UNIVERSIDAD POLITECNICA DE MADRIDESCUELA TECNICA SUPERIOR DE INGENIEROS AERONAUTICOS

The study and control of wall bounded

flows

El estudio y control de flujos de pared

Tesis Doctoral

Por

Mark Phil Simens

Ingeniero Mecanico

Madrid, January 2008

DEPARTAMENTO DE MOTOPROPULSION Y TERMOFLUIDODINAMICA

ESCUELA TECNICA SUPERIOR DE INGENIEROS AERONAUTICOS

The study and control of wall bounded

flows

El estudio y control de flujos de pared

Autor

Mark Phil Simens

Ingeniero Mecanico

Director de Tesis

Javier Jimenez Sendn

Doctor Ingeniero Aeronautico

Madrid, January 2008

2

Contents

Contents i

Acknowledgements vi

Preface ix

Nomenclature ix

Abstract xxii

1 Introduction 1

1.1 Wall bounded flows with and without strong APGs . . . . . . . . . . 2

1.1.1 Review of the scaling of the mean flow and Reynolds-stresses . 5

1.2 Near-wall flow in a turbulent channel . . . . . . . . . . . . . . . . . . 9

1.2.1 Objective for the first part of the thesis . . . . . . . . . . . . . 10

1.3 Control and its numerical study . . . . . . . . . . . . . . . . . . . . . 11

1.3.1 Control of separation on low-pressure turbine blades . . . . . . 11

1.3.2 High resolution versus low resolution . . . . . . . . . . . . . . 13

1.3.3 Objectives for the second part of the thesis . . . . . . . . . . . 14

1.4 Tools used for the thesis . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4.1 Organization of the thesis . . . . . . . . . . . . . . . . . . . . 15

2 Characterization of near-wall turbulence in terms of equilibriumand bursting solutions 17

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Computational methods . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Autonomous solutions . . . . . . . . . . . . . . . . . . . . . . 19

2.2.2 Plane Couette solutions . . . . . . . . . . . . . . . . . . . . . 21

2.3 Classification of solutions . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.1 The structure of the flow field . . . . . . . . . . . . . . . . . . 26

2.4 Comparison with turbulent flows . . . . . . . . . . . . . . . . . . . . 29

2.4.1 Flow type and Reynolds-number effects . . . . . . . . . . . . . 29

2.4.2 Comparisons with full turbulence . . . . . . . . . . . . . . . . 33

2.5 Bursting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

i

Contents

3 A numerical code to simulate turbulent boundary layers 45

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3 The spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.3.1 Fourth-order schemes and the DG matrix . . . . . . . . . . . 51

3.4 Pressure, mass conservation and boundary conditions . . . . . . . . . 57

3.5 Code verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.5.1 Individual discretization . . . . . . . . . . . . . . . . . . . . . 61

3.5.2 Time-dependent flows . . . . . . . . . . . . . . . . . . . . . . 63

3.5.3 Blasius profile . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.5.4 Three-dimensional turbulent separated boundary layer . . . . 67

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4 The control of laminar separation bubbles 71

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2 Numerical techniques and unperturbed flows . . . . . . . . . . . . . . 72

4.2.1 Problem description . . . . . . . . . . . . . . . . . . . . . . . 74

4.3 The influence of C, forcing frequency, Reynolds number and APGon flow development . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.3.1 Influence of C and forcing frequency on flow development . . 76

4.4 Frequency selection criteria . . . . . . . . . . . . . . . . . . . . . . . . 82

4.4.1 Kelvin-Helmholtz instability . . . . . . . . . . . . . . . . . . . 82

4.4.2 Vortex creation due to high amplitude forcing . . . . . . . . . 83

4.4.3 Influence of suction . . . . . . . . . . . . . . . . . . . . . . . . 89

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5 The study of a boundary layer under the influence of strong Ad-verse Pressure Gradients and wakes 93

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.2 Numerical method, geometry and boundary conditions . . . . . . . . 95

5.2.1 Design and resolution requirements of future numerical exper-iments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.3.1 Instantaneous results . . . . . . . . . . . . . . . . . . . . . . . 102

5.3.2 First order moments . . . . . . . . . . . . . . . . . . . . . . . 103

5.3.3 Second and higher order moments . . . . . . . . . . . . . . . . 111

5.3.4 Comparison with ITP laboratory data . . . . . . . . . . . . . 120

5.3.5 Momentum and energy balances . . . . . . . . . . . . . . . . . 123

5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6 Conclusions and future work 133

ii

Contents

A Additional numerical issues 137A.1 Stability in time and space . . . . . . . . . . . . . . . . . . . . . . . . 137A.2 Numerical schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

A.2.1 Boundary schemes . . . . . . . . . . . . . . . . . . . . . . . . 141

B Multi-grid solution method 147B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

B.1.1 Multi-grid ingredients . . . . . . . . . . . . . . . . . . . . . . 149B.2 (Non)-uniform restriction and interpolation . . . . . . . . . . . . . . . 151B.3 Creation of matrix A on coarse grids . . . . . . . . . . . . . . . . . . 153

B.3.1 The high-order schemes . . . . . . . . . . . . . . . . . . . . . 154B.3.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . 154

C Resumen 157C.1 Introduccion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157C.2 Primera parte. La caracterizacion de la region de la pared turbulenta 158

C.2.1 Objetivos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159C.3 Caracterizacion de la region de la pared turbulenta en termino de

soluciones en equilibrio y bursting . . . . . . . . . . . . . . . . . . 159C.4 Segunda parte. Estudio numerico de la separacion y su control . . . . 161

C.4.1 Codigo numerico para simular capas lmite turbulentas . . . . 162C.4.2 Control de burbujas de separacion laminar . . . . . . . . . . . 168C.4.3 El control de separacion en turbinas de bajo presion . . . . . . 169C.4.4 Estudio de una capa lmite bajo la influencia de fuertes gra-

dientes de presiones adversas y estelas . . . . . . . . . . . . . 175

Bibliography 181

Curriculum Vitae 191

iii

Contents

iv

Acknowledgments

After six years of hard work my PhD thesis has finished. In the first place I haveto thank my supervisor Professor J. Jimenez for this. Without his motivation,confidence and constructive criticism this thesis would not have been possible.

Thinking about the past six years I would like to mention my two closest col-leagues. Juan Carlos del Alamo and Oscar Flores have made the time I worked inthe Fluid Mechanics department unforgettable. Especially at the beginning of myPhD thesis Juan Carlos made a large effort to help me with my Spanish and to makeme feel at home. The last years of my thesis have been made bearable by Oscar.Related to the present thesis I would like to thank them for proofreading it.

I am also very pleased to know Miguel Hermanns and Marcos Vera. I will alwaysbe intrigued by the the difficult problems they solve using semi-analytical techniques.

Professor Theofilis and Professor Higuera have always shown a large interest inmy work, for which I am very grateful.

I would also like to thank the people working at ITP S.A. for their interest inmy work and in particular Roque Corral, Javier Crespo and Fernando Gisbert.

A lot of people have past by or are still present in the department. The ones Iwould like to thank are Sergio, Samuel, Guillem, Rafa, Cedric, Yoshi, Jorge, Guil-laume, Gwenael, Alvaro, and Ricardo.

I would also like to mention my parents and grandparents for their support.On a more personal level I would like to thank the person who has been most

important for me the last six years namely, Carmen.

v

vi

Preface

Funding

This thesis was funded by several public entities and private companies. In alpha-betical order they are

Airbus; the CAFEDA network.

CICYT contract DPI2003-03434.

EU TMR network (HPRN-CT-2002-00300).

ITP S.A.

The Wallturb consortium.

Publications, reports and presentations

The results presented in this thesis have been partly or completely published inconference proceedings, publicly available reports, or refereed journal publications.

Chapter 2 has been integrally published. The title is: Characterization of near-wall turbulence in terms of equilibrium and bursting solutions, written by J.Jimenez, G. Kawahara, M.P. Simens, M. Nagata and M. Shiba and publishedin volume 17 of the 2005 edition of Physics of Fluids.

Chapter 3 was made publicly available partly, as an ETSIA /MF-071 technicalnote.

The results in chapter 4 were published in various conference proceedings andwere presented in several conferences.

Some of the results in chapter 5 were presented at a meeting of the Wallturbconsortium.

Division of work between authors

The numerical Couette flow data in chapter 2 was calculated by G. Kawahara,M. Shiba and M. Nagata. Other numerical data was provided by J.C. delAlamo (JCA). The autonomous channel data results were calculated by M.P.Simens (MPS). The code used to obtain this data has a long history, whererecent changes have been made by O. Flores, J. Jimenez (JJ), A. Pinelli andMPS. The article was written by JJ.

vii

The numerical code presented in chapter 3 and used in chapter 5 was entirelyprogrammed by MPS. Except for a few exceptions, which include the RFFTsused, which were adapted from a publicly available code by JJ, and the tri-diaganol solver, written by JJ. The cosine transform used was programmed byR.A. Sweet, L.L. Lindgren and R. F. Boisvert. JJ aided in solving theoreticaldifficulties. Some useful remarks were made by G. Hauet (GH). The technicalnote ETSIA /MF-071 was written to a large extent by MPS with considerablecontributions from JJ.

The code used to obtain the results in chapter 4 was programmed by JCA. Itwas adjusted to implement different boundary conditions by MPS afterward.The conference proceedings were written with the collaboration of JJ.

