A.4 Lin-lin plots of a 2 versus λ/δ (left) and streamwise energy spectra plotted at wall distances y + = 41, 64, 88 and 125 (from top to bottom) at Re θ = 8100. . . . . . . . . . . . . . . 178 A.5 Lin-lin plots of a 2 versus λ/δ (left) and streamwise energy spectra plotted at wall distances y + = 41, 64, 88 and 125 (from top to bottom) at Re θ = 8100. . . . . . . . . . . . . . . 179 A.6 Lin-lin plots of a 2 versus λ/δ (left) and streamwise energy spectra plotted at wall distances y + = 90, 158, 195 and 268 (from top to bottom) at Re θ = 20600. . . . . . . . . . . . . . . 180 A.7 Lin-lin plots of a 2 versus λ/δ (left) and streamwise energy spectra plotted at wall distances y + = 305, 450, 630 and 740 (from top to bottom) at Re θ = 20600. . . . . . . . . . . . . . . 181 A.8 (a) Exponents p obtained from the best power-law fit of a 2 ∼ (λ/δ) p . (b) p + q versus y + . . . . . . . . . . . . . . . . . . . . 182 B.1 Sketch of the ramp model . . . . . . . . . . . . . . . . . . . . 185 B.2 Contour plot of the streamwise mean velocity along the ramp 185 B.3 Distribution of the skin friction coefficient, C f along the ramp 186 B.4 Distribution of the viscous scaled pressure gradient parameter 186 C.1 Picture of the SPIV set-up used to characterise the corner flow upstream of the model and on the -5 • ramp. . . . . . . . . . 188 C.2 Mean streamwise velociy U (colour plot), wall-normal (V) and spanwise (W) velocities . . . . . . . . . . . . . . . . . .

First and foremost, I express my sincere gratitude to my advisers Prof. Jean-Marc Foucaut and Dr. Jean-Philippe Laval for giving me the opportunity to conduct this research and increase my understanding of turbulent flows. The guidance, support and patience they provided during the course of this work is deeply appreciated.

I am indebted to Prof. Christos Vassilicos for his ideas in developing the model described in Chapter 3 and his help in shaping this chapter. I also thank Prof. Michel Stanislas and Dr. Christophe Cuvier for their help in setting up the experiments and Dr. Christophe Cuvier for analysing the PIV fields, and with whom I learned a lot scientifically and technically.

I would also like to thank Dr. Bérengère Podvin and Dr. Denis Sipp for having accepted to be the reporters of my thesis. I also thank Dr. Geritt Elsinga and Dr. Thomas Gomez for having accepted to be part of my thesis jury.

To my collegues at LML: Ilkay, Linh, Raoul, Robin and Florian; You have provided an invaluable set of resources both inside and outside the lab. Your company has really made my time in Lille enjoyable and memorable.

Finally, I would like to thank my parents and friends and for their love and support over the course of my life and their patience over the last three years.

Without financial support, the study would not be possible. I would like to thank Centrale Lille, the Nord-pas-de-Calais region and EuHIT for the financial support that made this study possible.

7

CONTENTS

Résumé

Le but de ce travail est d'étudier une couche limite soumise à un gradient de pression et de la comparer avec une couche limite de plaque plane à grands nombres de Reynolds. Dans ce cadre, l'accent est mis sur le comportement des structures cohérentes à grande échelle. En raison de leur grande longueur, ces structures ne sont pas faciles à extraire et à caractériser en utilisant des techniques de mesure standard. Pour cette raison, des dispositifs expérimentaux spécifiques utilisant la PIV dans les plans longitudinaux et parallèles à la paroi ont été conçus pour capturer les structures à grande échelle et pour mieux comprendre les mécanismes régissant la dynamique de ces écoulements. La présente thèse est organisée en quatre parties, pour un total de cinq chapitres. Ceux ci se résument comme suit:

• La première partie donne le contexte du présent travail. Le premier chapitre contient une courte introduction motivant l'objectif de la thèse,qui est une caractérisation expérimentale de l'écoulement en vue de sa modélisation et de sa prédiction. Les écoulements avec gradient de pression adverse sont importants pour de nombreux enjeux industriels, mais restent encore mal connus.

• Le chapitre 2 commence par une brève description de la turbulence de paroi: équations moyennées de Reynolds, équations de Prandtl qui régissent une couche limite turbulente, structure d'une couche limite turbulente sans gra-10 CONTENTS dient de pression (région interne, région externe, zone de recouvrement logarithmique, effet d'un gradient de pression sur la sous-couche visqueuse, équations de transport des tensions de Reynolds et de l'énergie cinétique turbulente. Suit ensuite une description détaillée de l'état de l'art des connaissances sur l'organisation des structures cohérentes dans une couche limite sans et avec gradient de pression.. L'accent est mis sur l'organisation de l'coulement en structures cohrentes. Pour la couche limite sans gradient de pression, le modèle de Townsend-Perry s'appuyant sur des tourbillons attachés à la paroi est détaillé. Pour un gradient de pression adverse, différentes théories et scalings pour le profil de vitesse moyenne sont présentés.

• La deuxième partie (le chapitre 3) revisite les résultats obtenus sur une couche limite plaque plane en sondant l'origine d'une décroissance spectrale en k -1

x dans la couche limite turbulente. Dans cette perspective, un modèle simpliste basé sur le modèle de Townsend-Perry est proposé. On s'intéresse particulièrement au spectre longitudinal en vitesse. Ce modèle est une généralisation du modèle de Perry-Townsend. On stipule que l'intensité des stries dépend de la longueur de celles-ci suivant une loi de puissance paramétré par un coefficient p (le modèle de Perry-Townsend étant obtenu à nouveau pour p = 0). L'intérêt de ce modèle généralisé est que le spectre énergétique de la fluctuation de vitesse longitudinale se comporte maintenant comme E 11 (k x ) ∼ k -1-p x , ce qui pourrait permettre de donner un cadre théorique conceptuel pour comprendre pourquoi E 11 ne se comporte pas strictement comme k -1

x , comme cela a été observé dans de nombreuses expériences. Ce modèle peut, en principe, être appliqué à n'importe quel écoulement turbulent de paroi.

• La troisième partie (le chapitre 4) se concentre sur l'amélioration de la compréhension de la turbulence en gradient de pression adverse en effectuant une caractérisation complète d'un écoulement académique au dessus d'une géométrie. Une description du dispositif est tout d'abord donnée. Les mesures de pression le long de la rampe permettent d'établir que la couche limite turbulente est soumise sur une partie de la rampe à un gradient de pression favorable, puis à un gradient de pression adverse. Le caractère bi-dimensionnel de l'écoulement moyen (pas de variation dans la direction transverse) est établi. Une analyse des statistiques turbulente dans la zone de gradient de pression favorable est également effectuée. L'accent est mis sur les caractéristiques des structures (longueur, loi d'échelle, contribution énergétique et distribution selon la normale à la paroi) ainsi que sur l'influence du gradient de pression adverse sur les structures des grandes échelles. L'analyse permet de comparer le comportement d'une couche limite en présence de gradient de pression adverse avec le cas d'une plaque plane à grands nombres de Reynolds.

• La dernière partie (le chapitre 5) présente les conclusions de l'étude et propose des directions de recherche futures, telles que le calcul du spectre de vitesse, l'extension à des gradients de pression plus intenses, l'analyse de la structure de l'écoulement dans la direction transverse. Plusieurs annexes fournissent des informations supplémentaires sur la méthodologie utilisée dans cette analyse et des résultats statistiques supplémentaires.

CONTENTS

Nomenclature 13

CONTENTS

Latin Symbols

C 1 , C 2 , p, q, r Values of the constants in the form -C 1 + C 2 (λ/δ) Event with u ′ > 0 and v ′ > 0 ; Q 2

Event with u ′ < 0 and v ′ > 0 ; Q 3

Event with u ′ < 0 and v ′ < 0 ; Q 4

Event with u ′ > 0 and v ′ < 0 ; [START_REF] Kim | Very large-scale motion in the outer layer[END_REF] showing the process that creates VLSM . .

CHAPTER 1 Introduction

A large majority of natural flows and flows encountered in engineering applications are in fact turbulent e.g. flows of water in rivers, seas, oceans, motions of the air in the Earth's atmosphere, in water pipes, the nozzles of jet engines, etc. Most of these turbulent flows are bounded by one or more solid surfaces along which a boundary layer develops. This thin layer is responsible for the skin friction drag of bodies and thus understanding and modelling of the boundary layer can help in improving the design of transport vehicles in order to increase safety, decrease fuel consumption and green house gas emissions.

In the study of turbulence, devising methods to segregate the complex turbulent motions into simplified events called "coherent structures" has been a challenging task. This is especially so in the past two decades as in the case of canonical turbulent wall flows, where a comprehensive research based on experimental and numerical analysis has been undertaken to understand the pattern of coherent structures. Over the last several years, the observation of large-scale coherent structures in turbulent boundary layers has sparked great experimental and numerical interest, as they are known to play a significant role in the production of turbulent kinetic energy [START_REF] Ganapathisubramani | Investigation of large-scale coherence in a turbulent boundary layer using two-point correlations[END_REF]). While the occurrence and length of these structures have been documented, the relation between these large-scale structures and the energy spectrum has not yet been fully understood. One of the goals of this thesis is to gain a better understanding by bridging the gap between both. To 26 CHAPTER 1. INTRODUCTION achieve this goal we show that the scaling of the streamwise energy spectrum in a wavenumber range directly affected by the wall, are determined by wallattached eddies.

Although many studies have been conducted on zero pressure gradient (ZPG) flows (see [START_REF] Smits | High-Reynolds number wall turbulence[END_REF] for a recent review on ZPG flows), most engineering devices like airfoils, compressors, diffusers, turbine blades to name a few, encounter boundary layers exposed to pressure gradients. So fundamentally, the study of APG boundary layers may act as a catalyst for deeper understanding of real turbulent flows. Furthermore, the APG boundary layer is usually the most difficult to predict using computational fluid dynamics (CFD). Most of these applications mentioned above occur at high Reynolds numbers, significantly higher than can be examined even in large scale laboratory experiments. Direct numerical simulation (DNS) can provide highly reliable data as all the relevant turbulent length and time scales are resolved. However, due to the limitations of computational power, only low or moderate Reynolds numbers and simple geometries have been investigated by DNS so far. For this reason, alternative numerical methods like Reynolds-Averaged Navier Stokes (RANS) models and Large-Eddy Simulations (LES) have been approached. RANS models have been primarily used in industrial flows. They are based on averaged equations describing the mean flow with all the scales of the turbulence being modelled. Although fast and reliable for simple flows, RANS models perform poorly in complex turbulent separated/vortical flows and are often calibrated for geometry specific flows. When these models calibrated with low and moderate Reynolds number for specific flows are used for more complex flows, the results are inaccurate. While LES provides a possible alternative of resolving the flow structures and achieving better predictions for high Reynolds numbers, requirement of a fine mesh in the inner region of the boundary layer to resolve the near wall flow accurately, makes LES computationally expensive as well. The study of adverse pressure gradient (APG) boundary layers is therefore still a significant challenge to advanced computational schemes.

Additionally, understanding of wall turbulence especially in a decelerating situation through experiments is also quite limited due to the lack of sufficiently high Reynolds number data and large test facilities. These are necessary to reach some state of an equilibrium boundary layer where theoretical approaches can be relevant. Also, the length of these large-scale structures, up to 14 times the boundary layer thickness [START_REF] Kim | Very large-scale motion in the outer layer[END_REF], [START_REF] Hutchins | Evidence of very long meandering features in the logarithmic region of turbulent boundary layers[END_REF], [START_REF] Lee | Very-large-scale motions in a turbulent boundary layer[END_REF]) requires a large field of view and a high spatial resolution. An experimental database at high Reynolds number subjected to an APG was built, in the framework of a EuHIT (European High-Performance Infrastructures in Turbulence) project to investigate the problem. The fundamental aim of this project was to improve the understanding of turbulence under APG by performing a complete characterisation of the flow. Emphasis is laid on the characteristics of the structures (length, scaling, energetic contribution and their wall normal distribution). The analysis is also extended to compare the behaviour of APG with the ZPG case at high Reynolds numbers.

Organization of the thesis

The present thesis is organized in four parts, for a total of five chapters. Its content is summarized as follows:

The first part gives the context of the present work. Chapter 1 introduces the background of the thesis and presents the motivation and objective of the study. Chapter 2 begins with a brief description of wall turbulence. This is followed by a literature review focussing on turbulence structures in wall-bounded flows, particularly the ZPG and APG turbulent boundary layer flows.

The second part (chapter 3) is devoted to the analysis of an experiment on ZPG turbulent boundary layer flow at high Reynolds number. A simple model relating the attached flow structures and the streamwise energy spectra is first presented. The experimental facility and measurement techniques used to validate the model are described followed by a discussion of the turbulence statistics and the large-scale structures.

Part three (Chapter 4) describes the results of an experiment dedicated to the study of APG turbulent boundary layer flow. The extensive set of CHAPTER 1. INTRODUCTION experiments that were performed in the framework of a EuHIT project to improve the understanding of APG flows is described first. Then, the characterization of the flow over the APG region is covered in detail along with the influence of the adverse pressure gradient on the large-scale structures.

The last part (chapter 5) summarizes the main conclusions of the study and highlights further research opportunities to continue this investigation. Several appendices provide additional information on the methodology used in this analysis and additional statistical results.

Publications and communications

Journal Articles

• Cuvier, C., Srinath, S., Stanislas, M., Foucaut, J.-M., Laval, J.-P., Kahler, C. J., Hain, R., Scharnowski, S., Schrder, A., Geisler, R., Agocs, J., Rose, A., [START_REF] Cuvier | Extensive characterisation of a high Reynolds number decelerating boundary layer using advanced optical metrology[END_REF].

Extensive characterisation of a high reynolds number decelerating boundary layer using advanced optical metrology. Journal of Turbulence, 18(10):929-972.

• Srinath. S., Vassilicos

CHAPTER 2

Literature review

Wall turbulence

Fluid flow can be classified as laminar or turbulent. In laminar flows, the motion of fluid particles is very orderly with the fluid moving in parallel layers. Turbulent flows on the other hand lose the orderly flow pattern with sharp and irregular space and time fluctuating motion. Transition to turbulence is generated due to the development of instabilities from a laminar flow. These instabilities continue to develop when inertial forces in the flow grow larger than the viscous forces, ultimately resulting in the flow field becoming unstable. The ratio of the inertial forces to the viscous forces is known as the Reynolds number (Re = LsUs ν , where U s and L s are a chosen length and velocity scale and ν is the kinematic viscosity of the fluid).

Regarded as one of the most complex problem of applied mathematics and physics, there is perhaps not a conventional way to define turbulence. Instead of giving a standard definition, [START_REF] Tennekes | A first course in turbulence[END_REF] listed out the main characteristics of turbulent flows :

• Non-linear: turbulence arises from the coupling between the viscous terms and the non-linear inertial terms in the Navier-Stokes equations.

• Chaotic: a small perturbation introduced at any point of the field can affect the entire flow making turbulent flows difficult to predict.

32

CHAPTER 2. LITERATURE REVIEW

• Rotational and three-dimensional: turbulence is characterised by high levels of fluctuating vorticity that are maintained through three-dimensional vortex stretching.

• Diffusive: turbulence causes rapid mixing and increased rates of momentum, heat and mass transfer along but also across mean streamlines.

• Dissipative: the kinetic energy gets converted into heat due to viscous shear stresses. This process is enhanced for turbulent flows because of the higher deformation rate.

• Multiscale: a wide range of length scales exists in turbulent flows ranging from the dimensions of the flow field to the action of molecular viscosity (Kolmogorov scales)

According to Hinze (1938), turbulence can be categorised into two types: (a) wall turbulence where flows remain "attached" to the surface and evolve entirely under the influence of it and (b) free shear flows where mean velocity gradients develop in the absence of boundaries. In this thesis, we focus on wall-bounded flows i.e the ZPG and APG turbulent boundary layer flows.

Reynolds Average Navier Stokes Equations

The Navier-Stokes equations (equation 2.1) govern the flow of incompressible fluids. Throughout this dissertation, we assume the flow to be incompressible and isothermal so that the fluid properties such as the density, ρ and dynamic viscosity, µ in the Navier-Stokes equation are not affected by temperature.

             ∂ ũi ∂x i = 0. ρ( ∂ ũi ∂t + ũj ∂ ũi ∂x j ) = - ∂ p ∂x i + µ ∂ 2 ũi ∂x 2 j . (2.1)
where ũi is i-th component of the instantaneous fluid velocity at the position (x 1 , x 2 , x 3 ), t is the time and p is the instantaneous pressure field.

Along this document, x 1 or just x denotes the streamwise, x 2 or y the wallnormal and x 3 or z the spanwise directions repectively. The equations are written using the Cartesian tensor notation and the Einstein summation convention. Also, u 1 or u, u 2 or v and u 3 or w can be used in reference to the streamwise, wall-normal and spanwise velocities respectively.

Through Reynolds decomposition, the instantaneous flow quantities can be expressed by the sum of their mean (U i , P ) and fluctuating parts (u i , p) as shown in equation 2.2.

   ũi = U i + u i . p = P + p i .
(2.2)

The Reynolds decomposition can be then introduced in the Navier-Stokes equation (equation 2.1) to obtain the basic equations for the averaged turbulent flow known as the Reynolds Averaged Navier Stokes equations (RANS) as shown in equation 2.3.

       ∂U i ∂x i = 0 ρ( ∂U i ∂t + U j ∂U i ∂x j ) = - ∂P ∂x i + µ ∂ 2 U i ∂x 2 j -ρ ∂u i u j ∂x j (2.3)
where u i u j is the Reynolds stress tensor. It is a symmetric tensor, the trace of which is equal to twice the turbulent kinetic energy, k.

k = 1 2 u ′ i u ′ i (2.4)

Turbulent boundary layer equation

The conservation equations can be reduced to simpler forms by examining the relative magnitude of the terms in the equations. The respective length and velocity scales are used to evaluate the order of magnitude of the various terms in the equations. Considering a 2D flow over a wall at high Reynolds number, it is common to introduce two length scales L and δ which are the parallel and normal length scales to the wall. Near the wall, as the length CHAPTER 2. LITERATURE REVIEW scale δ is much smaller than the longitudinal scale (δ << L), equation 2.3 reduces to the boundary layer equations (2.5).

                           ∂U ∂x + ∂V ∂y = 0. ρ(U ∂U ∂x + V ∂U ∂y ) = - ∂P ∂x + µ ∂ 2 U ∂y 2 -ρ ∂uv ∂y . ∂P ∂y + ρ ∂v 2 ∂y = 0.
(2.5)

ZPG turbulent boundary layer

In canonical wall-bounded flows (i.e. two-dimensional, smooth-wall, ZPG turbulent boundary layer and fully developed pipe and channel flows) at sufficiently high Reynolds number, two overlaping regions with different physical processes can be distinguished i.e. inner and outer regions [START_REF] Panton | Overview of the self-sustaining mechanisms of wall turbulence[END_REF]). For a ZPG turbulent boundary layer flow, the boundary layer equations are obtained by taking ∂P ∂x = 0 in equations 2.5. Conventionally, the turbulent flow can be characterized with two regions; An inner region and an outer region:

Inner region: This region extends from the wall until a height of about 0.1δ [START_REF] George | Lectures in turbulence for the 21st century[END_REF]; [START_REF] Pope | Turbulent flows[END_REF]) with δ being the boundary layer thickness. The flow is affected by the wall shear stress and in the case of a smooth wall, by the fluid viscosity only. Here, the convective terms are negligible compared to the viscous term and the turbulent shear stress. Thus for a ZPG turbulent boundary layer, equation 2.5 can be simplified to

µ ∂ 2 U ∂y 2 -ρ ∂uv ∂y = ∂ ∂y (µ ∂U ∂y -ρuv) = ∂τ ∂y = 0 (2.6)
Integrating equation 2.6 and applying the boundary condition at the wall, the equation for the inner region (equation 2.7) can be obtained with U τ the friction velocity given by U τ = τ y=0 ρ .

τ = µ ∂U ∂y -ρuv = τ y=0 = ρU 2 τ (2.7)
From equation 2.7, the streamwise mean velocity, U can depend only on y, U τ , ρ and µ. Dimensional analysis leads to a scaling of the mean velocity profile in the form

U + = f (y + ) (2.8)
where

U + = U Uτ and y + = yUτ ν
The inner region is generally thought to comprise of three layers:

• At y + ≤ 5, viscosity dominates the flow and thus ρuv is negligible. This layer is called the viscous sublayer or linear sublayer as the velocity profile exhibits a linear relation between U + and y + (U + = y + ).

• Above the linear sublayer, the two stresses are of comparible magnitude. This region is called the buffer layer and extends up to y + ≤ 30

• Beyond y + > 30 and up to y ≤ 0.1δ, the turbulent shear stress becomes dominant compared to the viscous stress.

Outer region: The region beyond y > 0.1δ is called the outer region and here, the streamwise velocity profile can be considered as independent of viscosity. The characteristic length and velocity scales are δ and U τ respectively. In this region, a velocity defect law (equation 2.9) is generally used to describe the mean velocity profile.

U e -U U τ = f y δ (2.9)
The 'wake law' proposed by [START_REF] Coles | The law of the wake in the turbulent boundary layer[END_REF] is generally used to describe this region. Recently, [START_REF] George | Recent advancements toward the understanding of turbulent boundary layers[END_REF] and [START_REF] George | Is there a universal log law for turbulent wall-bounded flows?[END_REF] questioned the validity of the outer region velocity defect law (equation 2.9) for boundary layer flows and proposed an alternative form where the velocity deficit is scaled by the freestream velocity U e rather than U τ . They argued that this alternative form is a similarity solution of the Reynolds-averaged equations giving rise to a power law equation of the overlap region. However, [START_REF] George | Recent advancements toward the understanding of turbulent boundary layers[END_REF] CHAPTER 2. LITERATURE REVIEW and George (2007) admit that the logarithmic law provides a good empirical description of the overlap region indistinguishable from the power-law solution.

Overlap region: At sufficiently high Reynolds number, an overlap region exists where the scaling laws for both the inner and outer regions simultaneously hold. It is often called the 'logarithmic region' and is reputed to be valid not only for boundary layers, but also for pipe and channel flows. Matching equations 2.8 and 2.9, [START_REF] Millikan | A critical discussion of turbulent flow in channels and circular tubes[END_REF] obtained an expression to describe the mean velocity profile. Also referred to as the log-law, in inner variables it is given by

U + = 1 κ ln y + + C (2.10)
where κ is the Von Karman constant and C is an additive constant. For ZPG boundary layers, the generally accepted values for these constants are κ = 0.41 and C = 5 [START_REF] Cuvier | Characterisation of a high Reynolds number boundary layer subject to pressure gradient and separation[END_REF]; [START_REF] George | Lectures in turbulence for the 21st century[END_REF]). Conventionally this overlap region is considered to be equivalent to an inertial sublayer, indicating that neither viscous nor energetic scales are relevant. [START_REF] George | Recent advancements toward the understanding of turbulent boundary layers[END_REF] introduced a region called the 'mesolayer' in the inner part of the overlap region (30 ≤ y + ≤ 300) and described it as the region where the viscous stresses are negligible, but in which viscosity acts directly on the turbulence scales producing the Reynolds stresses.

In summary, a typical mean streamwise velocity profile showing the different regions of a turbulent boundary layer is shown in figure 2.1.

Turbulent boundary layer subjected to a pressure gradient

For turbulent boundary layers where the pressure gradient, i.e. the first term on the right hand side of equation 2.5 is non-zero and positive, the flow is said to be subjected to an APG, which can lead to separation of the boundary layer from the surface. Flow separation in many applications is generally undesirable. An APG causes the boundary layer thickness, δ to increase in the streamwise direction. In addition to the boundary layer, other quantities [START_REF] George | Zero-pressure-gradient turbulent boundary layer[END_REF] are defined to characterise the boundary layer. The displacement thickness δ * (equation 2.11), the momentum thickness θ (equation 2.12) and the shape factor H (equation 2.13) generally increase while the skin friction coefficient C f (equation 2.14) generally decreases. Neglecting the turbulent shear stress near the wall and integrating equation, the viscous sublayer equation (2.17) is obtained.

δ * = δ 0 1 - U (y) U e dy (2.11) θ = δ 0 U (y) U e 1 - U (
U + = 1 2 ∂P + ∂x y +2 + y + (2.17)
where

∂P + ∂x = ν ρU 3 τ ∂P ∂x (2.18)
Equation 2.17 shows that the effect of the pressure gradient on the mean velocity profile starts very close to the wall. If the term involving the pressure gradient is negligibly small compared to the other terms, the equation reduces to the equation governing the inner part of a ZPG layer, i.e. when ∂P + ∂x y +2 ≪ 1 in equation 2.17, U + = y + can be used to describe the viscous sublayer. However, for strong APG flows at finite Reynolds, this term cannot be neglected.

The scaling of the mean velocity will be presented in section 2.3.2.

Reynolds shear stress transport

The transport equation for the Reynolds shear stress u i u j can be obtained by manipulating the Navier-Stokes equations. By decomposing the momentum equation for the total velocity and subtracting the equation of the mean, a transport equation for the momentum of the fluctuations is obtained. This is then multiplied by fluctuating velocity components, u j . The subscripts i and j are then exchanged. These last two equations are added together and the result is then averaged and simplified to obtain the transport equation for the Reynolds stress tensor given by equation 2.19 where

τ ij = µ ∂u i ∂x j + ∂u j ∂x i ρ ∂u i u j ∂t 1 + ρU k ∂u i u j ∂x k 2 = -ρu i u k ∂U j ∂x k -ρu j u k ∂U i ∂x k 3 + ∂(u i τ jk e + u j τ ik ) ∂x k 4 - ∂(u i pδ jk + u j pδ ik ) ∂x k 5 -ρ ∂u i u j u k ∂x k 6 + p ∂u i ∂x j + p ∂u j ∂x i 7 -τ ik ∂u j ∂x k -τ jk ∂u i ∂x k 8 (2.19)
The terms under the curly braces will be described in the next sub-section.

Turbulent kinetic energy transport

The transport equation for the turbulent kinetic energy is obtained from the Reynolds stress equation (2.19) by contracting the free indices (i.e by taking j = i).

