(qπy) (2.59b) Θ(x, y, t) = p,q≥1
Θ pq (t) cos(pπx) sin(qπy) (2.59c) where Θ(x, y, t) ≡ θ(x, y, t) + y is the (dimensionless) temperature deviation from the conduction state such that Θ = 0 on the top and bottom plates. The series coefficients are obtained using the discrete sine and discrete cosine transform as required. A schematic representation of the first couple modes in both directions is shown in figure 2.3. While this decomposition verifies the symmetries of the system, but does not satisfy the no-slip boundary conditions. [START_REF] Chandra | Dynamics and symmetries of flow reversals in turbulent convection[END_REF][START_REF] Chandra | Flow reversals in turbulent convection via vortex reconnections[END_REF] sought to explore the relation between selected Fourier modes inside a regime of consecutive reversals for Pr = 1 by exploring the natural symmetries of 2-D Rayleigh-Bénard cells. Each Fourier mode belongs to one of four possible groups: even-even (E), odd-odd (O), even-odd (M eo ), and odd-even (M oe ). The quadratic non-linear interaction between two such modes results on a mode with a particular symmetry. More precisely, they respect the following rules:

⊗ O E M eo M oe O E O M oe M eo E E M eo M oe M eo E O M oe E Table 2
.1: Symmetric matrix summarizing the quadratic non-linear interaction between even-even (E), odd-odd (O), even-odd (M eo ), and odd-even (M oe ) modes.

For instance O ⊗ O = E means that non-linear interaction between two odd modes results in an even mode.

Because of the above, the following symmetry operators

i) E → E, O → -O, M eo → M eo , M oe → -M oe ii) E → E, O → -O, M eo → -M eo , M oe → M oe iii) E → E, O → O, M eo → -M eo , M oe → -M oe
leave the Boussinesq equations in the Fourier space invariant [START_REF] Chandra | Flow reversals in turbulent convection via vortex reconnections[END_REF][START_REF] Verma | Flow reversals in turbulent convection with free-slip walls[END_REF]. As a result, if a set of modes {E, O, M eo , M oe } is a solution of the Boussinesq equations, then {E, -O, M eo , -M oe }, {E, -O, -M eo , M oe }, and {E, O, -M eo , -M oe } are also a solution of these equations. Additional symmetry operators may be defined by considering fluctuating modes with zero mean [START_REF] Verma | Flow reversals in turbulent convection with free-slip walls[END_REF].

In this sense, if we have a configuration that exhibits a flow reversal. Then, the flow structure observed after the reversal, is similar to the one obtained if one applies these symmetry operators to the structure observed before the reversal. This line of reasoning becomes particularly useful to identify the modes that play an active role during the re-organization and reorientations of the large-scale flows.

Analysis of spatial structure of the flow: Proper orthogonal decomposition (POD)

The proper orthogonal decomposition (POD) is a procedure for extracting an optimal basis of decomposition from a data set ψ(x, t) (see for instance [START_REF] Berkooz | The proper orthogonal decomposition in the analysis of turbulent flows[END_REF][START_REF] Podvin | Introduction à la décomposition orthogonale aux valeurs propres (pod)[END_REF][START_REF] Holmes | Turbulence, coherent structures, dynamical sy 2nd ed[END_REF]. A given field ψ(x, t) is expanded in a set of orthonormal functions φ k which satisfy the boundary conditions. We define a coherent structure by a function φ that maximizes the energy content φ|ψ 2 / φ|φ (2.60) where the operator •|• corresponds to the complex inner product operator, defined in R 2 (resp. R 3 ) as a|b = (a * b)dxdy (resp. a|b = (a * b)dxdydz) where a * indicates the complex conjugate of a. It is shown, that this implies that φ is an eigenfunction of the symmetric two-point correlation tensor R(x, y) = ψ(x, t)ψ(y, t)

R(x, y)φ(y)dy = λφ(x) (2.61)
The Hilbert-Schmidt theory implies there exists not one, but a denumerable infinity of solutions, or empirical eigenfunctions φ k , which compose an orthonormal basis, i.e.

φ j |φ k = δ jk .
The modal decomposition of ψ(x, t) into the empirical base functions φ k reads as

ψ(x, t) = ∞ k=1 α k (t)φ k (x) (2.62)
where the corresponding modal coefficients

α k α k (t) = ψ(x, t)|φ k (x) (2.63) are uncorrelated to each other α * j α k = δ jk λ k (2.64)
The eigenvalue λ k can be seen as the mean energy of the system projected on the φ k -axis in the function space.

λ k = | ψ|φ k | 2 (2.65)
This implies that the energy of the flow is such that

ψ(x, t)|ψ(x, t) = ∞ k=1 α k α k ψ(x, t)|ψ(x, t) = ∞ k=1 λ k (2.66)

The snapshot method

If we consider a discretized domain of size N x by N y in 2D (resp. N x by N y by N z in 3D), the discretized correlation tensor R ij (x, x ′ ) is of size 4N x N y for measurements in 2D (resp. 9N x N y N z for measurements in 3D). Because of this large size, it is often not possible to obtain a direct solution of equation (2.61). An alternative may be found in the snapshot method proposed by [START_REF] Sirovich | Turbulence and the dynamics of coherent structures part i: coherent structures[END_REF]. This method relies on the ergodicity hypothesis to approximate the correlation tensor as follows

R(x, y) = M m=1 ψ(x, t m )ψ(y, t m ) with t m = m∆t (2.67)
for large M . The time ∆t between snapshots is assumed to be sufficiently large for snapshots to be uncorrelated. Introducing hypothesis (2.67) into equation (2.61) results in a degenerate integral equation and solutions can be expressed as a linear combination of the fields ψ(x, t m ).

φ k (x) = M m=1 α km ψ(x, t m ) (2.68)
where the coefficients α km verify the following equation

K mn α kn = λ k α km (2.69)
with the matrix K mn defined as

K mn = 1 M ψ(x, t m )|ψ(x, t n ) (2.70) leaving one eigenvalue problem of size M 2 instead of 4N x N y in 2D (resp. 9N x N y N z in 3D).

Different formulations of the POD for convective flows

For thermal convection, different alternative formulations of the POD can be defined. Velocity and temperature fields may be treated separately or jointly. And one may apply the POD to the complete or the fluctuating fields. Consider the following options for two-dimensional problems All of which are admissible. The latter is formulated with the intent of capturing the velocity/temperature coupling and commonly is used in natural convection, see for instance [START_REF] Podvin | Low-order models for the flow in a differentially heated cavity[END_REF][START_REF] Bailon-Cuba | Aspect ratio dependence of heat transfer and large-scale flow in turbulent convection[END_REF][START_REF] Podvin | A large-scale investigation of wind reversal in a square Rayleigh-Bénard cell[END_REF]. Since both fields represent different physical quantities, for the joint-formulation we require to introduce arbitrary scale factors a and b. In our case, we use a = b = 1. Given the difficulty to obtain proper time-averaged fields that respect the natural symmetries of the system, we perform the POD on the complete fields.

ψ(x, t) =          θ (A) Temperature formulation (u, v) (B)
For each case, we assemble a matrix K mn

K mn =          1 M θ m θ n dx for formulation (A) 1 M (u m u n + v m v n )dx for formulation (B) 1 M (u m u n + v m v n + θ m θ n )dx for formulation (C) (2.72)
and solve an eigenvalue problem (2.69) providing λ k and α km . One may then proceed to assemble our orthonormal functions φ k (x) for each formulations. For the formulation (C), the POD analysis provides both temperature and velocity fields.

(φ

(u) k , φ (v) k , φ (θ) k )(x) = M m=1 α k (t m ) ψ(x, t m ) for formulation (C) (2.73)
For formulation (A), only modes corresponding to the temperature field are obtained, i.e. φ (θ)

k (x) = M m=1 α k (t m ) θ(x, t m ).
Thereafter, the educed POD modes for the velocity field may be computed to ensure the same modal coefficient [START_REF] Borée | Extended proper orthogonal decomposition: a tool to analyse correlated events in turbulent flows[END_REF] (φ

(u) k , φ (v) k )(x) = M m=1 α k (t m ) u(x, t m ) for formulation (A) (2.74)
For the velocity formulation (B), one obtains first (φ

(u) k , φ (v) k ) = M m=1 α k (t m ) u(x, t m )
, and then the educed temperature field,

φ (θ) k (x) = M m=1 α k (t m ) θ(x, t m )
for formulation (B) (2.75) to ensure the same modal coefficient. In formulations (A) and (B), it is important to observe that orthogonality is not preserved for the educed modes.

In the following chapters, we perform the POD analysis on a relatively large dataset using an implementation of the snapshot method written in modern FORTRAN, based on an existing and freely available F77 routine [START_REF] Pierce | Empirical Orthogonal Function (EOF) software[END_REF].

29

Chapter 3

Direct numerical simulations (DNS) of the Boussinesq equations

In this chapter, we present the numerical methodology used for the investigation of natural convection. In the first part, we provide a minimal description of the numerical method and the most distinctive features of the two numerical codes we used. In the second part, we present details of the validation process of both codes. The first test case corresponds to a differentially heated cavity [START_REF] Tric | A first incursion into the 3D structure of natural convection of air in a differentially heated cubic cavity, from accurate numerical solutions[END_REF] and allows to validate the incompressible flow solver with the buoyancy effect using no-slip conditions, adiabatic and iso-thermal boundary conditions. The second test case corresponds to a rectangular (3-D) Rayleigh-Bénard cell (S. [START_REF] Wagner | Aspect-ratio dependency of Rayleigh-Bénard convection in box-shaped containers[END_REF]. It is mostly focused on reproducing the onset of unsteady motion as function of the aspect ratio inside a rectangular cell. We conclude with a series of performance tests, demonstrating that the Basilisk code can be run efficiently and without modification in distributed memory parallel machines.

Presentation of the numerical codes: Fusion and Basilisk

The first computing code Fusion is written in modern Fortran and was developed over the past two decades within the LIMSI. This code is specific to natural convection and solves the 3D Navier-Stokes equations under the Boussinesq approximation on Cartesian grids using a second-order accuracy scheme. The code was modified to support shared-memory parallelism through OpenMP directives. The second computing code uses an extension of the C programming language known as Basilisk C. A number of pre-defined solver blocks can be put together to solve partial differential equations on adaptive Cartesian meshes. At the time of this writing, Basilisk C supports both the shared-memory (OpenMP) and the distributed-memory (MPI) parallelism models.

By the time we began this work, Basilisk was in early development at the Institut ∂'Alembert with the intention to become the successor of Gerris [START_REF] Popinet | Gerris: a tree-based adaptive solver for the incompressible euler equations in complex geometries[END_REF]. Results from Basilisk were cross-validated against Fusion during every stage of the development process to ensure the integrity of the numerical simulations. Additional work included developing a series of post-processing tools and a parallel I/O implementation of the XDMF format [START_REF] Clarke | A distributed computing environment for interdisciplinary applications[END_REF][START_REF] Clarke | Enhancements to the extensible data model and format (xdmf)[END_REF]XDMF Standard, 2016) which is designed to store and organize large amounts of data, briefly introduced in at the end of this chapter.

Code FUSION

FUSION is a finite volume code written in FORTRAN tailored for natural convection problems under the Boussinesq approximation. Numerical solutions are obtained using a finite volume method with second-order accuracy in space and in time. The Boussinesq equations are discretized in space using a central difference approximation on a staggered grid and in time using the second-order backward Euler method. To ensure the numerical stability, the viscous terms are considered implicitly, whereas the convective terms are evaluated explicitly and approximated using an second-order Adams-Bashforth scheme.

Due to the incompressibility condition and the pressure gradient involved, the velocitypressure coupling is more complex. We use a rotational incremental-pressure correction scheme (see for instance [START_REF] Chorin | On the convergence of discrete approximations to the Navier-Stokes equations[END_REF][START_REF] Guermond | An overview of projection methods for incompressible flows[END_REF]. In this scheme, the pressure term is treated explicitly to obtain a provisional velocity field u * which may not be divergence-free. We introduce a pressure potential Π

∂ 2 i Π = 3 2∆t ∂ i u i *
with ∂ n Π = 0 on the boundaries from which a correction to velocity and pressure fields may be evaluated.

The discretized equations have the following form

∂ i u i n+1 = 0 (3u i * -4u i n + u i n-1 ) 2∆t + ∂ j (2u j n u i n -u j n-1 u i n-1 ) = -∂ i p n + PrRa -0.5 ∂ 2 j u i * + Prθ n+1 δ i2 (3θ n+1 -4θ n + θ n-1 ) 2∆t + ∂ j (2u j n θ n -u j n-1 θ n-1 ) = Ra -0.5 ∂ 2 j θ n+1
In this way, the problem is reduced to the solution of one 3D Helmholtz-Poisson problem for each primitive variable (u,v,w, and θ) and one Poisson problem for the pressure correction term Π.

For each of the 3-D Helmholtz-Poisson problems we use the Douglas-Rachford Alternating Direction Implicit (ADI) method to separate the operators into one-dimensional components and split the scheme into one sub-step for each coordinate. Consider the following model equation of the form

1 -D∂ 2 j φ n+1 = Q (3.2)
Here φ represents the primitive variables φ = {u, θ}, D the corresponding diffusion coefficients D = { 2∆t 3 PrRa -0.5 , 2∆t 3 Ra -0.5 }, and Q the corresponding source terms. The main component the ADI method is the factorization of the right-hand side of our model equation as the product of one-dimensional operators in the following form [START_REF] Ferziger | Computational methods for fluid dynamics[END_REF].

1 -D∂ 2 j φ n+1 = 1 -D∂ 2 x 1 -D∂ 2 y 1 -D∂ 2 z φ n+1 -D 2 (∂ 2 x ∂ 2 y + ∂ 2 x ∂ 2 z + ∂ 2 y ∂ 2 z )φ n+1 + D 3 ∂ 2 x ∂ 2 y ∂ 2 z φ n+1 (3.3)
By dropping O(∆t 2 ), i.e. D 2 and D 3 terms, we obtain the following simplified expression

1 -D∂ 2 x 1 -D∂ 2 y 1 -D∂ 2 z φ n+1 = Q
Using a multi-step approach leaves a 1-D Helmholtz problem for each ADI sub-step

1 -D∂ 2 x φ n+ 1 3 = Q w/ BC on e x 1 -D∂ 2 y φ n+ 2 3 = φ n+ 1 3 w/ BC on e y 1 -D∂ 2 z φ n+1 = φ n+ 2 3 w/ BC on e z
We use an staggered grid arrangement [START_REF] Harlow | Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface[END_REF] with a central differentiation scheme (CDS) and linear interpolation for an overall second-order precision. For this arrangement one defines different type of grids. A cell centered grid for the scalar quantities, such as the temperature, pressure and density fields, and one additional grid for each velocity component. In this way, each ADI sub-step takes the form of a tridiagonal system that can be solved directly using the Thomas algorithm. The bidiagonalization algorithm is used for the Poisson problem, sometimes referred to as a fast Poisson solver [START_REF] Golub | A fast poisson solver for the finite difference solution of the incompressible navier-stokes equations[END_REF][START_REF] Strang | Computational science and engineering[END_REF][START_REF] Shishkina | Simulation of turbulent thermal convection in complicated domains[END_REF]. The time step is fixed to ensure that CFL=0.5 is verified

u ∆x + v ∆y + w ∆z ∆t ≤ CFL (3.4)
where ∆x, ∆y, and ∆y correspond to the dimensions of the control volume along the x, y, and z directions, respectively. A detailed explanation of the numerical method used in FUSION is given in the annex A.

Code BASILISK

Basilisk is an ensemble of solver-blocks written using an extension to the C programming language, called Basilisk C, useful to write discretization schemes in Cartesian grids (Stéphane Popinet, 2016). Instead of writing an entirely new code, existing blocks of code were combined to solve Boussinesq equations. The formulation of the Boussinesq equations uses the fractional-step method using a staggered in time discretization of the velocity and the scalar fields (see [START_REF] Popinet | Gerris: a tree-based adaptive solver for the incompressible euler equations in complex geometries[END_REF][START_REF] Popinet | An accurate adaptive solver for surface-tension-driven interfacial flows[END_REF]: one supposes the velocity field to be known at time n and the scalar fields (pressure, temperature, density) to be known at time n-1/2, and one computes velocity at time n+1 and scalars at time n+1/2.

We use a pressure correction scheme (see for instance [START_REF] Chorin | On the convergence of discrete approximations to the Navier-Stokes equations[END_REF][START_REF] Guermond | An overview of projection methods for incompressible flows[END_REF] for the velocity-pressure coupling. In this scheme, the pressure term is treated explicitly to obtain a provisional velocity field u * which may not be divergence-free. A correction term is obtained by projecting the provisional velocity into a divergence-free space, similar to Fusion. The discretized equations would have the following form

∂ i u i n+1 = 0 (3.5a) u i * -u i n ∆t + ∂ j (u j u i ) n+ 1 2 = PrRa -0.5 ∂ j (∂ j u i * + ∂ j u i n ) 2 -Prθ n+ 1 2 δ i2 (3.5b) u i n+1 = u i * -∆t ∂ i p n+ 1 2 (3.5c) θ n+ 1 2 -θ n-1 2 ∆t + ∂ j (u j θ) n = ∂ j (Ra -0.5 ∂ j θ) n+ 1 2 (3.5d)
Space is discretized using a Cartesian (multi-level or tree-based) grid where the variables are located at the center of each control volume (a square in 2-D, a cube in 3-D) and at the center of each control surface.

The velocity advection term ∂ j (u j u i ) n+ 1 2 and the temperature advection term ∂ j (u j θ) n are estimated by means of the Bell-Colella-Glaz second-order unsplit upwind scheme [START_REF] Bell | A second-order projection method for the incompressible Navier-Stokes equations[END_REF][START_REF] Popinet | Gerris: a tree-based adaptive solver for the incompressible euler equations in complex geometries[END_REF]. In this way, the problem is reduced to the solution of a 3D Helmholtz-Poisson problem for each primitive variable (u * , v * , w * , θ) and a Poisson problem for the pressure correction terms. Both the Helmholtz-Poisson and Poisson problems are solved using an efficient multilevel Poisson solver described in [START_REF] Popinet | Gerris: a tree-based adaptive solver for the incompressible euler equations in complex geometries[END_REF]. Basilisk uses a variable time-step to ensure the CFL condition (3.4) is verified. Observe that (3.4) is a necessary condition, but does not automatically ensures the stability of the numerical method. Additional restrictions on the time-step may be required. Additional details on the numerical method used in Basilisk are given in the annex B.

Implementation of a parallel Input/Output (I/O) model

I would like to present some of the work done for retrieving and storing the large amount of data generated from direct numerical simulations. This work was motivated by a healthy need to store and organize large sets of data which greatly facilitates the post-processing and by a growing trend to implement standard data formats in scientific computing. To this end, we implemented the eXtensible Data Model and Format (see [START_REF] Clarke | A distributed computing environment for interdisciplinary applications[END_REF][START_REF] Clarke | Enhancements to the extensible data model and format (xdmf)[END_REF] for outputs produced in both Fusion and Basilisk. It is worth noting that other formats, like netCDF, are also widely used. Each one has its own set of strengths and weaknesses. We choose to use the XDMF format since the libraries required are usually available in large computing centers. The XDMF format has a low entry barrier and supports the use of hierarchical structures inside the data files. Additional features include support for a large number of objects, file compression, a parallel I/O implementation through the MPI-IO or MPI POSIX drivers. A software library that provides a high-level API with interfaces to both C and Fortran, can be usually found already installed in a large number of computing centers.

The amount of I/O operations in large scale simulations may consume an important part of the execution time. One of the most common approaches consists in sending data to a single MPI task, which is responsible of writing data into the disk. This approach is not scalable. Another option consist in writing one file per MPI task, but this implies handling large numbers of files. Our implementation of parallel I/O uses PHDF5 (Parallel HDF5) and is divided in three parts. First, the MPI tasks collectively create/open a single parallel and Pr = 0.71 to be compared against [START_REF] Tric | A first incursion into the 3D structure of natural convection of air in a differentially heated cubic cavity, from accurate numerical solutions[END_REF]. (b) Rectangular Rayleigh-Bénard cell with aspect ratio Γ z = 1/10 for Ra = 10 6 and Pr = 0.786 to be compared against (S. Wagner, and Shishkina, 2013).

HDF5 file and create/open groups, sub-groups and so on. Second, each MPI task is set to write a part or a chunk of the data into each dataset independently. Once each task has finished writing into the file, the MPI tasks collectively close the datasets, before closing the file. This strategy has been implemented on the Ada and Turing parallel computing clusters (IDRIS, 2016a; IDRIS, 2016b).

Validation of the numerical codes: case studies

In order to validate the two numerical codes we compare their results against benchmark results for two different configurations within the Boussinesq approximation.

Case study 1: Differentially heated cavity

The first test case is a classical problem and has become a very popular to compare the different numerical algorithms available for incompressible flows. For this test case, commonly known as a differentially heated cavity, we will study both 2-D and 3-D cavities.

The fluid domain is a 2-D square (resp. 3-D square box) cavity of size 1, centered around (0, 0) (resp. (0, 0, 0)), see figure 3.1a. We consider no-slip boundary conditions in all borders, imposed temperature in the left and right walls, θ(±0.5, y) = ±0.5 in 2-D (resp. θ(±0.5, y, z) = ±0.5 in 3-D). Adiabatic boundary conditions are imposed on all other boundaries.

The fluid is initially at rest and a linear temperature gradient is imposed in the horizontal direction. We start the simulation on a very coarse regular Cartesian grid and run until a steady-state condition is met, i.e. the maximum change in the x-velocity component over two consecutive time steps is lower than 10 -8 . This final state is then used as an initial condition for a finer grid size and it is run until a new steady-state is obtained. This process is repeated until convergence is reached.

The benchmark solutions used for the validation of the two-dimensional differentially heated square cavity are [START_REF] Vahl | Natural convection of air in a square cavity: a benchmark solution[END_REF][START_REF] Quéré | Accurate solutions to the square thermally driven cavity at high Rayleigh number[END_REF][START_REF] Tric | A first incursion into the 3D structure of natural convection of air in a differentially heated cubic cavity, from accurate numerical solutions[END_REF], while solutions for the three-dimensional cavity are taken from [START_REF] Tric | A first incursion into the 3D structure of natural convection of air in a differentially heated cubic cavity, from accurate numerical solutions[END_REF].

Differentially heated 2-D square cavity

Numerical results used for validation inside a square cavity can be of several kinds: different global quantities, profiles measured along a vertical or horizontal line, or local measurements. Figure 3.2 displays ψ iso-lines superposed to the temperature field θ(x, t) evaluated at the steady state. These results are in agreement with figure 4 (Ra = 10 7 ) and figure 6 (Ra = 10 8 ) from [START_REF] Quéré | Accurate solutions to the square thermally driven cavity at high Rayleigh number[END_REF]. In addition, one may compare the peak value of the stream-function ψ max and its position. The placement of ψ max is indicated in figure 3.2. We measure the relative error of ψ max

e ψmax = ψ (N ) max -ψ max,ref ψ max,ref × 100 (3.6)
The evolution of e ψmax as function of the grid spacing is displayed on figure 3.3a for Ra = 10 7 and Ra = 10 is computed as function of the grid spacing for Ra = 10 7 and Ra = 10 8 (see figure 3.3b). Again, results are consistent with a global second order accuracy.

Other quantities used in benchmark results include the components of velocity field evaluated along a vertical or horizontal centerline (figures 3.4a,d and 3.4b,e), or the local heat-flux evaluated in one of the iso-thermal walls (figure 3.4c,f). Figure 3.4 displays the numerical results obtained using Basilisk (figures 3.4a to 3.4c) and Fusion (figures 3.4d to 3.4f), and as well as the benchmark results from [START_REF] Tric | A first incursion into the 3D structure of natural convection of air in a differentially heated cubic cavity, from accurate numerical solutions[END_REF]. Both codes are in good agreement with previous studies and with each other. , Ra = 10 8 . Marks correspond to results from [START_REF] Tric | A first incursion into the 3D structure of natural convection of air in a differentially heated cubic cavity, from accurate numerical solutions[END_REF]. Figures (a,b,c) correspond to results from Basilisk. Figures (d,e,f) correspond to results from Fusion. [START_REF] Tric | A first incursion into the 3D structure of natural convection of air in a differentially heated cubic cavity, from accurate numerical solutions[END_REF] as a function of the grid spacing. Dashed line correspond to ideal slope for a second order scheme.

Differentially heated 3-D cubic cavity

3D calculations with Fusion and Basilisk have been performed for four values Ra < 10 6 using different grids up to 512 3 , see table 3.1c. All converge into a steady-state flow, see for instance figure 3.1a. To test the convergence of Basilisk we use the following Nusselt numbers Nu(z) ≡ 0.5 -0.5 q x (-0.5, y, z)dy (3.8) Nu 3D ≡ 0.5 -0.5

Nu(z)dz (3.9) Nu mp ≡ Nu(z) with z = 0 (3.10)

Values of Nu 3D and Nu mp display a monotonic convergence towards the reference values consistent with a second order method (figure 3.5). We are also interested in comparing the departure from bi-dimensionality of the steady-state for the different codes. More precisely, we evaluate the relative difference between these terms (Nu 2D -Nu 3D )/Nu 2D , (Nu 2D -Nu mp )/Nu 2D , and (Nu 3D -Nu mp )/Nu 3D (figure 3.6), where Nu 2D is the Nusselt number obtained for a 2D square cavity. In addition to the convergence of Nu 3D and Nu mp , one may also consider the evolution of Nu(z) displayed on figure 3.7a. Another possibility consists in following the maximum amplitude of the velocity components measured over the vertical and horizontal centerlines: max (u(0, y, 0)) and max (w(x, 0, 0)). Let us consider the vertical (resp. horizontal) velocity profiles measured over the horizontal (resp. vertical) centerlines (figures 3.7b and 3.7c). This is confirmed in table 3.2, where the values of max (u(x, y, z)), max (v(x, y, z)), max (w(x, y, z)) are found.

Results from both codes are in good agreement with benchmark results and with each other. In most cases, results obtained from Basilisk were found to be more accurate than Fusion for an equivalent number of grid points. Ref.

Nu 2D -Nump Nu 2D
Ref.

Nu 3D -Nump Nu 3D

Figure 3.6: Relative heat transfer rates as a function of Ra for Pr = 0.71. Reference results from [START_REF] Tric | A first incursion into the 3D structure of natural convection of air in a differentially heated cubic cavity, from accurate numerical solutions[END_REF] shown in ⋄ marks, results from Fusion shown in marks and results from Basilisk in • marks. Nu 3D and Nu mp as defined in equation (3.8), while Nu 2D corresponds to the Nusselt number obtained for the 2D differentially heated cavity. [START_REF] Tric | A first incursion into the 3D structure of natural convection of air in a differentially heated cubic cavity, from accurate numerical solutions[END_REF].

Case study 2: Rectangular Rayleigh-Bénard cells

The second test-case focuses on a rectangular Rayleigh-Bénard cell near the onset of convection1 . The fluid domain is a box with equal height and width, while the depth is defined by the aspect ratio Γ z = d/H and ranges from 1/10 to 1/3, see figure 3.1b. We consider no-slip boundary conditions in all borders, adiabatic conditions on the side-walls and imposed temperature on the top and bottom walls θ(x, ±0.5, z) = ∓0.5.

The fluid is initially at rest and a linear temperature gradient is imposed in the vertical direction. To avoid waiting for the transient to end, we speed up the process as follows: we start the simulation on a very coarse regular Cartesian grid and run until statistical convergence is met (in practice, this corresponds to several hundred convective time units).

The final state is then used as an initial condition for a finer grid size and it is run again with the same criteria. This process is repeated until convergence is reached, see table 3.2.

In the following, we focus only on comparing results produced using Basilik and benchmark results. The same test case was performed using Fusion, but is not presented for simplicity since there are no notable differences between the results. The benchmark solutions used for the validation are results obtained using a fourth order accurate finite volume method by (S. [START_REF] Wagner | Aspect-ratio dependency of Rayleigh-Bénard convection in box-shaped containers[END_REF] inside rectangular Rayleigh-Bénard cells for for (Ra = 10 6 , Pr = 0.786) and three different aspect ratios: Γ z = 1/10, Γ z = 1/4, and Γ z = 1/3. For these values of (Ra, Pr) the influence of the aspect ratio on the flow dynamics and on the Nusselt was found to be quite strong: a steady state solution is observed for Γ z = 1/4, whereas Γ z = 1/10 and Γ z = 1/3, result in a unsteady flow: for Γ z = 1/10 one obtains a type of oscillatory solution, whereas for Γ z = 1/3 the solution is oscillatory and periodic (S. [START_REF] Wagner | Aspect-ratio dependency of Rayleigh-Bénard convection in box-shaped containers[END_REF].

Figures 3.8a and 3.8b display a slice taken at the vertical mid-plane of the instantaneous temperature field with the velocity vector field for Γ z = 1/10 and Γ z = 1/3 corresponding to unsteady-state solutions. For Γ z = 1/10, a (quasi-)two-dimensional flow is observed. This configuration is characterized by the presence of large thermal plumes along the side-walls, and compares well with figure 11a in (S. [START_REF] Wagner | Aspect-ratio dependency of Rayleigh-Bénard convection in box-shaped containers[END_REF]. For Γ z = 1/3, one observes a single-roll similar to figure 11b in (S. Wagner, and Shishkina, 2013). In addition, a 3-D plot (figure 3.8c) illustrates the roll axis is located in the horizontal plane but twisted from the z axis. This is also consistent with (S. [START_REF] Wagner | Aspect-ratio dependency of Rayleigh-Bénard convection in box-shaped containers[END_REF].

A time series is displayed along with the time-frequency spectra of Nu top and Nu bot in figures 3.9a to 3.9d. On the one hand, for Γ z = 1/10, the values of Nu top and Nu bot oscillate out-of-phase with each other but display the same dominant frequency f = 0.1026 ± 0.004 compared to f = 0.107 reported by (S. [START_REF] Wagner | Aspect-ratio dependency of Rayleigh-Bénard convection in box-shaped containers[END_REF], see figure 3.9b. On the other hand, for Γ z = 1/3 the values of Nu top and Nu bot oscillate in-phase with a single characteristic frequency f = 0.0482 ± 0.004 compared to f = 0.0500 reported by (S. [START_REF] Wagner | Aspect-ratio dependency of Rayleigh-Bénard convection in box-shaped containers[END_REF], see figures 3.9c and 3.9d.

An alternative validation of Basilisk consists in verifying numerically the exact timedependent relations between the rate-of-change of the kinetic energy d dt E k , of the potential energy d dt E p , of the available potential energy d dt E a , and the corresponding energy transfer rates (see §2)

Ra 0.5 Pr -1 d dt E k = Nu vol -(ǫ + 1) Ra 0.5 Pr -1 d dt E p = 1 2 (Nu bot + Nu bot ) -Nu vol Ra 0.5 Pr -1 d dt E a = Nu bot + Nu bot -Nu vol -Φ d Ra 0.5 d dt E θ = 1 2 (Nu bot + Nu bot ) -ǫ θ (3.11)
The left hand sides of equations 3.11 is approximated using a second-order finite differences approximation, see figure 3.10 for Γ z = 1/10 and Γ z = 1/3. Evaluating the time derivatives introduces an error function of the sampling frequency. The corresponding right hand side is evaluated using the instantaneous Nusselt numbers and the energy transfer rates and matches the left hand side satisfactorily. This agreement is useful to evaluate the adequacy of the spatial resolution. This check is similar to one proposed by R. Stevens, Verzicco, We conclude the validation of both numerical codes. Both codes are consistent with a global second order method and the reference literature. In the following, we discuss a more practical aspect of the numerical simulations, regarding the time and computing resources required to perform direct numerical simulations at Ra ∼ 10 7 -10 9 . In the following we focus on the code Basilisk which is better suited for large parallel computations.

Parallel performance of the numerical codes

There are two important reasons for using a parallel computer : to have access to more memory or to obtain higher performance [START_REF] Eijkhout | Introduction to High Performance Scientific Computing[END_REF]. While the available memory is the sum of the individual memory of the machines, the performance gains or speed-up in a parallel execution are more difficult to measure due to the additional overhead (redundant, idle, and extra computation and time spent in communications). Two metrics are used to quantify the performance of the code: a strong scaling and a weak scaling. On the one hand, the strong scaling is defined by the evolution of the restitution time as function of the number of processors for a fixed overall problem size. Ideally, the restitution time should decrease as a linear function of the number of processors. However, distributing a sequential problem over many threads may give a distorted picture since each processor may end up with very little data. On the other hand, the weak scaling is defined by the evolution of the restitution time as a function of the number of processors for a fixed size per processor. Ideally, if the number of grid points and the number of processors increase proportionally, the restitution time should remain constant. Weak scaling tests were carried on two computing clusters: Ada (IDRIS, 2016a) Turing (IDRIS, 2016b), while strong scaling test were only performed on Ada. Ada has 332 compute nodes, each of four 8-cores cadenced at 2.7 GHz (Intel Sandy Bridge E5-4650 processors). Its theoretical peak processing power is 233 TFLOPS. Each node has 128 GB of memory and nodes are interconnected by a high-speed InfiniBand network. In practice, the resources are accessible for up to 2,048 execution cores. Turing (IDRIS, 2016b) has 6 racks, each containing 1,024 compute nodes of 16-core processors cadenced at 1.6GHz (IMB Power BQC 16C). Its theoretical peak processing power is 1,258 TFLOPS. In practice, resources are accessible for up to 65,536 execution cores.

The test simulation used corresponds to a 2-D square (or 3-D cubic) Rayleigh-Bénard cell from the unstable conductive state with a random perturbation. Each simulation lasts 5 convective time units, i.e. hundreds of time steps. The cpu time per time step is then obtained by averaging over the number of time steps.

Strong scaling

For the strong scaling, the two configurations, 2-D and 3-D were tested: a square Rayleigh-Bénard cell from 128 2 up to 1024 2 grid points and a cubic Rayleigh-Bénard cell from 32 2 up to 256 2 grid points. Both configurations were chosen based on the type of simulations to be performed in this thesis. We define a speed-up factor S p , which compares the restitution time on a single processor u.t mono and on a parallel machine with p processors u. This approach works as long as the available memory on a single processor is large enough to solve the reference problem. The optimal value of S p is bounded by the value of p. The speed-up factor for 2D simulations peaks at S p = 70 for 128 MPI processes, i.e. E = 0.54 for a 1024 2 mesh, whereas for a 3D simulation one obtains a peak value S p = 278, i.e. E = 0.54 for a 256 3 mesh when using 512 MPI processes, see figure 3.11.

Weak scaling

For the weak scaling, only the cubic Rayleigh-Bénard cells were tested. However, measurements were performed on both computing clusters, Ada and Turing. In Ada we tested different domain sizes up to 512 3 grid points and using from 1 up to 512 cores. The architecture of Turing allowed to test domain sizes up to 1024 3 grid points and using from 1 up to 2,048 BlueGene compute nodes (or alternatively from 64 up to 32,768 cores). A good overall weak scaling is achieved in both machines as seen on figure 3.12.

Additional metrics, such as the user time per time-step per grid point, u.t/n x n y n z are also commonly used for estimating the compute and restitution times for a given simulationvalues range from 10 -5 s/step/point for sequential execution up to 4 • 10 -8 s/step/point for parallel runs on Ada (512 3 points and 512 processors) and up to 3 • 10 -9 s/step/point for parallel runs on Turing (1024 3 points and 2,048 nodes).

These results are used to define the range of Ra values that can be explored through direct numerical simulations with the available resources and how to use efficiently said resources. The numerical simulations presented in the following chapters were performed using the HPC resources of LIMSI and GENCI-IDRIS under generous allocations made by GENCI. Chapter 4

• 10 -1 1.895 • 10 -1 2.147 • 10 -2 1 • 256 2.055 2.251 1.673 • 10 -1 1.897 • 10 -1 2.153 • 10 -2 1 • 512 2.055 2.251 1.672 • 10 -1 1.898 • 10 -1 2.155 • 10 -2 1 • Ref. 2.054 2.250 1.672 • 10 -1 1.898 • 10 -1 2.156 • 10 -2 1 • 32 4.453 4.732 1.427 • 10 -1 2.207 • 10 -1 3.015 • 10 -2 1 • 64 4.361 4.638 1.404 • 10 -1 2.240 • 10 -1 3.043 • 10 -2 1 • 128 4.340 4.616 1.396 • 10 -1 2.242 • 10 -1 3.058 • 10 -2 1 • 256 4.336 4.612 1.392 • 10 -1 2.246 • 10 -1 3.061 • 10 -2 1 • 512 4.336 4.612 1.390 • 10 -1 2.247 • 10 -1 3.065 • 10 -2 1 • Ref. 4.337 4.612 1.388 • 10 -1 2.247 • 10 -1 3.064 • 10 -2

Reversal cycle in square Rayleigh-Bénard cells in turbulent regime

In the next two chapters we are interested in characterizing the large-scale flow dynamics inside two-dimensional (2-D) square Rayleigh-Bénard cells in the (weakly-) turbulent regime. For the range of Rayleigh numbers and Prandtl numbers considered, two types of flow regimes are observed intermittently: a regime consisting of consecutive flow reversals, when the large-scale circulation (LSC) switches sign; and a regime consisting of an extended cessation of the LSC.

In the present chapter, a statistical characterization of the regime of consecutive flow reversals is portrayed in terms of different global quantities. This work led to the preparation of an article [START_REF] Castillo-Castellanos | Reversal cycle in square Rayleigh-Bénard cells in turbulent regime[END_REF], which is reproduced verbatim in this chapter. The presence of a generic reversal cycle is evidenced by means of a time re-scaling procedure. This cycle is composed of three successive phases: acceleration, accumulation and release including a rebound event. Reversals are then viewed in terms of available potential energy and global kinetic energy, and their corresponding energy transfer rates.

A decomposition of the rate-of-change of the global angular impulse reveals a delicate balance between thermal and viscous forces. Based on this analysis, we identify two points of interest signaling a transition of flow regime before the flow reversal takes place during the accumulation phase.

In chapter 5 we continue the analysis of the reversal regime by performing a proper orthogonal decomposition (POD) analysis [START_REF] Podvin | A large-scale investigation of wind reversal in a square Rayleigh-Bénard cell[END_REF][START_REF] Podvin | Precursor for wind reversal in a square Rayleigh-Bénard cell[END_REF] combined to the statistical approach to the modal coefficients.

Paper: Reversal cycle in square Rayleigh-Bénard cells

Introduction

Decaying or stochastically forced two-dimensional turbulence in the presence of rigid boundaries is characterised by self-organised coherent structures. For a square domain [START_REF] Clercx | Spontaneous spin-up during the decay of 2d turbulence in a square container with rigid boundaries[END_REF][START_REF] Molenaar | Angular momentum of forced 2D turbulence in a square no-slip domain[END_REF]), a spontaneous spin-up is observed which leads to the formation of a single vortex structure. This structure can persist for very long periods of time, before suddenly breaking and re-organising itself, in some cases with a reversed rotation sense. A similar process occurs in the geomagnetic field under the form of polarity switches [START_REF] Wicht | Numerical models of the geodynamo: From fundamental cartesian models to 3D simulations of field reversals[END_REF][START_REF] Valet | Dynamical similarity of geomagnetic field reversals[END_REF]. In turbulent Rayleigh-Bénard (RB) convection experiments, this phenomenon is also observed: a large scale circulation (LSC), commonly referred to as the wind, changes sign intermittently. Several models have been proposed to describe this process in RB convection through either stochastic differential equations [START_REF] Sreenivasan | Mean wind and its reversal in thermal convection[END_REF][START_REF] Benzi | Flow reversal in a simple dynamical model of turbulence[END_REF][START_REF] Brown | Large-scale circulation model for turbulent Rayleigh-Bénard convection[END_REF]Podvin & Sergent 2015) or phenomenological and physically motivated assumptions [START_REF] Araujo | Wind reversals in turbulent Rayleigh-Bénard convection[END_REF][START_REF] Resagk | Oscillations of the large scale wind in turbulent thermal convection[END_REF][START_REF] Brown | Large-scale circulation model for turbulent Rayleigh-Bénard convection[END_REF].

The LSC structure, its global properties and the transition between dominant flow structures are found to be dependent on the cavity shape (Grossmann & Lohse 2003;Xi & Xia 2008b;[START_REF] Van Der Poel | Connecting flow structures and heat flux in turbulent Rayleigh-Bénard convection[END_REF]. Inside cylindrical cells, a change of sign of the LSC occurs either by a rotation-led reversal through an azimuthal rotation of the nearvertical circulation plane known as azimuthal meandering, or by a cessation-led reversal through the breakdown of the existing LSC before re-organising in a different spatial direction (see, for instance [START_REF] Niemela | The wind in confined thermal convection[END_REF]; [START_REF] Sreenivasan | Mean wind and its reversal in thermal convection[END_REF]; [START_REF] Brown | Large-scale circulation model for turbulent Rayleigh-Bénard convection[END_REF]; [START_REF] Funfschilling | Torsional oscillations of the large-scale circulation in turbulent Rayleigh-Bénard convection[END_REF]; Xi & Xia (2008a,b)). One approach to separate rotation-led from cessation-led reversal events consists in restricting the experimental study to a square box of small aspect ratio in the transversal direction [START_REF] Xia | Particle image velocimetry measurement of the velocity field in turbulent thermal convection[END_REF][START_REF] Sugiyama | Flow reversals in thermally driven turbulence[END_REF]Wagner & Shishkina 2013;[START_REF] Ni | Reversals of the large-scale circulation in quasi-2D Rayleigh-Bénard convection[END_REF]. Another viewpoint uses two-dimensional direct numerical simulations [START_REF] Sugiyama | Flow reversals in thermally driven turbulence[END_REF][START_REF] Petschel | Statistical analysis of global wind dynamics in vigorous Rayleigh-Bénard convection[END_REF]Chandra & Verma 2011;[START_REF] Verma | Flow reversals in turbulent convection with free-slip walls[END_REF]Podvin & Sergent 2015) since rotation-led reversals are not possible in such a configuration. However, it is not entirely clear whether 2D reversals and cessation-led reversals correspond to the same phenomenon. [START_REF] Sugiyama | Flow reversals in thermally driven turbulence[END_REF] have identified a region in the (Ra, Pr) space in which reversal events are observed experimentally inside quasi-two-dimensional cells as well as numerically in two-dimensional simulations. For this range of (Ra, Pr), the flow inside a square cell is mainly composed of a large diagonal roll and two counter-rotating corner-rolls. (Chandra & Verma 2013;[START_REF] Sugiyama | Flow reversals in thermally driven turbulence[END_REF]) pointed out the feeding of corner-rolls by plumes detached from horizontal boundary layers. Both papers proposed that the growth of corner-rolls ended by a sudden LSC transition.