All the statistics, results and interpretation of the post-processing presentedin chapter 5 was done by MPS. The set-up of the numerical experiment wasdone in collaboration with JJ. Helpful comments were made by Roque Corral.

viii

Nomenclature

Abbreviations

b.c.s Boundary conditions []

FD First derivative []

I Interpolation []

N S Navier-Stokes []

SD Second derivative []

APG Adverse Pressure Gradient []

FFT Fast Fourier Transform []

FPG Favorable Pressure Gradient []

GB Giga-Byte []

GHz Giga-Herz []

IBM International Business Machines []

PBS Portable Batch System []

RAM Random Access Memory []

UPM Universidad Politecnica de Madrid []

ZPG Zero Pressure Gradient []

Greek symbols

x Dimensionless domain size in x []

y Dimensionless domain size in y []

z Dimensionless domain size in z []

1,2,3 Implicit Runge-Kutta time integration coefficients []

pb Perturbation wavenumber [1m

]

ix

Non-dimensional pressure

1,2,3 Implicit Runge-Kutta time integration coefficients []

pb Perturbation wavenumber [1m

]

Displacement thickness [m]

99 Boundary layer thickness: height where u(y) = 0.99U [m]

f Displacement thickness measured at forcing slot position [m]

The error between a model and measurement []

0 Constant in wake-velocity profile 0.0222U,0cddw [m2

s]

Error between measured vortex and elliptic Gaussian model []

P Error due to the finite difference discretization of the Poisson equation []

ad Error due to the finite difference discretization of the advective terms []

ITP Error between ITP inlet data and numerical inlet data []

Dimensionless length scale []

k Kolmogorov micro-scale [m]

The ratio between up and u []

Circulation of vortex [m2

s]

1,2,3 Explitcit Runge-Kutta time integration coefficients []

pb Perturbation wavenumber [1m

]

von Karman constant []

u Third power of the ratio between up and u []

bl Boundary layer parameter []

Kinematic viscosity [m2

s]

Control volume over which velocity is integrated [m3]

Vorticity or radial frequency depending on context [1s, 1

rad]

(y) Perturbation function [m]

Approximately 3.14159265358979 []

x

Stream function [m2

s]

1,2,3 Explicit Runge-Kutta time integration coefficients []

Density [ kgm3

]

Shear stress [ kgms2

]

k Kolmogorov time-scale [t]

w Shear stress at the wall [kg

ms2]

Momentum thickness [m]

b Momentum thickness of shear layer formed by separation bubble [m]

k The angle with the kx-axis of k [rad]

s Momentum thickness at separation point [m]

dw Diameter of cylinder generating wake [m]

Uw Velocity of cylinder generating wake [m]

upb Perturbation velocity [ms]

Mathematical symbols

()T Transpose of a matrix []

() Variable in between l and l + 1 or n and n+ 1 []

()1 Inverse of a matrix []

()l Variable at Runge-Kutta substep l []

()n+1 Variable at timestep n + 1 []

()n Variable at timestep n []

() Quantity assumed to be at infinite distance to wall []

()x Variable in x-direction []

()y Variable in y-direction []

()z Variable in z-direction []

()0 Quantity assumed to be at an origin []

(),0 Quantity assumed to be at infinite distance to wall and at an origin []

xi

yy() Finite difference approximation of yy []

Divergence operator []

Gradient operator []

() Average variable []

xx() Second derivative in the x-direction []

x() First derivative in the x-direction []

yy() Second derivative in the y-direction []

y() First derivative in the y-direction []

zz() Second derivative in the z-direction []

z() First derivative in the z-direction []

() () Inner product of two vectors []

Vector []

() Variable in Fourier space []

i1 []

O Order symbol []

Roman symbols

h Grid size [m]

t Timestep [s]

x Grid size in the streamwise direction [m]

y Grid size in the wall-normal direction [m]

z Grid size in the spanwise direction [m]

t Turbulent dissipation [m2

s4]

Ma Mach number []

A Matrix given by: I tLRe []

a1 First coefficient in global mass conservation function [kgms2

]

a2 Second coefficient in global mass conservation function [kgms2

]

xii

A Area of a vortex [m2]

a Length of the longest semi-axis of an elliptic vortex [m]

AA Amplitude analytic function [1s]

ac Coefficient in global mass conservation function: p2(x, y) = ac(x2y2) [kgm

s2]

AD Left-hand matrix of divergence operators []

AG Left-hand matrix of gradient operators []

as Suction velocity amplitude [ms]

aAPG Coefficient used the in description of U(x) [1s]

Apb Perturbation amplitude [ms]

asf Structure function []

b Length of the shortest semi-axis of an elliptic vortex [m]

BD Right-hand matrix of divergence operators []

BG Right-hand matrix of gradient operators []

bs Length over which suction is applied [m]

C A constant value []

c Cord length of wing or turbine blade [m]

C Oscillatory momentum coefficient []

cd Drag coefficient []

Cf Dimensionless shear stress []

cs Position where suction is applied [m]

CFL Courant-Friedrichs-Lewy number []

D Matrix of finite difference approximation of the divergence operator []

Ds Length scale of the shedding frequency [m]

E(t) Kinetic energy as a function of time:

i,j u2i,j,t + v

2i,j,t [

m2

s2]

f Frequency [Hz]

f(x) General two or three-dimensional function []

xiii

fs Shedding frequency [1s]

fd Fourier discretization []

fo Fourth-order standard Pade scheme []

G Matrix of finite difference approximation of the gradient operator []

G Clauser parameter []

g(x) General two or three-dimensional function []

H Shape factor / []

I Unitary matrix []

i Grid index in the streamwise direction []

j Grid index in the wall-normal direction []

k Wavenumber [ 1m

]

K Relation between forcing and boundary layer mass flux: VfLf/Uf []

k Grid index in the spanwise direction []

k Non-dimensional wavenumber []

kmod Modified wavenumber [1m

]

L Matrix of finite difference approximation of second derivative operator []

Lb Length of separation bubble [m]

Lf Forcing slot length [m]

Li Turbulent integral scale [m]

Lv Approximate length over which suction is applied [m]

Lx Size of experimental domain in the streamwise direction [m]

Ly Size of experimental domain in the wall-normal direction [m]

Lz Size of experimental domain in the spanwise direction [m]

Lb0 Length of initial unforced separation bubble [m]

Lfb Distance between trailing edge forcing slot and separation point [m]

Lx,s Useful length along the plate to collect data [m]

xiv

Lx,s Useful domain length [m]

m Non-dimensional pressure [m]

N Matrix of finite difference approximation of non-linear operator []

n The coordinate normal to a surface [m]

Np Total number of grid points NxNyNz []

Nx Grid points in the streamwise direction []

Ny Grid points in the wall-normal direction []

Nz Grid points in the spanwise direction []

p, P Pressure [kgm3

s2]

p1 Pressure computed from the Poisson equation [kgm3

s2]

p2 Pressure computed to globally conserve mass [kgm3

s2]

r Right hand side of discretized N-S equations with the explicit terms []

r Radius of a vortex [m]

rm Modification of r []

Re Reynolds number []

Re Reynolds number based on momentum thickness []

Red Reynolds number based on cylinder length []

S Surface of control volume [m2]

S xu2 + xvyu [

1s2

]

so Second-order scheme []

St Strouhal number []

St Strouhal number based on U and b []

StF Strouhal number based on U and Lb []

t Time [s]

T Non-dimensional level of turbulence []

tf Forcing time [s]

xv

ts Time that vortex entrains fluid determined by fs and Ds [s]

tav Averaging time used to compile statistics [s]

tlu Boundary layer lift-up time [s]

two Wash-out time [s]

u, U Velocity in the streamwise direction [ms]

Ub(y) Blasius laminar velocity profile [ms]

Uc Convective streamwise velocity [ms]

UH(y) Streamwise Hiemenz velocity profile [ms]

Uw Velocity with which the cylinder bar moves [ms]

uwake Wake velocity [ms]

v, V Velocity in the wall-normal direction [ms]

vT Vector spanning nullspace []

Vf Forcing amplitude [ms]

VH(y) Wall-normal Hiemenz velocity profile [ms]

Vsuct Suction velocity generating adverse pressure gradient [ms]

w,W Velocity in the spanwise direction [ms]

x Streamwise direction [m]

x0 Virtual origin of boundary layer [m]

xg x-coordinate vortex center of gravity [m]

xw Length between wake generating cylinder and leading edge flat plate [m]

xsep Separation length [m]

y Wall-normal direction [m]

y0 Origin in stream function describing Stuart vortices [m]

yg y-coordinate vortex center of gravity [m]

z Spanwise direction [m]

k Wavenumber vector containing kx, ky, kz [1m

]

xvi

Velocity scales and related

()+ Quantities in wall-units []

()p Quantities scaled with up []

() Quantities scaled with u []

m Maximum shear stress in flow [kg

ms2]

B Parameter in the Perry and Schofield velocity profile: 2.86 U,x

us[m]

u Velocity scale combining up and u [ms]

u Friction velocity [ms]

um Velocity scale given as (uv)max [ms ]

up Velocity scale given as 3 1

dP0/dx [

ms]

us Perry and Schofield velocity scale: 8(

BL

)um [

ms]

ut Velocity scale [ms]

uZS Zagarola and Smiths velocity scale:

Ux,99[m

s]

xvii

xviii

Abstract

Turbulent wall-bounded flows are important, because many of the flows found inindustry and nature belong to this type of flows. One of the examples is the flow inthe low pressure turbine part of an engine. Efficiency can be increased by increasingthe curvature of the blades. However, this leads to higher risks of the flow separating,which is undesirable as it can cause vibration and efficiency losses. Another, related,area of interest is the reduction of skin friction on airplane wings.