ρ ∂k ∂t 1 + ρU k ∂k ∂x k 2 = -ρu i u k ∂U i ∂x k 3 + ∂u i τ ik ∂x k 4 - ∂u k p ∂x k 5 - 1 2 ρ ∂u i u i u k ∂x k 6 -τ ik ∂u i ∂x k 8 (2.20)
Both equations 2.19 and 2.20 can be explored for understanding the dynamics of turbulent motion. Each numbered term in equations 2.19 and 2.20 has a distinct role in the overall energy balance as described in [START_REF] George | Lectures in turbulence for the 21st century[END_REF].

Term (1) is the rate of change of kinetic energy per unit mass due to nonstationarity. Term (2) is the rate of change of kinetic energy per unit mass due to convection by the mean flow. Term (3) is the production term that takes kinetic energy from the mean flow to feed turbulence. Terms (4), ( 5) and (6) correspond to the viscous diffusion, diffusion by pressure fluctuations and CHAPTER 2. LITERATURE REVIEW diffusion by velocity fluctuations respectively. They transport the turbulent kinetic energy from one place to another. Term (7) in the Reynolds shear stress equation is the pressure-strain rate term. It redistributes energy among normal stresses and makes them more isotropic. Term (8) is the rate of dissipation of turbulence kinetic energy due to viscous stresses.

2.2 Zero pressure gradient turbulent boundary layer

Organisation of coherent structures

Wall turbulence is characterized by the presence of eddy structures or coherent motions over a wide range of scales (Adrian et al. (2000)). [START_REF] Robinson | Coherent motions in the turbulent boundary layer[END_REF] defined coherent motions as three dimensional region(s) of the flow over which at least one fundamental flow variable exhibits significant correlation with itself or with another flow variable over a range of space and/or time that is significantly larger than the smallest local scales of the flow. The presence of coherent structures in turbulent flow has been evidenced statistically in the non-zero Reynolds shear stress and two point space-time correlations and they are believed to play a role in the production and dissipation of turbulence in boundary layers. The importance of coherent structures was initially highlighted by [START_REF] Theodorsen | Mechanism of turbulence[END_REF]; [START_REF] Townsend | The properties of equilibrium boundary layers[END_REF]; [START_REF] Kline | The structure of turbulent boundary layers[END_REF]. Efforts have since been taken by various research groups to characterize and classify them using flow visualization and measurement techniques. In the following, a literature review of coherent structures is presented in two separate frames. First, the near wall coherent structures in the inner region that have been characterized extensively are presented. This is followed by a review of the large and very large-scale motions that have been found to populate the logarithmic region (refer 2.1.2.1) at high Reynolds numbers.

Near wall coherent structures

The organisation of turbulence in the near wall region has been studied extensively using dedicated experiments and more recently by DNS, through which full 3D time resolved information are available. The near wall cycle of turbulence can be explained in terms of well identified coherent structures that are listed below.

• Low-speed and high-speed streaks: They appear as alternating low and high momentum regions generally observed in the near wall region (0 < y + < 40) of the boundary layer. They were first studied from hydrogen bubble visualizations [START_REF] Kline | The structure of turbulent boundary layers[END_REF]). Within the sublayer, low speed streaks tend to be longer than their high-speed counterparts and this difference is more pronounced at high Reynolds numbers [START_REF] Robinson | Coherent motions in the turbulent boundary layer[END_REF]).

Apart from their length that varies widely, there has been an agreement on their characteristics [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF]). Low speed streaks appear below y + < 10 and are between 500 + and 2000 + long, 20 + and 40 + wide and up to 50 + high. The streaks interact with the outer portions of the flow through a process of gradual lift-up, followed by a sudden oscillation, bursting and ejection [START_REF] Kline | The role of visualization in the study of the structure of the turbulent boundary layer[END_REF]).

• Ejections and sweeps: [START_REF] Wallace | The wall region in turbulent shear flow[END_REF] defined the Reynolds shear stress events in terms of quadrant motions according to the signs of the streamwise and wall-normal velocity fluctuations. The second quadrant (Q 2 ) events (u < 0, v > 0) were termed "ejections" while the fourth quadrant (Q 4 ) events (u > 0, v < 0) were called "sweeps". The product of the Reynolds shear stress is negative in the second and fourth quadrants and thus these events produce turbulent energy (recall equation). From hotwire anemometry and conditional averaging, Wallace showed that sweeps contribute more to the production of turbulent kinetic energy compared to ejections in the region (y + < 15) and less as we move away from the wall y + > 15.

• Vortices [START_REF] Robinson | A review of quasicoherent structures in a numerically simulated turbulent boundary layer[END_REF] defined a vortex as a coherent structure that exhibits circular or spiral instantaneous streamlines in a plane normal to its core when viewed in a reference frame moving with the centre of the vortex core. It is shown by several researchers that the near-wall region of turbulent flows is populated with quasi-streamwise vortices (also called rolls) which are associated with ejections and sweeps (refer figure 2.2). Using conceptual ideas based on the vorticity transport equations, [START_REF] Theodorsen | Mechanism of turbulence[END_REF] postulated that boundary layers are populated by horseshoe structures originating from the stretching of perturbed spanwise vorticity. This structure is often described as having two counter-rotating streamwise oriented legs connected to a spanwise oriented head through a neck and straddling low speed fluid with their heads inclined at 45 • to the wall. In agreement with [START_REF] Theodorsen | Mechanism of turbulence[END_REF], [START_REF] Robinson | A review of quasicoherent structures in a numerically simulated turbulent boundary layer[END_REF] using DNS data showed that the wall layers contain counter-rotating vortices oriented and tilted downstream and Robinson suggested that these vortices create and lift the streaks in the sublayer by creating cross flows near the

ZERO PRESSURE GRADIENT TURBULENT BOUNDARY LAYER43

wall. The shape and size of hairpins near the wall strongly depend on the Reynolds number [START_REF] Head | New aspects of turbulent boundary-layer structure[END_REF]; [START_REF] Smith | A synthesized model of the near-wall behavior in turbulent boundary layers[END_REF]; [START_REF] Zhou | Formation of coherent hairpin packets in wall turbulence[END_REF][START_REF] Zhou | Mechanisms for generating coherent packets of hairpin vortices in channel flow[END_REF]; Adrian et al. (2000)).

A model of the organization of coherent structures in the inner region was built by [START_REF] Lin | Etude détaillée des structures cohérentes de la zone tampon de la turbulence de paroi à l'aide de données de PIV stéréoscopique[END_REF] from spatial correlations analyses and structure parameters (size, orientation, etc.). A recent review of near wall turbulence can be found in [START_REF] Stanislas | Near wall turbulence: an experimental view[END_REF]. These studies could be referred to get a current picture of the near-wall organization in ZPG flows.

Large and very large-scale motions

Large scale coherent motions in the inner and outer regions have been observed in boundary layer, pipe and channel flows. Usually categorised into large-scale motions (LSM) and very large-scale motions (VLSM), they are generally referred to elongated regions of velocity fluctuations having a streamwise extent of 1 ∼ 3δ for LSM and larger than 3δ for VLSM [START_REF] Kim | Very large-scale motion in the outer layer[END_REF]; [START_REF] Guala | Large-scale and verylarge-scale motions in turbulent pipe flow[END_REF]; Balakumar and Adrian (2007)) and exist in the logarithmic and lower wake regions of a turbulent boundary layer [START_REF] Hutchins | Evidence of very long meandering features in the logarithmic region of turbulent boundary layers[END_REF]; Dennis and Nickels (2011a); [START_REF] Lee | Very-large-scale motions in a turbulent boundary layer[END_REF]). Much focus has been laid on these LSM and VLSM as they contribute to a significant amount of turbulent kinetic energy and Reynolds shear stress in turbulent flows [START_REF] Ganapathisubramani | Characteristics of vortex packets in turbulent boundary layers[END_REF]; [START_REF] Jiménez | The large-scale dynamics of near-wall turbulence[END_REF]; [START_REF] Wu | A study of energetic large-scale structures in turbulent boundary layer[END_REF]). The study and organization of these structures is thus imperative for understanding the physics of wall turbulence and for the development of reliable turbulence models.

Early studies on the outer intermittent region of a turbulent boundary layer with zero pressure gradient by [START_REF] Kovasznay | Large-scale motion in the intermittent region of a turbulent boundary layer[END_REF] was carried out to throw some light on the shape and motion of the interface separating the turbulent and non-turbulent regions as well as on nature of the LSM existing in the turbulent regime. Implementing conditional averaging and space-time correlations showed LSM in the form of turbulent bulges (see figure 2.3), that they concluded to be three dimensional and elongated in the streamwise (2000). U c1 , U c2 and U c3 are the convection velocities of each hairpin packet respectively. direction, having a streamwise characteristic length of δ. From time scales of wall shear velocity correlations, [START_REF] Brown | Large structure in a turbulent boundary layer[END_REF] found LSM of the order of 2δ related to the bursting phenomenon near the wall. They found these structures to be inclined at 18 • similar to [START_REF] Falco | Coherent motions in the outer region of turbulent boundary layers[END_REF].

With the improvement in measuring techniques, Adrian et al. (2000) used

ZERO PRESSURE GRADIENT TURBULENT BOUNDARY LAYER45

Particle Image Velocimetry (PIV) in a streamwise-wall-normal plane of the near wall region of a ZPG turbulent boundary layer at three Reynolds numbers in the range 930 ≤ Re θ ≤ 6845. They attributed the origins of LSM to the streamwise alignment of hairpin vortices to form packets moving with the same convection velocity and inducing a low speed zone beetween their legs (see figure 2.4). These packets grow upwards in the streamwise direction at a mean angle of approximately 12 • contributing to about 25% of -uv, even though they occupy less than 4% of the total area. [START_REF] Ganapathisubramani | Characteristics of vortex packets in turbulent boundary layers[END_REF] observed that their streamwise coherence breaks down beyond the logarithmic layer. PIV measurements in the streamwise-spanwise plane at a Reynolds numbers in the range 1015 ≤ Re θ ≤ 7705 by [START_REF] Tomkins | Spanwise structure and scale growth in turbulent boundary layers[END_REF] in a ZPG turbulent boundary layer revealed that long low momentum regions with a streamwise length of over 2δ and a typical width of 0.3δ to 0.5δ are the dominant LSM in the upper buffer layer and in the logarithmic region. They also suggested that the streamwise separation between packets increases with the wall normal direction and ranges in 200 + -250 + at Re θ = 7750 and the spanwise spacing of counter rotating haipin legs ranges in 100 + -120 + at y + = 100 and y + = 220 Although the origin of VLSM is not certain, [START_REF] Kim | Very large-scale motion in the outer layer[END_REF] from hot-film measurements in pipe flows suggested that hairpin packets align coherently to form these VLSM. The pre-multiplied spectra k x E 11 of the streamwise velocity fluctuations showed a bimodal behaviour and the maximum associated with the low wavenumber mode was interpreted to be the VLSM extending up to 14 -16R (R is the pipe radius) in the outer layer of the fully developed turbulent pipe flow at different Reynolds numbers. They termed the structures VLSM (such as individual packets of hairpins and bulges) to distinguish them from LSM that have a typical scale of the order of δ. They suggest that these large structures may be a concatenation of hairpin packets (see figure 2.5). VLSM 20h long (h is the channel halfwidth) have also been observed in channel flows by del [START_REF] De Jesus | Large eddy simulations and RANS computations of adverse pressure gradient flows[END_REF][START_REF] Del Álamo | Scaling of the energy spectra of turbulent channels[END_REF]; Balakumar and Adrian (2007); [START_REF] Monty | Large-scale features in turbulent pipe and channel flows[END_REF]. In boundary layers, [START_REF] Hutchins | Evidence of very long meandering features in the logarithmic region of turbulent boundary layers[END_REF] used hot-wire rake measurements of an atmospheric surface layer at Re τ ≃ 6.6 × 10 5 and found Figure 2.5: Conceptual model of [START_REF] Kim | Very large-scale motion in the outer layer[END_REF] showing the process that creates VLSM. The hairpins align coherently in groups to form long packets and packets align coherently to form VLSM. very long meandering structures extending up to 20δ that they term superstructures, populating the log layer. However, when viewed from single point statistics, the meandering tendency masks the true length of these structures resulting in shorter length scales up to 6δ. Performing high-speed stereo PIV at Re θ = 4700, Dennis and Nickels (2011b) noticed structures 7δ long. In general, VLSM in pipe and channel flows are longer than the corresponding scales in boundary layer flows (Balakumar and Adrian (2007)) and in channel flows they grow with distance from the wall. By comparison in the log region, the structures were 1.6 times wider in the pipe and channel compared to those found in the boundary layer [START_REF] Monty | Large-scale features in turbulent pipe and channel flows[END_REF]). Furthermore, the low-speed regions were observed to be longer than their high-speed counterparts [START_REF] Dennis | Experimental measurement of large-scale three-dimensional structures in a turbulent boundary layer. Part 2. Long structures[END_REF]) and in channels, the low-speed regions were longer compared to pipe and boundary layer flows [START_REF] Sillero | Two-point statistics for turbulent boundary layers and channels at Reynolds numbers up to δ + ≈ 2000[END_REF]).

VSLM carry substantial portions of the Reynolds shear stress (30-50%)

ZERO PRESSURE GRADIENT TURBULENT BOUNDARY LAYER47

and kinetic energy (40-65%) in channel, pipe and boundary layer flows [START_REF] Jiménez | The large-scale dynamics of near-wall turbulence[END_REF]; [START_REF] Guala | Large-scale and verylarge-scale motions in turbulent pipe flow[END_REF]; Balakumar and Adrian (2007); [START_REF] Lee | Comparison of very-large-scale motions of turbulent pipe and boundary layer simulations[END_REF]) making them vital while modelling turbulence. Additionally, the VSLM maintain a footprint in the near-wall region seeming to modulate the small scales [START_REF] Hutchins | Evidence of very long meandering features in the logarithmic region of turbulent boundary layers[END_REF]; [START_REF] Mathis | Large-scale amplitude modulation of the small-scale structures in turbulent boundary layers[END_REF]).

These LSM and VSLM have been interpreted to account for the long correlation tails and responsible for the k -1

x scaling range in the turbulence spectrum [START_REF] Smits | High-Reynolds number wall turbulence[END_REF]) and thought to be the attached eddies discussed by [START_REF] Townsend | The structure of turbulent shear flow[END_REF], the subject of which will be discussed in the following section.

An overview of the Townsend-Perry theory

The traditional scaling of the components of the Reynolds shear stress tensor (u i u j ) in wall bounded flows was developed by [START_REF] Townsend | The structure of turbulent shear flow[END_REF]. This was, in part, inspired by an earlier work [START_REF] Townsend | The structure of the turbulent boundary layer[END_REF]) from hot-wire measurements in a turbulent boundary layer. In particular, he examined the terms in the equation for kinetic energy of the velocity fluctuations and noted that there was a surprisingly strong flow of turbulent energy towards the wall. He postulated that turbulence consists mostly of superimposed eddies and the bulk of the energy-containing eddies are, in a sense, attached to the wall, and that the dependence of scale with distance from the wall is not a local effect but due to this attachment of most of the eddies. In this paper he also suggested a possible form of these eddies where these eddies may be thought of as velocity fields of some representative vortex structures. Thus, in the sense of [START_REF] Townsend | The structure of turbulent shear flow[END_REF], any eddy with a size that scales with its distance from the wall may be considered attached to the wall. Eddies farther from the wall are larger in size and hence their velocity fields still extend to the wall. These eddies form the basis of the attached eddy hypothesis. The hypothesis itself is that the main energy containing motion of a turbulent wall bounded flow may be described by a random superposition of such eddies of different sizes, but with a similar velocity distributions. These eddies are considered as statistically representative structures in that their geometry and strength CHAPTER 2. LITERATURE REVIEW are derived from an ensemble average of many different structures of similar scale. [START_REF] Townsend | The structure of the turbulent boundary layer[END_REF] then went on to form expressions for the contributions of a random superposition of attached eddies of different sizes to the correlation fuctions and derived the distribution of eddy sizes with wall distance necessary to produce invariance of the Reynolds shear stress (-uv/U 2 τ ≈ 1) with distance from the wall as observed in the equilibrium layer. This analysis effectively leads to a population density of eddies that varies inversely with size and hence with distance from the wall. That is, the number of eddies of size y per unit wall area is A/y, where A is a constant. These distributions of eddies also lead to predictions for the variation of the other components of the Reynolds shear stress tensor:

u 2 /U 2 τ = B 1 + A 1 ln(δ/y) v 2 /U 2 τ = B 3 , -uv/U τ 2 = 1 w 2 /U 2 τ = B 2 + A 2 ln(δ/y) (2.21)
where the terms A 1 , A 2 , B 1 , B 2 and B 3 are constants which depend on the shape and details of the representative eddy.

In the past forty years, the turbulence spectrum of velocity fluctuations in wall turbulence has received considerable attention as it gives valuable insight into the behaviour of wall-bounded flows by indicating the distribution of energy across scales. Spectral scaling laws built on ideas initiated by [START_REF] Townsend | The structure of turbulent shear flow[END_REF], in particular the attached eddy hypothesis, have seen consistent development over the years [START_REF] Perry | On the mechanism of wall turbulence[END_REF]; [START_REF] Perry | A theoretical and experimental study of wall turbulence[END_REF]; [START_REF] Perry | Experimental support for the attached-eddy hypothesis in zero-pressure-gradient turbulent boundary layers[END_REF]; [START_REF] Marusic | Similarity law for the streamwise turbulence intensity in zero-pressure-gradient turbulent boundary layers[END_REF]; [START_REF] Marusic | Streamwise turbulence intensity formulation for flat-plate boundary layers[END_REF]). [START_REF] Perry | Asymptotic similarity of turbulence structures in smooth-and rough-walled pipes[END_REF], [START_REF] Perry | On the mechanism of wall turbulence[END_REF] and [START_REF] Perry | A theoretical and experimental study of wall turbulence[END_REF] showed how Townsend's attached eddy hypothesis implies that the energy spectrum E 11 (k x ) of the turbulent streamwise fluctuating velocity at a distance y from the wall scales as

E 11 (k x ) ∼ U 2 τ k -1
x in the range 1/δ ≪ k x ≪ 1/y where U τ is the friction velocity and δ is the boundary layer thickness. [START_REF] Perry | On the mechanism of wall turbulence[END_REF] also made the connection between the energy spectra and the stresses and pointed out that a k -1

x spectrum integrates to give a logarithmic dependence of the streamwise stress on the wall-normal position as given in equation 2.21. [START_REF] Nickels | Evidence of the k -1 1 law in a high-Reynolds-number turbulent boundary layer[END_REF] stressed the use of overlap arguments to deduce the -1 power law behaviour. That is, a k -1

x region in the spectra would exist where the inner scaling (based on y and U τ ) and outer scaling (based on δ and U τ ) are simultaneously valid over the same wavenumber range. [START_REF] Nickels | Some predictions of the attached eddy model for a high Reynolds number boundary layer[END_REF] stated that it is necessary to take measurements surprisingly close to the wall to observe a k -1

x behaviour and thought this was the reason why [START_REF] Morrison | Scaling of the streamwise velocity component in turbulent pipe flow[END_REF] and [START_REF] Mckeon | Asymptotic scaling in turbulent pipe flow[END_REF] did not observe any -1 region in their spectra as their measurements were not close enough to the wall. However, recent experiments by [START_REF] Vallikivi | Spectral scaling in boundary layers and pipes at very high Reynolds numbers[END_REF] do not show an overlap region and these authors infer that the k -1 x region cannot be expected even at very high Reynolds numbers.

Chapter 3 looks at the basis for the k -1

x range in flat plate turbulent boundary layers from a new perspective. Using Particle Image Velocimetry (PIV) and a simple model which can in principle be applied to various wallbounded turbulent flows, it will be shown how in the turbulent boundary layer, a power-law spectral range exists but is not a Townsend-Perry k -1

x range and how it can be accounted for by taking only streamwise lengths and intensities of wall-attached structures into account.

APG turbulent boundary layers 2.3.1 Equilibrium boundary layers

The challenge in attempting to study turbulent flows has been attributed to the complex, nonlinear or multiscale phenomenon of the flow. To reduce the complexity of the problem, [START_REF] Clauser | Turbulent boundary layers in adverse pressure gradients[END_REF] introduced the idea of an "equilibrium turbulent layer". In general terms, an equilibrium layer is one in which non-dimensional parameters such as H, C f etc., vary only slowly with distance from the origin. The self-preserving character of the outer regions of such layers then makes it possible to adopt plausible assumptions that would greatly simplify the analysis.

CHAPTER 2. LITERATURE REVIEW [START_REF] Clauser | Turbulent boundary layers in adverse pressure gradients[END_REF] defined a non-dimensional pressure gradient parameter, β = δ * (dP/dx)/τ w . For equilibrium boundary layers, β would then be a constant and Clauser expected such equilibrium layers to be dynamically similar at all stations in both the mean and fluctuating velocity fields. Working on the ZPG which is a boundary layer in the equilibrium sense of Clauser, where dP/dx = 0 and consequently β = 0, he showed that the mean velocity defect (or deficit) law given by equation 2.22 made the turbulent velocity data to fall together in a very satisfactory way.

u -U U τ = f y δ (2.22)
He introduced a form parameter, G given in equation 2.23 to give a measure of the velocity defect defined by

G = ∞ 0 U -u Uτ 2 dy ∞ 0 U -u Uτ dy (2.23)
which is related to the shape factor, H = δ * /θ by equation 2.24

G = H-1 H C f 2 1/2 (2.24)
Now, if G is to remain constant with downstream distance, the pressure force acting on the boundary layer must remain in a constant ratio to the skinfriction force, i.e., the parameter β must remain constant. Clauser reasoned that a velocity defect law of the same form should apply to equilibrium layers in pressure gradients for non zero constant values of β however concluding that equilibrium flows are hard to maintain and that most flows are nonequilibrium flows.

Since [START_REF] Clauser | Turbulent boundary layers in adverse pressure gradients[END_REF], there have been many studies that have extended his analysis and proposed new definitions for equilibrium turbulent boundary layers while improving certain forthcomings of his theory for equilibrium layers for flows near separation where U τ approaches zero which consequently made β approach infinity. Analysing Clauser's flow, [START_REF] Townsend | Equilibrium layers and wall turbulence[END_REF] developed a self-preserving flow such that the distribution of mean velocity was

U = U e -u 0 f y l 0 (2.25)
and the Reynolds stress was given by uv = u 2 0 g y l 0

(2.26) l 0 and u 0 were the length and velocity scales depending only on the downstream distance and the functions f and g were characteristics of the whole flow. According to Townsend, a turbulent flow is self-preserving if the motions at different sections of the flow differ only in scales of velocity and length. He then substituted these distributions in the usual boundary layer equation for the two-dimensional mean flow that gave an ordinary differential equation. He then concluded that the only possibility for this equation to be compatible with the scales of the equilibrium layer (l 0 and u 0 ) if

U e ∝ (x) -1 , l 0 ∝ (x)
(2.27)

U e ∝ (x) -0.23 , l 0 ∝ (x) (2.28)
The first possibility shown in equation 2.27 is an accelerating flow in a converging wedge while the second shown in equation 2.28 is a zero stress flow in an adverse pressure gradient. Thus [START_REF] Townsend | Equilibrium layers and wall turbulence[END_REF] showed that self-similarity for equilibrium boundary layers in pressure gradients is obtained if the freestream velocity variation has the form of a power law in the downstream direction, U e ∼ x m ('m' is an exponent describing the distribution of the freestream velocity) and the length scale varies linearly with the downstream co-ordinate (refer equation 2.28).

Extending the work of [START_REF] Clauser | Turbulent boundary layers in adverse pressure gradients[END_REF] and [START_REF] Townsend | Equilibrium layers and wall turbulence[END_REF], [START_REF] Mellor | Equilibrium turbulent boundary layers[END_REF] hypothesized an effective or eddy viscosity function presumed to be independent of pressure gradient and valid for the entire boundary layer. [START_REF] Clauser | Turbulent boundary layers in adverse pressure gradients[END_REF] discovered that in the outer 80% of the boundary layer, the CHAPTER 2. LITERATURE REVIEW eddy viscosity, ν e could be considered constant with respect to the coordinate perpendicular to the wall. Thus following [START_REF] Clauser | Turbulent boundary layers in adverse pressure gradients[END_REF], in the outer layer he defined ν e = RU δ * where R was an inverse of the Reynolds number that had to be determined empirically. For the overlap layer, Prandtl's mixing length theory was used to define the eddy viscosity, ν e = κ 2 y 2 ∂u ∂y . This eddy viscosity function predicted the whole range of equilibrium boundary layers (-0.5 ≤ β ≤ ∞) and overcame Clauser's limiting case for layers with U τ near 0 (i.e β = ∞).

Tennekes and Lumley (1972) assumed a high Reynolds number flow and analysed the momentum integral equation. This was done in order to linearize the equations in the velocity defect with which they proceeded to obtain a relation between the exponent 'm' and 'β' which read m = -β 1+3β with U ∼ x m . [START_REF] Skote | Direct numerical simulation of self-similar turbulent boundary layers in adverse pressure gradients[END_REF] argued that this linear analysis based on the assumption of asymptotically high Reynolds number was not applicable to finite Reynolds number flows and so they carried out a non-linear approach to obtain a useful relation between 'm' and 'β' such that m = -

β H(1+β)+2β
with U ∼ x m and δ * ∼ x. The pressure gradient parameter, β was found to be a constant when the freestream velocity varied according to this power law. [START_REF] Head | Equilibrium and near-equilibrium turbulent boundary layers[END_REF] used an integral method to calculate the turbulent boundary layer development where U ∼ x m for m = -0.15, -0.255 and -0.35. The results of this study showed that for the first case, a unique equilibrium profile developed while for the second case, multiple solutions existed as long as the momentum thickness at the initial reference location didn't exceed some critical value for both. No equilibrium condition was obtained in the third case. [START_REF] Perry | Mean velocity and shear stress distributions in turbulent boundary layers[END_REF] and [START_REF] Schofield | Equilibrium boundary layers in moderate to strong adverse pressure gradients[END_REF] argued that U τ and velocity scales based on the local pressure gradient are inappropriate for layers in moderate to strong adverse pressure gradients. Rather than using the wall shear stress they instead introduced a velocity scale U m that related to the maximum shear stress in the layer U m = τm ρ . This velocity scale was favoured as it accurately described the half-power distributions of the mean velocity profile that have been observed in moderate to strong adverse pressure gradient layers [START_REF] Perry | Mean velocity and shear stress distributions in turbulent boundary layers[END_REF]). The validity of this analysis was however restricted to equilibrium layers under a moderate to strong adverse pressure gradient flow and in which the maximum value of the shear stress was at least 1.5 times larger than the wall shear stress. [START_REF] Skåre | A turbulent equilibrium boundary layer near separation[END_REF] reported experimental results for an equilibrium boundary layer in a strong pressure gradient. The equilibrium region extended for about 1.0 m where the skin friction coefficient was maintained at a constant level of about C f = 5.7 × 10 -4 and the mean velocity profiles were documented to be self-similar. It was also shown that in this region, the length scales grow linearly with distance in accordance with the equilibrium requirements and that the non-dimensional pressure gradient, β, as well as the Clauser parameter G were constant. [START_REF] Townsend | Equilibrium layers and wall turbulence[END_REF] pointed out that equilibrium in the mean velocity might only be obtained if the turbulent stresses also show similarity. In this study, similarity was obtained in all the turbulent stresses as well as in the triple correlations. Although the Reynolds stress distribution across the layer was entirely different from the ZPG flow, the ratios between the different turbulent stress components were found to be similar, showing that the mechanism for distributing the turbulent energy between the different components remain unaffected by the mean flow pressure gradient. Close to the surface, the gradient of the mixing length was found to increase by almost twice as high as for the ZPG case. The gradient of the mixing length was found to increase from k m ∼ 0.41 at the beginning of the logarithmic layer to k m ∼ 0.78 where the layer merges with the wake. This did not influence the mean velocity which followed the law of the wall closely with the conventional von Karman constant of K = 0.41. This agreed with the findings of [START_REF] Perry | Velocity and temperature profiles in adverse pressure gradient turbulent boundary layers[END_REF] that the pressure gradient only defined the region of applicability of the logarithmic profile for a flow, but did not distort it.