The presence of such coherent structures has been investigated by computing the first Fourier modes (Chandra & Verma 2011;[START_REF] Verma | Flow reversals in turbulent convection with free-slip walls[END_REF], or by obtaining these modes from a proper orthogonal decomposition [START_REF] Bailon-Cuba | Aspect ratio dependence of heat transfer and large-scale flow in turbulent convection[END_REF]Podvin & Sergent 2015). Coherent structures are actually associated with a sum of various such modes: a large-scale monopole, a quadrupole and a vertically or horizontally stacked dipole. A study of the transition sequences between these first Fourier modes indicated the presence of a reversal path [START_REF] Petschel | Statistical analysis of global wind dynamics in vigorous Rayleigh-Bénard convection[END_REF]. To analyse such a process, one could adopt the perspective used in geomagnetic fields combining a careful selection of reversal records as well as a time rescaling [START_REF] Valet | Dynamical similarity of geomagnetic field reversals[END_REF][START_REF] Lhuillier | Statistical properties of reversals and chrons in numerical dynamos and implications for the geodynamo[END_REF]. In geomagnetism, this method led to the definition of three successive phases: a precursory event, a polarity switch and a rebound.

Another viewpoint is based on energetical considerations. For instance available potential energy is key to understand how mechanical energy is transported, stored, and dissipated in RB convection [START_REF] Winters | Available potential energy and mixing in density-stratified fluids[END_REF][START_REF] Hughes | Available potential energy in Rayleigh-Bénard convection[END_REF]. This approach could make more precise the idea of an avalanche mechanism (mentioned in [START_REF] Sreenivasan | Mean wind and its reversal in thermal convection[END_REF]), due to a localised accumulation of energy, which increases local gradients until a certain threshold is reached and energy is expelled as a single burst.

In the present paper, we propose for RB convection, a formulation similar to the one proposed in geomagnetism [START_REF] Valet | Dynamical similarity of geomagnetic field reversals[END_REF]): the main objective is to establish the existence of a generic reversal cycle and to identify in this cycle three phases (release, accumulation, and acceleration). This analysis combines a statistical analysis with a physical approach relying on the angular momentum as well as kinetic and potential energy to highlight the underlying physical mechanisms. In addition, we identify flow patterns corresponding to each phase of the generic cycle by using a conditional averaging. A threshold state in generic reversal cycles is identified from which the release is inevitable.

The paper is organised as follows. Section 2 introduces the model equations and global quantities: global angular impulse, available mechanical energy and corresponding conversion rates. A brief description of the numerical method and the spatial resolution is presented in section 3. In section 4, a filtering method is proposed that identifies two regimes, and then allows to perform a statistical study of reversals. The dynamics of a Reversal cycle in Rayleigh-Bénard cells 3 generic reversal mechanism is described as composed by three phases in section 5. These results are then analysed in terms of coherent flow structures, physical mechanisms in section 6 for particular realisations. In section 7, a stability analysis is applied on the generic cycle. Section 8 contains a brief comparison of the present analysis with previous works. Finally, some prospective works are mentioned in conclusion.

Model equations and analysis tools

Consider a fluid contained in a square cell, cooled at the top with constant temperature T top and heated at the bottom with constant temperature T bot > T top . The flow equations are based on the Boussinesq approximation. The flow regime is defined as a function of the Rayleigh and Prandtl numbers,

Ra ≡ gH 3 β (T bot -T top ) κν , Pr ≡ ν κ (2.1)
where g denotes gravity, H the cell height and β, κ, ν are respectively volumetric thermal expansion, thermal diffusivity and kinematic viscosity coefficients. The values of (Ra, Pr) used for direct numerical simulations (DNS) correspond to a weakly turbulent flow regime where reversals have been reported [START_REF] Sugiyama | Flow reversals in thermally driven turbulence[END_REF]. As far as notations are concerned, x (resp. u) and y (resp. v) stand for the horizontal and vertical directions (resp. velocities). Coordinate vector x = (x, y) is equal to (0, 0) at the cavity centre. One introduces the reduced temperature θ( x, t)

≡ (T -T 0 )/(T bot -T top ), with T 0 ≡ (T bot + T top )/2 as well as the only vorticity component ω( x, t) ≡ ∂ x v -∂ y u.
For a field a( x, t), the fields a( x) and σ(a)( x) denote the time average and standard deviation computed using the full long-term time series. In addition, quantity a vol (t) stands for the volume average of a( x, t).

Based on the cell height H as characteristic length scale and κ H Ra 0.5 as velocity scale, the dimensionless velocity u = (u, v) and reduced temperature θ satisfy the dimensionless system of equations

   ∇ • u = 0, ∂ t u + ∇ • [ u ⊗ u] = -∇p + PrRa -0.5 ∇ 2 u + Prθ e y , ∂ t θ + ∇ • [ uθ] = Ra -0.5 ∇ 2 θ (2.2)
No-slip condition for the velocity field is ensured on walls. On top (resp. bottom) walls, one imposes θ = -0.5 (resp. θ = 0.5) while adiabaticity ∂ x θ = 0 is satisfied on side-walls. From now on, quantities are written in dimensionless form only.

Global angular impulse

The global angular momentum

L 2D (t) ≡ - 1 2 x 2 ω( x, t)dxdy (2.3)
serves as a measure of organised rotation (see for instance [START_REF] Molenaar | Angular momentum of forced 2D turbulence in a square no-slip domain[END_REF]).

Figure 1 and one observes an extended cessation. Such complementary region is denoted here as the extended cessation (EC) regime.

In order to differentiate in a precise manner the CR regime from the EC regime, a filtering algorithm has been devised which is modeled after [START_REF] Lhuillier | Statistical properties of reversals and chrons in numerical dynamos and implications for the geodynamo[END_REF]Podvin & Sergent 2015). We identify the set of consecutive times r i at which L 2D changes sign. The time interval [r i , r i+1 ] is considered to be inside the CR regime if, during this interval, the value of |L 2D | reaches at least once the threshold value 1). A time interval where such threshold is not reached can be of two kinds corresponding to the darker blue areas or the white areas in figure 1. The first kind is sandwiched between two CR intervals and corresponds to a "rogue" reversal, which likely belongs to the CR regime but has been filtered out by our criterion (the criteria for the selection of events in the CR regime is rather stringent as seen from the "rogue" events displayed in figure 1). The second type, displayed in white, corresponds to an extended cessation.

|L 2D | + σ(|L 2D |) (light blue area in figure
For any interval [r i , r i+1 ], its duration τ 1,i ≡ r i+1 -r i is also computed. When both intervals [r i-1 , r i ] and [r i , r i+1 ] are inside a CR regime, the duration τ d,i of the jump occurring around time r i between a clockwise and counter-clockwise central vortex or vice versa can be evaluated. It is computed by identifying the times located just before and just after time

r i such that |L 2D | reaches the threshold value |L 2D | -σ(|L 2D |). τ d,i
is simply the time lapse between these two events.

The change of the global angular momentum L 2D may be better understood considering the relation directly obtained from the governing equations (2.2),

dL 2D dt = M + I a -I b    M(t) ≡ 1 2 Pr x 2 ∂ x θ dxdy I a (t) ≡ PrRa -0.5 [ x • n] ω dl I b (t) ≡ 1 2 PrRa -0.5 x 2 n • ∇ω dl (2.4)
where n stands for the outwards unit normal vector to the domain boundary and dl for a contour line differential element. Note that it is assumed that line integrals are performed in a counterclockwise direction. The angular momentum thus evolves because of a bulk forcing term M(t) known as the input torque [START_REF] Molenaar | Angular momentum of forced 2D turbulence in a square no-slip domain[END_REF]) and two boundary integrals terms I a (t) and I b (t). I b (t) is close but not identical to the integrated vorticity flux over the domain boundary. For a square cavity, the boundary term I a (t) simplifies to I a = 1 2 PrRa -0.5 ωdl. Vorticity on the boundary is related to the friction exerted by the fluid on the walls. This integral I a (t) is thus quantifying friction along the boundary.

Mechanical energy balance

Other global quantities are useful to characterise at each time the instantaneous state of the system: the global kinetic energy E kin (t) ≡ 1 2 u 2 dxdy and the global potential energy E pot (t) ≡ -Pr yθ dxdy. This latter quantifies the energy amount required to bring all fluid particles against gravity from their position at time t to the reference level y = 0. However, one can introduce a more pertinent instantaneous quantity namely the available potential energy (Sutherland 2010), which is defined here below. For a given time t, the fluid is characterized by an instantaneous temperature field θ(x, y, t) with a lower (resp. upper) bound at θ min (resp. θ max ). Let us consider a one-to-one mapping (x r (x, y), y r (x, y)) from the square onto the square. This may be interpreted as a reordering of fluid particles inside the square cell. Rearranging modifies the temperature field leading to a new field θ r (x, y, t) but this process is an adiabatic one i.e. θ r (x r , y r , t) = θ(x, y, t). As a consequence, the probability distribution function (PDF) of temperature in the rearranged state is identical to the PDF in the instantaneous state. The potential energy of the rearranged state can be measured. Among the set of such mappings, there exists a subset which corresponds to the lowest potential energy. It is easy to understand that all mappings belonging to this subset have identical rearranged temperature field θ r (y, t) with no x dependency and increasing (i.e. density decreasing) monotonically with height (figure 2). This field characterizes the background state. Although y r (x, y, t) is not a simple function of (x, y), the above remark implies that y r is a one-to one function of θ for a mapping in this subset. In practice (see Tseng & Ferziger (2001)), the field θ r (y, t) is computed by the following procedure. First, the PDF of the instantaneous temperature at time t which is denoted by P (θ), is directly evaluated numerically within the interval [θ min , θ max ] since θ(x, y, t) is known on the whole square box. Second, the conservation of temperature PDF with rearrangement imposes that y r (θ) be evaluated as a cumulative density function

y r (θ) -y bot = y r (θ) -y r (θ min ) = θ θ min P (θ) dθ.
( 2.5) This relation depends a priori on the domain geometry. For a square box of unit size, however, the proportionality is factor is reduced to unity. Third, function θ r (y, t) is then obtained by a simple inversion of y r (θ, t).

The background state is characterized by the lowest potential energy that can be reached by an adiabatic process starting from the instantaneous temperature field θ(x, y, t). This quantity called the background potential energy is equal to E bpot ≡ -Pr y r (x, y, t)θ(x, y, t) dxdy. By a simple change of variable and using A. Castillo et al adiabaticity θ r (x r , y r , t) = θ(x, y, t), one gets E bpot = -Pr y r θ r (y r ) dx r dy r = -Pr y r θ r (y r )dy r .

The difference E apot (t) ≡ E pot (t) -E bpot (t) > 0 in potential energy between the instantaneous state and its background companion is called the available potential energy and represents the potential energy which could be effectively transformed from the instantaneous field θ(x, y, t) into motion (Lorenz 1955;[START_REF] Winters | Available potential energy and mixing in density-stratified fluids[END_REF].

In analogy with L 2D , the process may be better grasped by considering the evolution of the energies E kin , E pot , E apot , through some exact relations (see [START_REF] Winters | Available potential energy and mixing in density-stratified fluids[END_REF]; [START_REF] Hughes | Available potential energy in Rayleigh-Bénard convection[END_REF]). For the kinetic energy E kin , the following relation holds (e ij denotes the symmetric velocity gradient tensor)

dE kin dt = PrRa -0.5 [Φ y -ǫ], Φ y ≡ Ra 0.5 vθ vol ǫ ≡ ∇ u : ∇ u vol = 2 e ij e ij vol (2.6)
The first bulk term Φ y (t) is a convective heat flux. More precisely let us introduce the volume-averaged Nusselt number Nu vol ≡ Ra 0.5 vθ vol -∂ y θ vol . It is easily found that, for RB cells, Φ y (t) = Nu vol (t) -1. The second bulk term ǫ(t) > 0 stands for the viscous dissipation rate. Finally one may write

dE kin dt = PrRa -0.5 [Nu vol -(ǫ + 1)] (2.7)
The potential energy E pot verifies instead the relation

dE pot dt = PrRa -0.5 [-Nu vol + Φ b1 ] (2.8)
which contains the bulk term Nu vol and a boundary term

Φ b1 (t) ≡ -y[ n • ∇θ] dl (2.9)
quantifying the conversion rate to E pot from external sources. More precisely, let us introduce the Nusselt number Nu top (t) ≡ -∂ y θdx evaluated at the top y = 0.5 as well as the Nusselt number Nu bot (t) ≡ -∂ y θdx evaluated at the bottom plates y = -0.5. For the present square cell, one easily verifies that

Φ b1 = 1 2 (Nu top + Nu bot ) and consequently dE pot dt = PrRa -0.5 [-Nu vol + 1 2 (Nu top + Nu bot )] (2.10)
Finally, the evolution equation for the available potential energy E bpot reads as

dE bpot dt = PrRa -0.5 [Φ d -Φ b2 ], Φ d ≡ ∂y r ∂θ ∇θ • ∇θ vol = ∇y r • ∇θ vol Φ b2 ≡ y r [ n • ∇θ] dl (2.11)
where the bulk term Φ d (t) quantifies the energy conversion rate due to diapycnal mixing.

Since by definition ∂y r ∂θ > 0, Φ d (t) is bound to be positive. The boundary term Φ b2 provides the conversion rate from external sources. For the present RB cells, since y r (x, 1/2, t) = -1/2 and y r (x, -1/2, t) = 1/2 and because adiabaticity of side walls it is clear that Φ b2 (t) = Φ b1 (t). Finally by substracting equation 2.

Numerical method

Simulations are carried out using a finite volume code using a semi-implicit scheme based on Bell-Colella-Glaz advection scheme [START_REF] Bell | A second-order projection method for the incompressible Navier-Stokes equations[END_REF], and a pressure-correction scheme for the velocity-pressure coupling, with a global second order precision. Numerical implementation is done using BASILISK C, details of which can be found in [START_REF] Popinet | Basilisk. Website[END_REF]. Simulations listed in table 1 have been performed on a uniform Cartesian grid with 512 points in each direction, with a variable time-step that verifies the condition CFL< 0.5. In the most unfavorable case (Ra = 10 8 , Pr = 4.3) the thermal boundary layers contain 10 points along the vertical direction.

Spatial resolution is verified evaluating numerical convergence of time-averaged Nusselt numbers obtained by different methods [START_REF] Stevens | Radial boundary layer structure and Nusselt number in Rayleigh-Bénard convection[END_REF]. Note that to perform the averaging, statistical sampling is obtained at regular intervals. We compare Nu vol , Nu top and Nu bot to the Nusselt numbers obtained from the thermal and viscous dissipations

Nu θ ≡ ∇θ • ∇θ vol , Nu ǫ ≡ ǫ + 1 (3.1)
All these quantities should be equal (Shraiman & Siggia 1990). The value of Nu shown in table 1 is the average value of Nu vol , Nu top , Nu bot , Nu θ , Nu ǫ while the maximum relative difference between any of them is shown as %Diff. These values converge within 2% of Nu for all (Ra, Pr) presented. We have also verified that our numerical results are wellconverged in a completely different way. This check has been performed by comparing results of the probability density functions (PDF) of the time interval τ 1 obtained by our code against benchmark results. This computation requires to get long term simulations and was done for (Ra = 5 • 10 7 , Pr = 4.3) since such parameter values had already been computed by a spectral code (Podvin & Sergent 2015). The data for this check are postponed at the end of section 4.

Temporal analysis and statistical characterisation

In the present work, we focus on turbulent RB systems for which flow reversals are observed. For Pr = 3.0, this dynamics is associated with the interval Ra ∈ Note that, these boundaries are not clearly established. For instance, some transitions were found for very long runs at (Pr = 4.3, Ra = 10 7 ) (not presented here) but it is difficult to assert whether or not the few cycles observed correspond to an established statistical steady state or to a transient behaviour. In the following, we consider only Table 2. Probabilities as a function of Ra and Pr. p cr (resp. p ec ) denotes the probability that the system be inside the CR (resp. EC) regime. p rr denotes the probability of a "rogue" reversal..

values inside the aforementioned range for which the number of events is large enough (see table 1): simulations are performed from 9,600 to 29,000 convective time units (see table 1), which gives from 50 to 160 events (except for Pr = 4.3 and Ra = 3 • 10 7 which is situated close to the boundary region where the reversal dynamics is established).

For a given couple (Ra, Pr), one computes the percentage of time or equivalently the probability that the system be in one of the three states: p cr in the CR regime, p ec in the EC regime and p rr inside a "rogue" reversal. The probability p rr of rogue events is always of a few percents (see table 2). For both Pr = 3.0 and Pr = 4.3, in the interval where the CR regime is observed, p cr first decreases and then increases with increasing Ra (see table 2).

The PDFs of τ 1 and τ d are measured based on the full simulation length and shown on figure 3 (resp. figure 4) for Pr = 4.3 (resp. Pr = 3). Using the filtering method of section 2.1, we separated the PDF of τ 1 into three contributions: one for intervals inside the CR regime (color blue), one for intervals from the EC regime (color red) and one corresponding to rogue events (color purple). This PDF shows that the distribution of τ 1 is not peaked inside the CR regime: intervals may have different durations. This is also valid for the EC regime. For Pr = 4.3, a similar probability distribution of τ 1 is observed for both values of Ra (figure 3) and a characteristic time-scale τ c ≈ 60 exists which separates EC from CR regimes. For Pr = 3.0, a change in the PDFs of τ 1 and τ d is observed as we increase the values of Ra (figure 4). For the lowest Ra displayed, a characteristic time τ c cannot be clearly defined. For the highest Ra displayed Ra = 10 8 , the EC regime completely disappears. For intermediate Ra, reversal events become evenly distributed over a narrow band of τ 1 (see figure 4 center and right) and a clear separation of time-scales between the EC and CR regimes is observed at τ c ≈ 50 which is at least one order of magnitude larger than the large eddy turn-over time, τ E ≡ 4π/|ω c | (ω c denoting vorticity measured at the cavity center). The PDF of the inter-switch intervals observed for cylindrical convection cell experiments is an exponential distribution [START_REF] Sreenivasan | Mean wind and its reversal in thermal convection[END_REF]). It is not seen here (figures 3 and 4) illustrating the fact that rotation-led reversals are not present here contrary to cylindrical cells.

From the PDF of transition durations τ d for CR regime only (see figures 3 and 4 bottom), the peak value tends to increase as Ra is increased for both Pr = 3.0 and Pr = 4.3. Concerning the numerical check, the average value τ 1 | cr during the CR regime and the average duration τ d of transition were both found to be in good agreement with published results for (Ra = 5 • 10 7 , Pr = 4.3): τ 1 | cr = 146 and τ d = 11.5 convective time units (Podvin & Sergent 2015).

Dynamics of the generic reversal

We have shown above that reversals cycles have different durations τ 1,i . A simple time-rescaling, however, can be used to identify features common to all reversal cyles.

Reversal cycle in Rayleigh-Bénard cells ) corresponding to "rogue reversals", and an additional part in ( ) corresponding to the EC regime.

5.1.

Averaging procedure and generic reversal as function of (Ra, Pr)

In order to identify similarities between different intervals in the CR regime, the following procedure is proposed to treat time-series of global quantities such as L 2D , E kin , and E apot . Once the intervals [r i , r i+1 ] inside the CR regime are properly identified, all these intervals with L 2D > 0 (resp.L 2D < 0) are stacked together so that they have a common origin at r i (resp. r i+1 ). If the time axis of each interval is re-scaled by τ 1 , one obtains figure 5 left. If the time axis of each interval is rescaled by its particular duration τ 1,i , these curves display a consistent dynamical pattern (see figure 5 Note that, while figure 5 displays only 10 reversals to avoid cluttered graphs, these events displayed are considered as representative of the entire set. Obviously, all of the events inside the CR regime are taken into account in our procedure but "rogue reversal" events are not. This procedure is similar to one used in the study of statistical properties of magnetic switches in the geodynamo problem [START_REF] Valet | Dynamical similarity of geomagnetic field reversals[END_REF][START_REF] Lhuillier | Statistical properties of reversals and chrons in numerical dynamos and implications for the geodynamo[END_REF]. For a sufficiently large number of recorded events, the average over these rescaled curves is expected to remove the noisy dynamics and to represent a generic reversal cycle (figure 5 right). This averaging technique, once applied to E kin and E apot , recovers the evolution of mechanical energies during the generic reversal cycle.

centre). A. Castillo et al -2 -1 0 1 2 -1 -0.5 0 0.5 1 L 2D /|L 2D | -2 -1 0 1 2 -1 -0.5 0 0.5 1 -2 -1 0 1 2 -1 -0.5 0 0.5 1 0 1 2 -1 -0.5 0 0.5 1 E kin /|E kin | 0 1 2 -1 -0.5 0 0.5 1 0 1 2 -1 -0.5 0 0.5 1 2 3 4 5 -1 -0.5 0 0.5 1 E apot /|E kin | 2 3 4 5 -1 -0.5 0 0.5 1 2 3 4 5 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 [t -r i ]/τ 1 -1 -0.5 0 0.5 1 [t -r i ]/τ 1,i -1 -0.5 0 0.5 1 [t -r i ]/τ 1,i
Figure 6 shows the L 2D /|L 2D | curves for generic reversal cycle at Pr = 3.0: as we increase Ra, the reversal cycle becomes more regular and the band representing the standard deviation narrows. The same averaging procedure can be applied to instantaneous temperature and velocity fields in order to obtain the evolution of a conditionally averaged temperature θ o (x, y, t) and velocity u o (x, y, t) fields during the reversal cycle (e.g. figure 7 for (Ra = 5 • 10 7 , Pr = 3.0)). Despite fluctuations between various realisations, dominant and persistent structures appear at specific times of the generic cycle. E apot . It corresponds to figures 7 v to vii. A second step follows from points (b) to (c) in which E kin suddenly decreases and E apot remains almost constant. This is associated with figures 7 viii to ix. After these two steps referred to as a rebound, a new increase of E kin is observed from points (c) to (d) concomitantly with a decrease of E apot until it reaches its minimum value.

L 2D /|L 2D | [t -r i ]/τ 1,i -2 -1 0 1 2 -1 -0.5 0 0.5 1 [t -r i ]/τ 1,i -2 -1 0 1 2 -1 -0.5 0 0.5 1 [t -r i ]/τ 1,i
(i) (ii) (iii) (iv) (v) +0.5 0.0 -0.5 (vi) (vii) (viii) (ix) (x)
E k in / | E k in | L 2 D / | L 2 D | E apot /|E kin | L 2D /|L 2D | -1.5 -1 -0.5 0 0.5 1 1.5 i ii iii iv--ix x E kin /|E kin | 0.4 0.6 0.8 1 1.2 i ii iii iv--ix x E apot /|E kin | b ′ b 2.2 2.4 2.6 2.8 3 a ′ c ′ d ′ e ′ a c d e a ′ i ii iii iv--ix x
Finally, the acceleration phase is located between points (d) and (e) and is characterised by an increase of |L 2D | and E kin to peak values, whereas E apot remains almost constant. During this period, the flow reorganises gradually into a single dominant vortex (figure 7x).

For the (Ra, Pr) considered inside the CR regime, we are able to recover a generic reversal cycle expressed in terms of the available mechanical energy (figure 9). Similarities between these curves for different (Ra, Pr) suggest an equivalent underlying mechanism behind flow reversals, even if the intensity of the rebound decreases with Pr. For (Ra = 5 • 10 7 , Pr = 3.0), the accumulation phase lasts longer (60%), while the release and acceleration phases have shorter and similar durations (respectively 18% and 22% of the reversal cycle). For the range of Ra considered these proportions are similar: for instance, the accumulation, release, and acceleration are observed to last 75%, 13%, and 12% of the reversal cycle for (Ra = 10 8 , Pr = 3.0).

Dynamics of a particular reversal

From now on, we focus on a single value of (Ra, Pr) (Ra = 5 • 10 7 , Pr = 3.0) in order to explore the nature of the reversal dynamics. To look at the small-scales effects, the analysis below considers particular realisations of reversal cycles rather than conditionally averaged fields.

For each particular reversal cycle, we define similarly to the generic curve, consecutive instants (a p )-(b p )-(c p )-(d p )-(e p ) which pinpoints the dynamical times: L 2D = 0 at instant (a p ); E kin reaches a local maximum at instant (b p ), E kin reaches a local minimum at instant (c p ); E apot reaches its minimum at instant (d p ); |L 2D | reaches its maximum at instant (e p ).

Time evolution of the available mechanical energy and flow structures

The spatial distribution of mechanical energy can be linked to flow structures observed during different phases of a reversal cycle. At the beginning of the accumulation phase (point (e p ')), a large diagonal vortex with small counter-rotating corner-flows is observed (top figure 10). On the one hand, it corresponds to a state of maximum kinetic energy condensed inside the central vortex (bottom figure 10). On the other hand, the integrand Pr (y r -y) of E apot is mostly distributed inside the corner-flows or along thin thermal boundary layers (top figure 11). The field ∇y r • ∇θ is used to highlight the contour of thermal plumes (bottom figure 11). This field is related to the spatial distribution of Φ d (see equation 2.11). Small-scale thermal plumes are observed to be detached from the thermal boundary layers. They are then swept by the central vortex. Plumes are channelled into the corner-flows, directly or after having been advected along the side-walls. The progressive growth of corner-flows is illustrated on figures 10 and 11. This is consistent with previous observations by [START_REF] Sugiyama | Flow reversals in thermally driven turbulence[END_REF]) that tied the time evolution of the corner-flows height. The steady increase in E apot coincides with a deceleration of the central vortex and to a build-up of thermal energy inside the cornerflows. Indeed, let us compute the contributions to E apot due to the region of thermal boundary layers (BL) and the bulk. The hot and cold BL are taken to have a constant thickness δ -1 θ = 2Nu. Contributions inside both BL (figure 12) amount to 30-40% of E apot and are fairly constant in time: standard deviation is less than 1%. On the contrary, contributions to E apot from outside BL are directly influenced by the reversal cycle: a steady increase is observed during the accumulation phase until the release phase (figure 12) . This suggests that the energy exchange observed during the release takes place only inside the bulk, while the boundary layers seem to be largely unaffected by the reversals. During the first part of the rebound (points (a p ) to (b p )), the opposing corner-flows have become large and strong enough to deform and finally split the central vortex (figure 13): opposing corner-flows then connect and form a single vortex (with opposite rotation with respect to the previous LSC). This allows the thermal energy stored inside corners to be rapidly released and transformed into kinetic energy. By the end of this exchange, the amount of thermal energy E apot stored outside the thermal boundary layers will be halved (figure 12). The complete rebound period (point (a p )-(c p )) can be better illustrated by highlighting the role of the thermal plumes and boundary layers.

To do so, one uses y r (x, y, t) which is a bijective function of temperature, and function ∇y r •∇θ in figure 14. Once the vortex reconnection has taken place, the blobs of hot (resp. cold) fluid which are now inside the main central vortex, are allowed to travel upwards (resp. downwards) directly into the bulk. This results in the exchange of potential energy (contained inside small scale structures) into kinetic energy (contained inside the central vortex). The newly formed circulation proceeds to rotate and is able to advect thermal blobs against the action of buoyancy forces (as seen between points (b p ) and (c p ) figure 14). Simultaneously as the thermals are released into the bulk, the surface separating hot and cold fluid increases which reinforces the mixing process (bottom figures 14).

The acceleration phase is illustrated on figure 15. Self-organisation of the LSC takes place, during which the kinetic energy and the angular impulse arrive to peak values. During that phase, the bulk contains less and less plumes: temperature becomes nearly homogeneous. By the end of this phase the flow settles to a large diagonal roll with small corner-flows.

Mechanical energy transfer rates

Figure 16 displays the evolution of energy transfer rates (given in equations 2.6 to 2.12) for a generic reversal cycle as well as for two particular reversal cycles. More precisely, the bulk terms Nu vol = Φ y + 1, ǫ + 1, Φ d , and boundary term Φ b1 = 1 2 (Nu top + Nu bot ) normalised by Nu are presented. First, note that, while the time-averaged quantities ǫ+1, b)) for the particular reversal (resp. for the generic curve), Nu vol reaches peak values which are several times larger than Nu. This is related to the release of thermal energy E apot (figure 8) and plumes (figure 14). Between points (b) and (c), the generic Nu vol abruptly decreases. In terms of a particular realisation, this is due to the rotation of the bulk acting against buoyancy forces (points (b p ) and (c p ) in figure 14) and may result in a negative heat transfer as seen in figure 16 for the particular reversal cycle. This is consistent with results from (Chandra & Verma 2013). During the acceleration and accumulation phases, Nu vol fluctuates less and slightly decreases.

In addition to Nu vol , the viscous dissipation rate ǫ governs the evolution of E kin (see equation 2.7). Viscous dissipation increases briefly during the re-organisation periods: once during the release phase around point (b), and again during the rotation of the central vortex between points (c) and (d). A gradual increase in ǫ is observed during the acceleration phase, between points (d) and (e), followed by a progressive decay during the accumulation phase. For a particular reversal cycle, contributions to the viscous dissipation rate ǫ are located (see figure 17) primarily along the vertical side-walls and along the horizontal plates where ascending (resp. descending) plumes collide.

In addition to Nu vol , E apot is governed by the mixing term Φ d (see equation 2.12). As seen in fields ∇y r • ∇θ on figures 14 and 15, contributions to Φ d are distributed along thin filaments which trace the contour of thermal plumes. Φ d is thus affected by small scales and fluctuates during all the process.

However, fluctuations are slightly larger during the second part of the release phase (points (b p ) to (d p )). An increase on such mixing fronts during the rotation of the bulk leads to the corresponding increase of Φ d . Note that during the accumulation phase, the generic curve for Φ d slightly decreases. Finally, the forcing boundary term Φ b1 does not fluctuate much on the whole cycle.

To summarize, the dominant mechanisms during the release phase are first the energy conversion Nu vol followed by the mixing Φ d during the rebound. The acceleration phase is characterized by the increase of the viscous dissipation ǫ. A constant decay of both dissipation ǫ and mixing Φ d processes is conversely observed during the accumulation phase.

Angular momentum transfer rates

Since the evolution of L 2D is characteristic of flow reversals, it is of interest to examine the angular impulse transfer rates M, (M -I b ) and I a of equation 2.4 (figure 18). In addition to points (a) to (e), we introduce points (f) and (g). These points are located inside the accumulation phase and coincide for point (f) with a change of sign for M and for point (g) with a change of sign for I a .

Time evolution of the input torque M has a maximum value during the rebound, followed by a local minimum near point (d), a slight increase during the acceleration phase and a monotonic decrease during the accumulation phase. The bulk term M(t) and the boundary term I b (t) are well-correlated and have similar orders of magnitude. This is why only the M(t) -I b (t) evolution is plotted.

Before point (e), the difference M(t) -I b (t) is larger than I a (t), from (e) to (f) it is the opposite ( dL 2D dt changes sign). One may discriminate three time periods: from (a) to (f), I a opposes L 2D (i.e. the central vortex), contrary to the torque M or (M -I b ); from (f) to (g), I a opposes L 2D similarly to M or (M -I b ); and finally, from (g) to (a), I a contributes to L 2D which is opposed by M or (M -I b ). During the accumulation phase, the integrand 0.5Pr[ x • x]∂ x θ of the input torque M(t) is spatially distributed as follows (figure 19): it is negligible inside the bulk, and concentrated in two regions, i.e. along vertical side-walls associated with the central vortex and inside corners-flows. In these two regions they are overall of opposite sign. At point (f), M(t) changes sign: the corner-flows become dominant in the integrand of M(t) and the overall torque is then opposing the central vortex (figure 18). Similarly, during the accumulation phase, the integrand of I a that is vorticity, is distributed in two sectors along the boundaries: one along the boundaries of the central vortex, and another of opposite sign along the top and bottom corners (figure 10). From point (g), I a has changed sign: this is related to the dominant contribution of the corner-rolls.

As a consequence, it is noteworthy that the input torque M(t) changes sign at point (f) long before the reversal time (point (a)). The interval from point (f) to point (g) can be seen as a transition period from a central vortex-dominated to a corner-rolls-dominated flow.

Mechanism of transitions

Linear stability approach

The significance of instant (g) where I a changes sign, is tentatively explained by two complementary approaches. In a first approach, one considers the following hypothesis: the change occurring in the accumulation phase near point (g), is due to a modification of the dynamics governing fluctuations around large scale structures. The large scale structures identified by fields θ o (x, y, t) and u o (x, y, t) are here obtained by an ensemble average over many realisations (see section 5.1). A linear stability analysis is thus defined. The base state at time t o is given by fields θ o (x, y, t o ) and u o (x, y, t o ) frozen at this particular time. The evolution of infinitesimal fluctuations θ ′ (x, y, t) and u ′ (x, y, t) is studied around this frozen base state. The linear stability analysis is performed by direct numerical simulations of the linearised Boussinesq equations,

   ∇ • u ′ = 0 ∂ t u ′ + ∇ • [ u ′ ⊗ u o + u o ⊗ u ′ ] = -∇p ′ + PrRa -0.5 ∇ 2 u ′ + Prθ ′ e y ∂ t θ ′ + ∇ • [ u ′ θ o + u o θ ′ ] = Ra -0.5 ∇ 2 θ ′ (7.1)
Each linearized simulation starts with random disturbances of velocity and temperature fields and is computed for several hundred time units. The perturbation kinetic energy u ′ i u ′ i vol or the square of the fluctuation temperature θ ′ θ ′ vol are monitored in time. In figure 20, each curve is related to a different base state t o . Positive values of growth rate σ are obtained in all cases.

For t o before transition point (g), the values of σ are quite small (∼ 10 -3 -10 -2 ) and the most amplified mode corresponds to a trace of the base state itself (see figure bottom left 20). This is possibly related to a slow variation in time of the large scale flow. In contrast, once point (g) is reached, the growth rate σ increases by a factor of 20, while a different most amplified mode appears. This amplified mode is reminiscent of the flow features observed in the bottom right corner of figure 10.

Non-linear approach with adiabatic boundary conditions

We have identified through a linear stability analysis a critical generic state around point (g). Let us now seek to relate this state to the "avalanche" mechanism: a mechanism due to a localised accumulation of energy inside the fluid which is followed by a sudden transition. One could argue that, when localised accumulation is sufficient, a reversal event takes place even if the external thermal forcing is thereafter suppressed. A second approach, which is non-linear, is based on this idea and confirms point (g) as a transition point associated to a threshold state. transition we follow the evolution of Fourier modes vmn of the vertical velocity for two initial conditions before and after point (g) as seen on figure 21. For initial conditions preceding I a = 0, an already weakened central vortex is unable to contain the corner-flows in place but does not break (the mode v11 which is related to the monopole decreases but remains negative). On the contrary, for initial conditions following the transition point (g), Fourier coefficient v11 changes sign and thus indicates a LSC reversal, before the subsequent decay.

Comparison with previous works

The present DNS results and their interpretation in terms of dynamical processes may be discussed by comparison with previous experimental data and models. Experimental data (Brown & Ahlers (2006); Xi & Xia (2008a,b)), and models like [START_REF] Brown | Large-scale circulation model for turbulent Rayleigh-Bénard convection[END_REF], 2008)) are devoted to the study of cylindrical 3D convection cells, where the LSC plane oscillates. They are mostly focused on the dynamics of the angle of the LSC plane which is a different phenomenon from the dynamics studied in this work. Similarly models like [START_REF] Sreenivasan | Mean wind and its reversal in thermal convection[END_REF]; [START_REF] Benzi | Flow reversal in a simple dynamical model of turbulence[END_REF]) which are based on nonlinear one-dimensional stochastic models, have been compared against experimental data in cylindrical cells. Mainly they attempt to recover the exponential distribution for the PDF of the inter-switch intervals. In 2D systems, such distribution is not observed and again this is probably due to the difference between cessations-driven and rotation-led reversals. [START_REF] Sreenivasan | Mean wind and its reversal in thermal convection[END_REF] states that reversals can be understood in terms of an imbalance between buoyancy effects and friction, where inertia is playing only a secondary role. In the present study, reversals do require the accumulation of potential energy and point (g) is associated to a threshold state pinpointed by the change of sign of I a a quantity related to wall friction. However, the model proposed in [START_REF] Sreenivasan | Mean wind and its reversal in thermal convection[END_REF] is a local one, which follows a single parcel of fluid, while our understanding of the dynamics depends on the existence of global flow structures. It is also mentioned in [START_REF] Sreenivasan | Mean wind and its reversal in thermal convection[END_REF] the possible role of side-wall thermal conductivity on reversals, which is excluded here. The model by [START_REF] Araujo | Wind reversals in turbulent Rayleigh-Bénard convection[END_REF] is based on the inertia of a plume to initiate the reversal. In our case, inertial effects appear during the rebound: as mentioned in §6.1, the newly formed circulation is able to advect thermal blobs against the action of buoyancy forces. However, this is a direct consequence of the reversal and not the triggering factor. Low order models could be based also on POD modes (Podvin & Sergent (2015, 2016)). This approach allows to rebuild the full dynamics of the velocity and temperature fields. The model proposed in Podvin & Sergent (2015) that uses 3 leading modes, is able to reproduce large-scale features of our DNS results when noise is introduced: reversal and cessation dynamics, and growth of corner flows during the accumulation phase. It is also able to reproduce the characteristic time scales as given in §4. However, the phases predominantly associated in the present work to small scales, such as the acceleration and second part of the release, are by construction not recovered by the model described in Podvin & Sergent (2015). Going back to the threshold state on point (g), the existence of such a point has also been identified through a large scale description of reversals by a POD approach [START_REF] Podvin | On a precursor mechanism for wind reversal in a square Rayleigh-Bénard cell[END_REF]). The scenario of the growth of corner flows leading to the release in the case of a square RB cell has been previously proposed by (Chandra & Verma (2013); [START_REF] Sugiyama | Flow reversals in thermally driven turbulence[END_REF]). Both papers pointed out the feeding of the corner flows by plumes detached from horizontal boundary layers. Our results agree with these findings. But we present this process in a more quantitative way through the use of field Pr(y r -y)θ (see figure 11): in this way, we are able to show that E apot is stored mainly in the corner flows. Neither of the two previous papers quantified the energy exchange between kinetic and potential energy during the release. In the present paper, the energy exchange is clearly demonstrated on figures 8 and 9. Furthermore, the figure 12 shows that the energy exchange takes place between the corner flows and the bulk.

Concluding remarks

In this paper we used long-term data from two-dimensional DNS in square RB cells inside the CR regime to perform a statistical characterisation of reversals. Once having removed the samples not contained in the CR regime, a simple time rescaling allowed us to identify a generic reversal cycle in terms of the evolution of three global quantities: the global angular impulse, the global kinetic energy, and available potential energy. Consistent dynamical features were found for different values of (Ra, Pr), and suggested the existence of a generic reversal mechanism, composed of three successive phases: accumulation, release, and acceleration.

The accumulation phase is characterised by a progressive build-up of thermal energy almost exclusively inside the corners thereby inducing them to grow. During the release phase, an energy exchange takes place from available potential energy to global kinetic energy: the opposing corner-flows connect to form a new central vortex and the thermal energy contained from small-scales is suddenly released into the bulk. A newly formed vortex may complete several turnovers before the temperature differences inside the bulk are reduced. Strong fluctuations in both the angular impulse and the kinetic energy are observed during this process, referred to as a rebound. During the acceleration phase, the global angular impulse and the kinetic energy increase as a result of spontaneous self-organisation of the flow into a large diagonal vortex and two small counter-rotating corner-flows. During this phase, increased mixing inside the central vortex results in a very homogeneous bulk temperature and almost constant potential energy. We complement this view in terms of the evolution of energy transfer rates.

Finally, in order to identify the presence of a transition between accumulation and release, we propose two approaches. First a linear stability analysis is performed around generic fields u o (x, y, t o ) and θ o (x, y, t o ) (obtained as the ensemble average over long term DNS results). A sharp increase in the exponential growth rate σ is shown before the beginning of the release phase. In a second approach, the same transition zone is put into evidence by suppressing the external thermal forcing and letting the system evolve from different initial conditions inside the accumulation phase. The presence of a reversal-type event is an indication that a sufficient amount of thermal energy already being stored in the system triggers the reversal.

This work can be easily extended for 2D cells with different geometries or different boundary conditions. This could provide a complementary view to improve the understanding of reversals in 2D convection, in particular to the role of corner flows. In a future work, we intend to compare more precisely the present findings with results from PODbased models proposed in Podvin & Sergent (2015, 2016). In RB 3D cells, the energy budget has already been used but it was considered only in a time-averaged sense. The instantaneous energy budget used here and the way it is related to flow structures can be extended to 3D cells. Indeed, this approach is currently being implemented by the authors in the particular case of a 3D cell which is very much confined in the transversal direction. Our statistical approach however might be difficult to use in cylindrical convection cells for two reasons: first, one needs a long time signal containing a sufficient number of events. This is a difficult thing to achieve in a fully resolved 3D DNS. Second, the situation is more complex in 3D than 2D since in cessation-led (resp. rotation-led) reversals, the LSC plane changes in time abruptly (resp. through azimuthal or torsional motions). This implies to identify or develop a new criteria to follow the evolution of the LSC. This is a limiting factor for expanding the method to 3D convection case in a cylindrical domain.