The design of wings or turbine blades is complicated, because the amount ofunderstanding of turbulent wall-bounded and/or separated transitional flows is in-sufficient to apply daily engineering design methods. The problem is that the physicsare not completely understood and the absence of valid models, that can aid in thedesign process.

Wall bounded flows are treated in this thesis from three different points of view.The main interest of the first part of this thesis is the study of Poiseuille channel andCouette flows. These flows, as studied here, are not typically found in real worldapplications. However their study is necessary to be able to develop models for realworld wall-bounded flows. Particularly the near-wall region of wall-bounded flows isknown to be difficult to model. The relation between different types of wall-boundedflows is studied.

A new code is developed to be able to study boundary layer flows, which isdescribed in the second part. This development is necessary since the code used forthe channel simulations cannot simulate boundary layers. The goal was to develop afourth-order, high resolution numerical scheme to simulate incompressible turbulentflows similar to the ones found on turbine blades.

The focus of the third part of this thesis is on the study of separated flow. Firstthe control of separated flow is discussed. Secondly, another important problem isstudied, namely the transition and turbulent development in flows that are laminar,separate and reattach after transitioning to turbulence. The information obtainedin the second study can be used by modelers to improve or to check the models usedin the industrial design of turbines. This part of the thesis has two objectives. Thefirst goal is to give physical mechanisms that control and suppress separation. Thesecond goal is to provide data on the development of the turbulent flow at and afterreattachment of the separation bubble.

The control of separation is studied by forcing a separation bubble on a flat plate.The forcing is applied at the wall upstream but close to the separation bubble. Thesesimulations are done in two-dimensions using a second-order finite difference codeon a staggered grid. The Reynolds number Re = 30, based on the momentum

xix

thickness at the inlet of the numerical domain, is relatively low.

The newly developed code is used to do three-dimensional simulations on a tran-sitional laminar separated bubble on a flat plate. In addition, and related to control,simulations are done with incoming wakes, mimicking real turbine configurations.Flows with various different adverse pressure gradients are simulated. The highestadverse pressure gradient is chosen similar to the one in a laboratory experiment.No information is available on the spectral content of the velocity perturbations,and therefore only the mean and rms profile are matched. The spectral distributionis chosen according to the most unstable one for a Blasius boundary layer. Thefrequency with which the wakes pass by the inlet plane is f = 84Hz, which is higherthan in the experiment to obtain statistics in a reasonable computational time. Atthe inlet the Reynolds number is Re = 114 in all the three-dimensional cases.

The two-dimensional simulations show that three different possibilities exist tocontrol separation. One is related to the instability of the shear layer formed betweenthe separation and the free-stream. The shear layer instability can be triggered withfairly low-amplitude forcing when St 0.012.

The second possibility is to use high-amplitude forcing, which results in largevortices being generated. These are very effective in mixing the low-speed fluidfrom the separation bubble with the high-speed fluid from the free-stream. Theeffective frequencies are related with the momentum thickness of the shear layerand the distance between the forcing slot and the separation bubble. The thirdpossibility is to use periodic suction without blowing. The latter two options workwith high-amplitudes forcings of around ten percent of the free-stream velocity only.Periodic suction is only effective when applied close to the unforced separation point.

The three-dimensional results have provided detailed information on the statis-tics of the flow. The statistics compare fairly well with the laboratory data, althoughtransition takes place much further downstream compared with the laboratory data.If properly scaled, the data also compares quite well to other data found in the liter-ature. The momentum, Reynolds-stress and energy balances are given. They showthe large importance of the turbulent transport term in the Reynolds stress andenergy balances. The incoming wakes have a remarkable positive effect as they de-crease the bubble length with a factor of at least six. They also cause a decrease inH = /, with a factor of 1.5 in the turbulent part of the domain, compared withthe unforced simulation.

This newly developed code is second-order, instead of fourth-order accurate.This reduction in order is necessary because the solution of the Pressure-Poissonequation with fourth-order accuracy resulted in a serious penalty in computationalspeed. However, the convective and viscous terms are all fourth-order accurate usingcompact finite difference schemes. Overall, this meant that the theoretical resolutionof the code is as if fourth-order compact schemes would have been used. The codeis second-order accurate in time.

The developed code is now used extensively to do other simulations of boundarylayers under different types of pressure gradients and with turbulent inflow profiles.

xx

It is expected that the new three-dimensional code will be used by the industrialpartners to gain insight in the physics that determine the effectiveness of incomingwakes in reducing the risk of open separation. Small modifications would make itpossible to study the influence of roughness on separation and reattachment.

The three-dimensional simulations, apart from demonstrating the capabilities ofthe code, have given useful information that can be used to fine-tune the modelsused in design processes. The two-dimensional simulations provided several controlstrategies that, after some engineering development, are to be implemented.

xxi

xxii

CHAPTER

ONE

Introduction

This thesis has two parts, which in both cases treat wall bounded flow. The firstpart deals with near-wall turbulent channel flow. The second one is about separatedflow, its control and turbulent flow under the influence of large adverse pressuregradients.

Turbulence is the motion of fluid characterized by its randomness of velocity andpressure in space and time. Turbulent channel flow is understood to be the turbulentflow between two flat, parallel, infinite plates. Channel flow as defined here may notbe of direct practical interest, but its study is useful, because the results are relatedto other more practical flows, such as pipes and boundary layers without pressuregradient. Furthermore it is of great importance to understand this type of flow tobe able to develop models for wall-bounded turbulence in general.

Separation is the phenomenon by which fluid initially close to a wall moves awayfrom it. This is caused by an external force that decelerates the flow. The fluidclose to the wall has a low velocity, and due to the deceleration, its velocity becomesnegative. This fluid with negative velocity acts as an obstacle for the oncomingflow that advects over it. If the velocity becomes again positive downstream of theobstacle a region is formed that contains the fluid with negative velocity, called aseparation bubble. Sometimes the separated region does not terminate before theend of the geometry, which is called open separation. The external force that decel-erates the flow is an adverse pressure gradient (APG), caused by steep curvatures inthe flow geometry. Two other concepts have to be introduced related to this topic.In the case where the flow is accelerated the flow is under the influence of a favorablepressure gradient (FPG). Zero pressure gradient (ZPG) flow is not accelerated nordecelerated. The study of separated flow and their control is important, as it hasseveral negative side-effects in industry in general and aviation in particular.

1

1.1. Wall bounded flows with and without strong APGs

1.1 Wall bounded flows with and without strong

APGs 1

This section is intended to give a general introduction on the relevant equations,analysis and scaling in turbulent research on wall bounded flows. This section isparticularly relevant for the chapters 2 and 5.

It is now generally accepted that incompressible wall-bounded flows can be de-scribed by the Navier-Stokes equation, written as

u = 0, (1.1)

tu + u u = p+1

Re2u. (1.2)

In the case of laminar flows, the important terms describing the flow are obtainedfrom an order of magnitude analysis of these equations. However, as these equationsdescribe the total flow, it is not possible at this point to distinguish between thefluctuating velocities and the average flow. Usually, to facilitate an order of magni-tude analysis that does take this into account, a decomposition is made of the totalvelocity

u(x, t) = U(x, t) + u(x, t), (1.3)

in an average U and a fluctuating part u. A similar decomposition of the pressureis made to obtain

p(x, t) = P (x, t) + p(x, t). (1.4)

After substituting this decomposition in Eqs. (1.1, 1.2), averaging leads to

(U + u

)= 0, (1.5)

t(U + u

)+(U + u

) (U + u

)=

(P + p

)+

1

Re2(U + u

). (1.6)

As U = U , u = 0 and Uu = 0, Eqs. (1.5, 1.6) become

U = 0, (1.7)

tU + U U + u u = P +1

Re2U . (1.8)

These approximations of Eqs. (1.1, 1.2) are important not only to analyze turbulentflows, but they are also used extensively to simulate turbulent flows.

The average u u 6= 0 gives rise to new terms, which are the derivatives ofthe the Reynolds-stresses given by

uu uv uw

vu vv vw

wu wv ww

. (1.9)

1This section is an overview of the literature on wall-bounded or general turbulent flows. Alarge part is based on [113]. Furthermore as density is constant it is included in the pressurep = p/.

2

1.1. Wall bounded flows with and without strong APGs

This tensor contains six independent components due to symmetries. The turbulentboundary layers which are considered here are periodic in z and statistically two-dimensional, implying that uw = vw = 0 and W = zU = zP = 0. Equations(1.7, 1.8) then simplify to give the average, two dimensional, stationary flow as

xU + yV = 0, (1.10)

x

(U

2+ uu

)+ y

(UV + uv

)= xP +

1

Re(xx + yy)U, (1.11)

x(UV + uv

)+ y

(V

2+ vv

)= yP +

1

Re(xx + yy)V . (1.12)

Order of magnitude estimates that lead to scaling parameters are presented later.The different terms in these equations have been calculated for a boundary layerunder the influence of an APG and are presented in section 5.3.5.