Scaling of the mean velocity

For the scaling of wall bounded turbulent flows, the velocity field is usually categorized into an inner region and an outer region where a suitable length and velocity scale may be defined. The near wall region is dominated by viscous forces and is scaled by a characteristic velocity known as the friction velocity U τ . The friction velocity is defined as based on the wall shear stress. The outer region is usually scaled by the free stream velocity (U e ). Since viscosity plays an important role near the wall, the length scale is constructed using the kinematic viscosity ν and friction velocity (U τ ) leading to a length scale given by ν/U τ . For the outer region the length scale is generally defined by the boundary layer thickness, δ.

However in the case of a boundary layer under an APG, U τ is not the relevant scale. This is true especially for strong APGs and low Reynolds numbers and obviously for a separating boundary layer since U τ reaches zero. In cases of a strong APG and separation, a velocity scale based on the pressure gradient can be defined as

u p ≡ ν 1 ρ dP dx 1/3
(2.29)

Using the inner length (ν/U τ ) and velocity scales U τ , the total shear stress in the inner region can be written as

τ + ≡ dU + dy + -uv + = 1 + ν U 3 τ 1 ρ dP dx y + (2.30)
From studies by Bradshaw (1967); Samuel and Joubert (1974); [START_REF] Skåre | A turbulent equilibrium boundary layer near separation[END_REF]; [START_REF] Spalart | Experimental and numerical study of a turbulent boundary layer with pressure gradients[END_REF] and [START_REF] Skote | Direct numerical simulation of self-similar turbulent boundary layers in adverse pressure gradients[END_REF], it was observed that the pressure gradient term in equation 2.30 is important for the shear stress distribution in the inner part of the boundary layer. The pressure gradient term decreases with increasing Reynolds number and is thus important for low Reynolds number flows. However, close to separation where U τ approaches zero, it is clear that the pressure gradient term becomes infinite even for large Reynolds numbers. [START_REF] Skote | Direct numerical simulation of a separated turbulent boundary layer[END_REF] used a velocity scale u p to avoid the singularity mentioned above. The authors developed an expression for the shear stress τ p ≡ τ ρu 2 p as a function of

y p ≡ yup ν τ p = y p + U τ u p 2 (2.31)
where y p is y scaled by u p /ν. τ p has an asymptotic form τ p = y p when separation is approached. Thus in this rescaled form, the singularity using the traditional scaling is avoided. In the viscous sublayer, the Reynolds shear stress approaches zero so equation 2.31 can be integrated to give

u p = U u p = 1 2 (y p ) 2 + U τ u p 2 y p (2.32)
where u p is the velocity scaled by u p For the overlap region, neither U τ nor u p as a velocity scale results in a self-similar expression. However, equation 2.30 can be formulated as

τ * ≡ 1 U 2 * ν ∂U ∂y -uv = 1 (2.33)
where u * is a velocity scale that depends on y and can be expressed in either viscous or pressure gradient units,

u 2 * = U 2 τ + u 3 p U τ y p = U 2 τ + u 2 p y p (2.34)
Based on DNS data, [START_REF] Manhart | Near-wall scaling for turbulent boundary layers with adverse pressure gradient[END_REF] proposed a new extended inner scaling for the wall layer of wall-bounded flows under the influence of a streamwise pressure gradient. This scaling accounted for the effect of both, the wall shear stress and the streamwise pressure gradient, which plays an important role for separated flows. Non-dimensional velocity U * and nondimensional length y * were defined by

U * = U u τ p , y * = yu τ p ν (2.35)
where u τ p = U 2 τ + u 2 p is a combined velocity using the classical friction velocity at the wall, U τ and an additional velocity based on the streamwise 56 CHAPTER 2. LITERATURE REVIEW pressure gradient, u p proposed by [START_REF] Simpson | A model for the backflow mean velocity profile[END_REF]. With this scaling, the velocity profile in the viscous region including the pressure gradient effects can be written in non-dimensional form as a function of only two non-dimensional parameters

U * = f (y * , α) (2.36) where α = U 2 τ u 2 τ p
is used to quantify the relative importance of each of the two velocity scales, i.e., α = 0 would correspond to a zero shear stress flow (flow with a separation point) and α = 1 would correspond to a zero pressure gradient flow.

Angele and Muhammad-Klingmann ( 2006) studied a weakly separating and reattaching turbulent boundary layer flow subjected to an APG using PIV. The authors then examined the applicability of various velocity scalings on the acquired data. They found that the mean velocity profiles in the outer layer of the boundary layer and to some extent the Reynolds shear-stress are self-similar when using both a velocity scale based on the pressure gradient, u p originally suggested by [START_REF] Mellor | Equilibrium turbulent boundary layers[END_REF] and the scaling suggested by [START_REF] Perry | Mean velocity and shear stress distributions in turbulent boundary layers[END_REF]. They thus reported these two scaling to be closely comparable contrary to the findings of Alving and Fernholz (1996). The authors also interpreted the similarity of the Reynolds shear stress as experimental evidence supporting the claimed relation between the Perry-Schofield velocity scale and the Reynolds shear-stress.

An alternative scaling approach was developed by [START_REF] Castillo | Similarity analysis for turbulent boundary layer with pressure gradient: outer flow[END_REF] for the outer region and is based on their Asymptotic Invariance Principle (AIP). AIP means that "in the limit as the Reynolds number dependence goes to infinity, the equations of motion become independent of Reynolds number; thus any function or scale must also be independent of Reynolds number ". They started their study with the turbulent boundary layer equations given by [START_REF] Townsend | Equilibrium layers and wall turbulence[END_REF] for high Reynolds number and pressure gradient. The authors applied a similarity type solution, which showed that the outer equations permitted full similarity. The appropriate length scale was chosen as δ, and the appropriate velocity scale was deter-

APG TURBULENT BOUNDARY LAYERS

57

mined by requiring that the differential equation should be independent of the downstream direction. It was concluded that U e is the appropriate velocity scale (for a flow with fixed upstream conditions

) if δ ∝ U -1 Λ e is a constant where Λ = -δ ρU 2
e dδ/dx dP∞ dx . Upon reviewing experimental data, they claimed that Λ only can have three different values, one for the case of a favorable pressure gradient (FPG), one for APG and one for ZPG, the constant however, depending on the upstream initial conditions. Zagarola and Smits (1998a) determined an empirical scaling parameter, U e δ * δ for the velocity profile in the outer region of a developing pipe or channel flow. Extending their study to turbulent boundary layers, [START_REF] Zagarola | A new mean velocity scaling for turbulent boundary layers[END_REF] showed that the velocity profiles successfully collapse to one for a ZPG turbulent boundary layer with this scaling. [START_REF] Castillo | Application of Zagarola/Smits scaling in turbulent boundary layers with pressure gradient[END_REF] showed that when the mean velocity deficit profiles are normalized by the [START_REF] Zagarola | A new mean velocity scaling for turbulent boundary layers[END_REF] scaling parameter in APG flows, the profiles collapse to a single curve.

Using PIV, [START_REF] Maciel | Self-similarity in the outer region of adverse-pressure-gradient turbulent boundary layers[END_REF] studied a very strong APG turbulent boundary layer leading to separation, transitioning from strong favourable pressure gradient to strong APG. They showed that the scaling of Zagarola and Smits (1998a) (U e δ * δ ) 2 scales all the Reynolds stresses in the outer region of the flows, while U e and U τ were not appropriate outer scales for the Reynolds stresses. They also reviewed APG data from several experiments and they reported that the Zagarola and Smits (1998a) scaling works well. [START_REF] Indinger | Mean-velocity profile of turbulent boundary layers approaching separation[END_REF] analysed the inner and outer scaling of mean velocity profiles and Reynolds shear stress profiles for turbulent boundary layers approaching separation. They found out that the outer scaling proposed by Zagarola and Smits (1998a) is the most suitable for the mean velocity profile, even for strong APGs. The shear stress scaling proposed by [START_REF] Castillo | Similarity analysis for turbulent boundary layer with pressure gradient: outer flow[END_REF] showed a reasonably good collapse of the data in the outer region. However very close to separation it failed due to the effect of backflow. [START_REF] Lögdberg | On the scaling of turbulent separating boundary layers[END_REF] focussed on the mean velocity distribution of turbulent boundary layers near, at and after separation. They found that the mean velocity defect profiles are self-similar in the regions between separation and the position of maximum reverse flow. They also found out that the Zagarola-Smits velocity scaling for mean velocity defect profiles is useful not only for the region near separation but also for cases of different adverse pressure gradients.

With regards to these results, since the Zagarola-Smits scaling can be used in all flow conditions of turbulent wall flows, it appears to be more universal.

In the present work, we also apply the Zagarola-Smits scaling on the acquired data from PIV (Chapter 4) and compare it to the Castillo-George scaling.

Turbulent structures in APG flows

The understanding of canonical turbulent wall bounded flows has steadily improved over the past several decades. However, information about coherent structures in APG turbulent boundary layers are lacking in literature as expressed recently in [START_REF] Lee | Effects of an adverse pressure gradient on a turbulent boundary layer[END_REF]; [START_REF] Rahgozar | Low-and high-speed structures in the outer region of an adverse pressure gradient turbulent boundary layer[END_REF] and [START_REF] Harun | Pressure gradient effects on the large-scale structure of turbulent boundary layers[END_REF]. The studies conducted have generally focussed on the statistical properties of these flows; an overview of which is presented below. [START_REF] Krogstad | Influence of a strong adverse pressure gradient on the turbulent structure in a boundary layer[END_REF] examined the turbulence satistics and structure of an APG equilibrium boundary layer. The equilibrium region extended for about 1m with the non-dimensional pressure gradient parameter held constant at β ≃ 20. They showed that the APG significantly affects the time scales of the turbulent events, and also alters the contributions to -uv from the various quadrants. In contrast to the ZPG boundary layer, strong contributions from the first and fourth quadrant develop in the region between the outer production peak and the wall. Events in these quadrants were shown to be characterized by in-rushing turbulent fluid which is reflected back from the wall. Hence, when an adverse pressure gradient is present, they concluded that the turbulent flow in the outer layer may interact more efficiently with the inner flow than is normally the case for the ZPG layer. Following this, [START_REF] Krogstad | Structure inclination angle in a turbulent adverse pressure gradient boundary layer[END_REF] studied the effect of a strong adverse pressure gradient on the turbulent structure in an equilibrium boundary layer using two-point correlations. They observed that the APG correlations are higher and shorter than those of the ZPG boundary layer. This was in agreement with the reduced vortex stretching expected by the strong negative dU/dx term in the APG flow. The two-point streamwise velocity correlations from APG were found to be inclined at a considerably higher angle with respect to the wall in the inner region than ZPG. A wavelet analysis of the correlation data showed that differences in the inclination angle existed at all length scales larger than the boundary layer thickness. They observed two maxima in the main production term; one close to the surface and a second which moved towards the middle of the layer with increasing pressure gradient. This outer peak significantly affected the turbulence field. They concluded that when APG is present, the turbulent flow in the outer layer might interact more efficiently with the inner flow than is normally the case for the zero pressure gradient layers. [START_REF] Laval | On the relation between kinetic energy production in adverse-pressure gradient wall turbulence and streak instability[END_REF] performed a direct numerical simulation (DNS) of a converging-diverging turbulent channel flow at a Reynolds number Re τ ≈ 617 based on the inlet. They found regions of strong kinetic energy production at both the upper and lower walls, just after the beginning of the APG region. While performing a stability analysis of these streaks and by looking at the velocity field, conditioned by the streaks they were able to explain that the peak of turbulent kinetic energy was a result of the enhancement of instability of streaks in the APG region. [START_REF] Shah | A specific behaviour of adverse pressure gradient near wall flows[END_REF] suggested that the outer peak found in the streamwise Reynolds stress was triggered by the APG and had its origin in an instability in the turbulent boundary layer which developed soon after the pressure gradient changed its sign. This may explain the reason for not being able to find a universal scaling for APG turbulent boundary layers. [START_REF] George | New insights into adverse pressure gradient boundary layers[END_REF] said that the position of the outer peak in the streamwise Reynolds stress depended on the initial conditions and the manner in which the APG is imposed. They also found out that far enough downstream, an inflection point of the mean velocity profile occurs very close to the observed peak in streamwise Reynolds stress. [START_REF] Lee | Effects of an adverse pressure gradient on a turbulent boundary layer[END_REF] found out that the coherent structures in the outer layer in APG turbulent boundary layers are more activated than those in ZPG turbulent boundary layer. This might be due to the higher turbulence intensities in APG flows and which leads to the development of a second peak in the turbulent energy. The pressure gradient which intensifies the low speed streaky structures are present in the outer regions of APG flows and can be related to the outer maximum observed in the turbulent kinetic energy. [START_REF] Lee | Coherent structures in turbulent boundary layers with adverse pressure gradients[END_REF] while investigating the response of coherent structures in the turbulent boundary layer subjected to APG found that the strength of the low-speed streaks decreased and the spanwise spacing between two adjacent streaks became irregular and increased four times larger than that of a ZPG flow. Also, dominance of low momentum regions in the buffer and inner log layers were found and the enhancement of the turbulent kinetic energy in the outer layer was attributed to the presence of large scale outer layer hairpin vortices. [START_REF] Monty | A parametric study of adverse pressure gradient turbulent boundary layers[END_REF] performed a parametric study on the effects of increasing APG in many cases with constant conditions of matching Reynolds number and viscous scaled sensor length. Studying the energy content of the flow, they found that the large scale events in the log region are strongly increased by the APG which contributes to the increasing streamwise turbulent profile across the boundary layer.

CHAPTER 3

Low and high momentum regions in a zero pressure gradient turbulent boundary layer

The current chapter looks at the basis and requirement for the k -1

x spectral range in a turbulent boundary layer. To this end, a modified but rather simple model, which can in principle be applied to various wall-bounded turbulent flows is proposed from a new perspective based on the work of Townsend-Perry. This proposed model is then validated using experimental results obtained from Particle Image Velocimetry (PIV) data with a large field of view and good spatial resolution

A model for the spectral signature of the attached eddies

As already mentioned in section 2.2.2, [START_REF] Perry | Asymptotic similarity of turbulence structures in smooth-and rough-walled pipes[END_REF] and [START_REF] Perry | A theoretical and experimental study of wall turbulence[END_REF] showed how Townsend's attached eddy hypothesis implies [START_REF] Perry | A theoretical and experimental study of wall turbulence[END_REF] also developed a flow structure model for this spectral range in terms of specific attached eddies of varying sizes randomly distributed in space and with a number density that is inversely proportional to size. Here we attempt to distill such a type of model to its bare essentials. In this section we show that it is sufficient to consider simple on-off representations of elongated streaky structures of varying sizes as long as their number density remains inversely proportional 62CHAPTER 3. LOW AND HIGH MOMENTUM REGIONS IN A ZPG TBL to size.

E 11 (k x ) ∼ U 2 τ k -1 x in the range 1/δ ≪ k x ≪ 1/y.
We assume that the attached eddies responsible for the k -1

x spectral range have a long streaky structure footprint on the 1D streamwise fluctuating velocity signals at a distance 'y' from the wall. We also assume that these streaky structures can be modeled as simple on-off functions and that it is sufficient to represent the streamwise velocity fluctuations u(x) at a given height y from the wall as follows

u(x) = nm a nm Π(ξ) (3.1)
where

Π(ξ) = 1 if -1 < ξ < 1 with ξ = x-xnm λn
and Π(ξ) = 0 otherwise. The on-off function Π(ξ) is our cartoon model of a streaky structure. Streaky structures of length λ n are present at random positions x nm and their intensity is given by the coefficients a nm . For each subscript n, the subscript m counts the spatial positions where cartoon structures of size λ n can be found. The sum in (3.1) is over all structures lengths λ n and all their positions x nm .

The energy spectrum of u(x) will be the average of the square of the Fourier transform of this model, given by equation 3.2

E 11 (k x ) = (2π) 2 L x |û(k x )| 2 (3.2)
where L x is the length of the record, û(k x ) is the Fourier transform of u(x), and the overbar signifies an average over realisations. The Fourier transform of Π( x-xnm λn ) being

Π(k x , λ n , x nm ) = 2ik -1 x e ikxxnm sin(k x λ n ) (3.3) it follows that, û(k x ) = 2ik -1 x nm
a nm e ikxxnm sin(k x λ n ) (3.4)

MODEL FOR THE STREAMWISE ENERGY SPECTRUM

63

which implies that the energy spectrum is given by

E 11 (k x ) = 4 (2π) 2 L x k -2 x nm
a nm e ikxxnm sin(k x λ n ) pq a pq e -ikxxpq sin(k x λ p ).

(3.5)

We introduce two assumptions which were also used by [START_REF] Perry | A theoretical and experimental study of wall turbulence[END_REF] in their more intricate model. The first assumption is that the positions and amplitudes of our cartoon stuctures are uncorrelated and that different positions are not correlated to each other either, i.e. e ikxxnm e ikxxpq = δ pn δ qm . As a result, the expression for the energy spectrum simplifies as follows:

E 11 (k x ) = 4 (2π) 2 L x k -2 x nm (a nm ) 2 sin 2 (k x λ n ). (3.6)
Let us say that there is a number N n of spatial positions where cartoon stuctures of size λ n can be found. The expression for E 11 (k x ) simplifies even further:

E 11 (k x ) = 4 (2π) 2 L x k -2 x n a 2 n N n sin 2 (k x λ n ) (3.7)
where a 2 n ≡ (a nm ) 2 is the same irrespective of position x nm . We now consider a continuum of different structure sizes λ rather than discrete length-scales λ n and the previous expression for E 11 (k x ) must therefore be replaced by

E 11 (k x ) = 4 (2π) 2 L x k -2 x dλa 2 (λ)N (λ) sin 2 (k x λ) (3.8)
in terms of easily understandable notation. At this point, we introduce a generalised form of the second assumption which was also used by [START_REF] Perry | A theoretical and experimental study of wall turbulence[END_REF]: we assume a power-law form for N (λ) in the range λ i < λ < λ o where λ i ∼ y and λ o ∼ δ, and N (λ) = 0 outside this range for simplicity. This power law form is

N (λ) = (-N M + N o (λ/δ) -1-D )/δ (3.9)
where N M and N o are positive dimensionless numbers which increase 64CHAPTER

LOW AND HIGH MOMENTUM REGIONS IN A ZPG TBL

proportionally to L x so as to keep number densities constant. The number N M is introduced to allow for the possibility of an upper bound on streaky structure size given by N (λ o ) = 0, i.e. N M = N o (λ o /δ) -1-D which should be small given that LSM and VLSM streaky structures have been observed with lengths greater than δ (Smits et al. ( 2011)). [START_REF] Vassilicos | Fractal dimensions and spectra of interfaces with application to turbulence[END_REF] proved that, if 0 ≤ D ≤ 1, then the set of points defining the edges of the on-off functions Π(ξ) is fractal and D is effectively the fractal dimension of this set of points. The case where this fractal dimension is D = 1 is the case where these points are space-filling. The population density assumption of [START_REF] Perry | A theoretical and experimental study of wall turbulence[END_REF] corresponds to D = 1 which is also the choice we make in this work. We now show that this choice can lead to E 11 (k x ) ∼ k -1

x in the range 1/λ o ≪ k x ≪ 1/λ i . We calculate the energy spectrum by carrying out the integral in (3.8). This requires a model for a 2 (λ) which, in this section, we chose to be a 2 (λ) = A 2 /δ for λ i < λ < λ o where A 2 is a constant. Using our models for N (λ) and a 2 (λ) and the change of variables λk x = l, (3.8) becomes

E 11 (k x ) = A 2 δ(C o (k x δ) -2+D -C m (k x δ) -2 ) (3.10)
where

C o = 4(2π) 2 N o L x λokx λ i kx dl sin 2 (l)l -1-D
and

C M = 4(2π) 2 N M L x (k x δ) -1 λokx λ i kx dl sin 2 (l)
which is bounded from above by

N M Lx λo-λ i δ . In the attached eddy range 1/λ o ≪ k x ≪ 1/λ i , C o ≈ 4(2π) 2 N o L x ∞ 0 dl sin 2 (l)l -1-D (3.11)
which means that C o is approximately independent of k x in this range.

Substituting the value D = 1 in equation (3.10), we get

E 11 (k x ) = A 2 (C o k -1 x -C M δ -1 k -2 x )
(3.12)

MODEL FOR THE STREAMWISE ENERGY SPECTRUM

65

which is well approximated by

E 11 (k x ) ≈ C o A 2 k -1 x (3.13) for wavenumbers k x δ ≫ C M /C o (i.e. C o k -1 x ≫ C m δ -1 k -2 x ). Note that C M /C o is much smaller than 1 because N M is much smaller than N o and that equation (3.13) is valid in the range 1/λ o ≪ k x ≪ 1/λ i
where λ o scales with but is much larger than δ. For a good correspondence with the scalings of the Townsend-Perry attached eddy model one needs to take λ i ∼ y and A 2 ∼ U 2 τ .

A straightforward generalisation

It is worth generalising the previous section's model by assuming that a 2 (λ) is not constant but varies with λ in the range λ i < λ < λ o , for example as a 2 (λ) = (A 2 /δ)(λ/δ) p where p is a real number with bounds which we determine below. The arguments of the previous section can be reproduced till equation (3.8) which now becomes

E 11 (k x ) = A 2 δ[c o (k x δ) -2+D-p -c m (k x δ) -2 ] (3.14)
where

c o = 4(2π) 2 N o L x λokx λ i kx dl sin 2 (l)l -1-D+p and c M = 4(2π) 2 N M L x (k x δ) -1-p λokx λ i kx dl l +p sin 2 (l) which is bounded from above by Nm (1+p)Lx [( λo δ ) 1+p -( λ i δ ) 1+p ]. In the attached eddy range 1/λ o ≪ k x ≪ 1/λ i , c o ≈ 4(2π) 2 N o L x ∞ 0 dl sin 2 (l)l -1-D+p (3.15) which means that c o is approximately independent of k x in this range if 0 < D -p < 2.
Substituting the value D = 1 in equation (3.14), we obtain the following 66CHAPTER 3. LOW AND HIGH MOMENTUM REGIONS IN A ZPG TBL leading order approximation in the parameter range -1 < p < 1:

E 11 (k x ) ≈ c 0 A 2 δ(k x δ) q (3.16)
where

p + q = -1 (3.17) for wavenumbers k x δ ≫ (c M /c o ) 1 1-p . Note that c M /c o is much smaller than 1 if p is not too close to 1 because N M is much smaller than N o .
The spectral shape (3.16) is potentially significantly different from what the classical Townsend-Perry attached eddy model predicts. We emphasize that in this and the previous sections we have developed a simple model based on on-off functions representing long streaky structures which returns a wavenumber dependency of E 11 (k x ) which is either identical to the Townsend-Perry spectral shape if p = 0, or different but in some ways comparable if p = 0. In the remainder of this chapter we present experimental evidence in support of D = 1 and equations (3.16)-(3.17) rather than (3.13)

Experimental validation of the model

In the previous section, we developed a model for attached flow structures in a turbulent boundary layer. To test its behaviour at a fairly high Reynolds number, we had the possibility to set up experiments at the boundary layer wind tunnel facility at LML (Laboratoire de Mécanique de Lille). The facility and the experiment that was set-up will be presented in the following section.

Description of the experiment

The LML wind tunnel facility

The front view and top view of the LML boundary layer wind tunnel is pictured in figure 3.1. The wind tunnel is powered by a fan and driven by a 37kW electric motor that allows the free-stream velocity at the entrance of the test section to be varied continuously from 3 m/s up to 10 m/s with a stability better than 0.5%. The level of turbulence is less than 0.3% and the temperature is regulated within 0.15 • C with the help of a heat exchanger located in the plenum chamber.

The test section is 20.6 m long in the streamwise direction (X direction) and has a constant cross-section, 2 m wide (Z direction) and 1m high (Y direction). The test section is fully transparent and this provides easy access to optical measurements and allows for excellent model viewing. The boundary layer is tripped at the wind tunnel entrance on both the top and bottom 68CHAPTER 3. LOW AND HIGH MOMENTUM REGIONS IN A ZPG TBL walls to fix the boundary layer transition. It is tripped at the bottom wall using a cylinder rod, 4 mm in diameter followed by a 93 mm wide grit size 40 sandpaper while the upper wall was tripped only using sandpaper of the same grit classification as the bottom wall.

The very long test section allows the Reynolds number based on the momentum thickness to reach up to Re θ = 20600 with a boundary layer thickness of about 0.24m at 19.6m from the trip (refer [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF] for more details), allowing detailed near wall measurements. In the present studies, the wind tunnel was used in a closed loop configuration.