0 2 4 6 8 •10 -2 a' c' d' e' f'g' a' c' d' e' f'g' to E k , Ea E k Ea E k + Ea

Additional remarks added after publication

Figure 8 of the present paper illustrates a competing behavior between the kinetic E k and the total and available potential energies, E p and E a respectively. This behavior may give the impression that the sum of E k and E a to be nearly constant during reversals as expected in the inviscid limit. This is not the case, see figure 4.1. It is convenient to note, that this difference may be accounted for by fluctuations of the internal energy, and to a smaller degree to the Boussinesq approximation, which neglects the heating due to viscous dissipation of energy.

Chapter 5

Coherent structures during flow reversals and cessations

For turbulent Rayleigh-Bénard cells, modal analysis can be used to separate the effects of large-and small-scale flow in order to highlight the spatio-temporal dynamics of the large-scale circulation. So far two types of modal analyses have been used: Fourier analysis [START_REF] Chandra | Dynamics and symmetries of flow reversals in turbulent convection[END_REF][START_REF] Chandra | Flow reversals in turbulent convection via vortex reconnections[END_REF] and proper orthogonal decomposition (POD) analysis [START_REF] Verdoold | Prime modes of fluid circulation in largeaspect-ratio turbulent Rayleigh-Bénard convection[END_REF][START_REF] Bailon-Cuba | Aspect ratio dependence of heat transfer and large-scale flow in turbulent convection[END_REF][START_REF] Podvin | Proper orthogonal decomposition investigation of turbulent Rayleigh-Bénard convection in a rectangular cavity[END_REF][START_REF] Podvin | A large-scale investigation of wind reversal in a square Rayleigh-Bénard cell[END_REF][START_REF] Podvin | Precursor for wind reversal in a square Rayleigh-Bénard cell[END_REF]. The POD basis is orthogonal, is derived exclusively from observations and the resulting modes are arranged 'optimally' regarding their energetic contents. Unlike the Fourier modes, the POD can be applied to different geometries and boundary conditions.

We chose the latter approach to provide a low-order description of two-dimensional Rayleigh-Bénard dynamics inside the regime of consecutive reversals and the regime of extended cessations in terms of the evolution of the most energetic structures, see §2.6.2. This chapter was built upon existing works [START_REF] Podvin | A large-scale investigation of wind reversal in a square Rayleigh-Bénard cell[END_REF][START_REF] Podvin | Precursor for wind reversal in a square Rayleigh-Bénard cell[END_REF] and complements the statistical approach presented in §4.

This chapter is divided in five main parts. In the first part we compare different POD formulations on the reconstruction of primitive fields as presented in [START_REF] Podvin | A large-scale investigation of wind reversal in a square Rayleigh-Bénard cell[END_REF]. Additionally we compare the reconstruction of the mechanical energy during a generic reversal in order to select the formulation of the POD and the number of modes for a low-order description. Next, DNS data is divided on two subsets: one subset which corresponds to the regime of consecutive reversals only, and one subset which excludes this regime. The second part of the chapter is devoted to the POD analysis of the consecutive reversals exclusively. This is seen as complementary to the statistical interpretation of the generic reversals. This approach is extended to regime of cessations in the third part of the chapter. Then we compare both regimes in the fourth part, by fixing the set of POD modes. Finally, in the fifth chapter, we follow the energy contents of the different modes as we increase Ra to provide insight on the evolution of the large-scale dynamics.

Comparison of the different POD formulations for convection problems

The idea behind any decomposition is to be able to represent a given quantity as the sum of a series of simpler expressions. If all of the terms are retained, one is expected to recover the original data, regardless of the method used. For a finite number of POD modes, one formulation might reconstruct more accurately the evolution the quantities we are interested in. Consider the three different formulations of the POD which were presented earlier on §2.6.2: formulation (A) based on the temperature field, formulation (B) based on the velocity vector field, and formulation (C) which combines the temperature and velocity fields.

For convection problems a mixed formulation is generally preferred [START_REF] Sirovich | Turbulence and the dynamics of coherent structures part i: coherent structures[END_REF][START_REF] Park | Turbulent thermal convection in a finite domain: part ii. numerical results[END_REF]). In the study of flow reversals inside square Rayleigh-Bénard cells, [START_REF] Podvin | A large-scale investigation of wind reversal in a square Rayleigh-Bénard cell[END_REF] compared different formulations and obtained similar coherent structures for each one. Based on these observations, three-mode and five-mode models were proposed [START_REF] Podvin | Introduction à la décomposition orthogonale aux valeurs propres (pod)[END_REF]Sergent, 2015 and[START_REF] Podvin | Introduction à la décomposition orthogonale aux valeurs propres (pod)[END_REF]Sergent, 2017, respectively) that reproduce many of the key features of the large-scale flow reversals.

In the following sections, we validate our approach by comparing against reference results [START_REF] Podvin | A large-scale investigation of wind reversal in a square Rayleigh-Bénard cell[END_REF][START_REF] Podvin | Precursor for wind reversal in a square Rayleigh-Bénard cell[END_REF]. Having validated our POD code, we proceed to compare, for a mixed formulation the influence of the low-order truncation on the reconstruction of flow structures and global quantities which characterize the reversal process. In addition, we compare the results obtained from a six-mode reconstruction of the kinetic and potential energies during a generic reversal.

Comparison of POD modes for the different formulations

In order to validate our approach we compare to published results by [START_REF] Podvin | A large-scale investigation of wind reversal in a square Rayleigh-Bénard cell[END_REF][START_REF] Podvin | Precursor for wind reversal in a square Rayleigh-Bénard cell[END_REF]. Converged results were obtained for (Ra = 5 • 10 7 , Pr = 4.3)from 1200 snapshots taken every 5 convective time units. Results change very little when using a larger number of snapshots or a different time interval between snapshots. These snapshots correspond to the time series displayed in figure 5.1a.

An initial comparison is based on the POD spectra and the eigenvalues λ i (see equation (2.65)) displayed on figure 5.1b for the first 20 modes. For each formulation (A,B,C) the contributions from mode k = 1 are at least 3 orders of magnitude larger than contributions from mode k = 20. For formulation (A), apart from the first and second modes, the energetic content from each mode was found to be quite small and evenly distributed among the different modes. As a consequence, if we consider the cumulative energy defined as k i=1 λ i / ∞ i=1 λ i , the amount of energy contained inside a fixed number of modes is smaller for formulation (A) than for formulations (B) and (C). For instance, the cumulative energy inside the first 6 modes in formulation (A) is only of 0.80, whereas this value is close to 0.90 (resp. 0.88) for formulations (B) (resp. (C)), see figure 5.1c. From this point of view, formulation (A) is deemed less favorable for a low-order description than formulations (B) and (C). k and the second row corresponds to the stream-function of the velocity modes, i.e. (φ (u) k , φ (v) k ). The bottom row corresponds to contributions from each mode to the convective heat-flux in the vertical direction, i.e. φ (v) k φ (θ) k . Results in figures 5.2a and 5.2b have been normalized by a factor such that φ (θ) j φ (θ) k

+ φ (u) j φ (u) k + φ (v) j φ (v) k
= c jk , as to facilitate the comparison. For formulation (C) c jk corresponds to the Kronecker delta δ jk . However for formulations (A) and (B), since the educed modes are not orthogonal c jj = 1 and c jk = 0 for j = k.

We validate our implementation of the POD by comparing our results with those from [START_REF] Podvin | A large-scale investigation of wind reversal in a square Rayleigh-Bénard cell[END_REF]: For formulations (B) and (C), we recover the same type of flow structures and the energetic content of each mode is in good agreement. For the formulation (A), the two most energetic modes are identical. For modes k ≥ 3, we obtain a similar set of modes although in a different order. If we consider a POD spectra in which several modes have a similar energetic content, the order of these modes could be influenced by the sampled interval. For instance, the third to fifth modes in formulation (A). As observed by Podvin and Sergent, 2015, all three formulations produce the same type of coherent structures with only a different distribution of the energy contents and can be grouped together based on their natural symmetries S x , S y , R π described in 2.5, see table 5.1. For consistency, we will use the same notation as [START_REF] Podvin | A large-scale investigation of wind reversal in a square Rayleigh-Bénard cell[END_REF][START_REF] Podvin | Precursor for wind reversal in a square Rayleigh-Bénard cell[END_REF].

Consider the results obtained using formulation (C). As shown by [START_REF] Podvin | A large-scale investigation of wind reversal in a square Rayleigh-Bénard cell[END_REF][START_REF] Podvin | Precursor for wind reversal in a square Rayleigh-Bénard cell[END_REF], the first (denoted L) and fourth (denoted L * ) most energetic modes have the same symmetry group: both are antisymmetric with respect to the S x and S y symmetries, and symmetric with respect to R π . The mode L corresponds to a large-scale single-roll mode with small recirculation cells in the corners, whereas mode L * corresponds roughly to three vertically stacked rolls. The second (denoted Q) most energetic mode is symmetric with respect to S x and S y and corresponds to a four-roll mode. This mode bears a strong resemblance to the mean fields [START_REF] Podvin | A large-scale investigation of wind reversal in a square Rayleigh-Bénard cell[END_REF]. The sixth most energetic mode is similar to Q, but displays a pattern consisting of eight rolls: four in the center and four in the corners. This mode is characterized by having blobs of hot (resp. cold) fluid inside the bulk just opposite to the cold (resp. hot) thermal boundary layers. Since this mode belongs to the same symmetry group as Q, we denote this mode as Q * . The third and fifth most energetic modes are symmetric with respect to either S x or to S y . Mode S y consists on a vertically stacked double roll mode, which is symmetric with respect to the horizontal axis, hence the name. Conversely, mode S x consists on two horizontally stacked rolls, which is symmetric with respect to the vertical axis. All of these modes have a positive overall contributions to the vertical heat-flux. However, the modes S x and Q * display zones with important negative heat-flux near the corners.

We would like to compare how low-order reconstructions using the different formulations are able to describe the main flow features of the generic reversal and recover the evolution of different global quantities which characterize the reversal process, namely the global kinetic, total potential and available potential energies. Before making such comparison, it is necessary to fix a number of modes for the low-order reconstructions. In the following section we focus on influence of the low-order truncation on the reconstructed fields and the global quantities.

Influence of the low-order truncation on the reconstructed fields using the mixed POD formulation (C)

In order to illustrate the influence of the truncation on a low-order reconstruction, let us compare the DNS results against reconstructions obtained using the mixed POD modes for (Ra = 5 • 10 7 , Pr = 4.3). Reconstructed fields are obtained using the three, six, nine, and twelve most energetic modes: modes k = 1 to k = 6 are displayed on figure 5.2, while modes k = 7 to k = 12 are displayed on figure 5.3. We are primarily interested in the evolution of the global kinetic and the total potential energies. First, one reconstructs the primitive fields ( θ, ũ, ṽ) using a fixed number of POD modes, then compute E k and E p from these reconstructed fields. Last, one evaluates the absolute deviation from the DNS %Error

(E k ) = 100 E k -E k E k %Error(E p ) = 100 E p -E p E p (5.1)
as function of time. Note that contributions from each mode to E k are always positive, whereas the contribution to E p can be both positive and negative. Furthermore, contributions to E p from modes antisymmetric with respect to S x are zero, for instance modes L and L * .

Consider the time series of Error(E k ) and Error(E p ) displayed in figure 5.4 during a particular reversal. A time-series of L 2D is displayed for reference. Let us focus on the three-mode reconstruction. During the first part of the accumulation phase (see plateau in L 2D ), there is little difference from the reconstruction of E k to the DNS. However, by the end accumulation phase (• mark in figure 5.4a), the value of Error(E k ) begins to increase with respect to the other reconstructions until the end of the acceleration. The picture for E p is similar but deviations are accentuated positively or negatively from the end of the accumulation until the end of the acceleration. A six-mode reconstruction works better during both phases (compare • and • marks in figures 5.4a and 5.4b) and has a consistent behavior with the nine-and twelve-mode reconstructions.

The larger deviations during the release and acceleration phases suggest an increased activity of the smaller scales. To illustrate this we compare four snapshots issued from the DNS against their respective POD reconstructions (figures 5.5a to 5.5d): the first snapshot takes place during the accumulation phase; the second one takes place during the release phase, just before the corner-rolls are merged; the third snapshot is taken during the rebound; and the last one during the acceleration phase corresponds to the rotation of the central vortex. For the first and last snapshots, one is able to recover most of the flow features using a three-mode reconstruction (figures 5.5a and 5.5d). However, during the release this reconstruction fails to reproduce the merging of the corner flows (figure 5.5b). In addition, during the rebound, the three-mode reconstruction fails to recover the detachment of the central vortex from the upper and lower walls (figure 5.5c). For all of the snapshots, the small-scale plumes are smoothed out and the blobs of hot/cold fluid inside the corner-rolls are barely visible. Furthermore, thermal boundary layers are missing from a three-mode reconstruction during the rebound and during the acceleration (figures 5.5c and 5.5d).

It is clear that a three-mode reconstruction is insufficient to recover the energetic and some key features of a particular flow reversal. However, it is more difficult to state how many modes are actually required. At least six modes are required to prevent unphysical results such as the vanishing of the boundary layers. Incremental improvements are observed as one includes additional modes, but it is difficult to extract a general trend. For this reason it may be interesting to compare the reconstructions of the generic reversal instead of a particular one. In the following section, we take this approach to compare the different formulations of the POD using a six-mode reconstruction of the generic reversal.

Comparison of a six-mode reconstruction of the energetics of a generic reversal for the different formulations of the POD

In order to illustrate the differences between the POD formulations, let us consider the generic reversal cycles obtained by the DNS results against six-mode reconstructions. We use the following procedure to build a generic reversal from the reconstructed fields:

1. Project primitive fields and obtain modal coefficients α k (t) for each formulation.

2. Evaluate a six-mode reconstruction of the primitive fields ψ = ( θ, ũ, ṽ).

3. Evaluate the global kinetic energy Ẽk (t), the potential energy Ẽp (t), and the available potential energy Ẽa (t), based on the reconstructed fields ψ = ( θ, ũ, ṽ). For practical purposes we approximate the reference height as ỹr = y r ( θ).

4. Apply the conditional average presented in §4 to each quantity based on the amplitude of mode L (by analogy to L 2D ).

(a) Results obtained using each formulation are displayed in a different color on figure 5.6.

0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 f g E k /E k [t -r i ]/τ 1,i (b)
Using formulation (C), one is able to recover the approximate shape and order of magnitude of E k throughout the cycle, although the reconstruction only contains 91% of E k . For E p the picture is slightly more complicated and two major deviations from the generic cycle are observed (figure 5.6b). The first deviation is observed during the rebound, where the value of E p is underestimated. The second deviation is observed during the accumulation phase: E p grows linearly but stagnates near the transition points (f) and (g), before decreasing instead of increasing faster as observed from the DNS. This suggests that, near the end of the accumulation phase, thermal energy is being stored inside smaller scales, absent from the six-modes reconstruction.

For formulation (B velocity-based), the reconstruction of E k is roughly the same as in formulation (C), whereas for formulation (A temperature-based), the reconstruction of E p now follows closely the generic cycle. This improvement is reflected on the amount of energy contained inside these modes, see table 5.2. The improvement also suggests that one of the six temperature-based modes from figure 5.2, which is not included in the first six mixed-modes, becomes the carrier of thermal energy just before the reversal. Now consider using formulation (A) to evaluate E k and formulation (B) to evaluate E p .

The reconstruction of the potential (resp. kinetic) energy using only the velocity (resp. temperature) based modes, fares much worse than results produced from formulation (C).

Before moving on to the next section, let us consider the case of a reconstruction of the available potential energy E a (see figure 5.7a). Observe that E a has the same shape as the reconstructions of E p , but underestimates its value by as much a 20% with respect to the DNS. This deficit is nearly constant all along the cycle and is observed in all three formulations, even for the temperature-based one. Let us remind that E p depends on the probability distribution of the temperature field which is required to compute y r . A low-order reconstruction does not preserve the bounds nor the probability distribution of the temperature field (see figure 5.7b), making very difficult to compute E a accurately.

Formulation ψ2 / ψ E k /E k E p /E p E a /E a A 80% - 99% 88% B 90% 91% - - C 88% 91% 97% 85%
Table 5.2: Comparison of the generic reversal cycle using different formulations of the POD. Six-mode reconstruction of the combined energy ψ2 / ψ 2 with ψ = ( θ, ũ, ṽ) (resp. ψ = (θ, u, v)), and reconstructions of the kinetic, potential, and available potential energies measured with respect to DNS results.

(a) 4.3). Layout is similar to figure 5.6. Right: Probability density and cumulative density functions of an instantaneous temperature field θ(x, t) corresponding to the release phase of a particular reversal, compared to a six-mode reconstruction using formulation (A).

1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 f g E a /E k [t -r i ]/τ 1,i (b)
Using separate expansions for the temperature and velocity field gives better results but comes at the expense of losing the coupling between the velocity and temperature fields. If we opt for a single expansion, then formulation (C) gives better results. This comparison makes a good case for using a mixed formulation for POD analysis inside Rayleigh-Bénard cells. This comparison does not change our previous conclusion that six-modes are sufficient to provide a simplified description of the reversal process. In the following we focus on results obtained using formulation (C).

Dynamics inside of the 'regime of consecutive reversals' using mixed POD modes

Let us separate our data into two sub-sets using the filtering procedure presented in §4.

For (Ra = 5 • 10 7 , Pr = 4.3) the first set is composed of 3150 snapshots taken every 4 convective time units belonging to the 'regime composed of continuous reversals' (CR), while the second one excludes snapshots from this regime. For consistency, we repeat this procedure for (Ra = 5 • 10 7 , Pr = 3.0) using 1500 snapshots inside the CR regime. In this section, we focus on the POD analysis performed on the CR subset.

k i=1 λ i / ∞ i=1 λ i .
The POD spectra and the eigenvalues λ i (see equation (2.65)) are displayed on figure 5.8 for the first 20 modes. Contributions from the most energetic mode are at least 3 orders of magnitude larger than contributions from mode k = 20, whereas the the cumulative energy contained inside the first 6 modes is close to 0.90 for both values of (Ra, Pr), compared to 0.88 when using the complete series.

The resulting POD modes are displayed in figure 5.9. In both cases, we recover a similar set of coherent structures. Since the complete series features prominently the CR regime, it comes as no surprise that CR modes and modes obtained from the entire set resemble each other (compare figures 5.2c and 5.9a). The filtered modes are cleaner and more symmetric. Modes L, Q, L * and Q * are nearly identical to those from the complete set, but the two other modes are slightly different from S y and S x . In the following we refer to these modes as S and S * , since S is antisymmetric (resp. symmetric) with respect to S x (resp. S y ) whereas S * is antisymmetric (resp. symmetric) with respect to S y (resp. S x ), making both modes antisymmetric with respect to R π .

In the following sections, we focus on the time evolution of these modes using a six-modes reconstruction of the generic reversal cycle.

Time evolution of the leading POD modes of the CR subset

We project the modes displayed in figure 5.9 into the temperature and velocity fields to obtain the modal coefficients α k for the entire set. The evolution of the amplitude of mode L is nearly identical to L 2D (not shown), thus allowing us to apply the filtering procedure presented in §4 and separate the CR regime, the EC regime, and intermediate flow regimes.

The curves of the modal coefficients during each reversal are stacked together to obtain conditionally-averaged (or generic) coefficients for each POD mode, see figures 5.10 and 5.11, where each mode is observed to display a characteristic behavior.

Let us consider the time evolution for Pr = 3.0. First, note that L and L * are the only modes that display a reversal at the same time. The mode L mirrors the evolution of L 2D and alternates between a positive and a negative plateau states. The evolution of mode L * is tied to the sign of L and to the different phases of a generic reversal (figures 5.10( 1) and

(a) The evolution of the fully symmetric modes Q and Q * are tied to the phases of a generic cycle, but not to the sign of mode L. Mode Q is mostly positive and is reminiscent of the evolution of E p described in §4: Mode Q increases progressively during the accumulation phase (figure 5.10( 3)), exhibits a sudden drop during the reversal, which is followed by a rebound. On the contrary, the value of Q * is centered around zero, but is highly correlated to value of Q: the value of Q * decreases as Q increases and vice versa. Both Q and Q * are strongly related to the thermal boundary conditions and enforce the global unstable temperature stratification. In this sense, Q * serves to compensate for changes in Q. Observe that Q * changes sign during the accumulation around the transition point (g), whereas

k=1 k=2 k=3 k=4 k=5 k=6 L Q S L * Q * S * (b) k=1 k=2 k=3 k=4 k=5 k=6 L Q L * S Q * S *
-0.2 0 0.2 L ( (1)
) -0.2 0 0.2 L * (3) -0.2 0 0.2 Q (4) -0.2 0 0.2 Q * (5) 0 0.5 1 1.5 -0.2 0 0.2 f' g' f' g' (t -ri)/τ1,i S ( 2
) 0 0.5 1 1.5 -0.2 0 0.2 f' g' f' g' (t -ri)/τ1,i S * Figure 5 6
.10: Evolution of modal coefficients for (Ra = 5 • 10 7 , Pr = 3.0). Conditional averages are obtained using the procedure described in §4. Generic curves displayed in solid lines, shaded region indicates one standard deviation with each phase of the reversal in color: release (orange), acceleration (green), and accumulation (blue). Marks • correspond to the fields in figures 5.13(i) to 5.13(x). where (i) is indicated by a • mark.

( the sign switch in L * is closer to the transition point (f). After both modes switch signs, they contribute to the accumulation of thermal energy inside the corner rolls and to the detachment of the large diagonal vortex.

) -0.2 0 0.2 L ( 1
) -0.2 0 0.2 L * (3) -0.2 0 0.2 Q (4) -0.2 0 0.2 Q * (5) 0 0.5 1 1.5 -0.2 0 0.2 f' g' f' g' (t -ri)/τ1,i S ( 2
) 0 0.5 1 1.5 -0.2 0 0.2 f' g' f' g' (t -ri)/τ1,i S * 6
α k L -0.2 0 0.2 α k Q -0.2 0 0.2 α k L * -0.2 0 0.2 α k S -0.2 0 0.2 α k Q * -0.2 0 0.2 α k S * (b) (b) -0.2 0 0.2 10 -1 10 0 10 1 α k L -0.2 0 0.2 α k Q -0.2 0 0.2 α k L * -0.2 0 0.2 α k S -0.2 0 0.2 α k Q * -0.2 0 0.2 α k S * (b)
The horizontally and vertically stacked dipoles (modes S and S * ) play no dominant role during the generic cycle: their generic curves are essentially zero, but have a larger fluctuations during the release phase. Similar trends in the evolution of the modal coefficients are observed for Pr = 4.3. However, the placement of the sign switch in modes L * and Q * relative to the transition points (f) and (g) is less clear (see figure 5.11).

In order to illustrate the variability of the modal coefficients, let us consider their probability density functions as well. To focus on the behavior of the CR regime, figure 5.12 displays conditional probability density functions (PDF) superimposed to the PDF of the entire set. Inside the CR regime, the antisymmetric modes (L and L * ) display a bimodal probability distribution, whereas the symmetric modes (Q and Q * ) have asymmetric distributions. Additionally, modes S and S * are consistent with a double exponential distribution centered around zero. This suggest their evolution in time to be driven by a random (stochastic) process, which may explain why the generic curve is flat. This is consistent with observations for Pr = 1 [START_REF] Verma | Flow reversals in turbulent convection with free-slip walls[END_REF].

To verify if the evolution of these six POD modes is representative of the reversal process, we perform a reconstruction of the generic reversal using the POD modes and the generic modal coefficients presented here and compare it against the conditionally-averaged fields presented in §4.

(i) (ii) (iii) (iv) (v) +0.5 0.0 -0.5 (vi) (vii) (viii) (ix) (x) +0.5 0.0 -0.5 (vi) (vii) (viii) (ix) (x)
(i) (ii) (iii) (iv) (v) +0.5 0.0 -0.5 (i) (ii) (iii) (iv) (v) +0.5 0.0 -0.5 (vi) (vii) (viii) (ix) (x) +0.5 0.0 -0.5 (vi) (vii) (viii) (ix) (x)

A six-modes reconstruction of the reversal cycle based on the generic modal coefficients

We use the generic modal coefficients from figure 5.10 and the corresponding POD modes to obtain a six-mode reconstruction of a generic reversal for (Ra = 5 • 10 7 , Pr = 3.0). Reconstructed fields are compared to the generic fields obtained in §4, see figure 5.13.

Each field correspond to one of the marks in figure 5.11, where the first snapshot (i) is indicated with a • mark. A similar exercise is performed for Pr = 4.3 using the generic modal coefficients from figure 5.11 and the corresponding POD modes.

For both values of Pr, the reconstructed fields reproduce the main features of the accumulation (i to iv) and acceleration (ix to x) phases identified by the conditional averages of the primitive variables §4: a large diagonal roll squeezed by growing counter-rotating corner-rolls and large thermal plumes located along the opposite side-walls. However, the release of thermal energy into the bulk is not recovered with the same level of detail (v to vi): the breakup of the large-scale circulation and the formation of a new one are clearly visible, however, the continuous rotation of the newly formed circulation and the temperature field inside the bulk are missing. This was expected, since small-scale plumes are not recovered by the reconstructed fields by construction. Indeed, the conditional average of the primitive variables preserves the most representative spatial small-scales, as seen from the comparison for Pr = 4.3 in figure 5.14.

Both approaches are seen as complementary to each other and give a different perspective of the same physical process with consistent results.

Dynamics outside of the 'regime of consecutive reversals' using mixed POD modes

In the previous section we separated our data into sub-sets: a first one belonging to the CR regime, and a second one excluding the CR regime composed of 1900 snapshots taken every 2 convective time units for (Ra = 5 • 10 7 , Pr = 4.3). In this section, we concentrate on the POD analysis performed on the second set which contains snapshots taken from the 'regime of extended cessations' (EC) and from an intermediate regime, referred to as rogue reversals in §4. The main idea is to determine if there exists a particular type of flow structure strictly related to cessations, that is hidden or distorted due to the presence of the CR regime. This is a challenging thing to do for several reasons. For the selected values of (Ra, Pr) the amount of time spent inside extended cessations is only a small fraction of the total length. The limited number of snapshots available inside this regime also makes very difficult to verify the convergence of our POD analysis. In order to validate our results, we repeated this procedure for (Ra = 5 • 10 7 , Pr = 3.0) using a total of 1600 snapshots. Since cessations are less common for this value of (Ra, Pr) and our observations are shorter, snapshots in the subset excluding reversals are taken every convective time unit.

The resulting POD spectra is shown in figure 5.15 for the first 20 modes. Results for both values of (Ra, Pr) are consistent with each other: cumulative energy contained inside the first 6 and 20 POD modes of 0.88 and 0.94, respectively, and at least 2 decades of decay are observed between modes k = 1 and k = 20.

Figure 5.16 displays the corresponding POD modes. A different set of energetic structures is observed inside and outside the CR regime. Outside the CR regime, the most energetic mode is a double-roll mode similar to mode S y . Similarly, the fourth most energetic mode is similar to S x . It is reminded that, S y and S x correspond to the double-roll patterns extracted from the complete series (see figure 5.2c). The second most energetic mode corresponds to a central vortex which is detached from the top and bottom walls. This mode is a combination of modes L and L * as shown in [START_REF] Podvin | Precursor for wind reversal in a square Rayleigh-Bénard cell[END_REF]. The third and fifth most energetic modes are distorted versions of the modes Q and Q * , or combinations of these modes. The sixth most energetic mode was not shown previously and it is composed of two central rolls surrounded by six smaller counter-rotating rolls. This mode is approximatively antisymmetric with respect to S x and S y , the same symmetry group of modes L and L * . This mode is not present in the first twelve modes displayed in figures 5.2 and 5.3. Some of the POD modes obtained from the snapshots including and excluding the CR regime are similar, while others are specific to one regime or the other.

A POD analysis of the complete series is shown to mix these modes.

Comparison inside and outside the 'regime of consecutive reversals'

In order to compare the CR and EC regimes we fix a set of modes φ k that allows for a direct comparison. In this case, φ k corresponds to the POD modes obtained inside the CR regime.

Time evolution of the modal coefficients inside the EC regime using the leading POD modes of the CR regime

Figure 5.17 displays the modal coefficients obtained by projection into the POD modes displayed in 5.9. Inside the EC regime (shaded region), the mode S becomes dominant and alternates between positive and negative plateau states. Mode S * displays stronger fluctuations but lacks a well defined behavior. The coefficient for mode L oscillates around The time evolution of mode S during the EC regime is reminiscent of the evolution of mode L in the CR regime. This suggests we can follow the evolution of mode S to characterize the EC regime. For this reason we compute the inter switch time τ 2 of mode S, similarly as the interval τ 1 of mode L presented §4. For the sake of completeness, we extend this analysis to S * . Consider a set of consecutive times s i (resp. t j ) at which the modal coefficient S (resp. S * ) changes sign. One evaluates the time interval τ 2,i = s i+1s i (resp. τ 3,j = t j+1t j ). The PDFs of τ 1 , τ 2 , and τ 3 for (Ra = 5 • 10 7 , Pr = 3.0) are compared on figure 5.18. Inside the CR regime, one of the main features of τ 1 is the absence of short inter-switch timescales, while this is the opposite in the EC regime. In contrast to τ 1 , inside the CR regime the distributions of τ 2 and τ 3 exhibit short inter-switch times, while inside the EC regime, the distributions are more dispersed. It appears that the distributions of τ 2 and τ 3 fit well to a Log-normal distribution

P (τ ) = 1 τ 1 σ τ √ 2π exp - (ln τ -µ τ ) 2 2σ 2 τ (5.2)
inside the CR regime. Here µ τ (resp. σ τ ) is the mean (resp. standard deviation) of the natural logarithm of the variable τ .

One alternative used in the characterization of the reversals of the geo-dynamo involves 98 considering a Gamma distribution [START_REF] Lhuillier | Statistical properties of reversals and chrons in numerical dynamos and implications for the geodynamo[END_REF]. For such process, the PDF of the inter-switch interval has the following form

P (τ, k) = k µ k τ k-1 Γ(k) exp -k τ µ (5.3)
where Γ(k) is the Gamma function of the shape parameter k, and µ the expected mean value of τ . A Gamma process has a memory of previous events when k = 1. A special case is observed for k = 1 and corresponds to a Poisson process [START_REF] Parker | The magnetic field of the earth: paleomagnetism, the core, and the deep mantle[END_REF].

A data fit of τ 1 inside the CR regime returns a shape parameter k(τ 1 ) = 16.35 with an expected mean value µ(τ 1 ) = 77.05. In contrast, a data fit of τ 2 returns a shape parameter k(τ 2 ) = 0.97 and expected mean value µ(τ 2 ) = 4.78, whereas τ 3 yields a shape parameter k(τ 3 ) = 1.07 and µ(τ 3 ) = 3.78. We repeated this procedure for (Ra = 5•10 7 , Pr = 4.3) seen on figure 5.19 and obtained the shape parameters k(τ 2 ) = 5.00, k(τ 2 ) = 1.12 and k(τ 3 ) = 1.10, with the expected mean values µ(τ 1 ) = 138.09, µ(τ 2 ) = 4.56 and µ(τ 3 ) = 3.70, respectively. These observations would suggest that reversals of mode L are influenced by previous events, whereas the evolution of mode S and mode S * follows Poissonian statistics. This behavior is very different from other systems which display field reversals, such as the cessation-led reversals inside 3D cells [START_REF] Brown | Rotations and cessations of the large-scale circulation in turbulent Rayleigh-Bénard convection[END_REF] and the reversals of Earth's magnetic field [START_REF] Lhuillier | Statistical properties of reversals and chrons in numerical dynamos and implications for the geodynamo[END_REF].

In this sense, the 'regime of consecutive reversals' contains elements of a deterministic system [START_REF] Araujo | Wind reversals in turbulent Rayleigh-Bénard convection[END_REF][START_REF] Castillo-Castellanos | Reversal cycle in square Rayleigh-Bénard cells in turbulent regime[END_REF] and noise. The evolution of certain POD modes (L, Q, L * , and Q * ) displays a consistent dynamical pattern for each reversal, while the modes S and S * appear to be driven by a random process. It is through these modes that 'randomness' appears to be introduced into the reversal process.

Our observations are consistent with results from [START_REF] Podvin | Precursor for wind reversal in a square Rayleigh-Bénard cell[END_REF] in which a three-mode model using modes L, Q, and L * , results in periodic reversals.

In §5.2.2 we mentioned the existence of well-defined relations between different modes inside the CR regime. Our observations suggest there exists, at least in the average sense, a particular relation between different pairs of modes: modes L and L * , Q and Q * , S and S * , and so on. In the following section we focus on how the interaction between different pairs of modes changes inside and outside of the CR regime.

A note on the coexistence between modes

We use a bivariate histogram to illustrate the relation between different pairs of modes.

In order to highlight the differences between the CR and EC regimes, we have produced separate histograms for each in figures 5.20 to 5.23. For the CR regime, we also display the generic curves and identified each of the phases and functional instants which characterize the reversal cycle.

Inside the CR regime, the accumulation (resp. release) phase corresponds to the upper (resp. lower) branches in the (L,Q) plane, with the acceleration phase connecting the upper and lower branches. Two attractive fixed points are observed in the (L,S) and (L,S * ) planes. Outside of the CR regime, two attractive fixed points are observed in the (L,Q) and (L,S) planes. These observations are consistent with the phase portraits in the (L,Q) and (L,S) planes and to the near-heteroclinic cycle identified by [START_REF] Podvin | A large-scale investigation of wind reversal in a square Rayleigh-Bénard cell[END_REF]. Observations are also consistent with the phase space projections of Fourier modes by [START_REF] Petschel | Statistical analysis of global wind dynamics in vigorous Rayleigh-Bénard convection[END_REF][START_REF] Verma | Flow reversals in turbulent convection with free-slip walls[END_REF]) in a similar system using stress-free boundaries and Pr = ∞.

A strong relation between modes L and L * is also shown to exist. As noted by [START_REF] Podvin | Precursor for wind reversal in a square Rayleigh-Bénard cell[END_REF], the quadrants LL * > 0 are predominantly occupied during the CR regime. As the intensity of L decreases near the end of the accumulation phase, the mode L * does just the opposite, see figure 5.20 [START_REF] Araujo | Wind reversals in turbulent Rayleigh-Bénard convection[END_REF]. During reversals, the inversion of mode L passes systematically through rapid inversions in mode L * . During cessations the relation between both modes changes as both modes are locked in the quadrants LL * < 0, see 5.21 [START_REF] Araujo | Wind reversals in turbulent Rayleigh-Bénard convection[END_REF]. These attractors would correspond to a clockwise or counter-clockwise central vortex that is fully detached from the top and bottom walls [START_REF] Podvin | Precursor for wind reversal in a square Rayleigh-Bénard cell[END_REF]. This is also consistent with the second most energetic mode obtained from the EC subset displayed in figure 5.16.

A relation between modes Q and Q * is shown to be quite robust and does not appear to be affected by the change of regime between reversals and cessations, see figures 5.20( 6) and 5.21 [START_REF] Bejan | Convection heat transfer[END_REF]. Inside the CR regime, the generic cycle makes evidence the presence of lower and upper branches corresponding to the release and accumulation phases, respectively.

Modes S and S * belong to the same symmetry group. During the CR both modes oscillate randomly around zero (figure 5.20( 5)), whereas during the EC regime we observe three attractive points in the (S,S * ) plane.

So far we have established a relation between the intensity of the leading modes and specific parts of the reversal process. Dominant L and Q modes are related to the accumulation phase. The intensity of modes L * and Q * increase during the end of the accumulation phase and during the transition. On the other hand, modes S and S * are predominantly active during cessations. In this sense, the strength of one mode relative to the others might provide some insight on the large-scale dynamics. In the following section, we follow this relative strength as function of Ra. Here α k α k corresponds to the mean part of the combined energy projected on the φ k -axis, whereas φ k corresponds to the first 6 POD modes obtained from the CR subset for Ra = 5 • 10 7 . The total combined energy ψ|ψ is displayed in black lines (Tot.). The values of Ra where flow reversals are observed are enclosed in the hatched area, whereas the combination of the EC and CR regimes is observed between the red dashed lines.

Evolution of the leading POD modes as function of Ra

For each Pr, we fix a set of modes φ k (x, t). Here, φ k corresponds to the POD modes obtained from the CR subset for Ra = 5 • 10 7 . For this comparison we perform a phasespace projection of the instantaneous velocity and temperature fields on the φ k -axis

α k (t) = ψ(x, t)|φ k (x, t)
with ψ = (θ, u, v) (5.4) where .|. is the inner product operator defined in §2.6. For each value of (Ra, Pr), we compute α k α k , which corresponds to the mean part of the combined energy ψ|ψ contained inside the mode φ k . In general, the selected POD modes are deemed as representative of the (Ra, Pr) considered here: the amount of combined energy contained inside these six modes represents between 80% and 95% of ψ|ψ for all Ra.

If we follow the evolution of α k α k as function of Ra, it is possible to identify different regions based on the relative intensity of the different modes (figure 5.24). Each region is associated to a particular type of dynamics displayed on figure 5.25.

As both values of Pr display many similarities on their behavior, let us consider the case for Pr = 3.0. For values below Ra = 3•10 5 , mode L is clearly dominant and the corresponding flow is a steady-state single roll. A second region is observed between Ra = 3 • 10 5 and Ra = 3 • 10 6 , where the mode Q becomes slightly dominant over mode L. However, the value of Q decreases slowly as we increase Ra, while the remaining modes are shown to increase. Inside this region we observe periodic or chaotic reversals sans the accumulation phase, i.e. no plateau is visible in L 2D . An almost periodic oscillatory solution is observed for Ra = 3•10 5 . This is exclusively composed of the release phase of the CR regime, similar to periodic oscillatory solutions of the flywheel mechanism by [START_REF] Lappa | Some considerations about the symmetry and evolution of chaotic Rayleigh-Bénard convection: The flywheel mechanism and the "wind" of turbulence[END_REF][START_REF] Chandra | Flow reversals in turbulent convection via vortex reconnections[END_REF]. A third region appears for Ra ≥ 3 • 10 6 , where the mode L becomes clearly dominant again. Inside this region, we observe intermittent flow reversals containing the three phases described by the generic cycle. Mode Q is the second most energetic mode. All other modes are smaller by an order of magnitude except in a particular range of Ra where the mode S becomes comparable to Q. Inside this range, extended cessations coexists with the consecutive reversals. Above this range, cessations disappear almost completely as the plateau duration increases progressively, as presented in §4. Above a certain value of Ra, it becomes increasingly difficult to observe any reversals at all.

It is difficult to compare this sequence of regimes to the description provided by [START_REF] Chandra | Flow reversals in turbulent convection via vortex reconnections[END_REF] using Fourier modes of the vertical velocity. Several reasons may account for these differences: different values of Pr are used, a different decomposition, and the fact that our simulations may be too short to ensure the statistical convergence.

Conclusions of this chapter

In this chapter we used a POD analysis to characterize the reversal regime inside a square Rayleigh-Bénard cells. A reconstruction of the mechanical energy from three different formulations was compared against the generic reversal curves. In agreement with the existing literature, the mixed velocity-temperature formulation was found to provide the best results. A six-mode representation of flow reversal retains the most important features of the reversal process and is deemed as a good compromise.

We performed a POD analysis of the CR and EC regimes separately. Inside the CR regime, we followed the evolution of the six most energetic POD modes during the generic cycle. Two of these modes exhibit a reversal (L and L * ), two others (Q and Q * ) follow a dynamical pattern but do not reverse, whereas the other two (S and S * ) oscillate around zero. Both regimes (EC and CR) are characterized by the same set of coherent structures, except for one mode present in the EC regime and not observed in the CR regime. This made possible to perform a direct comparison of the dynamics of the CR and EC regime. Finally, we followed the energy contents of the different modes as function of Ra to illustrate the limited range in which consecutive flow reversals are observed and the presence of extended cessations within this range. Influence of imposing a stress-free surface on a square Rayleigh-Bénard cell

Complementary material: Simulation Parameters

In the study of classic Rayleigh-Bénard cells, one assumes the Oberbeck-Boussinesq (OB) approximation and considers the same type of boundary conditions imposed on the top and bottom plates. In a such case, the reflection with respect to the horizontal plane S y (see §2. 5) is a natural symmetry of the system [START_REF] Golubitsky | Symmetries and pattern selection in Rayleigh-Bénard convection[END_REF]. Recent studies in natural convection have been focused on systems which lack this reflection symmetry either due to the presence of non-Oberbeck-Boussinesq (NOB) effects [START_REF] Zhang | Non-boussinesq effect: thermal convection with broken symmetry[END_REF][START_REF] Horn | On non-Oberbeck-Boussinesq effects in threedimensional Rayleigh-Bénard convection in glycerol[END_REF]Ahlers, Brown, et al., 2006;[START_REF] Sugiyama | Flow organization in twodimensional non-Oberbeck-Boussinesq Rayleigh-Bénard convection in water[END_REF][START_REF] Xia | Flow reversals in Rayleigh-Bénard convection with non-Oberbeck-Boussinesq effects[END_REF] or trough the use of asymmetric boundary conditions, for instance due to the use of a bottom rough plate (J. C. Tisserand, et al., 2011;[START_REF] Salort | Thermal boundary layer near roughnesses in turbulent Rayleigh-Bénard convection: Flow structure and multistability[END_REF][START_REF] Liot | Boundary layer structure in a rough Rayleigh-Bénard cell filled with air[END_REF] or due to the use of different mechanical conditions on the top and bottom plates [START_REF] Van Der | Effect of velocity boundary conditions on the heat transfer and flow topology in two-dimensional Rayleigh-Bénard convection[END_REF]. Other physical systems where field reversals are present, are sensitive to asymmetric boundary conditions. For instance, periodic or random reversals of the magnetic field generated by a fluid dynamo in the Von Karman Sodium (VKS) experiment are observed only when different angular rotation velocities are imposed on the top and bottom lids [START_REF] Berhanu | Magnetic field reversals in an experimental turbulent dynamo[END_REF][START_REF] Gallet | Reversals of a largescale field generated over a turbulent background[END_REF]. For the Rayleigh-Bénard studies, the influence of the asymmetry is centered on the flow structures and on different global quantities. Deviations from the symmetric case are best observed on the mean temperature profiles, since the mean temperature at the center is no longer expected to correspond to the median temperature. For global quantities, the system response ranges from a modest increase in heat-flux to a transition to a different flow regime [START_REF] Van Der | Effect of velocity boundary conditions on the heat transfer and flow topology in two-dimensional Rayleigh-Bénard convection[END_REF]. One may also study the influence of asymmetry on stability. For instance, [START_REF] Xia | Flow reversals in Rayleigh-Bénard convection with non-Oberbeck-Boussinesq effects[END_REF] focuses on the impact of NOB effects on the stability of flow structures during flow reversals. As the asymmetry increases, flow reversals are found to cover a wider range of the parameter space (Ra, Pr) than before.

