Scuola di Ingegneria Industriale e dell’Informazione Corso ...per canale turbolento a numero di Reynolds viscoso pari a Re ˝ = 200, Re ˝ = 500 e Re ˝ = 1000, sono stati considerati

POLITECNICO DI MILANO

Scuola di Ingegneria Industriale e dell’Informazione

Corso di Laurea Magistrale in Ingegneria Aeronautica

Second-order structure function tensor budgets

for channels at different values of Reynolds

number

Relatore: Prof. Maurizio QUADRIOCo-relatore: Dr. Ing. Davide GATTI

Tesi di laurea di:Marco VillaniMatr. 878932

Anno Accademico 2018 - 2019

Abstract

The present work describes the processes of production, dissipation andtransport of the Reynolds stress components in turbulent channel flow forthree values of the Reynolds number, considering simultaneously the spaces ofscales and positions. The tool used to perform the analysis is the AnisotropicGeneralized Kolmogorov Equation (AGKE), a statistical budget equationfor the second-order structure function 〈δuiδuj〉, where δui is the incre-ment of the ith velocity component at position X and separation r, i.e.δui = (ui(X + r/2, t)− ui(X − r/2, t)).For the computation of the AGKE terms, three DNS-generated databases forturbulent channel flow at friction Reynolds number Reτ = 200, Reτ = 500and Reτ = 1000 were considered.The analysis highlighted the presence of three peaks in the production of〈δuδu〉. The first, common to all the three Reynolds numbers, involves thestructures of the near-wall cycle. The second, observed for Reτ = 500 andReτ = 1000 but not for Reτ = 200, was associated to the presence of attachededdies. The third, observed only for Reτ = 1000, was linked to the so-calledlarge- and very large-scale motions. In the region of the first peak, near to thewall, the presence of transport processes of 〈δuiδuj〉 that do not scale in wallunits highlighted the interaction with the outer scales. In the second peak,proportionality between the scale and the wall-normal distance of some trans-port processes and the inactivity at the wall of motions transporting 〈δvδv〉were observed features. Moreover, an observed transfer of 〈δwδw〉 towardsscales more and more spanwise-oriented for increasing wall-normal distanceswas associated to the presence of hairpin vortices in the flow. Finally, somefeatures of this second peak were found to be visible also for Reτ = 200,suggesting a continuous transition from low- to high-Reynolds number flows.The third peak of production of 〈δuδu〉 was also characterized, together withthe associated transport of energy.

i

ii ABSTRACT

The complete description provided by the AGKE in the compound space ofscales and positions makes it an effective tool to study turbulent flows inpresence of inhomogeneities. This description is even more important forhigh Reynolds numbers, for which the energy is distributed among a largernumber of scales in different points of the physical space. In particular, itappears to be beneficial from both a theoretical and a modelling perspective.For example, it may be used to study flows to which drag reduction tech-niques have been applied, or to refine existing models used in large-eddiessimulations, reproducing the effect of the small unresolved scales on the re-solved motion.

Sommario

Il presente lavoro descrive i processi di produzione, dissipazione e traspor-to delle componenti del tensore degli sforzi di Reynolds in correnti turbo-lente delimitate da pareti per tre valori del numero di Reynolds, consi-derando simultaneamente lo spazio delle scale e delle posizioni. Lo stru-mento utilizzato per effettuare l’analisi e la Anisotropic Generalized Kol-mogorov Equation (AGKE), un’equazione di bilancio statistica per la fun-zione di struttura del second’ordine 〈δuiδuj〉, in cui δui e l’incremento del-la i-esima componente di velocita nella posizione X e separazione r, i.e.δui = (ui(X + r/2, t)− ui(X − r/2, t)). Tale equazione permette di coglierei processi di trasporto di 〈δuiδuj〉 che avvengono simultaneamente tra flut-tuazioni di diversa scala e diverse regioni della corrente.Per il calcolo dei termini dell’AGKE tre databases, prodotti mediante DNSper canale turbolento a numero di Reynolds viscoso pari a Reτ = 200,Reτ = 500 e Reτ = 1000, sono stati considerati.Tale analisi ha evidenziato la presenza di tre picchi nella produzione di〈δuδu〉. Il primo, comune a tutti e tre i numeri di Reynolds, coinvolge lestrutture del ciclo di parete. Il secondo, osservato a Reτ = 500 e Reτ = 1000ma non a Reτ = 200, e stato associato alla presenza di vortici attached. Ilterzo, osservato solo per Reτ = 1000, e stato attribuito ai cosiddetti motidi grande e grandissima scala. Nella regione del primo picco, vicina a pare-te, la presenza di processi di trasporto di 〈δuiδuj〉 che non scalano in unitaviscose ha evidenziato l’interazione con le scale esterne. Nel secondo picco,la proporzionalita tra la scala e la distanza da parete di alcuni processi ditrasporto e l’inattivita a parete dei moti che trasportano 〈δvδv〉 sono caratte-ristiche osservate. Inoltre, l’individuato trasferimento di 〈δwδw〉 verso scalecon orientazione sempre piu trasversale alla corrente all’aumentare della di-stanza da parete e stato associato alla presenza di hairpin vortices. Infine,alcune caratteristiche di questo secondo picco sono state osservate anche per

iii

iv SOMMARIO

Reτ = 200, supportando l’idea di una transizione continua da bassi ad altinumeri di Reynolds. Il terzo picco di produzione di 〈δuδu〉 e stato inoltrecaratterizzato, insieme al trasporto di energia a questo associato.L’AGKE, offrendo una descrizione esaustiva nello spazio composto delle sca-le e delle posizioni, risulta costituire uno strumento efficace per lo studio dicorrenti turbolente in presenza di disomogeneita. Tale descrizione acquistaimportanza ad elevati numeri di Reynolds, per i quali l’energia e distribuitatra un maggiore numero di scale in diversi punti dello spazio fisico. In par-ticolare, puo avere un impatto rilevante sia da un punto di vista teorico siaper la modellazione. Per esempio, potrebbe essere impiegata per lo studio dicorrenti alle quali sono state applicate tecniche di drag reduction, o per affi-nare modelli esistenti utilizzati nelle simulazioni LES, riproducendo l’effettodelle piccole scale non risolte su quelle risolte.

Contents

Abstract i

Sommario iii

1 Introduction and aim of the work 1

2 Fundamentals and state of the art 32.1 Coherent structures . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Near-wall coherent structures . . . . . . . . . . . . . . 42.1.2 The attached eddies hypothesis . . . . . . . . . . . . . 42.1.3 Large and very large-scale motions . . . . . . . . . . . 5

2.2 Structure function equations in wall-bounded turbulence . . . 6

3 Numerical methods and procedures 9

4 Results 154.1 The inner peak of ξ11 . . . . . . . . . . . . . . . . . . . . . . . 154.2 The second peak of ξ11 . . . . . . . . . . . . . . . . . . . . . . 30

4.2.1 From Reτ = 200 to Reτ = 1000: a dynamic view . . . . 354.3 The third peak of ξ11 . . . . . . . . . . . . . . . . . . . . . . . 47

5 Conclusions and future developments 53

v

vi CONTENTS

List of Figures

3.1 Sketch of the channel flow in a Cartesian reference system.Lx, Lz and 2h are the size of the computational domain in thestream-wise, span-wise and wall-normal directions. . . . . . . . 10

3.2 Comparison between the data used in this work ( ) and thoseused in [32] ( ) for Reτ = 1000: velocity profile U (on theleft) and single-point statistics 〈u2〉 (on the right), both givenin wall units. . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3 Incremental averages of ξ11 for Reτ = 1000 on the r+x = 0

plane. The colours represent the number of averaged fields n,spanning from 1 to 37, as shown by the colorbar. On the left,moving from the outside to the inside, the plotted isolines areξ11 = 0.2, ξ11 = 0.3, ξ11 = 0.4 and ξ11 = 0.5 respectively. Onthe right, the two largest sets of isolines are ξ11 = 0.0001, theone at Y + ∼ 200 is ξ11 = 0.006 and the sets in the inner layersare, moving from the outside to the inside, ξ11 = 0.006 andξ11 = 0.3 respectively. . . . . . . . . . . . . . . . . . . . . . . . 11

4.1 Three peaks of ξ11 for Reτ = 1000: isosurfaces ξ11 = 0.5( ), ξ11 = 0.006 ( ) and ξ11 = 0.0001 ( ). Of these two lastisosurfaces, only the part for Y + > 30 is illustrated. A detailof the isosurfaces ξ11 = 0.5 is also represented. . . . . . . . . . 16

4.2 Inner peak of ξ11: isosurface ξ11 = 0.5 for Reτ = 200 ( ),Reτ = 500 ( ) and Reτ = 1000 ( ). . . . . . . . . . . . . . . . 16

4.3 Dominant components of ∇ · Φ11 balancing the inner peak

of ξ11: isosurfaces T//11 = −0.05 ( ), −∂Φdiff

rz

∂rz= −0.03 ( ),

−∂Ψturb

∂Y= −0.14 ( ), −∂Ψdiff

∂Y= −0.15 ( ) and −∂Φry

∂ry= −0.25

( ) for Reτ = 200, Reτ = 500 and Reτ = 1000 respectively. . . 17

vii

viii LIST OF FIGURES

4.4 Transport processes in the inner layer for Reτ = 200. On theleft: isosurfaces Π11 = −0.13 ( ), T

//11 = 0.05 ( ), T

//11 = −0.05

( ) and −∂Ψturb

∂Y= −0.02 ( ). On the right: isosurface Ψ = 0

( ) and flux lines of 〈δuδu〉: recall that these lines are locallyparallel to the flux vector Φ11 = (Φrx ,Φrz ,Ψ) and colouredwith its magnitude. . . . . . . . . . . . . . . . . . . . . . . . . 18

4.5 Top row: isosurface Ψ = 0 ( ) and flux lines of 〈δuδu〉. Bot-tom: isosurface Ψ = 0 ( ) and flux lines of the scale energy〈δuiδui〉 (where repeated index implies summation). Left col-umn: Reτ = 200. Right column: Reτ = 1000. . . . . . . . . . 19

4.6 Isosurface Φrz = 0 of 〈δuδu〉 for Reτ = 200 (left) and Reτ =1000 (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.7 Isosurfaces Π11 = −0.13 ( ), Π22 = 0 ( ), Π33 = 0.1 ( ) forReτ = 200, Reτ = 500 and Reτ = 1000 respectively. . . . . . . 20

4.8 Isosurfaces T//33 = 0.006 ( ), T

//33 = −0.01 ( ) and flux lines of

〈δwδw〉 for Reτ = 200, Reτ = 500 and Reτ = 1000 respec-tively. For Reτ = 1000, a detailed view of the flux lines isincluded. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.9 Wall-normal transport and scale transfer of 〈δwδw〉: isosur-

faces −∂Ψturb

∂Y= −0.009 ( ) and −∂Φry

∂ry= −0.03 ( ) for Reτ =

200, Reτ = 500 and Reτ = 1000 respectively. . . . . . . . . . . 22

4.10 Isosurfaces T//22 = −0.002 ( ), T

//22 = 0.002 ( ) and −∂Φdiff

rz

∂rz=

−0.003 ( ) and flux lines of 〈δvδv〉 for Reτ = 200, Reτ = 500and Reτ = 1000 respectively. For Reτ = 1000, a detailed viewof the flux lines is included. . . . . . . . . . . . . . . . . . . . 23

4.11 Main wall-normal derivatives of ∇·Φ22 for Reτ = 200, Reτ =500 andReτ = 1000 respectively. Isosurfaces−∂Ψturb

∂Y= −0.006

( ) and −∂Ψpress

∂Y= −0.006 ( ). . . . . . . . . . . . . . . . . . . 24

4.12 Isosurface 〈−δuδv〉 = 1.9 for Reτ = 200 ( ), Reτ = 500 ( )and Reτ = 1000 ( ). . . . . . . . . . . . . . . . . . . . . . . . 24

4.13 Isosurfaces P12 = 0.17 ( ) and Π12 = −0.17 ( ) for Reτ = 200,Reτ = 500 and Reτ = 1000 respectively. . . . . . . . . . . . . . 25