PIV set-up

At LML, the possibility of conducting stereo PIV experiments exists, thereby gaining access to all the three components of the velocity. However, to test our model in section 3.1 , it is of interest to have a large field of view through which we aim to capture the long structures in the boundary layer along the direction of flow. Thus it would suffice to have a standard (Two-dimensional two-component) 2D2C PIV set-up with cameras arranged in series overlooking a large field of view; the experiment of which is described below:

The tests on the flat plate boundary layer were conducted at two free stream velocities of 3 m/s and 10 m/s corresponding to Reynolds numbers Re θ = 8100 and Re θ = 20600 respectively. To capture the large streamwise wall-normal field, four 12bits Hamamatsu cameras having a resolution of 2048x2048 pixels were installed in series to observe a region between 19.26 m and 20.42 m from the inlet which is 1.16 m long (≈ 3.36δ and ≈ 3.85δ for Re θ = 8100 and Re θ = 20600 respectively) and 0.3 m high (≈ 0.86δ and ≈ 1δ for Re θ = 8100 and Re θ = 20600 respectively). Nikon lenses of 50 mm focal length were set on the cameras and the magnification obtained was M = 0.05. The aperture was set at f # = 5.6, which allowed a particle image diameter slightly larger than one pixel. The Software HIRIS was used to acquire the images of the four cameras simultaneously. The flow was seeded with 1µm Polyethylene glycol and illuminated by a double-pulsed NdYAG laser at 400 mJ/pulse. A set of mirrors and lenses allowed generating a light The calibration needed to obtain the magnification for each camera and the merging regions was obtained by placing a wall normal plate with crosses covering the entire field of view desired. The images obtained of this target by the four cameras was then processed with a home-made software to obtain the coefficients of a pinhole model for each camera. Then the four camera target pictures are merged into one (see figure 3.3) for checking the calibration parameters. The magnification of each camera together with the coordinates of the four extreme points of the fields of view of each camera are also given by the software with the help of the pinhole models obtained. The meshing program then used the entire picture of the target and these output parameters.

Several sources of uncertainty are present in a 2D2C PIV set-up (Adrian (1991); [START_REF] Richard | Theory of cross-correlation analysis of piv images[END_REF]; [START_REF] Raffel | Particle image velocimetry: a practical guide[END_REF]; [START_REF] Foucaut | Characterization of different piv algorithms using the europiv synthetic image generator and real images from a turbulent boundary layer[END_REF], etc.). The main sources are the out of plane motion of particles, the nonuniform displacement of the particles in the interrogation windows (i.e. due to velocity gradient), the non-uniform particles concentration and the particles displacement too large compared to the interrogation window size.

To obtain good PIV measurements, [START_REF] Richard | Theory of cross-correlation analysis of piv images[END_REF] and Fou- caut et al. (2004b) have given the following recommendations: the particle image diameter has to be around 2 pixels, the particles concentration has to be about 0.04 particles per pixel (to obtain a minimum of 10 particles per interrogation window), the maximum difference in particle displacement in the interrogation windows has to be less than half the particle's image diameter and the out of plane displacement has to be less than 20% of the laser sheet thickness.

EXPERIMENTAL VALIDATION OF THE MODEL

The estimation of PIV accuracy is a complex topic. [START_REF] Kostas | Application of double spiv on the near wall turbulence structure of an adverse pressure gradient turbulent boundary layer[END_REF] and [START_REF] Herpin | Stereoscopic piv measurements of a turbulent boundary layer with a large spatial dynamic range[END_REF] estimated the accuracy with the merging regions of the PIV field, i.e., the PIV uncertainty can be obtained from the same velocity overlooking a common region by two independent PIV systems. In the merging region, the random PIV uncertainty with a 95% confidence index is estimated by σ u = ±(u syst1 -u syst2 ) RM S for the streamwise component (u) and by σ v = ±(v syst1 -v syst2 ) RM S for the wall-normal component (v), where RM S refers to the root mean square value.

With respect to the mean velocity components, as the random error goes to zero by definition by taking enough samples, only the bias error affects the accuracy. The PIV uncertainty on the mean streamwise velocity U is then estimated by ∆U = ±(u syst1 -u syst2 ) and on the wall-normal velocity component by ∆V = ±(v syst1 -v syst2 ).

The uncertainties on the mean streamwise velocity and streamwise turbulence intensities are plotted as a percentage of the freestream velocity, U ∞ = 10 m/s in figures 3.4 and 3.5, respectively. The position of the merging regions between the respective cameras in the field of view is shown. Globally, the uncertainty is quite good at 1% or lower. Similar estimates for the uncertainty at U ∞ = 3m/s was obtained and is not shown here.

It is important to ensure that a good connection of the respective velocity fields is made between cameras. As we will see later on, a threshold on the turbulence intensities will be applied and if two cameras report different intensity levels, having a single threshold would distort the detected structures, if present, between these two cameras. Plotting the turbulence intensity field merging the cameras could give us a clear picture to gauge the connection in this region. This is presented in figure 3.6 where the streamwise turbulence The LML database for the current study is validated through an analysis of some of the single point statistics such as the mean flow, the turbulence intensity and the energy spectra of streamwise fluctuating velocity. For each flow configuration, these quantities were obtained through an ensemble average over the number of records acquired, and also through an average along the quasi-homogeneous directions of the measurement planes (the streamwise direction for the XY plane). Note that the homogeneity assumption along the streamwise direction can be assumed as the flow is fairly constant over the field of interest of this experiment ( refer figure 3.6).

Figure 3.7 shows profiles of the mean streamwise velocity U and the streamwise turbulence intensity u ′ obtained from PIV at Re θ = 8100 and Re θ = 20600 and compared with the hot-wire anemometry results of [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF]. The mean velocity profiles are in good agreement with the hot-wire data and are well resolved from y + = 30 and y + = 90 upwards for Re θ = 8100 and 20600 respectively. Comparisons of the profiles of u ′+ (u ′ scaled with inner variables) show a fairly good match with the hot-wire data. A plateau of u ′+ is present in the range 100 < y + < 300 for the higher Reynolds number case. Close to the wall, the u ′+ values obtained from our PIV are slightly underestimated, in particular for Re θ = 20600, demonstrating some filtering of the PIV at this resolution (Foucaut et al. (2004a)). To compute from PIV the energy spectra used in this paper, we used the method of Foucaut et al. (2004a). As seen in figure 3.8 for the particular case of wall distance y + = 200 at Re θ = 20600, the agreement between the spatial spectrum from the PIV and the temporal spectrum from the hotwire anemometry of [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF] is good up to wavenumbers k x ≈ 500 corresponding to length-scales of 2 mm.

Results and discussion

Two-point spatial correlation

While instantaneous flow fields may give some insights about the presence and formation of various flow structures, quantitative information about the average shape and size of these structures may be obtained from the two-point spatial correlation function introduced by [START_REF] Stanislas | Double spatial correlations, a new experimental insight into wall turbulence[END_REF] without invoking Taylor's frozen turbulence hypothesis [START_REF] Taylor | The spectrum of turbulence[END_REF]). The twopoint correlation coefficient, R uu is given by: R uu = u(x 0 , y 0 ).u(x 0 + ∆x, y 0 + ∆y) (u 2 (x 0 , y 0 )). (u 2 (x 0 + ∆x, y 0 + ∆y))

(3.18)

where x 0 and y 0 are the positions of the fixed points in space, ∆x and ∆y are the distances between the moving point and the fixed point and u is the streamwise turbulent velocity fluctuation. As seen in figures 3.9 and 3.10, the spatial correlation extends over a distance of up to 5δ at y + = 200 when looking at the 0.1 correlation value for both the Reynolds numbers. The correlation function appears shaped approximately as an ellipse and is strongly elongated in the streamwise direction with a slight inclination to the horizontal. The overall streamwise extent of the correlation grows with distance from the wall in the logarithmic region but drops (and also broadens in the wall normal direction) beyond that region in agreement with [START_REF] Ganapathisubramani | Investigation of large-scale coherence in a turbulent boundary layer using two-point correlations[END_REF].

76CHAPTER 3. LOW AND HIGH MOMENTUM REGIONS IN A ZPG TBL

-3 -2 -1 0 1 2 3 0 0.5 1 ∆x/δ y/δ y + =4500; y /δ =0.622 -3 -2 -1 0 1 2 3 0 0.5 1 ∆x/δ y/δ y + =3500; y /δ =0.496 -3 -2 -1 0 1 2 3 0 0.5 1 ∆x/δ y/δ y + =2000; y /δ =0.284 -3 -2 -1 0 1 2 3 0 0.5 1 ∆x/δ y/δ y + =1000; y /δ =0.143 -3 -2 -1 0 1 2 3 0 0.5 1 ∆x/δ y/δ y + =400; y /δ =0.057 -3 -2 -1 0 1 2 3 0 0.5 1 ∆x/δ y/δ y + =200; y /δ =0.027

EXPERIMENTAL VALIDATION OF THE MODEL

77

-3 -2 -1 0 1 2 3 0 0.5 1 x/δ y/δ y + =1500; y /δ =0.545 -3 -2 -1 0 1 2 3 0 0.5 1 x/δ y/δ y + =1000; y /δ =0.377 -3 -2 -1 0 1 2 3 0 0.5 1 x/δ y/δ y + =600; y /δ =0.222 -3 -2 -1 0 1 2 3 0 0.5 1 x/δ y/δ y + =400; y /δ =0.152 -3 -2 -1 0 1 2 3 0 0.5 1 x/δ y/δ y + =200; y /δ =0.077 -3 -2 -1 0 1 2 3 0 0.5 1 x/δ y/δ y + =150; y /δ =0.059
Two-point correlations computed using hot-wire rakes obtained by [START_REF] Tutkun | Two-point correlations in high Reynolds number flat plate turbulent boundary layers[END_REF] at a relatively similar Reynolds number (Re θ ≈ 19100) show a similar picture (refer figure 3.11) when the reference wall-normal position in the computation was placed at y/δ of 0.063, which is close to y/δ of 0.057 in figure 3.9 in this study. The time series data from [START_REF] Tutkun | Two-point correlations in high Reynolds number flat plate turbulent boundary layers[END_REF] was converted to space using Taylor's frozen field hypothesis. Their correlations shows a similar shape and also extends from the near-wall region to about 0.8δ, as shown in figure 3.9.

Detection of streamwise large-scale structures

In section 3.1 we developed a spectral model of the streamwise turbulence fluctuating velocity based on the concept of elongated streaky structures which are part of attached eddies and can be modeled as simple on-off functions. In this and the next subsections we use our PIV data to test this concept and assess its potential as an hypothesis for understanding near-wall turbulence spectra. A two-dimensional Gaussian filter was found to be sufficient to capture and connect the structures while retaining their overall shape. The standard deviation of the Gaussian filter was three pixels which corresponds to approximately 0.015δ for both Reynolds numbers, i.e. 105 wall units for Re θ = 20600 and 33 wall units for Re θ = 8100. The result of this operation on figure 3.12(a) leads to figure 3.12(b). Further details on the Gaussian filters that were tested has been included in Appendice A.1.

Detection function:

The existence of well-defined elongated and tilted wall-attached regions of relatively high (positive or negative) u values is clear from 3.12, where a sample field of instantaneous streamwise fluctuating velocity components, u is shown. It is these regions that we need to target in relation to the elongated streaky structures of our model. Thus, u is chosen as our detection function.

Thresholding:

To educe on-off functions such as the ones required by our model, we apply a threshold u th on the gaussian-filetered u * to obtain binary images which distinguish between u * < u th and u * > u th . Effects of the threshold on the statistics of educed structures were investigated in the range 0.1u ′ 300 + < |u th | < u ′ 300 + where u ′ 300 + is u ′ at y + = 300. The number of structures educed by the algorithm described in subsection 3.2.2.2 for the three negative threshold values -0.2u ′ 300 + , -0.4u ′ 300 + and -0.6u ′ 300 + are reported in table 3.1. A threshold u th equal to -0.4u ′ 300 + (refer figure 3.14) was finally chosen to detect low momentum structures in the present study as it corresponds to the value that leads to least threshold- dependency of our statistics for a negative u th . This thresholding operation leads to figure 3.12(c) when applied to figure 3.12(b). The white structures in figure 3.12(c) correspond to u * < u th .

Structures extraction:

One more step is required before comparing with our model. White structures which cut through the vertical borders of the figure are discarded because their streamwise extent is unknown; and white stuctures which are not attached to the bottom wall (at y = 0 but in fact as close to y = 0 as allowed by our PIV data) are also discarded because we are concerned with wall-attached structures only. With this extra step, figure 3.12(c) gives rise to figure 3.12(d).

All the steps leading from raw fluctuating streamwise velocity fields to the binary fields which we use in our statistical analysis are depicted in figure 3.12. The current study's effort is concentrated on wall-attached elongated structures of negative streamwise fluctuating velocity as in figure 3.12(d), but the analysis can be repeated equally well on structures of positive streamwise fluctuating velocity and is reported in Appendix A.3. The general behaviours of positive and negative fluctuating streamwise velocity structures are similar, the negative velocity structures being slightly longer in agreement with

Lengths of wall-attached streamwise velocity structures

We now need to obtain statistics of wall-attached elongated streaky structures represented as on-off functions in our model and as binary structures in the final stages of our structure eduction method. We first label the connected components of the binary images using image processing tools. Then we compute the streamwise length λ of each labelled structure at a distance y from the wall, i.e. the difference between the smallest and the largest values of streamwise coordinate x in this labelled structure at height y. Finally we compute the average value α of the streamwise fluctuating velocity component u inside this labelled structure at height y. Thus we obtain a pair (λ, α) for each labelled structure at each height y considered. This procedure is illustrated in figure 3.13 where the streamwise extent λ and the corresponding amplitude α of two labelled structures at wall distance y/δ = 0.03 is shown. elongated streaky structures of size λ has a decreasing power-law dependence on λ in a certain range of λ values. Following [START_REF] Perry | A theoretical and experimental study of wall turbulence[END_REF], we expect the spatial distribution of such structures to be space-filling, which implies (see [START_REF] Vassilicos | Fractal dimensions and spectra of interfaces with application to turbulence[END_REF]) that the exponent of this power law should be -2.

y + =52 y + =88 y + =125 y + =198 y + =306 -C 1 +C 2 (λ/δ) -2
2 in the form -C 1 + C 2 (λ/δ) -2
of the PDF of λ/δ. The fit is over a range of λ/δ with the lower and upper bounds for the two Reynolds numbers indicated in the table. tween 0.3δ and 0.5δ and lengths λ longer than 3.5δ occur very rarely.

We tested for finite size effects of the field of view by computing the PDF on smaller domains, namely 3.5δ and 3δ long in the streamwise direction but same in the wall normal direction. As shown in figure 3.15 there is no significant differences caused by the three fields of view except that the smallest field returns a slightly more noisy PDF. Indeed, a reduced field of view leads to a smaller number of detected wall-attached elongated binary structures and therefore to reduced statistical convergence.

Figures 3.15 and 3.16 show a power law dependence on λ between about 0.5δ and 3δ with power law exponent -2, i.e. D = 1, in all cases. Given the form of N (λ) hypothesised in section 3.1, we fit the PDF of λ/δ with a functional form

-C 1 + C 2 (λ/δ) -2 (where C 1 /N M = C 2 /N o ).
The fit is shown in figures 3.15 and 3.16 and is effectively the same for both Reynolds numbers and all values of y + in the mean flow's approximate log region. The constants C 1 and C 2 are reported in table 3.2. They are indeed fairly constant over the range of wall distances and for both Reynolds numbers. It is worth noting, though, that the lower bound of the range where the PDF of λ/δ is well approximated by -C 1 + C 2 (λ/δ) -2 seems to increase slightly with increasing y + . The dependence on threshold on the results obtained in figures 3.15 and 3.16 was tested and is reported in Appendix A.2 86CHAPTER

LOW AND HIGH MOMENTUM REGIONS IN A ZPG TBL

Energy spectra of streamwise fluctuating velocities

Figure 3.17 shows log-log plots of premultiplied energy spectra of streamwise fluctuating velocities u(x) which have been obtained from our PIV data at various normalised wall distances y + for both Reynolds numbers. These plots might suggest that

E 11 (k x ) ∼ U 2 τ k -1
x in a range of wavenumbers 0.25/δ k x 0.63/y for y + larger than about 88 and smaller than the value of y + where this range of wavenumbers no longer exists. The apparent k -1

x wavenumber range is close to a decade long at y + = 90 for Re θ = 20600 and shorter for higher wall normal distances and for the lower Re θ = 8100. One would be justified to conclude that this is indeed experimental support for the Townsend-Perry k -1

x spectrum if the only available theoretical glasses through which to look at these spectral plots were those of the Townsend-Perry attached eddy model. However the situation is subtler and, in effect, quite different.

A closer look at the spectra in the lin-log plot of figure 3.18 suggests the possibility for small corrections to this conclusion, particularly at the lower of the two Re θ values, but the result in equations (3.16)-(3.17) of our model in section 3.1 may pave the way for a significantly different interpretation. This model leads to E 11 (k x ) ∼ (k x δ) q with p + q = -1 if D = 1. Support for D = 1 has been obtained and reported in the previous subsection in the range of lengths λ between about 0.5δ and 3δ. It is therefore worth taking a closer look at our energy spectra in the corresponding wavenumber range. For our data, this wavenumber range turns out, in fact, to be comparable to the wavenumber range 0.25/δ k x 0.63/y mentioned in the previous paragraph as a candidate for Townsend-Perry scaling. Specifically, k x = 2/δ corresponds to k x y = 0.25, 0.41, 0.58, 0.91 and 1.41 in increasing order of the y + values in figures 3.17 and 3.18 for Re θ = 8100; and to k x y = 0.15, 0.33, 0.53, 0.78 and 1.1 in increasing order of the y + values in figures 3.17 and 3.18 for Re θ = 20600. The wavenumber range 0.5/δ ≤ k x ≤ 2/δ where the analysis is carried out is therefore not radically different for our data from the wavenumber range 0.25/δ < k x < 0.63/y where one would interpret our spectra to have a Townsend-Perry scaling for y + ≥ 88. In figures 3.19 to 3.22 we plot a 2 versus λ/δ where a 2 is the average of α 2 /∆λ conditional on the streamwise length of a labelled structure being between λ and λ + ∆λ (α and λ being obtained as explained in the first paragraph of subsection 3.2.2.3). The upper values of λ/δ in these plots are all below about 2.3 because we do not have enough samples of educed structures beyond λ/δ ≈ 2.3 to obtain values of a 2 which are statistically converged. The lower values of λ/δ in these plots are all close to δ/2 because the range where the PDF of λ/δ has been found in the previous subsection to be well approximated by -C 1 + C 2 (λ/δ) -2 is bounded from below by about 1/2 in all our y + and Re θ cases. In figures 3.19 to 3.22 we also plot E 11 (k x ) in the corresponding wavenumber range 0.5/δ ≤ k x ≤ 2/δ which, as

EXPERIMENTAL VALIDATION OF THE MODEL

89

discussed in the previous paragraph, may be close to the wavenumber range 0.25/δ < k x < 0.63/y that one could interpret as a Townsend-Perry range. We do not have enough data and high enough Reynolds numbers to clearly distinguish between these two ranges in the present work.

As an aside for the moment, note that the large-scale motions (LSMs) and very large-scale motions (VLSMs), which have been found to exist in the logarithmic and lower wake regions of a turbulent boundary layer [START_REF] Kovasznay | Large-scale motion in the intermittent region of a turbulent boundary layer[END_REF]; [START_REF] Brown | Large structure in a turbulent boundary layer[END_REF]; [START_REF] Hutchins | Evidence of very long meandering features in the logarithmic region of turbulent boundary layers[END_REF]; Dennis and Nickels (2011a); Lee and Sung ( 2011)), generally refer to elongated regions of streamwise velocity fluctuations having a streamwise extent from about 2δ to 3δ for LSMs and larger than 3δ for VLSMs [START_REF] Kim | Very large-scale motion in the outer layer[END_REF]; [START_REF] Guala | Large-scale and verylarge-scale motions in turbulent pipe flow[END_REF]; Balakumar and Adrian ( 2007)) as described in chapter 2. The LSMs near the wall and the VLSMs have been interpreted to account for long correlation tails such as the ones in figure 3.12 and responsible for the k -1

x scaling range of the turbulence spectrum [START_REF] Smits | High-Reynolds number wall turbulence[END_REF]. The range of scales we concentrate on, in figures 3.19 to 3.22, just about includes some LSMs at its upper range.

Returning now to figures 3.19 to 3.22, we have included best fits of power law curves in the plots of a 2 versus λ/δ and of E 11 versus k x . These best fits are indicated in the inserts of each plot and provide an estimation of the exponents p and q in a 2 ∼ (λ/δ) p and E 11 (k x ) ∼ k q

x . Figure 3.23 summarizes the information with plots of p, q and p + q as functions of y + . It is perhaps remarkable that p + q is very close to -1 (see figure 3.23) as predicted by equations (3.16)-(3.17) for all examined values of y + and for both Reynolds numbers Re θ . Whereas this subsection's initial interpretation in terms of the Townsend attached eddy model is limited to y + larger or equal to 88 (based on the log-log plots of figure 3.17), the lin-lin plots of figures 3.19 to 3.22 present a different and consistent picture which covers both Reynolds numbers and all our y + positions, including y + smaller than 88.

Discussion

It is important to stress that the support for equations (3.16)-(3.17) in figures 3.19 to 3.22 cannot be obtained without the crucial last step of our structure detection algorithm in subsection 3.2.2.2 which discards structures that are not attached to the wall. The structures which do not touch the wall are in fact less elongated and less intense (i.e. smaller a 2 ) on average. We have checked that if we only consider them, we do not find anything close to p + q = -1, i.e. equation (3.17).

The attached eddy concept introduced by Townsend ( 1976) is therefore important for explaining E 11 (k x ) but the results of our analysis suggest that the Townsend-Perry model does not hold because the turbulent kinetic energy content in these wall-attached flow structures does not just scale with U τ . At different y inside such a structure, the level of turbulent kinetic energy depends both on U τ and on the streamwise length of the structure at that height. Furthermore, this dependence varies with height: a 2 decreases with increasing λ/δ very close to the wall, in the buffer layer, and increases with increasing λ further up. As a 2 transits smoothly from one dependence to the other, a particular height exists where a 2 is independent of λ and therefore depends only on U τ . At that very particular height, E 11 (k x ) ∼ k -1

x . However this is not a Townsend-Perry spectrum, it is just the spectrum at that particular distance from the wall where the turbulent kinetic energy inside the streaky structures transits from a decreasing to an increasing dependence on the length of these structures. Our conclusion agrees with [START_REF] Nickels | Some predictions of the attached eddy model for a high Reynolds number boundary layer[END_REF] in their statement that it is necessary to take measurements close to the wall to observe a k -1

x behaviour, in fact at y + between 100 and 200 as they also found. However, these authors were not in possession of equations (3.16)-(3.17) and therefore did not measure a 2 at various heights and for various values of λ/δ which now allows us to see that the k -1

x behaviour at the edge of the buffer layer is not the Townsend-Perry spectrum but just a transitional instance of a more involved spectral structure. In fact, the spectral picture which emerges from our analysis is a unified picture which brings together the buffer and inertial layers in a seemless way.

In figure 3.24 we plot examples of measured streamwise velocity fluctuations and the on-off signals with which we model them at various heights from the wall. Our model on-off signals are clearly a drastic simplification of the data but one gets the impression from these plots that they capture the sharpest gradients in the signal and therefore much of its spectral content at the length-scales considered here. The lengths of the non-zero parts of the model signals correspond to λ and the actual values of the on-off signal in these non-zero parts correspond to the average value α of the streamwise fluctuating velocity component inside each part.

It is clear that a wider range of Reynolds numbers needs to be examined to establish the scalings of the lower and upper bounds of the range of wavenumbers where equations (3.16)-(3.17) holds. One might expect the upper bound to scale as 1/y because of the recent evidence [START_REF] Hultmark | Turbulent pipe flow at extreme Reynolds numbers[END_REF]; Laval et al. ( 2017)) that a Townsend-like approximately logarithmic (or very weak power law) dependence on y exists for the rms streamwise turbulent velocity in the outer part of the inertial range of wall-distances. If one assumes the lower bound to scale as 1/δ and therefore an energy spectrum E 11 (k x ) of the form (i)

E 11 (k x ) ∼ U 2 τ δ for 0 ≤ k x ≤ B 1 /δ, (ii) E 11 (k x ) ∼ U 2 τ δ(k x δ) -1-p for B 1 /δ ≤ k x ≤ B 2 /
y (where B 1 and B 2 are dimensionless constants and p may be a function of y as in figure 3.23a comparatively negligible energy at wavenumbers k x > B 2 /y, then we should have

(u ′+ ) 2 ∼ 1 + B p 1 p (B -p 1 -(B 2 δ/y) -p ). (3.19) If p ≡ 0 then this expression for (u ′+ ) 2 becomes (u ′+ ) 2 ∼ 1 + (ln B 1 -ln(B 2 δ/y)) (3.20)
which is the Townsend logarithmic dependence on y [START_REF] Townsend | The structure of turbulent shear flow[END_REF]; [START_REF] Perry | Asymptotic similarity of turbulence structures in smooth-and rough-walled pipes[END_REF]; [START_REF] Perry | A theoretical and experimental study of wall turbulence[END_REF]). This logarithmic dependence (3.20) results from the assumption that the upper bound of the range of wavenumbers where equations (3.16)-(3.17) may hold with p ≡ 0 scales as 1/y. Slightly non-zero values of p give slight deviations from this logarithmic dependence, 96CHAPTER

LOW AND HIGH MOMENTUM REGIONS IN A ZPG TBL of the form (3.19).

Using the values of p obtained in this work and plotted versus y + in figure 3.23 for our two values of Re θ , it is not possible to fit (3.19) to the data in the lower plot of figure 3.7 from y + = 41 to 306 in the Re θ = 8100 case and from y + = 90 to 742 in the Re θ = 20600 case as shown in figure 3.25. These are the y + ranges where equations (3.16)-(3.17) has been established for our data and they should therefore also be the ranges where (3.19) holds if the spectral model of the previous paragraph is good enough. However, in spite of the three adjustable dimensionless constants (B 1 , B 2 and an overall constant of proportionality), (3.19) cannot fit the entire y + range for which this model has been designed, that is a y + range which includes both the p < 0 and the p > 0 regions.