The equation for the time evolution of the kinetic energy and the Reynolds-stresses can be obtained after taking the matrix product

uT(tu + u u = p +

1

Re2u

). (1.13)

The trace of the resulting tensor gives the equation for the time evolution of thekinetic energy, while the individual components of the tensor give the time-evolutionof the Reynolds-stresses as given in Eq. (1.9).

The time evolution of the Reynolds-stresses is obtained by substituting in Eq.(1.13) the Eqs. (1.3, 1.4), taking the average and subtracting the average stressequations. In the case of turbulent boundary layers, the interesting equations arethe ones describing the time evolution of the diagonal components of Eq. (1.9) and

of uv. The average equations can be obtained by multiplying Eq. (1.8) with UT.

The equations for the time evolution of the Reynolds-stresses are deduced from

1

2t(U + u)2 =

(U + u)( (U + u)(U + u) + x(P + p) +

1

Re2(U + u)

), (1.14)

1

2t(V + v)2 =

(V + v)( (V + v)(U + u) + y(P + p) +

1

Re2(V + v)

), (1.15)

1

2tw2 =

w( w(U + u) + z(P + p) +

1

Re2w

), (1.16)

t(U + u)(V + v) =

(V + v)( (U + u)(U + u) + x(P + p) +

1

Re2(U + u)

). (1.17)

3

1.1. Wall bounded flows with and without strong APGs

After averaging and subtracting the average-energy equation, one obtains the fourtransport equations for the Reynolds-stress budgets, being

tuu = (1.18)

2u u U uu U uuu 2uxp +1

Re2uu 2

Reu u,

tvv = (1.19)

2v u V vv U vvu 2vyp +1

Re2vv 2

Rev v,

tww = (1.20)

ww U wwu 2wzp +1

Re2ww 2

Rew w,

tuv = uv U (1.21)

uu U uvu vxp uyp +1

Re2uv 2

Reu v.

The statistically averaged convective terms, uu U , give the advection of theReynolds-stresses by the mean flow. In channel and ZPG boundary layers theseterms are normally small compared to the production and dissipation terms, becausein these flows only the y() terms are large, and V 0. In APG boundary layersthey may be similar in magnitude.

The production terms, u u U , are important terms because they describethe energy transfer from the mean flow gradients to the Reynolds-stresses. In chan-nel flows the only production term is uvyU . In APG boundary layers there areprobable more production terms that become important as V > 0 substantially, andx() 6= 0.

The turbulent transport terms, uu u, give the average transport of theReynolds-stresses by the turbulent motion. It is shown in chapter 5, that thesetransport terms are important in typical turbine flows.

A term that is difficult to measure in laboratory experiments, but which is im-portant is the pressure transport term up. This term gives the energy transportbetween the different Reynolds-stresses. In channel flow where the mean flow ex-changes energy only with uu, the other Reynolds-stress terms receive their energydue to the pressure gradient term, which extracts energy from the streamwise stress,uu.

There are two terms related to viscosity. One is the viscous dissipation 1/Re2uuterm, which is unimportant expect very close to the wall where the second derivativebecomes large. The second term is the viscous diffusion term uu, describingthe important turbulent dissipation.

4

1.1. Wall bounded flows with and without strong APGs

1.1.1 Review of the scaling of the mean flow and Reynolds-

stresses

Boundary layers under the influence of strong APGs are different compared to ZPGboundary layers in the sense that u becomes small. In the limit u 0, there isa problem scaling with u as the scaled velocities have infinite values. Moreover,the experimental results in [102], show that the maximum for the Reynolds-stressesscaled with u are all found at y

+ 1100. This in sharp contrast to what is foundin ZPG boundary layers [106], for which the maxima are found at around y+ 20.This is a second indication that boundary layer flow under the influence of strongAPGs should be scaled differently. However, there is little consensus, much less thenon the scaling used in ZPG flows, on how to scale these flows. Here a review is givenof what has been used. The theoretically most promising scalings are then used toscale the data in chapter 5.

General scaling of the mean profile

Consider Eqs. (1.10, 1.11), and Eq. (1.12). For the moment it is assumed that thereexists a velocity scale ut that scales the mean streamwise velocities, as well as theReynolds-stresses. Furthermore, it is assumed that a length scale /ut exists andthat normally Lxut/ 1. An order of magnitude analysis of Eq. (1.10) gives

xU + yV utLx

+V ut

= 0 V Lx. (1.22)

The orders of magnitude of the momentum equations then become

UxU + V yU u2tLx, UxV + V yV

utL2x

, (1.23)

As v2 uv u2t the Reynolds-stress terms in Eq. (1.12) are of the order

yv2 O(u3t

), xuv O

(u2tLx

), (1.24)

and the viscous terms are of order

yyV O(u2tLx

), xxV O

(2

L3x

), (1.25)

using V Lx

.

Multiplying all terms with /u3t the following order of magnitude equation isobtained for the V momentum balance

2

L2xu2t

u3tyP 1

Lxut+

Lxut+

3

L3xu3t

. (1.26)

5

1.1. Wall bounded flows with and without strong APGs

This identifies that to balance the equation

yP yv2, (1.27)

which, as will be shown later, is actually true for the data presented in this thesis onstrong APG boundary layers. Equating the two terms in Eq. (1.27) the following

yP = yv2, (1.28)

is obtained. After integrating this equation in y and differentiating with respect tox

x(P P0) + xv2 = 0, (1.29)is found, where P 0 is the value of the pressure at the wall.

The terms in Eq. (1.11) can now estimated, and are

UxU + V yU = O(u2tLx

), yuv yyU = O

(u3t

), (1.30)

xu2 = O(u2tLx

), xxU = O

(utL2x

), xP dP0/dx = O

(u2tLx

). (1.31)

If Lxut/ 1 and xP dP0/dx are assumed, the following equation is obtained

yuv + yyU = dP0/dx. (1.32)

These equation can be integrated from y = 0 to y, which gives

uv + yU = ydP0/dx+ u2 . (1.33)

From this equation two velocity scales can be obtained. A first scale can be definedby considering that uv = yU = 0 for y > . This scale is the well known frictionvelocity scale u .

The second scale that can be formed is for flows that are on the verge of sepa-rating, and thus have u2 = 0. An interesting deduction of this scale was developedin [103]. In this work they scale Eq. (1.33) with u and /u to obtain

+ y+U+ uv+ = 1 + y

+

u3dP0/dx. (1.34)

This can be written as

+ = 1 +

(upu

)3, (1.35)

which, after multiplying both sides with (u/up)2, becomes

p = yp +

(uup

)2, (1.36)

6

1.1. Wall bounded flows with and without strong APGs

with

u3p = dP0/dx, up =3

dP0/dx. (1.37)

This scaling prevents singularities when u 0, and it was first deduced in [109].This alternative velocity scale is only correct when up u .

To be able to integrate Eq. (1.34) and obtain the velocity in the viscous sub-layerthe nice argument developed in [113] was used. They define the non-dimensionalKolmogorov micro-scale +k to be

+k (y+

) 14 , (1.38)

where is a constant. The integral scale or, what is equivalent, the largest eddiesshould scale with the distance to the wall. This gives the dimensionless integralscale

L+i y+. (1.39)In [113] it is shown that below y+ / 5, the integral scale becomes smaller than theviscous Kolmogorov scale. Energy is assumed to be injected at the integral scales.If these scale becomes smaller than the viscous Kolmogorov scale, it means thatthis energy will be dissipated almost directly. Then turbulence cannot sustain itself.Therefore the turbulent Reynolds-stresses can be neglected compared with the vis-cous stresses in this viscous sub-layer. The velocity profile can then be approximatedby integrating Eq. (1.36)

yp

0

(ypU = y

p +(uup

)2)dyp up = 1

2(yp)2 +

(uup

)2yp. (1.40)

Far from the viscous sublayer, where the length scale is /u , an outer flow velocityprofile is found. The length scale for the outer flow region could be 99, althoughsome other possibilities are discussed later. The outer flow region can be written asthe following velocity defect law

U,x U()ut

= f(), =y

99. (1.41)

In between these two layers the overlap region exists. This overlap region is knownalso as the logarithmic region or the inertial sublayer. Here the length should scalewith y as it is too far from the boundary to scale with the viscous length scale,and too far from the outer flow to scale with 99. This dimensional argument has amathematical counterpart based on matched asymptotic expansion. First considerthe inner layer for y+ , which gives

dU

dy=u2t

dg

dy+, (1.42)

while the outer flow for 0 givesdU

dy=ut

df

d. (1.43)

7

1.1. Wall bounded flows with and without strong APGs

The first derivative is a function of y+ while the second derivative is a function of .They can only be equal in the overlap region, as they should, when they are bothequal to a constant

u2t

dg

dy+=ut

df

d=

1

. (1.44)

After integration

U(yut/)

ut=

1

lnyut

+ A, (1.45)

U,0 U()ut

=1

lny

+ C. (1.46)

The velocity scale ut can be u or up. In [103] a similar mixture between these twoscales, like in the viscous sublayer, was developed for the overlap region. The ideais to use

u2 = u2 +

u3puy+ = u2 + u

2py

p, (1.47)

as a velocity scale. After matching an outer flow Eq. (1.43) with the viscous sublayerthe equation to be solved becomes

y(dU

dy

)=

1

. (1.48)

Here y is given by

y =

(y+)2 + (yp)3. (1.49)

After integration of Eq. (1.48) the velocity

u+ =1

(ln y+ 2 ln

1 + y+ + 1

2+ 2

(1 + y+ 1

))+ A, (1.50)

can be obtained, or

up =1

(22 + yp + ln yp 2 ln

(2 + yp +

))+ C. (1.51)

Here

u =

(upu

)3, =

uup. (1.52)

For u = 0 this gives the well-known logarithmic law for channel flows and ZPGboundary layer flows.