4.14 Flux lines of 〈−δuδv〉. Left column: sets I and II. Right col-umn: sets III and IV. From top to bottom, the rows illustrateReτ = 200, Reτ = 500 and Reτ = 1000 respectively. . . . . . . 26

LIST OF FIGURES ix

4.15 Main derivatives of ∇ · Φ12 for Reτ = 200, Reτ = 500 andReτ = 1000 respectively. Isosurfaces T

//12 = −0.02 ( ), T

//12 =

0.02 ( ), −∂Φry

∂ry= −0.04 ( ), −∂Ψturb

∂Y= −0.01 ( ) and−∂Ψpress

∂Y=

0.1 ( ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.16 First and second peak of ξ11 for Reτ = 500 (left) and Reτ =1000 (right): isosurfaces ξ11 = 0.5 ( ) and ξ11 = 0.006 ( ). . . . 31

4.17 Terms of ∇ · Φ11 for Reτ = 1000 contributing to the second

peak of ξ11: isosurfaces T//11 = −0.004 ( ), −∂Φry

∂ry= −0.002 ( ). 31

4.18 Isolines of ξ11 ( ) and T//11 ( ) on the r+

x = 0 plane forReτ = 1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.19 Isosurfaces T//33 = −0.002 ( ) and T

//33 = 0.002 ( ) for Reτ =

1000. On the left: 3D view. On the right: 2D view from above. 32

4.20 Isosurfaces T//22 = −0.0004 ( ), T

//22 = 0.0004 ( ) and−∂Ψpress

∂Y=

−0.002 ( ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.21 Isosurfaces Π22 = 0.009 ( ) and flux lines of 〈δvδv〉 for Reτ =1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.22 Terms of ∇ · Φ11 for Reτ = 200, Reτ = 500 and Reτ =

1000 respectively: isosurfaces T//11 = −0.002 ( ) and −∂Φry

∂ry=

−0.002 ( ). Only for Reτ = 200, the isosurface −∂Φry

∂ry=

−0.0002 ( ) is illustrated. . . . . . . . . . . . . . . . . . . . . 36

4.23 Evolution of ∂Ψturb

∂Yfor 〈δuδu〉: isosurfaces −∂Ψturb

∂Y= 0.004 for

Reτ = 200, −∂Ψturb

∂Y= 0.001 for Reτ = 500 and −∂Ψturb

∂Y=

0.00055 for Reτ = 1000 respectively. . . . . . . . . . . . . . . . 37

4.24 Isosurfaces T//33 = −0.002 ( ) and T

//33 = 0.002 ( ) for Reτ =

200, Reτ = 500 and Reτ = 1000 respectively. . . . . . . . . . . 38

4.25 Terms of ∇ ·Φ22 for Reτ = 200, Reτ = 500 and Reτ = 1000respectively. Isosurfaces T

//22 = −0.0004 ( ), T

//22 = 0.0004 ( )

and −∂Ψpress

∂Y= −0.002 ( ). . . . . . . . . . . . . . . . . . . . . 39

4.26 Isosurfaces Π22 = 0.009 ( ) and flux lines of 〈δvδv〉 for Reτ =200, Reτ = 500 and Reτ = 1000 respectively. . . . . . . . . . . 40

x LIST OF FIGURES

4.27 Analysis of the correlations for Reτ = 200. On the left: iso-surfaces 〈uu′〉 = 4 ( ), 〈uu′〉 = −0.4 ( ), 〈−uv′〉 = 0.6 ( ) and〈−uv′〉 = −0.13 ( ). On the right: coherent streamwise ( )and wall-normal ( ) velocity field induced by the dominantquasi-streamwise vortex, educed as described in [10]. Contourlevels at (0.2:0.2:0.8) of the maximum (solid line) and the min-imum (dashed line) of the respective component (0.0058 and-0.0077 for u and 0.0035 and -0.0035 for v) are plotted on ay+−z+ plane passing through the center of the vortex, locatedat z+ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.28 Analysis of the correlation 〈−uv′〉 for different Reynolds num-bers. Isosurface 〈−uv′〉 = −0.13 for Reτ = 200 ( ), Reτ = 500( ) and Reτ = 1000 ( ). Isosurface 〈−uv′〉 = 0.6 for Reτ = 200( ), Reτ = 500 ( ) and Reτ = 1000 ( ). . . . . . . . . . . . . . 41

4.29 Analysis of the correlation 〈uu′〉 for different Reynolds num-bers. Isosurfaces 〈uu′〉 = −0.4 for Reτ = 200 ( ), Reτ = 500( ) and Reτ = 1000 ( ). Isosurface 〈uu′〉 = 4 for Reτ = 200( ), Reτ = 500 ( ) and Reτ = 1000 ( ). A zoomed view of thesame picture is provided in the inset. . . . . . . . . . . . . . . 42

4.30 Correlations 〈ww′〉 and 〈vv′〉 for Reτ = 1000: isosurfaces〈ww′〉 = −0.04 ( ) and 〈vv′〉 = −0.01 ( ). On the right, a2D view is shown. . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.31 On the left, section of two counter-rotating streamwise-alignedvortices. On the right: scheme of a hairpin vortex, which canbe split in the parts (1), (2) and (3), detailed in the text. . . . 43

4.32 Analysis of the correlations for Reτ = 200 (on the left) andReτ = 500 (on the right): 〈ww′〉 = −0.04 ( ) and 〈vv′〉 =−0.01 ( ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.33 3D (left) and 2D (right) view of the isosurfaces ξ11 = 0.0001for Reτ = 1000. The portion of the isosurface having Y + < 25has not been represented in this figure. . . . . . . . . . . . . . 47

4.34 Isosurfaces −∂Φturbrx

∂rx= −0.0004 ( ), −∂Φturb

rz

∂rz= −0.0003 ( ),

−∂Ψturb

∂Y= −0.0002 ( ) and −∂Φry

∂ry= −0.0002 ( ) for Reτ = 1000. 48

4.35 Approximate invariance of the isosurface ξ11 = 0.0001 alongr+x for Reτ = 1000. The portion of the isosurface for Y + < 25

has not been represented in this figure. . . . . . . . . . . . . . 48

LIST OF FIGURES xi

4.36 ξ11 at Y + = 201 (continue), Y + = 246 (dotted-dashed) andY + = 300 (dashed) for Reτ = 1000. . . . . . . . . . . . . . . . 49

4.37 −〈δuδv〉 (left) and ξ12 (right) for Reτ = 1000. Y + = 201(continue), Y + = 246 (dotted-dashed) and Y + = 300 (dashed). 49

4.38 〈δvδv〉 (left) and ξ22 (right) for Reτ = 1000. Y + = 201 (con-tinue), Y + = 246 (dotted-dashed) and Y + = 300 (dashed). . . 49

4.39 ξ22 at Y + = 415 (continue), Y + = 465 (dotted-dashed) andY + = 520 (dashed) for Reτ = 1000. . . . . . . . . . . . . . . . 50

4.40 Components of Φ22: Φrz ( ), Φrx ( ) and Ψ ( ). Y + = 201(continue), Y + = 246 (dotted-dashed) and Y + = 300 (dashed). 50

xii LIST OF FIGURES

List of Tables

3.1 Parameters of the numerical simulations used in the presentwork. Lx, Ly and Lz are the dimensions of the computationaldomain (Fig. 3.1). Nx and Nz are the number of Fouriermodes adopted in both the statistically homogeneous direc-tions and Ny is the number of points in the wall-normal direc-tion. ∆x+ and ∆z+ are the resolutions in the streamwise andspanwise directions, ∆y+

min is the minimum wall-normal dis-tance between two adjacent points: all these three are givenin wall units. N is the number of the averaged fields and∆t+ = uτh/ν is non-dimensional time step between two adja-cent fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

xiii

xiv LIST OF TABLES

Chapter 1

Introduction and aim of thework

Wall-bounded turbulent flows have always been interesting from both a the-oretical and an applied perspective. In particular, a growing interest hasrecently been devoted to the study of high-Reynolds numbers flows, whichare defined by a sufficient separation of scales and are characterized by thepresence of coherent structures larger than those partecipating to the near-wall cycle [1]. In particular, these can be distinguished into attached eddies,having dimensions proportional to their wall-normal distances, and large- andvery large- scale motions. These large organized motions were shown to con-tribute significantly to the turbulent kinetic energy and the Reynolds stressproduction, and were suggested to have a strong influence on the behaviourof the near-wall turbulence [2]. However, these flows still lack a completecharacterization: this is due not only to the technical difficulties in perform-ing simulations and experiments at large Reynolds numbers, but also to thecomplex nature of wall-bounded flows, which are characterized by processestaking place simultaneously in two spaces: the physical space, denoted bythe usual (X, Y, Z) coordinates, and the space of scales, which is associatedto the characteristic dimensions of the eddies. This makes the complete com-prehension of wall-bounded flows a very demanding challenge even for thesimplest case, i.e. the plane channel flow. Over the last century, the investi-gation of these phenomena has focused separately on the two aforementionedspaces, neglecting their intrinsic interconnection. The description of homo-geneous isotropic turbulence ([3], [4]) addressed the processes taking place inthe space of scales. On the other hand, regarding the physical space, spatial

1

2 CHAPTER 1. INTRODUCTION AND AIM OF THE WORK

fluxes of turbulent kinetic energy along the wall-normal direction were dis-cussed ([5]). The first step to provide a unifying view in the compound spaceof scales and positions was the derivation of the so called Generalized Kol-mogorov Equation (GKE), a budget equation for the second order structurefunction 〈δuiδui〉, with δui = ui(X +r/2, t)−ui(X−r/2, t) (repeated indeximplies summation) [6]. This tool allowed to point out that the Richardsonscenario is modified in presence of spatial fluxes ([7], [8], [9]). In [10], ananalogous equation was written for the generic second-order structure func-tion 〈δuiδuj〉, addressing the anisotropic behaviour of channel flows, whichcannot be captured if the only scale energy 〈δuiδui〉 is considered. Throughthe application of this equation to the turbulent channel flow with frictionReynolds number Reτ = 200, some statistical properties of the flow werederived. From these, different coherent motions, such as those partecipatingto the wall-cycle or attached eddies-like structures, were identified.In this work, the AGKE is exploited to study the turbulent channel flow forlarger values of the friction Reynolds number, i.e. Reτ = 500 andReτ = 1000.Through the analysis of its terms, different high-Reynolds numbers proper-ties are identified. In particular, in the near-wall region, features derivingfrom the interaction with the outer scales are captured while, in the logarith-mic region, transport phenomena associated to attached eddies and large-and very large-scale motions are highlighted.The present work is organized as follows. In chapter 2 the required back-ground is presented. Chapter 3 provides a description of the datasets, thesimulations, the computation of the AGKE terms and the post-processingof the data. Chapter 4 shows the most important results. In chapter 5, theconclusions and the potential future developments are discussed.

Chapter 2

Fundamentals and state of theart

This chapter is organized as follows: first, the different types of coherentstructures present in wall-bounded turbulence, with particular emphasis onthe indefinite fully developed channel flow, are discussed. Secondly, theanisotropic generalized Kolmogorov equation (AGKE) is introduced, and theoutcomes obtained from its application to the Reτ = 200 case are briefly re-called.

2.1 Coherent structures

Despite the lack of a generally accepted definition of a coherent structure,it can be thought as a deterministic and repeatable feature immersed ina random background. Different types have been observed through bothexperiments and simulations: the inner layer, i.e. the region near to the wallwhere the mean velocity profile is assumed to be determined only by theviscous scales ([5]), is populated by low-speed streaks and quasi-streamwisevortices, which were already found studying low-Reynolds number flows. Inthe logarithmic layer, in which the effect of both the viscous and the outerscales is supposed to vanish, it is largely accepted the presence of attachededdies, whose features were first postulated by Townsend [11]. Moreover,the recent achievements in studying high Reynolds number flows allowedthe observation of very large structures, which do not belong to any of theaforementioned cathegories and may extend even farther from the wall.