In this chapter, we consider an asymmetric cell: stress-free (resp. no-slip) mechanical boundary condition imposed on the top (resp. bottom). This configuration is relevant for different systems, such as the convective boundary layer in the atmosphere. In the following the asymmetric cells are denoted as ST cells, whereas classic RB cells are denoted as NT cells.

Model equations and numerical method

The governing equations are again the Boussinesq equations (2.22), with the same boundary conditions as in the previous chapters, except that a free-slip condition is now imposed on the top plate y = 0.5.

For the 2-D numerical simulations, we consider values of Ra ∈ [10 4 , 5 • 10 8 ] with Pr = 3.0 and Pr = 4.3. This range of values covers the onset of convective motion, the apparition of periodic and oscillatory solutions, and the transition to a (weakly-)turbulent regime.

Simulations have been performed using Basilisk on a regular Cartesian grid, with a variable time-step that verifies the condition CFL< 0.5. Initial conditions correspond to no-motion and a linear temperature gradient. Spatial resolution is verified by following the procedure described in §3.2.2: one checks numerically the exact relations between the instantaneous rates-of-change (see equation (3.11)) and energy transfer rates. The second check concerns the statistical convergence of time-averaged Nusselt numbers proposed by (R. [START_REF] Stevens | Radial boundary layer structure and Nusselt number in Rayleigh-Bénard convection[END_REF]. The average values of Nu vol , Nu top , Nu bot , Nu ǫ , Nu θ , Nu d , and Nu as well as the maximum relative difference %Diff are listed in tables 6.1a and 6.1b (see §2.2.7 for definitions). These quantities are shown to converge within 2% when averaged over the entire length of the simulation except for the highest Ra.

In the following, for visualization we use the reference height y r (x, t), instead of the temperature field θ(x, t). Field y r (x, t) is indeed a one-to-one mapping of the temperature field, which ensures that y r (θ(x, t)) < 0 for 'cold' fluid parcels θ(x, t) < θ xy and vice versa, thus composing a good candidate for (visually) separating thermal boundary layers from the bulk (see figure 6.1). . However, the choice depends as well on the initial conditions: if we start from an initial condition corresponding to a single-roll and increase Ra, we keep the single-roll structure. The presence of this type of multi-stable solutions is reflected on the different global quantities. This multi-stability is commonly observed in classic Rayleigh-Bénard cells (see for instance [START_REF] Mizushima | Sequential transitions of the thermal convection in a square cavity[END_REF][START_REF] Lappa | Thermal convection: patterns, evolution and stability[END_REF].

Overview

Whenever convective motion is present for this range of Ra, the system reaches after a transition period a steady-state1 with a lower mean temperature θ xy than the median temperature (θ top + θ bot )/2 (figure 6.1c). This is allowed since the top/bottom symmetry no longer exists. This effect is further explored in the following sections.

Oscillatory one-and four-roll pattern

In the first type of oscillatory solution, a single-roll and four-roll pattern appears cyclically resulting in a periodic reversal of the large-scale circulation. A close inspection of the instantaneous fields (top part of figure 6.2) reveals the counter-rotating vortex is larger on the lower half than on the upper half of the cell. This difference shifts the rotation center of the central vortex slightly upwards. This type of solution is known to exist in classic RB cells at similar Ra, see [START_REF] Mizushima | Sequential transitions of the thermal convection in a square cavity[END_REF][START_REF] Shimizu | Onset of reversal and chaos in thermally driven cavity flow[END_REF][START_REF] Lappa | Some considerations about the symmetry and evolution of chaotic Rayleigh-Bénard convection: The flywheel mechanism and the "wind" of turbulence[END_REF], and is clearly reminiscent of the release phase of a generic flow reversal presented in chapter 4. To illustrate the difference between the solutions observed inside free-slip (ST) and no-slip (NT) cells, figure 6.2 displays a time-series of the angular impulse L 2D , the spatially averaged vorticity C and the normalized Nusselt numbers at the top and bottom plates, Nu top and Nu bot , for both cells. The evolution of L 2D is similar in both cells, but the ST cell displays a larger amplitude and shorter period: the characteristic frequency of L 2D being f = 0.081 against f = 0.072 for the NT cell. In addition, for the modified cells C is non-zero and has the same dominant frequency as L 2D (f = 0.072) but is shifted in time with respect to L 2D by 2.8 convective time units. The evolution of Nu top is different from the NT case, while Nu bot is similar. For the ST and NT cells, Nu top and Nu bot have twice the frequency of L 2D and similar triangular shapes. However, in the ST cell the fluctuations of Nu top are twice as large as those of Nu bot . Also, Nu top is delayed 0.85 convective time units with respect to Nu bot , instead of being in phase as in the NT case.

Oscillatory double-roll pattern

A second type of oscillatory solution is observed on the ST cells. This solution is characterized by two opposing thermal plumes locked along the right wall and a single ascending plume along the left wall. This solution does not respect the top/bottom symmetry. The left/right symmetry is not respected either, but a mirror opposite configuration may be observed. Dominant flow structures correspond to two vertically stacked rolls, pushing back and forth one another, with one smaller corner roll which appears every half cycle, see bottom part of figure 6.3 at t = 25. This solution is not perfectly periodic as indicated by the noise in the spectra. A similar periodic solution is reported to exist in classic Rayleigh-Bénard cells for (Ra = 10 7 , Pr = 1.0) [START_REF] Chandra | Flow reversals in turbulent convection via vortex reconnections[END_REF], Supplementary

Intermittency and transition towards turbulence

Both types of oscillatory solutions are observed inside a narrow band from 1.5 • 10 5 < Ra < 3 • 10 5 . For Pr = 3.0, continuous reversals were observed for Ra = 1.5 • 10 5 and Ra = 2.5 • 10 5 , while the stacked rolls were found for Ra = 2.0 • 10 5 , see figure 6.4. For Ra = 3 • 10 5 , both oscillatory solutions are observed intermittently: continuous reversals are seen on the amplitude of L 2D (figure 6.5), but also on the characteristic frequencies of the other quantities.

Weakly turbulent flow regime (3

• 10 5 < Ra < 5 • 10 8 )
Once a (weakly-)turbulent regime is established, one observes the appearance of dynamical flow patterns which are reminiscent of both oscillatory solutions described earlier.

Resulting flow structures are very similar to those ones previously observed in the NT cells, complete with the presence of flow regimes analogous to the regime of 'consecutive reversals' (CR) and the regime composed of 'extended cessations' (EC), see figure 6.6.

The former regime is associated to a dominant central vortex and counter-rotating cornerrolls, similar to the CR regime presented on the previous chapter. The latter regime is associated to a dominant double-roll pattern which is sustained for relatively long periods of time, before spontaneously re-organizing in the opposite direction (figure 6.6a), similar to the EC regime.

As the value of Ra increases, a series of consecutive reversals are observed with increased regularity, while the extended cessations become less frequent and eventually disappear altogether (figure 6.6b). For higher Ra, an increasingly large amount of time is spent inside the plateau state, making the flow reversals difficult to observe (figure 6.6b). The sequence of coexistence of the CR and EC regime, followed by a progressive disappearance of the EC regime, then by an increasingly longer plateau state is consistent with observations inside NT cells presented in §5.5.

While the dominant structures are very similar in nature, the resulting flow dynamics are faster and more complex than for NT cells. In order to perform a direct comparison between the NT and ST cells, a statistical approach is required.

Statistical characterization: global quantities and spatial distribution

Lower bulk temperature

The modified cells are characterized by having a lower bulk temperature than their no-slip counterpart. This is observed for all flow regimes (steady, oscillatory, and turbulent). For instance for (Ra = 5 • 10 7 , Pr = 4.3), bulk temperature is up to 10% lower than the median temperature (θ bot + θ top )/2, see figure 6.7. To quantify this difference, let us define an asymmetry coefficient

χ = 0.5 -θ xy θ xy + 0.5 (6.1)
For the modified cells, the asymmetry coefficient χ > 1 for all of the values of Ra considered (figure 6.7). Three slopes are observed in this figure each associated to the temporal dynamics (steady-state, periodic oscillatory and turbulent flow regime respectively). In each case, the value of χ appears to increase with Ra. This asymmetry was also inside a periodic box with asymmetric (no-slip/free-slip) conditions and Pr = 1 [START_REF] Van Der | Effect of velocity boundary conditions on the heat transfer and flow topology in two-dimensional Rayleigh-Bénard convection[END_REF]. It seems to us that the asymmetry is stronger for our configuration. In the case of an asymmetric cell due to a rough bottom plate and smooth top plates, a similar effect was also observed, but for much higher Ra.

Increased heat-flux and turbulent motion

Going from NT to ST, the average Nusselt number Nu increases from 20% to 40%, whereas the average Reynolds number Re based on the mean velocity increases from 15% to 20% (figure 6.8). For both quantities, this increase is reflected on the dependence on Ra on the multiplicative coefficient rather than on the scaling exponents, see comparison between NT and ST in figure 6.8. Note that scaling exponents β Nu = 0.30 and β Re = 0.60 for Pr = 4.3, were obtained by performing a power law fit of the NT cells.

Let us now consider the temporal fluctuations of the Nusselt around the mean, more precisely the probability density functions of the instantaneous Nusselt integrated along the fluid domain Nu vol and the Nusselt numbers evaluated at the bottom and top plates, Nu bot and Nu top , (figure 6.9). Values are normalized by Nu to facilitate the comparison between the NT and ST cells. For the modified cell, the PDFs of Nu vol and Nu bot have the same overall shape with slightly longer tails. For Nu top , both configurations display a

Changes in the spatial distributions of different quantities

Let us consider the spatial distribution for different fields. In order to facilitate the comparison between the ST and NT cells, let us focus on results for (Ra = 5 • 10 7 , Pr = 4.3).

Vertical heat-flux at the top and bottom plates

For the contributions to Nu bot and Nu top , we consider the local heat flux q y evaluated on the top (y = 0.5) and bottom (y = -0.5) plates as a function of x (figure 6.10). For the bottom plate, the profiles are very similar in both cases. For the top plate, different profiles are observed: unlike the NT case, the ST profile decreases monotonically as one gets away from the center. This might be related the effect of top/bottom asymmetry prominently during the extended cessations or due to changes in the upper corner flows due to the free-slip condition.

Thermal and viscous dissipation rates

Because of the increase in Nu and the relations in (2.50), the total thermal dissipation ǫ θ and total viscous dissipation ǫ increase as well when going from NT to ST cells. For both cells, the thermal dissipation rate ǫ θ is highly localized near the top and bottom plates inside thermal boundary layers (figure 6.11). Also, the thermal boundary layer for the ST cells is thinner at the top plate than at the bottom plate.

In both cells, the viscous dissipation rate ǫ is very localized near the no-slip walls, particularly along the side-walls (figure 6.12). Additional, less intense contributions are observed near the corners. For the ST cell, a dissipation layer is observed just below the top plate, in which the dissipation has a comparable intensity to the corners. The viscous dissipation inside the bulk region (-0.4 < x < 0.4 and -0.4 < y < 0.4) is similar for both cells: 46% (resp. 45%) of ǫ for the ST (resp. NT) cells. Since the viscous dissipation barely changes in the bulk, the absence of dissipation at the top plate, should be compensated by remaining no-slip walls.

Temporal and spatial fluctuations in the modified cell near the (no-slip) bottom plate are similar to those in both plates of the no-slip cell. This suggests a disconnection between the top and bottom plates, which interact directly with the bulk flow but not with each other, 121 consistent with observations from other asymmetric systems such as (J. C. Tisserand, et al., 2011). In order to explore such hypothesis, let us take a closer look at the differences in the flow structure near the top and bottom plates.

Comparison between the lower and upper half-cells

Consider the mean temperature and temperature fluctuation profiles in both the ST and NT cells near the bottom plate (figure 6.13). As we mentioned earlier, the ST cell displays a larger temperature difference θ botθ xy across the bottom thermal boundary layer, but its thickness δ θ,bot is comparable in both ST and NT cells. Near the bottom plate, the vertical profiles of temperature fluctuations as well as the thermal dissipation are similar in the ST and NT cases. The vertical profiles near the top plate are different. For the ST cell, both the temperature difference θ xyθ top , thickness δ θ,top are smaller than in the NT cell. However, the slope of the mean temperature profile near the top wall is similar in both cells (figure 6.14). Temperature fluctuations near the top plate are smaller than fluctuations near the bottom plate for the ST cell, while contributions from the thermal dissipation are much more localized near the top plate.

The vertical profiles based on the velocity also support the separation between the upper and lower half-cells: the vertical profiles of the spatio-temporal velocity fluctuations close to the bottom plate, and the viscous dissipation profile in the lower half-cell are very similar for both NT and ST cells (figure 6.15). For the modified cell, the spatio-temporal fluctuations of the horizontal velocity are maximum and the contributions to the viscous dissipation are minimum at the top plate (figure 6.16). However, the contributions from the dissipation layer are larger than in the no-slip case. Since the contributions from the bulk change little, this increase is likely to come from the viscous boundary layers at the side-walls.

Let us define

∂ j u i ∂ j u i x /(Nu -1)
ST NT Figure 6.16: Same as in figure 6.15 but for the upper half-cell.

123 Tisserand, et al., 2011;[START_REF] Salort | Thermal boundary layer near roughnesses in turbulent Rayleigh-Bénard convection: Flow structure and multistability[END_REF]. This is equivalent to defining two virtual cells, which are symmetric (no-slip/no-slip and free-slip/free-slip), have an homogeneous bulk temperature and have the same temperature difference across the thermal boundary layers as the asymmetric cell (no-slip/free-slip). The Rayleigh number associated to the no-slip plate is defined as The resulting Ra n , Ra s , Nu n and Nu s are given in figure 6.17. Let us discuss these results separately. One the one had, results for the free-slip plate now reflect the three different regimes associated to the temporal evolution (steady, oscillatory and turbulent regimes).

Ra n = 2β(T bot -T xy )gH 3 κν = 1 2(0.5 -θ xy ) Ra (6.
For values of Ra s > 10 7 , a power law Nu s ∼ Ra 0.34 s is observed. On the other hand, results for the no-slip plate follow a power law scaling Nu n ∼ Ra 0.30 n which is comparable to previous measurements in the NT cells, but the multiplicative factor of the power law is 15% larger. These measurements appear to be insensitive to the change in regime on the free-slip plate and suggest the behavior of both plates to be rather independent from each other. However, since results from a symmetric free-slip/free-slip cell with adiabatic no-slip sidewalls are not available at this time, we are unable to verify if the Nu s ∼ Ra 0.34 s scaling is representative of such cells, or if it corresponds to a transition between different flow regimes. Additional simulations for higher Ra would help clarify this point. To conclude this chapter, let us consider the influence of imposing the free-slip condition on the dynamics of the large-scale flow by extending the statistical characterization presented on §4 to the modified cells. More precisely, we are interested in the changes between the regime of 'consecutive reversals' and the regime of 'extended cessations' for (Ra = 5 • 10 7 , Pr = 4.3).

We follow the procedure described in §4: first we consider the evolution of the global angular impulse L 2D (figure 6.18), then we identify a set of consecutive times r i at which L 2D changes sign and define a time interval τ i,1 = r i+1r i . We use the amplitude-based filter to separate the CR and EC regimes and compute percentage of time spent in each regime, p cr and p ec = 1p cr , as well as the duration of the jump τ d .

The number of sign switches of L 2D increases significantly from 375 events in the NT cell to over 1160 events on the modified cell in the same interval of 29,000 convective time units. In addition, the number of sign switches associated to flow reversals increases from 161 to 358 in the same period. Additionally, the amount of time spent inside the CR regime decreases from p cr = 83% to p cr = 79%, suggesting that flow reversals are shorter and more frequent.

Consider the probability distribution of τ 1 inside the CR and EC regimes (figure 6.19).

The ST cells display an average inter-switch interval of τ 1 = 151 convective time units, which is comparable to τ 1 = 159 for the NT cell. However, the values are not distributed in the same way, in general reversals in the modified cell are faster and more frequent: the peak probability distribution of the inter-switch interval (resp. duration of the jump) is τ 1 = 53 compared to τ 1 = 103 (resp. τ d = 3.2 compared to τ d = 11.2) for the NT cell.

For the ST cell, the EC regime is characterized by a short inter-switch interval (τ 1 < 50), whereas the values of τ 1 are distributed over a wider range (10 < τ 1 < 400) in the CR regime. Both regimes overlap each other such that a separation time-scale τ c may not be defined.

In order to illustrate the variability of the results, consider the evolution of L 2D , E k , and E p , displayed in figure 6.20 for a dozen reversals representative of the entire series to avoid cluttered graphs: roughly half of the events assigned to the CR regime display the dynamical pattern of the generic cycle presented in §4, while the other half does not appear to follow any particular trend. While it is still possible to use the conditionally averaged curves and identify a generic reversal cycle with roughly the same features as in the no-slip case, the standard deviation from the generic cycle is larger.

Conclusions of this chapter

We have studied the influence of mechanical boundary conditions on Rayleigh-Bénard convection. The main differences between the classic and modified cells, concern the change in the bulk temperature and an increase in both Nusselt and Reynolds numbers.

Our observations support a simplified description where the system can be viewed as being composed of two half-cells interacting directly with the bulk flow but not with each other.

The lower half-cell is similar in many aspects (temporal fluctuations, boundary layers) to the no-slip cell, except for the larger temperature gradient. The upper half cell lacks a kinetic boundary layer, which leaves a thinner thermal boundary layer exposed to the turbulent bulk flow.

The dynamics of the large scale flow are reminiscent of the regime of consecutive reversals and of extended cessations observed in the no-slip cells, but the separation between both regimes is less clear. Modified cells are characterized by the same type of flow structures than the no-slip case, but their temporal evolution is quite different. Reversals are faster and more frequent, whereas the extended cessations are more frequently observed. The reversal mechanism is identical for both cells, but larger deviations from the generic reversal are commonly observed.

Complementary material: Simulation Parameters

(a) Confined cells with adiabatic side-walls

Preamble

Under the Boussinesq approximation, the 3-D RB system is fully defined by the Rayleigh and Prandtl numbers, by the cell geometry (in particular the aspect ratio along the transversal direction Γ z ) and the boundary conditions. The influence of geometrical parameters on the large scale circulation has been the subject of many studies in cylindrical vessels or rectangular boxes. In the latter case, numerical or experimental studies (S. [START_REF] Wagner | Aspect-ratio dependency of Rayleigh-Bénard convection in box-shaped containers[END_REF][START_REF] Huang | Confinement-induced heattransport enhancement in turbulent thermal convection[END_REF][START_REF] Kaczorowski | Turbulent flow in the bulk of rayleighbénard convection: aspect-ratio dependence of the small-scale properties[END_REF][START_REF] Ni | Reversals of the large-scale circulation in quasi-2D Rayleigh-Bénard convection[END_REF] have focused on the influence of the aspect ratio Γ z on global properties as well as the large-scale flow. In the numerics, for instance, the global Nusselt and Reynolds numbers display a strong dependence on Γ z for Ra ∼ 10 5 -10 7 (S. [START_REF] Wagner | Aspect-ratio dependency of Rayleigh-Bénard convection in box-shaped containers[END_REF]. This geometrically induced heat-transfer enhancement was confirmed experimentally and numerically by (S. D. [START_REF] Huang | Confinement-induced heattransport enhancement in turbulent thermal convection[END_REF] and was explained by the formation of vertical turbulent channels instead of a large-scale roll. Experimentally, the influence of confinement on the dynamics of these large-scale circulation has been also discussed by [START_REF] Ni | Reversals of the large-scale circulation in quasi-2D Rayleigh-Bénard convection[END_REF].

Studies by [START_REF] Chong | Condensation of coherent structures in turbulent flows[END_REF][START_REF] Chong | Exploring the severely confined regime in rayleigh-bénard convection[END_REF] identified three different flow regimes denoted as I, II, and III, see figure 7.1. The differences between the regimes can be illustrated by considering an horizontal slice inside the bottom boundary layer. Regime I is the classical regime in which thermal plumes are arranged into sheetlike structures with no preferential alignment, except close to the side-walls (figure 7.1a).

When the characteristic size of the plumes is comparable to the transversal size of the box, plumes do not split but in turn transform into sheets of constant width oriented along the transversal direction. This corresponds to regime II (figure 7.1b). Inside regime III we observe clear indications of a (quasi-)two-dimensional flow consistent with Hele-Shaw cells: a transversal velocity close to zero and a Poiseuille-type velocity profile (figures 7.1c and 7.1d). To illustrate these changes in 3-D we consider instantaneous isotherms for different aspect ratios displayed on figure 7.2. During transition from II to III the large thermal structures locked near the corners which were separated (figure 7.2b) tend to merge and form a single plume (figures 7.2c and 7.2d). Note that, the separation between regimes II and III (Γ z = 29.37Ra -0.31 ) corresponds to the aspect ratio at which the average Nusselt number reaches a maximum value while keeping Ra constant and increasing Γ z [START_REF] Chong | Exploring the severely confined regime in rayleigh-bénard convection[END_REF].

Thermal and solutal convection in Hele-Shaw cells has been the subject of several experimental [START_REF] Hartline | Thermal convection in a hele-shaw cell[END_REF][START_REF] Ozawa | Flow and temperature measurement of natural convection in a Hele-Shaw cell using a thermo-sensitive liquid-crystal tracer[END_REF], numerical [START_REF] Bizon | Plume dynamics in quasi-2d turbulent convection[END_REF] and analytical [START_REF] Graham | Time-periodic thermal convection in heleshaw slots: the diagonal oscillation[END_REF] studies. Such systems model fluid filtration in saturated porous media [START_REF] Gorin | On analogy between convective heat and mass transfer processes in a porous medium and a hele-shaw cell[END_REF][START_REF] Nield | Heat transfer through a porous medium[END_REF]. The most striking feature is again, the suppression of a system-wide circulation in favor of the formation of large plumes (or clusters of plumes) along the vertical direction. The scaling exponents of the Nusselt and Reynolds numbers deviate from the exponents observed in cubic Rayleigh-Bénard convection. To the best of my knowledge, this case is not considered by the unifying theory of thermal convection [START_REF] Grossmann | Scaling in thermal convection: a unifying theory[END_REF][START_REF] Grossmann | On geometry effects in Rayleigh-Bénard convection[END_REF].

In the present chapter, we identify the main dynamical features of confined cells through the use of fully resolved 3-D DNS in regime III. Note that in this regime, the observed largescale dynamic is very different from the one of pure two-dimensional turbulent convection presented in chapters 4 and 5 of this thesis. In the subsequent chapter we compare results from the 3-D DNS with two simplified models, which are used to explore the parameter space and to study long-term dynamics for which the use of 3-D DNS is prohibitively expensive.

(a) Γ z = 1/2 (b) Γ z = 1/4 (c) Γ z = 1/8 (d) Γ z = 1/16 (e) Γ z = 1/32 (f) Γ z = 1/64

Numerical approach

We explore here regime III, where we fixed Pr = 4.38 and varied Ra from 10 6 to 10 9 . Different aspect ratios 1/64 ≤ Γ z ≤ 1/8 were also considered. Direct numerical simulations were performed using the code Basilisk described in section 3.1.2. It is reminded that one solves the Boussinesq equations

∂ i u i + ∂ z w = 0 (7.1a) ∂ t u i + ∂ j (u j u i ) + ∂ z (wu i ) = -∂ i p + PrRa -0.5 (∂ j ∂ j u i + ∂ 2 z u i ) + Prθδ iy (7.1b) ∂ t w + ∂ j (u j w) + ∂ z (ww) = -∂ z p + PrRa -0.5 (∂ j ∂ j w + ∂ 2 z w) (7.1c) ∂ t θ + ∂ j (u j θ) + ∂ z (wθ) = Ra -0.5 (∂ j ∂ j θ + ∂ 2 z θ) (7.1d)
Here the Einstein notation is used only for the in-plane components, i.e. indices i, j = x, y.

The domain is defined as [-0.5, 0

.5] × [-0.5, 0.5] × [-Γ z /2, Γ z /2], with Γ z ≪ 1.
Velocity is zero on all walls. On the top (resp. bottom) walls, the reduced temperature is imposed to be θ(y = 0.5) = -0.5 (resp. θ(y = -0.5) = 0.5). Side-walls are adiabatic i.e. ∂ n θ = 0.

Results were obtained on uniform Cartesian grids with a variable time-step that verifies CFL < 0.5. As a general rule, the distribution of grid points must be able to resolve the steep gradients in the velocity and temperature fields near the walls [START_REF] Grötzbach | Spatial resolution requirements for direct numerical simulation of the Rayleigh-Bénard convection[END_REF]. This is a difficult thing to do using regular Cartesian grids for confined cells since the number of points along the transversal direction is usually small. A second, more subtle limitation is a slow time evolution since velocities are quite small, making fully resolved direct simulations increasingly onerous for Γ z < 1/64. Spatial resolution is verified to be adequate by following the procedure described on §3.2.2: one checks numerically whether the exact relations between the instantaneous rates-ofchange (see equation (3.11)) and energy transfer rates are verified. A small residual value is an indication that spatial resolution is adequate. The second check concerns the statistical convergence of time-averaged Nusselt numbers proposed by (R. [START_REF] Stevens | Radial boundary layer structure and Nusselt number in Rayleigh-Bénard convection[END_REF].

Dynamics of the confined regime in RB cells

Large-scale plume dynamics

Flow dynamics is mostly defined by the competition between friction and buoyancy forces.

For severely confined cells, the frictional drag on the front and back walls (z = ±Γ z /2) favors the formation of large plume clusters along the vertical direction (also called turbulent channels) instead of a system-scale circulation observed in pure 2-D or in conventional 3-D convection. Figure 7.3 illustrates the evolution of flow structures as we decrease Γ z for (Ra = 10 7 , Pr = 4.38). The plume width increases with the confinement. The resulting flow is mainly controlled by the motion of the turbulent channels. Some channels cross the domain unopposed from top to bottom or vice versa, as seen near the center of the cavity in figures 7.3a to 7.3c. Other channels may collide with each other as they traverse the domain. Such head-on collisions are typically found along the vertical side-walls, in which the plumes are deviated in the horizontal direction.

For (Γ z = 1/8, Ra = 10 7 ) the top-bottom symmetry is not observed and side-wall plumes collide at a nearly constant height close to top or bottom plates. The interaction of sidewall plumes with the central plume creates a periodic sweeping motion, which is reflected on the the evolution of Nu bot , Nu top , and Nu vol . A dominant frequency of f = 0.12 is clearly observed for Nu vol (see figure 7.3a).

For (Γ z = 1/16, Ra = 10 7 ) the collision of the side-wall plumes takes place near the midheight of the cavity. The interaction between the side-wall plumes and the central plume leads to a chaotic motion with a dominant frequency f = 0.036. The side-wall plumes can be moving in-phase or out-of-phase. During the in-phase motions the side-plumes keep the central plume locked in place, whereas during the out-of-phase motions the side-plumes push back and forth the central plume. In some instances, the coordinated motion pushes the central plume into a wall which then merges with a side-wall plume.

For (Γ z = 1/32, Ra = 10 7 ) one observes two types of periodic solutions: a flow with a central plume locked in place (see figure 7.3c), and a flow without a central plume (see figure 7.3d). In the former case, the motions of the side-wall plumes are coordinated, pumping hot fluid from the bottom to the top or vice versa. In the latter case, the sidewall plumes move out-of-phase to form a large-scale circulation and prevent other plumes to settle. Both solutions have the same dominant frequency f = 0.025, suggesting the oscillating motion of the side-plumes remains unchanged.

For the most confined cell (Γ z = 1/64, Ra = 10 7 ), one arrives to a steady single-roll configuration with one ascending and one descending plume (not shown in figure 7.3). from the one in figure 7.3b. Here the side-wall plumes move back and forth independently from one another. One ascending and one descending central plume are observed almost simultaneously, which repel each other to the sides as they collide, thus creating a systemwide circulation.

Dynamics for (Ra = 10 8 , Γ z = 1/64) are determined by the interaction between the side-wall plumes and one ascending (or descending) central plume (figure 7.5). This configuration results in a chaotic motion with characteristic frequency f = 0.1. This solution similar to the one displayed on figure 7.3b.

When Ra is further increased, additional channels are allowed to form (figures 7.6 and 7.7). For (Ra = 2 • 10 8 , Γ z = 1/64) in addition to having side-wall plumes of different heights, we identify one ascending and one descending channels near the center of the cavity (figure 7.6). Unlike configurations with a single central plume, this solution is nearly centro-symmetric. The interaction between multiple turbulent channels leads to rather complex dynamics characterized by a dominant frequency near f = 0.05.

For (Ra = 10 9 , Γ z = 1/64) thermal plumes are clustered into four ascending and three descending turbulent channels (figure 7.7). Intervals of light (lower than average) and intense (higher than average) heat-transfer alternate roughly every 100 convective time units (see Nu vol (t) in figure 7.7). Light periods often coincide with the merging of hot (resp. cold) plumes into large clusters close to the lower (resp. upper) thermal boundary layers. These clusters detach from the boundary layers and make their way into the bulk. The intense periods coincide with the collision of said clusters against the upper (resp. lower) plate. The formation of multiple turbulent channels is consistent with observations from [START_REF] Bizon | Plume dynamics in quasi-2d turbulent convection[END_REF]. Near the onset of convection, the sequence of conductive, stationary, and oscillatory flow patterns is also consistent with experiments in Hele-Shaw cells [START_REF] Ozawa | Flow and temperature measurement of natural convection in a Hele-Shaw cell using a thermo-sensitive liquid-crystal tracer[END_REF].

Finally, note a strong similarity between the turbulent channels and the type of convective structures observed in similar systems where the third dimension is partially suppressed due to the action of stabilizing forces. Examples include strong magnetic fields, a rapid rotation around the vertical axis, the presence of an additional density gradient [START_REF] Chong | Confined Rayleigh-Bénard, Rotating Rayleigh-Bénard, and Double Diffusive Convection: A unifying view on turbulent transport enhancement through coherent structure manipulation[END_REF], or geometric confinement like in porous media [START_REF] Otero | High-rayleigh-number convection in a fluid-saturated porous layer[END_REF][START_REF] Hewitt | Ultimate regime of high rayleigh number convection in a porous medium[END_REF][START_REF] Hewitt | High rayleigh number convection in a threedimensional porous medium[END_REF]. The loss of three-dimensionality may be illustrated through the various dissipative terms in (3.11): the viscous dissipation rate ǫ, the thermal dissipation rate ǫ θ and the diapycnal mixing Φ d . Each of these terms may be split in two parts: one part due to contributions from gradients along the transversal direction z, and another due to gradients along the in-plane components x and y

ǫ (z) ≡ (∂ z u i )(∂ z u i ) + (∂ z w) 2 xyz ǫ (xy) ≡ ǫ -ǫ (z) (7.2a) ǫ (z) θ ≡ (∂ z θ) 2 xyz ǫ (xy) θ ≡ (∂ j θ)(∂ j θ) xyz (7.2b) Φ (z) d ≡ (∂ z y r )(∂ z θ) xyz Φ (xy) d ≡ (∂ j y r )(∂ j θ) xyz (7.2c)
The relative contributions from ǫ (z) /ǫ, ǫ (xy) θ /ǫ θ , and Φ

d /Φ d provide a good measurement of the bi-dimensionalization of the flow structures (see table 7.2). First, ǫ (z) corresponds to the energy lost due to frictional drag along the front and back walls (z = ±Γ z /2). As we decrease the aspect ratio Γ z , ǫ (z) contributes anywhere from 70% up to 98% of the total viscous dissipation ǫ (see table 7.2). Second, ǫ (xy) θ represent from 70% to nearly 100% of the total thermal dissipation, whereas Φ (xy) d represent from 50% to nearly 100% of the total mixing rate Φ d . This is expected in confined cells where adiabaticity of the front and back walls imposes small temperature gradients along z.

Flow reversals inside the laterally confined geometry

A pure 2-D turbulent convection can be described by the superposition of a large-scale flow over a incoherent turbulent background (see chapters 4 and 5). This large-scale flow is mainly characterized by a single-roll structure which is sustained for extended periods of time. However, a spontaneous re-organization takes place (known as a flow reversal), changing the sense of rotation of the roll, thus restoring the statistical symmetry of the system. In 3-D rectangular cells, flow reversals have been observed for different values of Ra and Pr. For Γ z = 0.3, this was seen by [START_REF] Sugiyama | Flow reversals in thermally driven turbulence[END_REF], for Γ z = 0.2 by [START_REF] Vasiliev | Reversals of large-scale circulation at turbulent convection in rectangular boxes[END_REF], and for 0.84 ≤ Γ x ≤ 1.10 and 0.33 ≤ Γ z ≤ 0.25 by [START_REF] Ni | Reversals of the large-scale circulation in quasi-2D Rayleigh-Bénard convection[END_REF] In the present work, we report a reversal process for more confined 3-D cells, which corresponds to a different large-scale flow.

-1 0 1 i ii iii iv v vi -viii ix x -1 0 1 û12 -1 0 1 -1 0 1 û13 -1 0 1 -1 0 1 û14 -1 0 1 0 500 1,000 1
A configuration with one ascending plume near the center of the cavity (fig. 7.8a.i) may be sustained for over 1,000 convective time units. Since the central plume collides against the top plate, cold fluid is pushed towards the side-walls and suppresses the formation of additional large-scale plumes. Due to a coordinated motion of the side-wall plumes, the central plume (the direction of its motion is indicated by a red arrow) is displaced (fig. 7.8a.ii) and allows a second plume to appear on its left (fig. 7.8a.iii). Thereafter, the central plume swings back towards the left, sweeping the newly formed plume towards the side-wall leading to the destruction of both central plumes (fig. 7.8a.iv). In the absence of a single central plume, ascending and descending plumes form simultaneously and actively repel each other (fig. 7.8a.v-a.viii). They coexist until one is swept away and the other becomes dominant (figs. 7.8a.ix-a.x).

This sequence is quite different from the reversals observed in pure 2-D. This is illustrated based on a half-range Fourier decomposition of the horizontal velocity, see §2.5. In pure 2-D, the reversal of the large diagonal vortex occurs after being squeezed by counterrotating corner-rolls and it is characterized by the evolution of the first 2-D Fourier modes (in particular modes [START_REF] Adami | Surface tension and buoyancy in vertical soap films[END_REF][START_REF] Adami | Surface tension and buoyancy in vertical soap films[END_REF], [START_REF] Adami | Surface tension and buoyancy in vertical soap films[END_REF][START_REF] Ahlers | Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection[END_REF], [START_REF] Adami | Surface tension and buoyancy in vertical soap films[END_REF]2) and(2,2)). Reversals are evidenced by a change in sign in the mode [START_REF] Adami | Surface tension and buoyancy in vertical soap films[END_REF][START_REF] Adami | Surface tension and buoyancy in vertical soap films[END_REF]. Let us consider a similar approach for the laterally confined cells and follow the evolution 2-D Fourier modes in the transversal mid-plane z = 0 (see figure 7.8b). In laterally confined 3-D cells, the mode û1,1 is not dominant and oscillates around zero, whereas the leading modes correspond to û2,2 , and û1,2 instead. The value of û2,2 is mostly positive, while û1,2 and û1,4 switch signs during the flow reversal near t = 100. In this sense, û1,2 is a natural choice to characterize the presence of flow reversals in this geometry. Furthermore, near t = 550, t = 1000, and to a lesser degree for t = 1500 one observes failed reversals: these coefficients drop to zero, before returning to the same plateau.

In 2-D reversals, the whole process is driven by localized and progressive accumulation of thermal energy inside the corner-rolls, which eventually overcomes the central vortex. In the confined geometry, the reversal is characterized by strong oscillations in E k , E a and Nu vol (figure 7.8c), but it is less clear if there is a progressive build-up of thermal energy as in pure 2-D. No noticeable change in size of the thermal structures was observed. Note that reversal are rare and it is difficult to perform a detailed study using DNS. This constitutes one of the motivations for developing the reduced order models presented in the next chapter.

Statistical characterization of the confined regime: number of plumes and average plume width

We take a statistical approach to characterize the turbulent flow and the persistent largescale structures. In the context of the confined regime, we are interested in the number and size of large-scale turbulent channels that controls the flow dynamics. Given the loss of three-dimensionality of the thermal structures along the transversal direction, we can focus our observations on the transversal mid-plane z = 0.

From figure 7.9a, two type of plumes can be distinguished. Near the bottom plate we observe three narrow ascending plumes (in orange) just detached from the boundary layers, and two wider descending plumes (in blue) which are about to impact against the bottom boundary layer. Similarly, near the upper plate we observe two narrow descending plumes (in blue) and one wider ascending plume (in orange). A geometric characterization of such individual plumes is a complex task. Different alternatives are available, see for instance [START_REF] Shishkina | Analysis of sheet-like thermal plumes in turbulent rayleigh-bénard convection[END_REF]. Here, a simple method is proposed taking advantage of the alignment of sheet-like plumes perpendicular to the front and back walls (z = ±Γ z /2) in the confined regime. To compute the position of the narrow plumes just detached from the boundary layer, one focuses on the height y = ±(0.5 -1/Nu), i.e. twice the thermal boundary layer thickness. Since plumes are the main carriers of heat across the domain, it is reasonable to associate a plume to each local maxima in φ 2 z (x, t) where φ(x, t) = vθvθ x denotes the deviations from the mean instantaneous vertical heat-flux. Hence, the location of the plumes is estimated by searching for the number and positions of these local maxima (indicated in figure 7.9 by marks).

The number of plumes at a given time depends on length Γ x (here kept at Γ x = 1), and on the geometry of the plumes, which depends on Ra and Γ z . For (Ra = 10 8 , Γ z = 1/64) one identifies from 3 to 4 plumes near the bottom plate and from 3 to 4 others near the top plate (figures 7.9a to 7.9c), whereas for (Ra = 10 9 , Γ z = 1/64) one identifies from 6 to 10 plumes near the bottom plate and a similar amount near the top plate (figures 7.9d to 7.9f). Ejected plumes are not evenly distributed but have instead the tendency to form clusters (see figures 7.9e and 7.9f). Hence the distance between contiguous ejected plumes, denoted by d 0 , is not necessarily equal to the length of the cavity divided by the number of plumes. Figure 7.10a displays the probability distribution of d 0 for (Γ z = 1/64, Pr = 4.38) and different values of Ra.

For Ra = 5•10 7 , plume spacing near the bottom plate peaks near d 0 = 0.5, i.e. the distance between the side-walls and the central plume, while near the top plate a peak is observed near d 0 = 0.95, i.e. the distance between the side-wall plumes. For Ra = 10 8 , a central plume is observed at the bottom and two side-wall plumes at the top, but additional plumes form near both plates, which are pushed towards the center or towards the side-walls. The plume spacing d 0 is distributed over a wider range due to the horizontal displacements of the plumes, while a second smaller length scale reflects the motion of the new plumes (see top part of figure 7.10a). Unlike Ra = 5 • 10 7 and Ra = 10 8 , a transition towards a symmetric configuration is observed for Ra = 2 • 10 8 and Ra = 10 9 . This results in a similar distributions of d 0 at the top and bottom plates. Finally, for Ra = 10 9 the motion of several turbulent channels that are formed, pushed and merged continuously with each other is reflected on a smoother distribution of d 0 .

One may estimate a characteristic width of the ejected (narrow and intense) plumes. This is done by computing the time averaged spatial auto-correlation function of φ f corr (r) = φ(x, y, z, t)φ(x + r, y, z, t) x φ (x, y, z, t)φ(x, y, z, t) x at y = ±(0. and measuring the distance r 0 as the first zero-crossing of f corr (r 0 ) = 0, i.e. the distance at which the plumes are no longer correlated to themselves (figure 7.10b). The distance r 0 is then used to compute an integral length scale of the vertical heat-flux L 0

L 0 = r 0 -ro f corr (r) dr (7.4)
The value L 0 is considered to be representative of the average plume width. Note that the value of L 0 is comparable to δ

(1) θ = 1/(2Nu) (see table 7.3). This is consistent with observations in cylindrical cells [START_REF] Shishkina | Analysis of sheet-like thermal plumes in turbulent rayleigh-bénard convection[END_REF] in which the thickness of sheet-like plumes was found to be of the same order of the thermal boundary layer.

Statistical characterization of the confined regime: mean profiles and boundary layer thickness

Turbulent convection is divided spatially into thermal and kinetic boundary layers and a turbulent background away from the boundaries. Traditionally, in order to provide a simplified description of the turbulent flow, one considers vertical profiles of different quantities. In our particular case, the influence of the aspect ratio Γ z is also explored through the use of profiles of the viscous dissipation in the transversal direction.