A most suspect part of the spectral model used to derive (3.19) is its low wavenumber part. [START_REF] Vassilicos | The streamwise turbulence intensity in the intermediate layer of turbulent pipe flow[END_REF] showed that the second peak or plateau part of the u ′+ profile can be reproduced by a spectral range between the very low wavenumber range where E 11 (k x ) ∼ U 2 τ δ and the wavenumber range where E 11 (k x ) ∼ U 2 τ δ(k x δ) -1-p . In fact, [START_REF] Vassilicos | The streamwise turbulence intensity in the intermediate layer of turbulent pipe flow[END_REF] also showed that this extra intermediate spectrum is necessary for a sufficiently fast growth of the integral scale with distance from the wall. A complete model of E 11 (k x ) would therefore require the spectral range introduced by [START_REF] Vassilicos | The streamwise turbulence intensity in the intermediate layer of turbulent pipe flow[END_REF] as well as the spectral range studied here. It is interesting that equations (3.16)-(3.17) has been established in this work for length-scales up to about 3δ which therefore includes LSMs, suggesting that VLSMs might be linked with the low wavenumber spectral range introduced by [START_REF] Vassilicos | The streamwise turbulence intensity in the intermediate layer of turbulent pipe flow[END_REF]. x . (c) p + q versus y + . These fits are obtained over the range of scales investigated in figures 3.19 to 3.22 and the resulting exponents are plotted with the 95% confidence intervals for these fits. The y + positions and the two Reynolds numbers Re θ are those in figures 3.19 to 3.22.

Introduction

In the previous chapter, the case of a ZPG turbulent boundary layer was studied. Although theoretically speaking, it simplifies the study of the turbulence problem, it seldom occurs in practical situations as most engineering applications encounter boundary layers exposed to pressure gradients. At present, turbulence models fail to predict the flow behaviour of an APG boundary layer [START_REF] Manceau | Recent progress in the development of the Elliptic Blending Reynolds-stress model[END_REF]). This is due to the fact that the near wall region of all turbulence models is mostly based on the physics of the ZPG boundary layer and still, little is known about the physics of these APG boundary layers [START_REF] Stanislas | Near wall turbulence: an experimental view[END_REF]). Furthermore, understanding of wall turbulence especially in a decelerating situation is still quite limited due to the lack of sufficiently high Reynolds number data and large facilities. These are necessary to reach some state of an equilibrium turbulent boundary layer where theoretical approaches can be relevant [START_REF] George | New insights into adverse pressure gradient boundary layers[END_REF]). Also, the length of these structures extending up to 14δ (in ZPG turbulent boundary layers [START_REF] Hutchins | Evidence of very long meandering features in the logarithmic region of turbulent boundary layers[END_REF]) requires a large field of view and a high spatial resolution. In a bid to overcome these challenges, an extensive set of experiments [START_REF] Cuvier | Extensive characterisation of a high Reynolds number decelerating boundary layer using advanced optical metrology[END_REF] were conducted in the boundary layer wind tunnel at LML under the framework of EuHIT (European High-Performance Infras-102 CHAPTER 4. APG TURBULENT BOUNDARY LAYER FLOW tructures in Turbulence). The following section of this chapter describes the experiment that was performed. This includes the rationale behind the choice of the APG ramp, the metrology used such as pressure measurements and Particle Image Velocimetry (PIV). The final section presents the characterisation of the flow developing along the APG ramp model and the results are compared to a ZPG case whenever possible to compare the different physics between the two type of flows.

Description of the experiment

To thus improve our understanding on APG flows, a set of experiments under the framework of EuHIT were conducted at the LML boundary layer wind tunnel.

In addition to the main 2C2D PIV measurements in a streamwise wallnormal plane with 16 sCMOS cameras on the -5 • flat plate generating the APG, the flow above the 1.5 • flat plate generating a FPG was also characterised through 2C2D PIV at two free-stream velocities (U ∞ = 5 and 9 m/s). Furthermore, to complement these large field of view measurements, time resolved near wall velocity profiles were obtained in order to determine the wall-shear stress and its fluctuations at some specific locations along the wall. This would also enable us to get an overall picture of the turbulence quantities at these measurement stations right from the wall up to the edge of the boundary layer.

The aim of the subsequent subsections of this chapter is to describe the various metrologies used followed by providing the main results of this unique experiment with an emphasis on the statistics and the boundary conditions. One objective is in fact to provide turbulence modellers with information pertaining to a carefully characterised flow which will then allow them either to validate their RANS model on a challenging test case or to be able to compare their LES predictions of the flow organisation with the experiment. For that purpose, pressure distributions along the ramp model and along the upper wall were measured. The upstream boundary conditions are also provided and the 2D character of the flow has been checked by performing measurements near the side walls and complementary measurements have been performed in the streamwise plane of symmetry, upstream of the APG in order to provide the modellers with as complete picture of the flow as possible.

The ramp model

Before deciding to design experiments on such a large scale, several points pertaining to nature of APG flow we wish to study needs to be addressed:

• The strength of the APG we wish to study (either strong or mild).

• The angle the plate should be set at, to generate the desired APG case.

• The length of the plate for the flow to develop and reach a state of an equilibrium APG turbulent boundary layer.

Addressing the first point, the case of a strong adverse pressure gradient quickly leads to separation and the flow in this region is dominated by pressure effects. The case of mild APGs acting over a considerable streamwise distance results in history effects becoming important and this poses a challenge for turbulence modellers. One common physical fact that has been observed by several authors and recently summarized by [START_REF] Stanislas | Near wall turbulence: an experimental view[END_REF] has been the development of an intense second peak of turbulence which, as opposed to the standard near wall peak, spreads and moves away from the wall as it progresses downstream. Thus, it was decided to develop a ramp model that would lead to a mild APG study case.

To design the desired ramp and to answer points 2 and 3, numerical simulation using the commercial CFD package, StarCCM+ was performed enabling us to have a first estimation of the resulting flow by varying the angle of the plate on the ramp. Test were conducted for a range of ramp angles from -3 • to -5 • with respect to the horizontal, in 1 • increments to analyse the development of pressure coefficient and the skin friction coefficient in particular. A -5 • ramp was finally chosen as this ensured a mild APG flow in equilibrium that would not separate. The details of the simulation pertaining converging part, having a contraction ratio of 0.75. It was followed by a 2.2 m flat plate inclined at +1.5 • to the horizontal that generated a small FPG and a second plate of length 3.5 m inclined at -5 • to the horizontal where an APG region developed. This 3.5 m flat plate is made of 7 individual pieces connected together. Of these, the first is a 240 mm long Aluminium plate; the following five are in plexiglass (four of 625 mm in length and one of 515 mm in length) and the last is an Aluminium plate (210 mm long and 2 mm thick). The adjustments between plates are better than 0.1 mm and the surface quality of this 3.5 m plate is below ±0.1 • on 20 cm. To minimize leaks, the junctions between plates are taped by a 0.05 mm thick black Aluminium tape. Also to minimize the vortex that developed on the sidewalls, a gasket is fixed between the sidewalls and the ramp model. Each plexiglass plate is equipped with a 240 mm by 625 mm insert to allow specific near wall measurements during the test campaign.

The exact ramp coordinates in the (X, Y, Z) global wind tunnel frame are given below (X = 0 corresponding to wind tunnel entrance). X r is defined as X r = X -X le , with X le the ramp leading edge position (X le = 9400 mm). The converging part is defined by equation 4.1 with X r and Y in mm,

         Y = - 500 1200 3 X 3 r + 750 1200 2 X 2 r for 0 ≤ X r ≤ 1200mm Y = 250mm for 1200 ≤ X r ≤ 1330mm (4.1)
At the end of the converging part, the articulation has a radius of 10 mm which can be modelled by a sharp corner. This articulation is followed by a flat plate at +1.5 • from the floor on a length of 2140 mm. At the end of this flate plate, the articulation corresponds to a circular arc of radius 10.25 mm with the centre at X r = 3470 mm, Y = 295.77 mm and a -6.5 • rotation to join the 3486 mm flat plate which follows at -5 • from the floor. the connection with the wind tunnel floor at the end of this plate was done with a curvature radius of about 7.27 mm centered at X r = 6949.04 mm, Y = 5.14 mm.

CHAPTER 4. APG TURBULENT BOUNDARY LAYER FLOW

The ramp was also equipped with 51 pressure taps; 27 pressure taps for the streamwise pressure distribution and 24 for four transverse pressure distribution stations. Two transverse stations located on the 1.5 • plate and two on the -5 • plate. The large field experiment carried out to characterise the flow along the APG ramp is described below.

Static pressure distribution

The pressure distribution on the model was measured for two free-stream velocities (U ∞ = 5 m/s and 9 m/s, both measured at the entrance of the wind tunnel). This was done using scanivalves and a Furness FCO 14 manometer having a range of 0 to 10 mm H 2 O and an uncertainty of ±0.5% of the reading value. The pressure tap number 17, located just before the 3.5 m plate was chosen as reference as it corresponded to the smallest pressure on the model. The pressure coefficient, C p = P -P 17 1 2 ρU 2 ∞ was computed with P 17 being the reference pressure, ρ the density of air and U ∞ the free stream velocity upstream of the ramp located 10 cm downstream of the test section entrance. In the following figures, ′ s ′ will refer to the curvilinear abscissa along the ramp, with the leading edge of the ramp set as origin as defined in [START_REF] Cuvier | Characterisation of a high Reynolds number boundary layer subject to pressure gradient and separation[END_REF].

The pressure coefficients along the model for the two free stream velocities (5 and 9 m/s) are shown in figure 4.3(a) while figure 4.3(b) shows the corresponding distributions of the pressure gradient. As seen in figure 4.3(a), the flow accelerates in the converging part 0 ≤ s ≤ 1360 mm of the ramp causing a decrease in the pressure coefficient until the suction peak at s = 1146 mm corresponding to pressure tap '6'. This suction peak then induces a locally strong adverse pressure gradient (see Figure 4.3(b)). Behind this suction peak, a region of pressure recovery occurs after which the flow begins to accelerate again due to the favourable pressure gradient caused by the 2.14 m long flat plate inclined at 1.5 • to the wind tunnel floor. A second suction peak is observed at the articulation of the -5 • plate at s = 3500 mm close to the reference pressure tap 17. The 3.5 m long plate inclined at -5 • to the wind tunnel floor then causes a region of relatively constant adverse pressure Spanwise pressure measurements were carried out at four locations to check the spanwise homogeneity and the two dimensionality of the flow. The first two stations were located on the 1.5 • plate at s = 1727 mm and s = 3010 mm while the last two stations were located on the -5 • plate at s = 4301 mm and s = 6176 mm. As seen in figure 4.4 where z is the transverse direction with z = 0 corresponding to tunnel centreline, the pressure distribution is almost constant at the 4 stations for both velocities except slight variations close to z = -800 mm and +800mm. This confirms the two dimensionality of the flow for at least -600 ≤ z ≤ 600 mm.

To provide complete boundary conditions, the pressure distribution on the top wall of the wind tunnel was measured using a wooden window equipped with three pressure taps. The distribution was acquired by moving this wooden window from port to port along the upper wall. The effect of the ramp becomes visible around X = 8 m (See Figure 4.5). Thus, the mean velocity and Reynolds stress profiles at X = 6.8 m (2.6m upstream of the ramp meading edge), which can serve as inlet boundary conditions for computational fluid dynamics (CFD) are then not affected by the presence of the model. The smoothness of the C p curve shows the very good repeatability of the flow. The pressure distribution starts with a slight favourable pressure gradient (see Figure 4.5(b)) due to the boundary layer development on each wall of the test section. It is -0.44 Pa/m for U ∞ = 9 m/s and -0.19 Pa/m for U ∞ = 5 m/s close to the values found by [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF]. Then a trend similar to the pressure distribution on the model is observed. The presence of the contraction and the flat plate at 1.5 • creates a flow acceleration which induces a strong FPG until X = 12.54 m after which the flow switches to adverse pressure gradient caused by the 3.5 m plate. It should be noted that the pressure gradient contains some fluctuations due to measurement uncertainties.

Uncertainties

To ensure accurate measurements, the time constant of the Furness was set at its maximum value and a 2-3 minute waiting period was observed for the measurement to stabilize. The zero offset was checked before every reading to avoid voltage drift over time. The uncertainty for both the pressure rT where P a is the atmospheric pressure, r the ideal gas constant (r = 287J/kg/K) and T the temperature, the error on C p is given by

△C p C p = △(P -P 17 ) P -P 17 2 + △P a P a 2 + △T T 2 + 2 △U ∞ U ∞ 2 + △r r 2 (4.2)
The estimation of △(P -P ref erence ) is given by the Furness uncertainty which should be ±0.5% of the value (P -P ref erence ) . As (P -P ref erence ) is read on the Furness scale, the accuracy is slightly lower. The real Furness uncertainty is estimated at ±0.6% of the measuring value. The other uncertainties are △P a = 100P a, △T = ±0.2 • C and △U ∞ = ±0.5% of U ∞ . The uncertainty on r was neglected. The uncertainty on C p was thus found to be 1.2% of the local value.

The uncertainty on the pressure gradient could be estimated using the same method mentioned above. As ( dCp ds ) i+1/2 = Cp i+1 -Cp i ∆s

= P i+1 -P i 1 2 ρU 2
∞ ∆s (i is the position of the pressure tap), the error on dCp ds is given by equation 4.3.

∆ dCp ds dCp ds 2 = ∆(P i+1 -P i ) P i+1 -P i 2 + ∆(s i+1 -s i ) s i+1 -s i 2 + ∆P a P a 2 + ∆T T 2 + 2 ∆U ∞ U ∞ 2 + ∆r r 2 (4.3)
As the real Furness uncertainty is estimated at ±0.6% of the measuring value, the uncertainty ∆(P i+1 -P i ) is estimated with a quadratic mean of the two errors as 0.6 100 (P i+1 -P ref erence ) 2 + (P i -P ref erence ) 2 . The uncertainty was found to be 6.5% on the pressure gradient.

was applied with the Soloff reconstruction method [START_REF] Soloff | Distortion compensation for generalized stereoscopic particle image velocimetry[END_REF]). The analysis was done with four passes starting with 48 x 64 pixels and ending with 16 x 24 pixels which was found to be the optimal final interrogation window size. Image deformation was used to improve the quality of the results. The final interrogation window size corresponds to 2.4 x 2.4 mm 2 in the physical space. The mesh spacing was 1 mm in both directions corresponding to an overlap of about 60%. This resulted in 180 points in the wall normal direction and 299 in the transverse direction. The maximum displacement was about 11 pixels in the external region.

Figure 4.7 shows the mean streamwise velocity field for both free-stream velocities investigated. The local free-stream velocity was found to be nearly constant (difference between minimum and maximum less than 0.3%),however the difference is more pronounced on the boundary layer thickness, δ as the extraction of this quantity is very sensitive to small variations in the local free-stream velocity. At 5 m/s, the mean BL thickness was 10.7 cm with a minimum value of 9.9 cm and a maximum of 11.3 cm. At 9 m/s, δ was 10.0 cm with a minimum of 9.4 cm and a maximum of 10.5 cm. These slight variations were attributed to the remains of the wake of the turning vanes in the settling chamber.

Figure 4.8 shows the mean streamwise velocity profile in the plane of symmetry for the two cases. The profiles obtained from the TRHM PIV have also been included to get a complete picture of the mean streamwise velocity in the boundary layer. The mean velocity profile at 9 m/s has been compared to hot wire data, obtained previously by [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF]. A small adjustment in the friction velocity, U τ was necessary to connect and match the SPIV and the TRHM PIV data. A friction velocity U τ = 0.203 m/s was used for SPIV compared to 0.204 m/s for TRHM PIV at 5 m/s and 0.351 m/s and 0.350 m/s respectively at 9 m/s. This was because the two experiments were not conducted at the same time (about one month between both experiments) which could change the viscosity marginally due to variations in atmospheric pressure. The collapse with the hot-wire data is good for the 9 m/s case as it corresponds to almost the same Reynolds number based on the momentum thickness (Re θ ≈ 7750 compared to Re θ ≈ 7500). The boundary layer characteristics are provided in Table 4.1.

Also, a characterisation of the corner vortices developing on the side walls of the wind tunnel was performed using a spanwise/wall-normal SPIV plane. The effects of the corner flow is found to be negligible at about 200 mm from the side wall of the wind tunnel. Additional details about this experiment has been included in Appendix C.

U ∞ (m/s) U e (m/s) δ (mm) δ * (mm) θ (mm) H Re θ U τ (m/

Flat plate FPG flow

PIV Analysis

The flow above the 2.14 m long flat plate inclined at 1.5 • with respect to the horizontal was characterised through 2D2C PIV for the two free-stream velocities (U ∞ = 5 and 9 m/s). The aim was to characterise the accelerating flow upstream of the APG. The light sheet was introduced in the same manner as the large field 2D2C PIV in the APG region also using the same optics. The mirror inside the wind tunnel located downstream of the model was raised by 50 cm. The light sheet was 1 mm in thickness on the 2.14 m of the 1.5 • plate.

The experiment was conducted in two set-ups. The first was with two sCMOS cameras close to the end of the 1.5 • plate with an overlap of 10 mm between both fields of view forming a global field of view of 46.5 cm long and 19.8 cm in height above the wall. This field of view was positioned to get a 7 mm overlap with the large field APG-TBL set-up. The second set-up (refer figure 4.9) also used two sCMOS cameras, however without any overlap between them. The field of view of each camera was 23.6 cm along the wall and 19.5 cm in the wall-normal direction. The field of view of the first camera began at 1.5 • plate and the second was located close to the middle of this plate. For both cases, the cameras were equipped with Nikon 105 mm lenses at f # = 5.6. The time interval between frames was tuned to obtain 12 pixels of displacement in the free-stream region to ensure sufficient velocity dynamic range for the turbulence intensity measurements. For the first set-up, 30,000 PIV fields were recorded for the two velocities and 10,000 for the second set-up. The data were processed in the same way similar to the previous subsection by the modified version of the Matpiv toolbox by LML. The final interrogation window size corresponded to 2.25 mm by 2.25 mm in the physical space. The grid spacing was selected as 0.94 by 0.94 mm leading to an overlap of 58%.

Statistical results

Figure 4.10 shows the evolution of the mean streamwise velocity profiles for both velocities along the 1.5 • plate normalised with the respective reference velocities, U ∞ . The wall-normal distance, ′ y ′ is normalised by the boundary layer thickness, δ 0 taken at s = 1.362 m, which is 101 mm at 5 m/s and 95 mm at 9 m/s (refer tables 4.2 and 4.3). The first profile at s = 1.362 m located just after the beginning of the 1.5 • plate exhibits a small peak close to y/δ 0 = 0.15. The external velocity is about 1.4 U ∞ due to the contraction upstream. At the second station, the peak is still visible but is strongly attenuated and is located further away from the wall (y/δ 0 = 0.4). After this station, the peak disappears. The mean velocity is increasing continuously from the beginning of the plate to its end due to the FPG encountered, with an external velocity reaching 1.55 U ∞ at the last station. At the last station located very close to the ramp articulation, the acceleration is highly marked due to the strong FPG generated by the sudden change of flow direction. Tables 4.2 and 4.3 show the boundary layer characteristics on the 1.5 • plate at the stations depicted in figure 4.10 for U ∞ = 5 and 9 m/s respectively. The evolution of the shape factor indicates a similar behaviour of the boundary the converging part of the model (see Figure 4.3). In the outer part, a second plateau is clearly visible in the first profile, close to y/δ 0 ∼ 0.5, which progressively transforms into a vanishing kink at the following stations. This second plateau is probably linked to the external turbulence of the incoming boundary layer which is seriously attenuated by the strong FPG encountered in the converging part of the ramp. At s = 3.497 m, both the kinks are nearly smoothed out.

Figure 4.12 shows the evolution of the wall-normal turbulence intensity profiles for both velocities. At the first station, the shape is similar to the streamwise component, however the plateaus are replaced by well-defined peaks. At the following stations, the outer peak spreads out and nearly disappears while the one closer to the wall stays fairly visible.

Figure 4.13 shows the evolution of the Reynolds shear stress profiles for both velocities. The behaviour is strongly influenced by the wall-normal component of the fluctuations (see Figure 4.12). Figure 4.14 shows the 16 sCMOS cameras placed in series beside the wind tunnel test section. Out of these 16 cameras, 8 (number 1-8) were provided by UniBw, 6 (number 9-14) by DLR Gottingen, and 2 cameras (number 15 and 16) by LML. All the cameras were fixed on the same 3.5 m long X95 bench and mounted on Manfrotto 410 articulations to allow the tuning of the cameras normal to the side glasses and parallel to the ramp surface. The cameras were mounted at 90 • so that the side of the sensor corresponding to 2560 pixels was aligned with the wall-normal direction. This was done in order to obtain the best resolution of 25 cm in height. The field of view of each camera was 230 mm along the wall and 273 mm in the wall normal direction. A common region of about 10 to 20 mm was set between each camera to obtain a continuous field. All but three cameras were equipped with macro planar 100 mm Zeiss lens at a working distance of 1680 mm. To avoid the shadow of the side pillars of the wind tunnel (see Figure 4.14), camera 6 and 13 (with camera 1 being the first, upstream) were equipped with 85 mm Zeiss lens at a working distance of 1445 mm while camera 14 with a 50 mm Zeiss lens operating at a working distance of 1050 mm. The f number was #4 for all cameras except for cameras 6, 15 and 16 where it was fixed at 2.8, 4.8 and 4.8 respectively due to presence of the wind tunnel frame. The fields of view for all cameras were of the same order of approximately 230 mm in the streamwise direction and 255 mm in the wall normal direction. The LML BMI YAG laser system placed 11.6 m from the inlet of the test section with 2 x 200 mJ 532 nm was used to generate the light-sheet with a height of approximately 260 mm and a thickness of approximately 1 mm over a length of about 3 m. This laser light sheet was produced by passing the laser beam through a spherical lens with a focal length of 7500 mm and then through a diverging cylindrical lens having a focal length of -250 mm. This light sheet was introduced to the tunnel using a mirror placed at 45 • downstream to the ramp model that directed the laser sheet vertically upwards. The second mirror was located inside the test section, 87 cm downstream of the end of the ramp. This mirror directed the laser sheet along the 3.5m long APG section (refer figure 4.15). The backside of the mirror was equipped with an aerodynamic profile to limit the vibration due to vortex shedding. A PEG smoke generator was used for all measurements to seed the whole flow with particles having a mean size of 1µm.

A total of 30 000 samples were recorded at a frequency of 4 Hz for two free stream velocities of 5 and 9 m/s.

The modified version of MatPIV toolbox by LML, under Matlab was used to process the acquired images. The magnification for each camera and the merging regions, was obtained by placing a calibration target across the field (refer figure 4.14). The images obtained of this target by the sixteen cameras were then processed to merge the target pictures of these cameras into one picture. The magnifcation of each camera together with the coordinates in this picture of the four extreme points of the fields of view of each camera was obtained. This entire target picture and these output parameters was used to create the mesh. The mean background images were first mapped using a basic pinhole model and the reflection (i.e wall position) was manually fitted with a line. A mesh was then built above this line in the mapped images (spacing of about 1.07 mm by 1.07 mm corresponding to 10 pixels by 10 pixels) and projected on each camera with the pinhole models. The analysis was then done with these projected grids. Four passes were used (first pass 64x64 pixels, second and third of 32x32 pixels and a final pass of 24x24 pixels with a mean overlap of 65%). Image deformation was used before the final pass with a cubic b-spline interpolation of the grey level and bilinear interpolation for the displacement to improve the result quality. Also background division was used to limit the effect of the laser reflection and the camera noise. The final grid then had 3250 points along the wall and 238 points in the wall-normal direction corresponding to a region 3.41 m long and 0.25 m high with a grid spacing of 1 mm. b) shows the evolution of the mean streamwise velocity normalised with the reference free-stream velocity U ∞ along the -5 • plate for both velocities studied. In this section, s = 0 now corresponds to the beginning of the 3.5 m long flat plate inclined at -5 • with respect to the horizontal. Due to the enlargement of the test section along the APG plate, the flow decelerates continuously but does not separate. As expected for a boundary layer subjected to an APG, the thickness of the boundary layer increases rapidly such that the ratio between the local free-stream velocity at the beginning and at the end of the 3.5 m long field is 1.45.

To obtain quantitative information on the evolution of the mean streamwise velocity, profiles for both free-stream velocities along the APG plate normalised by U ∞ and δ 0 are shown in Figure 4.17. As the flow has been accelerated on the convergent and the 2.14m long flat plate, the velocity is globally higher than the freestream for the first profile at s = 3.5m, located at the beginning of the ramp. The external velocity is then progressively reduced by the APG. It is also interesting to note the similarity in the profiles for the two test cases.

The velocity components are generally scaled based on the friction veloc- ity (U τ ) and therefore it is important to use a proper method to estimate U τ .

We shall now see how the friction velocity, U τ was determined.

Determination of friction velocity (U τ )

The Clauser chart method [START_REF] Clauser | Turbulent boundary layers in adverse pressure gradients[END_REF]) has been traditionally used to determine the friction velocity, U τ in wall-bounded flows. It relies on the existence of the log-law region near the wall, i.e by plotting U + vs y + on a semi logarithmic scale along the x-axis, the overlap region of the boundary layer plots as a straight line. This straight line relationship is known as the log-law as seen in Section 2.1, given by the relation

U (y) U τ = 1 κ log yU τ ν + B (4.4)
The presence of the logarithmic region in APG flows is still a contentious issue (Anderson (2011), [START_REF] Monty | A parametric study of adverse pressure gradient turbulent boundary layers[END_REF]). In addition to ZPG flows, [START_REF] Monty | A parametric study of adverse pressure gradient turbulent boundary layers[END_REF] showed that the skin friction determined from the Clauser chart method agreed with that obtained from oil-interferometry for mild APG boundary layer flows at Re θ = 11860. Up to a β < 3, the skin friction obtained between both methods showed a good match. However beyond β ≈ 3, the difference between both methods became significant, up to approximately 10% difference in C f where

C f = τ w 1 2 ρU 2 ∞ = 2 U 2 τ U 2 ∞ (4.5)
Thus, this study suggests that the Clauser chart method could be used for mild APG flows but should be used with caution for strong APG flows as inaccurate values of U τ could lead to inaccurate conclusions concerning the scaling arguments for pressure gradient boundary layers.