3

4 CHAPTER 2. FUNDAMENTALS AND STATE OF THE ART

2.1.1 Near-wall coherent structures

Among the several observed coherent structures in wall-bounded flows ([12]),two of them have been identified to be part of the self-sustaining mechanismof turbulence, i.e. the wall-cycle: the low speed streaks (LSSs) and thequasi-streamwise vortices (QSVs) [13]. The quasi-streamwise vortex is a vor-tical structure highly elongated in the streamwise direction. It lies in thebuffer layer and it is slightly tilted in both the vertical and horizontal planes.Pulling down high-speed fluid and lifting upwards low-speed fluid, it gener-ates streaks of streamwise velocity at its sides. Moreover, the horizontal tilt-ing of the QSVs has been found to be responsible for the redistribution of theturbulent energy contained within the streaks to the wall-normal and span-wise components via pressure-strain effects. After generating, the low-speedstreaks grow linearly until they become unstable, leading to the formation ofstreamwise vorticity from which further QSVs may be generated. Note thatif the generation of streaks from QSVs is considered known, the same doesnot hold for the inverse, for which different mechanisms have been proposed[14].

2.1.2 The attached eddies hypothesis

The attached eddy hypothesis (AEH) is one of the most accepted modelsof turbulent eddies. The concept was first postulated by Townsend (1976)([11]) purely on the basis of intuition and kinematic considerations and ap-plies only to the logarithmic layer. According to this theory, the most rele-vant turbulent eddies within the flow are those with diameters proportionalto the wall-normal distance, namely the attached eddies. Introducing thisconcept of similarity, Townsend found that the density of attached eddieswas required to be inversely proportional to their size in order to make themodel consistent. Most importantly, he derived the following logarithmicdependencies for the second order statistics:

〈u2〉u2τ

= A1 +B1 log

(h

y

)(2.1)

〈w2〉u2τ

= A2 +B2 log

(h

y

)(2.2)

〈v2〉u2τ

= A3 (2.3)

2.1. COHERENT STRUCTURES 5

where h is the length size of the flow, and A1, B1, A2, B2, A3 are constantsdependent on the shape and velocity distribution of the representative eddy.Over the last years the attached eddy model has been further developed,showing compatibility with the experimental results. The logarithmic varia-tion of 〈u2〉 was found to be equivalent to the constancy of the premultipliedpower spectrum kxΦuu. However, there is unconvincing evidence of this con-stancy at laboratory-scale Reynolds numbers. Recall that 1D spectra are theintegrated contributions across all spanwise wavelengths (not only the self-similar ones): thus, they contain the mixed contribution of a range of eddiesthat may or may not be attached. The limitations of the spectral analysisled some authors ([15] and [16]) to employ the structure function (discussedin section 2.2), spatial analog of the Fourier spectrum, as a better tool foranalyzing self-similarity.Since the AEH has been presented as a theory applied to an average field,at this point a question arise: do attached eddies actually exist or are theypurely a statistical construct? Although this issue is still debated, someauthors (for example [17]) found experimental evidence of a hairpin-typevortex structure as the candidate representative attached eddy. Moreover, inthe works of ([18], [19], [20], [21]) some properties of attached eddies, as theself-similarity, are derived from the Navier-Stokes equations.

2.1.3 Large and very large-scale motions

Visualization, experiments and numerical studies on wall flows revealed theexistence of the so called large- and very large-scale motions (LSMs andVLSMs), which cannot be explained through the attached eddy model [22],[23], [24]. Spectral analysis indicated that both LSMs and VLSMs contributesignificantly to the turbulent kinetic energy and Reynolds stress productionfor large Reynolds numbers. For example, Balakumar and Adrian ([2]) foundthat 40-65 % of the kinetic energy and 30-50 % of the Reynolds shear stressare accounted for in the long modes with streamwise wavelengths λx/δ > 3.Furthermore, starting from the work of Rao, Narasimha and Narayanan([25]), several studies suggested that motions in the logarithmic and outerlayer strongly affect the near-wall turbulence, and that this influence tendsto increase with the Reynolds number. In particular, two interaction mech-anisms were identified: the footprint of the outer motions within the innerlayer ([26], [27]) and the modulation of the near-wall cycle by long-wavelengthmotions farther from the wall ([28]).


Both the distinction between LSMs and VLSMs and their origin are still de-bated topics. LSMs are believed to be created by the vortex packets formedwhen multiple hairpin vortices travel at the same convective velocity [2].These hairpin structures were found to be aligned in the streamwise direction,with their heads lying along a line inclined w.r.t. the wall. The characteristicstreamwise scale of LSMs is approximately 2-3h. On the other hand, VLSMswere found having length up to 30 times the channel half-height ([29]) andto scale in outer units. In [24] these appear to be due to a pseudostreamwisealignment of the LSMs, while the authors of [19] suggest linear or non-normalprocesses to be the cause.

2.2 Structure function equations in wall-bounded

turbulence

The physics of wall-bounded turbulence is composed of processes taking placeboth in the space of scales and in the physical space. Richardson energycascade from the energetic to the dissipative scales ([3], [4]) is an exampleof the first, while the transport of turbulent kinetic energy both towardsand away from the wall operated by the turbulent convection belongs tothe second. Regarding the space of scales, an approach consists of using adouble-point statistics as the velocity correlation or the velocity spectrum inorder to infer the shape and size of eddies. Alternatively, a balance equa-tion for the single-point statistics k, i.e. the turbulent kinetic energy, canbe employed to study the processes taking place in the physical space. Inparticular, this dynamic equation allows to capture the production, dissipa-tion and transport occurring as a consequence of directions of inhomogeneitycharacterizing wall-bounded flows. However, both the aforementioned ap-proaches lack something: specifically, the single-point budget equation is notcapable of capturing the multi-scale nature of turbulence, while the spec-tral decomposition and the two-point spatial correlation do discern betweenscales but do not provide direct information on their role in the processesof production, dissipation and transfer, therefore missing the dynamic de-scription of turbulent interactions. The anisotropic generalized Kolmogorovequation (AGKE), which is a budget equation for the double-point statistics〈δuiδuj〉, provides both kinds of informations, since it describes production,dissipation and transport of the turbulent stresses in the compound space

2.2. STRUCTURE FUNCTION EQUATIONS INWALL-BOUNDED TURBULENCE7

of scales and positions. Indeed the structure function 〈δuiδuj〉 depends on 6independent variables, which reduce to 4 in the case of the fully developedchannel flow (rx, ry, rz, Y ). This domain is limited by the kinematic con-straint ry < 2Y , due to the finite height of the channel. Before recalling theAGKE, it is relevant to point out the relation between the structure functionand the spatial correlation function:

〈δuiδuj〉(X, r, t) = Vij(X, r, t)−Rij(X, r, t)−Rij(X,−r, t) (2.4)

where

Vij(X, r, t) = 〈uiuj〉(X +

r

2, t)

+ 〈uiuj〉(X − r

2, t)

(2.5)

is the sum of the correspondent single-point statistics evaluated at time t andat the points X + r/2 and X − r/2 respectively, and

Rij(X, r, t) =⟨ui

(X +

r

2, t)uj

(X − r

2, t)⟩

(2.6)

is the two-point spatial correlation function. For sufficiently large values of|r|, the correlation vanishes, so that 〈δuiδuj〉 reduces to Vij and the AGKEreduces to the budget equation for the single-point Reynolds stresses atX ± r/2. Moreover, if the subspace ry = 0 is considered, then

Vij(X, t) = 2〈uiuj〉(X, t) (2.7)

since the directions x and z are homogeneous. Thus, the dependence of〈δuiδuj〉 on the scales is left to the correlation Rij.Let us now recall the AGKE, tailored to the infinite fully developed channelflow:

∂Φk,ij

∂rk+∂Ψij

∂Y= ξij (2.8)


where:

Φk,ij = 〈δuiδujδU〉δk1︸︷︷︸mean transport

+ 〈δukδuiδuj〉︸︷︷︸turbulent transport

−2ν∂

∂rk〈δuiδuj〉︸︷︷︸

viscous diffusion

k = 1, 2, 3(2.9)

Ψij = 〈v∗δuiδuj〉︸︷︷︸turbulent transport

+1

ρ〈δpδuj〉δi2 +

1

ρ〈δpδui〉δj2︸︷︷︸

pressure transport

−ν2

∂

∂Y〈δuiδuj〉︸︷︷︸

viscous diffusion

(2.10)

ξij = −〈v∗δuj〉δ(

dU

dy

)δi1 − 〈v∗δui〉δ

(dU

dy

)δj1︸︷︷︸

production (Pij)

+

−〈δvδuj〉(

dU

dy

)∗δi1 − 〈δvδui〉

(dU

dy

)∗δj1︸︷︷︸

production (Pij)

+

+1

ρ

⟨δp∂δui∂Xj

⟩+

1

ρ

⟨δp∂δuj∂Xi

⟩︸︷︷︸

pressure strain (Πij)

− 4ε∗ij︸︷︷︸dissipation (Dij)

(2.11)

δij is the Kronecker delta, and the asterisk superscript f ∗ denotes the averageof the generic quantity f between positions X ± r/2.Interesting insights on the physics of the channel flow for Reτ = 200 wereachieved in [10] thanks to the adoption of the anisotropic generalized Kol-mogorov equation. The transfer of the diagonal components 〈δuiδui〉 con-firmed that in the buffer layer almost the entire turbulent kinetic energy iscarried by the low-speed streaks and quasi-streamwise vortices. Moreover,it illustrated the evolutionary process of some eddies presenting propertiessimilar to those of the attached eddies. According to this process, these ed-dies were found to finally break down into smaller detached eddies and getdissipated via viscous effects. Further insights were obtained considering thesecond-order structure function of the Reynolds shear stresses 〈−δuδv〉: first,a transfer of this quantity from the centerline towards the wall was found totake place through both incoherent motions and sort of coherent streamwisevortices; secondly, another transfer from the vortices in the viscous sublayer,i.e. y+ < 10, to the structures associated to the wall-cycle was highlighted.

Chapter 3

Numerical methods andprocedures

The analysis presented in the next chapters results from the post-processingof three DNS-databases produced for a fully developed plane channel flowfor three Reynolds numbers. The bulk Reynolds number is defined as Reb =Ubh/ν, where Ub is the velocity averaged through the channel section, h isthe semi-height of the channel and ν is the kinematic viscosity of the fluid.Differently, Reτ = uτh/ν is the friction Reynolds number, where uτ =

√τw/ρ

is the so-called friction velocity expressed in terms of the average shear stressτw at the wall and the density ρ. The three databases analysed were producedfor Reτ = 200, Reτ = 500 and Reτ = 1000 respectively. Details of the firstand the last can be found in [10] and [30], while the Reτ = 500 one wasproduced for this analysis. In detail, all these simulations derive from a DNScode written in CPL language by Luchini and Quadrio [31]. Here, the NavierStokes equations are projected in the divergence-free space v − η and solvedusing a pseudo-spectral method. The size of the computational domain is4πh × 2h × 2πh for Lx, 2h and Lz respectively (Fig. 3.1). In the wall-normal direction an hyperbolic tangent distribution for the Ny points hasbeen exploited in order to obtain a more refined grid near the wall. Otherdetails of the simulations, as the resolutions along the three directions orthe number of resolved fields N , are given in Table 3.1. For each field, thepressure was computed by solving the Poisson equation. The results of thesimulations for Reτ = 1000 were compared with those used in [32]: Fig.3.2 illustrates this comparison for the mean velocity profile and the variance〈u2〉.

9

10 CHAPTER 3. NUMERICAL METHODS AND PROCEDURES

Figure 3.1: Sketch of the channel flow in a Cartesian reference system. Lx,Lz and 2h are the size of the computational domain in the stream-wise,span-wise and wall-normal directions.

Figure 3.2: Comparison between the data used in this work ( ) and thoseused in [32] ( ) for Reτ = 1000: velocity profile U (on the left) and single-point statistics 〈u2〉 (on the right), both given in wall units.

11

Figure 3.3: Incremental averages of ξ11 for Reτ = 1000 on the r+x = 0 plane.