8

1.2. Near-wall flow in a turbulent channel

Alternative mean velocity scalings for APG boundary layers

The failure of the classic mean velocity scale u has lead to several other proposalsto scale the mean velocity profile. In [91] and [96], a velocity scale is proposed basedon the maximum kinematic shear stress

u2m =m

= (uv)max, (1.53)

which only applies when um

32u , approximately. The defect law describing the

outer flow is obtained as

U,x U(y/B)us

= f( yB

), us = 8

(B

L

)um, B = 2.86

U,xus

, (1.54)

where L is the distance from the wall to the local maximum kinematic shear stress.The cited authors describe that the viscous sublayer is the same as in flows withoutAPGs. However they find a small logarithmic layer that tangentially joins a powerlaw described by

U(y/)

U,x= 0.47

(us

U,x

) 32 ( y

) 12

+ 1 usU,x

. (1.55)

If both sides are multiplied with U,x/us this becomes

U(y/)

us= 0.47

(us

U,x

) 52 ( y

) 12 1 + U,x

us. (1.56)

In their interpretation this is the innermost portion of the velocity-defect law de-scribing the outer flow.

Another scaling for the outer mean flow velocity-defect was proposed in [132] as

U,x U(y/)U,x

99=

(y

99

), UZS =

Ux,99. (1.57)

In [12] this scaling was shown to reasonably collapse mean velocity data for a widerange of APG flows.

1.2 Near-wall flow in a turbulent channel

The near-wall region of turbulent channel flow is important from the modelingpoint of view. The relevant parameter in this region is the friction velocity u =((yU)y=0)

1/2, where is the kinematic viscosity and (yU)y=0 is the shear at thewall. Length scales are scaled with /u to give he dimensionless distance to the

9

1.2. Near-wall flow in a turbulent channel

wall y+ = uy/. In this region the flow is characterized by the presence of stream-wise velocity streaks and streamwise vortices (x). The spanwise distance betweenthe streaks is about z+ 100 and they have a length of about x+ 1000. Thestreamwise vortices are associated with the streaks and have a longitudinal spacingof about x+ 400. It has been shown in [56, 57] that there exists a near-wall cyclewhich maintains turbulent flow without the influence of the outer flow. This regionis confined to y+ . 50.

Methods, such as LES, have been developed to reduce resolution requirementsto simulate turbulent flows. These methods only resolve the large scale turbulentmotion, that carry the kinetic energy and the Reynolds-stresses. A model is usedto estimate, from the large scale turbulent motion, the dissipation of kinetic energycontained in the unresolved small scales of the turbulence. This method worksreasonably well as long as the large scales are resolved correctly.

In wall-bounded flows the stress-carrying structures range from sizes of the orderof the channel height in the outer flow, to the viscous structures in the near-wallregion. The resolution requirements are thus determined by the small scales of thenear-wall region, unless an approximate model is found for them. The problem isthat, as the Reynolds number increases, the proportion between the near-wall regionand the whole channel decreases. This means that there exists smaller scales thatneed resolving. This dependence on the Reynolds number implies that resolutionrequirements without a near-wall model are not much lower than the resolutionrequirements using LES. For LES or any other method to really save time andresources, its resolution has to depend only on the characteristic length scales of thegeometry. For example the radius of a pipe or the height of a channel [55, 50].

However, the lack of physical knowledge about the near-wall region impedescorrect modeling. Nowadays, a qualitative idea of near-wall physics exists [57, 56,120]. However, the results in [56, 57] were obtained in somewhat special channelsand the relation between real channels and other typical wall bounded turbulentflows was not clear. Another question that remains is how the cycle maintains itself.

The study of this problem was done using a similar approach for the simulationsas in [51]. However, here a minimization of the size of the numerical box was searchedfor, to isolate the fundamental structures of the near-wall region. Furthermore,data of full turbulent channel flows were used [23], and minimal channels [54] werecomputed, to compare with the solutions that were obtained from the minimizationprocedure. The comparison with other wall bounded flows was possible due to thecooperation with the Department of Aeronautics and Astronautics at the Universityof Kyoto. They calculated several time periodic [60] and steady [82] plane Couetteflows.

1.2.1 Objective for the first part of the thesis

The objective of the first part of this thesis is a more complete comprehensionof the turbulent near-wall region in wall bounded flows. Different types of wallbounded flows have been analyzed, to see if the nonlinear equilibrium solutions

10

1.3. Control and its numerical study

found in the near-wall region are universal, for example [57]. Furthermore, theobserved temporal intermittency has been addressed and was related to the nonlinearequilibrium solutions. This work has been published in [52].

1.3 Control and its numerical study

Control is defined as the manual or automatic regulation of a system to satisfy acertain objective. The next two examples illustrate its necessity.

Turbines convert kinetic and thermal energy of burnt gasses into a useful torque,using their blades. A turbine blade is a curved surface which causes the flow onits suction side to be decelerated by an APG. As the APG increases, so does thetheoretical efficiency. However, increasing curvature also means that the flow has ahigher risk of separating, which would result in a loss of efficiency and in the pos-sibility of undesirable vibrations. Turbine blades are nowadays designed to preventseparation or to have short separation bubbles. Increasing the curvature would onlybe efficient if separation can be prevented.

The flow over airplane wings separates during take off and landing, due to alarge angle of attack. Separation can be very severe, which in aviation is called stall,resulting in an airplane that does not react to the controls anymore. Nowadays, incivil aviation, separation and stall is controlled by large, expensive and complicateddevices.

In the first example, control should be used to allow turbine blades with highercurvature. The objective in civil aviation would be for more advanced, lighter andsimpler control techniques, to replace the devices used nowadays.

In this thesis the system to be controlled is the separated flow over a geometry.The objective of control in the case of separated flow is to diminish the negativeeffects of separation. This can be obtained, for example, by shortening the separationbubble.

The regulation can be obtained in several ways. In the first place one may dis-tinguish between passive and active control. Passive control uses time independentdevices. Active control includes all types of control that are functions of time. Bothtypes of control can use small perturbations which, in their vicinity, do not causelarge changes in the flow. On the other hand large perturbations are also possible.They rely on large-scale changes in the flow to obtain the control objective. Closed-loop control and optimal control react to changes in the flow, while open-loop controldoes not. Passive control is always open-loop control, while active flow control ismore flexible. A more complete overview can be found in [17, 30].

1.3.1 Control of separation on low-pressure turbine blades

In this thesis the control that will be studied is related, primarily, to flow over low-pressure turbine blades. The fluid has expanded in the high-pressure stages, before itreaches the low-pressure turbine stage reducing its density . The Reynolds number

11

1.3. Control and its numerical study

Re = UL/, based on a characteristic velocity U , a length scale L and viscosity, will thus be lower in this stage. Therefore the flow on the blades in this stage istypically laminar. Under the influence of a strong APG this leads to separation thathas detrimental effects on the operation of the turbine, because it increases drag,and decreases lift.

There are at least two ways to control the separation bubble on the blade. Thefirst option is to implement the control in the wall on the blade. In this thesis thisis implemented as open-loop active control, as the primary interest is the physicalresponse to specified controls. The second option is to use the wakes which comeperiodically from the upstream stator blades. The latter option is also open-loopactive control and can be implemented more easily than the first one. Forcing atthe wall may be feasible with some engineering development.

Open-loop-active control (from now on OLAC) can be studied by laboratoryexperiments or by numerical experiments. Numerous laboratory studies on OLACcan be found in the literature. Most of them [67, 100, 30, 46] conclude that atwo-dimensional instability of the shear layer, formed due to separation, is a goodpossibility to control the separation bubble. However laboratory experiments onthe effectiveness of control are difficult, because laminar separation bubbles are verysensitive to external fluctuations [31].

Another approach is used here, namely two-dimensional numerical experimentson the control of separated incompressible flows. This seems a reasonable setupfor the experiments, as the flow is laminar until separation. Moreover it has beenassumed here that shear layer instabilities play an important role in the control ofseparation bubbles. Their related instability is predominantly two-dimensional, aswell as the shear layer roll-up process [35].

The incompressible approximation is valid as long as [95]

1

2Ma2 1, (1.58)

were Ma is the Mach number. In low pressure turbines Ma 0.6, which meansthat the flow can be considered reasonably incompressible.

The simulations are done on a flat plate. A wide range of separation bubblesare generated by different suction velocities [3] imposed at the upper part of thenumerical domain. Although a flat plate is not able to capture all the physicspresent on a turbine blade, it is expected that this geometry will represent the effectof the control accurately. The forcing is imposed at the wall of the flat plate in theform of a sinusoidal zero-mass blowing wall jet. Wakes could also be imposed, butusing the sinusoidal forcing offers a broader control parameter space. This is usefulto find an optimal control, and it aids in the understanding of the flow physics. Itis expected that understanding the flow physics by forcing at the wall will help tounderstand how to use the wakes for control.

12

1.3. Control and its numerical study

100

101

102

103

106

108

1010

1012

1014

Re

Np

Figure 1.1: Estimation of the total number of points needed to simulate a turbulentboundary layer for : second central scheme; : fourth order standard Padescheme and : spectral method. Here xyz = 1 105 is used, which is quite aconservative estimate of the boundary layer size which one wants to simulate.