Vertically averaged profiles

Consider the thermal vertical profiles displayed on figure 7.11: the spatio-temporal fluctuations θ rms xz and the plane and time-averaged thermal dissipation rate ∂ j θ∂ j θ xz . Two regimes are identifiable. In the first one, fluctuation profiles are similar to the typi- cal profiles found on unconfined turbulent convection cells [START_REF] Kerr | Rayleigh number scaling in numerical convection[END_REF]. This is the case for (Γ z = 1/64, Ra = 10 9 ) (figure 7.11a) but also for (Γ z = 1/32, Ra = 10 8 ) and (Γ z = 1/32, Ra = 10 9 ) (not shown): temperature fluctuations reach a peak value close to the top and bottom plates, indicating the presence of thermal boundary layers, whereas a nearly constant temperature fluctuation is observed in the bulk. Additionally, thermal dissipation is localized near the top and bottom plates. The second regime corresponds to configurations where a central plume is observed to remain in place for long periods of time. This is the case for (Ra = 10 8 , Γ z = 1/64) (figure 7.11b) but also for (Γ z = 1/64, Ra = 5 • 10 7 ) (not shown). In such cases, the vertical profiles are asymmetric by exchanging the lower and upper half-cells inside the bulk. In addition to the features of the first regime, the temperature fluctuations display a peak in the bulk, more precisely at an intermediate height where the ascending and descending plumes collide. A small increase in thermal dissipation is also observed at the same height.

These two regimes have a counterpart in the vertical velocity profiles of the spatio-temporal fluctuations u rms xz (y), the plane-averaged viscous dissipation rate ∂ j u i ∂ j u i xz and the viscous dissipation rate ∂ j u i ∂ j u i x at the median plane z = 0 (figure 7.12). In the first regime, the horizontal velocity fluctuations display a peak value close to the top and bottom plates and comparatively smaller values in the bulk (figure 7.12a). Unlike the model profiles by [START_REF] Kerr | Rayleigh number scaling in numerical convection[END_REF], the transversally averaged dissipation profile ∂ j u i ∂ j u i xz is nearly constant in the bulk due to frictional drag along the front and back walls. If we consider the dissipation profile at the median plane instead of the transversally averaged one, the viscous dissipation is localized near the top and bottom plates. For the second regime, additional peaks in the velocity fluctuations and in the viscous dissipation are observed at intermediate heights where the ascending and descending side-wall plumes collide (figure 7.12b).

Mean thermal and kinetic boundary layers

In this section, we focus on the relation between the thermal and kinetic boundary layers. In confined convection, the thickness of the thermal boundary layer depends not only on Ra and Pr, as in the conventional convection, but also on Γ z . The situation for the kinetic boundary layer is different: for the confined case, its thickness depends mostly on Γ z . This implies that the ratio between thermal and kinetic boundary layers is no longer defined by Pr as conventional (unconfined) convection, and leads to a situation where the thermal boundary layer interacts directly with the bulk even for Pr > 1, which is not possible in conventional convection.

Thermal boundary layer thickness at the top and bottom plates

We provide different estimates of the thermal boundary layer thickness presented in §2.4: namely, thickness δ defined as the closest distance between the position where ∂ j θ∂ j θ xz (y) = Nu and the top and bottom plates. To measure such thicknesses, we could have used the profiles at the transversal mid-plane (z = 0) instead of the transversally averaged profiles, since ∂ j θ∂ j θ x (y, z) for

Ratio between the thermal and kinetic boundary layers at the plates

For conventional (unconfined) cells, the ratio between δ p and δ θ is mostly defined by the value of Pr: δ θ is expected to be smaller than δ p for Pr > 1 and vice versa. However, for very confined cells, the thickness δ θ increases as the convective transfer, i.e. Nusselt (see table 7.1) is hampered by the confinement, while δ p depends mostly on Γ z and not on Ra (see table 7.5). In this sense, the ratio δ p /δ θ is expected to become smaller as Γ z decreases regardless of the value of Pr. This a major difference between the unconfined and confined regimes, which is consistent with observations from [START_REF] Chong | Confined Rayleigh-Bénard, Rotating Rayleigh-Bénard, and Double Diffusive Convection: A unifying view on turbulent transport enhancement through coherent structure manipulation[END_REF]. This behavior is independent of the two regimes mentioned earlier on figures 7.11 and 7.12, which are related to behavior of thermal plumes and the asymmetry of the vertical profiles. Both regimes are observed for δ p /δ θ > 1 and δ p /δ θ < 1 alike. Examples of the second regime include (Ra = 5 • 10 7 , Γ z = 1/8) and (Ra = 10 8 , Γ z = 1/8) for δ p /δ θ > 1, and (Ra = 10 7 , Γ z = 1/8) and (Ra = 10 8 , Γ z = 1/16) for δ p /δ θ < 1.

Kinetic boundary layers on the front and back walls

The second major difference from unconfined convection concerns the kinetic boundary layers on the front and back walls which become dominant for geometrical reasons. To define the boundary layer thickness, we consider the transversal profiles of the timeaveraged viscous dissipation ∂ j u i ∂ j u i xy (figure 7.16) and compute the positions z where [START_REF] Adami | Surface tension and buoyancy in vertical soap films[END_REF]. The smallest distance between these points and the front or back walls is taken as the thickness δ w .

∂ j u i ∂ j u i xy = ∂ j u i ∂ j u i xyz = (Nu -
Locally, the velocity field along the side-walls may be considered as a wall-bounded flow. For such a velocity field, the thickness δ w is expected to increase as the Reynolds number decreases and vice versa. This Reynolds number is based on the mean velocity magnitude At some level of confinement, the value of δ w loses its Reynolds dependence being bounded by the depth of the cavity. This process is illustrated in figure 7.16. As Γ z decreases, the dissipation becomes increasingly parabolic: dissipation peaks at the front and back walls and is close to zero at the transversal mid-plane. Furthermore, the boundary layer thickness 'saturates' to a constant value δ w = 0.211Γ z (indicated by a red dotted line in figure 7.16). This value can be equally obtained by assuming the velocity profile to be parabolic. For such a parabolic profile, the viscous dissipation at the front and back walls (z/Γ z = ±0.5) tends to ∂ j u i ∂ j u i xy ≈ 3(Nu -1) and the dissipation ǫ (bl,w) inside the region enclosing the front and back kinetic boundary layers accounts for 80.7% of the total dissipation ǫ. This peak dissipation at the front and back walls (see figure 7.14d) and the value of the ratio ǫ (bl,w) /ǫ (see table 7.6) are observed in our simulations.

The transition towards δ [START_REF] Adami | Surface tension and buoyancy in vertical soap films[END_REF] p /δ

(1) θ < 1 is observed around Re d ∼ 2 and often coincides with the saturation the boundary layer thickness at the front and back walls to a constant value δ w = 0.211Γ z indicating the transition to a parabolic velocity profile in the transversal direction (table 7.6). In the following section, we explore how these deviations from the conventional (unconfined) convection are reflected on the behavior of the Nusselt and Reynolds numbers. The scaling exponents β Nu-1 and β Re are estimated by fitting our data and presented on table 7.7. These exponents are consistent with values found in previous studies [START_REF] Chong | Exploring the severely confined regime in rayleigh-bénard convection[END_REF].

For confined cells, scaling exponents display a dependence on Γ z and hence deviate from the [START_REF] Grossmann | Scaling in thermal convection: a unifying theory[END_REF] theory. For Γ z = 1/64 one obtains β Nu-1 = 0.61 and β Re = 0.80 (red lines on figures 7.17a and 7.17b), which are different from β Nu-1 = 0.30 and β Re = 0.52 observed in cubic cells (black lines in figures 7.17a and 7.17b). The latter are compatible with [START_REF] Grossmann | Scaling in thermal convection: a unifying theory[END_REF].

Chong and Xia, (2016) followed the evolution of scaling exponents with respect to Γ z up to Γ z = 1/128 and concluded that these exponents were approaching the asymptotic values (β Nu-1 = 0.61 and β Re = 0.80), which we found here for Γ z = 1/64. However, for Hele-Shaw cells with aspect ratio Γ z = 1/240 using free-slip boundaries, higher exponents have been reported (β Nu-1 = 0.872 and β Re = 0.96) [START_REF] Bizon | Plume dynamics in quasi-2d turbulent convection[END_REF]. While cells with no-slip and free-slip conditions may have different dynamics, it is reasonable to wonder if the same high exponents exists for no-slip conditions for aspect ratios smaller than Γ z = 1/128.

Since the kinetic boundary layer thicknesses δ p and δ w become increasingly thin with Γ z , it is quite difficult to perform fully resolved direct numerical simulations in such confined configurations. In order to explore this severely confined regime, it is convenient to take a different approach. In the following chapter, we present a reduced model based on the Hele-Shaw cells which is compared to DNS results. The resulting data is used to explore the long term evolution of the large scale flow and the evolution of different global quantities for Γ z ≪ 1, before presenting a scaling theory (see section 8.5) suggesting the asymptotic scaling exponents have different values from those of [START_REF] Chong | Exploring the severely confined regime in rayleigh-bénard convection[END_REF].

and take the average along the transversal direction (for the present chapter means z ) of the Boussinesq equations similar to the technique of von Kármán and Pohlhausen.

∂ i u i = 0 (8.2a)

∂ t u i + ∂ j u j u i = -∂ i p + PrRa -0.5 ∂ j ∂ j u i + ∂ z u i ] ε -ε + Pr θ δ iy (8.2b) ∂ t θ + ∂ j u j θ = Ra -0.5 ∂ j ∂ j θ (8.2c)
If we neglect inertial forces on the fluid inside a very confined cell ε ≪ 1, one may consider transversal velocity to be negligible and the pressure to be independent of the transversal direction. The resulting in-plane velocity components are expected to have a Poiseuille-like profile. One may also suppose for a confined system with adiabatic walls the temperature field to have no dependency on the transversal direction. In the first model, we assume the primitive variables to have the following form θ(x, y, z, t) = θ(x, y, t) (8.3a) p(x, y, z, t) = p(x, y, t) (8.3b) u(x, y, z, t) = v(x, y, t)g(z) (8.3c) where

g(z) = 3 2 1 - z ε 2 and v • e z = 0 (8.4)
Introducing ansatz (8.3a) to (8.3c), one evaluates the operator which leads to the following system of equations These equations are reminiscent of those developed by [START_REF] Gondret | Shear instability of two-fluid parallel flow in a hele-shaw cell[END_REF] in the study of the Kelvin-Helmholtz instability. In order to compare results from this model against the DNS, it is convenient to express the viscous and thermal energy dissipation rates that is consistent with the simplified model. This is done by introducing (8.3a) to (8.3c) in equations (7.2a) and (7.2b) ǫ ≡ ǫ (xy) + ǫ (z) with ǫ

∇ xy • v = 0 ∂ t v + 6 5 v • ∇ xy v = -∇ xy p + PrRa -0.5 ∇ 2 xy v -λ 1 v + Pr θe y ∂ t θ + v • ∇ xy θ = Ra -0.

A 'better' 2-D model with inertial corrections

In this section, we present a second reduced model which takes into account inertial effects. This is done in the spirit of the work by Ruyer-Quil, 2001 that extended the model by [START_REF] Gondret | Shear instability of two-fluid parallel flow in a hele-shaw cell[END_REF] Corrections are based on a classical perturbation technique lying on the existence of a basic state to which deviations are applied: the velocity, pressure and temperature fields are developed in terms of a small formal parameter. The profile of the basic state along z corresponds to the solution profile at order zero: a parabolic profile for the velocity field, a constant profile for the temperature and pressure fields. The model equations may be obtained using two alternative methods: the gradient expansion method and the weighted residuals method [START_REF] Ruyer-Quil | De la modélisation et des instabilités des films liquides tombants[END_REF]. For a given order of approximation, both methods yield the same equations. In this work, we chose the weighted residuals approach. The complete computations and approximations required are detailed in annex F.

From this analysis, one obtains the expressions for θ and u

θ = θ (x, y, t)f 0 (z) + θ(1,2) (x, y, t)f 2 (z) (8.8a) u = u (x, y, t)g 0 (z) + ã(1,2) (x, y, t)g 2 (z) + ã(1,4) (x, y, t)g 4 (z) (8.8b)
in terms of the transversally averaged fields θ , and u , as well as the inertial correction terms θ(1,2) , ãi (1,2) , and ãi (1,4) . The functions f 0 (z) = 1 and g 0 (z) = 3 2 (1 -( z ε ) 2 ) are given by the solution of the zeroth-order formulation and functions f 2 (z), f 2 (z), g 2 (z), and g 4 (z) are the following polynomial functions f 2 (z) = 1 60 (7 -30(z/ε) 2 + 15(z/ε) 4 ) (8.9a)

g 0 (z) = 3 2 (1 -(z/ε) 2 ) (8.9b) g 2 (z) = - 1 8 (7 -21(z/ε) 2 + 35(z/ε) 4 ) (8.9c) g 4 (z) = 1 16 (11 -165(z/ε) 2 + 385(z/ε) 4 -231(z/ε) 6 ) (8.9d)
The amplitude of the inertial correction term for the temperature field is expressed by

θ(1,2) = - ε 2 Ra 0.5 2 [ u • ∇ xy θ ] (8.10a)
while the amplitudes of the corrections for the velocity field read as follows with ∇ xy = e x ∂ x + e y ∂ y , and λ 1 being the same friction coefficient (8.6) as in the previous model and λ 2 being a correction coefficient for the buoyancy term λ 2 = 105 2ε 2 Ra -0.5 (8.12) This expression resembles equations (8.5a) to (8.5c) but with different coefficients and one supplementary correction term for the buoyancy force.

ã(1,2) = - ε 2 Ra 0.
Once again, in order to be able to compare results from the model against the DNS, it is convenient to find an expression for the viscous and thermal energy dissipation rates that are consistent with the model with inertial corrections. It is important to determine whether the reduced models are able to reproduce the main features of the large-scale flow described in the 3-D DNS for 1/64 ≤ Γ z ≤ 1/8. As seen in chapter 7, the confined Rayleigh-Bénard system may posses multiple stable states with different dynamics for some aspect ratios. In order to facilitate the comparison, both models use the same initial condition, which is obtained by averaging the initial condition of the DNS along the transversal direction. The system is let evolve for several hundreds of convective time units before taking any measurements. For the least confined geometry Γ z = 1/8 (figure 8.1a), the fields already display some of the elements of the DNS: the presence of plumes along the side-walls and a single central plume spanning the height of the cavity. However, both models fail to recover the periodic sweeping motion of the central plume observed in the DNS, returning a chaotic solution with no dominant frequency instead (figure 8.2a). For Γ z ≤ 1/16 (figures 8.2b to 8.2d) both models are able to reproduce both the size, height and shape of thermal plumes. The Freq.

(c) Freq.

(d) Freq.

(c) Freq.

(d) Freq. taken at consecutive times every 5 convective units in order to give a sense of the plumes motion. The same type of flow dynamics is recovered in each case.

The configuration with two stable central plumes (one ascending, one descending) displayed on figure 8.3c for Ra = 2 • 10 8 was initially obtained using the reduced model. We wanted to check if such solution indeed existed for the DNS and the model with inertial corrections. For the DNS we reconstructed the primitive initial fields using instantaneous fields obtained from the model and assuming a constant temperature profile and a parabolic profile for the velocity. For the model with inertial corrections, we used the same instantaneous fields as initial conditions. Both simulations were let evolve for several hundreds of convective time units. In each case, the same configuration was retrieved. This shows that the models can be used to help exploring the parameter space. However, one should be aware that the type of solutions obtained by both models is limited to symmetric solutions with respect to z. Both models are qualitatively well suited to recover the main features of the large-scale flow for the most confined configuration (Γ z = 1/64).

In the following section, we attempt to perform a quantitative comparison for this value of Γ z .

Let us end with a side note: the approximate size and shape of the thermal structures presented in figure 8.5 for Γ z = 1/128 are in good agreement with the instantaneous fields taken at the vertical mid-plane displayed in figures 5(g) to 5(i) of Chong and Xia, 2016.

Number of plumes and average plume width with the reduced models

As seen on figures 8.3, for the most confined cell (Γ z = 1/64), the number, shape, and size of the thermal structures are consistent to the DNS. For Ra = 5 • 10 7 and Ra = 10 8 a persistent configuration is observed with three ascending (a central plume and the sidewall plumes) and two descending channels. For Ra = 2 • 10 8 one observes three ascending To quantify this aspect, the probability distribution of the inter-plume distance d 0 is compared for the two models and the DNS in figure 8.6, and the mean spatial autocorrelation of the vertical heat-flux in figure 8.7. Both quantities are in good agreement with the DNS for up to Ra ≤ 2 • 10 8 . For Ra = 10 9 the correlation distance r 0 and the integral length scale L 0 are underestimated by as much as 15%.

Mean vertical profiles

To illustrate the range of validity in Ra of the models, figures 8.8a to 8.8d display different vertical profiles for (Γ z = 1/64, Pr = 4.38). Profiles associated to the temperature correspond to mean temperature θ xz , spatio-temporal fluctuations of the temperature θ rms xz , and plane-and time-averaged thermal dissipation ∂ j θ∂ j θ xz , whereas the profiles associated to the velocity field correspond to the (spatio-temporal) fluctuations of the horizontal and vertical velocities, u rms xz and v rms xz respectively, and to the plane-and time-averaged viscous dissipation

∂ j u i ∂ j u i xz .
The reduced model is able to recover the shape of the vertical profiles of the DNS. When using the model with inertial corrections, these results slightly improve for Ra ≤ 2 • 10 8 (figures 8.8a to 8.8c). The most visible change being in the viscous dissipation profile near the top and bottom plates, where the reduced model predicts a peak in dissipation not seen on the DNS. For the model with inertial corrections, the position of the peak values in θ rms xz (resp. u rms xz ) close to the top and bottom plates is in good agreement with the DNS. These profiles are used to estimate the thickness of the thermal (resp. kinetic) boundary layers. For the higher Ra (Ra = 10 9 ), deviations from the DNS become noticeable, but both models still manage to recover the correct orders of magnitude and the approximate shapes (see figure 8.8d).

Average Nusselt and Reynolds numbers

The last comparison concerns the time-averaged Nusselt (see §2.50) and time-averaged Reynolds number (see equation (7.6)) for different Γ z . For the DNS, the time-averaged Nusselt Nu ref , its maximum relative difference %Diff were evaluated in the previous chapter (see table 7.1), and the time-averaged Reynolds number Re ref is shown in figure 7.17. For the reduced models, we evaluate Nu vol , Nu top , Nu bot , Nu ǫ , and Nu θ and compute again their average value Nu and maximum relative difference %Diff. A time-averaged Reynolds number based on the root mean squared velocity Re is also evaluated. For the 'simple' model, the expressions required to compute these quantities are (8.7a) and (8.7b). For the model with inertial corrections, these expressions are detailed in annex F. In particular, the Reynolds number is obtained using equation (F.44). There are incremental improvements when using the model with inertial corrections, most notably on the average Reynolds number and the viscous dissipation but also on the fluctuation profiles. In the following, we only consider results from the model with inertial corrections.

Using the reduced model with inertial corrections to explore the confined regime

One may use the reduced model with inertial corrections to study the long term evolution of the large-scale flow in order to characterize flow reversals in the confined regime. To illustrate the feasibility of such studies, we perform simulations using a 512 2 grid for (Γ z = 1/16, Ra = 2 • 10 7 , Pr = 4.38). A flow reversal was observed for this configuration using 3-D DNS in section 7.3.2. The initial condition for the reduced model corresponds to instantaneous fields obtained from the DNS which are averaged over the transversal direction. The system was then evolved for over 35,000 convective time units: several disruptions of the large-scale flow were observed, in particular 4 flow reversals. Each reversal is indicated in the time series (figure 8.9) by marks (i) to (iv). ( that, the odd-modes, û1,1 and û1,3 , have a symmetric PDF, û1,2 and û1,4 display a bimodal PDF, and the PDF of û2,2 is skewed symmetric. The relation between the different modes might be related to the natural symmetries of the Rayleigh-Bénard cells, as in the pure 2-D case.

) (2) (3) (4) 1
Figure 8.10 displays a sequence of snapshots describes the evolution of the thermal structures during 100 convective units in which the reversal (iv) is observed. First, due to the action of the side-wall plumes, the central plume is displaced to the sides, thus allowing a second plume to appear (figures 8.10(1) to 8.10( 4)). Instead of changing directions and plunging against the left wall, like in the reversal described in figure 7.8, both plumes continue their motion and merge into the right side-wall plume (figure 8.10( 5)). Then, ascending and descending plumes form simultaneously (figures 8.10(6) to 8.10( 8)) and coexist until one becomes dominant (figures 8.10(9) to 8.10(10)). Reversals (i) to (iii) follow a similar process, although some variations, like the direction of the plumes motion and the duration of the reversal are observed. Despite these differences, the sequence is considered as representative of the ensemble of flow reversals observed. Our observations suggests the driving force behind flow reversals to be the interaction between the large-scale plumes along the side-walls and near the center of the cavity, consistent with previous observations from DNS results.

In this sense, the model appears to be well adapted for the study the long-term evolution of the large-scale flow. Exploration of the parameter space is required in order to identify a configuration with more frequent reversals in order to verify these observations.

Scaling exponents of the Nusselt and Reynolds numbers in the severely confined regime

One may use the reduced models with inertial corrections to perform simulations for very small values of Γ z focusing on the Nusselt and Reynolds scaling exponents in the confined regime. Indeed, one interesting question raised by [START_REF] Chong | Exploring the severely confined regime in rayleigh-bénard convection[END_REF] concerns the evolution of these scaling exponents towards some asymptotic values. The use of such models could provide some insight on this question by considering values Γ z which are difficult to access for fully resolved DNS. This is done in §8.5.2, but before we would like to introduce a theoretical prediction for the scaling exponents in the severely confined regime.

A theoretical approach to the scaling exponents in the severely confined regime

Scaling exponent for the Nusselt number

Let us revisit some central elements of Rayleigh-Bénard convection. Under a severe confinement, one may argue that characteristic velocity scales should differ those of equation ( 1.3): the velocity scale of the heat conduction and the momentum dissipation are based on the depth of the cavity d instead of the height H, while keeping the scale of the buoyancy unchanged

U ′ cond = κ d U ′ visc = ν d U buoy = β∆TgH (8.16)
Using the ratio between these velocities, we define a new dimensionless Rayleigh number

Ra d ≡ U 2 buoy U ′ cond U ′ visc = β∆TgHd 2 κν (8.17)
which is related to the classical Rayleigh number as

Ra d = Γ 2 z Ra (8.18)
The parameter Ra d is reminiscent of the Darcy-Rayleigh number used in convection inside fluid-saturated porous media, if the Darcy permeability coefficient (the square of a length scale characterizing the porous medium) is replaced by K ∼ d 2 , where d is the depth of a Hele-Shaw cell [START_REF] Nield | Heat transfer through a porous medium[END_REF]. The similarities between both systems can be seen in figure 8.11: the large-scale plumes observed in confined convection is analogous to the 'dripping' patterns observed in convection inside porous media [START_REF] Otero | High-rayleigh-number convection in a fluid-saturated porous layer[END_REF].

In the classical Rayleigh-Bénard, Malkus' and Howard's arguments based on the marginal stability of the boundary layer lead to the classical β Nu-1 = 1/3 exponent, see [START_REF] Otero | High-rayleigh-number convection in a fluid-saturated porous layer[END_REF].

Conventional convection

Severe confinement

δ θ = H/(2Nu)
where δ θ is independent of H.

δ θ = H/(2Nu)
where δ θ is independent of H. hence β Nu-1 = 1. This value is larger than the β Nu-1 = 0.872 observed by [START_REF] Bizon | Plume dynamics in quasi-2d turbulent convection[END_REF] and β Nu-1 = 0.61 observed by [START_REF] Chong | Exploring the severely confined regime in rayleigh-bénard convection[END_REF].

β∆Tgδ 3 θ κν = δ θ H 3 Ra ∼ 10 3 β∆Tgδ θ d 2 κν = δ θ H Ra d ∼ 10 3 Nu ∼ Ra 1/3 Nu ∼ Ra d = Γ 2 z Ra

Scaling exponent for the Reynolds number

The scaling exponent for the Reynolds number may be deduced by following the reasoning from Grossmann-Lohse (GL) scaling theory. The basic element of the GL theory are the exact relations for the viscous dissipation rates derived from the Boussinesq equations, ǫ = (Nu -1) [START_REF] Shraiman | Heat transport in high-Rayleigh-number convection[END_REF]. This term is decomposed into contributions from the different boundary layers and the viscous bulk and estimates for each term are obtained from dimensional analysis.

Let us focus on the contributions from the kinetic boundary layers at the front and back walls. In the GL scaling theory, the contributions from the side-walls are estimated from dimensional analysis assuming a balance between the advective and viscous terms

ǫ bl,w ∼ U 2 δ 2 w δ w Γ z (8.20)
with δw Γz being the fraction of volume occupied by the kinetic boundary layers at the sidewalls.

Using DNS results for the most confined cells Γ z = 1/32 and Γ z = 1/64, we measured the thickness of the kinetic boundary layer at the front and back walls δ w and the relative contributions to the viscous dissipation from the region enclosing the boundary layers ǫ bl,w (see table 7 (8.25) This leads to the theoretical exponents β Nu-1 = β Re = 1, which are not identical but closer to the values β Nu-1 = 0.872 and β Re = 0.96 obtained by [START_REF] Bizon | Plume dynamics in quasi-2d turbulent convection[END_REF] for Γ z = 1/240 using stress-free boundaries. In order to verify if such scaling may be observed, we perform simulations from reduced model with inertial corrections for up to Γ z = 1/256. [START_REF] Chong | Exploring the severely confined regime in rayleigh-bénard convection[END_REF] to collapse our data.

A first group which corresponds to Ra/Ra * < 0.005 and Nu > 5 exhibits β Nu-1 = 0.82 and β Re = 0.91 (dashed red lines in figure 8.13). These scaling exponents are consistent with equation (8.24). A second group which corresponds to Ra/Ra * > 0.005 yields the scaling exponents β Nu-1 = 0.44 and β Re = 0.69 (dashed green lines in figure 8.13). In contrast, these exponents do not verify equation (8.24). A third group of outliers corresponds to points with Nu < 5 and Ra/Ra * < 0.005. For Ra/Ra * < 0.005 and Nu > 5, the exponents β Nu-1 = 0.82 and β Re = 0.91, are larger than the asymptotic values estimated by [START_REF] Chong | Exploring the severely confined regime in rayleigh-bénard convection[END_REF], but differ from the values β Nu-1 = 0.827 and β Re = 0.96 obtained by [START_REF] Bizon | Plume dynamics in quasi-2d turbulent convection[END_REF] as well as the theoretical exponents β Nu-1 = β Re = 1 proposed in section 8.5.1.

There are two possibilities for the differences in the scaling exponents. The first one is that the exponents β Nu-1 = 0.82 and β Re = 0.91 do not correspond yet to the asymptotic regime. To test this affirmation, additional simulations for Γ z < 1/256 should be considered. The second possibility, is that our scaling theory is too simple. In this sense, it could be interesting to extend the Grossmann & Lohse unifying scaling theory as to include the aspect ratio dependency which may provide a better prediction of the scaling exponents confined regime.

Conclusions of chapters 7 and 8

In this part we used data from 3-D DNS in rectangular Rayleigh-Bénard cells for small aspect ratio to highlight the main features of the confined regime III. As one increases the level of confinement, the temperature field becomes independent of the transversal coordinate while the velocity field becomes increasingly parabolic. The confined regime is characterized by the formation of large-scale turbulent channels in the vertical direction and the non-linear interaction between these channels is seen to play a fundamental role on the dynamics of the large-scale flow.

Turbulent convection inside the confined regime may be divided spatially in different regions: thermal and kinetic boundary layers along the top and bottom plates, kinetic boundary layers along the side-walls, and a bulk flow away from the boundaries. Under a severe confinement, the thickness of the kinetic boundary layer at the top and bottom plates becomes comparable to the depth of the cavity. In contrast, the thermal boundary layer becomes increasingly thicker as the convective heat-flux is hindered by the front and back walls. This leads to a situation where the thermal boundary layer is directly exposed to the bulk flow, even for Pr > 1. Another important difference from conventional (unconfined) convection concerns the thickness of the kinetic boundary layer at the front and back walls, which 'saturates' to a constant value consistent with a parabolic velocity profile.

This configuration is well suited for dimensionality reduction. By analogy to previous works inside Hele-Shaw cells, we extended an reduced model to take into account inertial corrections and performed quantitative comparison of both models to 3-D DNS with good results. Two potential applications of the model with inertial corrections are explored. The first potential use is in the study the long term evolution and the dynamics of reorientations of the large-scale flow. The second potential use concerns the evolution of global quantities for very confined cells. Our observations suggests the presence of a change in flow regime and different scaling laws between confined and severely confined geometries.

General conclusions

The work exposed in this manuscript is a contribution to the numerical study of largescale flow dynamics in Rayleigh-Bénard convection. Using a DNS code we simulated the long term evolution for various Rayleigh-Bénard configurations and developed a series of analysis tools for their characterization. A particular attention was given to spontaneous reorientations and reconfigurations of the large-scale flow (or flow reversals) over long periods of time by considering a pure 2-D flow inside a square cell, then by considering a slim rectangular cavity. For such long simulations, it is imperative to identify a meaningful quantity, global or local, to guide our observations. For the 2-D case, a pertinent choice is the global angular impulse. We devised a filtering procedure based on the amplitude of this quantity to separate the different flow regimes, which allowed us to characterize each regime separately. A first regime composed of consecutive reversals displays a well defined and consistent dynamical pattern. Using a time rescaling and conditional statistics allowed us to identify a generic reversal mechanism in terms of accumulation and exchange of mechanical energy. This work resulted in a publication [START_REF] Castillo-Castellanos | Reversal cycle in square Rayleigh-Bénard cells in turbulent regime[END_REF]. A second regime composed of extended cessations of the large-scale circulation, required to choose a different quantity. A good candidate was obtained from a POD analysis. A statistical characterization suggested this regime to be driven by random stochastic processes.

We extended this analysis to a configuration with a free-slip condition imposed on the top and a no-slip bottom plate. Observations suggest a disconnection between the thermal boundary layer at the top and bottom plates. Both interact directly to the bulk flow but not with each other. This configuration is characterized by the same type of flow structures as in the classical case, but their time evolution is entirely different and the clear separation between the regime of consecutive reversals and the regime of extended cessations no longer exists. Finally, by considering the plates separately we identified a change of regime in the free-slip plate.

In the last part, we used 3-D DNS to study the flow dynamics inside a slim rectangular cavity. By analogy to other confined systems, such as Hele-Shaw cells and thin liquid films, we proposed a reduced model with inertial corrections to mimic the three-dimensional flow. Results from this model agree well with our own 3-D DNS and with existing publications. We successfully tested the feasibility of studying the dynamics of flow reversals inside confined cells by performing long simulations using this model for one configuration and identified a handful of flow reversals. The nature of these reversals is different from pure 2-D reversals. The reduced model also allowed us to identify inside regime III a transition between two different dynamics, which depends on the ratio between the kinetic and thermal boundary layer thicknesses. A scaling theory for such confined regime was proposed.

Perspectives

In this work we identified a generic reversal mechanism inside square cells in which accumulation of thermal energy inside the counter-rotating corner-rolls is a key process. Since Prandtl number is related to diffusion of thermal energy, it might be interesting to examine the influence of this number on the reversal cycle. Another interesting point is to consider the influence of corner-rolls by studying a case in which such rolls are weaker or nonexistent but still experience flow reversals. This would allow us to verify if the main ingredient of flow reversals is related to the localized accumulation of energy somewhere in the flow, or specifically depends on the existence of corner-rolls. For instance, it could be interesting to study from our energetical point of view a 2-D cell with free-slip conditions imposed on all boundaries. Indeed [START_REF] Verma | Flow reversals in turbulent convection with free-slip walls[END_REF] showed that this system exhibits reversals without corner-rolls.

Our statistical approach can be extended to 3-D geometries provided one is able to follow the plane of large-scale circulation and that one is able to obtain a long time signal containing a sufficient number of events. However, extending the statistical approach to cubic and cylindrical containers is more challenging because of the meandering of the LSC plane which can also be subject to torsional or sloshing motions. In this work, we were able to contour both limitations when considering a slim rectangular cavity, effectively restricting the large-scale flow to a plane and by proposing a reduced model which is computationally less expensive than using 3-D DNS.

Large-scale flow dynamics inside laterally confined cells are to be further explored. To explore the apparent significance of the ratio between the thickness of the thermal and kinetic boundary layers, a parametric study of the aspect ratio and the Prandtl number would be desirable. These observations could eventually allow to include the aspect ratio dependency in the unifying scaling theory of Grossmann & Lohse. Such approach would allow for a smooth transition between the different regime and provide a better prediction of the scaling exponents inside the confined regime.

Part VI

Appendices

i • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • ⊙ • • x 1 x 2 • • x nx • • x nx-1 ⊙ xu 1 ⊙ xu 0 ⊙ xu nx-1 ⊙ xu nx x i x i+1 x i-1 xu i xu i+1 xu i-1
+ i = x i+1 -x i for i = 1, • • • , n x -1 ∆x - i = x i -x i-1 for i = 2, • • • , n x
and define the length of the segment ∆x i

∆x i =          ∆x 2 for i = 1 1 2 (∆x + i + ∆x - i ) for i = 2, • • • , n x -1 ∆x nx-1 for i = n x
In similar fashion, we define a quantity [∆xu 1 , ∆xu 2 , • • • , ∆xu nx ] as the length of the segment between the indicated points as follows. First, one introduces a forward and backward spacing ∆xu + i and ∆xu - i , respectively

∆xu + i = xu i+1 -xu i for i = 1, • • • , n x -1 ∆xu - i = xu i -xu i-1 for i = 2, • • • , n x
and we proceed to evaluate the length of the segment ∆xu i

∆xu i =    0 for i = 1, n x 1 2 (∆x + i + ∆x - i ) for i = 2, • • • , n x -1
The same process is repeated on the y and z directions.

Staggered grid arrangement

Four different grids are defined in function of these sets of points. We define the following quantities in each grid:

-Grid for the scalar quantities of the cell, cell face, or cell vertex, accordingly, is associated to the index [0, 0, 0] in the local stencil. Neighboring cells are referred to by their relative placement with respect to the current cell: The neighboring cell in the positive x-direction would correspond to index [START_REF] Adami | Surface tension and buoyancy in vertical soap films[END_REF]0,0], and so on. The stencils automatically define certain variables associated to the current grid, such as the position (x, y, z) of the current cell.

B.1.2 Fields and Iterators

The main difference with respect to other numerical codes, is the introduction of predefined types for scalar, vector, and tensor fields, and field iterators meant to reduce duplicate code. Consider for instance, the following scalar field a(x) in R 2 , declared as follows

B.2 Overall temporal discretization scheme

A solution to the Boussinesq equation system is obtained by combining a centered formulation of the incompressible Navier-Stokes equations (see centered.h), a diffusion solver for the temperature field (see diffusion.h) and one tracer advection event (see tracer.h). The numerical scheme resulting from this combination has a global second-order accuracy, as shown in [START_REF] Bell | A second-order projection method for the incompressible Navier-Stokes equations[END_REF].

The formulation of the incompressible Navier-Stokes equations uses the fractional-step method using a staggered in time discretization of the velocity and the scalar fields (see [START_REF] Popinet | Gerris: a tree-based adaptive solver for the incompressible euler equations in complex geometries[END_REF][START_REF] Popinet | An accurate adaptive solver for surface-tension-driven interfacial flows[END_REF]: one supposes the velocity field to be known at time n and the scalar fields (pressure, temperature, density) to be known at time n -1/2, and one computes velocity at time n + 1 and scalars at time n + 1/2,

∂ i u i n+1 = 0 (B.1a) (u i n+1 -u i n ) ∆t + ∂ j (u j u i ) n+ 1 2 = -∂ i p n+ 1 2 + ν∂ j (∂ j u i n+1 + ∂ j u i n ) 2 + a i n+ 1 2 (B.1b) θ n+ 1 2 -θ n-1 2 ∆t + ∂ j (u j θ) n = ∂ j (κ∂ j θ) n+ 1 2 (B.1c)
where ν = PrRa -0.5 corresponds to the kinematic viscosity, κ = Ra -0.5 to the thermal diffusive coefficient. Since the external forcing term a i n+ 1

2 in our case corresponds to the buoyancy force a i n+ 1 2 = -Prθ n+ 1 2 δ i2 , it is compulsory to start by solving the equation for temperature (B.1c), before the equations (B.1a) and (B.1b).

In the following, we use the lowercase (resp. uppercase) variables, for instance u i and θ (resp. U i and Θ) to identify the quantities evaluated at the cell center (resp. at the center of the cell face). The velocity component at the center of the face U i n is evaluated from the velocity u i n from both cells sharing this face through a simple interpolation. In addition, subscript (d) refers to the cell faces.

B.2.1 Solution of the energy equation

The solution of equation (B.1c) is obtained in two steps. First, the advection terms are evaluated using the Bell-Collela-Glaz (BCG) second-order un-split upwind scheme. Second, an Helmholtz-Poisson problem for the energy equation is solved using a multilevel Poisson solver to obtain θ n+ 1 2 .

Approximation of the scalar advection term

Consider the advection term ∂ j (u j θ) n integrated over a control volume Ω, which is approximated as the sum of the fluxes in each control surface (4 in 2D or 6 in 3D).

Ω ∂ j (u j θ) n dΩ = ∂Ω (n j u j n )θ n dS ≈ (d) S (d) (n j U j n )Θ n (B.2)
where n j corresponds to the (outward) normal unit vector at the face center.

where u y n and u z n are evaluated as follows

B.2.2 Solution of the velocity-pressure coupling

The velocity-pressure coupling is treated using a classical time-splitting projection method (see [START_REF] Chorin | On the convergence of discrete approximations to the Navier-Stokes equations[END_REF]. First, the pressure term is ignored in order to obtain a provisional velocity field u * which may not be divergence-free.

(u i * -u i n ) ∆t + ∂ j (u j u i ) n+ 1 2 = ∂ j (ν(∂ j u i * + ∂ j u i n )) + a i n+ 1 2 (B.8a)
The momentum equation has the general form of an Helmholtz-type equation, which can be solved using a multi-level Poisson-Helmholtz solver if the advection term is provided, see below.

Second, a correction is sought by projecting the provisional velocity u * into a divergencefree space, using a pressure such that This problem can be solved using the multi-level Poisson-Helmholtz solver (see poisson.h).

u i n+1 -u i * ∆t = -∂ i p n+

Approximation of the advection term

Consider the advection term integrated over a control volume Ω, which is approximated as the sum of the fluxes in each control surface (4 in 2D or 6 in 3D).

Ω ∂ j (u j u i ) n+ 1 2 dΩ = ∂Ω ((n j u j )u i ) n+ 1 2 dS ≈ (d) S (d) (n j U j n+ 1 2 ) U i n+ 1 2 (B.9)
where n j corresponds to the (outward) normal unit vector at the face center.

A priori, both U i where D xyz indicates a discrete Laplacian, then correcting the velocity field

U i n+ 1 2 = U i n+ 1 2 -∂ i φ (B.18)
We repeat this prediction-correction process for each component of the velocity field and for each face, in order to evaluate the non-linear term using equation (B.9).

The advection scheme implemented inside tracer.h, gbc.h, and centered.h follows closely the equivalent BCG scheme used in Gerris. The sequence of events required to solve one generic time step is represented in figure B. [START_REF] Ahlers | Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection[END_REF]. Additional details may be found in [START_REF] Popinet | Gerris: a tree-based adaptive solver for the incompressible euler equations in complex geometries[END_REF]. 217

Instead of solving the general problem E(U) = 0, the N + 1 residuals yield a linear system of equations for the amplitudes φ (n) . The choice of the weight functions w m determines the particular kind of weighted residual method being used. Note that a judicious choice of test functions simplifies the resulting equation system [START_REF] Ruyer-Quil | Improved modeling of flows down inclined planes[END_REF].

When the weight and test functions are identical, i.e. w m = f m , the WRM is the same as a Galerkin method.

F.2 Introduction to a reduced model with inertial corrections

In the following we introduce a rescaled space variable z ≡ z ε defined in the interval [-1, 1]. The equation of continuity (7.1a) suggests the transversal velocity component w to be rescaled as w = ε w with w ≡ -z -1

(∂ i u i )dz (F.3)
The non-dimensionalization of velocity in system (7.1a)- (7.1d) is based on the free-fall velocity. For confined cells this characteristic velocity is no longer representative: velocities are smaller than one and the time scales are much longer. Indeed, the root mean squared velocity is such that (see table in Leading to the following system of equations

∂ i u i + ∂ z w = 0 (F.6a) α 2 ∂ τ u i + ∂ j (u j u i ) + ∂ z (w u i ) = -∂ i p + αPrRa -0.5 ∂ j ∂ j u i + ε -2 ∂ 2 z u i + Prθδ iy (F.6b) α 2 ∂ τ w + ∂ j (u j w) + ∂ z (w w) = -ε -2 ∂ z p + αPrRa -0.5 ∂ j ∂ j w + ε -2 ∂ 2 z w (F.6c) α ∂ τ θ + ∂ j (u j θ) + ∂ z (w θ) =Ra -0.5 ∂ j ∂ j θ + ε -2 ∂ 2 z θ (F.6d)
The characteristic temperature scale is still valid and the temperature field remains bounded in the interval [-0.5, 0.5]. However, outside the boundary layers the temperature is expected to be smaller than one. Since the pressure gradient and the thermal forcing are inextricably intertwined in the following we assume Far from the BL : p = αp θ = αθ (F.7)

where p and θ are terms of order one. Introducing these terms into equations (F.6a)-(F.6d) yields a system of equations where the orders of magnitude of each term is deduced in terms of α, ε, Ra and Pr. Inside Hele-Shaw cells, the terms highlighted in blue are expected to lead: the viscous and thermal diffusion operators in the transversal direction, i.e. ∂ 2 z u i and ∂ 2 z θ, as well as the buoyancy force and the pressure gradient. This imposes the following constraints on the values of ε and Ra ε -2 Ra -0.5 ≫ α or equivalently αε 2 Ra 0.5 ≪ 1 (F.10)

Using the DNS results presented in chapter 7, we perform a data fit to obtain a very rough approximation for the velocity scale α in terms of Ra and ε, α ≈ 0.083ε 0.92 Ra 0.17 As a first approximation, consider removing the terms of order α from equations (F.8a) to (F.8d), and neglect the terms in Ra -0.5 with respect to ε -2 Ra -0.5 . This zeroth-order approximation has the following form

∂ z w = -∂ i u i (F.12a)
∂ 2 z u i = ε 2 Ra 0.5 Pr -1 ∂ i p -Prθδ iy (F.12b)

∂ z p = 0 (F.12c)

∂ 2 z θ = 0 (F.12d)
From equation (F.12d) and the adiabatic boundary conditions one deduces that the temperature profile must be constant along the transversal direction, i.e. θ(x, y, z, τ ) = θ(x, y, τ ). Equation (F.12c) implies the pressure field to be independent of the transversal coordinate, i.e. p(x, y, z, τ ) = p(x, y, τ ). Equation (F.12b) implies the velocity to have a parabolic profile in order to satisfy the no-slip boundary conditions on (z = ±1). The solution at the zeroth-order has the following form This procedure leads to the same transversal profiles as in the simplified model 8.1, but does not lead to the same dynamical equations. In this case, this leads to a Poisson equation for the pressure field.