The colours represent the number of averaged fields n, spanning from 1 to37, as shown by the colorbar. On the left, moving from the outside to theinside, the plotted isolines are ξ11 = 0.2, ξ11 = 0.3, ξ11 = 0.4 and ξ11 = 0.5respectively. On the right, the two largest sets of isolines are ξ11 = 0.0001,the one at Y + ∼ 200 is ξ11 = 0.006 and the sets in the inner layers are,moving from the outside to the inside, ξ11 = 0.006 and ξ11 = 0.3 respectively.

The terms involved in the AGKE have been computed using a specific solver([33]) written in CPL language, already validated for the Reτ = 200 case[10]. However, for the Reτ = 500 and Reτ = 1000 cases, only the subspacery = 0 has been computed, due to computational and memory issues. Thecorrespondent file generated had a size of approximately 425 GB. The AGKEsolver requires, as input, the fluctuating velocity field, the mean velocityprofile and the pressure field of all the DNS-database fields. It provides asoutput the flux vector Φij, the two-points statistics 〈δuiδuj〉 and the sourceterm ξij, together with its components Pij, Πij and Dij. Moreover, differentlyfrom the previous version, also the single terms composing the fluxes wereextracted, i.e. mean transport, turbulent transport, pressure transport andviscous diffusion. The computation has been carried out on the ForHLRcluster at Steinbuch Centre for Computing (SCC) at the Karlsruher Institutfur Technologie (Germany).For the Reτ = 200 case, the statistical convergence of the data was verifiedby ensuring that the residual of equation 2.8 was negligible compared tothe dissipation, production and pressure strain terms. For Reτ = 500 and

Reτ = 1000 this was not possible, since the term∂Φry

∂rywas not computed

directly, because of the restriction to the subspace ry = 0. However, theconvergence was verified through a series of incremental averages: consideringthe generic function fk(rx, rz, Y ) associated to the kth field, its incremental


Reτ Reb (Lx, Ly, Lz)/h Nx, Ny, Nz ∆x+,∆z+ ∆y+min N ∆t+

200 3176 4π, 2, 2π 256, 256, 256 9.8, 4.9 0.46 200 0.62500 9092 4π, 2, 2π 512, 250, 512 12.3, 6.1 0.96 37 11000 19996 4π, 2, 2π 1024, 500, 1024 12.3, 6.1 0.96 37 0.6

Table 3.1: Parameters of the numerical simulations used in the present work.Lx, Ly and Lz are the dimensions of the computational domain (Fig. 3.1).Nx and Nz are the number of Fourier modes adopted in both the statisticallyhomogeneous directions and Ny is the number of points in the wall-normaldirection. ∆x+ and ∆z+ are the resolutions in the streamwise and spanwisedirections, ∆y+

min is the minimum wall-normal distance between two adja-cent points: all these three are given in wall units. N is the number of theaveraged fields and ∆t+ = uτh/ν is non-dimensional time step between twoadjacent fields.

average is a series defined as:

F (n)(rx, rz, Y ) =1

n

n∑k=1

fk(rx, rz, Y ) n = 1, .., N (3.1)

where N is the number of fields computed. Operatively, the first step was tocompute the terms of the AGKE for each single field. Then, the incrementalaverages of some terms were performed. Fig. 3.3 illustrates the 37 incre-mental averages of the source term ξ11 on the r+

x = 0 plane for Reτ = 1000.Evidently, in the inner layer (on the left), convergence is achieved even if theaverage is performed on a very small number of fields. This does not hold inthe logarithmic and outer layers (on the right): here, for small values of n,i.e. when the number of averaged fields is still low, the regions delimited bythe corresponding isolines both reduce and enlarge as n increases. However,when n becomes large enough, the lines are shown to superimpose, provingconvergence is achieved.

In the following, all the variables are given in viscous units, i.e. normalizedwith uτ and ν. However, for the sake of conciseness, only the coordinatesin the AGKE space (r+

x , r+y , r

+z , Y

+) show the superscript +. For the samepurpose, the notation 〈uiu′j〉 is preferred for the spatial correlation 〈ui(X −r/2)uj(X + r/2)〉.Some further considerations are required for an easier understanding of the

13

results presented in chapter 4. In section 2.2, the AGKE was introduced inits classical form. If we take the derivatives in equation 2.8 to the right side,we get:

ξij −∂Φk,ij

∂rk− ∂Ψij

∂Y= 0 (3.2)

This allows the interpretation of these derivatives as if they were “sourceterms” analogous to ξij: thus, regions where these quantities are negativeare denoted as donor, i.e. an extraction of 〈δuiδuj〉 takes place, while regionswhere these are positive are referred to as receiver, meaning there is a depositof 〈δuiδuj〉. In [32] an equation analogous to 3.2 was derived for the spectra.However, the AGKE has the advantage of capturing not only the donor andreceiver regions of these processes but also, through the fluxes, the directionof these transports in every point of the compound space of scales and posi-tions.If equations 2.9, 2.10 and 2.11 are exploited, we get:

Pij + Πij−Dij−∂Φmean

k,ij

∂rk−∂Φturb

k,ij

∂rk−∂Φdiff

k,ij

∂rk−∂Ψturb

ij

∂Y−∂Ψpress

ij

∂Y−∂Ψdiff

ij

∂Y= 0

(3.3)

Notice that −∂Φmeank,ij

∂rk= 0 if the r+

y = 0 subspace is considered. Since the

AGKE was computed only on the subspace r+y = 0, the term

∂Φry

∂rywas

computed a posteriori exploiting equation 3.2, assuming the residual to benegligible (hypothesis which was tested analyzing the incremental averages).This does not allow to split this quantity into its turbulent and viscous com-ponents, analogously to what has been done for the other terms of ∇ ·Φij .

Finally, in order to ease the notation, one more definition is introduced. T//ij

is defined as the turbulent transfer across the “parallel” separations, i.e. r+x

and r+z . In symbols:

T//ij = −(

∂Φturbrx

∂rx+∂Φturb

rz

∂rz) (3.4)

The computation of the derivatives of the fluxes was performed a posteriori,using a Fourier method in the homogeneous directions, a finite differencescheme in the wall-normal direction. For the 2D and 3D visualizations ofthe results, the softwares MATLAB and PARAVIEW were employed, re-spectively. For the visualization of the fluxes, flux lines have been frequently


used: for the component 〈δuiδuj〉, these are the lines locally tangent to theflux Φij = (Φrx ,Φrz ,Ψ) and coloured with its magnitude.

Chapter 4

Results

In this chapter, the main results from the analysis of the AGKE for thethree different Reynolds numbers are presented. One relevant outcome ofthe present work has been the identification of three peaks of the source ξ11.All these peaks are found on the r+

x = 0 plane. The first one, located in theinner layer, is common to all the three Reynolds numbers and will be ana-lyzed in section 4.1. The second is found only for Reτ = 500 and Reτ = 1000in the logarithmic layer at larger spanwise separations: it will be studied issection 4.2. Moreover, in 4.2.1 it is shown how some features of this secondpeak can be found also for Reτ = 200, suggesting a continuous view fromthe low- to the high-Reynolds number wall bounded flows. Finally, in section4.3 the third peak of ξ11, found for very large spanwise separations only inthe Reτ = 1000 case, is introduced and analyzed. In Fig. 4.1 the regionsassociated to these three peaks of ξ11 are illustrated.

4.1 The inner peak of ξ11

Dominant terms of the flux divergence It has been already shown in [10]as the inner layer is characterized by the presence of a source peak for boththe scale energy 〈δuiδui〉 and the streamwise component 〈δuδu〉 placed atY + =∼ 12, r+

x ∼ 0 and r+z ∼ 40. In Fig. 4.2 this peak is illustrated for

Reτ = 200, Reτ = 500 and Reτ = 1000. Although no qualitative differ-ences are evident, a tendency of the isosurface to shrink along r+

x for largerReynolds numbers is visible. The poor scaling in wall units has already been

15

16 CHAPTER 4. RESULTS

Figure 4.1: Three peaks of ξ11 for Reτ = 1000: isosurfaces ξ11 = 0.5 ( ),ξ11 = 0.006 ( ) and ξ11 = 0.0001 ( ). Of these two last isosurfaces, only thepart for Y + > 30 is illustrated. A detail of the isosurfaces ξ11 = 0.5 is alsorepresented.

Figure 4.2: Inner peak of ξ11: isosurface ξ11 = 0.5 for Reτ = 200 ( ), Reτ =500 ( ) and Reτ = 1000 ( ).

4.1. THE INNER PEAK OF ξ11 17

Figure 4.3: Dominant components of ∇ · Φ11 balancing the inner peak of

ξ11: isosurfaces T//11 = −0.05 ( ), −∂Φdiff

rz

∂rz= −0.03 ( ), −∂Ψturb

∂Y= −0.14 ( ),

−∂Ψdiff

∂Y= −0.15 ( ) and −∂Φry

∂ry= −0.25 ( ) for Reτ = 200, Reτ = 500 and

Reτ = 1000 respectively.


Figure 4.4: Transport processes in the inner layer for Reτ = 200. On the left:isosurfaces Π11 = −0.13 ( ), T

//11 = 0.05 ( ), T

//11 = −0.05 ( ) and −∂Ψturb

∂Y=

−0.02 ( ). On the right: isosurface Ψ = 0 ( ) and flux lines of 〈δuδu〉: recallthat these lines are locally parallel to the flux vector Φ11 = (Φrx ,Φrz ,Ψ) andcoloured with its magnitude.

observed for both the single-point 〈u2〉 ([34]) and the correlation 〈uu′〉 ([35]),components of 〈δuδu〉 according to equation 2.4. However, the reason of thedecreasing streamwise extent of ξ11 with increasing Reτ is unclear.The energy produced by the inner peak is transported in the (r+

x , r+z , Y

+)space by the fluxes. As explained in section 3, while the fluxes describe thedirection of this transport, the derivatives appearing in ∇·Φ11 (equation 3.2)illustrate the amount of energy locally extracted or deposited by these pro-cesses. The largest terms of ∇ ·Φ11 in the inner peak-region are illustratedin Fig. 4.3. Of these, the derivatives along Y + and r+

y are the dominant

ones: regarding the first, while ∂Ψdiff

∂Y(in orange) scales in wall units, ∂Ψturb

∂Y

(in yellow) scales approximately in outer units, suggesting the presence of

larger structures interacting with the near-wall dynamics. Instead,∂Φry

∂ry(in

gray) reduces for increasing Reτ , consistently with the trend shown by ξ11.

It is recalled that∂Φry

∂rycannot be split in its constituent components, since

only the subspace ry = 0 has been computed. Smaller but still contributing

to the inner peak are T//11 and ∂Φdiff

rz

∂rz: both these isosurfaces tend to reduce

their extents for increasing Reynolds numbers.

Transport processes of 〈δuδu〉 Until this point, we have identified thetransport terms which are dominant in a neighbourhood of the inner peak:these terms extract the energy which is here produced. Now, we provide aview of how this energy is transported in the space of scales and positions


Figure 4.5: Top row: isosurface Ψ = 0 ( ) and flux lines of 〈δuδu〉. Bottom:isosurface Ψ = 0 ( ) and flux lines of the scale energy 〈δuiδui〉 (where re-peated index implies summation). Left column: Reτ = 200. Right column:Reτ = 1000.

Figure 4.6: Isosurface Φrz = 0 of 〈δuδu〉 for Reτ = 200 (left) and Reτ = 1000(right).


Figure 4.7: Isosurfaces Π11 = −0.13 ( ), Π22 = 0 ( ), Π33 = 0.1 ( ) forReτ = 200, Reτ = 500 and Reτ = 1000 respectively.

and exchanged among the other diagonal components 〈δvδv〉 and 〈δwδw〉.In order to do so, let us consider simultaneously the information providedby the fluxes and their derivatives, as illustrated in Fig. 4.4 for Reτ = 200.First, consider the largest (in absolute value) negative and positive isosur-

faces of T//11 : the first (donor region) is located at r+

x = 0, the second atr+z = 0: 〈δuδu〉 is then transferred from the first to the second, consistently

with the flux lines. This transfer, which peaks at Y + ∼ 15, is coherent withthe break-up of streamwise-elongated streaks, due to instabilities: the changein the separation vector moving from the donor to the receiver region is dueto a tendency of streak-instabilities to become phase coherent in the span([32]). However, recall from Fig. 4.3 that an important role is played bythe wall-normal transport: in particular, the energy produced in the innerpeak is directed both towards and away from the wall, as both the flux linesand the isosurface Ψ = 0 show (right part of Fig. 4.4). However, the energyproduced in the inner peak is not only transported along Y + and transferredacross scales, but it is also redistributed by the pressure strain (dark gray iso-surfaces), which has its absolute minimum (negative) on the r+

x = 0 plane. Asimilar redistribution takes place also on the r+

z = 0 plane, where another lo-cal minimum of Π11 can be found. Both these minima are found for Y + ∼ 20.