From equation 4.4, U (y) can be measured directly from experiments and assuming that ν, κ and B are constants, the only undetermined term is U τ . This can be determined by using a least squared fit on the data between the region 30 ≤ y + ≤ 200 [START_REF] Cuvier | Characterisation of a high Reynolds number boundary layer subject to pressure gradient and separation[END_REF]), beyond which a departure from the log-law was observed across the APG plate (refer figure 4.19). Figure 4.18 shows the development of U τ across the APG ramp for both test cases. The solid lines indicate the value of U τ for both test cases obtained using the Clauser chart method on the large field streamwise (LFStW) 2D2C PIV The friction velocity was obtained from the TRHM PIV by the wallshear rate measurement technique developed by [START_REF] Willert | High-speed Particle image velocimetry for the efficient measurement of turbulence statistics[END_REF] that retrieves actual velocity data from the imaged sequences to estimate the mean wallshear rate. The method relies on the one-dimensional cross-correlation of single, streamwise rows of pixels that are taken at the same wall-normal distance but separated by a few time steps (here 2-5 time steps). A onedimensional Gaussian peak fit at the location of maximum correlation then provides sub-pixel accurate displacement information, which, in conjunction with magnification factor, M and time difference ∆t, would yield an estimate of the streamwise velocity for the given wall distance. Then, estimates of the mean streamwise velocity for each wall distance are obtained by averaging the instantaneous velocity estimates over the entire length of the sequences. Finally, a least squares fitting to the linear portion of this data provided an The values of U τ estimated from TRHM PIV and LFStW PIV for the test cases at U ∞ = 5 m/s and U ∞ = 9 m/s are reported in tables 4.4 and 4.5. Also from figure 4.18, a good agreement on U τ (within 3%) from the two methods is noted for both the test cases. This suggests that the Clauser chart technique could indeed be used to determine U τ across the APG ramp with a fairly high accuracy.

We could make use of both techniques (TRHM PIV and LFStW PIV) to get an entire picture of the turbulent boundary layer at the three stations. Figure 4.19 shows the entire inner-scaled mean streamwise velocity, from the viscous sublayer to the wake region at these three stations for the test case at U ∞ = 5 m/s. ZPG data from [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF] at Re θ = 8100 is plotted with symbols for comparison. The near wall and part of the overlap region shown in red were obtained from the TRHM PIV while the overlap and outer regions shown in blue were obtained from the LFStW PIV. A good agreement between both methods is noted. It is then interesting to observe a logarithmic region (30 y + 300) indicated by a solid black line on all APG profiles but with a limited extent compared to the ZPG flow. The mean velocity of the APG case also drops below the classical log law in the overlap region while it rises higher than the ZPG case (refer figure 4.19) in agreement with [START_REF] Monty | A parametric study of adverse pressure gradient turbulent boundary layers[END_REF].

Boundary layer parameters

The evolution of the mean streamwise velocity along the -5 • plate was shown earlier in figure 4.16. From these mean velocity profiles, boundary layer parameters can be obtained and are given in tables 4.6 and 4.7. The boundary layer thickness, δ is defined by the distance from the wall where the mean streamwise velocity, U reaches 99% of the local free-stream velocity, U e . The displacement thickness (δ * ) and momentum thickness (θ) were obtained by integrating the profile using the trapezoidal rule. The shape factor is given by H is the non-dimensional pressure gradient parameter.

= δ * θ . Re θ = Ueθ ν is
Although the profiles of the mean streamwise velocity are quite similar between the last station of the FPG (figure 4.10) and the first station of the APG (figure 4.16), a significant difference appears in the value of the shape factor. This indicates that the strong variations of the pressure gradient observed in figure 4.3 close to the articulation of the APG ramp significantly affects the near wall region. The reduction of the external velocity by a factor of about 1.3 along the 3.5m plate is accompanied by a notable increase of the boundary layer thickness by a factor of about 4. Figure 4.20 shows the evolution of δ, δ * and θ along the APG ramp for the two test cases. In addition to the growth of δ, the integral parameters (δ * , θ) also increase and the resulting values of the shape factor, H across the APG ramp indicates that the boundary layer is far from separation as the value of the shape factor reported in the literature for separation is about 2.7 (Alving and Fernholz (1996); [START_REF] Castillo | Separation criterion for turbulent boundary layers via similarity analysis[END_REF]; [START_REF] Kline | Correlation of the detachment of two-dimensional turbulent boundary layers[END_REF])

Scaling of the mean flow

From a modelling perspective, the possibility of self-similarity of a boundary layer subjected to a pressure gradient has been an important issue. Methods for universally describing turbulent flows have long been sought after with a view to improve modelling and computational accuracy (Anderson et al. (2004)). [START_REF] Castillo | Similarity analysis for turbulent boundary layer with pressure gradient: outer flow[END_REF] addressed this issue by extending the equilibrium-type similarity analysis of [START_REF] George | Zero-pressure-gradient turbulent boundary layer[END_REF] Through their analysis, they concluded that the local free-stream velocity, U e is the appropriate scale for the mean velocity (refer section 2.3.2). Furthermore, [START_REF] Castillo | Similarity analysis for turbulent boundary layer with pressure gradient: outer flow[END_REF] showed how the mean deficit profiles of developing APG flows can be scaled with U e δ * /δ of Zagarola and Smits (1998a). In this section, the scaling of the velocity deficit is presented in two different ways. Firstly, it is scaled with the CG scaling [START_REF] Castillo | Similarity analysis for turbulent boundary layer with pressure gradient: outer flow[END_REF]), the free-stream velocity (U e ) and secondly with the ZS scaling (Zagarola and Smits (1998a)), U e δ * /δ. Finally, the equilibrium pressure gradient parameter developed by CG is tested on the APG data to corroborate the existence of self-similarity of boundary layers subjected to APG.

Contrary to internal flows, where the outer length scale δ is exactly defined through the geometry, the counterpart for semi-confined flows such as boundary layers "can hardly be exactly defined" [START_REF] Robinson | A review of quasicoherent structures in a numerically simulated turbulent boundary layer[END_REF]). The problem is commonly circumvented by using integral quantities such as the displacement thickness δ * , e.g. through usage of the Rotta-Clauser outer scale ∆ = U + ∞ δ * (Vinuesa et al. ( 2016)). For complex turbulent boundary layer configurations such as the current study involving pressure gradients, the problem becomes even more apparent and even commonly used wake descriptions in the case of ZPG boundary layers are unable to accurately describe the data. Thus in addition to the standard definition of δ 99 used in the previous section, δ 95 (distance from the wall where the streamwise velocity U reaches 95% of the external velocity U e ) is also used as the outer length scale with perhaps less determination error following CG.

Figure 4.21 shows the velocity deficit scaled with the freestream velocity, U e for the cases at U ∞ = 5 m/s and U ∞ = 9 m/s respectively. Both flows don't show a perfect collapse. The trends of the deficit profiles scaled with the CG scaling, U e for both cases are that there is an increase in magnitude with downstream location, such that the first streamwise profile has the lowest magnitude and the profile located furthest downstream, has the largest magnitude. However, it is important to keep in mind that using the CG scaling, U e , the deficit profiles are not expected to collapse into a single curve. The profiles should converge towards an asymptote as the local Reynolds number increases [START_REF] Castillo | Similarity analysis for turbulent boundary layer with pressure gradient: outer flow[END_REF]). In their study, CG also showed how the scaling of ZS collapsed the velocity profiles for the developing cases of pressure gradient flows. As described in section 2.3.2, ZS empirically determined a new scaling U e δ * /δ for the velocity deficit in the outer region of a developing pipe or a channel flow (Zagarola and Smits (1998a)). They later showed that the velocity profiles collapsed for ZPG turbulent boundary layer on a smooth surface with this scaling [START_REF] Zagarola | A new mean velocity scaling for turbulent boundary layers[END_REF]). [START_REF] Castillo | Effect of upstream conditions on the outer flow of turbulent boundary layers[END_REF] showed that the ∞ dδ/ds dP∞ ds ). Figure 4.22 shows the velocity deficit data scaled with the ZS scaling, U e δ * /δ for the test cases at U ∞ = 5 m/s and U ∞ = 9 m/s respectively. As mentioned earlier, applying the ZS scaling to mean velocity deficit profiles can remove the Reynolds number dependence, and it is expected to result in a collapse of profiles. Compared to the CG scaling shown in figure 4.21, we observe a much better collapse of the profiles for both test cases. The profiles corresponding to the streamwise location of 0.5m and 1m do not fall within the bandwidth of collapse compared to the other streamwise stations. This might be as a result of the flow in this region still being under transition from FPG to APG and thus, the region on the -5 • ramp encompassing these two streamwise locations do not have a typical APG behaviour; i.e, whilst all these flows are APG flows, FPG behaviour is also evident and flow development is causing this spread. Only after the flow reaches equilibrium, a good collapse of the data is observed from 1.5m up to 3m. The pressure gradient parameter, Λ is used to further confirm this equilibrium flow state.

y δ 99 U e -U U e x = 0.5m x = 1.0m x = 1.5m x = 2m x = 2.5m x = 3m
As discussed in section 2.3.2, CG defined the boundary layer equilibrium state differently compared to [START_REF] Clauser | Turbulent boundary layers in adverse pressure gradients[END_REF], β = ∂P ∂s δ * ρU 2 τ = constant. In the sense of CG, a boundary layer under equilibrium is one where the pressure gradient parameter, Λ = constant and when (Λ = 0), δ ∼ U -1/Λ e . The value of the constant, Λ depends on the upstream conditions [START_REF] Castillo | Similarity analysis for turbulent boundary layer with pressure gradient: outer flow[END_REF]).

The constant, Λ is obtained by fitting the slope of U e vs δ in a log-log plot. In order to get a better fit, the evolution of δ along the APG ramp that was shown earlier in figure 4.20, was smoothed out using a least square spline interpolation. The profiles of δ 95 and δ 99 obtained after a spline interpolation are shown in black in figures 4.23 which were then used to plot U e vs δ 95/99 . From figure 4.24, it is clear that the significant region of constant Λ are in excellent agreement with the equilibrium similarity requirement of CG. The region exhibiting the equilibrium range is shown by the line in magenta. A constant of Λ = 0.27 and Λ = 0.38 is obtained when δ 95 and δ 99 are used, implying that the value of Λ is also sensitive to the definition of δ. From figures 4.24, the range on the APG ramp over which flow equilibrium is exhibited can be estimated between 1 m to 3.4 m on the APG ramp as indicated in figure 4.24. For the first meter of the ramp, a region of constant Λ isn't observed. This might be due to the fact that the flow is still in transition from FPG to APG and needs a certain distance to develop into a complete APG flow. This would then explain why the mean velocity profiles scaled by the ZS variable didn't show a reasonable collapse upto this region on the ramp. Thus, the behaviour of the Λ parameter according to CG should be a constant to enable the successful application of scaling given by ZS. Looking at the contour plots, a good continuity is observed between the fields of view of the 16 cameras assembled in this set-up. This is a testament to the high quality of data acquired. As generally observed in all APG flows with and without separation [START_REF] Cuvier | Characterisation of a high Reynolds number boundary layer subject to pressure gradient and separation[END_REF]; [START_REF] Simpson | Turbulent boundary-layer separation[END_REF]; [START_REF] Webster | Turbulence characteristics of a boundary layer over a two-dimensional bump[END_REF]; [START_REF] Wu | Numerical investigation of the turbulent boundary layer over a bump[END_REF]), a region of high streamwise turbulence intensity reaching about 13% of U ∞ develops above the wall, that both spreads and moves away from it while developing downstream. The profiles at the first station is similar to the profile located at the last streamwise position on the FPG (see figure 4.11) with a slight reduction of level which is more apparent near the wall, at the highest Reynolds number. At the second station (s = 0.7 m), a weak outer peak located at about y/δ 0 = 0.15, is induced by the change in sign of the pressure gradient. Moving downstream, this peak moves away from the wall with the peak magnitude decreasing along the APG. At the last station, it is localised at about y/δ 0 = 0.7 corresponding to about 30% of the local boundary layer thickness δ. The inner near wall peak of the streamwise turbulence intensity is not observed as it is located too close to the wall to be captured by the PIV measurement even for the 5 m/s case. As illustrated in figure 4.31, the TRHM PIV resolves the near wall peak of the streamwise velocity fluctuations, which is present at its usual position of y + ∼ 15. The presence of this near wall peak is an indication of the existence of near wall streaks all along the APG region. In this inner units representation, the near wall peak is observed to change very little at the measurement stations, while the outer peak develops and reaches a level comparable to the near wall peak at station 4. Based on Re θ , the APG profile at station 3 is similar to the ZPG case plotted in black. It is clear that the turbulence intensity is highest throughout the flow for the APG case compared to the ZPG flow. The contrasting difference occurs in the outer region, where an outer peak is not observed in the ZPG data. This outer peak is related with the large-scale structures' energy of the flow [START_REF] Hutchins | Evidence of very long meandering features in the logarithmic region of turbulent boundary layers[END_REF]; [START_REF] Monty | A parametric study of adverse pressure gradient turbulent boundary layers[END_REF]) suggesting consequences on the coherent structure statistics between both flows.

The profiles of the wall normal component of the turbulence intensity are shown in 4.32. In the inner region, the profiles collapse fairly well for the APG flows but do not collapse onto the ZPG profile. Additionally, the outer peak increases and also moves out from the wall whereas no such peak is The Reynolds shear profiles are shown in figure 4.33. Similar to the two other stress components, only the outer peak is clearly visible and is located at a wall position similar to the two other components. It is growing as well downstream along the APG ramp. In the inner region of a boundary layer subjected to an APG, the Reynolds shear stress can be expressed by equation

u ′ v ′ + = dU + dy + -1 - ν U 3 τ 1 ρ dP ds y + (4.6)

Skewness and flatness

The third moment of a quantity, such as u 3 scaled by ( u 2 ) 3 , describes the skewness S(u) or asymmetry of the probability distribution of u (refer equation 4.7). The probability density function is symmetric about the origin, S(u) = 0, if u 3 = 0. A positive value of S(u) implies that large positive values of u are more frequent than large negative values. For a Gaussian distribution, S(u) = 0. The fourth moment or Flatness, F (u) of the u distribution is given by u 4 scaled by ( u 2 ) 4 , and is a measure of the frequency of occurence of events far from the axis (refer equation 4.8). If these are relatively frequent, F (u) will take greater values than the Gaussian value of 3 [START_REF] Dengel | An experimental investigation of an incompressible turbulent boundary layer in the vicinity of separation[END_REF]).

S(u) = u 3 u 2 3/2
(4.7)

F (u) = u 4 u 2 2 (4.8)
Figure 4.34 shows the skewness as a function of wall normal distance in inner scaling. The crosses in black were obtained from the hot wire ZPG data of [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF] corresponding to a Re θ of 11500. For completeness, data from TRHM PIV was used to throw light on the near wall region inaccessible from the LFStW PIV measurements. Station 3 at U ∞ = 5 m/s corresponding to Re θ of 10630 was selected to compare the behaviour of APG with the ZPG flow.

For the zero pressure gradient case in figure 4.34, the skewness is negative in the lower part of the traditional logarithmic region (20 < y + < 80). Further from the wall (0.017 < y/δ < 0.06 or 80 < y + < 400), the pdf of streamwise velocity closely follows a Gaussian distribution (i.e., S ∼ 0). From figure 4.34, it becomes clear that the influence of the pressure gradient is to increase the skewness over most part of the boundary layer, which was also reported by [START_REF] Nagano | Higher-order moments and spectra of velocity fluctuations in adverse-pressure-gradient turbulent boundary layer[END_REF]. For the inner region, [START_REF] Nagano | Higher-order moments and spectra of velocity fluctuations in adverse-pressure-gradient turbulent boundary layer[END_REF] suggested that the rise in skewness is caused by 'structural changes in the near-wall' region due to pressure gradient. [START_REF] Monty | A parametric study of adverse pressure gradient turbulent boundary layers[END_REF] proposed that for high Reynolds number (Re τ ≈ 2000), the change in skewness with pressure gradient is due to the increased large-scale influence in the near-wall region that earlier thought to be associated with increased β. They also pointed out a similar conjecture made by [START_REF] Metzger | A comparative study of near-wall turbulence in high and low reynolds number boundary layers[END_REF] who compared low Reynolds number laboratory data with that from a high Reynolds number atmospheric boundary layer. At lower Reynolds number, it was found that in the inner region of y + < 100, the skewness was negative, while for the high Reynolds number data from the geophysical flow, 2011) stated that in general, the flatness increases with pressure gradient. In the outer region till about y/δ ≈ 0.4 or y + = 1000, the flatness is nearly invariant comparing both the APG and ZPG flows. A rise in flatness is often attributed to a rise in intermittency [START_REF] Dengel | An experimental investigation of an incompressible turbulent boundary layer in the vicinity of separation[END_REF]). This interpretation leads to the conclusion that there is weakly increasing intermittency in the near-wall region as well as the near wake region where F (u) rises to quite high values in a boundary layer subjected to an APG.

Turbulence production

For a 2D stationary flow, the production terms for the Reynolds stress u i u j are given by equation 4.9 while those for the turbulent kinetic energy are given by equation 4.10. ∞ /δ 0 (δ 0 defined in section 4.3.2). These terms correspond to the production terms accessible in half the streamwise Reynolds stress ( 1 2 u 2 ) and also present in the production terms of the turbulent kinetic energy.

-u i u ∂U j ∂x -u i v ∂U j ∂y -u j u ∂U i ∂x -u j v
The production of the turbulent kinetic energy is dominated by -uv ∂U ∂y . From figure 4.36 it becomes apparent that the most striking difference between the APG flow and the ZPG case is that strong turbulent production does not only occur in the wall region but is also found in the outer part of the boundary layer. This would then affect the terms in the turbulent energy budget for e.g., the transport of the kinetic energy by diffusion receives its main contribution from ∂vk ∂y . The second order production terms (-uu ∂U ∂xvv ∂V ∂y ) is not taken into account in ZPG flows as they don't contribute to production. However, in the case of APG flows this isn't the case, since the streamwise derivatives increase (-vv ∂V ∂y could be written as vv ∂U ∂x ). As shown in figure 4.37, the term -uu ∂U ∂x is present in the outer part of the current boundary layer, continously growing and should therefore be not neglected. The case of the ZPG isn't ∂y of 1 2 u 2 . The effect of the pressure gradient in the case of a mild APG is noticeable. The APG leads to an increased inner peak in the production profile and a moderate increase in production in the outer region as Reynolds number increases. Furthermore, there is a weak emergence of a second peak in the case of APG which moves outward. [START_REF] Skåre | A turbulent equilibrium boundary layer near separation[END_REF] also observed 2 peaks in the production term while studying a strong APG flow with β ≈ 20. In their study, they claimed that the inner peak was due to the mean strain as the wall was approached while the outer peak in the turbulent stresses which they noted was caused by the strong APG in their experiment. In figure 4.38, the profile marked in black shows the production term obtained from the large field PIV (refer chapter 4) at Re θ ≈ 8100 in a ZPG turbulent boundary layer. Although the near wall region below y + < 40 isn't captured by the measurement, it is expected to be lower than the APG case. Globally though, it is clear that production in an APG turbulent boundary layer is higher than that of a ZPG.

Quadrant analysis

In the previous section we observed that the production increases due to the APG. The production in a turbulent boundary layer is strongly linked to the negative contributions of uv and for that reason, [START_REF] Wallace | The wall region in turbulent shear flow[END_REF] introduced the idea of quadrant analysis of the u and v velocity fluctuations to examine the structure of turbulence. They classified the product of these fluctuations into four categories: Q1(u > 0, v > 0), Q2(u < 0, v > 0), Q3(u < 0, v < 0) and Q4(u > 0, v < 0) which later were called the quadrants of the Reynolds shear stress plane.

In the log region of a ZPG flow, Q2 is the most contributive part of the Reynolds shear stress followed by Q4 and these motions dominate the contributions of Q1 and Q3. This behaviour was also observed in pipe flow [START_REF] Nagano | Statistical characteristics of wall turbulence with a passive scalar[END_REF]) and is considered to be a characteristic of canonical wall CHAPTER 4. INFLUENCE OF APG ON THE LSS flows. On the other hand in the APG flow, the situation is remarkably different from that in the ZPG flow. This is shown in figure 4.39 where the weighted joint pdf of -uv is plotted. The plots give the individual contribution to the total Reynolds shear stress of each quadrant. Figure 4.39a illustrates the contributions of sweep motions becomes larger than that of ejections in the log region. The locations of the peak values in Q2 and Q4 specify the values of u and v that contribute the most to the Reynolds shear stress at this location in the flow; however, these are not the most extreme values of the velocity fluctuations [START_REF] Wallace | The wall region in turbulent shear flow[END_REF]). Furthermore, the increased activity in these quadrants is coupled with higher contributions from both Q1 and Q3 motions. Q1, known as outward motion, represents transport of high speed fluid away from the wall, while the inward motion from Q3 motions tends to bring low speed fluid back to the surface. [START_REF] Krogstad | Influence of a strong adverse pressure gradient on the turbulent structure in a boundary layer[END_REF] found that these quadrants are at least twice as high near the wall when the APG is applied than when the gradient is absent. [START_REF] Nagano | Structure of turbulent boundary layer subjected to adverse pressure gradient[END_REF] also found the contribution of sweep motions larger than that of ejections in an APG flow compared to a ZPG flow (refer figure 4.40) and noted the outward and wallward motions increase near the wall. They concluded that this indicated a change in coherent structures between ZPG and APG flows. For the outer region (refer figure 4.39b), contributions from Q2 are clearly dominant which is also the trend observed in ZPG flows [START_REF] Krogstad | Influence of a strong adverse pressure gradient on the turbulent structure in a boundary layer[END_REF]).

Influence on large-scale structures

Spatial velocity correlations

It was shown in Chapter 3 on ZPG flows that the correlation function could be used to investigate the large-scale structures. In the following section, the correlation function will be used to analyse the effect of the mild pressure gradient in the near-wall and outer regions on the streamwise and wall-normal correlations.

Figure 4.41 shows the outer scale two-point spatial correlation for the streamwise velocity fluctuation R uu at three wall-normal locations, at station 3 for U ∞ = 5 m/s. Similar to the ZPG case (refer figure 3.10), the correlation iso-contours are plotted as functions of ∆x/δ and ∆y/δ. A lack influence of APGs on the streamwise length of large-scale structures. [START_REF] Lee | Large-scale motions in turbulent boundary layers subjected to adverse pressure gradients[END_REF] recently suggested that the strength of the APG, characterised by β may be important in determining the large-scale features in the outer layer. In his review, he observed that when β is suficiently large, e.g. β = 20 [START_REF] Krogstad | Influence of a strong adverse pressure gradient on the turbulent structure in a boundary layer[END_REF]) and β = 8.5 [START_REF] Rahgozar | Coherent structures in a turbulent boundary layer under a strong adverse pressure gradient[END_REF]), the streamwise lengths of the large-scale motions are shorterned by the APG. On the other hand, they are more energized with longer streamwise coherence for relatively small values of β, e.g. β = 1.74 [START_REF] Harun | Pressure gradient effects on the large-scale structure of turbulent boundary layers[END_REF]), β = 0.82 [START_REF] Monty | A parametric study of adverse pressure gradient turbulent boundary layers[END_REF]), β = 1.68 [START_REF] Lee | Effects of an adverse pressure gradient on a turbulent boundary layer[END_REF]).

Structure detection

The case of a ZPG boundary with a 1.16 m long field of view was studied in chapter 3. In that case, turbulence statistics was assumed to be quasihomogeneous. However in the case of an APG, we saw in the preceeding chapter that parameters such as U τ and δ vary significantly along the direction of flow. Thus, before directly applying the structure detection tools described in section 3.2.2.2 for ZPG on the APG data, few steps are necessary to modify our detection algorithm.

In figure 4.31, the growth of an outer peak in the streamwise turbulence intensity was observed. Thus, a threshold u th equal to -0.4u ′ 300 + where (u ′ 300 + is u ′ at y + = 300) would no longer be a constant along the streamwise direction of the ramp. In order to take this into account, at each grid point along the streamwise direction, the local u ′ at y + = 300 is used for the threshold.

Once the threshold has been applied on the streamwise fluctuating velocity fields, one more step is needed before the results can be analysed. The variation of δ and its effect on the structures needs to be taken into account. Said otherwise, the structures detected at the beginning of the ramp might not have the same physics with that found towards the end. Thus, it was decided to create ranges with almost constant value of δ. A variation of 20% in δ was chosen to have a sufficiently long region to capture the large-scale structures. This results for U ∞ = 9 m/s to four ranges. Through the detection procedure, the midpoints of the structures are stored. Instead of neglecting the structures that touch the side borders of the investigated region that was carried out for the ZPG case where the data were available on a limited domain, structures that have their midpoints between this region are kept. Finally, the size of the filter used on the APG data was chosen to be equal to the ZPG case in y + . The value of U τ varies only by about 20% in the equilibrium region, therefore the filter would vary by the same extent within this region.

Figure 4.43 shows a power law dependence on λ between about 0.5δ upto 6δ in all cases in what can be defined as the log region. For all cases, the PDF of λ/δ with a functional form -C 1 + C 2 (λ/δ) -r . The values taken by the exponent r for the wall distances plotted in figure 4.43 are reported in table 4.9. The fit shown in figure 4.43 with an exponent of -1 to show the behaviour of the streamwise length distribution. The constants C 1 and C 2 are remain fairly constant over the range of wall distances (refer table 4.9). Returning now to the ZPG case, the difference in the distribution of the streamwise length of the attached structures is apparent as shown in figure 4.44. While the most probable length appears to be 0.5δ for both, their streamwise extents aren't distributed in the same way. From figure 3.16, it was observed that in the case of a ZPG turbulent boundary layer, the streamwise lengths of the long structures followed a power law, with an exponent -2 over a long range of y + . However, from figure 4.43 and table 4.9, it is clear that the streamwise lengths of structures in an APG boundary layer aren't distributed in a similar manner as the exponent r decreases as we move away from the wall. While the reason for this difference is not known at present, certain hints could be obtained from the two-point spatial correlation function (refer figure 4.41). As mentioned earlier, the largest structures appear longer in an APG compared to a ZPG flow and are also inclined differently at distances from the wall. These differences in the characteristics of the large structures may be assumed to be the reason that the distribution of streamwise lengths aren't the same in both flows. Future study on this behaviour would be needed to have a conclusive evidence.

