Studies ofthe rotating-disk boundary-layer flow

by

Shintaro Imayama

December 2014

Technical Reports from

Royal Institute of Technology

KTH Mechanics

SE-100 44 Stockholm, Sweden

Akademisk avhandling som med tillstand av Kungliga Tekniska Hogskolan i

Stockholm framlagges till o↵entlig granskning for avlaggande av teknologie

doktorsexamen den 30 januari 2015 kl 10.15 i sal F3, Lindstedsv. 26, Stock-

holm.

©Shintaro Imayama 2014

Universitetsservice US–AB, Stockholm 2014

Shintaro Imayama 2014, Studies of the rotating-disk boundary-layer flow

Linne FLOW Centre, KTH Mechanics, Royal Institute of TechnologySE–100 44 Stockholm, Sweden

AbstractThe rotating-disk boundary layer is not only a simpler model for the study of

cross-flow instability than swept-wing boundary layers but also a useful simplificationof many industrial-flow applications where rotating configurations are present. Forthe rotating disk, it has been suggested that a local absolute instability, leading toa global instability, is responsible for the small variation in the observed laminar-turbulent transition Reynolds number however the exact nature of the transition isstill not fully understood. This thesis aims to clarify certain aspects of the transitionprocess. Furthermore, the thesis considers the turbulent rotating-disk boundary layer,as an example of a class of three-dimensional turbulent boundary-layer flows.

The rotating-disk boundary layer has been investigated in an experimental ap-paratus designed for low vibration levels and with a polished glass disk that gavea smooth surface. The apparatus provided a low-disturbance environment and ve-locity measurements of the azimuthal component were made with a single hot-wireprobe. A new way to present data in the form of a probability density function(PDF) map of the azimuthal fluctuation velocity, which gives clear insights into thelaminar-turbulent transition region, has been proposed. Measurements performedwith various disk-edge conditions and edge Reynolds numbers showed that neither ofthese conditions a↵ect the transition process significantly, and the Reynolds numberfor the onset of transition was observed to be highly reproducible.

Laminar-turbulent transition for a ‘clean’ disk was compared with that for adisk with roughness elements located upstream of the critical Reynolds number forabsolute instability. This showed that, even with minute surface roughness elements,strong convectively unstable stationary disturbances were excited. In this case, break-down of the flow occurred before reaching the absolutely unstable region, i.e. througha convectively unstable route. For the rough disk, the breakdown location was shownto depend on the amplitude of individual stationary vortices. In contrast, for thesmooth (clean-disk) condition, the amplitude of the stationary vortices did not fixthe breakdown location, which instead was fixed by a well-defined Reynolds number.Furthermore, for the clean-disk case, travelling disturbances have been observed atthe onset of nonlinearity, and the associated disturbance profile is in good agreementwith the eigenfunction of the critical absolute instability.

Finally, the turbulent boundary layer on the rotating disk has been investigated.The azimuthal friction velocity was directly measured from the azimuthal velocityprofile in the viscous sublayer and the velocity statistics, normalized by the inner scale,are presented. The characteristics of this three-dimensional turbulent boundary-layerflow have been compared with those for the two-dimensional flow over a flat plate andclose to the wall they are found to be quite similar but with rather large di↵erencesin the outer region.

Descriptors: Fluid mechanics, laminar-turbulent transition, convective instability,

absolute instability, secondary instability, hot-wire anemometry.

iii

Shintaro Imayama 2014, Studier av gransskiktsstromning over en roterande

skiva

Linne FLOW Centre, KTH Mechanics, Royal Institute of TechnologySE–100 44 Stockholm, Sweden

SammanfattningGransskiktet over en roterande skiva ar ett grundlaggande stromningsfall och en

modell av stromning over en svept flygplansvinge men ocksa en anvandbar forenklingav flera industriella stromningsfall med roterande komponenter. Nara centrum argransskiktsstromningen laminar och stabil, men med okande radie blir den insta-bil och slutligen turbulent. En sk lokal absolutinstabilitet, som ger upphov till englobal instabilitet, har foreslagits vara orsaken till omslaget fran laminar till turbu-lent stromning i ett sadant gransskikt, men den precisa karaktaren av omslaget arinte klarlagd. Denna studie avser att klarlagga vissa aspekter av omslagsprocessenoch dessutom behandlar den det turbulenta gransskiktet pa den roterande skivan.

Gransskiktsstromningen studerades i en experimentell uppstallningen som arkonstruerad for att ha laga vibrationer och dar skivan ar en polerad (slat) glasskiva.Detta ger sammantaget mycket sma yttre storningar. Hastighetskomponenten i ro-tationsriktningen mattes med varmtradsteknik. Matresultaten presenterades i en nyform dar sannolikhetsfordelningen av hastighetsfluktuationerna visades som funktionav Reynolds tal, vilket ger en tydlig bild av de olika delarna av omslagsprocessenmellan laminar och turbulent stromning. Matningar som genomfordes med olikaforhallanden vid den yttre kanten av skivan liksom olika Reynolds tal visade att in-get av dessa paverkade omslagsprocessen namnvart och Reynolds tal for starten avomslaget befanns vara i stort sett oberoende av dessa forhallanden.

Omslaget fran laminar till turbulent stromning for en skiva med rahetselementsom placerades uppstroms det kritiska Reynoldska talet for absolut instabilitet jam-fordes med resultaten for en slat disk. Aven mycket sma rahetselement visade sigge starka stationara storningar. For detta fall intra↵ar omslaget innan det kritiskaReynoldska talet for absolutinstabilitet, genom vad som kallas en konvektiv insta-bilitet. For skivan med rahetselement, visade det sig att omslaget beror av de indi-viduella stationara virvlarnas amplitud. Detta till skillnad mot den slata skivan daramplituden av de stationara virvlarna inte var av betydelse for laget av omslaget,utan att det istallet var fixerat vid ett specifikt, val definierat Reynolds tal, vilkettyder pa att omslaget harror fran en absolutinstabilitet. For den slata skivan, sa ob-serverades instationara storningar i samband med att ickelinjariter forst observerades.Variationen av amplituden av dessa storningar genom gransskiktet overensstamde valmed den (linjara) egenfunktion som galler for den kritiska absoluta instabiliteten.