Figure 4.8: Isosurfaces T//33 = 0.006 ( ), T

//33 = −0.01 ( ) and flux lines

of 〈δwδw〉 for Reτ = 200, Reτ = 500 and Reτ = 1000 respectively. ForReτ = 1000, a detailed view of the flux lines is included.


Figure 4.9: Wall-normal transport and scale transfer of 〈δwδw〉: isosurfaces

−∂Ψturb

∂Y= −0.009 ( ) and −∂Φry

∂ry= −0.03 ( ) for Reτ = 200, Reτ = 500 and


These processes occurring in the inner layer do not change qualitatively forReτ = 500 and Reτ = 1000 (not illustrated). However, an interesting differ-ence among the three Reynolds numbers is the location of the origin pointof the flux lines. It has been already shown that for Reτ = 200 this is foundon the r+

x = 0 plane at r+z ∼ 50, i.e. near to the maximum of ξ11. This is not

the case for Reτ = 500 and Reτ = 1000, for which the origin point is foundat larger spanwise separations (top part of Fig. 4.5). Recall that this is apoint in which all the components of the flux vector are zero [8]. Then, Fig.4.6 illustrates that the displacement of this point is associated to a shift ofthe isosurface Φrz = 0 to larger values of r+

z . Thus, the previously describedtransfer across scales appears to involve larger spanwise separations for in-creasing Reynolds numbers. Notice that this shift of the origin point of theflux lines does not hold if the whole scale energy 〈δuiδui〉 is illustrated: thisfeature is reported in the bottom part of the same Fig. 4.5 for Reτ = 200and Reτ = 1000.

Inter-component transfers It has already been shown that Π11 has twonegative peaks on the r+

x = 0 and on the r+z = 0 planes respectively. Fig. 4.7

illustrates also the pressure-strain terms of 〈δvδv〉 and 〈δwδw〉. The mod-


Figure 4.10: Isosurfaces T//22 = −0.002 ( ), T

//22 = 0.002 ( ) and −∂Φdiff

rz

∂rz=

−0.003 ( ) and flux lines of 〈δvδv〉 for Reτ = 200, Reτ = 500 and Reτ = 1000respectively. For Reτ = 1000, a detailed view of the flux lines is included.


Figure 4.11: Main wall-normal derivatives of ∇ ·Φ22 for Reτ = 200, Reτ =500 and Reτ = 1000 respectively. Isosurfaces −∂Ψturb

∂Y= −0.006 ( ) and

−∂Ψpress

∂Y= −0.006 ( ).

Figure 4.12: Isosurface 〈−δuδv〉 = 1.9 for Reτ = 200 ( ), Reτ = 500 ( ) andReτ = 1000 ( ).


Figure 4.13: Isosurfaces P12 = 0.17 ( ) and Π12 = −0.17 ( ) for Reτ = 200,Reτ = 500 and Reτ = 1000 respectively.

ifications of the isosurfaces of Π11 and Π33 w.r.t. Reτ highlight that thesequantities do not scale in wall units. This can be found also in the cor-respondent single-point statistics and is due to the scaling of the pressurefluctuations themselves: specifically, when scaled in inner units, their inten-sity increase logarithmically with the Reynolds number [36]. Although therepresented isosurface of Π22 does not show any modification, minor changesw.r.t. Reτ can be found also for this term, even if smaller than those for thepressure-strain terms of the other two diagonal components: this behaviouris the same shown by the pressure-strain terms of the single-point quantities[37]. However, the position of the peaks of Π11 (negative) and Π33 (positive)remains the one evident for Reτ = 200. Notice that the maxima of Π33 arevery near to the minima of Π11. Moreover, a part of the isosurface of Π33

which lies on the r+x = 0 plane is characterized by negative values of Π22:

thus, while near the r+z = 0 plane 〈δwδw〉 receives energy only from 〈δuδu〉,

near the r+x = 0 one also 〈δvδv〉 contributes through the so-called splat-effect

([32]), which occurs for Y + < 12, approximately indipendently on r+x and

r+z . It is furtherly observed that Π33 is positive everywhere in the domain,

i.e. 〈δwδw〉 only receives energy from the other two diagonal components,consistently with the correspondent single-point quantity.


Figure 4.14: Flux lines of 〈−δuδv〉. Left column: sets I and II. Right column:sets III and IV. From top to bottom, the rows illustrateReτ = 200, Reτ = 500and Reτ = 1000 respectively.


Figure 4.15: Main derivatives of ∇ · Φ12 for Reτ = 200, Reτ = 500 andReτ = 1000 respectively. Isosurfaces T

//12 = −0.02 ( ), T

//12 = 0.02 ( ),

−∂Φry

∂ry= −0.04 ( ), −∂Ψturb

∂Y= −0.01 ( ) and −∂Ψpress

∂Y= 0.1 ( ).

Transport processes of 〈δwδw〉 The energy introduced by Π33 both on the

r+x = 0 and on the r+

z = 0 planes is transferred away by T//33 , which is negative

(donor) in these regions for all the three values of Reτ , as shown in Fig. 4.8.The correspondent isosurfaces (pink) are shown to scale considerably well inwall units, extending to slightly larger separations for increasing Reynoldsnumbers. This extracted energy is transported towards the receiver region,where it is deposited. The correspondent isosurface (in blue) does not scalein wall units, extending to larger wall-normal distances as Reτ increases. Italso shows a tendency to warp, as it will be detailed in section 4.2. The afore-mentioned transport is consistent with the flux lines of 〈δwδw〉, illustratedin the same figure, in which a zoomed view is also provided for Reτ = 1000.Their origin point is found on the r+

x = 0 plane at r+z ∼ 25 very near to the

wall, i.e. Y + ∼ 8. From here, a set of lines is driven towards larger r+z , where

it takes energy from the donor region near to the r+x = 0 plane and describes

an ascending spiral until the small dissipative scales. Another set is driventowards larger r+

x , taking energy from the other donor region and describingan ascending spiral in the opposite verse of the first set, again tending to thesmall dissipative scales on the Y + axis. Finally, there is a set of flux lineswhich is driven to the wall, where dissipation takes place. The transport


along the wall-normal direction is dominated by the turbulent component∂Ψturb

∂Y, which is illustrated in Fig. 4.9 together with the transport across

wall-normal separations∂Φry

∂ry. Evidently, these quantities do not scale in wall

units, increasing with the Reynolds number. It is interesting to observe thatthese two derivatives show very similar isosurfaces, both having their max-

ima on the plane r+x = 0 for Y + ∼ 7. Precisely, the maximum point of

∂Φry

∂ry

is found for r+z ∼ 50, that of ∂Ψturb

∂Yfor r+

z ∼ 60: these points do not change

significantly with the Reynolds number. The similar behaviour of ∂Ψturb

∂Yand

∂Φry

∂rysuggests a strong relation between the wall-normal distances Y + and

the wall-normal scales r+y involved in the transport of 〈δwδw〉, which is con-

sistent with the generally accepted organization of eddies in “hierarchies”,depending on their distance from the wall [17]. The viscous diffusion ∂Ψdiff

∂Y

(not illustrated) is relevant only for Reτ = 1000.

Transport processes of 〈δvδv〉 The same analysis is performed for the〈δvδv〉 component. Fig. 4.10 illustrates the main transport terms across r+

x

and r+z , together with some flux lines. These lines have origin on the r+

z = 0plane: this is visible here only for Reτ = 200, but will be shown to hold alsofor Reτ = 500 and Reτ = 1000. Consistenly, a donor isosurface of T

//22 lies on

the same plane. The flux lines are driven towards the wall, towards larger r+z

and lower r+x . However, around Y + = 50, a part of them is directed towards

larger separations, another part to smaller ones. This split is due to both theturbulent and the viscous transfers across scales. Finally, a portion of theselines is driven towards the wall, while another is directed upwards, ending onthe Y + axis for r+

x = r+z = 0. These wall-normal transports are mainly due to

∂Ψturb

∂Yand ∂Ψpress

∂Y, illustrated in Fig. 4.11. Regarding the differences among

the three Reynolds numbers, the isosurfaces of T//22 undergo small modifica-

tions. ∂Ψturb

∂Yscales considerably well in wall-units, while this does not hold

for the pressure-transport: this is probably due to the nature of the pressurefluctuations, which mix contributions from the whole range of scales [35]. Inparticular, several differences can be found: first, the maximum is larger inthe Reτ = 1000 case. Secondly, the illustrated isosurface of ∂Ψpress

∂Ytends to

show a Y + − r+z linearity near to the r+

x = 0 plane for increasing values ofReτ . Moreover, the same isosurface is seen to disappear very near to the wallfor intermediate spanwise separations, i.e. for r+

z ∼ 100: this suggests that,while for Reτ = 200 all the scales are donor of 〈δvδv〉, which is transported


towards the wall, for larger Reynolds numbers this is true only for the verysmall and very large spanwise scales. This phenomenon is assumed to bedue to the interaction with the outer scales. Further insights in the origin

of the flux lines will be given in section 4.2. It is pointed out that also∂Φry

∂ry

(not shown) is an important derivative in the inner layer: it is positive for

Y + > 20 approximately and shows a very similar behaviour to ∂Ψturb

∂Y, as al-

ready illustrated and discussed for 〈δuδu〉 and 〈δwδw〉. It scales considerablywell in wall units and its maximum is found in (r+

x , r+z , Y

+) = (90, 50, 35) forall the three Reynolds numbers considered.

The off-diagonal term 〈−δuδv〉 and ξ12 Let us now consider the off-diagonal component 〈−δuδv〉. Note that it is not definite in sign, so thisterm and the associated fluxes cannot be interpreted in terms of energy andenergy transport, respectively. Thus, the concepts of production and dissi-pation apply only after considering the sign of 〈−δuδv〉. However, in thesubspace r+

y = 0, differently from the r+x = 0 one, this quantity is positive

in the whole domain. From Fig. 4.12, 〈−δuδv〉 appears not to scale in wall-units. This Reτ dependence can be found both in the correspondent singlepoint statistics and in the correlation, which will be discussed in section 4.2.1.In particular, the isosurface of 〈−δuδv〉 tends to a Y + − r+

z linearity for in-creasing Reynolds numbers, which will be given an explanation in section4.2.Let us now consider as 〈−δuδv〉 is produced and dissipated. The sourceterm ξ12 is dominated by the production term P12 and by the pressure-strainterm Π12 (see equation 2.8), since the viscous dissipation is negligible forall the three Reynolds numbers considered. Production and pressure-strainof 〈−δuδv〉 are illustrated in Fig. 4.13. Clearly, both these terms do notscale in wall-units, increasing with Reτ . The maximum point of P12 is onthe r+

x = 0 plane, and it moves from (r+z , Y

+) = (36, 18) for Reτ = 200, to(r+z , Y

+) = (33, 13) for both Reτ = 500 and Reτ = 1000. Instead, Π12 showsa minimum on the same plane for (r+

z , Y+) = (65, 16), (r+

z , Y+) = (69, 15)

and (r+z , Y

+) = (71, 14) for the same Reynolds numbers, taken with the sameorder. Moreover, notice that a second negative peak of Π12 is present on ther+z = 0 plane for all the values of Reτ .