However, while figure 4.44 shows a difference of structure's length between the two flows, it is important to keep in mind certain caveats. The fields of view of the ZPG (1.16 m) and APG (3.41 m) set-up are not the same and hence extracting structures under similar conditions is difficult as the statistics of very long structures can be affected even if this effect has been tested on the ZPG case. While the threshold for the APG was chosen to be the value of u ′ at y + = 300, the same criterion set for ZPG, from figure 4.31 it is evident that the streamwise turbulence intensity profiles do not behave the same way for both flows. Thus, while the trends seen in the correlation of Additionally, similar to the ZPG case, we plot a 2 versus λ/δ where we have included best fits of power law curves in the plots. These best fits are indicated in the inserts of each plot and provide an estimation of the exponents p such as a 2 ∼ (λ/δ) p . Although we do not have access to the exponent q from the spectra due to the non-homogeneous nature of the APG along the direction of flow , it would be worthwhile to observe how p evolves. Compared to the ZPG flow, it is perhaps remarkable that p is still negative (figure 4.45) for y + up to 400 beyond which p becomes positive as seen for y + = 450. Furthermore, this dependence varies with height like in the ZPG case: a 2 decreases with increasing λ/δ up to y + = 400, and increases with increasing λ further up. This suggests that the shorter structures are more energetic compared to the ZPG flow and retain this behaviour at distances greater to the wall. Only after y + = 450, we observe a transition similar to the ZPG flow.

CHAPTER 5

Conclusion and outlooks

Summary of findings

The aim of the present work was to study a boundary layer subjected to a pressure gradient and to compare it with a ZPG at high Reynolds numbers. Within this framework, focus is laid on the behaviour of large-scale coherent structures. Due to their large streamwise extent, these structures are not easy to extract and characterize using standard measurement techniques. For this reason, specific experimental set-ups using PIV in the streamwise/wallnormal planes was designed to capture the large-scale structures and to gain more insight into the mechanisms governing the dynamics of these flows. The achievements of the present investigation can be divided into two parts. The first part revisits the results obtained on a ZPG turbulent boundary layer while the second focuses on the effect of the boundary layer subjected to a pressure gradient.

To begin with, a well-resolved PIV data of a flat plate turbulent boundary layer in a large field of view at two medium to high Reynolds numbers, Re θ = 8100 and Re θ = 20600 is obtained (refer Chapter 3). The database is validated though the analysis of single point statistics and power spectra that were compared with reference hot-wire data showing good agreement with the reference profiles. The aim here is to probe the origin of a k -1

x spectral range in a turbulent boundary layer. To this end, a simple model which can A direct inspection of the streamwise energy spectrum (refer figure 3.17) would suggest E 11 (k x ) ∼ U 2 τ k -1 x in the range 2π/(4δ) < k x < 0.63/y. However, a closer look assisted by relation (3.16)-(3.17) reveals a significantly subtler behaviour. This relation introduces a specific data analysis which involves the extraction of wall-attached elongated streaky structures from PIV data. The concurrent analysis of streamwise energy spectra and of the relation between the turbulence levels inside streaky structures and the length of these sructures offers strong support for (3.16)-(3.17) over a significant range of length-scales. This range covers LSMs and is comparable to the range where one might have expected the Townsend-Perry attached eddy model spectra to be present. Even though k -1

x spectra are not, strictly speaking, validated by our data, the streaky structures which account for the scalings of E 11 (k x ) do need to be wall-attached for relation (3.16)-(3.17) to hold. Our conclusions agree with the experiments of [START_REF] Vallikivi | Spectral scaling in boundary layers and pipes at very high Reynolds numbers[END_REF] which suggest that the Townsend-Perry k -1

x spectrum cannot be expected even at very high Reynolds numbers. The revised Townsend-Perry streamwise energy spectral form (3.16)-(3.17) with p = p(y + ) given by figure 3.23 appears to extend the validity of the attached eddy concept and its revised consequences to a wider range of Reynolds numbers and a wider range of wall distances.

Finally, we stress that relation (3.16)-(3.17) is predicated on these wallattached streaky structures being space-filling, i.e. D = 1 in the notation of section 3.1. The pdf of the streamwise length of the educed streaky structures does indeed follow a power law with exponent -1 -D = -2 over the range of scales which corresponds to the one where (3.16)-(3.17) holds. This work has shed some new light on the streamwise turbulence spectra of wall turbulence by revealing that some of the inner structure of wall-attached eddies is reflected in the scalings of these spectra via p(y + ). An important implication of this inner structure is that the friction velocity is not sufficient to scale the spectra.

Chapters 4 dedicated to the study of flows subjected to a mild pressure gradient far from separation at external velocities. The fundamental aim of 5.1. SUMMARY OF FINDINGS 167 this project is to improve the understanding of turbulence under APG by performing a complete characterisation of the flow. Emphasis is laid on the characteristics of the structures (length, scaling, energetic contribution and their wall normal distribution). The analysis is also extended to compare the behaviour of APG with the ZPG case at fairly high Reynolds numbers.

Analysis of the single point statistics show that the boundary layer behaves differently compared to a ZPG boundary layer even when subjected to a mild pressure gradient. Although the wake region grows substantially, the logarithmic law of the wall remains valid along the APG ramp for such mild pressure gradients. The study confirms the occurence of an enhanced outer peak of turbulence develops on all the Reynolds stresses suggesting an apparent consequence on the characteristics of the large-scale coherent structures while the near wall peak of the streamwise component indicating the presence of near-wall streaks is still visible.

Of primary interest was to build an APG ramp long enough for the flow to develop and reach a state of quasi-equilibrium where scaling laws can be relevant. Using Castillo and George's pressure parameter, Λ lead to the determination of an equilibrium region along the APG ramp. It is shown that in this zone of equilibrium, the mean velocity deficit profiles display an acceptable collapse. The value of Λ is seen to be constant from s = 1 m to s = 3.4 m along the APG ramp but clearly dependent on the definition of δ.

The two-point correlations show the presence of large structures existing in a boundary layer subjected to a mild pressure gradient. The APG increases the structural inclination angle and the streamwise length of the correlation is decreased in the inner layer. The same method used in ZPG flows to detect the large-scale structures on the fluctuating streamwise velocity component has been applied on this flow with APG. The evolution of the lengths of wall-attached streamwise velocity structures along the wall-normal direction show a power law dependence on λ, however, with a power law exponent (refer table 4.9) different to that of a ZPG boundary layer (-2). The exponent p also behaves differently when compared to the ZPG flow at comparable wall distances suggesting that the structural organization in an APG boundary layer is different to the classical ZPG case.

Future work

The findings of this thesis provide new avenues for further research and improvements. Based on the results in chapter 3 focussing on the ZPG boundary layer, as a short term perspective, future work must now probe the inner structure of wall-attached eddies, attempt to explain it and extend analysis to higher Reynolds numbers so as to establish with certainty the ranges of the power laws (exponents p and q in equation 3.16-3.17). The model developed could also be tested with DNS data and is currently being investigated by a PhD student at LML but at a fairly lower Reynolds number. The structure detection methodology employed here is carried out by discarding structures that cut through the vertical borders because their streamwise extent is unknown. This step could be negated by carrying out the experiment with a larger field of view that is currently possible at LML. As mentioned earlier in the case of the ZPG boundary layer, the pdf of the streamwise length followed a power law with an exponent -2 over the range of scales where equation 3.17 i.e p + q = -1 holds. However in the case of an APG boundary layer the power r and p changes and thus we could guess that the spectra behaves differently and equation 3.17 is no longer valid. Therefore, in the immediate future, in addition to understanding the cause of this behaviour, measures to obtain the spectra in a decelerating flow should be explored. This is particularly not evident as the streamwise homogeneity of the flow can no longer be exploited and thus the spatial spectrum obtained from PIV as in the ZPG case in chapter 3 is not valid. Although not performed due to time constraints, as a first approximation, obtaining a time spectra from hot-wire is planned. Furthermore, LML is also currently trying to study APG flows along a 2D bump at low Reynolds numbers using DNS in a configuration similar to that described in chapter 4. Thus, a specific study comparing the different spectra could be carried out. Another idea could be to apply similarity mapping through which the velocity field is scaled and transformed to a homogeneous field. This was done in a study by [START_REF] Wänström | Spatial decompositions of a fully-developed turbulent round jet sampled with Particle Image Velocimetry[END_REF] on jets that resulted in streamwise homogeneity in similarity coordinates. This experiment being a case of a mild APG may serve as an opportunity to test this method and subject to satisfactory results, it could be applied on a stronger APG.

With a focus on the streamwise component of the structures in the current study, the spanwise structure of the flow in APG should also be characterized. This can be made possible in the immediate future through the readily availability database of spanwise/wall normal SPIV measurements at two streamwise positions of s = 0.595 m and s = 2.192 m on the APG ramp. Also, the interaction between turbulence structures such as ejections, sweeps (see section 4.4.3.4) in the regions of the boundary layer are quite complex and difficult to comprehend in APG flows. Thus the manner in which these structures interact in the boundary layer at high Reynolds numbers could be studied with the help of the current database. Furthermore, while the presence of LSMs in APG boundary layer has been evidenced though this study, the existence of hairpins and hairpin packets has not been explored. The role of hairpin vortices and packets have been previously carried out for ZPG boundary layers (see section 2.2.1.1), however, there isn't a clear picture in APG flows. The size, swirl intensity, orientation and the effect of the pressure gradient on the hairpin and hairpin packets could be explored. Furthermore, studies in ZPG flows have shown the existence of LSM present in the outer region having an influence on the near wall region [START_REF] Hutchins | Evidence of very long meandering features in the logarithmic region of turbulent boundary layers[END_REF]; Mathis et al. ( 2009)) It would be worth probing how the pressure gradient affects the modulation of the small scaled by the large scales.

To understand the effects of a strong pressure gradient on the single point statistics and on the coherent structures, the APG experiments can be conducted with the same model with a steeper angle of the ramp as a medium term perspective. This could be carried out at a location further upstream on the wind tunnel floor thereby facing in a smaller boundary layer thickness to keep a large ration of the APG length with respect to δ and therefore the possibility of the flow to remain in equilibrium. Finally, with the improvement of measurement techniques, it could be possible to have access to 3D data at a very large scale for both the APG and ZPG flows using the recent 3D time resolved Shake-the-box technique associated quite well in capturing the overall shape of the coherent structures and looks similar to what we expect while visually inspecting the thresholded PIV velocity fields.

A.2 Effects of threshold levels

Our results have no significant dependence on threshold u th in the range -0.2u ′ 300 + to -0.6u ′ 300 + . An example of this lack of threshold dependence can be seen in the PDFs of λ/δ which we plot in figure A.2. We also reported in table 3.1 the number of structures educed by the algorithm described in subsection 3.2.2.2 for the three negative threshold values -0.2u ′ 300 + , -0.4u but we checked that they remain very similar without deviations from our conclusions if the threshold u th is chosen in the range -0.2u ′ 300 + to -0.6u ′ 300 + .

A.3 Behaviour of positive fluctuating streamwise velocity structures

As mentioned in subsection 3.2.2.2, this detection analysis can be repeated equally well on structures of positive streamwise fluctuating velocity. We provide examples of results obtained with u th = 0.4u ′ 300 + in figure A.3 and table A.1. There are indeed no significant differences in the results for the low and high speed attached flow regions, except for a lower but consistent value of C 1 and for a consistently lower value of C 2 in the lower Re θ case. Figures 3.19 to 3.22 can be reproduced for this positive threshold u th = 0.4u ′ 300 + and show the exact same trend (refer figures A.4 to A.7) with p increasing while q is decreasing with increasing y + as shown in figure A.8. However, whereas p takes values similar to those for u th = -0.4u ′ 300 + in the lower Re θ case, it does not do so in the higher Re θ case. As a result p + q is quite close to -1 in the lower Re θ case but less so, and in fact closer to -1.1 on average, for the higher Re θ . A steady-state, two-dimensional, incompressible pressure based solver is used with the default settings of the StarCCM+ package. The linear pressurestrain model developed by [START_REF] Gibson | Ground effects on pressure fluctuations in the atmospheric boundary layer[END_REF] was used for this computation. Although not the most advanced model available, it is typical of models generally available in commercial packages. Other models may have produced better results for the stresses or mean flow or skin friction coefficient, however the idea here was to obtain a quick first guess of the flow behaviour using a commercially available software package. At the walls, the RST model used the enhanced wall treatment option which employs a twolayer formulation for the near wall region. The closure modelling techniques employed by StarCCM+ can be found in the StarCCM+ users manual.

Figure B

.1 shows a sketch of the geometry over which the flow was to be simulated. The flat plate generating the FPG was set at an inclination of 1.5 • while the plate generating the APG was set at -5 • with respect to the wind tunnel floor. The dimensions of this geometry was set up to have the same physical dimensions as the wind tunnel flow, from the upstream to the final downstream location. A mesh was generated using the trimmer and prism layer meshers present in StarCCM+'s meshing tool. The prism layer mesher was used to capture the complex and intricate features near the wall. The inlet boundary conditions (U + , k + and ǫ + ) were set from experimental data measured 7m (i.e 2.4m upstream of the ramp described in Chapter 4) from the entrance of the wind tunnel on the flat plate. The no-slip condition was used as the boundary condition for the top and bottom wall. The boundary conditions at the inlet and outlet were set as velocity inlet and pressure outlet respectively.

  . . . . . . . . . . . . . . . . . . . . 46 3.1 Schematic of the LML boundary layer wind tunnel . . . . . . 67 3.2 Set-up of the optics for the PIV experiment . . . . . . . . . . 69 3.3 Picture of the calibration target reconstructed from the four camera images. . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.4 PIV uncertainty on the streamwise velocity component, U at U ∞ = 10 m/s . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.5 PIV uncertainty on the turbulence intensity u ′ at U ∞ = 10 m/s 72 3.6 Streamwise turbulence intensity field (u ′ /U ∞ ) for Re θ = 20600 where U ∞ = 10 m/s. . . . . . . . . . . . . . . . . . . . . . . . 73 3.7 Mean and rms streamwise fluctuating velocity profiles . . . . . 74 3.8 Comparison of the streamwise energy spectra obtained from PIV and hot-wire anemometry . . . . . . . . . . . . . . . . . . 75 3.9 Two-point spatial correlation function of the streamwise velocity fluctuations with varying wall distance at Re θ = 20600 . 76 3.10 Two-point spatial correlation function of the streamwise velocity fluctuations with varying wall distance at Re θ = 8100 . 77 18 LIST OF FIGURES 3.11 Two-point correlation function of the streamwise velocity fluctuations at Re θ = 8100 from Tutkun et al. (2009) . . . . . . . 78 3.12 Wall-attached elongated streaky structure eduction method applied on a sample instantaneous streamwise velocity field at Re θ = 20600 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.13 The average streamwise fluctuating velocity α and the streamwise length λ of the detected wall-attached structures at y/δ = 0.03 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.14 The threshold parameter, u th = ±0.4u′ 300 + . . . . . . . . . . . 82 3.15 PDFs of streamwise lengths λ (see figure 3.13) for varying domain lengths at wall distance y + = 195 for Re θ = 20600 . . 83 3.16 PDFs of streamwise lengths λ of wall-attached structures (see figure 3.13) at selected wall distances . . . . . . . . . . . . . . 84 3.17 Log-log plots of pre-multiplied streamwise energy spectra at selected wall distances . . . . . . . . . . . . . . . . . . . . . . 87 3.18 Same as figure 3.17 in lin-log plots. . . . . . . . . . . . . . . . 88 3.19 Lin-lin plots of a 2 versus λ/δ (left) and streamwise energy spectra plotted at wall distances y + = 41, 64, 88 and 125 (from top to bottom) at Re θ = 8100. . . . . . . . . . . . . . . 90 3.20 Lin-lin plots of a 2 versus λ/δ (left) and streamwise energy spectra plotted at wall distances y + = 150, 198, 246 and 306 (from top to bottom) at Re θ = 8100. . . . . . . . . . . . . . . 91 3.21 Lin-lin plots of a 2 versus λ/δ (left) and streamwise energy spectra plotted at wall distances y + = 90, 158, 195 and 268 (from top to bottom) at Reθ = 20600. . . . . . . . . . . . . . . 92 3.22 Lin-lin plots of a 2 versus λ/δ (left) and streamwise energy spectra plotted at wall distances y + = 305, 450, 630 and 740 (from top to bottom) at Re θ = 20600. . . . . . . . . . . . . . . 93 3.23 (a) Exponents p obtained from the best power-law fit of a 2 ∼ (λ/δ) p . (b) Exponents q obtained from the best power-law fit of E 11 ∼ k q x . (c) p + q versus y + . . . . . . . . . . . . . . . . . 97 LIST OF FIGURES 3.24 An example of a detected wall-attached flow structure for Re θ = 20600 and the u(x) signal through this structure at three different y + positions . . . . . . . . . . . . . . . . . . . . 3.25 Fit of u ′+ from equation 3.19 over the PIV data shown in figure 3.7 (bottom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Sketch of the ramp model showing the main dimensions. . . . 4.2 The ramp model placed in the LML boundary layer wind tunnel104 4.3 (a) Distribution of the streamwise pressure coefficient and (b) the streamwise pressure gradient along the ramp . . . . . . . . 4.4 Spanwise pressure distribution along the ramp. (a) 9 m/s (b) 5 m/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 (a) Distribution of the streamwise pressure coefficient and (b) the streamwise pressure gradient along the roof . . . . . . . . 4.6 Picture of the SPIV set-up used to obtain inlet boundary conditions at x = 6.8 m. . . . . . . . . . . . . . . . . . . . . . . . 4.7 Mean streamwise velocity field U in m/s for (a) U ∞ = 5 m/s and (b) U ∞ = 9 m/s . . . . . . . . . . . . . . . . . . . . . . . 4.8 Mean streamwise velocity profile obtained 2.6 m upstream of the ramp, along the plane of symmetry for both free stream velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Picture of the 2D2C set-up used to characterise the FPG flow on the 1.5 • plate. . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Evolution of the mean streamwise velocity profiles along the 1.5 • plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Evolution of the streamwise turbulence intensity profiles along the 1.5 • plate . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12 Evolution of the wall-normal turbulence intensity profiles along the 1.5 • plate . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13 Evolution of the Reynolds shear stress profiles along the 1.5 • plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.14 The set of 16 sCMOS cameras aligned in series and mounted on the side of the wind tunnel . . . . . . . . . . . . . . . . . . LIST OF FIGURES 4.15 Photo of the large field 2D2C PIV experimental set-up showing the laser sheet parallel to the direction of flow . . . . . . . . . 126 4.16 Mean streamwise velocity (U/U ∞ ) fields along the -5 • APG ramp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 4.17 Evolution of the mean streamwise velocity profiles along the -5 • APG ramp . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.18 Distribution of the friction velocity, U τ along the -5 • plate at U ∞ = 5 m/s and U ∞ = 9 m/s respectively . . . . . . . . . . . 130 4.19 Mean streamwise velocity profiles at the three stations: 0.343 m (station 1), 1.733 m (station 3) and 2.358 m (station 4), on the -5 • plate at U ∞ = 5 m/s . . . . . . . . . . . . . . . . . . 131 4.20 Evolution of δ, δ * , θ along the -5 • APG ramp at (a) U ∞ = 5 m/s and (b) U ∞ = 9 m/s. . . . . . . . . . . . . . . . . . . . . 135 4.21 Mean velocity deficit scaled with the free-stream velocity U e at U ∞ = 9 m/s . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.22 Mean velocity deficit scaled with the ZS scale U e δ * δ at U ∞ = 9 m/s, with the wall normal distance, 'y' normalised by (a) δ 95 and (b) δ 99 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.23 Evolution of the boundary layer thickness, δ along the -5 • APG ramp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.24 Pressure gradient parameter, Λ along the -5 • APG ramp at U ∞ = 5 m/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 4.25 Streamwise turbulence intensity (u ′ /U ∞ ) fields along the -5 • APG ramp (a) U ∞ = 5 m/s (b) U ∞ = 9 m/s . . . . . . . . . . 141 4.26 Evolution of the streamwise turbulence intensity (u ′ /U ∞ ) profiles along the -5 • APG ramp (a) U ∞ = 5 m/s, (b) U ∞ = 9 m/s. δ 0 as mentioned in section 4.3.2. . . . . . . . . . . . . . . 142 4.27 Wall-normal turbulence intensity (v ′ /U ∞ ) fields along the -5 • APG ramp (a) U ∞ = 5 m/s (b) U ∞ = 9 m/s . . . . . . . . . . 143 4.28 Evolution of the wall-normal turbulence intensity (v ′ /U ∞ ) profiles along the -5 • APG ramp (a) U ∞ = 5 m/s, (b) U ∞ = 9 m/s. δ 0 as mentioned in section 4.3.2 . . . . . . . . . . . . . . 144 LIST OF FIGURES 4.29 Reynolds shear stress (u ′ v ′ /U 2 ∞ ) fields along the -5 • APG ramp (a) U ∞ = 5 m/s (b) U ∞ = 9 m/s . . . . . . . . . . . . . 4.30 Evolution of the Reynolds shear stress (u ′ v ′ /U 2 ∞ ) profiles along the -5 • APG ramp (a) U ∞ = 5 m/s, (b) U ∞ = 9 m/s. δ 0 as mentioned in section 4.3.2 . . . . . . . . . . . . . . . . . . . . 4.31 Distribution of the streamwise turbulence intensity . . . . . . 4.32 Distribution of the wall-normal turbulence intensity . . . . . . 4.33 Distribution of the Reynolds shear stress . . . . . . . . . . . . 4.34 Distribution of skewness at station 4 for U ∞ = 5 m/s . . . . . 4.35 Distribution of flatness at station 4 for U ∞ = 5 m/s . . . . . . 4.36 Production term -uv ∂U ∂y of 1 2 u 2 . . . . . . . . . . . . . . . . . 4.37 Production term -u 2 ∂U ∂x of 1 2 u 2 . . . . . . . . . . . . . . . . . 4.38 Turbulence production term -uv ∂U ∂y of ( 1 2 u 2 ) . . . . . . . . . . 4.39 Weighted probability density functions of -uv . . . . . . . . 4.40 Weighted probability density functions of -uv in the log region157 4.41 Two-point spatial correlation function of the streamwise velocity fluctuations (R uu ) with varying wall distance . . . . . . 4.42 Streamwise two-point spatial correlation function (R uu ) at station 3 along the APG ramp . . . . . . . . . . . . . . . . . . . 4.43 PDFs of streamwise lengths λ of wall-attached structures at selected wall distances for U ∞ = 9 m/s in the third region . . . 4.44 PDFs of streamwise lengths λ of wall-attached structures . . . 4.45 Plots of a 2 versus λ/δ (left) and streamwise energy spectra plotted at wall distances y + = 100, 200 and 400 (from top to bottom) at U ∞ = 9 m/s. . . . . . . . . . . . . . . . . . . . . . A.1 Image 11 from the instantaneous PIV snapshots . . . . . . . . A.2 PDFs of streamwise lengths λ of wall-attached structures at y + = 195 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 PDFs of streamwise lengths λ of wall-attached structures at selected wall distances . . . . . . . . . . . . . . . . . . . . . . List of Tables 3.1 Number of structures detected over a set of thresholds for Re θ = 20600 and Re θ = 8100 . . . . . . . . . . . . . . . . . . 81 3.2 Values of the constants C 1 and C 2 in the form -C 1 +C 2 (λ/δ) -2 of the PDF of λ/δ . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.1 Inlet boundary layer characteristics. . . . . . . . . . . . . . . . 116 4.2 Boundary layer characteristics at different stations on the 1.5 • plate at U ∞ = 5 m/s. . . . . . . . . . . . . . . . . . . . . . . . 116 4.3 Boundary layer characteristics at different stations on the 1.5 • plate at U ∞ = 9 m/s. . . . . . . . . . . . . . . . . . . . . . . . 116 4.4 Values of U τ at U ∞ = 5 m/s . . . . . . . . . . . . . . . . . . . 132 4.5 Values of U τ at U ∞ = 9 m/s . . . . . . . . . . . . . . . . . . . 132 4.6 Boundary layer characteristics at different stations on the -5 • plate at U ∞ = 5 m/s. . . . . . . . . . . . . . . . . . . . . . . . 133 4.7 Boundary layer characteristics at different stations, 's' on the -5 • plate at U ∞ = 9 m/s. . . . . . . . . . . . . . . . . . . . . 134 4.8 The start, end, length and δ mid of the four regions obtained through the structure detection algorithm . . . . . . . . . . . 161 4.9 Values of C 1 , C 2 and r in the form -C 1 + C 2 (λ/δ) -r of the PDF of λ/δ from figure 4.43 at U ∞ = 9 m/s . . . . . . . . . . 162 A.1 Values of the constants C 1 and C 2 in the form -C 1 +C 2 (λ/δ) -2 of the PDF of λ/δ. . . . . . . . . . . . . . . . . . . . . . . . . 176

Figure 2 . 1 :

 21 Figure 2.1: Sketch showing the various regions of a turbulent boundary layer flow adapted from George et al. (1997)

Figure 2

 2 Figure 2.2: (a) Conceptual model of the kinematical relationships between ejection/sweep motions, streaks and quasi-streamwise vortices proposed by Robinson (1991) for low Reynolds number boundary layer flows. (b) sketch of a horseshoes vortex by[START_REF] Theodorsen | Mechanism of turbulence[END_REF]. (c) attached vortices concept of[START_REF] Townsend | The structure of turbulent shear flow[END_REF]. Figure is taken from[START_REF] Rahgozar | Coherent structures in a turbulent boundary layer under a strong adverse pressure gradient[END_REF]

  Figure 2.2: (a) Conceptual model of the kinematical relationships between ejection/sweep motions, streaks and quasi-streamwise vortices proposed by Robinson (1991) for low Reynolds number boundary layer flows. (b) sketch of a horseshoes vortex by[START_REF] Theodorsen | Mechanism of turbulence[END_REF]. (c) attached vortices concept of[START_REF] Townsend | The structure of turbulent shear flow[END_REF]. Figure is taken from[START_REF] Rahgozar | Coherent structures in a turbulent boundary layer under a strong adverse pressure gradient[END_REF]

Figure 2

 2 Figure 2.3: Large-scale turbulent bulges visualised by in the streamwise wallnormal plane by illuminating oil vapour with a light sheet performed by Falco(1977)

  Figure 3.1: Schematic of the LML boundary layer wind tunnel

Figure 3

 3 Figure 3.2: Set-up of the optics for the PIV experiment

Figure 3 .