F.2.2 First-order formulation

We would like to go further than the zeroth-order approximation and compute the deviations from such profiles by taking into account the terms of order α from equations (F.8a)-(F.8b). Let us consider the temperature, pressure and velocity fields may be written in the following form θ = θ (0) + αθ (1) + O(α 2 ) (F.16a) p = p (0) + αp (1) + O(α 2 ) (F.16b)

u i = u i (0) + αu i (1) + O(α 2 ) (F.16c)
The resulting series has the following form Having selected the weight and test functions for each problem, we may proceed to solve for the amplitudes.

F.3.3 Amplitudes for the temperature field

Thanks to the choice of test and weight functions, equations (F.27) is simplified reducing the summation to one term.

• For m = 0 one obtains an equation for ϑ (0) α ∂ τ ϑ (0) 1|1 + ∂ j (a j (0) ϑ (0) ) 1|g 0 = Ra -0.5 ∂ 2 j ϑ (0) 1|1 (F.34a)

• For m = 2, one obtains an expression for ϑ (1,2) as function of a i (0) and ϑ (0)

ϑ (1,2) f ′′ 2 |f ′′ 2 =ε 2 Ra 0.5 ∂ j (a j (0) ϑ (0) ) f ′′ 2 |g 0 (F.34b)

• The remaining amplitudes are zero.

ϑ (1,m) =0 for m = 0, 2 (F.34c)

The temperature field takes the form θ(x, y, z, τ ) = αθ(x, y, z, τ ) = αϑ (0) + α 2 ϑ (1,2) f 2 (F.35)

or alternatively as θ(x, y, z, τ ) = θ (x, y, τ ) + α 2 ϑ (1,2) (x, y, τ )f 2 (z) (F.36)

F.3.4 Amplitudes for the velocity field

Thanks to the choice of test and weight functions (F.27) is also simplified but a summation remains. However, because of (F.35), this summation reduces to one term.

• For m = 0 one obtains an equation for a i (0)

α ∂ τ a i (0) g 0 |g 0 + ∂ j (a j (0) a i (0) ) g 0 |g 2 0 = -∂ i p (0) g 0 |1 (F.37a) + PrRa -0.5 ∂ 2 j a i (0) g 0 |g 0 + ε -2 PrRa -0.5 a i (0) g 0 |g ′′ 0 + Prϑ (0) g 0 |1 δ iy + αPrϑ (1,2) g 0 |f 2 δ iy • For m = 2 one obtains an expression for a i [START_REF] Adami | Surface tension and buoyancy in vertical soap films[END_REF]2) α ∂ τ a i (0) g 2 |g 0 + ∂ j (a j (0) a i (0) ) g 2 |g 2 0 =PrRa -0.5 ∂ 2 j a i (0) g 2 |g 0 (F.37b)

+ αε -2 PrRa -0.5 a i (1,2) g 2 |g ′′ 2 + αPrϑ (1,2) g 2 |f 2 δ iy

• For m = 4 one obtains an expression for a i [START_REF] Adami | Surface tension and buoyancy in vertical soap films[END_REF][START_REF] Araujo | Wind reversals in turbulent Rayleigh-Bénard convection[END_REF] ε -2 PrRa -0.5 a i (1,4) g 4 |g ′′ 4 =∂ j (a j (0) a i (0) ) g 4 |g 2 0 -Prϑ (1,2) g 4 |f 2 δ iy (F.37c)

• The remaining amplitudes are zero a i (1,m) =0 for m = 0, 2, 4 (F.37d)

The velocity field has the following form u i (x, y, z, τ ) = αu i (x, y, z, τ ) = αa i (0) g 0 + α 2 a i (1,2) g 2 + α 2 a i (1,4) g 4 (F.38) or alternatively u i (x, y, z, τ ) = u i (x, y, τ )g 0 (z) + α 2 a i (1,2) (x, y, τ )g 2 (z) + α 2 a i (1,4) (x, y, τ )g 4 (z) (F.39)

F.4 Resulting equation system

All of the above leads to a dynamical equation for the average fields: a (0) , ϑ (0) and p (0) , and to an expression for the inertial correction terms: ϑ (1,2) , a i (1,2) , and a i (1,4) .

F.4.1 Equation system for the transversally averaged fields

Using (F.34b) and equations (F.21), (F.34a) and (F.37a) gives a dynamical equation for a (0) , ϑ (0) and p (0) . However, it is much more convenient to write these expressions in terms of more physical quantities such as the mean temperature θ = α ϑ (0) , velocity u = α a (0) , and pressure p = α p

F.4.2 Equations for the inertial correction terms

For the evolution of the mean fields θ and u , the equations (F.40a) to (F.40c) are sufficient. However, one may be interested on computing the inertial corrections in order to perform a a reconstruction of the temperature and velocity fields (F.36) and (F.39), or to compute different global quantities.

θ(x, y, z, t) = θ + α 2 ϑ (1,2) f 2 (F.42a) u(x, y, z, t) = u g 0 + α 2 a (1,2) g 2 + α 2 a (1,4) g 4 (F.42b)

where the correction terms are given by the following expressions:

• One correction for the temperature field evaluated from (F.34b)

α 2 ϑ (1,2) = - ε 2 Ra 0.5 2 [ u • ∇ xy θ ] (F.43a)
• One correction for the velocity field is obtained from (F.37b)

1 35 ∂ t u + 2 35 [ u • ∇ xy u ] = 1 35
PrRa -0.5 ∇ 2 xy u + α 2 Prε -2 Ra -0.5 a (1,2) + α 2 2 315 Prϑ (1,2) e y (F.43b) which is combined to (F.40b) and (F.43a) to get rid of the time derivative and gives the following expression in terms of u , p , and θ α 2 a (1,2) = -ε 2 Ra 0. • An additional correction for the velocity field evaluated from (F.37c) α 2 a (1,4) = -2ε 2 Ra 0.

F.5 Mechanical energy budget for the model with inertial corrections

We are interested in describing the evolution of different global quantities in terms of the amplitudes to the velocity field a (n) and the amplitudes of the temperature field ϑ (n) .

F.5.1 Global kinetic energy and transfer rates

Using (F.42b) we may approximate the kinetic energy as e k = 1 2 u g 0 + α 2 a i (1,2) g 2 + α 2 a i (1,4) g 4 2

Integrating over the fluid volume leaves the following expression (1,2) xy + (F.44) α 4 7 45 a (1,2) • a (1,2) xy -

E k (t) = 3 5 u • u xy - α 2 5 u • a
1 9
a (1,2) • a (1,4) xy + 11 117 a (1,4) • a (1,4) xy

In practice, the terms of order α 4 in equation (F.44) may be safely neglected. A Reynolds number based on the root mean squared velocity, Re ≡ Ra 0.5 Pr -1 √ 2E k may be evaluated using equation (F.44). Now consider the rate-of-change for the kinetic energy as d dt E k (t) = PrRa -0.5 (Φ yǫ). Each term can be subsequently written in terms of the amplitudes a (n) and ϑ (n) .

Vertical heat-flux

Using (F.42a) and (F.42b) we may approximate the vertical heat flux as vθ = u y g 0 + α 2 a y (1,2) g 2 + α 2 a y (1,4) g 4 θ f 0 + α 2 ϑ (1,2) f 2

Integrating over the fluid volume, gives an expression for Φ y Φ y =Ra 0.5 u y θ xy + α 2 Ra 0.5 4 105 u y ϑ (1,2) xy (F.45)

α 4 Ra 0.5 2 45 a y (1,2) ϑ (1,2) xy -2 315 a y (1,4) ϑ (1,2) xy where terms of order α 4 may be neglected.

Viscous dissipation rate

Consider the following expression (∇u) 2 = (∇ xy u) 2 + (∂ z u) 2 where each part can be written in terms of the amplitudes a (n) and ϑ (n) .

(∇ xy u) 2 = ∇ xy u g 0 + α 2 ∇ xy a (1,2) g 2 + α 2 ∇ xy a (1,4) g 4 2 (∂ z u) 2 = u g ′ 0 + α 2 a (1,2) g ′ 2 + α 2 a (1,4) g ′ ∇ xy u : ∇ xy a (1,2) xy (F.46) + α 4 14 45 ∇ xy a (1,2) : ∇ xy a (1,2) xy -

2 9
∇ xy a (1,2) : ∇ xy a (1,4) xy + 22 117 ∇ xy a (1,4) : ∇ xy a (1,4) xy (1,2) • a (1,2) xy + α 4 11 ε 2 a (1,4) • a (1,4) xy (F.47) where terms of order α 4 may be neglected. Combining these expressions gives the total viscous dissipation ǫ = ǫ (xy) + ǫ (z) .

ǫ (z) = 3 ε 2 u • u xy + α 4 7 ε 2 a

F.5.2 Total potential energy and transfer rates

The total potential energy E p = -Pr yθ vol may be rewritten as E p = -Pr y θ xy (F.48)

Now consider the rate-of-change d dt E p (t) = PrRa -0.5 (Φ y -Φ i -Φ b1 ). Similarly to Φ y , the terms Φ i and Φ b1 can be written as function of the amplitudes a (n) and ϑ (n) .

Φ i = ∂ y θ xy (F.49) Φ b1 = - 1 Γ x y∂ n θ ds (F.50)
with ds being an infinitesimal element of the boundary ∂A, where dA = dxdy and the loop integral is counterclockwise.

F.5.3 Global thermal fluctuations and transfer rate

Using the equation (F.42a) we may approximate the temperature fluctuations (1,2) ϑ (1,2) xy (F.52)

θ 2 = θ f 0 + α 2 ϑ (1,
where terms of order α 4 may be neglected. Now consider the rate-of-change d dt E θ (t) = -Ra -0.5 (ǫ θ -Φ b3 ). Each term can be written as function of the amplitudes ϑ (n) .

Thermal dissipation rate

Consider the following expression (∇θ) 2 = (∇ xy θ) 2 + (∂ z θ) 2 Each part can be written in terms of the amplitudes ϑ (n) .

(∇ xy θ) 2 = ∇ xy θ f 0 + ∇ xy ϑ (1,2)

f 2 2 (∂ z θ) 2 = θ f ′ 0 + ϑ (1,2) f ′ 2 2
Expanding and integrating both expressions over the volume gives (1,2) • ∇ xy ϑ (1,2) xy + 8α 4 105 ϑ (1,2) ϑ (1,2) xy (F.53)

ǫ θ = ∇ xy θ • ∇ xy θ xy + 4α 4 525 ∇ xy ϑ
where terms of order α 4 may be neglected.

Boundary flux

A similar procedure is used to evaluate Φ b3

Φ b3 = 1 Γ x θ ∂ n θ ds + 1 Γ x 4α 4 525
ϑ (1,2) ∂ n ϑ (1,2) ds with ds being an infinitesimal element of the boundary ∂A, where dA = dxdy and the loop integral is counterclockwise. A closer inspection of equation (F.43a) reveals that ϑ (1,2) = 0 on the boundaries such that Φ b3 = 1 Γ x ϑ (0) ∂ n ϑ (0) ds (F.54)

Figure 1 . 1 :

 11 Figure 1.1: (a) A picture of a "Montgolfiere" (Wikimedia commons, 2011). (b) Granular structure of the photosphere (NASA / JAXA, 2015). (c) Convection in a "soap" bubble[START_REF] Seychelles | Thermal convection and emergence of isolated vortices in soap bubbles[END_REF].

Figure 1 . 2 :

 12 Figure 1.2: (a) Schematic representation of the Rayleigh-Bénard problem and (b) a shadowgraph of rising and falling plumes in a clockwise circulation[START_REF] Shang | Measured local heat transport in turbulent rayleigh-bénard convection[END_REF].

Figure 1 . 3 :

 13 Figure 1.3: Left: Phase diagram in Ra-Pr plane indicating the different turbulent regimes of the Grossmann-Lohse scaling theory in a cylindrical cell of aspect ratio Γ = 1, Right:Comparison between simulations, experiments, and the prediction of the Grossmann-Lohse theory (R.[START_REF] Stevens | The unifying theory of scaling in thermal convection: the updated prefactors[END_REF].

Figure 1 . 4 :

 14 Figure 1.4: Long-term evolution of the LSC inside a cylinder: (a) Azimuthal meandering of the LSC plane[START_REF] Funfschilling | Plume motion and large-scale circulation in a cylindrical rayleigh-bénard cell[END_REF]; (b) Reversal of the LSC inside a cylindrical cell(Xi and Xia, 2008); (c) Torsional oscillations of the LSC plane[START_REF] Funfschilling | Torsional oscillations of the large-scale circulation in turbulent Rayleigh-Bénard convection[END_REF]; and (d) off-center oscillations of the LSC plane[START_REF] Zhou | Oscillations of the largescale circulation in turbulent rayleigh-bénard convection: the sloshing mode and its relationship with the torsional mode[END_REF].

Figure 1 . 5 :

 15 Figure 1.5: Configurations used in the study of flow reversals: (a) numerical studies on a pure 2-D geometry (image from Chandra and Verma, 2011), and (b) experimental and numerical studies on a slim rectangular geometry (image from Vasiliev and Frick, 2011).

  From this point on, quantities are written in dimensionless form only and tildes (˜) are omitted.

Figure 2 . 1 :

 21 Figure 2.1: Configurations considered for this work: (a) a 2-D square with top and bottom no-slip conditions; (b) a 2-D square with top stress-free surface and bottom no-slip condition; and (c) a slim rectangular geometry of aspect ratio Γ x = 1 and Γ z < 1.

Figure 2 . 2 :

 22 Figure 2.2: Temperature field θ(x, y, t), corresponding temperature field θ r (y, t) at the reference state and height y r (x, y, t) for a square RB cell. Reference state is obtained by rearranging the fluid into a stable stratified configuration, see text.

  Velocity formulation (au, av, bθ) (C) Joint Temperature-Velocity formulation(2.71)

Figure 3 . 1 :

 31 Figure 3.1: Iso-contours of the temperature field corresponding to each of the test cases selected. (a) Differentially heated cavity for Ra = 10 5 and Pr = 0.71 to be compared against[START_REF] Tric | A first incursion into the 3D structure of natural convection of air in a differentially heated cubic cavity, from accurate numerical solutions[END_REF]. (b) Rectangular Rayleigh-Bénard cell with aspect ratio Γ z = 1/10 for Ra = 10 6 and Pr = 0.786 to be compared against (S. Wagner, andShishkina, 2013).

Figure 3

 3 Figure 3.2: 2-D Cavity: Streamlines superposed to θ(x, t) obtained using Basilisk for Pr = 0.71 at the steady-state. Position of ψ max is indicated using a × symbol and compared to the position given in the reference results (Le Quéré, 1991), displayed here as •.

8 Figure 3

 83 Figure 3.3: 2-D Cavity: Convergence of ψ max and Nu obtained using Basilisk compared with respect to (Le Quéré, 1991) as a function of grid spacing. Dashed line correspond to ideal slope for second order scheme.

Figure 3 . 4 : 2 -

 342 Figure 3.4: 2-D Cavity: Profiles measured at the steady state for Pr = 0.71. Lines correspond to different Ra: Ra = 10 3 , Ra = 10 4 , Ra = 10 5 , Ra = 10 6 , Ra = 10 7, Ra = 10 8 . Marks correspond to results from[START_REF] Tric | A first incursion into the 3D structure of natural convection of air in a differentially heated cubic cavity, from accurate numerical solutions[END_REF].Figures (a,b,c) correspond to results from Basilisk. Figures(d,e,f) correspond to results from Fusion.

Figure 3 . 5 :

 35 Figure 3.5: 3D Cavity (Basilisk): Convergence of Nu 3D and Nu mp compared with respect to[START_REF] Tric | A first incursion into the 3D structure of natural convection of air in a differentially heated cubic cavity, from accurate numerical solutions[END_REF] as a function of the grid spacing. Dashed line correspond to ideal slope for a second order scheme.

Figure 3 . 7 :

 37 Figure 3.7: Profiles measured at the steady state for Pr = 0.71. Solid lines indicates results obtained using Basilisk, × marks indicate results obtained using Fusion and marks correspond to reference results from[START_REF] Tric | A first incursion into the 3D structure of natural convection of air in a differentially heated cubic cavity, from accurate numerical solutions[END_REF].

Figure 3 . 8 :

 38 Figure 3.8: Flow structure for (Ra = 10 6 , Pr = 0.786) with Γ z = 1/10 and Γ z = 1/3 using Basilik. Figures (a,b) display a slice at the vertical mid-plane of the instantaneous temperature field with velocity vectors. Figure (c) displays iso-thermal contours.

Figure 3 . 9 :

 39 Figure 3.9: Rayleigh-Bénard convection using Basilisk for (Ra = 10 6 , Pr = 0.786) and different aspect ratios. Top row corresponds to Γ z = 1/10, bottom row to Γ z = 1/3. Figures (a,c) display a time series for Nu bot and Nu top , while (b,d) display the corresponding time-frequency spectra.

Figure 3 . 10 :

 310 Figure 3.10: Rayleigh-Bénard convection using Basilisk for (Ra = 10 6 , Pr = 0.786) and two different aspect ratios: (a) Γ z = 1/10, and (b) Γ z = 1/3. From top to bottom: Time derivatives d dt E k , d dt E p , and d dt E a . Left hand sides of equations (3.11) are evaluated using finite differences (continuous lines), whereas the right hand sides are displayed in • marks.

Figure 3 . 11 :

 311 Figure 3.11: Strong scaling: Speed-up factor S p as a function of the number of processors p for a constant grid size. Measurements correspond to a 2D (resp. 3D) RB cell for a grid of size n x n y with n y = n x on the left panel (resp. n x n y n z with n y = n z = n x on the right panel). Ideally, both curves should increase linearly.

Figure 3 . 12 :

 312 Figure 3.12: Weak scaling: (a) restitution time for a single time-step as function of the number of processors for two values of (n x n y n z /p); (b) restitution time for a single timestep as function of the number BlueGene compute nodes bg for two values of (n x n y n z /bg). Ideally, both curves should remain constant.

Figure 1 .

 1 Figure 1. Time evolution of L 2D (t)/|L 2D | shown for (Ra = 5 • 10 7 , Pr = 3). Light blue areas correspond to a consecutive reversal (CR) regime, while blank areas correspond to an extended cessation (EC) regime. Some events (darker blue areas) may not be clearly assigned to CR régime, see text. The two continuous lines correspond to the thresholds used by the filtering procedure. Value of the normalized standard deviation: σ(|L 2D |)/|L 2D | = 0.499.

θFigure 2 .

 2 Figure 2. Temperature field θ(x, y, t) at a given instant t (left), the corresponding background state θ r (y, t) (center) and height y r (x, y, t) (right) for a square RB cell.

  [5 • 10 6 , 3 • 10 8 ]. For Pr = 4.3, it corresponds to the interval Ra ∈ [3 • 10 7 , 4 • 10 8 ] (Sugiyama et al. 2010).

Figure 3 .

 3 Figure 3. Probability density function (PDF) of τ 1 (top) and τ d (bottom) for Pr = 4.3, (left figure) Ra = 5 • 10 7 , (right figure) Ra = 10 8 . The PDF value is represented by • marks. For τ 1 it is the sum of three conditional PDFs: one in () corresponding to the CR regime, a smaller one in ( ) corresponding to "rogue reversals", and an additional part in ( ) corresponding to the EC regime.

Figure 4 .

 4 Figure 4. Probability density function (PDF) for τ 1 (top) and τ d (bottom) for Pr = 3.0 and Ra = 10 7 (left), Ra = 5 • 10 7 (centre) and Ra = 10 8 (right). Layout is similar to figure 4.

Figure 5 .

 5 Figure 5. From top to bottom: Normalised global angular impulse L 2D /|L 2D |, normalised kinetic energy E kin /|E kin |, and normalised available potential energy E apot /|E kin | for (Ra = 5 • 10 7 , Pr = 4.3). Left panel: Each reversal cycle is centred and its time is rescaled by τ 1 (only 10 reversals are displayed and each colour is a different reversal); Centre panel: Each reversal cycle is centred and its time is rescaled by τ 1,i ; Right panel: Average value of rescaled curves obtained from the complete time-series (thick lines) and curves corresponding to one standard deviation (dashed lines).

Figure 6 .

 6 Figure 6. Curves corresponding to reversals for Pr = 3.0 as obtained using the procedure described in figure 5. The average L 2D /|L 2D | is shown in thick lines and one standard deviation in dashed lines. From left to right: a) Ra = 10 7 , b) Ra = 5 • 10 7 , and c) Ra = 10 8 .

Figure 7 .

 7 Figure 7. Conditionally averaged fields u o (x, y, t) and y r (θ o (x, y, t)) at different instants during the generic reversal cycle for (Ra = 5•10 7 , Pr = 3.0). Fields are obtained as the ensemble average over 83 particular reversal cycles. Streamlines of the velocity field u o are superposed over the color map of field y r (θ o ). Solid and dashed streamlines indicate the two senses of rotation.

Figure 8 .Figure 9 .

 89 Figure 8. Generic reversal cycle for (Ra = 5 • 10 7 , Pr = 3.0). Left panel: system trajectory in phase space (L 2D /|L 2D |, E kin /|E kin |, E apot /|E kin |); Right panel from top to bottom: average curves for L 2D /|L 2D |, E kin /|E kin | and E apot /|E kin |. Marks (i-x) and (a-e) indicate particular instants (see text for explanations). The accumulation phase is shaded and indicated by a blue colour bar. The release (resp. acceleration) phase is indicated by an orange (resp. green) colour bar. The same colour code is used in the phase space trajectory position.

Figure 10 .

 10 Figure 10. Instantaneous fields for a particular reversal cycle during the accumulation phase at regular time intervals at (Ra = 5 • 10 7 , Pr = 3.0). The first snapshot follows point (e p ') the last precedes point (a p ). Streamlines are superposed either to vorticity ω (top figures), or to kinetic energy 1 2 u i u i (bottom figures).

Figure 11 .Figure 12 .

 1112 Figure 11. Instantaneous fields for a particular reversal cycle during the accumulation phase. They are identical to those of figure 10 but streamlines are superposed either to the field Pr(y r -y)θ (top figures) or to the field ∇y r • ∇θ (bottom figures). Snapshots correspond to the same instants as in figure 10.

Figure 13 .

 13 Figure 13. Instantaneous fields for a particular reversal cycle during the early release phase (point (a p )-(b p )) at regular time intervals (Ra = 5 • 10 7 , Pr = 3.0). Streamlines are superposed to the field 1 2 u i u i . First snapshot follows point (a p ), last precedes point (b p ).

Figure 14 .

 14 Figure 14. Instantaneous fields for a particular reversal cycle during the release phase at (Ra = 5•10 7 , Pr = 3.0). Snapshots correspond to points (a p ) to (d p ). Streamlines are superposed to either the field y r (x, z, t) (top figures), or to field ∇y r • ∇θ (bottom figures).

Figure 15 .

 15 Figure 15. Instantaneous fields for a particular reversal cycle taken at regular intervals during the acceleration phase at (Ra = 5 • 10 7 , Pr = 3.0). The first snapshot follows point (d p ); the last corresponds to point (e p ). Streamlines are superposed either to vorticity ω (top figures), or to ∇y r • ∇θ (bottom figures).

Figure 16 .

 16 Figure 16. Time evolution of the energy transfer rates. From top to bottom: normalised Nu vol /Nu, (ǫ + 1)/Nu, Φ d /Nu, , Φ b1 /Nu for (Ra = 5 • 10 7 , Pr = 3.0). Figure for L 2D /|L 2D | is given as a reference curve. The left figure corresponds the generic reversal: the curve in corresponds to the average, while the hatched region represent one standard deviation from this average. The center and right figures correspond to particular reversals displayed in alongside the standard deviation.

Figure 17 .

 17 Figure 17. Instantaneous fields for a particular reversal cycle taken at regular intervals during the early release phase at (Ra = 5 • 10 7 , Pr = 3.0). The first (resp. last) snapshot follows point (a ′ p ) (resp. precedes point (b ′ p )). Streamlines are superposed to field 2e ij e ij the value of which is given by the colour. Corresponding values of ǫ and Φ d below each snapshot.

Figure 18 . 10 Figure 19 .

 181019 Figure 18. Time evolution of the angular impulse transfer rates for (Ra = 5 • 10 7 , Pr = 3.0). From top to bottom: Evolution of M/|M|, (M -I b )/|M|, and I a /|M|. Figure for L 2D /|L 2D | is given as a reference curve. The left figure corresponds to the generic reversal: the average curve is displayed in and the hatched region represents one standard deviation from this average. The center and right figures depict two particular reversal cycles.

5 (Figure 20 .

 520 Figure 20. Linear stability analysis of the generic fields for (Ra = 5•10 7 , Pr = 3.0). (i) Placement of the different values of t o with respect to the generic L 2D and E apot curves. (ii) Evolution of the energy of the normalised velocity and temperature perturbations for different base states (or equivalently t o ). Growth rate σ measured for different values of t o : △ σ = 0.009, σ = 0.005, σ = 0.005, σ = 0.006, • σ = 0.011, • σ = 0.250; (iii) Streamlines corresponding to field u o (x, y, t o ) superposed to base state θ o (x, y, t o ) shown for two values of t o : • located between points (f) and (g) and • located on point (g). (iv) Disturbance field θ ′ θ ′ (x, y, t) measured at the end of the curves corresponding to • and •.

  First, let us consider the conditionally averaged fields (θ o (x, y, t o ) and u o (x, y, t o )) obtained at time t o for the standard RB problem, see figure 7. Starting from such fields labeled by time t o , we perform a direct numerical simulation of the non-linear Boussinesq equations changing boundary conditions from isothermal to adiabatic on the top and bottom plates. This effectively suppresses the external thermal forcing since Φ b1 = Φ b2 = 0 and limits the amount of thermal energy contained inside the cavity. Note that, different initial conditions i.e. different t o lead to different trajectories for the adiabatic problem. Evolution of the normalised kinetic energy E kin /|E kin | and the normalised angular impulse L 2D /|L 2D | can be seen in figure 21, where each curve corresponds to different t o (shown as • marks) inside the accumulation phase. Dashed (resp. solid) lines indicate negative (positive) values of L 2D /|L 2D |. For simulations preceding the transition point (g) a decay in both the angular impulse and the kinetic energy are observed. On the contrary, from point (g) on, a change from negative to positive values of L 2D /|L 2D | simultaneously as peak values in E kin /|E kin | are observed, before the ensuing decay. To illustrate this Reversal cycle in Rayleigh-Bénard cells

Figure 4 . 1 :

 41 Figure 4.1: Evolution of the kinetic energy E k and the available potential energy E a during a generic reversal for Ra = 5 • 10 7 and Pr = 3.

Figure 5 . 1 :

 51 Figure 5.1: (a) Time series of L 2D during the sampled interval for (Ra = 5 • 10 7 , Pr = 4.3). (b) Eigenvalues obtained for the first 20 POD modes; (c) 100 k i=1 λ i / ∞ i=1 λ i . POD analysis based on temperature formulation (A); on the velocity formulation (B); or the joint velocity-temperature formulation (C).

Figure 5 . 2 :

 52 Figure 5.2: Six most energetic POD modes obtained using formulations (A), (B), and (C) for (Ra = 5 • 10 7 , Pr = 4.3). For each formulation the top row corresponds to the temperature modes, the middle row to the stream-function of the velocity modes and the bottom row to the advective heat-flux in the vertical direction, i.e. φ(v) k φ(θ) k . Results by a common factor as to facilitate the comparison, see text.

Figure 5 . 3 :

 53 Figure 5.3: Modes k = 7 to k = 12 obtained using formulation (C) for (Ra = 5 • 10 7 , Pr = 4.3). Continues from figure 5.2.

Figure 5 . 4 :

 54 Figure 5.4: Evolution of the error in the low-order reconstructions of E k and E p during a particular reversal for (Ra = 5 • 10 7 , Pr = 4.3). Color indicates the number of POD modes: three-modes (red lines), six-modes (blue lines), nine-modes (green lines) and twelve-modes (purple lines). Inset displays the evolution of L 2D for reference. Hollow • (resp. solid •) marks indicate results to a three-modes (resp. six-modes) reconstruction corresponding to snapshots displayed on figure 5.5.

Figure 5 . 6 :

 56 Figure 5.6: Kinetic and potential energy during a generic reversal: DNS results (black lines) are compared to the conditional averaged values obtained from the first six POD modes using different formulations for (Ra = 5 • 10 7 , Pr = 4.3): (A) in red, (B) in blue, and (C) in purple. Shaded region represents one standard deviation from the DNS results in three phases: release (orange), acceleration (green), and accumulation (blue). The transition points, (f) and (g) defined in the previous chapter are shown for reference.

Figure 5 . 7 :

 57 Figure 5.7: Left: Available potential energy during a generic reversal compared to the conditional averaged values obtained from the first six POD modes for (Ra = 5 • 10 7 , Pr =4.3). Layout is similar to figure 5.6. Right: Probability density and cumulative density functions of an instantaneous temperature field θ(x, t) corresponding to the release phase of a particular reversal, compared to a six-mode reconstruction using formulation (A).

Figure 5 . 8 :

 58 Figure 5.8: POD analysis of the filtered subset corresponding to the CR regime for Ra = 5 • 10 7 . (a) Eigenvalues obtained for the first 20 POD modes; (b) 100 k i=1 λ i / ∞ i=1 λ i .

Figure 5 . 9 :

 59 Figure 5.9: POD modes obtained from a filtered subset including only the regime of 'consecutive reversals' for (a) (Ra = 5 • 10 7 , Pr = 4.3) and (b) (Ra = 5 • 10 7 , Pr = 3.0). Layout is identical to that of figure 5.2.

Figure 5 . 11 :

 511 Figure 5.11: Same as in figure 5.11 for (Ra = 5 • 10 7 , Pr = 4.3). Marks • corresponds to the fields in figures 5.14(i) to 5.14(x) where (i) is indicated by a • mark.

Figure 5 . 12 :

 512 Figure 5.12: Conditional PDF of the modal coefficients α k (t) inside the CR regime displayed in color, PDF of the complete series shown in black. (a) (Ra = 5 • 10 7 , Pr = 3.0) and (b) (Ra = 5 • 10 7 , Pr = 4.3).

Figure 5 . 13 :

 513 Figure 5.13: Generic fields (top) compared to reconstructed fields using coefficients from figure 5.10 (bottom) for (Ra = 5•10 7 , Pr = 3.0). Streamlines are superposed to a color-map of field y r (x, t). Snapshots correspond to a mark in figure 5.10.

Figure 5 . 14 :Figure 5 . 15 :

 514515 Figure 5.14: Same as in figure 5.13 for (Ra = 5 • 10 7 , Pr = 4.3). Each snapshot correspond to a mark in figure 5.11.

Figure 5 . 17 :

 517 Figure 5.17: Time series of modal coefficients α k (t) for (a) (Ra = 5 • 10 7 , Pr = 3.0) and (b) (Ra = 5 • 10 7 , Pr = 4.3). Shaded (resp. white) regions are used to identify an EC (resp. CR) regime on the time series.

Figure 5 .

 5 Figure 5.18: Conditional PDF of τ 1 , τ 2 , and τ 3 for (Ra = 5 • 10 7 , Pr = 3.0), where τ 1 , τ 2 , and τ 3 correspond to the inter-switch interval in modes L, S, and S * , respectively. (a) belongs to the CR regime, (b) belongs to the EC regime. For τ 2 and τ 3 , a continuous (resp. dashed) line corresponds to a data fit to a Log-normal (resp. Gamma) distribution. Coefficients given in text.

Figure 5 .

 5 Figure 5.19: Same as in figure 5.18 but for (Ra = 5 • 10 7 , Pr = 4.3).

Figure 5 .Figure 5 .

 55 Figure 5.20: Bivariate histogram of modal coefficients for (Ra = 5 • 10 7 , Pr = 3.0) using only contributions from CR regime. Generic reversal represented with a solid line with arrows indicating the direction and color indicates the phase: release between points (a) and (d) in orange, acceleration between points (d) and (e) in green, accumulation between points (e) and (a) in blue. (a), • (b), △ (c), ▽ (d), ♦ (e), ⊛ (f), ⊗ (g).

Figure 5 .

 5 Figure 5.22: Bivariate histogram of modal coefficients for (Ra = 5 • 10 7 , Pr = 4.3) using only contributions from CR regime. Generic reversal represented with a solid line with arrows indicating the direction and color indicates the phase: release between points (a) and (d) in orange, acceleration between points (d) and (e) in green, accumulation between points (e) and (a) in blue. (a), • (b), △ (c), ▽ (d), ♦ (e), ⊛ (f), ⊗ (g).

Figure 5 .

 5 Figure 5.23: Bivariate histogram of modal coefficients for (Ra = 5 • 10 7 , Pr = 4.3) excluding contributions from the CR regime.

Figure 5 .

 5 Figure 5.24: Evolution of α k α k as function of Ra and two different Pr. Here α k α k corresponds to the mean part of the combined energy projected on the φ k -axis, whereas φ k corresponds to the first 6 POD modes obtained from the CR subset for Ra = 5 • 10 7 . The total combined energy ψ|ψ is displayed in black lines (Tot.). The values of Ra where flow reversals are observed are enclosed in the hatched area, whereas the combination of the EC and CR regimes is observed between the red dashed lines.

Figure 5 .

 5 Figure 5.25: Time series of the global angular impulse L 2D for different Ra and (a) Pr = 3.0 and (b) Pr = 4.3. From top to bottom: periodic reversal without accumulation phase, chaotic reversals without accumulation phase, reversals with accumulation phase, consecutive reversals and extended cessations, consecutive reversals without extended cessations, long stable plateaus.

3 )

 3 of the flow dynamics for 10 5 < Ra < 10 8 inside free-slip (ST) cells 6.2.1 Steady-state solutions (Ra < 10 5 , Pr = 4.Let us consider the results for Pr = 4.3. For this range of Ra, one obtains a single-roll and double-roll solutions (figure 6.1). The choice between one state or another depends on the value of Ra: from the same initial condition (no-motion and unstable temperature gradient with added noise), a single-roll state was obtained for Ra = 10 4 and Ra = 8 • 10 4 , whereas double-roll modes were obtained for Ra = 3 • 10 4 and Ra = 5 • 10 4

7 Figure 6 . 1 :

 761 Figure 6.1: Steady-state solutions inside free-slip cells (ST) for Pr = 4.3: (a) Ra = 10 4 and (b) Ra = 3 • 10 4 : streamlines are superposed over the color map of field θ(x, y) (resp. y r (x, y)). Fig (c) displays the vertical temperature profile θ x for both Ra.

6. 2 . 2

 22 Oscillatory and intermittent flows (10 5 < Ra ≤ 3 • 10 5 )The transition towards unsteady flows is observed near Ra = 10 5 . Two types of oscillatory solutions are identifiable for both Pr = 4.3 and Pr = 3.0: a flow regime composed of a central vortex and another regime composed of two vertically stacked rolls. The apparent significance of these solutions will be discussed on the following sections.

Figure 6 . 2 :Figure 6 . 3 :

 6263 Figure 6.2: Oscillatory solutions for (Ra = 3 • 10 5 , Pr = 4.3) for a free-slip (ST) cell. Top: Streamlines superposed to field y r (x, t) corresponding to the oscillatory one-and four-roll pattern. Snapshots correspond to marks indicated on the time series below. Middle: Time series of L 2D , C, Nu top and Nu bot (see §2.2 and 2.3), and corresponding frequency spectra. Bottom: Time series and frequency spectra for the NT cell shown for reference.

Figure 6 . 4 :Figure 6 . 5 :

 6465 Figure 6.4: Time-frequency spectra of Nu bot and Nu top for Pr = 3.0 and different Ra: Ra = 1.5 • 10 5 , Ra = 2 • 10 5 , Ra = 2.5 • 10 5 , and Ra = 3 • 10 5 .

Figure 6 . 6 :

 66 Figure 6.6: Instantaneous streamlines superposed over field y r (x, t) and corresponding time series of L 2D inside ST cells (a) (Ra = 5 • 10 7 , Pr = 4.3), (b) (Ra = 3 • 10 8 , Pr = 4.3) and (c) (Ra = 5 • 10 8 , Pr = 3.0).

Figure 6 . 7 :

 67 Figure 6.7: Mean temperature profile θ x (left) and probability distribution of the mean temperature θ(x) (center) for (Ra = 5•10 7 , Pr = 4.3). Asymmetry coefficient χ as function of Ra for Pr = 4.3 (right).

Figure 6 . 8 :

 68 Figure 6.8: Simulations for Pr = 4.3 (Left) Average Nusselt Nu and Reynolds Re number as function of Ra and (right) compensated Nu and Re.

Figure 6 . 9 :Figure 6 . 10 :

 69610 Figure 6.9: Probability density functions of the instantaneous Nusselt numbers Nu vol , Nu bot , and Nu top for (Ra = 5 • 10 7 , Pr = 4.3).

Figure 6 . 11 :Figure 6 . 12 :

 611612 Figure 6.11: Spatial distribution of contributions to the thermal dissipation ∂ j θ∂ j θ(x) for (Ra = 5 • 10 7 , Pr = 4.3) inside (a) ST and (b) NT cells. Solid lines correspond to ∂ j θ∂ j θ = 100.

1 ∂Figure 6 . 13 :

 1613 Figure 6.13: Vertical profiles for the mean temperature θ x , temperature fluctuations θ rms x , and normalized thermal dissipation ∂ j θ∂ j θ x in the lower half-cell for (Ra = 5 • 10 7 , Pr = 4.3). A horizontal line indicates the thickness δ θ,bot .

1 ∂Figure 6 . 14 :∂Figure 6 . 15 :

 1614615 Figure 6.14: Same as in figure 6.13 but for the upper half-cell. A horizontal line indicates the value of δ θ,top .

  vT -κ∂ y T 2κ(T bot -T xy )/H = 2(0.5θ xy ) Nu (6.3) Similarly for the free-slip plate, we define a Rayleigh number Ra s = 2β( T xy -T top )gH 3 vT -κ∂ y T 2κ( T xy -T top )/H = 2( θ xy + 0.5) Nu (6.5)

Figure 6 . 17 :Figure 6 .

 6176 Figure 6.17: Evolution of Nu n (resp. Nu s ) as function of Ra n (resp. Ra s ) for Pr = 4.3. Values of Nu as function of Ra for the NT cell are displayed as reference.

Figure 6 .

 6 Figure 6.19: PDF of τ 1 and τ d for (Ra = 5 • 10 7 , Pr = 4.3) inside a (a) NT and (b) ST cell. The PDF value is represented by • marks. For τ 1 it is the sum of two conditional PDFs: one shown by a thick blue line corresponding to the CR regime, the other shown by a fine red line corresponding to the EC regime.

Figure 6 .

 6 Figure 6.20: Normalized global angular impulse L 2D /|L 2D |, kinetic energy E k /E k and potential energy E p /E k for (Ra = 5 • 10 7 , Pr = 4.3) inside a free-slip (ST) cell. Layout is similar to figure 5 in chapter 4: Each curve is centered and its time is rescaled by τ 1 or by τ 1,i , accordingly. The leftmost figures correspond to average value of rescaled curves (thick lines) and curves corresponding to one standard deviation (dashed lines).

Figure 7 . 1 :

 71 Figure 7.1: (Left panel) Different regimes of heat transport under geometrical confinement borrowed from Chong and Xia, 2016 for Pr = 4.38. Regime I represents the classical three-dimensional case; Regime II is the plume-controlled regime; and regime III the severely confined regime. A blue line (Γ z = 12.42Ra -0.21 ) separates regimes I and II, an orange one (Γ z = 29.37Ra -0.31 ) separates regimes II and III, and a red line (Γ z = 21.96Ra -0.5 ) indicates the onset of convection. Note that for a given Γ z , the light area inside III separates the Ra at the onset of convection to five times this value. (Right panel) Horizontal slice of the temperature and velocity field in a plane in proximity of the bottom boundary layer (x, y = -0.48, z). Each slice corresponds to a mark on the left panel: (a) Γ z = 1/2, (b) Γ z = 1/4, (c) Γ z = 1/8, and (d) Γ z = 1/16.

Figure 7 . 2 :

 72 Figure 7.2: Instantaneous isotherms for {±0.45, ±0.35, ±0.25, ±0.15} and (Ra = 10 7 , Pr = 4.38).Figure (a) corresponds to regime I, figure (b) corresponds to regime II, and figures (c-f) correspond to regime III.

  Figure 7.2: Instantaneous isotherms for {±0.45, ±0.35, ±0.25, ±0.15} and (Ra = 10 7 , Pr = 4.38).Figure (a) corresponds to regime I, figure (b) corresponds to regime II, and figures (c-f) correspond to regime III.

Figures 7 .