Flux lines of 〈−δuδv〉 Fig. 4.14 illustrates the flux lines of 〈−δuδv〉.These can be grouped in four sets, depending on their origin and end. Both


the lines of sets I and II have their origin at large wall-normal distances, i.e.Y + > 150, and descend towards the wall: the first set terminates exactly onthe wall, while the second deviates towards smaller r+

z and terminates on apoint on the r+

z = 0 plane. This point has coordinates (r+x , Y

+) = (35, 17) forReτ = 200, and reduces its wall-normal distance with increasing Reynoldsnumber, until reaching the plane Y + = 0 for Reτ = 1000. This behaviourwill be justified in the following. Moreover, it is observed that the origin ofthese lines appears a definite point on the r+

x = 0 plane only for Reτ = 200,while it is difficult to identify for the other two Reynolds numbers. The ori-gin of the sets of lines III and IV, instead, is found very near to the wall, i.e.Y + = 5: from this point, lines belonging to set III move to larger streamwiseand spanwise separations before dissipating at the wall, while those of setIV converge in the same end point of set II. In [10] a similar transfer wasfound in the r+

x = 0 subspace and was associated to the interactions betweenthe uv−structures in the viscous sublayer and the turbulent structures of thenear-wall cycle.

Flux divergence of 〈−δuδv〉 Further insights on the transport processesof the off-diagonal term can be derived by considering the dominant terms of∇ ·Φ12, illustrated in Fig. 4.15. The main components contributing to the

positive peak of ξ12 are∂Φry

∂ry, ∂Ψturb

∂Yand T

//12 . Notice that the receiver region

of T//12 is located near to the end point of the sets of flux lines II and IV

on the r+z = 0 plane, consistently with the aforementioned transports. The

pressure-transport term −∂Ψpress

∂Yis the one displaying the largest (positive)

values. Clearly, it does not scale in wall units: in particular, its modulusincreases with Reτ . This implies a more intense sink of 〈−δuδv〉 takes place,which might explain the previoulsy mentioned shift of the II and IV lines’end point onto the Y + = 0 plane. Moreover, it is curious that the maximumvalue of ∂Ψpress

∂Yis found, for all the three values of Reτ , at r+

z ∼ 55 and atlarge streamwise separations, i.e. r+

x ∼ 150.

4.2 The second peak of ξ11

The inner peak of ξ11 is a common feature to all the three Reynolds numbersconsidered in the analysis. However, the cases Reτ = 500 and Reτ = 1000show a second peak of the same quantity, first found in [8] for the whole

4.2. THE SECOND PEAK OF ξ11 31

Figure 4.16: First and second peak of ξ11 for Reτ = 500 (left) and Reτ = 1000(right): isosurfaces ξ11 = 0.5 ( ) and ξ11 = 0.006 ( ).

Figure 4.17: Terms of ∇ · Φ11 for Reτ = 1000 contributing to the second

peak of ξ11: isosurfaces T//11 = −0.004 ( ), −∂Φry

∂ry= −0.002 ( ).

Figure 4.18: Isolines of ξ11 ( ) and T//11 ( ) on the r+

x = 0 plane for Reτ =1000.


Figure 4.19: Isosurfaces T//33 = −0.002 ( ) and T

//33 = 0.002 ( ) for Reτ =

1000. On the left: 3D view. On the right: 2D view from above.

Figure 4.20: Isosurfaces T//22 = −0.0004 ( ), T

//22 = 0.0004 ( ) and −∂Ψpress

∂Y=

−0.002 ( ).


Figure 4.21: Isosurfaces Π22 = 0.009 ( ) and flux lines of 〈δvδv〉 for Reτ =1000.

scale energy 〈δuiδui〉, located on the r+x = 0 plane but at larger spanwise

separations and wall-normal distances w.r.t. the first, as illustrated in Fig.4.16. Notice that the presence of this peak also in the source of 〈δuiδui〉 andnot only in that of 〈δuδu〉 implies that this is not just an effect of the redis-tribution operated by Π11, but a real production of energy is taking place.

Flux divergence of 〈δuδu〉 As for the inner peak, it is of interest to under-stand which physical processes are involved in the transport of this energy,once it has been generated. Fig. 4.17 illustrates the terms of ∇ ·Φ11 whichare dominant in this second peak of ξ11: the main contributions are given by∂Φry

∂ryand T

//11 . Regarding the first, outside of the inner layer it is reasonable to

consider this term dominated by its turbulent component, i.e.∂Φry

∂ry≈ ∂Φturb

ry

∂ry,

since this has been seen to occur for ∂Φrx

∂rxand ∂Φrz

∂rz.∂Φry

∂ryshows a local max-

imum on the r+x = 0 plane: thus, it defines the wall-normal distance and

the spanwise scale of the second peak of ξ11. The isosurface of T//11 has two

interesting features. First, it shows an approximately linear relation betweenthe Y + and the r+

z coordinates. This proportionality between the scale ofthe coherent structures and their wall-normal distance supports the presenceof attached eddies. Secondly, the isosurface of T

//11 , differently from that of

∂Φry

∂ry, extends to the wall: this is consistent with [16], in which the linear

subrange associated to the attached eddies was found to extend until smallscales, and is evident in Fig. 4.18: here, it is illustrated that some isolines of


T//11 enclose both the first and the second peak, suggesting a similar nature

of the two. This similarity is in line with [10], in which features associated tothe attached eddies were found for Reτ = 200, value of the Reynolds numberfor which no second peak of ξ11 can be observed. This concept will be fur-therly expanded in section 4.2.1. Furthermore, considering again Fig. 4.18,notice that the correspondent positive (receiver) region of T

//11 is found near

the r+z = 0 plane, exactly as already shown for the near-wall region (Fig. 4.4).

Flux divergence of 〈δwδw〉 The analysis of the main processes takingplace in this attached eddies-region of the (r+

x , r+z , Y

+) domain is illustrated

in Fig. 4.19 for 〈δwδw〉. The dominant term in ∇ ·Φ33 is T//33 , which shows

two donor regions. That on the r+x = 0 plane illustrates a Y + − r+

z linear-ity. This linear trend, as the one of 〈δuδu〉, is consistent with the previouslymentioned attached eddy hypothesis. Also the isosurface lying on the r+

z = 0plane shows a linear tendence between Y + and r+

x , but less evident than thefirst, consistently with what observed in [32] for the spectra. It is also inter-

esting to consider the particular shape of the receiver region of T//33 (blue),

viewed from above in the right part of the same figure. Very near to the wall,this consists of a thin surface lying along the r+

x = 3r+z line. Moving away

from the wall, this surface twists, aligning to the r+x = r+

z line for Y + ∼ 180.The explanation for the twist is given in [38], in which a statistical analysisrevealed that the signature of large-scale coherent structures exhibits increas-ing meandering behaviour with distance from the wall. The same feature wasobserved in [39], in which it was associated to the presence of hairpin eddies.Consistently, in section 4.2.1 of the present work, the occurrence of hairpinvortices is inferred through an analysis of the correlations. Above Y + ∼ 180,the receiver region of T

//33 shrinks along r+

x and r+z , turning into a slender

surface lying on the Y + axis, suggesting that the transfer is directed to thesmall isotropic scales. This might appear in contrast to the previously citedworks, in which the inclined features of w are seen to extend to larger wall-normal distances: however, less intense isosurfaces of T

//33 (not illustrated) do

not show this shrinking, but extend to larger Y + preserving a 45◦ inclination.

Both ∂Ψ∂Y

and∂Φry

∂ryhave been found to contribute negligibly to the transport

of 〈δwδw〉 in this region for all the Reynolds numbers considered (not shown).

Flux divergence of 〈δvδv〉 Fig. 4.20 illustrates the main terms of ∇ ·Φ22

in the same region. The donor isosurface of T//22 (pink) lies near to the r+

z = 0


plane and extends to large values of r+x also near to the wall. On the contrary,

the receiver region (blue) shows an approximately conical shape, tending tovery small separations for reduced wall-normal distances: this is a featurecharacteristic of 〈δvδv〉, which is not affected by large-scale motions in prox-imity of the wall (inactive motions [32]). Moreover, the Y + − r+

z linearityis a further argument supporting the existence of attached eddies. Thereis also another slender receiver region on the Y + axis, which implies that apart of 〈δvδv〉 is transferred to the small isotropic scales over a large rangeof wall-normal distances. The same figure illustrates the pressure transport∂Ψpress

∂Y(green), which is null for 〈δuδu〉 and 〈δwδw〉. This quantity plays a

relevant role, as can be seen from the intensity of the isosurface, one order ofmagnitude larger than those of T

//22 . Moreover, also this term shows an ap-

proximately conical shape, consistently with the the discussion just made forT//22 . A similar trend was also highlighted in [35] for the pressure correlation,

whose length and width were found to increase approximately linearly awayfrom the wall. The other terms of ∇ ·Φ22 were found to contribute negligi-bly. Further insights can be extracted from Fig. 4.21, which illustrates someflux lines of 〈δvδv〉 and the pressure-strain term Π22. For Reτ = 1000, theorigin point of these flux lines is found far from the wall, i.e. at Y + > 300:from here, they descend to the inner layer approximately along a straightline. This is a difference w.r.t. both 〈δuδu〉 and 〈δwδw〉, for which the originpoint of the flux lines was found near to the wall, and may suggest a stronginterdependence of the inner and outer processes for 〈δvδv〉. Interestingly,the isosurface of Π22 for Reτ = 1000 is approximately a cone which hasY + as simmetry axis and terminates with a plane at Y + ∼ 300. From thisshape, two considerations can be derived. First, the usual linearity, in thiscase between Y + and r+, where the second is defined as r+ =

√r+x

2 + r+z

2.Secondly, Π22 is isotropic along the homogeneous directions, since for eachwall-normal distance this quantity does not vary in the locus of points withconstant r+.

4.2.1 From Reτ = 200 to Reτ = 1000: a dynamic view

After describing the features of the second peak of ξ11, it is shown in thefollowing how some of these can be seen also in the Reτ = 200 case, anda continuous view of the transition from low- to high-Reynolds numbers isprovided.


Figure 4.22: Terms of ∇ · Φ11 for Reτ = 200, Reτ = 500 and Reτ = 1000

respectively: isosurfaces T//11 = −0.002 ( ) and −∂Φry

∂ry= −0.002 ( ). Only

for Reτ = 200, the isosurface −∂Φry

∂ry= −0.0002 ( ) is illustrated.


Figure 4.23: Evolution of ∂Ψturb

∂Yfor 〈δuδu〉: isosurfaces −∂Ψturb

∂Y= 0.004 for

Reτ = 200, −∂Ψturb

∂Y= 0.001 for Reτ = 500 and −∂Ψturb

∂Y= 0.00055 for


Transport processes of 〈δuδu〉 It has already been shown in Fig. 4.17

that T//11 and

∂Φry

∂ryare the main terms transporting the energy produced in

the second peak of ξ11. Fig. 4.22 illustrates the same quantities for all theReynolds number considered. It is interesting to observe that for Reτ = 200an isosurface of T

//11 is present with the same shape and intensity of that for

Reτ = 1000 previously described. This surface extends from the wall up to

Y + ∼ 120. Moreover, a region of large∂Φturb

ry

∂ryon the r+

x = 0 plane, similar

to that found in the second peak, is present at Y + = 100. However, thisisosurface is one order of magnitude lower than the one found for Reτ = 500and Reτ = 1000. Interestingly, for Reτ = 200 and Reτ = 500, very nearto the wall other isosurfaces of T

//11 can be found, which do not appear for

Reτ = 1000.Also the turbulent wall-normal transport ∂Ψturb

∂Y, shown in Fig. 4.23, shows an

interesting behaviour w.r.t. variations in the Reynolds number. As alreadymentioned, this quantity contributes negligibly to the second peak of ξ11, as-suming slightly negative values in that region. However, for Reτ = 200, this


Figure 4.24: Isosurfaces T//33 = −0.002 ( ) and T

//33 = 0.002 ( ) for Reτ = 200,

Reτ = 500 and Reτ = 1000 respectively.

derivative takes significant negative values around Y + = 100, where the weak

peak of∂Φry

∂ry(recall Fig. 4.22) takes place. This, together with the reduced

intensity of∂Φry

∂ry, prevents the appearance of a second peak for Reτ = 200.