 3 Figure 3.2 shows the optical set-up of the PIV experiment. A mirror inclined at 45 • was used to direct the beam from the laser output perpendicularly upwards through a telescope with a converging spherical lens of 800 mm focal length and 200 mm after which a diverging spherical lens of 600 mm focal length was placed to obtain a parallel laser beam of 6 mm in diam-

Figure 3

 3 Figure 3.4: PIV uncertainty on the streamwise velocity component, U at U ∞ = 10 m/s

Figure 3 . 6 :

 36 Figure 3.6: Streamwise turbulence intensity field (u ′ /U ∞ ) for Re θ = 20600 where U ∞ = 10 m/s.

Figure 3

 3 Figure 3.7: (a) Mean and (b) rms streamwise fluctuating velocity profiles at Re θ = 8100 (U ∞ = 3 m/s) and Re θ = 20600 (U ∞ = 10 m/s) obtained with PIV and compared with the hot wire anemometry results of Carlier and Stanislas (2005).

Figure 3

 3 Figure 3.8: Comparison of the streamwise energy spectra obtained from PIV and hot-wire anemometry at y + = 200 for Re θ = 20600. The hot-wire anemometry was made by Carlier and Stanislas (2005) at 19.6m from the test section entrance of the same wind tunnel.

Figure 3

 3 Figure 3.9: Two-point spatial correlation function of the streamwise velocity fluctuations with varying wall distance at Re θ = 20600. The contours range from 0.1 to 1 with an increment of 0.1.

Figures 3 .

 3 Figures 3.9 and 3.10 show the correlation map of the streamwise velocity fluctuations at various wall distances in the streamwise-wall-normal plane at Re θ = 20100 and Re θ = 8100 respectively. The correlation iso-contours

Figure 3 .

 3 Figure 3.10: Two-point spatial correlation function of the streamwise velocity fluctuations with varying wall distance at Re θ = 8100. The contours range from 0.1 to 1 with an increment of 0.1.

Figure 3

 3 Figure 3.12: Wall-attached elongated streaky structure eduction method applied on a sample instantaneous streamwise velocity field at Re θ = 20600. From top: (a) Raw instantaneous streamwise fluctuating velocity component field (b) The same field after application of a Gaussian filter (c) Binary image obtained after thresholding negative streamwise fluctuating momentum regions. (d) Final image obtained after cleaning as described in subsection 3.2.2.2

  Figure 3.14: Figure 3.7(b) reproduced to explain the threshold parameter, u th = ±0.4u ′ 300 + . The horizontal dash lines indicate u ′ at the respective Re θ while the vertical dash line in magenta indicates y + = 300

Figure 3

 3 Figure 3.15: PDFs of streamwise lengths λ (see figure 3.13) for varying domain lengths at wall distance y + = 195 for Re θ = 20600

  A total of 14493 and 19576 wall-attached binary structures were detected at Re θ = 20600 and Re θ = 8100 respectively. The model in section 3.1 assumes that the number density of wall-attached 84CHAPTER 3. LOW AND HIGH MOMENTUM REGIONS IN A ZPG TBL

Figure 3 .

 3 Figure 3.16: PDFs of streamwise lengths λ of wall-attached structures (see figure 3.13) at selected wall distances for Re θ = 20600 (top) and Re θ = 8100 (bottom). The fits shown here are for y + = 195 at Re θ = 20600 and y + = 198 at Re θ = 8100.

  Figure 3.17: Log-log plots of pre-multiplied streamwise energy spectra at selected wall distances for Re θ = 8100 (top) and Re θ = 20600 (bottom). Vertical lines indicate the wavenumbers corresponding to the length scale 1 δ

Figure 3

 3 Figure 3.18: Same as figure 3.17 in lin-log plots.

Figure 3 Figure 3 Figure 3 Figure 3

 3333 Figure 3.19: Lin-lin plots of a 2 versus λ/δ (left) and streamwise energy spectra plotted at wall distances y + = 41, 64, 88 and 125 (from top to bottom) at Re θ = 8100.

Figure 3

 3 Figure3.23: (a) Exponents p obtained from the best power-law fit of a 2 ∼ (λ/δ) p . (b) Exponents q obtained from the best power-law fit of E 11 ∼ k qx . (c) p + q versus y + . These fits are obtained over the range of scales investigated in figures 3.19 to 3.22 and the resulting exponents are plotted with the 95% confidence intervals for these fits. The y + positions and the two Reynolds numbers Re θ are those in figures 3.19 to 3.22.

Figure 3

 3 Figure 3.24: An example of a detected wall-attached flow structure for Re θ = 20600 and the u(x) signal through this structure at three different y + positions. The red line in the repeated binary image indicates the y + position where the signal u(x) is recorded (y + = 195, 450, 1110). The black/red line in the u(x) versus x/δ plots is the raw (un-filtered) PIV fluctuating streamwise fluctuating velocity outside/inside the detected flow structures. The dashed blue line is our model signal, equal to 0 outside and to the average value of u inside the detected structures.

  Figure 3.25: Fit of u ′+ from equation 3.19 over the PIV data shown in figure 3.7(b). Solid lines correspond to the fit given by equation 3.19 and circles are obtained from the PIV data at the two Re θ .

Figure 4

 4 Figure 4.3: (a) Distribution of the streamwise pressure coefficient and (b) the streamwise pressure gradient along the ramp

  Figure 4.5: (a) Distribution of the streamwise pressure coefficient and (b) the streamwise pressure gradient along the roof

  Figure 4.7: Mean streamwise velocity field U in m/s for (a) U ∞ = 5 m/s and (b) U ∞ = 9 m/s in a spanwise-wall normal plane at X = 6.8 m (2.6 m upstream of the ramp).

Figure 4

 4 Figure 4.9: Picture of the 2D2C set-up used to characterise the FPG flow on the 1.5 • plate.

Figure 4 .

 4 Figure 4.14: The set of 16 sCMOS cameras aligned in series and mounted on the side of the wind tunnel. The target used for calibration is placed along the centerline of the ramp

Figure 4 .

 4 Figure 4.15: Photo of the large field 2D2C PIV experimental set-up showing the laser sheet parallel to the direction of flow

  Figure 4.16: Mean streamwise velocity (U/U ∞ ) fields along the -5 • APG ramp (a) U ∞ = 5 m/s (b) U ∞ = 9 m/s.

Figure 4 .

 4 Figure 4.16 (a) and (b) shows the evolution of the mean streamwise velocity normalised with the reference free-stream velocity U ∞ along the -5 • plate for both velocities studied. In this section, s = 0 now corresponds to the beginning of the 3.5 m long flat plate inclined at -5 • with respect to the horizontal. Due to the enlargement of the test section along the APG plate, the flow decelerates continuously but does not separate. As expected for a boundary layer subjected to an APG, the thickness of the boundary layer increases rapidly such that the ratio between the local free-stream velocity at the beginning and at the end of the 3.5 m long field is 1.45.

Figure 4 .

 4 Figure 4.17: Evolution of the mean streamwise velocity profiles along the -5 • APG ramp (a) U ∞ = 5 m/s, (b) U ∞ = 9 m/s. δ 0 is 101 mm at 5 m/s and 95 mm at 9 m/s (see section 4.3.2)

  Figure 4.18: Distribution of the friction velocity, U τ along the -5 • plate at U ∞ = 5 m/s and U ∞ = 9 m/s respectively. Symbols: '+' has been obtained from time resolved high magnification (TRHM) PIV and solid lines have been obtained from the Clauser chart method using the large field streamwise (LFStW) PIV data.

Figure 4

 4 Figure 4.19: Mean streamwise velocity profiles at the three stations: 0.343 m (station 1), 1.733 m (station 3) and 2.358 m (station 4), on the -5 • plate at U ∞ = 5 m/s. Profile with the symbol 'o' corresponds to a ZPG flow case from Carlier and Stanislas (2005) at Re θ = 8100

Figure 4

 4 Figure 4.20: Evolution of δ, δ * , θ along the -5 • APG ramp at (a) U ∞ = 5 m/s and (b) U ∞ = 9 m/s.

Figure 4 .

 4 Figure 4.21: Mean velocity deficit scaled with the free-stream velocity U e at U ∞ = 9 m/s, with the wall normal distance, 'y' normalised by (a) δ 95 and (b) δ 99

Figure 4 .

 4 Figure 4.22: Mean velocity deficit scaled with the ZS scale U e δ * δ at U ∞ = 9 m/s, with the wall normal distance, 'y' normalised by (a) δ 95 and (b) δ 99

Figure 4

 4 Figure 4.23: Evolution of the boundary layer thickness (a) δ 95 (b) δ 99 along the -5 • APG ramp as a function of the streamwise coordinate (blue). The black line is a smoothed profile obtained by least square spline interpolation.

Figure 4 .

 4 Figure 4.24: Pressure gradient parameter, Λ and δ along the -5 • APG ramp at U ∞ = 5 m/s. (a) δ 95 (b) δ 99 . The line in magenta shows the region exhibiting the equilibrium range (Λ = constant) and the vertical black lines indicate the ranges over which equilibrium is exhibited

  Figure 4.25: Streamwise turbulence intensity (u ′ /U ∞ ) fields along the -5 • APG ramp (a) U ∞ = 5 m/s (b) U ∞ = 9 m/s

Figures

  Figures 4.26(a) and (b) show the evolution of the streamwise turbulence intensity profiles in the APG region at the same stations as in figure 4.16.The profiles at the first station is similar to the profile located at the last streamwise position on the FPG (see figure4.11) with a slight reduction of level which is more apparent near the wall, at the highest Reynolds number. At the second station (s = 0.7 m), a weak outer peak located at about y/δ 0 = 0.15, is induced by the change in sign of the pressure gradient. Moving downstream, this peak moves away from the wall with the peak magnitude

  Figures 4.26(a) and (b) show the evolution of the streamwise turbulence intensity profiles in the APG region at the same stations as in figure 4.16.The profiles at the first station is similar to the profile located at the last streamwise position on the FPG (see figure4.11) with a slight reduction of level which is more apparent near the wall, at the highest Reynolds number. At the second station (s = 0.7 m), a weak outer peak located at about y/δ 0 = 0.15, is induced by the change in sign of the pressure gradient. Moving downstream, this peak moves away from the wall with the peak magnitude

Figure 4 .

 4 Figure 4.26: Evolution of the streamwise turbulence intensity (u ′ /U ∞ ) profiles along the -5 • APG ramp (a) U ∞ = 5 m/s, (b) U ∞ = 9 m/s. δ 0 as mentioned in section 4.3.2.

Figures

  Figures 4.27(a) and (b) show the evolution of the wall-normal turbulence

  Figures 4.27(a) and (b) show the evolution of the wall-normal turbulence

Figure 4 .

 4 Figure 4.28: Evolution of the wall-normal turbulence intensity (v ′ /U ∞ ) profiles along the -5 • APG ramp (a) U ∞ = 5 m/s, (b) U ∞ = 9 m/s. δ 0 as mentioned in section 4.3.2

Figure 4 .

 4 Figure 4.30: Evolution of the Reynolds shear stress (u ′ v ′ /U 2 ∞ ) profiles along the -5 • APG ramp (a) U ∞ = 5 m/s, (b) U ∞ = 9 m/s. δ 0 as mentioned in section 4.3.2

Figure 4 Figure 4 CHAPTERFigure 4

 444 Figure 4.31: Distribution of the streamwise turbulence intensity profiles at the measurement stations 1, 3 and 4 on the -5 • APG ramp for U ∞ = 5 m/s symbols: Red line shows data from TRHM PIV, blue line from LFStW PIV and black 'o' corresponds to a ZPG flow at Re θ = 8100 from[START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF]

Figure 4

 4 Figure 4.34: Distribution of skewness at station 4 for U ∞ = 5 m/s scaled in (a) inner and (b) outer units. Hot-wire (HW FP) profiles from Carlier and Stanislas (2005) for ZPG flows at Re θ = 11500. Vertical line in magenta indicates the boundary layer thickness δ

Figure 4 .Figure 4

 44 Figure 4.35 shows the flatness, F, of the streamwise velocity fluctuations

  Figure 4.36: Production term -uv ∂U ∂y of 1 2 u 2 along the (a) APG at U ∞ = 5 m/s (b) APG at U ∞ = 9 m/s (c) ZPG at U ∞ = 10 m/s corresponding to Re θ ≈ 20600

  Figure 4.37: Production term -u 2 ∂U ∂x of 1 2 u 2 along the APG at (a) U ∞ = 5 m/s (b) U ∞ = 9 m/s

Figure 4 .

 4 Figure4.38 shows the turbulence production term -uv ∂U ∂y of 1 2 u 2 . The effect of the pressure gradient in the case of a mild APG is noticeable. The APG leads to an increased inner peak in the production profile and a moderate increase in production in the outer region as Reynolds number increases. Furthermore, there is a weak emergence of a second peak in the case of APG which moves outward.[START_REF] Skåre | A turbulent equilibrium boundary layer near separation[END_REF] also observed 2 peaks in the production term while studying a strong APG flow with β ≈ 20. In their study, they claimed that the inner peak was due to the mean strain as the wall was approached while the outer peak in the turbulent stresses which they noted was caused by the strong APG in their experiment. In figure4.38, the profile marked in black shows the production term obtained from the large field PIV (refer chapter 4) at Re θ ≈ 8100 in a ZPG turbulent boundary layer. Although the near wall region below y + < 40 isn't captured by the measurement, it is expected to be lower than the APG case. Globally though, it is clear that production in an APG turbulent boundary layer is higher than that of a ZPG.

Figure 4

 4 Figure 4.38: Turbulence production term -uv ∂U ∂y of ( 1 2 u 2 ). Symbols: Red line -TRHM PIV, blue line -LFStW PIV. Data correspond to Re θ = 8860 (station 1 U ∞ = 5 m/s), 10630 (station 3, U ∞ = 5 m/s), 14560 (station 4, U ∞ = 9 m/s). Black line corresponds to PIV data at Re θ = 8100 (Refer chapter 3).

  Sections 4.4.2 and 4.4.3 dealt with a detailed characterisation of a turbulent boundary layer subjected to a pressure gradient. Through two-point correlation functions and structure detection techniques, the study is continued looking at how the coherent structures are influenced in the presence of a pressure gradient.

  Figure 4.39: Weighted probability density functions of -uv in an APG flow at station 3 for U ∞ = 5 m/s at (a) y + = 100 (b) y/δ = 0.4. In the contour maps, blue and red lines represent positive and negative values respectively, starting at 0.01 with a 0.01 interval between successive contour lines.

Figure 4 .Figure 4

 44 Figure 4.40: Weighted probability density functions of -uv in the log region (y + = 50, y/δ = 0.1). the interval between contour lines is 0.01. (a) ZPG flow (P + = 0) (b) APG flow with P + = 3.08 × 10 -2 . Plot reproduced from Nagano et al. (1998)

  Values of C 1 , C 2 and r in the form -C 1 + C 2 (λ/δ) -r of the PDF of λ/δ from figure 4.43 at U ∞ = 9 m/s

Figure 4

 4 Figure 4.43: PDFs of streamwise lengths λ of wall-attached structures at selected wall distances for U ∞ = 9 m/s in the third region

Figure 4

 4 Figure 4.45: Plots of a 2 versus λ/δ (left) and streamwise energy spectra plotted at wall distances y + = 100, 200 and 400 (from top to bottom) at U ∞ = 9 m/s.

166

  CHAPTER 5. INFLUENCE OF APG ON THE LSSin principle be applied to various wall-bounded turbulent flows is proposed from a new perspective based on the work of Townsend-Perry.

Figure A. 3 :FigureFigureFigureFigure

 3 Figure A.3: PDFs of streamwise lengths λ of wall-attached structures at selected wall distances for (a) Re θ = 20600 and (b) Re θ = 8100 . The fits shown here are for y + = 195 at Re θ = 20600 and y + = 198 at Re θ = 8100.

Figure B. 2

 2 Figure B.2 shows the evolution of the mean streamwise velocity along the ramp. The plate inclined at 1.5 • generated a slight favourable pressure gradient causing an acceleration of the flow. The -5 • plate generating the APG then decelerates the flow continuously but does not separate. This is also supported by looking at the plots of the skin friction coefficient, C f along the ramp in figure B.3 which indicates that the flow is far from incipient separation. As expected for a boundary layer subjected to an APG, the thickness of the boundary layer, δ increased continuously along the APG

Table 3 .

 3 Figures 3.15 and 3.16 show the probability distribution function (PDF) of lengths λ at various wall distances. The most probable length λ lies be-2: Values of the constants C 1 and C

	Re θ		20600			8100
	y +	90	195 305 450 630	52	88	125 198 306
	C 1	0.03 0.03 0.03 0.03 0.03 0.04 0.04 0.04 0.03 0.03
	C 2	0.32 0.35 0.35 0.37 0.37 0.32 0.35 0.35 0.35 0.33
	lower bound 0.49 0.55 0.54 0.58 0.78 0.54 0.53 0.53 0.56 0.65
	upper bound 3.8 3.8 3.8 3.8 3.8 3.4 3.4 3.4 3.4 3.4

Table 4

 4

	s)	β

Station s (m) U e (m/s) δ (mm) δ * (mm) θ (mm) H .2: Boundary layer characteristics at different stations on the 1.5 • plate at U ∞ = 5 m/s. Station s (m) U e (m/s) δ (mm) δ * (mm) θ (mm) H

Table 4 .

 4 TRHM PIV U τ (m/s) LFStW PIV U τ (m/s) 5: Values of U τ from TRHM (time resolved high magnification) PIV and LFStW (Large field streamwise) PIV at the measurement stations measured from the start of the APG ramp at U ∞ = 9 m/s

	Station 1 (s = 0.48 m)	0.26	0.261
	Station 3 (s = 1.73 m)	0.21	0.204
	Station 4 (s = 2.35 m)	0.19	0.186
	Table 4.4: Values of U τ from TRHM (time resolved high magnification) PIV
	and LFStW (Large field streamwise) PIV at the measurement stations mea-
	sured from the start of the APG ramp at U ∞ = 5 m/s
		TRHM PIV U τ (m/s) LFStW PIV U τ (m/s)
	Station 1 (s = 0.48 m)	0.46	0.445
	Station 3 (s = 1.73 m)	0.36	0.355
	Station 4 (s = 2.35 m)	0.32	0.327

  the Reynolds number based on the momentum s (m) U e (m/s) δ (mm) δ * (mm) θ (mm) H

						Re θ	β
	0	7.70	66	3.9	2.9	1.34 1470 -0.26
	0.7	7.22	135	18.7	13.4	1.40 6430 1.42
	1.4	6.81	161	26.1	18.0	1.45 8180 2.04
	2.1	6.47	183	33.9	22.8	1.49 9840 2.01
	2.8	6.13	205	44.2	28.4	1.56 11600 3.68
	3.4	5.87	231	58.7	35.2	1.67 13790	-
	Table 4.6: Boundary layer characteristics at different stations on the -5 •
	plate at U ∞ = 5 m/s.
	thickness and β = ∂P ∂s	δ * ρU 2 τ

Table 4 .

 4 for the outer part 134 CHAPTER 4. APG TURBULENT BOUNDARY LAYER FLOW s (m) U e (m/s) δ (mm) δ * (mm) θ (mm) H 7: Boundary layer characteristics at different stations, 's' on the -5 • plate at U ∞ = 9 m/s. of ZPG boundary layers to include boundary layers with pressure gradient.

	Re θ	β

Table 4 .

 4 8: The start, end and length of the four regions obtained through the structure detection algorithm at U ∞ = 9 m/s. δ mid corresponds to the value of δ at the middle of the respective regions.end and length of these regions is reported in table 4.8. The value of δ at the midsection of these regions is also tabulated. The strong variation in δ upstream of the ramp explains why the first region is of shorter length compared to the other three. Also, regions 3 and 4 fall under the equilibrium zone established through the Castillo and George pressure parameter (refer section 4.4.2.4). With an interest of studying APG flows under equilibrium, regions 3 and 4 are of interest. Covering a longer distance, region 3 is taken to analyse the statistics of the structures detected.

	The beginning,

  ′ 300 + and -0.6u ′ 300 + . Figures 3.19 to 3.22 have been obtained for u th = -0.4u ′

	300 +

Acknowledgements

Résumé

Nomenclature

Acknowledgements

CHAPTER 5. INFLUENCE OF APG ON THE LSS

with appropriate tracers like soap bubbles.

of convergence can be observed, linked to the fact that the homogeneity of the flow can not be used anymore to increase the number of samples. The spatial correlation extends over a streamwise distance of 5δ at the various wall-normal locations based on a 0.1 correlation value. The general trend that can be observed from figure 4.41 is that the correlation function appears approximately as an ellipse and is strongly elongated in the streamwise direction. Compared to a ZPG flow, the correlations are slightly more inclined in the APG case at y + = 200 and they become broader and less inclined when moving from the wall. The largest structures also seem longer in APG.

To gauge the effect of the pressure gradient, a slice of R uu at ∆y = 0 or the auto-correlation at four selected wall distances as shown in 4.42 are compared to the ZPG flow at Re θ = 8100 described in Chapter 3. The plots in figure 4.42 indicate that the streamwise length of the correlations are comparable at all investigated wall-normal positions. The correlation with APG being shorter at y + = 200 in agreement with [START_REF] Nagano | Statistical characteristics of wall turbulence with a passive scalar[END_REF] and [START_REF] Monty | A parametric study of adverse pressure gradient turbulent boundary layers[END_REF] and as shown in figure 4.41, the correlation of the longest structures seem longer in the outer region at y + = 1000 corresponding to y/δ ≈ 0.4 for this Reynolds number.

There isn't a clear picture in the turbulence community regarding the The convolution kernel for the two-dimensional Gaussian filter was created using the function, "fspecial" from Matlab. The default value for the kernel size is [3 3] pixels (A pixel here refers to the grid spacing between velocity vectors in the PIV images) and the standard deviation, 'σ = 0.5'. The value of σ dictates the shape and size of the kernel. Truncating the Gaussian kernel at [3 3] pixels creates a sharp border that will cause unwanted artefacts in the output image thus losing the properties of a Gaussian filter. Cutting a Gaussian at a point where it is close to zero happens around 3σ meaning that the size of the Gaussian kernel should be 6σ, cutoff at 3σ on either side of the origin. Proceeding on this track, two Gaussian filters: a moderate filter having a σ = 1 (kernel [6 6] pixels) and a strong filter having a σ = 3 (kernel [18 18] pixels) was chosen in the present study. The respective filtered images were then compared to a case without any filter being applied, to ensure that there is no significant bias in the results and to also ensure that the filter doesn't modify the overall shape of the turbulent structures at large. To visually examine the effect of filtering, the moderate and strong Gaussian filters were tested on a few PIV images.

APPENDIX B RANS modeling of APG flow

In order to prepare the experiment described in Chapter 4, computational modelling using Reynolds-Averaged Navier-Stokes (RANS) was performed using the commercial CFD package, StarCCM+. Compared to ZPG turbulent boundary layer flows, additional complexities due to the variation of pressure gradient, curvature effects and flow separation in the case of APG turbulent boundary layer flows make the modelling of APG flows a challenge. A number of studies have been performed in the last few decades with an objective to predict the APG flow behaviour using RANS modelling [START_REF] Menter | Two-equation eddy-viscosity turbulence models for engineering applications[END_REF]; [START_REF] Shih | A new k-ǫ eddy viscosity model for high Reynolds number turbulent flows[END_REF]; [START_REF] Menter | Ten years of industrial experience with the SST turbulence model[END_REF]; de [START_REF] De Jesus | Large eddy simulations and RANS computations of adverse pressure gradient flows[END_REF]; [START_REF] Manceau | Recent progress in the development of the Elliptic Blending Reynolds-stress model[END_REF]). The standard k -ǫ model has generally failed to capture the physics of flows with pressure gradients. k -ω models have given better results than the standard k -ǫ models in APG flows and predicted the location of separation better. The recent work by de Jesus et al. ( 2013) presented comparative results from Large Eddy Simulations (LES) and RANS compared to available DNS data, in channel flows with an APG created by a two-dimensional bump. Although the RANS models that were tested did not perform on par with LES, results from the Reynolds-stress transport (RST) model showed the best performance among the RANS models evaluated. The transport equations in RST model contain the main physical mechanisms that drive turbulence. In particular, the production terms that are sufficient to explain many phenomena do not require modelling. It was thus decided to carry out the simulation of the APG flow using a RST model. Two Hamamatsu 2k by 2k cameras fitted with Nikon 50 mm lenses were used in Scheimpflug conditions and symmetric forward scattering. Figure C.1 shows a picture of the set-up used. The light sheet, about 2.5 mm in thickness, was introduced into the wind tunnel through the opposite wall and was normal to the wind tunnel floor. It was tuned tangent to the wall surface to minimize reflections. The cameras' aperture was f # = 5.6. As before, tests were performed for the two velocities of the experiment (5 and 9 m/s measured at the wind tunnel entrance). For each test case, 4000 velocity fields were acquired to ensure good convergence for the mean flow. The data was processed in similar way to the upstream conditions. The multigrid/multipass cross-correlation analysis [START_REF] Willert | Digital particle image velocimetry[END_REF]; [START_REF] Soria | An investigation of the near wake of a circular cylinder using a video-based digital cross-correlation particle image velocimetry technique[END_REF]) was also done with four passes starting with 48 x 64 pixels and ending with 16 x 24 pixels which was found to be the optimal size of the interrogation window. The final interrogation window corresponded to 3.5

APPENDIX D

Two-point correlation of wall-normal velocity component in APG ramp flow (R vv ) Figure D.1 shows the two-point spatial correlation for the wall-normal velocity fluctuation R vv at four wall-normal locations, at station 3 for U ∞ = 5m/s. First, there are clear differences between R uu (refer figure 4.41) and R vv , in that, the correlations for R vv are found to have limited spatial extent and they almost aligned with the y axis and slightly elongated in the vertical direction. This is attributed to the inhibiting influence of the wall on the wall-normal fluctuations, v [START_REF] Krogstad | Influence of a strong adverse pressure gradient on the turbulent structure in a boundary layer[END_REF]; [START_REF] Krogstad | Structure of turbulent boundary layers on smooth and rough walls[END_REF]).

Comparing R vv for ZPG and APG flows in figure 4.39, a striking similarity at y + = 200 is observed. The correlation for APG continues to have a slight overhang region in the streamwise direction up to 400 wall units while an overhang region is present in both the upstream and downstream directions for all wall distances in the ZPG flow. Also notable is the limited spatial extent of R vv for APG compared to ZPG. This is clearly visible when the autocorrelation of R vv is plotted in figure D.2. Up to y + = 200, R vv shows a similar behaviour and as we move further away from the wall, the streamwise extent of the correlation increases for a ZPG flow compared to a flow with a pressure gradient.