 7 Figures 7.4 to 7.7 display the dynamics for Γ z = 1/64 and different Ra. The plume width is shown to decrease as we increase Ra. The relative plume size with respect to the size of the cavity seems to play an important role in the overall dynamics. As such, similar flow regimes may be observed for different combinations of Ra and Γ z with a similar plume size. For (Ra = 5 • 10 7 , Γ z = 1/64) we observe two types of solutions. In the first one (figure7.4a), the coordinated motion of the side-wall plumes prevents the center plume from moving and one obtains a periodic oscillatory solution similar to the one displayed on figure 7.3c. The second kind of solution (figure7.4b) is a chaotic one, but different

  Ra = 10 7 , Γz = 1/32, periodic oscillatory without center plume

Figure 7 . 3 :

 73 Figure 7.3: DNS results for (Ra = 10 7 , Pr = 4.38) and different Γ z . (Left) Time series of Nu vol (black), Nu top (blue) and Nu bot (red); (Middle) Time-frequency spectra of Nu vol ; (Right) Snapshot of the reference height y r (x, y, 0, t) (see §2.2.4). Horizontal lines correspond to y = ±(0.5 -1 Nu ).

  Ra = 5 • 10 7 , Γz = 1/64, chaotic motion without center plume

Figure 7 . 4 :

 74 Figure 7.4: DNS results for (Γ z = 1/64, Ra = 5 • 10 7 , Pr = 4.38). The two sequences (a) and (b) correspond to different stable solutions, see text. First row: Time series of Nu vol (black), Nu top (blue) and Nu bot (red); a close-up view of the time-series inside the interval indicated in dashed lines ; and time-frequency spectra of Nu vol . Second row: snapshots of the reference height y r (x, y, 0, t) (see §2.2.4), where times corresponding to snapshots are indicated in the time series by a • mark. Thick horizontal lines correspond to y = ±(0.5 -1 Nu ) for reference.

Figure 7 . 5 :

 75 Figure 7.5: Same as in figure 7.4 but for Ra = 10 8 .

Ra = 2 •

 2 10 8 , Γz = 1/64, chaotic motion with two center plumes

Figure 7 . 6 :

 76 Figure 7.6: Same as in figure 7.4 but for Ra = 2 • 10 8 .

Figure 7 . 7 :

 77 Figure 7.7: Same as in figure 7.4 but for Ra = 10 9 . One additional row of snapshots is displayed to illustrate the sequence from below average to above average heat-flux.

  Time-series (and close-up view) of the Half-range Fourier coefficients ûpq

Figure 7 . 8 :

 78 Figure 7.8: Flow reversal observed for DNS at Ra = 2 • 10 7 and Γ z = 1/16. (a) A sequence of snapshots of field y r (x, y, 0, t) during the reversal. (b) Time-series of the half-range Fourier coefficients of the horizontal velocity ûpq . Modal coefficients are multiplied by a factor 10 2 . The dashed box indicates the time interval where the reversal occurs. This interval is displayed on the right indicating the times corresponding to the snapshots. (c) Time-series (and its close-up view) of the normalized kinetic and the available potential energies, E k and E a . 144

Figure 7 . 9 :

 79 Figure 7.9: Plume characterization for (Γ z = 1/64, Pr = 4.38) for (a,b,c) Ra = 10 8 and (d,e,f) Ra = 10 9 . Figures(a,d) display the reference height y r (x, y, 0, t 0 ), whereas figures (b,e) and (c,f) correspond to φ 2 z (x, t) with φ = vθvθ x for 5 consecutive instants (t 0 to t 4 ), each in a different color. Measurements taken at y = ±(0.5 -1/Nu). The number of plumes is deduced from local maxima in φ 2 z (x, t) (each shown as ).

9 Figure 7 .

 97 Figure 7.10: (a) Probability distribution of the inter-plume spacing d 0 measured near the top (blue) and bottom (red) boundary layers for Pr = 4.38 and Γ z = 1/64. From top to bottom: Ra = 5 • 10 7 (w/central plume), Ra = 10 8 , Ra = 2 • 10 8 , and Ra = 10 9 . (b) Spatial auto-correlation for Pr = 4.38 and Γ z = 1/64 used to illustrate the plume width.

Figure 7 . 11 :

 711 Figure 7.11: DNS results for (Γ z = 1/64, Pr = 4.38) and different Ra. Spatio-temporal fluctuations θ rms xz (y), and time-averaged thermal dissipation rate ∂ j θ∂ j θ xz (y).

  (y) × 100 ∂j ui∂j ui x (y, 0)/(Nu -1) ∂j ui∂j ui xz (y)/(Nu -1)

Figure 7 . 12 :

 712 Figure 7.12: DNS results for (Γ z = 1/64, Pr = 4.38) and different Ra. Spatio-temporal fluctuations u rms xz (y), time-averaged viscous dissipation ∂ j u i ∂ j u i xz (y) averaged along the z direction, and time-averaged viscous dissipation ∂ j u i ∂ j u i x (y, 0) at the transversal mid-plane (z = 0).

  distance between the closest peak in θ rms xz to the top and bottom plates; and thickness δ (3) θ

  are normalized by Γ z for reference.

Figure 7 . 14 :

 714 Figure 7.14: Spatial distribution of the viscous dissipation ∂ j u i ∂ j u i x (y, z) near the bottom for Ra = 10 8 , Pr = 4.38 and for: (a) Γ z = 1/8, (b) Γ z = 1/16, (c) Γ z = 1/32, and (d) Γ z = 1/64. Boundary layer thicknesses at the plates δ p and side-walls δ w shown for reference.

  θ . Color red indicates the kinetic boundary is nested inside the thermal one.

Figure 7 . 16 :

 716 Figure 7.16: DNS results for (Ra = 10 8 , Pr = 4.38) and different Γ z . Time-averaged dissipation profiles in the transversal direction (purple lines). The shaded area of size δ w /Γ z corresponds to the kinetic boundary layers on the front and back walls. The value δ w /Γ z is compared to the value δ w /Γ z = 0.211 (red dotted lines).

  the kinetic BL is nested inside the thermal one (shown in purple) or vice versa; average thickness of the kinetic boundary layer at the front/back walls δ w and contributions to the viscous dissipation from the front/back kinetic boundary layers. Grid spacing in the transversal direction ∆z is shown to illustrate the incertitude in the measurement of δ w . and on the depth of the cavity, that is Re d = Γ z Ra 0

7. 6 Figure 7 .

 67 Figure 7.17: (a) Average Nusselt and (b) Reynolds numbers as a function of Ra and Γ z for Pr = 4.38. Symbols indicate the aspect ratios: • Γ z = 1/8, Γ z = 1/16, Γ z = 1/32, and △ Γ z = 1/64. A black (resp. red) line corresponds to the scaling for Γ z = 1 (resp. Γ z = 1/64).

∇=

  xy v : ∇ xy v xy and ǫ (z) ∇ xy θ • ∇ xy θ xy and ǫ (z) θ = 0 (8.7b)

Figure 8 .

 8 1 displays one snapshot of the reference height field y r (x, y, t) obtained from the reduced model (top), the model with inertial corrections (middle) and from 3-D DNS (bottom) for (Ra = 10 7 , Pr = 4.38) and different aspect ratio Γ z . The sequence from left to right shows that the laminarization of the flow observed in the DNS when Γ z is decreased, is reproduced correctly by both models.

Figure 8 . 1 :Figures 8 .

 818 Figure 8.1: Comparison of the reduced model, the model with inertial corrections, and DNS results for Ra = 10 7 and Pr = 4.38. Instantaneous reference height field y r (x, y, t) for different Γ z : (a) Γ z = 1/8, (b) Γ z = 1/16, (c) Γ z = 1/32 almost periodic state with central plume, (d) Γ z = 1/32 almost periodic state without central plume, and (e) Γ z = 1/64 steady state.

Figure 8 . 2 :

 82 Figure 8.2: Comparison of reduced and DNS for Ra = 10 7 and Pr = 4.38: Time-frequency spectra of Nu vol . Reduced model in red, model with inertial corrections in blue, and DNS in black. (a) Γ z = 1/8, (b) Γ z = 1/16, (c) Γ z = 1/32 almost periodic state with central plume, (d) Γ z = 1/32 almost periodic state without central plume.

Figure 8 . 3 :

 83 Figure 8.3: Comparison of the reduced model, the model with inertial corrections, and DNS results for Γ z = 1/64 and Pr = 4.38. Instantaneous reference height field y r (x, y, t) for different Ra. The sequence from left to right is taken at the same regular time intervals.

Figure 8 . 3 :

 83 Figure 8.3: Continues from previous page.

Figure 8 . 4 :Figure 8 . 5 :

 8485 Figure 8.4: Comparison of reduced and DNS for Γ z = 1/64 and Pr = 4.38: Time-frequency spectra of Nu vol . Reduced model in red, model with inertial corrections in blue, and DNS in black. (a) Ra = 5 • 10 7 , (b) Ra = 10 8 , (c) Ra = 2 • 10 8 , (d) Ra = 10 9 .

Figure 8 . 6 :

 86 Figure 8.6: Comparison of the 2-D models against DNS results. Probability distribution of the inter-plume distance d 0 measured near the top (blue) and bottom (red) boundary layers for Pr = 4.38 and Γ z = 1/64. From top to bottom: Ra = 5 • 10 7 , Ra = 10 8 , Ra = 2 • 10 8 and Ra = 10 9 .

Figure 8 . 7 :

 87 Figure 8.7: Comparison of the 2-D models against DNS results. The top (resp. bottom) row displays the spatial auto-correlation function measured near the upper (resp. lower) boundary layers for Pr = 4.38, Γ z = 1/64, and different Ra.

Figure 8 . 9 :

 89 Figure 8.9: Flow reversals obtained using the model with inertial corrections for Γ z = 1/16, Ra = 2 • 10 7 and Pr = 4.38. Each reversal is indicated by a mark (i) to (iv). Time-series of the Fourier coefficients of the horizontal velocity ûpq (left) and corresponding probability density distributions (right). Modal coefficients are multiplied by a factor 10 2 .

Figure 8 . 10 :

 810 Figure 8.10: Flow reversal observed using the reduced model with inertial corrections for Ra = 2 • 10 7 and Γ z = 1/16. Figure displays a sequence of snapshots of field y r (x, y, t) superposed to the velocity vector field u (x, y, t) during the flow reversal identified as (iv) in figure 8.9.

Figure 8 .

 8 Figure 8.11: a) Snapshot of the reference height y r (x, y, 0, t) for Ra = 2 • 10 8 , Pr = 4.38, and Γ z = 1/64. b) Snapshot of the temperature field for Ra d = 1581 from 2-D Darcy-Oberbeck-Boussinesq equations. Image (b) is taken from[START_REF] Otero | High-rayleigh-number convection in a fluid-saturated porous layer[END_REF].

Figure 8 . 12 :

 812 Figure 8.12: Model with inertial corrections: compensated Nusselt and Reynolds numbers using exponents β Nu-1 = 0.61 and β Re = 0.81. A thick solid line indicates the scaling for a cubic cell. Symbols indicate the value of Γ z : Γ z = 1/16, Γ z = 1/32, △ Γ z = 1/64, ⋄ Γ z = 1/128, • Γ z = 1/192, and ⊗ Γ z = 1/256.

Figure 8 . 13 :

 813 Figure 8.13: Model with inertial corrections: Compensated Nusselt and Reynolds numbers using exponents β Nu-1 = 0.82 and β Re = 0.91 as a function of Ra/Ra * = 1.85•10 -5 Γ 3.23z Ra proposed by[START_REF] Chong | Exploring the severely confined regime in rayleigh-bénard convection[END_REF]. Symbols indicate the value of Γ z : Γ z = 1/16, Γ z = 1/32, △ Γ z = 1/64, ⋄ Γ z = 1/128, • Γ z = 1/192, and ⊗ Γ z = 1/256.

Figure 8 . 14 :

 814 Figure 8.14: Ratio between the kinetic δ (1) p and thermal δ (1) θ boundary layer thicknesses as function of (a) Ra and (b) Ra/Ra * = 1.85 • 10 -5 RaΓ 3.23 z . Symbols indicate the value of Γ z : ⋄ Γ z = 1/128, • Γ z = 1/192, and ⊗ Γ z = 1/256.

Figure A. 1 :

 1 Figure A.1: Distribution of points x i and xu i along the x-direction. Boundary domains are shown in thick black lines.

θFigure B. 2 :

 2 Figure B.2: Stencils for different scalar and vector fields used in Basilisk C. The hatched region indicates the current cell, while a ⊗ symbol marks the placement of the [0, 0, 0] index relative to this cell.

  b(x), defined as the discrete gradient of field a(x) in R 2 , b = ∇a evaluated at the center of the cell as vector b []; // Declare field foreach () { // Iterate over each cell b .x [] = ( a [1 ,0] -a [ -1 ,0]) / sq ( Delta ) ; // Gradient in X b .y [] = ( a [0 ,1] -a [0 , -1]) / sq ( Delta ) ; // Gradient in Y }

2 1 2

 1 represent the same quantity, but are treated in a different way. In this context, U i n+ represents the "advectee" and U i n+1 velocity into a divergence-free space which requires solving the following Poisson problem D

∂

  i u i + ∂ z w = 0 (F.8a) α ∂ τ u i + ∂ j (u j u i ) + ∂ z (w u i ) = -∂ i p + PrRa -0.5 ∂ j ∂ j u i (F.8b) + ε -2 PrRa -0.5 ∂ 2 z u i + Prθδ iy α ∂ τ w + ∂ j (u j w) + ∂ z (w w) =ε -2 ∂ z p + PrRa -0.5 ∂ j ∂ j w (F.8c) + ε -2 PrRa -0.5 ∂ 2 z w α ∂ τ θ + ∂ j (u j θ) + ∂ z (w θ) = Ra -0.5 ∂ j ∂ j θ + ε -2 Ra -0.5 ∂ 2 z θ (F.8d)This is completed by the mechanical and thermal boundary conditions at the front and back wallsu i = w = 0 at z = ±1 (F.9a)∂ n θ = 0 at z = ±1 (F.9b)

  (see figure F.1). Introducing this expression into (F.10), one obtains an upper boundary in Ra and Γ z for the range of validity of the Hele-Shaw approximation. Using this for a fixed value of Ra, the aspect ratio Γ z must verify Γ z = 2ε ≪ 4.689Ra -0.2295 (F.11) This limit may be better represented on the phase diagram displayed on the bottom right part of figure F.1. Ra Γ z α αε 2 Ra 0.5

Figure F. 1 :F. 2 . 1

 121 Figure F.1: (Left) Table providing the values of α and αε 2 Ra 0.5 obtained from DNS. The latter must remain small in order to satisfy the formulation proposed. Unwanted values are indicated in red. (Top right) Value of α compared to a data fit α = 0.083ε 0.92 Ra 0.17 estimated using DNS data for different Ra and Γ z . (Bottom right) Phase diagram reproduced from figure 7.1. Marks indicate the values of (Ra, Γ z ) considered and the thick line corresponds to the limit Γ z = 4.689Ra -0.2295 .

  u i (x, y, z, τ ) = ε 2 1 3 Ra 0.5 Pr -1 -∂ i p(x, y, τ ) + Prθ(x, y, τ )δ iy g 0 (F.13) in addition to the continuity equation (F.12a) imposes w = 0 and ∂ i u i = 0 (F.15)

11 -

 11 165z 2 + 385z 4 -231z 6 )

••

  Continuity equation: from equation (F.21) ∇ xy • u = 0 (F.40a) • Momentum equation: from equation (F.37a) and (FTemperature equation: from equation (F.34a) ∂ t θ + u • ∇ xy θ = Ra -0.5 ∇ 2 xy θ (F.40c) with ∇ xy = e x ∂ x + e y ∂ y , a friction coefficient λ 1 and a correction coefficient λ 2

4 2 5 ∇

 45 Expanding and integrating both expressions over the volume givesǫ (xy) = 6 xy u : ∇ xy u xyα 2 2 5

  Var and root mean square φ rms . For instance φ Var yz and φ rms yz (x, t) corresponds to the spatial variance and the root mean squared value evaluated over y and z-directions.

			φ(x, y, z, t) dxdz A xz	(2.27b)
	{φ} x (x, t) ≡ φ(x, t) -φ x (y, z, t)	φ x (y, z, t) ≡	φ(x, y, z, t) dx L x	(2.27c)
	For any such spatial averages, we define a spatial variance φ

t)φ xyz (t) φ xyz (t) ≡ φ(x, y, z, t) dxdydz V (2.27a) {φ} xz (x, t) ≡ φ(x,

t)φ xz (y, t) φ xz (y, t) ≡ φ Var yz (x, t) ≡ {φ} yz {φ} yz yz φ rms yz (x, t) ≡ φ Var yz 1/2

  ). In this way, it is possible to write d dt E k in terms of the instantaneous volume Nusselt number Nu vol as

		d dt	E k = PrRa -0.5 (Nu vol -(ǫ + 1))	(2.36)
	Conversion rates for the total potential energy
	If we multiply by y the temperature equation and average over the fluid volume, we obtain
	the following relation for the conversion rate of E p
	d dt	E

p = -PrRa -0.5 Ra 0.5 θu y xyz -∂ y θ xyz Nu vol =Φy+1

  which contains the volume averaged Nusselt Nu vol , and a boundary term Φ b1 which quantifies the energy conversion rate from external sources to E p . For the present geometry and considering adiabatic side-walls, from equation(2.30) one easily verifies that Φ b1 = 1 2 (Nu bot + Nu top ) and obtain a relation between Nu bot , Nu top , and Nu vol If we multiply by y r the temperature equation and average over the fluid volume, we obtain the following relation for the conversion rate of E b

	d dt	E p = PrRa -0.5 -Nu vol +	1 2	(Nu bot + Nu top )	(2.38)
	Conversion rates for the background potential energy
		d dt	E
						(2.37)
					-Φ b1

+ V -1 y∂ n θ dS b = -Pr y r ∂ t θ xyz -Pr θ∂ t y r xyz

(

2.39)

  by equation(2.42) one getsd dt E a = PrRa -0.5 (Φ b1 + Φ b2 -Nu vol -Φ d ) (2.44)Since iso-thermal conditions are imposed on the top and bottom walls, one may prefer to write d dt E a as function of the instantaneous Nusselt numbers instead.

	d dt	E

a = PrRa -0.5 (Nu top + Nu top -Nu vol -Φ d )

(2.45)

  n y n z /bg = 2 19 n x n y n z /bg = 2 20

				(a) Cubic RB cell @ Ada			(b) Cubic RB cell @ Turing
		8						8
			n x n y n z /p = 2 15					n
			n x n y n z /p = 2 18
		6						6
	u.t	4					u.t	4
		2						2
		0	2 1	2 3	2 5	2 7	2 9	0	2 1	2 3	2 5	2 7	2 9	2 11
				Processors p					BlueGene nodes bg
												t(p) and
	a parallel efficiency E as
				S p =	u.t mono u.t(p)				E =	S p p			(3.12)

x

Complementary material: Simulation parameters and numerical convergence

	(a) Differentially-heated square cavity (2D)
		Ra	Pr	Grid	α	min ∆x	max ∆x
	De Vahl Davis, 1983	10 3 -10 4 0.71	40	-	0.025
		10 5 -10 6 0.71	80	-	0.0125
	Le Quéré, 1991	10 6	0.71 N = 73 -9.5 • 10 -4 2.1 • 10 -2
		10 7	0.71 N = 81 -4.6 • 10 -4 1.9 • 10 -2
	Tric et al., 2000	10 3 -10 7 0.71 N = 51 -3.8 • 10 -4 3.1 • 10 -2
	Fusion	10 3 -10 8 0.71	128 2	-	7.9 • 10 -3
				256 2	-	3.9 • 10 -3
				512 2	-	2 • 10 -3
		10 8		768 2	-	1.3 • 10 -3
	Basilisk	10 3 -10 8 0.71	128 2	-	7.8 • 10 -3
				256 2	-	3.9 • 10 -3
				512 2	-	2 • 10 -3
				1024 2	-	9.8 • 10 -4
	(b) Differentially-heated cubic cavity (3D)
		Ra	Pr	Grid	α	min ∆x	max ∆x
	Tric et al., 2000	10 3 -10 6 0.71 N = 81	-	2.31 • 10 -3 3.92 • 10 -2
	Fusion (OpenMP)	10 3 -10 6 0.71	64 3		1.5	1.1 • 10 -2	1.9 • 10 -2
				128 3		1.5	5.5 • 10 -3	9.3 • 10 -3
				192 3		1.5	3.7 • 10 -3	6.2 • 10 -3
				256 3		1.5	2.7 • 10 -3	4.6 • 10 -3
	Basilisk (MPI)	10 3 -10 6 0.71	32 3		-	3.1 • 10 -2
				64 3		-	1.6 • 10 -2
				128 3		-	7.8 • 10 -3
				256 3		-	3.9 • 10 -3
				512 3		-	2 • 10 -3
	(c) Rayleigh-Bénard cells (3D) for Pr = 0.786
		Ra	Γz		Grid	γ	min ∆x	max ∆x
	Wagner and Shishkina, 2013 10 5	1/10-1/3	64 2 × 32	-	-	-
		10 6	1/10-1/3	96 2 × 32	-	-	-
	Fusion (OpenMP)	10 5 -10 6 1/10-1/3	64 2 × 32	1.5 1.1 • 10 -2 1.9 • 10 -2
				128 2 × 64	1.5 5.5 • 10 -3 9.3 • 10 -3
				192 2 × 96	1.5 3.7 • 10 -3 6.2 • 10 -3
	Basilisk (MPI)	10 5 -10 6 1/10	160 2 × 16	-	6.3 • 10 -3
				320 2 × 32	-	3.1 • 10 -3
				640 2 × 64	-	1.6 • 10 -3
			1/4		64 2 × 16	-	1.6 • 10 -2
				128 2 × 32	-	7.8 • 10 -3
				256 2 × 64	-	3.9 • 10 -3
				512 2 × 128	-	2 • 10 -3
			1/3		48 2 × 16	-	2.1 • 10 -2
					96 2 × 32	-	1 • 10 -2
				192 2 × 64	-	5.2 • 10 -3
				288 2 × 96	-	3.5 • 10 -3

Table 3 .

 3 1: Simulation parameters used for the different test cases. Here N indicates the number of spectra elements used in the benchmark results. The parameter γ corresponds to the stretching coefficient along the vertical direction, see appendix A.

				(a) Fusion
	Ra	Grid Nu 3D Nu mp	max(u)	max(v)	max(w)
	1 •	64	1.071 1.088 1.123 • 10 -1 1.121 • 10 -1 5.444 • 10 -3
	1 •	128	1.071 1.087 1.121 • 10 -1 1.121 • 10 -1 5.434 • 10 -3
	1 •	192	1.071 1.087 1.121 • 10 -1 1.121 • 10 -1 5.432 • 10 -3
	1 •	256	1.071 1.087 1.120 • 10 -1 1.121 • 10 -1 5.432 • 10 -3
	1 •	Ref. 1.070 1.087 1.120 • 10 -1 1.121 • 10 -1 5.471 • 10 -3
	1 •	64	2.069 2.266 1.678 • 10 -1 1.901 • 10 -1 2.171 • 10 -2
	1 •	128	2.058 2.254 1.675 • 10 -1 1.900 • 10 -1 2.160 • 10 -2
	1 •	192	2.056 2.251 1.675 • 10 -1 1.901 • 10 -1 2.159 • 10 -2
	1 •	256	2.056 2.250 1.675 • 10 -1 1.901 • 10 -1 2.157 • 10 -2
	1 •	Ref. 2.054 2.250 1.672 • 10 -1 1.898 • 10 -1 2.156 • 10 -2
	1 •	64	4.440 4.724 1.398 • 10 -1 2.246 • 10 -1 3.100 • 10 -2
	1 •	128	4.364 4.642 1.394 • 10 -1 2.247 • 10 -1 3.080 • 10 -2
	1 •	192	4.349 4.626 1.392 • 10 -1 2.248 • 10 -1 3.075 • 10 -2
	1 •	256	4.344 4.620 1.391 • 10 -1 2.248 • 10 -1 3.073 • 10 -2
	1 •	Ref. 4.337 4.612 1.388 • 10 -1 2.247 • 10 -1 3.064 • 10 -2
	1 •	64	9.193 9.439 1.285 • 10 -1 2.386 • 10 -1 2.645 • 10 -2
	1 •	128	8.808 9.048 1.279 • 10 -1 2.372 • 10 -1 2.593 • 10 -2
	1 •	192	8.719 8.956 1.276 • 10 -1 2.371 • 10 -1 2.576 • 10 -2
	1 •	256	8.685 8.922 1.274 • 10 -1 2.368 • 10 -1 2.569 • 10 -2
	1 •	Ref. 8.640 8.877 1.270 • 10 -1 2.367 • 10 -1 2.556 • 10 -2
				(b) Basilisk
	Ra	Grid Nu 3D Nu mp	max(u)	max(v)	max(w)
	1 •	32	1.077 1.096 1.112 • 10 -1 1.113 • 10 -1 5.757 • 10 -3
	1 •	64	1.074 1.092 1.116 • 10 -1 1.116 • 10 -1 5.637 • 10 -3
	1 •	128	1.072 1.090 1.118 • 10 -1 1.118 • 10 -1 5.552 • 10 -3
	1 •	256	1.072 1.089 1.118 • 10 -1 1.119 • 10 -1 5.506 • 10 -3
	1 •	512	1.071 1.089 1.119 • 10 -1 1.119 • 10 -1 5.482 • 10 -3
	1 •	Ref. 1.070 1.087 1.120 • 10 -1 1.121 • 10 -1 5.471 • 10 -3

  10 -1 2.366 • 10 -1 2.559 • 10 -2 1 • Ref. 8.640 8.877 1.270 • 10 -1 2.367 • 10 -1 2.556 • 10 -2

	1 •	32	9.383 9.619 1.331 • 10 -1 2.287 • 10 -1 2.635 • 10 -2
	1 •	64	8.840 9.076 1.301 • 10 -1 2.370 • 10 -1 2.561 • 10 -2
	1 •	128	8.685 8.921 1.285 • 10 -1 2.355 • 10 -1 2.563 • 10 -2
	1 •	256	8.648 8.884 1.277 • 10 -1 2.364 • 10 -1 2.561 • 10 -2
	1 •	512	8.640 8.877 1.274 •

Table 3 .

 3

2: Numerical convergence of characteristic values for Pr = 0.71 obtained using (a) Fusion and (b) Basilisk. Nusselt numbers Nu 3D and Nu mp defined in equation

(3.8)

, and maximum velocity components: max (u(x, y, z)), max (v(x, y, z)), max (w(x, y, z)).

Table 1 .

 1 For various Prandtl and Rayleigh numbers, the table provides the simulation length in convective time units, the number of reversal events, the average Nusselt number and maximum relative difference between Nu vol , Nu top , Nu bot , Nu θ , and Nu ǫ .

	Pr	Ra	Time Events Nu	%Diff
	3.0	10 7	9,600 48	12.5	0.30
		3 • 10 7	9,600 77	17.6	0.53
		5 • 10 7	9,600 83	20.7	0.69
		10 8	12,000 83	25.6	1.44
	4.3	3 • 10 7	19,000 22	18.4	0.24
		5 • 10 7	29,000 161	21.0	0.70
		10 8	19,000 156	25.9	1.38

8 by equation 2.11, one gets dE apot dt = PrRa -0.5 [Nu bot + Nu top -Nu vol -Φ d ] (2.12)

  A. Castillo et al

	Pr	Ra	p cr	p rr	p ec	Pr	Ra	p cr	p rr	p ec
	3.0	10 7	85%	1%	14%	4.3	-	-	-	-
		3 • 10 7	73%	3%	24%		3 • 10 7	95%	1%	4%
		5 • 10 7	77%	3%	20%		5 • 10 7	83%	1%	16%
		10 8	96%	4%	0%		10 8	89%	2%	9%

  /|L

				t o				Between points (f) and (g)
					g	0.1
	2d | L 2d /|L 2d |		f		vmn	0 0.05		m=1,n=1 m=2,n=2
						-0.05
			(t -t o )
						-0.1
							0	20	40
		2						t -t o
	kin | kin |							On point (g)
	E kin /|E E kin /|E	1	f	t o	g	0.1 0.05		m=1,n=1 m=2,n=2
		0			vmn	0
			20			-0.05
			(t -t o )	40	-0.1	0	20	40
								t -t o

Figure 21. Left panel: normalized angular impulse |L 2D |/|L 2D | (top) and normalized kinetic energy E kin /|E kin | (bottom) for (Ra = 5 • 10 7 , Pr = 3.0) as a function of t o characterizing the base state and time (t -t o ). Each curve represents different initial conditions i.e. different t o , where solid lines indicate when a flow reversal takes place (see text). Right panel: Evolution of modal coefficients vmn from two initial conditions: before and at point (g).

Table 5 .

 5 Mode-L Mode-Q Mode-S y Mode-L * Mode-S x Mode-Q * 1: Hierarchy of modes in each POD formulation for (Ra = 5 • 10 7 , Pr = 4.3). For each mode, we indicate if the mode is symmetric (S) or antisymmetric (AS) with respect to S x , S y , R π described in 2.5.

	Formulation (C)	1	2	3	4	5	6
	Formulation (B)	1	3	2	4	5	6
	Formulation (A)	2	1	5	3		4
	Symmetry S x	AS	S		AS	S	S
	Symmetry S y	AS	S	S	AS		S
	Symmetry R π	S	S		S		S

  Ra Grid Length Nu bot Nu top Nu vol Ra Grid Length Nu bot Nu top Nu vol

					(a) Pr = 3.0
							Nu ǫ	Nu d	Nu θ	Nu	%
	1 •	512	4,500	6.60	6.60	6.60	6.60	6.59	6.59	6.60 0.2
	3 •	512	4,500	8.82	8.82	8.84	8.82	8.80	8.81	8.82 0.4
	5 •	512	4,500	10.11	10.10 10.12 10.10 10.08 10.09 10.10 0.4
	8 •	512	4,500	11.61	11.62 11.63 11.61 11.55 11.58 11.60 0.7
	1 •	512	9,600	12.54	12.54		12.52		12.50 12.50 0.3
	2 •	512	9,600	15.52	15.52		15.49		15.46 15.50 0.4
	3 •	512	9,600	17.68	17.69		17.64		17.59 17.60 0.5
	4 •	512	9,600	19.42	19.42		19.36		19.30 19.30 0.6
	5 •	512	9,600	20.75	20.75		20.68		20.61 20.70 0.7
	8 •	512	5,000	23.95	23.94 24.00 23.87 23.42 23.57 23.79 2.5
	1 •	512	20,000	25.65	25.65 25.67 24.98 24.98 25.35 25.38 2.8
	3 • 1024	4,600	35.19	35.18 35.23 35.13 34.69 34.93 35.06 1.5
	5 • 1024	4,800	41.01	41.01 41.05 40.93 40.06 40.52 40.76 2.5
					(b) Pr = 4.3
							Nu ǫ	Nu d	Nu θ	Nu	%
	1 •	512	4,500	6.83	6.83	6.84	6.84	6.83	6.83	6.83 0.2
	3 •	512	4,500	9.10	9.09	9.11	9.09	9.07	9.08	9.09 0.4
	5 •	512	4,500	10.48	10.48 10.50 10.48 10.45 10.46 10.47 0.5
	8 •	512	3,500	12.16	12.16 12.17 12.15 12.08 12.12 12.14 0.7
	1 •	512	9,600	13.07	13.07 13.08 13.03		13.05 13.06 0.3
	3 •	512	19,000	18.37	18.37 18.38 18.31		18.04 18.29 0.2
	5 •	512	29,000	21.01	21.01 21.03 20.86		20.95 20.97 0.7
	8 •	512	5,000	24.13	24.11 24.17 24.05 23.56 23.72 23.96 2.6
	1 •	512	19,000	25.88	25.88 25.90 25.19 25.21 25.58 25.61 2.8
	3 • 1024	4,600	35.64	35.62 35.68 35.58 35.13 35.35 35.50 1.6
	5 • 1024	4,800	41.37	41.35 41.40 41.28 40.29 40.82 41.09 2.8

Table 5 . 3

 53

: Convergence of Nusselt number; Rayleigh number, grid size, and lenght of simulation. Time-averaged values of Nu, averaged Nusselt number Nu and % maximum relative difference. 107 Part

III Influence of boundary conditions: stress-free surface on a square Rayleigh-Bénard cell Chapter 6

Table 6 .

 6 1: Convergence of Nusselt number for (a) Pr = 3.0 and (b) Pr = 4.3. Rayleigh number, grid size, and length of simulation. Time-averaged values of Nu, average Nusselt number Nu, % maximum relative difference and mean temperature θ xy . A * symbol indicates a steady-state solution, while † and ‡ indicate one of two periodic solutions, see text.

	129

Influence of boundary conditions: severe lateral confinement on a rectangular Rayleigh-Bénard cell Chapter 7

  The average values of Nu vol , Nu top , Nu bot , Nu ǫ , Nu θ , Nu d , and Nu as well as the maximum relative difference %Diff are listed in table7.1 (see §2.2.7 for definitions). For the finer grids, these quantities are shown to converge within 3% of Nu for the range of (Ra, Pr) presented. In addition, the average Nusselt number Nu is in good agreement with published results from[START_REF] Chong | Exploring the severely confined regime in rayleigh-bénard convection[END_REF]

	Γ z	Ra	t.u.	Grid Nu top Nu bot Nu vol	Nu ǫ	Nu d	Nu θ	Nu %Diff
	1/8 1/16 1/32 5 • 10 6 10 6 5 • 10 6 10 7 5 • 10 7 10 8 10 6 5 • 10 6 10 7 5,000 ‡ 750 750 † 1,600 1,800 * 1,000 2 • 10 7 2,500 5 • 10 7 1,750 10 8 2,250 * 10 7 1,000 1024 2 × 32 256 2 × 32 256 2 × 32 10.70 10.72 10.70 10.61 10.60 10.62 10.66 5.18 5.18 5.17 5.16 5.16 5.17 5.17 512 2 × 64 14.87 14.87 14.87 14.83 14.81 14.82 14.85 512 2 × 64 28.44 28.44 28.45 28.30 28.19 28.17 28.33 512 2 × 64 35.00 35.01 35.01 34.75 34.62 34.53 34.82 4.12 4.12 4.11 4.07 4.11 4.11 4.11 256 2 × 16 7.17 7.17 7.16 7.08 7.14 7.14 7.15 256 2 × 16 9.42 9.44 9.43 9.40 9.40 9.41 9.41 512 2 × 32 512 2 × 32 15.40 15.40 15.40 15.33 15.31 15.33 15.37 512 2 × 32 22.76 22.76 22.76 22.63 22.58 22.59 22.68 512 2 × 32 29.48 29.49 29.50 29.28 29.06 29.11 29.32 5.33 5.33 5.33 5.27 5.32 5.32 5.32 512 2 × 16 6.52 6.54 6.53 6.52 6.52 6.52 6.52 5 • 10 7 750 1024 2 × 32 14.48 14.49 14.47 14.43 14.44 14.45 14.46 10 8 1,000 1024 2 × 32 19.94 19.88 19.91 19.85 19.85 19.86 19.88 10 9 1,000 1024 2 × 32 62.11 62.04 61.92 61.58 61.34 61.17 61.69 1/64 10 7 3.69 3.67 3.68 3.65 3.68 3.68 3.68 * 1024 2 × 16 5 • 10 7 7.84 7.87 7.85 7.76 7.85 7.85 7.84 1,200 1024 2 × 16 10 8 1,200 1024 2 × 16 12.61 12.59 12.60 12.47 12.58 12.60 12.58 2 • 10 8 1,000 1024 2 × 16 19.39 19.42 19.51 19.29 19.31 19.35 19.38 10 9 2,000 1024 2 × 16 46.83 46.85 46.85 46.28 46.31 46.30 46.57	0.5 1.0 0.4 1.0 1.4 1.0 1.2 0.4 0.6 0.8 1.5 1.0 0.3 0.4 0.5 1.5 1.2 1.0 1.1 1.1 1.2

Table 7 .

 7 1: Time-averaged Nusselt numbers and maximum relative difference between them for the finest grid used. Here t.u. indicates the length of the simulation. Symbol † indicates a periodic solution, ‡ an intermittent solution, and * a steady-state.

Table 7 .

 7 3: Geometric characterization of plumes for y = ±(0.5 -1/Nu): Average number of plumes (No.), average plume spacing d 0 , mean correlation distance r 0 and integral scale L 0 . Values normalized by δ

				Plumes near top BL		Plumes near bottom BL
	Ra	Γz No. d0/δ θ (1)	r0/δ θ (1)	L0/δ θ (1)	No. d0/δ θ (1)	r0/δ θ (1)	(1) θ L0/δ
	1 •	1/8	3.6	10.87	4.21	0.90	3.6	10.92	4.90	1.50
	5 •	1/8	4.7	14.49	8.69	2.99	4.7	14.50	9.47	3.23
	1 •	1/8	4.8	16.55	10.56	2.72	5.0	16.22	11.10	2.74
	1 • 1/16	3.0	8.96	1.05	0.46	2.6	11.09	1.05	0.50
	5 • 1/16	3.2	20.25	4.92	1.45	3.3	19.36	6.96	0.90
	1 • 1/16	4.5	16.33	4.64	1.52	4.1	18.54	5.90	1.64
	1 • 1/32	2.7	7.17	1.17	0.53	2.8	7.08	1.17	0.54
	5 • 1/32	3.5	11.42	1.09	0.55	4.0	9.45	1.00	0.47
	1 • 1/32	4.5	11.04	1.18	0.54	4.5	11.10	1.25	0.59
	1 • 1/32	6.8	21.09	15.62	3.03	7.2	19.96	16.58	2.68
	1 • 1/64	2.0	6.95	1.34	0.61	2.0	6.96	1.35	0.61
	5 • 1/64	2.9	7.92	1.07	0.51	2.9	7.94	1.09	0.51
	1 • 1/64	3.2	11.01	1.17	0.55	3.5	9.96	1.12	0.51
	1 • 1/64	8.0	13.24	1.68	0.78	7.7	13.84	1.77	0.78
				(1) θ = 1/(2Nu) for reference.

Table 7 . 4 :

 74 Different estimations for the thickness of the thermal boundary layers: δ

				Bottom	Top	Bottom	Top
	Γz	Ra	δ θ /Γz (1)	δ θ /Γz (2)	δ θ /Γz (2)	δ θ /Γz (3)	δ θ /Γz (3)
	1/8	1 • 10 7	0.27	0.26	0.27	0.52	0.45
	1/8	5 • 10 7	0.14	0.13	0.13	0.35	0.35
	1/8	1 • 10 8	0.11	0.10	0.10	0.34	0.34
	1/16	1 • 10 7	0.85	1.05	1.27	1.02	1.20
	1/16	5 • 10 7	0.35	0.36	0.36	0.77	0.77
	1/16	1 • 10 8	0.27	0.23	0.23	0.83	0.77
	1/32	1 • 10 7	2.45	3.20	3.17	2.52	2.55
	1/32	5 • 10 7	1.11	1.70	2.42	1.83	1.89
	1/32	1 • 10 8	0.81	0.95	1.11	1.64	1.61
	1/32	1 • 10 9	0.26	0.20	0.20	1.42	1.39
	1/64	5 • 10 7	4.04	6.28	6.22	4.22	5.59
	1/64	1 • 10 8	2.54	4.16	5.53	3.84	4.16
	1/64	2 • 10 8	1.65	3.41	3.34	3.16	3.09
	1/64	1 • 10 9	0.69	0.66	0.66	2.41	2.34
							(1) θ ,
	δ

Table 7 .

 7 5: Estimation for the thickness of the kinetic boundary layers at the top and bottom plates δ

			Kinetic BL thickness	Ratio between thermal and kinetic BL thicknesses
			Bottom	Top		Bottom	Top	Bottom	Top
	Γz	Ra	δ p /Γz (1)	δ p /Γz (1)	δ p /δ (1) θ (1)	δ p /δ (2) θ (1)	δ p /δ (2) θ (1)	δ p /δ (3) θ (1)	δ p /δ (3) θ (1)
	1/8	1 •	0.45	0.30	1.4	1.7	1.1	0.9	0.7
	1/8	5 •	0.29	0.29	2.0	2.2	2.2	0.8	0.8
	1/8	1 •	0.26	0.26	2.2	2.5	2.5	0.8	0.8
	1/16	1 •	0.58	0.70	0.8	0.6	0.6	0.6	0.6
	1/16	5 •	0.67	0.67	1.9	1.9	1.9	0.9	0.9
	1/16	1 •	0.45	0.48	1.7	1.9	2.1	0.5	0.6
	1/32	1 •	1.02	1.02	0.4	0.3	0.3	0.4	0.4
	1/32	5 •	0.80	0.98	0.8	0.5	0.4	0.4	0.5
	1/32	1 •	0.86	0.73	1.0	0.9	0.7	0.5	0.5
	1/32	1 •	0.52	0.52	2.0	2.5	2.5	0.4	0.4
	1/64	5 •	1.22	1.22	0.3	0.2	0.2	0.3	0.2
	1/64	1 •	1.09	1.22	0.5	0.3	0.2	0.3	0.3
	1/64	2 •	0.97	0.97	0.6	0.3	0.3	0.3	0.3
	1/64	1 •	0.91	0.91	1.3	1.4	1.4	0.4	0.4

Table 7 .