The different intensity of the isosurfaces among the three values of Reτ inFig. 4.23 aims at pointing out an interesting feature: for Reτ = 200, theisosurface ∂Ψturb

∂Y= −0.004 is only partially confined in the inner layer. The

portion which lies outside consists of two approximately planar surfaces con-nected in the point (r+

x , r+z , Y

+) = (0, 120, 100). As Reτ increases, these twoisosurfaces not only reduce their intensity, but also increase their recipro-cal distance: thus, the connection point between them becomes a line onthe plane r+

x = 0. It is thus highlighted the transition which drives to theY +−r+

z linearity characteristic of the attached eddies and to the appearanceof the second peak of ξ11.

Transport processes of 〈δwδw〉 Let us perform the the same analysis

for 〈δwδw〉: Fig. 4.24 illustrates the isosurfaces of T//33 found analyzing the

second peak (Fig. 4.19) for all the three values of Reτ considered. It is


Figure 4.25: Terms of ∇ · Φ22 for Reτ = 200, Reτ = 500 and Reτ = 1000respectively. Isosurfaces T

//22 = −0.0004 ( ), T

//22 = 0.0004 ( ) and −∂Ψpress

∂Y=

−0.002 ( ).


Figure 4.26: Isosurfaces Π22 = 0.009 ( ) and flux lines of 〈δvδv〉 for Reτ =200, Reτ = 500 and Reτ = 1000 respectively.


Figure 4.27: Analysis of the correlations for Reτ = 200. On the left: isosur-faces 〈uu′〉 = 4 ( ), 〈uu′〉 = −0.4 ( ), 〈−uv′〉 = 0.6 ( ) and 〈−uv′〉 = −0.13( ). On the right: coherent streamwise ( ) and wall-normal ( ) velocityfield induced by the dominant quasi-streamwise vortex, educed as describedin [10]. Contour levels at (0.2:0.2:0.8) of the maximum (solid line) and theminimum (dashed line) of the respective component (0.0058 and -0.0077 for uand 0.0035 and -0.0035 for v) are plotted on a y+−z+ plane passing throughthe center of the vortex, located at z+ = 0.

Figure 4.28: Analysis of the correlation 〈−uv′〉 for different Reynolds num-bers. Isosurface 〈−uv′〉 = −0.13 for Reτ = 200 ( ), Reτ = 500 ( ) andReτ = 1000 ( ). Isosurface 〈−uv′〉 = 0.6 for Reτ = 200 ( ), Reτ = 500 ( )and Reτ = 1000 ( ).


Figure 4.29: Analysis of the correlation 〈uu′〉 for different Reynolds numbers.Isosurfaces 〈uu′〉 = −0.4 for Reτ = 200 ( ), Reτ = 500 ( ) and Reτ = 1000( ). Isosurface 〈uu′〉 = 4 for Reτ = 200 ( ), Reτ = 500 ( ) and Reτ = 1000( ). A zoomed view of the same picture is provided in the inset.

Figure 4.30: Correlations 〈ww′〉 and 〈vv′〉 for Reτ = 1000: isosurfaces〈ww′〉 = −0.04 ( ) and 〈vv′〉 = −0.01 ( ). On the right, a 2D view isshown.


Anticorrelation of 〈ww′〉 on the r+x = 0 plane

Figure 4.31: On the left, section of two counter-rotating streamwise-alignedvortices. On the right: scheme of a hairpin vortex, which can be split in theparts (1), (2) and (3), detailed in the text.

Figure 4.32: Analysis of the correlations for Reτ = 200 (on the left) andReτ = 500 (on the right): 〈ww′〉 = −0.04 ( ) and 〈vv′〉 = −0.01 ( ).


apparent a strong similarity of the negative (donor) isosurfaces among thethree Reynolds numbers: in particular, the previously seen two-lobed config-uration is the same found in the inner layer (Fig. 4.8) and the Y + − r+

z andY +−r+

x linearities can be found also for Reτ = 200. Moreover, the previously

discussed twisting of the positive (receiver) region of T//33 is visible also for

Reτ = 200 and Reτ = 500. These observations give further support to thecommon nature shared by the first and the second peak of ξ11. Interestingly,for Reτ = 200 and Reτ = 500, a transfer of 〈δwδw〉 occurs also near to the

wall, as can be seen from the other receiver regions of T//33 at Y + ∼ 20.

Transport processes of 〈δvδv〉 In Fig. 4.25, the isosurfaces of T//22 and

∂Ψpress

∂Yalready discussed for Reτ = 1000 are illustrated also for Reτ = 500

and Reτ = 200. The positive isosurface of T//22 shows a tendency to linearity

also for these lower Reynolds numbers. However, differently from the higherReynolds numbers, no positive isosurface along the Y + axis is present in theReτ = 200 case: this is due to the total absence of scale separation, whichprevents the existence of small isotropic scales receiving 〈δvδv〉. The maindonor isosurface for Reτ = 200 appears to involve a larger range of r+

z as Y +

increases, suggesting the presence of ejections-like structures. For larger Reτ ,this isosurface moves to the r+

z = 0 plane, tending to a more parallelepipedalshape, having r+

x >> r+z . Finally, in the Reτ = 200 case there is a further

donor region in the inner layer lying on the r+x = 0 plane at r+

z ∼ 100. Thisis much shorter along r+

x than the other, i.e. it has r+x < 100 approximately.

This isosurface has been found to shift to larger r+z and reduce its intensity

as Reτ increases, so that it is not visible in the figure for Reτ = 500 andReτ = 1000. Fig. 4.26 illustrates the flux lines and Π22 for all the threeReynolds numbers. At this point, it is evident the shift of the origin pointof these lines from Y + ∼ 110 for Reτ = 200, to Y + ∼ 300 and Y + ∼ 350 forReτ = 500 and Reτ = 1000 respectively. Moreover, the flux lines departingfrom this point tend to a straight line for larger Reynolds numbers. Finally,the similarity of the isosurfaces of Π22 among the three values of Reτ supportsthe hypothesis of a continuous trend from low to high Reynolds numbers.

Analysis of the correlations Let us now study the transition from Reτ =200 to Reτ = 1000 through the analysis of the correlations 〈uu′〉 and 〈−uv′〉.For Reτ = 200 (Fig. 4.27), the maxima (positive) of 〈uu′〉 and 〈−uv′〉 areboth found for r+

z ∼ 0. Moreover, also the minima (regions of anticorrela-


tions, negative) of the two quantities share the same spanwise separation, i.e.r+z ∼ 60. In [10], it is pointed out how this configuration can be associated

to the presence of quasi-streamwise vortices, whose induced velocity field isillustrated on the right of Fig. 4.27. Positive values of both the correlationsare obtained when we consider two points located at the same side of thevortex, i.e. at r+

z ∼ 0, while negative ones derive from considering points atthe opposite sides, i.e. at r+

z ∼ 60, which is consistent with the characteristicdimension found in [35]. Moreover, the maximum and minimum of 〈uu′〉are located at Y + ∼ 15, wall-normal distance at which the break-up of thestreaks has been seen to occur. In Fig. 4.28, the cross-correlation 〈−uv′〉 isillustrated for the three values of Reτ considered. While the positive regionfor increasing Reτ simply extends to larger Y +, the negative one shows amore interesting behaviour. For Reτ = 500, an evident Y + − r+

z linearityis shown, i.e. larger vortices are found as larger wall-normal distances areconsidered, consistently with the presence of attached eddies in this regionof the (r+

x , r+z , Y

+) space. When the Reynolds number is furtherly increased,the isosurface breaks down in two parts, corresponding respectively to thenear-wall and second peaks of ξ11.

Fig. 4.29 illustrates the isosurfaces close to the maxima (positive) andminima (negative) of the auto-correlation 〈uu′〉 for the three values of Reτ .The region of positive correlation undergoes minor changes. On the contrary,moving from Reτ = 200 to Reτ = 500, the isosurface of anticorrelation is stillconfined in the inner layer, but reduces its extent. For Reτ = 1000, instead, itdoes not appear in the inner layer, but only in the region in which the secondpeak of ξ11 takes place. In [35], the anticorrelation peak (negative) of 〈uu′〉for Reτ = 2000 was also found outside of the inner layer. This highlights aninteresting difference w.r.t. 〈−uv′〉 in which, for Reτ = 1000, the isosurfaceshows both the near-wall contribution and the one from the second peak ofξ11.

Other considerations about the structure of the coherent motions for thethree Reynolds numbers can be made considering the regions of anticorre-lation of 〈vv′〉 and 〈ww′〉, illustrated in Fig. 4.30 for Reτ = 1000. Thelargest isosurface of 〈vv′〉 shows the usual Y + − r+

z linearity. Moreover, thisisosurface shrinks on the Y + axis very near to the wall: this confirms thepreviously discussed inactivity of some motions in the near-wall region forthe v component of velocity, due to the impermeability condition. This sur-


face of anticorrelation of 〈vv′〉 lies on the plane r+x = 0, confirming that these

attached structures have the shape of QSVs. An estimate of the averagetilt angle of these structures with respect to the streamwise direction in aplane parallel to the wall has been derived by considering the intersectionof this isosurface of 〈vv′〉 with a plane orthogonal to the Y + axis. Consid-ering a wall-normal distance Y + ∼ 20, a tilt angle θ ∼ 6◦ was found, nearto the values found in literature employing different criteria [13]. Moreover,Fig. 4.30 illustrates an isosurface of negative 〈ww′〉 on the r+

x = 0 plane:this suggests the presence of pairs of counter-rotating rollers aligned in thestreamwise direction, as schematized on the left of Fig. 4.31. Consistentlywith the scheme, the spanwise separation of the anticorrelation region of〈ww′〉 is larger than the one of 〈vv′〉, since the second is the spanwise widthof one vortex, while the first represents the distance between the two.In the following, it is suggested that these pairs of attached rollers are in facta part of the legs of hairpin vortices. A scheme of a hairpin structure is illus-trated in the right part of Fig. 4.31, in which three parts can be identified.The rollers previously described make up part (1), i.e. the streamwise-alignedportion of the legs. Notice that the correspondent isosurface of 〈ww′〉 extendsto large streamwise separations near to the wall, i.e. for Y + ∼ 50, well repre-senting the influence of these large streamwise scales on the inner layer. Theother isosurface of negative 〈ww′〉 on the r+

z = 0 plane might correspond topart (2) of this structure, i.e. the part of the legs inclined w.r.t. the wall.Part (3) of the hairpin vortex is represented by the head, which is spanwise-oriented: thus, 〈vv′〉 takes negative values, i.e. anticorrelation, if we considertwo points at opposite sides of the vortex. This region of anticorrelation of〈vv′〉 is visible in Fig. 4.30 on the r+

z = 0 plane, above the isosurface of neg-ative 〈ww′〉, at approximately the same streamwise separation r+

x ∼ 1000.This interesting feature was observed also in [35].It is interesting to understand how this hairpin structure changes for lowervalues of the Reynolds number, as illustrated in Fig. 4.32. Here, the iso-surface of 〈vv′〉 on the r+

z = 0 plane is connected to that on the r+x = 0

one, emphasizing how these two isosurfaces may be due to a single coherentstructure as the hairpin one previously described. Furthermore, both theisosurfaces of anticorrelation of 〈vv′〉 and 〈ww′〉 scale approximately in outerunits: this is particularly evident for the isosurface of 〈vv′〉 on the r+

z = 0plane, whose top part is found at the centre of the channel for all the threeReynolds numbers considered. The presence of this coherent structure inthe logarithmic and outer layer was first suggested by Theodorsen ([40]) and

4.3. THE THIRD PEAK OF ξ11 47

Figure 4.33: 3D (left) and 2D (right) view of the isosurfaces ξ11 = 0.0001for Reτ = 1000. The portion of the isosurface having Y + < 25 has not beenrepresented in this figure.

furtherly investigated through a statistical approach in [41], [42], [43] andconsidered in the more recent works of [39] and [38] in relation to the in-clined features of w discussed for Fig. 4.19. It is finally pointed out thatanother isosurface of anticorrelation of 〈ww′〉 (not illustrated) has been ob-served to lie on the r+

z = 0 plane for Y + ∼ 50 and very large streamwiseseparations, i.e. r+

x > 4000. This feature is likely to be associated to a span-wise meandering of the previously discussed rolled structures.