1.3.2 High resolution versus low resolution

Recall that two and three-dimensional simulations are done to study control. Thetwo-dimensional simulations are done using a second order-low resolution code, whilethe three-dimensional simulations are done using a second order high-resolutioncode. The use of a high-resolution code is important to limit the required resources,especially RAM memory. This can be seen by estimating the amount of points nec-essary to do a boundary layer simulation at a certain Re = U/. The estimationis made using the following assumptions

u

= 0.15U, 99 = 10, h = 8k (Fourier discretization),

h = 4k (4th discretization), h = 2k (2

nd discretization), (1.59)

where is the momentum thickness and 99 Li with Li the integral length scale,and a Reynolds number ReLi = u

Li/. This gives trivially ReLi = 1.5Re usingthe estimation for the streamwise rms-velocity, u

, while the Kolmogorov scale k is

given by k = Li/Re3

4

Li. The grid sizes, h, are necessary to correctly resolve the

turbulent flow considering the resolution of the numerical schemes. The amount ofpoints Nx needed to discretize a length Lx = xLi then becomes

Nfd =Lx8k

=x(1.5Re)

3

4

8, Nfo =

x(1.5Re)3

4

4, Nso =

x(1.5Re)3

4

2, (1.60)

for Fourier (fd), fourth order standard Pade (fo), and second order (so) discretizationrespectively. The estimate of points needed to do a simulation in three dimensionsis shown in Fig. 1.3.2. The available memory of computers is only limited and the

13

1.4. Tools used for the thesis

factor of two between the second order and fourth order scheme makes the differencebetween being able to do a well-resolved simulation or not. Note that the need touse twice as many points also leads to a much reduced time-step.

1.3.3 Objectives for the second part of the thesis

The two main objectives of the second part of this thesis are:

The development of a high-resolution computational code capable of simulat-ing turbulent separated boundary layers, and the transition of laminar sepa-ration bubbles to turbulence.

A better understanding of the active open-loop control of separation bubbles.

The computational code should be versatile within the boundary layer configuration.This means the ability of imposing a variety of boundary conditions, and favorableor adverse pressure gradients without major configurations. The code should beparallelisable and should have a good parallel scalability.

To study active open-loop control we strive to know thoroughly the importantparameters that determine the success of the control. Furthermore information needsto be obtained about the instantaneous flow-field as a function of this parameters.The flow-field is then related to the effectiveness in controlling the separation bubble.

1.4 Tools used for the thesis

The channel simulations were performed using a code developed, over the years,in the department of Fluid Mechanics of the School of Aeronautics of Madrid[18, 56, 51, 57]. The Navier-Stokes equations are written using the wall-normalvorticity and the Laplacian of the wall-normal velocity. The discretization is pseudo-spectral with Fourier expansions in the directions parallel to the wall, while Cheby-chev polynomials are used in the wall-normal direction.

Two new finite difference codes were used in this thesis. For the control studies atwo-dimensional, second-order code based on [87] was used that was partly developedin [24]. This code was parallelized using MPI.

A second code was especially developed as part of this thesis, because the ex-isting code was not high-resolution. This development was also necessary, as thepseudo-spectral code does not permit to impose the necessary boundary conditionsto simulate separated boundary layers. A numerical code, using a mixture of theFORTRAN 77 and FORTRAN 90 programming language, was developed. The MPImessage passing interface [38] was used to parallelize the code. Compact finite dif-ference schemes [71, 79] were used to discretize the Navier-Stokes equations writtenin primitive variable form. The conservation of mass was assured using a fractionalstep method based on [90, 26]. Two- and three-dimensional versions were developed,but only the latter was parallelized.

14

1.4. Tools used for the thesis

The simulations were done on several parallel machines. First of all the clusterof the Fluid Mechanics department of the School of Aeronautics of Madrid. Thiscluster consists of 64 Dual Intel Xeon(TM) processors nodes of 3.06 GHz. Each nodeis mounted in a 1U chassis. The nodes have no hard disk. A front-end node, thathas a two hard disks and the same characteristics as the nodes, takes care of themanagement of the nodes. Initially every node had 2GB of RAM-memory which waslater extended to 4GB of memory for 32 nodes. High speed communication betweenthe nodes is done by Myrinet [44] which uses optical fibre technology. The operatingsystem used is a Debian [42] distribution of Linux. PBS [41] is used as a queuingsystem. Compilation of the code was done by the Intel Fortran Compiler [43]. Thesecond machine used was MareNostrum at the BSC [39], which was for some timethe fourth largest machine in the world, considering computational power. Thismachine initially had 4812 IBM PowerPc 970FX processors of 2.2 GHz distributedover 2406 dual nodes. All the nodes have 4 GB of RAM memory. It uses theSUSE [45] version of Linux as an operating system. The high speed communicationbetween the nodes is again done by Myrinet. Compilation was done by the xlfcompiler of IBM. On this machine a maximum of 341 processors were used.

The third machine used was Magerit at the UPM [40], which had the samecharacteristics as MareNostrum but initially only had 360 processors. These aredistributed over 180 JS20 blades. The three-dimensional compact finite differencecode was run on all three machines. The two-dimensional codes and the pseudo-spectral code where only run on machines of the Fluid Mechanics department.

1.4.1 Organization of the thesis

This thesis has six chapters and three appendices.The first chapter gives an introduction to the problems that are studied and the

methods that are used.The second chapter discusses the near-wall region of turbulent flow. Results for

different types of wall bounded flows are presented and classified. Then a comparisonwith full turbulent flows is given and Reynolds number effects are shown. Then thetemporal intermittency of the near-wall region is studied, which is related to typicalflow scales of structures found in this region.

In the third chapter the numerical code to simulate the three-dimensional in-compressible Navier-Stokes equations is described. In this chapter the numericalmethod, implementation and numerical validation is presented. Various ways toobtain global mass conservation are given.

The fourth chapter is about the control of separation bubbles. Two and three-dimensional simulations are presented, from which quantitative and qualitative in-formation is obtained. Information is obtained about the optimal frequency, forcingslot length and forcing amplitude. The flow as a result of the different forcingparameters is detailed. Results are obtained for two different Reynolds numbers(Re0 = 30, Re0 = 100) based on the momentum thickness at the inlet. The latteris of the order of the Reynolds numbers found in turbines.

15

1.4. Tools used for the thesis

The fifth chapter treats the three-dimensional simulations of a separation bubbleon a flat plate. The pressure gradient applied in this simulation has industrialrelevance, and is compared with experiments provided by industry. This chapteralso contains the results for a separation bubble under the influence of incomingwakes. The model and approximations involved in imposing the wakes, and thepressure gradient are discussed. Then a comparison is made with laboratory datato validate the results. Different velocity scales are used to collapse the data. Finallythe momentum balances, the Reynolds-stress balances and the energy balance aregiven.

The sixth chapter ends this thesis with conclusions and recommendations. Twoappendices are given. The first appendix treats numerical details related to theimplementation of the code. The second appendix treats the solution of a Poissonequation using multigrid accelerated line-Gauss iteration. The third appendix givesa short summary in Spanish.

16

CHAPTER

TWO

Characterization of near-wall turbulence in

terms of equilibrium and bursting

solutions

2.1 Introduction

Wall-bounded flows have been important in turbulence research ever since the fa-mous 1883 experiments by Reynolds. This is particularly true of the immediatenear-wall layer which, because its Reynolds number is locally low, is usually con-sidered to be a good candidate for an approximate description in terms of simpledeterministic structures.

Nonlinear equilibrium solutions of the three-dimensional NavierStokes equa-tions, with characteristics which suggest that they may be useful in such a descrip-tion, have been obtained numerically in the past few years for plane Couette flow[82, 121], plane Poiseuille flow [114, 120, 121], and an autonomous wall flow [57]. Allthose solutions look qualitatively similar [119, 59], and take the form of a wavy low-velocity streak flanked by staggered quasi-streamwise vortices of alternating signs,resembling the spatially-coherent objects educed from the near-wall region of trueturbulent flows [110, 48]. The mean and fluctuation intensity profiles of the equilib-rium structures are also reminiscent of the experimental values [57, 121]. In thosecases in which their stability has been investigated, the equilibrium solutions areunstable saddles in phase space at the Reynolds numbers at which turbulence isobserved. They are not therefore expected to be found as such in real turbulencebut, since the velocity of the system in phase space vanishes at fixed points, whetherstable or not, any turbulent flow could spend a substantial fraction of its lifetime intheir neighborhood.

Although the observed similarities suggest that all those structures are relatedto each other and to self-sustaining wall turbulence, the nature of those relations isunclear. The first goal of this paper is to clarify that point by comparing as many aspossible of the known equilibrium solutions, among themselves and with real near-wall turbulence. This comparison will also include the time-periodic saddle orbitsidentified by [60], which not only approximate the profiles of average velocity andof the intensities of near-wall turbulence, but also part of its temporal structure.