 7 6: Most conservative estimate of the ratio δ

	Γz	Ra	Re d	δ δ	(1) p (1) θ	δw Γz	∆z Γz	ǫ (bl) ǫ	Γz	Ra	Re d	δ δ	(1) p (1) θ	δw Γz	∆z Γz	ǫ (bl) ǫ
	1/8	1 • 10 7	9.7		1.4	0.15	0.02	0.52	1/32	1 • 10 7	0.5	0.4	0.22	0.03	0.79
	1/8	5 • 10 7	26.5		2.0	0.13 0.02	0.50	1/32 5 • 10 7	1.8	0.8	0.21 0.03	0.75
	1/8	1 • 10 8	38.9		2.2	0.12	0.02	0.49	1/32	1 • 10 8	2.9	1.0	0.21	0.03	0.74
	1/16 1 • 10 7	2.4		0.8	0.21 0.03	0.69	1/32 1 • 10 9	15.5	2.0	0.21 0.03	0.71
	1/16	5 • 10 7	7.7		1.9	0.19	0.03	0.62	1/64	5 • 10 7	0.3	0.3	0.23	0.06	0.81
	1/16 1 • 10 8	12.1		1.7	0.19 0.03	0.61	1/64 1 • 10 8	0.6	0.5	0.23 0.06	0.80
									1/64	2 • 10 8	1.1	0.6	0.23	0.06	0.80
									1/64 1 • 10 9	3.7	1.3	0.22 0.06	0.77

  xy as the operator ∇ xy = e x ∂ x + e y ∂ y and λ 1 being a friction coefficient on the

				(8.5a)
				(8.5b)
		5 ∇ 2 xy θ	(8.5c)
	with ∇ front and back walls	λ 1 =	3 ε 2 PrRa -0.5	(8.6)

  The accuracy of the models is quantified by %Error(Nu) (resp. %Error(Re)) as the deviation from the reference Nusselt Nu ref (resp. Reynolds Re ref ) obtained from 3-D DNS results. Nu for the models, often coincides with small deviations from the reference results obtained by the DNS (see table8.1 and 8.2). Inside the range of validity of the models, both models recover the main features of the confined regime, dominated by large-scale coherent plumes and the Nusselt and Reynolds numbers are estimated within a few percent (see tables 8.1 and 8.2) which implies similar scaling laws as the DNS for these configurations.

	%Error(Nu) = (Nu -Nu ref )/Nu ref × 100 %Error(Re) = (Re -Re ref )/Re ref × 100	(8.14) (8.15)
	A good convergence in

Table 8 .

 8 1: Time-averaged Nusselt number and the maximum relative difference between definitions of the Nusselt number %Diff for the reduced models using a 512 2 grid. Measurements are obtained over 1,000 convective time units. Results from the DNS is noted here as Nu ref , while the deviation from the DNS is noted as %Error(Nu). A † indicates steady-state solutions.

Table 8 .

 8 2: Time-averaged Reynolds number for the reduced models using a 512 2 grid. Measurements are obtained over 1,000 convective time units. Results from the DNS is noted here as Re ref , while the deviation from the DNS is noted as %Error(Re). A † indicates steady-state solutions.

	The average time

Longer observations may be obtained from the reduced model, which allow us to compute different quantities, such as the probability distributions for each modal coefficient. Note

table 8

 8

.3. Using the same arguments 1 , but considering a local Ra d as the relevant control parameter

Table 8 .

 8

	3: Malkus' and Howard's theory applied to conventional convection (left) and
	convection under severe confinement (right). The latter is consistent with the ultimate
	regime in porous media.

8.5.2 Scaling exponents in the severely confined regime using the re- duced model with inertial corrections

  We compute scaling exponents obtained from simulations of the reduced model with inertial corrections for Γ z < 1/16, see table8.4. Since the kinetic boundary layer thickness is comparable to Γ z , this value becomes a limiting factor for these simulations. Here, we use a 1024 2 grid for Γ z = 1/64, Γ z = 1/128, Γ z = 1/192, and Γ z = 1/256Figures 8.12a and 8.12b display the compensated Nusselt and Reynolds numbers as function of Ra for different Γ z from 1/16 to 1/256. Fitted scaling exponents (see table8.4) are comparable to those obtained from DNS results and to published results[START_REF] Chong | Exploring the severely confined regime in rayleigh-bénard convection[END_REF]: β Nu-1 = 0.61 and β Nu-1 = 0.80 for Γ z = 1/128. The relation between the scaling exponents (8.24) also appears to be verified for the most confined cells (Γ z < 1/32) in both the DNS and the model. However, a close inspection of figure8.12 for the more confined cells Γ z < 1/128, suggests there is a change of slope. This change is seen in a compensated plot when represented as function of Ra/Ra * (see figure8.13), where we introduce 1/Ra * = 1.85 • 10 -5 Γ3.23

z as proposed by

  Figure B.3: Flowchart to the generic time-step, providing a minimal description of the contents of each event.

	stability	Update ∆t to verify CFL condi-tion	θ n-1 2 , p n-1 2 , u n
	tracer advection	Advection step of the tempera-ture equation (and other tracers)	U n , Θ n
		Predict velocity at staggered
	advection term	time, project in divergence-free	U n+ 1 2 , U n+ 1 2 , U n+ 1 2
		space, evaluate advection term
	viscous term	Implicit solution of viscous term, update face velocity
		Diffusion step of the tempera-
	acceleration	ture equation, update buoyancy forcing term, obtain provisional	a n+ 1 2 , u *
		velocity
	projection	Obtain pressure field and cor-rected velocity field	p n+ 1 2 , u n+1

  5 385Pr [ u • ∇ xy u ] + ε 4 Ra 3465 [ u • ∇ xy θ ] e y (F.43d)

We will revisit this configuration in the turbulent flow regime in chapters §7 and §8

† Email address for correspondence: anne.sergent@limsi.fr

We use the L∞ norm of the change in the horizontal velocity observed during two consecutive convective time units and fix 10 -8 as our tolerance criteria to define a steady-state solution.

This reasoning was already applied to obtain the ultimate regime of convection in porous media[START_REF] Doering | Bounds for heat transport in a porous layer[END_REF][START_REF] Otero | High-rayleigh-number convection in a fluid-saturated porous layer[END_REF]

2004; D. R.[START_REF] Hewitt | Ultimate regime of high rayleigh number convection in a porous medium[END_REF].

represents the "advector". The "advectee" is treated in the same way as any other advected field, whereas the "advector" receives an special treatment to ensure the conservation of mass. Both procedures will be detailed shortly.

Acknowledgements

This work was granted access to the HPC resources of GENCI-IDRIS under allocations 2014-and 2015-2a0326 made by GENCI. We acknowledge fruitful discussions with C. Nore, B. Podvin, and M.K. Verma.

A close-up of the vertical profiles near the top plate allows to compare the values of δ [START_REF] Adami | Surface tension and buoyancy in vertical soap films[END_REF] θ , δ (2) θ , and δ [START_REF] Ahlers | Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection[END_REF] θ as we increase Ra for Γ z = 1/64 (figure 7.13). In most cases, δ [START_REF] Adami | Surface tension and buoyancy in vertical soap films[END_REF] θ was found to be the smallest value and δ [START_REF] Ahlers | Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection[END_REF] θ was the largest one, see table 7.4. Since our main concern is the relative thickness of the thermal and kinetic boundary layers, we will take the low end estimate of the thickness, i.e. δ [START_REF] Adami | Surface tension and buoyancy in vertical soap films[END_REF] θ , to define the thermal boundary layer thickness (hatched region in figures 7.13b to 7.13d).

Kinetic boundary layers on the top and bottom plates

In turbulent Rayleigh-Bénard convection, a kinetic boundary layer develops in the vicinity of the plates and of the side-walls (figure 7.14). In this section we focus on the kinetic boundary layer of thickness δ p on the top and bottom walls. Different estimations for δ p were presented in §2. [START_REF] Araujo | Wind reversals in turbulent Rayleigh-Bénard convection[END_REF]. First, a thickness δ [START_REF] Adami | Surface tension and buoyancy in vertical soap films[END_REF] p defined based on the distance between the closest peak in u rms xz to the top and bottom plates (see §2. 4.2i). Second, a thickness δ

(2) p based on the closest distance between the position where ∂ j u i ∂ j u i x (y, 0) = (Nu -1) to the top and bottom plates. The plane z = 0 is chosen in order to reduce the influence of the side-walls on the measurements on the top and bottom plates. and is comparable to Γ z . In the following, we use the thickness δ p = δ [START_REF] Adami | Surface tension and buoyancy in vertical soap films[END_REF] p to define the kinetic boundary layer thickness at the top and bottom plates (blue hatched region in figures 7.15b to 7.15d).

Chapter 8 A reduced model for confined cells with adiabatic side-walls

Natural convection in the confined regime (III) is characterized by multiple large-scale turbulent channels along the vertical direction and the flow dynamics is defined by the non-linear interactions between these large-scale channels. In particular, over long periods of time, these structures observe a series of re-configurations. A complete study of the long-term behavior of these structures is deemed too expensive for 3-D DNS. An open question concerns the asymptotic scaling of the Nusselt and Reynolds numbers suggested by [START_REF] Chong | Exploring the severely confined regime in rayleigh-bénard convection[END_REF]. In order to verify this scaling, additional simulations for Γ z ≪ 1 are required. This is difficult because of the increasingly thin kinetic boundary layers as Γ z → 0. This configuration, however, is well suited for dimensionality reduction.

In this chapter, we focus on two reduced models for convection cells under severe confinement. The first model assumes a Poiseuille-like velocity and constant temperature profiles in the transversal direction. This approach has been used extensively for convection problems inside Hele-Shaw cells [START_REF] Hartline | Thermal convection in a hele-shaw cell[END_REF][START_REF] Vorontsov | Natural convection in a hele-shaw cell[END_REF][START_REF] Graham | Time-periodic thermal convection in heleshaw slots: the diagonal oscillation[END_REF][START_REF] Bizon | Plume dynamics in quasi-2d turbulent convection[END_REF]. A second model is also based on the assumption of a Poiseuille-like velocity profile but takes into account inertial corrections, which play an important role near boundaries. It represents a small but noticeable improvement over the first model. Similar work has been done for Hele-Shaw cells [START_REF] Gondret | Shear instability of two-fluid parallel flow in a hele-shaw cell[END_REF][START_REF] Ruyer-Quil | De la modélisation et des instabilités des films liquides tombants[END_REF]. By the end of this chapter, we expect to show how such models may assist in the exploration of the parameter space and provide some insight on the dynamics of the confined regime.

A 'simple' 2-D model for laterally confined cells

In this section, we consider a simple model for convection inside Hele-Shaw cells. In the following we introduce the small parameter

Part V

Conclusions and Perspectives

Appendix A

Code FUSION

A.1 Temporal discretization scheme

The time discretization of the equation system is performed by first introducing an increasing sequence if [t 0 , t

We use a second-order backward Euler method to approximate the time derivatives

Equations are discretised using a semi-implicit method, with viscous terms treated implicitly and advective terms explicitly and approximated as follows

The discretized equations would have the following form

In order to solve equations (A.4a) and (A.4b) one must know beforehand the solution for θ n+1 from equation (A.4c). This equation has the form of a Helmholtz-Poisson problem and can be treated independently.

Due to the incompressibility condition and the pressure gradient involved, the velocitypressure coupling is more complex. We use a rotational incremental-pressure correction scheme (see for instance [START_REF] Chorin | On the convergence of discrete approximations to the Navier-Stokes equations[END_REF][START_REF] Guermond | An overview of projection methods for incompressible flows[END_REF]. In this scheme, the pressure term is treated explicitly to obtain a provisional velocity field u * which may not be divergence-free. A correction is obtained by projecting the provisional velocity into a divergence-free space and assume the difference to derived from an scalar function Π, called the pressure potential term.

Solution of the energy equation

In order to obtain θ n+1 we need to solve a Helmholtz-Poisson problem; This type of problem can be written in the general form .5) with Λ = 3 2∆t and Q θ n (x) being the source term corresponding to field θ.

In order to solve directly this three-dimensional Helmholtz system, we use the Douglas-Rachford Alternating Direction Implicit method to separate the operators into one-dimensional components and split the scheme into one sub-step for each coordinate. The main component the ADI method is the factorization of the right-hand side of equation (A.5) as the product of one-dimensional operators in the following form [START_REF] Ferziger | Computational methods for fluid dynamics[END_REF].

Introducing equation (A.7) into equation (A.5) with Γ = 1 Λ Ra -0.5 and dropping O(∆t 2 Ra -1 ) terms, leaves the following expression

We use a multi-step approach to this problem and solve a one-dimensional Helmholtz problem for each direction.

1 .8) where θ n+ 1 3 and θ n+ 2 3 are the intermediate solutions from the first and second sub-steps, respectively. Observe that each ADI sub-step takes the form of a 1D Helmholtz problem.

Solution of the velocity-pressure coupling: prediction step

We treat the pressure term explicitly in equation (A.4b) leaving the following expression

with Q i n (x) being the source term corresponding to the velocity field

We then proceed to use ADI method to separate the operators into one-dimensional components and split the scheme into one sub-step for each coordinate .11) with each ADI sub-step taking the form of a one-dimensional Helmholtz problem.

Solution of the velocity-pressure coupling: projection step

In the projection step, we assume the difference (u n+1u * ) to be derived from a scalar function Π, called the pressure potential term

, with u n n+1 = u n * = 0 on the boundaries (A.12)

Taking the divergence and introducing (A.12) one obtains a Poisson equation

Solution of the velocity-pressure coupling: correction step

After having obtained a solution for Π n+1 , the corrected velocity field is evaluated as

Note that for the velocity field u n+1 satisfy equation (A.4a) the pressure term must be such that

Adding equations (A.9) and (A.12) leaves .16) where

* may be deduced from equation (A.12). Note that this condition ensures only that the imposed normal velocity be preserved for u n+1 and the splitting error manifest only in the form of an inexact tangential boundary condition on the velocity [START_REF] Guermond | An overview of projection methods for incompressible flows[END_REF].

A.2 Spatial discretization

We use an staggered grid arrangement [START_REF] Harlow | Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface[END_REF] with a central differentiation scheme (CDS) and linear interpolation for an overall second-order precision. For this arrangement one defines different type of grids. A cell centered grid for the scalar quantities, such as the temperature, pressure and density fields, and one additional grid for each velocity component.

Generation of grid points

The spatial discretization of the system is performed by first introducing an increasing sequence of points [

Let us consider the points x i : These points can be either regularly spaced

where Γ x is the aspect ratio on the x-direction.

Or these points can be spaced following a given function, like the hyperbolic tangent function.

where γ is a coefficient for the stretching function.

We then introduce a second sequence of points along each direction:

Distance between points

We will define a quantity [∆x 1 , ∆x 2 , • • • , ∆x nx ] as the length of the segment between the indicated points as follows. First, for each cell we define a forward spacing ∆x + i and a -Grid for the velocity component u y

Additionally, for any given node, for instance θ i,j,k , one may introduce the following notation for the neighboring points in each direction, For a given point

where each subindex alluding the cardinal directions in the plane: east and west along x, north and south along y; and front and back along the transversal direction z.

The finite volume method

The principle of the finite volume method lies on the integration of the model equation over each of the control volumes. Observe that in a staggered arrangement, different control volumes exist for each grid. Consider for instance, field φ defined on the scalar grid

where dΩ = dxdydz represents a differential element of the control volume Ω p = ∆x p ∆y p ∆z p and dS a differential element of the control surface S composed of six faces defined as: S e = S w = ∆y p ∆z p , S n = S s = ∆x p ∆z p , and S b = S f = ∆x p ∆y p .

In order to transform our model equation into an algebraic system, we need to approximate the volume (resp. surface) integrals using a simple second-order approximation considering the values at the center of the contol volume (resp. surface) as the mean value of the contol volume (resp. surface).

A.3 Solution for the energy equation

Consider the first ADI sub-step from equation (A.8) integrated over the scalar control volume as follows

Equations for the bulk nodes

Consider the interior points, or bulk points, which correspond to i = 3, • • • , n x -2. Introducing the approximations to the volume and surface integrals one obtains

Considering Ω p = ∆x p ∆y p ∆z p and S e = S w = ∆y p ∆z p , we can rearrange the terms in the following tridiagonal form

Before proceeding to solve this system, we have to apply the boundary conditions to the boundary nodes on both ends.

Equations for the boundary nodes

In order to implement the boundary condition additional nodes are placed outside the fluid domain, known as virtual or ghost nodes. These values are a function of the interpolation scheme selected, the values presented in this section correspond to linear interpolation and second order CDS.

Let us consider a node θ p located on x nx-1 , such that the neighboring node θ e corresponds to a ghost point on x nx and the boundary is located on xu nx-1 . On the boundary xu nx-1 , we consider the following Robin boundary condition

with b, c, and g being the parameters of the boundary condition. The value required at the ghost node θ e would read as

If we introduce this expression into equation (A.24) one obtains

Now consider a node θ p located on x 2 , such that the neighboring node θ w corresponds to a ghost point on x 1 and the boundary is located on xu 1 , the value required at the ghost 201 point would be

and the resulting equation system would read as

Resulting linear equation system

The resulting system is composed by equation (A.24) for the bulk

with the matrix coefficients defined as

This algebraic equation system may be solved using the tridiagonal matrix algorithm to obtain θ n+ 1 3 .

This procedure is repeated for the second ADI sub-steps by replacing Q θ n p by θ n+ 1 3 and introducing the corresponding boundary conditions accordingly.

with the matrix coefficients defined as

and so on for the last ADI sub-step by replacing θ n+ 1 3 with θ n+ 2 3 .

A.4 Solution for the provisional velocity

Solution of the provisional velocity field takes place on the staggered grid, one for each component of the velocity field. For the first two ADI sub-steps the procedure is the same as the one used in the temperature equation. For the last ADI sub-step, in which the velocity component is aligned to the direction of the ADI sweep.

Consider for instance, the first ADI sub-step from equation (A.11) along the x-direction.

For the velocity components perpendicular to the ADI sweep, i.e. v (resp. w), one integrates over the control volume Ω p = ∆x p ∆yv p ∆z p (resp. control volume Ω p = ∆x p ∆y p ∆zw p ).

and the corresponding control surfaces.

Equations for the bulk nodes

Consider the interior points, or bulk points, which correspond to i = 3,

Introducing the approximations to the volume and surface integrals one obtains

For convenience, the terms are rearranged as follows For convenience, let us consider a general Robin boundary conditionon said boundary

Conversely, if we consider a node v p located on x 2 , such that the neighboring node θ w corresponds to a ghost point on x 1 and the boundary is located on xu 1 , the value required at the ghost point would be

and the resulting equation

Resulting linear equation system

The resulting system is composed by equation (A.34) for the bulk nodes i = 3, • • • , n x -2, equation (A.37) for boundary node i = n x -1 and equation (A.39) for boundary node i = 2. This algebraic equation system may be solved using the tridiagonal matrix algorithm to obtain v n+ 1 3 . This procedure is repeated for the second ADI sub-step sweeping along the z-direction by replacing Q y n p by v n+ 1 3 and introducing the corresponding boundary conditions.

Last ADI sub-step

In last ADI-substep the velocity component is aligned to the direction on which the ADI sub-step is applied, i.e. for v when sweeping in the y-direction. In this case, the domain boundary overlaps with the center of the boundary nodes and we may only impose an homogeneous Dirichlet condition.

For the bulk nodes, j = 2, • • • , n y -2 one has

The value v * p = 0 is directly imposed on the boundary nodes j = 1 and j = n x -1. The values imposed on the ghost nodes are v * w = -v * e for j = 1 and j = n x -1. Finally, since the grid overlaps the domain boundaries, our equation system is modifies to ensure that v * p = 0. We do this by imposing a very large number on the matrix coefficients,

(A.41)

The resulting system is composed by equation (A.40) for the bulk nodes j = 2, • • • , n y -2, equation (A.41) and equation (A.39) for the boundary nodes j = 1 and j = n y -1. This algebraic equation system may be solved using the tridiagonal matrix algorithm to obtain v * .

A.5 Solution for the pressure potential

Integration of equation (A.13) over the scalar control volume gives

Introducing the the approximations from the finite volume method one obtains

where D 2 xyz represents the discrete Laplacian operator in 3D which can be written as a Kronecker sum of discrete Laplacians in each direction (see for instance Demmel, 1997)

x , D 2 y , and D 2 z being the one-dimensional discrete Laplacian operators in the x, y and z-direction, respectively. Each discrete Laplacian must be consistent with the boundary conditions imposed on each direction.

The right-hand term RHS p is evaluated as

In this section, we will present the bidiagonalization algorithm used for the Poisson problem, sometimes referred to as a fast Poisson solver, see for instance [START_REF] Golub | A fast poisson solver for the finite difference solution of the incompressible navier-stokes equations[END_REF][START_REF] Strang | Computational science and engineering[END_REF][START_REF] Shishkina | Simulation of turbulent thermal convection in complicated domains[END_REF].

Since this method is presented in terms of the Kronecker product, it is convenient to recall the definition of the Kronecker product and some of its most important properties.

Kronecker product

For any given matrix A of size m × n and matrix B of size p × q, then C = A ⊗ B is a matrix of size (mp) × (nq) .47) or as expressed in index notation as

The Kronecker product verifies the following properties

), as the eigendecomposition of D x (resp. D y ). Here, the matrix R (resp. S) is an orthogonal matrix whose columns are the eigenvectors of D x (resp. D y ), whereas the matrix Λ x (resp. Λ y ) is a diagonal matrix composed with the eigenvalues of D x (resp. D y ).

The first term of equation (A.44) becomes

We multiply this expression by SS -1 and use the mixed-product property from equation (A.49a) and the inversion rule equation (A.49b)

Supossing D 2

x to be real an symmetric matrix we can write

Similarly, we can write the second term of equation (A.44) as

By multiplying both sides of equation (A.43) by a factor (R⊗S⊗I) T and using expressions (A.51) and (A.52) we obtain a tridiagonal system of the form

with Π n+1 i,j,k being the pressure potential projected on the eigenspace D l,m,q Π n+1 l,m,q -U l,m,q Π n+1 l,m,q+1 -L l,m,q Π n+1 l,m,q-1 = RHS l,m,q (A.56)

with the corresponding coefficients defined as

while L l,m,q = 0 for q = 2 and U l,m,q = 0 for q = n z -1.

v. Finally, one multiplies Π n+1 l,m,q by (R ⊗ S ⊗ I) to obtain Π n+1 i,j,k .

A.6 Corrected pressure and velocity fields

The correction for the velocity terms is obtained from equation (A.12)

whereas the pressure field is obtained from equation (A.15)

Code BASILISK

Basilisk is an ensemble of solver-blocks written using an extension to the C programming language, called Basilisk C, useful to write discretization schemes in Cartesian grids (Stéphane Popinet, 2016). Instead of writing an entirely new code, existing blocks of code were combined to solve Boussinesq equations. Before presenting the numerical method used, it is convenient to mention some of the main features of this code. Three type of grids are available to us in Basilisk C: regular Cartesian grids (one level only), multi-level grids, and tree-based adaptative grids. Each discretized cell is a square in 2D and a cube in 3D. Code written in Basilisk C is meant to be the same for different types of grids. For tree-based grids, cells are organized hierarchically in levels, from a root cell placed at the lowest level to a leaf cell located at highest level. Each 'parent' cell may nest 4 'child' cells (resp. 8 cells) for 2D (resp. 3D) cells, see figure B.1. Each cell has at least one direct neighbour at the same level in each direction. Variables can be defined at the center of each cell (when declared using scalar, vector, tensor), on the center of the cell faces (using face vector), or at the vertex of the cell (using vertex scalar). In the present work, we limit ourselves to multi-level grids.

B.1 Main features B.1.1 Regular, Multi-grid or Adaptative grids

The use of pre-defined stencils simplifies the access to neighboring points. Assuming the shaded cell in figure B.2, corresponds to the current cell, the symbol × placed at the center

B.1.3 Events and building blocks

The second main characteristic of Basilisk C is the use of events. An event allows to evaluate a code block or perform any given action at the specified interval, be it every n steps, or every fixed amount of time. The syntax to declare a new event name, every 1 time unit from t=1 to t=5, would read as follows event name ( t = 1; t <= 5; t += 1) { do_something () ; } Certain pre-defined solvers are available, with each one having events of their own. Consider for instance, the centered formulation of the incompressible Navier-Stokes equation included in centered.h (the numerical method will be introduced in the following section). For each time step, one evaluates the code contained inside the following events: dtmax, stability, VOF, tracer advection, properties, advection term, viscosity, acceleration, and projection events.

Instead of modifying these existing routines and trying to adapt them to a particular problem, we seek to re-use the existing code and use these events to make the required changes. Consider the event properties contained inside centered.h event properties ( i ++ , last ) { boundary ({ alpha , mu , rho }) ; } Suppose that we define an event which is also named properties inside our main routine # include " centered . h " event properties ( i ++) { do_something ( alpha , mu , rho ) ; }

The code contained inside the new event will be introduced just before the event of the same name inside the included file centered.h. If we include several files that contain an event with the same name, the code will be executed in the order of the inclusion. Instead of modifying the source code, we are piggybacking on the existing code. This building-block approach is specially well suited for split-step methods.

In the following section, we will provide a brief description of the numerical method used to solve the Bousinesq equation in Cartesian multi-level grids.

Notation-wise, we refer to θ [0,0,0] as the value of the temperature at the center of the current cell. The values θ [0,-1,0] , θ [0,0,0] , and θ [0,1,0] correspond to the temperature evaluated at the center of three contiguous cells along the y-direction. Similarly, θ [0,0,-1] , θ [0,0,0] , and θ [0,0,1] indicate the temperature evaluated at the center of three contiguous cells along the z-direction.

The process may be explained using the case of cell face that is perpendicular to the x-axis and located west of the current cell center [0,0,0]. Note that this process has to be repeated in each face, before proceeding to evaluate the total temperature flux using equation (B.2). Based on the face velocity U x n , we identify the cell located upstream as C and located downstream as N d . We may define an index k, k = -1 for (U x n-1 2 > 0) and k = 0 otherwise, such that C = [k, 0, 0].

Since the temperature field (and other scalar fields for that matter) Θ n is staggered in time and in space with respect to the centered field θ n-1 2 , the value Θ n has to be approximated. Two possible values of Θ n may be constructed (one from each cell sharing the same face). Since our main focus is the advection process, we consider the value obtained from the cell C located upstream using the BCG advection scheme.

We perform a Taylor series expansion around the upstream cell C

where s = sign(U x n ). The temporal derivative is approximated in terms of spatial derivatives using the energy equation (discarding the diffusive terms) as

Combining both expressions leads to

where ∂ x θ is approximated by

and ∂ j θ denotes the spatial derivatives ∂ j θ evaluated numerically as follows. The gradient normal to the cell face, i.e. u x ∂ x θ, is approximated by

For the other directions we do something different

In the following, we refer to u i [0,0,0] as the value of the ith-velocity component at the center of the current cell. The values u i [0,-1,0] , u i [0,0,0] , and u i [0,1,0] indicate the ith velocity component evaluated at the center of three contiguous cells along the y-direction. Similarly, we consider u i [0,0,-1] , u i [0,0,0] , and u i [0,0,1] indicate the velocity component evaluated at the center of three contiguous cells along the z-direction.

Consider the previous example with a control surface perpendicular to the x-direction located west of the current cell center [0, 0, 0]. Based on the face velocity U x n , we identify the cell located upstream as C and located downstream as N d . We may define an index k, k = -1 for (U x n > 0) and k = 0 otherwise, such that C = [k, 0, 0].

Evaluation of the "advectee", i.e. U i

Since the required face velocity field U i

2 is staggered in time and in space with respect to the known velocity field u i n , the value U i n+ 1

2 has to be approximated. For given a cell face, two values of U i n+ 1

2 may be constructed. Following a similar reasoning to the one used on the previous section, one selects the value obtained from the upstream cell C using a BCG advection scheme.

We perform a Taylor series expansion around the upstream cell C

where s = sign(U x n ). The evolution of u i is assumed to be governed by a non-conservative form of Euler equation in order to approximate the temporal derivative in terms of the spatial derivatives of u i

where g i n contains both the pressure gradient and the external forcing term,

n both evaluated at the cell face. Since scalar quantities may not be available at this instant, as a first approximation, we assume that

This leaves the following expression [START_REF] Bizon | Plume dynamics in quasi-2d turbulent convection[END_REF] and ∂ j indicates the spatial derivatives ∂ j evaluated numerically as follows. The gradient normal to the cell face, i.e. u x ∂ x u i , is approximated by

For the other directions we do something different

where u y n+ 1 2 and u z n+ 1 2 are evaluated as follows

Evaluation of the "advector", i.e. U i

To obtain the field "advector" U i n+ 1

2 , we follow a very similar procedure up to equation (B.10d).

However, the quantities in the expression above are evaluated in a different way.

For the gradient normal to the cell face one may use a CDS scheme,

or the gradient may also be evaluated using a generalized minmod limiter (or any other kind of slope limiter for that matter), in which the gradient at the center of the neighboring cells in the x-direction may also be required. For our particular problem, we use CDS.

In contrast, the gradient normal to the cell face i.e. u x ∂ x u i , is now approximated as follows

which is slightly different from equation (B.12a).

For the other directions we do something different from equations (B.12c) and (B.12c) as well

The final step required to compute the advection term using equation (B.10d), is to ensure the "advector" to be divergence-free. This is done by projecting the provisional predicted

Appendix C

Parallel I/O model using XDMF

C.1 Presentation of the XDMF Format

The eXtensible Data Model and Format (XDMF) was developed to exchange scientific data from different high performance computing (HPC) codes (see [START_REF] Clarke | A distributed computing environment for interdisciplinary applications[END_REF][START_REF] Clarke | Enhancements to the extensible data model and format (xdmf)[END_REF]. XDMF data may be categorized by function: format data and model data. The format describes the rank and dimensions of arrays, while the model data describes what to use data for. Data may also be organized by size as light data and heavy data. Light data is stored using eXtensible Markup Language (XML) to describe the data model and the data format. Heavy data is composed of large datasets stored using the Hierarchical Data Format (HDF5). Consider the following minimal example composed of one Xdmf element, one Domain element and one Temporal collection with one Grid element per time instant.

Example C.1: Temporal collections <? xml version = " 1.0 " ? > <! DOCTYPE Xdmf SYSTEM " Xdmf . dtd " [] > < Xdmf Version = " 2.0 " > < Domain Name = " Rayleigh -Benard Cell " > < Grid Name = " Snapshots " Grid Type = " Collection " Collecti onType = " Temporal " > < Grid Name = " Time =0 " Grid Type = " Uniform " > < Time Value = " 0.00 " / > ... </ Grid > < Grid Name = " Time =5 " Grid Type = " Uniform " > < Time Value = " 5.00 " / > ... </ Grid > ... </ Grid > </ Domain > </ Xdmf > Each one of the Grid elements may contain one or more Topology, Geometry, and Attribute elements. Topology describes the general organization of the data, i.e. 2D or 3D, structured or unstructured, and so on. Geometry describes the mesh points. An Attribute element may define values associated with the mesh, such as a scalar, vector or tensor fields.

The following example, presents a domain with a structured mesh (3D) of size NX*NY*NZ. The mesh is defined by three arrays, one for each direction, all of them stored inside an external HDF5 file called "heavy data.h5". Finally, one attribute defines a scalar field called "Temperature" for each grid point, stored in the same HDF5 file. Additional fields may be added by following this template.

Example C.2: A Cartesian grid containing one scalar field ... < Grid Name = " Time =0 " Grid Type = " Uniform " > < Time Value = " 0.00 " / > < Topology Topology Type = " 3 DRectMesh " N u m b e r O f E l e m e n t s = " NZ NY NX " / > < Geometry Geometry Type = " VXVYVZ " > < DataItem Dimensions = " NX " NumberType = " Float " Precision = " 8 " Format = " HDF " > heavy_data . h5: / grid / vertex / CoordinateX </ DataItem > ... </ Geometry > < Attribute Name = " Temperature " Attribute Type = " Scalar " Center = " Point " > < DataItem Dimensions = " NZ NY NX " NumberType = " Float " Precision = " 8 " Format = " HDF " > heavy_data . h5: / snapshot / time_0 .0/ Temperature </ DataItem > </ Attribute > ... </ Grid > ...

C.2 Presentation of the HDF5 format

The second half of the XDMF format, i.e. the heavy data, is composed of relatively large datasets stored using the Hierarchical Data Format (HDF5), see [START_REF] The | Hierarchical Data Format[END_REF] As the name implies, data is organized following a hierarchical structure. HDF5 files can be read without any prior knowledge of the stored data. The list of software capable of reading HDF5 files includes Visit, Paraview, Matlab, and Tecplot, to name a few.

The type, rank, dimension and other properties of each array are stored inside the file in the form of meta-data. The following example uses a set of command-line tools to explore the contents of an existing file, Example C.3: HDF5 file contents organized in groups and subgroups.

where y bot is the reference height at the bottom plate, H(x) the Heaviside function, dV = dxdydz and dV ′ = dx ′ dy ′ dz ′ . Now consider the following definitions

We define a variable u = θ(x, y, z, t)θ(x ′ , y ′ , z ′ , t)

Finally, one combines this expression into the definition of y r (x, y, z, t)

which becomes zero if the temperature field is indeed univoque.

V θ∂ t y r dV = 0 (D.1f)

Appendix E

Demonstrations related to the vorticity functions

For a two-dimensional configuration r ∈ R 2 , vorticity has only one component perpendicular to the plane (xy) ω = ωe z . By taking the curl of the momentum equation one obtains the following dynamical equation for the vorticity ω,

The global angular impulse is defined as

From the dynamical equation of the vorticity one has the following expression

with I, II, and M given as

The first term I is transformed as follows

where ∂ j (r k r k ) ∂ j ω may be rewritten as follows

Finally, we may write I as

where ds is an infinitesimal element of the boundary ∂A, dA = dxdy, and the loop integral is counterclockwise.

The second term II is transformed through the following relation

Leaving the following expression for II II = 1 2 r k r k u n ω dsr j u j ω dxdy (E.12)

Since ω = ε li ∂ l u i , one may write

Applying the divergence theorem in two-dimensions

Considering impermeable boundaries, i.e. u n = 0, simplifies this expression to

where ds is an infinitesimal element of the boundary ∂A, dA = dxdy, and the loop integral is counterclockwise. Furthermore, imposing no-slip boundary conditions results in the same expressions from [START_REF] Molenaar | Angular momentum of forced 2D turbulence in a square no-slip domain[END_REF].

Derivation of a 2-D model with inertial corrections using the weighted residuals method

In this section we present a reduced model which takes into account inertial effects. This is done in the spirit of [START_REF] Ruyer-Quil | Inertial corrections to the darcy law in a hele-shaw cell[END_REF]) that extended the work by [START_REF] Gondret | Shear instability of two-fluid parallel flow in a hele-shaw cell[END_REF] using a weighted residuals method (WRM) (see for instance [START_REF] Finlayson | The method of weighted residuals and variational principles[END_REF].

F.1 Method of weighted residuals

Before presenting the model itself, let us present a general description of the weighted residuals method closely following [START_REF] Kalliadasis | Falling liquid films[END_REF]. Consider a problem that can be formally written as E(U) = 0, for some set of variables U. In our particular case, U corresponds to velocity and temperature fields and E to a simplified form of the nondimensionalized Boussinesq equations (7.1a)- (7.1d). We proceed to search a solution of U in the form of a series expansion

where f n (z) with n = 0, 1, • • • , N are called test functions. This set of functions forms a complete basis in the functional space S of functions of z inside the interval [-ε, ε] and that satisfy the boundary conditions of the problem. It is reminded that in our case ε = Γ z /2. In addition this space disposes of an inner product a|b = ε -ε (a b)dz.

The method of weighted residuals determines the amplitudes φ (n) and is divided in three steps. First, one introduces a series of N + 1 weight functions w m (z), m = 0, • • • , N . Second, one projects the original equation E(U) = 0 onto the N + 1 weight functions, leading to N + 1 residual problems

The terms θ (0) , p (0) , and u i (0) have identical profiles as the zeroth-order solution

but their amplitudes may be different from the zeroth-order formulation.

To define precisely each term in equations (F.17a) and (F.17c), all contributions to the transversally averaged fields u , p and θ are assumed to be concentrated in the zeroth-order terms,

In turn, this imposes gauge conditions for θ (1) , u i (1) , and p (1) θ (1) = 0 u i (1) = 0 p (1) = 0 (F.19)

Introducing equation (F.17c) into the continuity equation (F.8a) and integrating over z from -1 to z gives the following expression for the transversal velocity w w

where G 0 = (2+3z-z 3 )

2

. Since w verifies the no-slip boundary condition on z = 1 and u i

satisfies the gauge condition (F.19), the only non-trivial solution is

which implies that w is at least of order α, w = αw (1) (F.22)

Because of this, momentum equation along z (F.8c) is greatly simplified. Integrating the resulting equation between -ε and z gives an expression for the pressure field p = p (0) -αPrRa -0.5 ∂ j u j

(F.23)

where ∂ j u j | (z=-1) = 0 due to the no-slip condition.

Because of all of the above, the expanded system reads at the first-order as ) )

+ Pr(θ (0) + αθ (1) )δ iy (F.24a)

We decide to keep the diffusion terms along x and y, PrRa -0.5 ∂ j ∂ j u i (0) and PrRa -0.5 ∂ j ∂ j θ (0) , as to preserve the boundary layers on the top and bottom plates, but we neglect the same terms associated to the first order, i.e. αPrRa -0.5 ∂ j ∂ j u i (1) and αRa -0.5 ∂ j ∂ j θ (1) . For the first order terms, we keep only the second derivatives along the transversal direction, which is ε -2 larger than the previous terms. In addition, we discarded the second term in equation (F.23) which generates in equation (F.24a) a term that is ε 2 smaller than the transversal diffusion. Since we neglect these terms, the model equation reads ) )

+ Pr(ϑ (0) + αθ (1) )δ iy (F.25a)

In the following section we present a solution to the first-order formulations from equations (F.25a) and (F.25b).

F.3 Solution of the first-order formulation by the WRM

The next step of the derivation of the reduced model consists in using the resulting equations to deduce an expression for θ (1) (x, y, z, τ ) and u i (1) (x, y, z, τ ) in terms of the zerothorder terms ϑ (0) (x, y, τ ), p (0) (x, y, τ ) and a i (0) (x, y, τ ), then to obtain a system of partial differential equations for the zeroth-order terms. This can be done using one of two different approaches: a gradient expansion method and weighted residuals. Both approaches are known to give the same system of equations.

The gradient expansion approach benefits from the fact that θ (1) (resp. u i (1) ) appears only once as ∂ 2 z θ (1) (resp. ∂ 2 z u i (1) ) in equation (F.25b) (resp. (F.25a)). Integrating both equations twice with respect to z and imposing the respective gauge condition yields directly the expressions for θ (1) and u i (1) .

In the weighted residual approach we search a solution for θ (1) and u i (1) in the form of a series expansion θ (1) = N n=1 ϑ (1,n) (x, y, τ )f n (z) with f ′ n (z = ±1) = 0 (F.26a)

where ϑ (1,n) (resp. a i (1,n) ) are amplitudes to determine and f n (z) (resp. g n (z)) are called test functions which form a complete basis that satisfies the thermal (resp. mechanical) boundary conditions. The amplitudes are determined by introducing a series of weight functions w θ m (resp. w u m ) and projecting the original equations onto the weight function in order to define a series of residual problems R m = 0. The last part consists in solving the vanishing residuals R m = 0 which yields a linear equation system for the amplitudes ϑ (1,n) (resp. a i (1,n) ).

At this point one is not required to make a specific choice of test functions, save for f 0 and g 0 . Similarly, a specific choice of weight functions is not yet required.

We proceed to evaluate the inner product on (F.25b) by a series of N + 1 weight functions w u m (to be specified later) leaving the following equation system for the velocity problem α ∂ τ a i (0) w u m |g 0 + ∂ j (a j (0) a i (0) ) w u m |g 2 0 = -∂ i p (0) w u m |1 (F.27) + PrRa -0.5 ∂ 2 j a i (0) w u m |g 0 + ε -2 PrRa -0.5 a i (0) w u m |g ′′ 0 + Prϑ (0) w u m |1 δ iy + αε -2 PrRa -0.5 (0) . As such, the first-order amplitudes are pegged to the evolution and respond instantaneously to changes in these terms.

Likewise, we evaluate the inner product on equation (F.25b) by a series of N + 1 weight functions w θ m and write the following equation system for the temperature amplitudes α ∂ τ ϑ (0) w θ m |1 + ∂ j (a j (0) ϑ (0) ) w θ m |g 0 = Ra -0.5 ∂ 2 j ϑ (0) w θ m |1 (F.28)

+ αε -2 Ra -0.5

N n=1

ϑ (1,n) w θ m |f ′′ n with m = 0, • • • , N . In the two following sections we will select a set of test and weight functions, which reduces the complexity by removing the summations from equations (F.27) and (F.28).

F.3.1 Choice of test f n functions and weight functions w θ m for the temperature

In order to simplify the summations from equation (F.28), test and weight functions are selected so that the following relation is satisfied One alternative is to use test functions f m that satisfy f ′′ m |f ′′ n = c n δ mn and to use as weight functions w θ 0 = 1 and w θ m = f ′′ m for m ≥ 1. In order to find such a basis, let us consider a set F n of linearly independent functions that verify F ′ n (±1) = 0. For instance, let us consider a basis F n such that F ′ n = z n (z 2 -1). The second derivate of F n reads as F ′′ n = z n-1 ((n + 2)z 2n) for n ≥ 1. One then proceeds to apply a Gram-Schmidt orthogonalization procedure to F ′′ n to obtain an orthogonal basis over the interval [-1, 1] for n ≥ 1. The first orthogonalized function reads as

234 and the n-th orthogonalized function reads as

with proj a (b) = b|a a|a a being the projection of b on a. The resulting orthogonal basis f ′′ n (z) is integrated twice taking care to ensure that both f ′ n (±1) = 0 and 1|f n = 0 for n ≥ 1 are verified to obtain the following set of optimal test functions f n (z)

(1 -9z 2 + 15z 4 -7z 6 ) so on and so forth. In this case, each test function must satisfy g n (±1) = 0 so that expression (F.26b) verifies the no-slip boundary condition. We also require to verify the gauge condition 1|g n = 0 for n ≥ 1. One alternative is to choose the weight functions as w u m = g m (Galerkin method) and seek to define a complete basis that verifies g m |g ′′ n = d l δ mn and g n (±1) = 0. This is obtained as linear combination of Legendre polynomials in the following form [START_REF] Luke | The special functions and their approximations[END_REF][START_REF] Livermore | Galerkin orthogonal polynomials[END_REF] g n (z) = 1 4