Summing up, some characteristic features of the terms of ∇ ·Φij in the

second peak-region, such as donor and receiver regions of T//11 and T

//33 , and

Y +−r+ linearities of the wall-normal transport and the transfer across scales,can be found also for lower Reynolds numbers. However, some differences inthe behaviour of other terms, such as ∂Ψturb

∂Yfor 〈δuδu〉, prevent the forma-

tion of a similar peak at these low Reynolds numbers. Moreover, the study ofthe correlations highlighted the transition from the first to the second peakof ξ11, which appear to have similar nature. Considering in particular 〈vv′〉and 〈ww′〉, the previously discussed streamwise-aligned attached eddies weresuggested to form a part of the legs of a hairpin vortex extending until themid-height of the channel.

4.3 The third peak of ξ11

The first and second peaks of ξ11 have been found for both Reτ = 500 andReτ = 1000. However, a third peak has been observed, both for 〈δuδu〉 andfor the whole scale energy 〈δuiδui〉, only in the Reτ = 1000 case, as illus-trated in Fig. 4.33. Like the second peak, it lies on the r+

x = 0 plane and


Figure 4.34: Isosurfaces −∂Φturbrx

∂rx= −0.0004 ( ), −∂Φturb

rz

∂rz= −0.0003 ( ),

−∂Ψturb

∂Y= −0.0002 ( ) and −∂Φry

∂ry= −0.0002 ( ) for Reτ = 1000.

Figure 4.35: Approximate invariance of the isosurface ξ11 = 0.0001 along r+x

for Reτ = 1000. The portion of the isosurface for Y + < 25 has not beenrepresented in this figure.


Figure 4.36: ξ11 at Y + = 201 (continue), Y + = 246 (dotted-dashed) andY + = 300 (dashed) for Reτ = 1000.

Figure 4.37: −〈δuδv〉 (left) and ξ12 (right) for Reτ = 1000. Y + = 201(continue), Y + = 246 (dotted-dashed) and Y + = 300 (dashed).

Figure 4.38: 〈δvδv〉 (left) and ξ22 (right) for Reτ = 1000. Y + = 201 (con-tinue), Y + = 246 (dotted-dashed) and Y + = 300 (dashed).


Figure 4.39: ξ22 at Y + = 415 (continue), Y + = 465 (dotted-dashed) andY + = 520 (dashed) for Reτ = 1000.

Figure 4.40: Components of Φ22: Φrz ( ), Φrx ( ) and Ψ ( ). Y + = 201(continue), Y + = 246 (dotted-dashed) and Y + = 300 (dashed).


extends along r+x until large streamwise separations. However, it is located

at much larger spanwise separations, i.e. r+z > 2000, and its intensity is sig-

nificantly lower than that of the second peak. Thus, it can be associated tothe presence of very large-scale motions (VLSMs) in the logarithmic layer.Fig. 4.34 illustrates isosurfaces of the main terms of ∇ ·Φ11 contributing tothis third peak of ξ11. The terms ∂Φrx

∂rx, ∂Φrz

∂rzand ∂Ψ

∂Yare dominated by their

turbulent component (result not illustrated). The same is assumed to hold

for∂Φry

∂ry. Among these four terms,

∂Φturbrx

∂rxappears negligible at the spanwise

separation of these very large structures, i.e. r+z > 2000. This does not hold

for the second peak of ξ11, in which this derivative has been found to representa dominant contribution. Another difference between the two peaks is therole of ∂Ψturb

∂Y, which is negligible only in the second. Notice that ∂Ψturb

∂Yand

∂Φturbry

∂rydetermine the wall-normal distance at which this third peak of pro-

duction takes place, while∂Φturb

rz

∂rzdetermines the spanwise separation.

∂Φturbrz

∂rzremarks clearly the distinction between the second and the third peak: thefirst is associated to the part of the isosurface found for r+

z ∼ 500, while thesecond corresponds to separations r+

z > 2000. In particular, this second partappears to be furtherly splittable in two separated surfaces, at r+

z ∼ 2000 andr+z ∼ 3000 respectively: however, the achievement of convergence at these

extremely large spanwise separations has been seen to require a number offlow fields larger than that used in this work, as shown in Fig. 3.3. Finally,it is pointed out the much more pronounced variability of quantities such as

ξ11 and∂Φturb

rz

∂rzalong r+

z than along r+x . This appears consistent with the work

of Moin and Kim ([44]), in which the one-dimensional streamwise velocitycorrelation in the flow direction was found to extend over much longer dis-tances than in the transversal one.

Origin of the third peak The following part of the work aims at identify-ing the origin of this third peak of production of 〈δuδu〉. Since, as apparentfrom Fig. 4.35, the source isosurfaces are approximately invariant alongthe streamwise separation, an average has been performed from r+

x = 0 tor+x = 1900. The resulting averaged ξ11 is plotted in Fig. 4.36 for three wall-

normal distances. These three values of Y + were chosen inside the isosurfaceξ11 = 0.0001 illustrated in Fig. 4.33. Three local maxima can be found:the first, at r+

z ∼ 500, and the second, at r+z ∼ 2100, are associated respec-

tively to the second and the third peak of ξ11. The third local maximum,


found at r+z ∼ 2900, is not considered in the following, because of the related

uncertainty previously discussed. Recall that the production of 〈δuδu〉 is pro-portional to 〈−δuδv〉, according to equation 2.11: consistently, the maximumpoints of the two quantities are the same, as results from the comparison be-tween Fig. 4.36 and the left part of Fig. 4.37. Also the source of 〈−δuδv〉,reported on the right of Fig. 4.37, shows two local maxima at r+

z ∼ 500 andr+z ∼ 2100 (together with a third one between these two), although these are

slightly pronounced. Reasoning as before, it is recalled that the productionP12 is proportional to 〈δvδv〉, which is illustrated in Fig. 4.38. Although thisquantity changes significantly with Y +, the maximum point at r+

z ∼ 2100 isstill present. However, this does not hold for its source ξ22, illustrated on theright of the same figure: thus, the energy 〈δvδv〉 present in this region is notproduced here, but derives from a transport. If we now consider the sourceterm ξ22 at larger wall-normal distances (Fig. 4.39), we find again the peak atr+z ∼ 2000, suggesting that 〈δvδv〉 is first produced at these wall-normal dis-

tances and then transported towards the wall. This hypothesis is consistentwith the negative value of Ψ (i.e. directed downwards), illustrated in Fig.4.40, where it appears to be, in absolute value, the dominant flux component.

Chapter 5

Conclusions and futuredevelopments

In the present work, the Anisotropic Generalized Kolmogorov Equation (AGKE)has been employed to study the turbulent channel flow for values of the fric-tion Reynolds number Reτ = 200, Reτ = 500 and Reτ = 1000 respectively.The AGKE is a statistical budget equation for the second-order structurefunction 〈δuiδuj〉; tailored to the channel flow, it describes the processesof production, dissipation and transport in the compound space of scalesr = (rx, ry, rz) and positions Y .

In the present research, the interpretation of the terms composing theAGKE has allowed the derivation of insights not only on the inner but alsoon the outer coherent structures relevant in high-Reynolds number flows.Specifically, an important result has been the identification of three peaksin the production of 〈δuδu〉. The first, already discussed in [10], is linked tothe coherent motions of the near-wall cycle. The second, introduced in [8]and present only for Reτ = 500 and Reτ = 1000, has been here attributedto attached eddies. The third, documented in this work for the first time forReτ = 1000, has been associated to the presence of large- ad very large-scalemotions.Regarding the inner layer, some transport terms have been found not to scalein wall units, due to the interaction between the inner and outer structures.Moreover, a shift in the origin of the flux lines of 〈δuδu〉 to larger spanwiseseparations has been found for increasing Reynolds numbers, suggesting thatthe spanwise scales involved in the transport of 〈δuδu〉 are larger than those

53

54 CHAPTER 5. CONCLUSIONS AND FUTURE DEVELOPMENTS

of the whole scale energy 〈δuiδui〉, remarking the effectiveness of the AGKEin capturing the anisotropic behaviour of the turbulent channel flow.Regarding the second peak, in this region a linear relation between the scalesand the wall-normal distance has been found for different quantities. More-over, the “inactivity” of these attached motions at the wall for the component〈δvδv〉 was confirmed. The meandering behaviour of the transfer of spanwiseenergy 〈δwδw〉 has been another observed feature, linked to the presence ofhairpin vortices. Interestingly, some features of this second peak of ξ11 havebeen identified also for Reτ = 200, suggesting a continuous transition fromlow- to high-Reynolds number flows.Finally, the analysis of the third peak has revealed that the transport pro-cesses involved are different from those related to the second one, highlightingthe different nature of these two. Furthermore, the origin of the third peakhas been traced back to a vertical transport of 〈δvδv〉 occurring at largespanwise separations and directed towards the wall.

The link between the terms of the AGKE and the properties of the co-herent structures involved in the transfer of 〈δuiδuj〉 makes this tool verypromising for the study of inhomogeneous flows. The benefits are even largeras higher Reynolds numbers are considered, due to the increasing range ofinteracting scales.A very promising investigation is considering configurations modified by dragreduction techniques. Finally, providing a complete description in the com-pound space of scales and positions, the AGKE may be used to refine themodels used in large-eddies simulations aimed at reproducing the physics ofthe small unresolved scales and their effect on the resolved motions.

Acknowledgements

First and foremost, I intend to thank my family, who has always supported me with the love and

positivity I needed to face the most difficult challenges: my Mom, to whom I owe my determination

and persistency, my Dad, whose values have always inspired all my choices, and my Grandma, who

always believed in me more than I could do myself. My Uncle, my Aunt and my Cousins deserve a

thank too, giving me precious advices in my most important decisions.

I am deeply thankful also to Professor Maurizio Quadrio, to which I owe the passion for turbulence

and the chance to take part to this unique experience.

Special thanks to Davide: he loves what he does, as I had the chance to feel every time I needed

help, and he always made me feel at home.

I also owe a “Vielen Dank!” to the colleagues of the ISTM department, particularly Erik, Olivier,

Jan, Teresa, Sandra and Georg (I’ll never forget the Fußpills, Erik!).

I would also like to thank the friends of the gangs of Valleggia and Mioglia, who taught me how to

survive to the most intense nights and gave me the energy I needed.

A big thought goes also to the Timoteo and the Bluevelvet, whose enthusiasm taught me more than

I can explain in these lines.

I am deeply grateful also to my ex-colleagues Pier, Euge, Gabri and Jac, with the huge laughs and

the loads of sushi before the lectures: missed you guys.

A nostalgic thank goes also to my first colleagues and friends in Milan: the Aerodynamic project I

shared with you was the best teamwork I could ever dream of: Andre, Giuse, Lore, Il Cala, Santi,

Laura, Luca, Braghis and the legendary Berna, the nights of San BabilHouse and of the “Sandali”

will stay always vivid.

My homemate, Daniele, deserves a huge thank, too: the cooking, the reminders of the deadlines I

always forget, and the weird healthy habits of the housekeeper are the nicest memories.

I express my gratitude also to the colleagues of the ASP: the projects, the impossible deadlines, the

fun: you are an inspiration, always.

Finally, huge thanks go to the people who made my Erasmus: the SprachCafe, the Oxford Pub,

the Sully’s dinners, the nights at the Kroko and the magic of the Stammtisch. Ema, Sully, Chloe,

Juanpe, Paola, Sara, Margaux, Tingting, Sevde, Berta, Pascal, Evam, Kevin, Comodo, Zac, Casey,

Umaira, Hannah and the little Lady Natalia, looking forward to seeing you in Italy.

A final thought goes to all the people who were close to me in these years, which were fast, stressful,

hard and truly amazing. With love.

Marco

55

56 CHAPTER 5. CONCLUSIONS AND FUTURE DEVELOPMENTS

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Documents

Scuola di Ingegneria Industriale e dell’Informazione Corso ...per canale turbolento a numero di Reynolds viscoso pari a Re ˝ = 200, Re ˝ = 500 e Re ˝ = 1000, sono stati considerati