17

2.1. Introduction

In fact, the second problem that we will consider is the possible relation betweentemporal intermittency in the near-wall layer and such time-dependent simple so-lutions of the Navier-Stokes equations. The term burst was originally introducedto describe fluid eruptions observed near the wall in the early visualizations of tur-bulent boundary layers [61]. It was initially hypothesized that bursts were due tothe intermittent break-up of the near-wall streaks, but even the original authorslater acknowledged that their visualizations could be consistent with permanent ad-vecting objects [86], and the term became eventually associated with the ejectionsobserved by stationary velocity probes. With the advent of numerical simulations,it became apparent that the streaks were long-lived streamwise velocity structures,and that the sweeps and ejections identified in the analysis of single-point datawere mostly due to the passing of shorter quasi-streamwise vortices, intermittentin space but not necessarily in time [94]. The question of whether the observedtemporally-intermittent sublayer events were visualization artifacts or really existedin the near-wall layer was bypassed by this explanation.

The well-documented mutual dependence of the near-wall streaks and vortices isconsistent both with equilibrium models sustained by steady nonlinear interactions,such as in the structures mentioned above, and with temporal cycles in which bothtypes of structures periodically create each other. The difficulty of following forlong times individual structures in fully turbulent flows complicates the experimen-tal or numerical distinction between essentially permanent objects and intrinsicallytime-dependent processes with a long period, but intermittent breakdown of near-wall turbulence is observed in minimal-flow numerical simulations for which spatialintermittency is not an issue [54, 57]. The same is true of autonomous wall flowsin which the observation is simplified by the small wall-normal dimensions of thesimulation domain [56].

By comparing periodic solutions such those in [60], minimal simulations, andfully-turbulent ones, we will try to clarify whether intermittent behavior, as dis-tinguished from the vortex-passing bursts, is found in fully turbulent flows, andwhether it can be explained in terms of simple time-periodic solutions.

We will also address the question of whether the characteristics of such equilib-rium or periodic solutions can explain the wavelength-selection properties of near-wall turbulence, such as the well-known mean streak separation of z+ 100 or theequally intriguing x+ 300 streamwise separation found in turbulent flows betweenvortex pairs within the same streak [54].

To simplify the discussion, stationary or travelling permanent waves, and solu-tions which can be reduced to limit cycles in some frame of reference, will be referredto as simple from now on. Of those, the permanent waves and the turbulent flowswhose statistics are roughly similar to them, will be denoted as quiescent. Solutionswith stronger vorticity, usually corresponding to a fast evolution in phase space, willbe called excited.

Some of the older solutions required recomputing for the purpose of this paper,using numerical methods which are occasionally slightly different from the original

18

2.2. Computational methods

ones. Those methods are described in section 2.2. The comparison between thedifferent equilibrium and periodic solutions is made in section 2.3, and their relationwith fully-developed turbulence is discussed in section 2.4. Temporal intermittencyis discussed in section 2.5, and conclusions are offered in section 2.6. A preliminaryversion of part of the present manuscript appeared previously as [58].

2.2 Computational methods

2.2.1 Autonomous solutions

The permanent traveling-wave solutions described below as autonomous are com-puted using a slightly modified version [51] of the numerical scheme used in [56, 57].The flow is established in a numerical domain with spatial periodicities Lx and Lzin the streamwise and in the spanwise directions, over a wall located at y = 0.The streamwise, wall-normal and spanwise velocity components are u, v and w.The NavierStokes equations are integrated in the form of evolution equations forthe wall-normal vorticity y and for 2v, using a pseudospectral code with Fourierexpansions in the two wall-parallel directions and Chebychev polynomials in thewall-normal direction [64]. At each time step the right-hand sides of the two evolu-tion equations are multiplied by a damping mask 1 t F (y), where

F (y) = 0 if y 1, F (y) = 1/ if y 2 = 1.5 1, (2.1)

and the two limits of F (y) are connected smoothly by a cubic spline. This mask canbe interpreted as a linear dissipation for each of the two evolution variables. Thedecay time is chosen so that all the vorticity fluctuations are effectively dampedabove y (1 + 2)/2. The equations are not modified below the mask lower limit1. Irrotational fluctuations are not affected anywhere, and the outer edge of theNavierStokes layer is bounded by a potential core which prevents the formationof viscous boundary layers at the mask boundary. No-slip, impermeable boundaryconditions are imposed at the wall.

While the flows in [56, 57] were integrated at constant mass flux in a channel,the present computations were initially carried out at constant driving stress in asemi-infinite domain. The velocities were matched to outer potential fluctuationsextending to infinity from the edge, y = h > 2, of the computational domain [18].This driving mechanism is free from the complications of a second wall across thepotential layer, and in particular from the effect of a mean pressure gradient, andshould in principle be preferable to simulations involving two-walled channels. Thetotal shear stress, for example, is constant across the Navier-Stokes layer instead ofvarying linearly across the channel, and the only Reynolds number in the problemis +1 . The superscript

+ denotes wall variables normalized with the kinematicviscosity and with the friction velocity u .

This driving mechanism had been successfully used to simulate autonomous wallflows in large computational boxes [51] but, in the present case, it failed to reproduce

19

2.2. Computational methods

L+x L+z

+1 U

+c u

+max v

+max

+x,max

A1 189 180 42.0 12.6 2.71 0.616 0.116A2 189 180 45.6 12.4 2.84 0.612 0.117A3 168 180 38.4 13.2 2.54 0.592 0.124A4 168 180 42.0 12.8 2.59 0.598 0.121A5 151 180 42.0 13.2 2.51 0.578 0.123

Table 2.1: Parameters of the autonomous simulations used in the text. Lx and Lz arethe box dimensions, Uc is the phase velocity, and u

max, v

max and

x,max are the maximum

fluctuation intensities used below to characterize solutions.

the simple solutions found by [57] in a pressure-driven channel. The flow passed di-rectly from fully-chaotic (minimal) turbulence to laminar decay upon minor changesin the parameters.

It was therefore decided to reintroduce some pressure effects. The basic structureof the code is maintained, and in particular the driving mechanism for the mean flowis still a constant shear far from the wall, instead of a fixed imposed mean pressuregradient. The mean velocity profile is linear far from the wall, rather than parabolic,but the potential fluctuations in the masked region are required to match a no-stressimpermeable boundary at y = H > h instead of decaying at y . All the casespresented in this paper were computed with H = 2h and with the viscosity adjustedso that h+ = 120. This modification introduces a small fluctuating pressure gradientwhich maintains the instantaneous mass flow constant across the domain (0, H).It was found to be sufficient to restore the existence of steady travelling waves.Their computational parameters are summarized in table 2.1. They were computedusing 48 49 48 spectral modes, before dealiasing. The resulting resolution isx+ z+ 4, with a maximum grid spacing y+ 3 below the mask.

The significance of this observation is not clear, although it is not surprising thatthe properties of constant-mass and constant-stress simulations should differ in smallcomputational domains. Note that the solutions discussed in this section differ fromothers used in this paper in that they are obtained from an initial-value problem, andare therefore stable with a nonzero basin of attraction. This explains their sensitivityto the stress boundary condition, which changes their stability although not theirqualitative character. The present traveling waves do not differ visually from thosein [57], even though the codes are different. The chaotic solutions obtained fromthe constant-stress boundary conditions are also indistinguishable from the chaoticsolutions obtained by raising the mask either in the present code or in [57]. Theyall agree with minimal Poiseuille flows below roughly half the mask height [57, 51].Because of the presence of a required fluctuation of the spatially-constant pressuregradient, these flows are classified below as part of the Poiseuille family, even if thetemporal average of their mean pressure gradient is zero.

20

2.2. Computational methods

2.2.2 Plane Couette solutions

Nagatas steady solutions [82] for incompressible plane Couette flow were recom-puted using a method similar to that in [27]. The flow is described by its deviation(u, v, w) from the linear profile, and is obtained numerically by solving steady non-linear equations for the streamwise velocity u0(y) = u, where denotes averagingover a wall-parallel plane, and for y and 2v, as in the previous section. The solu-tions are expressed as double Fourier expansions in the two wall-parallel directions.The wall-normal expansions are expressed in terms of the centered dimensionlesscoordinate y = y/h 1, where y = 0 is the mid plane of the channel, and h is halfthe wall separation. They use the basis functions

(1 y2)Tl(y) (2.2)

for u0 and y, and(1 y2)2 Tl(y) (2.3)

for v, where Tl(y) is the lth-order Chebychev polynomial. They satisfy the bound-

ary conditions

u0 = y = v =v

y= 0 at y = 1. (2.4)

The collocation method with grid points y = cos[m/(M + 1)], (m = 1, 2, ,M)is used to construct a system of quadratic equations for the FourierChebychevFourier coefficients, which is solved by the NewtonRaphson method. The arc-lengthmethod [27] is used to track the nonlinear solutions, with the three parameters Re,Lx and Lz being changed independently.

It is well known that a laminar plane Couette flow is linearly stable for all finiteReynolds numbers. Nagatas upper and lower solution branches appear subcriticallyatRe = Uwh/ 125 from a saddlenode bifurcation, where Uw is half the differenceof the two wall velocities. In general, the upper-branch (or lower-branch) solutionsgenerated from the bifurcation have a larger (or smaller) deviation from a laminarstate. All the solutions have two spatial symmetries [80, 81, 82]: a reflection withrespect to the plane of z = 0 plus a streamwise shift by Lx/2,

(u, v, w)(x, y, z) = (u, v,w)(x+ Lx/2, y,z), (2.5)

and a rotation by around the line x = y = 0 plus a spanwise shift by Lz/2,

(u, v, w)(x, y, z) = (u,v, w)(x,y, z + Lz/2). (2.6)

The steady traveling-wave solution found by [120] for plane Poiseuille flow also hassymmetry (2.5). Figure 2.1 and table 2.2 summarize the properties and the compu-tational parameters of the solutions used below as representative of this flow. All thecases were computed using 16 16 complex Fourier coefficients in the wall-paralleldirections, and 33 Chebychev polynomials in y. This represents a resolution of atleast 15, 9 and 5 wall units respectively in the x, y and z directions, which is better

21

2.2. Computational methods

50 100 150

2

3

4

Lz+

Wal

l she

ar s

tres

s

Figure 2.1: Dimensionless wall shear rate for the solutions in table 2.2, versus thespanwise period L+z . , Re = 200; , 300; , 400; , 600. In all cases Lx = 2h.The lines are drawn from the tracking algorithm, and the points distinguished by symbolsare used later for a more detailed study. Solid symbols are classified as upper branch;open ones as lower branch. The two circled cases are those in figure 2.7.

than most direct simulations of turbulence [64, 66]. The most marginal direction isy, and grid convergence was tested at Re = 400 by reducing the resolution to 16polynomials. The changes in figure 2.1 were within the size of the symbols.

Note that both the upper and the lower branches have higher dimensionless shearrates at the wall, 1 + (h/Uw) du0/dy|y=1, than the unit shear of the laminar state.The range of existence of the solutions in figure 2.1 is always close to L+z = 100,as in the observed mean separation of sublayer streaks. The limits of that range,L+z 50 150, are also in good agreement with the range of streak spacings,+z 50 200 found in real boundary layers [104]. Two time-periodic solutionsfor a plane Couette flow are taken from [60], using Re = 400, Lx = 1.755h andLz = 1.2h, which are essentially the same as those of the minimal plane Couetteturbulence studied in [32]. The latter approximately satisfies the two symmetries(2.5) and (2.6) spontaneously [19, 32, 60], but they have been explicitly imposed onthe time-periodic solutions. Their parameters are summarized in table 2.3.

Figure 2.2 shows the wall shear rate of Nagatas steady waves as a function ofthe streamwise wavelength, and also those of the two periodic solutions. There is nosteady Nagata solution for the conditions of the periodic orbits, but the wall shearrate of the orbit O1 is roughly the same as that of Nagatas upper branch, while thatof the orbit O2 is closer to Nagatas lower branch. Hereafter, the former solutionwill be referred to as the upper periodic solution, while the latter will be calledthe lower periodic solution, although there does not appear to be a continuousconnection between the two. The upper cycle exhibits a full regeneration cycleof near-wall coherent structures, and approximates well the low-order turbulencestatistics of minimal plane Couette flow (see figure 2.3). It is interesting to note

22

2.3. Classification of solutions

Re h+ L+z u+max v

+max

+x,max

L1 200 1818 5161 2.593.19 0.290.40 0.120.12L2 300 2122 5382 3.104.32 0.210.30 0.080.10L3 400 2425 5392 3.205.09 0.170.30 0.070.08L4 600 2929 5591 3.495.62 0.140.28 0.050.06U1 200 2122 6885 2.262.42 0.610.70 0.150.16U2 300 2829 71110 2.462.68 0.640.83 0.130.13U3 400 3535 76132 2.793.42 0.530.89 0.110.13U4 600 4448 87151 3.153.61 0.610.94 0.090.11

Table 2.2: Approximate parameter ranges for the lower-branch (L) and upper-branch(U) Nagata solutions used in the paper. h+ is half the wall separation in wall units. Allthe cases use Lx = 2h.

L+x L+z h

+ T+p u+max v

+max

+x,max

O1 190 130 34.4 188 3.18 0.741 0.125O2 154 105 27.9 299 4.62 0.231 0.084

Table 2.3: Parameters of the two periodic solutions in [60]. T+p is the temporal periodof the solution, and the turbulence properties are averaged over each cycle. Both cyclesare traversed clockwise in the representation in figures 2.5 and 2.11, and counterclockwisein the representation in figure 2.14.

that both the upper and lower branches of Nagata steady waves have minimumstreamwise wavelengths which are relatively independent of the Reynolds number. Inthe case of the upper branch this minimum length is approximately consistent withthe wavelength, +x 250 300, below which wall turbulence cannot be sustained[54, 56].

The solution loci in figure 2.2 are open towards long wavelengths. Given thenumber of modes that can be used in practice by our continuation algorithm, wave-lengths longer than those in the figure would be numerically inaccurate. Solutioncurves closed towards large Lx were found in [15], and closed solutions (not shown)were also found by us for narrower boxes (Lx = 0.8). It may very well be that thecurves in figure 2.2 close at some longer wavelength.

2.3 Classification of solutions

In this section, we look at the similarities and differences among the simple solutionsavailable in the literature for Couette and Poiseuille flows, including the equilibriumand periodic solutions described in the preceding section. Figure 2.3 shows the

23

2.3. Classification of solutions

100 200 3001

2

3

4

LxW

all s

hear

rat

e+

Figure 2.2: Wall shear rate for the Nagata solutions versus the streamwise wavelengthL+x . , Re = 400. Lz = 1.2h; , Re = 600. Lz = h. The upper-long(lower-short) vertical thick segment represents the wall shear variation of the time-periodicsolution O1 (O2) in table 2.3.

streamwise and wall-normal r.m.s. velocity profiles, u and v, for the upper-branchNagata equilibrium solutions, for the upper cycle of [60], and for the autonomoussteady waves. The figure also includes the statistics of a fully-turbulent minimalCouette simulation. Figure 2.4 does the same for the lower-branch Nagata solu-tions and for the lower cycle. All the profiles in figure 2.3 agree roughly amongthemselves, as do those in figure 2.4, but the two families are very different fromeach other. The upper-branch solutions are very close to minimal near-wall turbu-lence, which is included in figure 2.3, and are characterized by a relatively strongwall-normal velocities, and by weaker streamwise fluctuations. The lower solutionshave much weaker wall-normal velocities and stronger streamwise fluctuations. Sincenear-wall turbulence is known to be dominated by streamwise-velocity streaks andby quasi-streamwise vortices [94], and since the latter are responsible for most of thegeneration of wall-normal velocity, the relative magnitudes of v and of u can re-spectively be used as indicators of the strength of the vortices and of the streaks [51].We will therefore characterize the upper-branch solutions as vortex-dominated, andthe lower-branch ones as streak-dominated.

The characteristics of the fluctuation profiles of the different solutions are sum-marized in figure 2.5, where each solution is represented by a single point whosecoordinates are the maximum values, umax and v

max, of its intensity profiles. Most

solutions fall into one of the two classes discussed above. In the upper-left corner ofthe plot we have the vortex-dominated solutions, and in the lower-right corner thestreak-dominated ones. In the former class we find the three upper-branch familiesdiscussed above, and in the latter the two lower-branch Couette solutions. The un-stable permanent wave obtained by [114] in a Poiseuille flow, and the heteroclinicconnection identified by the same authors [115] are streak-dominated solutions. Thefigure also includes the two permanent-wave solutions given by [121] for Poiseuilleflow, which are classified as lower- or upper-branch according to the original refer-

24

2.3. Classification of solutions

0 20 400

2

4

6

y+

u+

(a)

0 20 400

0.5

1

y+

v+

(b)

Figure 2.3: R.m.s. velocity profiles for: , the autonomous solutions from table 2.1;, upper-branch solutions of case U3 in table 2.2; , upper cycle O1 in table 2.3.

The heavy solid line is the minimal Couette simulation C1 in table 2.4. (a) Streamwisecomponent. (b) Wall-normal component.

ence, although they are too close to the turning point to differ too much from eachother. All that can be said about them is that they are ordered in the right way inthis representation with respect to the classification used for the other solutions.

It is remarkable that two families are enough to classify all the solutions discussedhere, which include Poiseuille, autonomous, and Couette flows, and steady, traveling-wave, and temporally-periodic solutions computed by a variety of techniques. Thehomogeneity within each class is indeed only approximate. It is clear from figure 2.3that, while the maxima of u is almost the same in the autonomous and in theCouette flows, the peaks are located farther from the wall in the former than inthe latter. It is also clear in figure 2.5 that v tends to be stronger in Couetteflows than either in the autonomous or in the Poiseuille solutions. The classificationinto two families is on the other hand relatively independent of the parameters ofthe solutions. Figure 2.6 shows several Nagata equilibrium solutions obtained byvarying Lz for a fixed Lx, using different Reynolds numbers. The separation intotwo families persists even when the Reynolds number is increased by a factor ofthree, and a similar result (not shown) is obtained when the box length is increasedby 50%.

Similar classifications can be obtained using other variables, but not all of themare as clear as the previous one. Consider for example the substitution of vmax bythe maximum x,max of the intensity profile of the streamwise vorticity, which couldbe considered a more direct indicator of the strength of the streamwise vortices. Themaximum of interest for us is the one around y+ 1520, which in fully-developedturbulence corresponds to the quasi-streamwise vortices [64]. There is usually asecond maximum of x at the wall itself, which is related to the interactions of thevortices with the transverse no-slip condition, and sometimes one more near the cen-ter of the flow, which is associated with the outer structures. Occasionally, specially

25

2.3. Classification of solutions

0 10 20 300

2

4

6

y+

u+

(a)

0 10 20 300

0.5

1

y+

v+

(b)

Figure 2.4: R.m.s. velocity profiles for: , lower-branch solutions of case L3 intable 2.2; lower cycle O2 in table 2.3. (a) Streamwise component. (b)