Slutligen har det turbulenta gransskiktet over en roterande skiva undersokts.Friktionshastigheten bestamdes fran hastighetsprofilen i det viskosa underskiktet.Medelhastigheten och fluktuationerna, normaliserade med de viskosa skalorna jam-fordes med det tva-dimensionella gransskiktet over en plan platta. Nara ytan befannsstrukturen vara likartad men tydliga skillnader konstaterades i den yttre regionen.

Nyckelord: stromningsmekanik, laminart-turbulent omslag, konvektiv instabiltet,

absolutinstabilitet, sekundarinstabilitet, varmtradsanemometri

iv

Preface

This doctoral thesis in engineering mechanics is based mainly on exper-

imental work to investigate an area of fluid mechanics. The thesis discusses

both the laminar-turbulent transition and the turbulent boundary layer of the

rotating-disk flow. The thesis is divided into two parts; the first part consists

of an introduction, states the governing equations, gives an overview of pre-

vious studies, and presents the experimental apparatus and the measurement

techniques in detail, as well as a summary of the results and the contributions

of the thesis author to the papers in the second part. The second part con-

tains six papers, four of them are published. The format of the papers may

vary from the published format to align with the formatting of this thesis (and

some minor corrections of the published papers have been made). The thesis

is also available as a PDF file at the KTH library.

December 2014, Stockholm

Shintaro Imayama

v

vi

Contents

Abstract iii

Abstract (in Swedish) iv

Preface v

Part I. Overview and summary

Chapter 1. Introduction 1

Chapter 2. Studies of the rotating-disk flow 5

2.1. The governing equations 5

2.2. Convective instability and absolute instability 11

2.3. Overview of previous studies and remaining problems 12

Chapter 3. Experimental methods 22

3.1. Experimental set-up of the rotating-disk system 22

3.2. Measurement techniques 31

Chapter 4. Main contributions and conclusions 41

Chapter 5. Papers and the author’s contributions 48

Acknowledgements 52

References 55

Part II. Papers

Paper 1. A new way to describe the transition characteristics of

a rotating-disk boundary-layer flow 63

vii

Paper 2. An experimental study of edge e↵ects on rotating-disk

transition 79

Paper 3. On the laminar-turbulent transition of the rotating-disk

flow: the role of absolute instability 107

Paper 4. Experimental study of the rotating-disk boundary-layer

flow with surface roughness 151

Paper 5. Linear disturbances in the rotating-disk flow: a comparison

between results from simulations, experiments and theory 185

Paper 6. The turbulent rotating-disk boundary layer 209

viii

Part I

Overview and summary

CHAPTER 1

Introduction

This thesis discusses the incompressible boundary layer over a rotating disk

in still surroundings. The rotating-disk boundary layer is a simple model of a

three-dimensional boundary-layer flow since it is created just by the disk rota-

tion. The laminar boundary layer is known as the ‘von Karman boundary layer’

and belongs to a family of rotating boundary-layer flows, including the so-called

Bodewadt, Ekman and von Karman boundary layers (BEK boundary layers),

which are exact solutions of the Navier-Stokes equations. These rotating-disk

flows are distinguished by the Rossby number Ro, which is written as

Ro =

⌦

⇤f

� ⌦

⇤d

⌦

⇤a

, (1.1)

with

⌦

⇤a

= (⌦

⇤f

+ ⌦

⇤d

)/4 + [(⌦

⇤f

+ ⌦

⇤d

)

2/16 + (⌦

⇤f

� ⌦

⇤d

)

2/2)]1/2,

where ⌦

⇤f

and ⌦

⇤d

are the fluid angular velocity outside the boundary layer and

the disk angular velocity, respectively (Arco et al. 2005). Here, superscript ⇤denotes a dimensional quantity. Ro throughout this study is �1 as ⌦

⇤f

= 0 and

therefore ⌦

⇤a

= ⌦

⇤d

. Figure 1.1(a) shows the flow visualization of the rotating-

disk flow by Kohama (1984). At the centre of the disk, the flow is stable since

the Reynolds number, which is the ratio of inertial forces to viscous forces, is

small. Here, in this study, the Reynolds number of the rotating-disk boundary-

layer flow is defined as the nondimensional radius, which is given as

R = r⇤r

⌦

⇤

⌫⇤, (1.2)

where r⇤ is the local radius of the disk, ⌦

⇤is the angular velocity of the disk

and ⌫⇤ is the kinematic viscosity of the fluid. In this study, various Reynolds

numbers are defined to describe the flow characteristics. Table 1 shows a sum-

mary of the various Reynolds numbers used in this study. The definition of

each is given in the relevant sections.

1

2 1. INTRODUCTION

(a) (b)

Figure 1.1. Visualization studies of three-dimensional

boundary-layer flows. (a) The rotating-disk (anti-clockwise)

boundary-layer flow by Kohama (1984)

1. (b) The swept-wing

boundary-layer flow by Crawford et al. (2013).

Reynolds number The description

R Local Reynolds number.

RCA

Critical Reynolds number for

absolute instability.

Rt

Transition Reynolds number.

Redge

Edge Reynolds number.

Table 1. Definitions of various Reynolds numbers.

The rotating-disk flow has an inflection point in the radial velocity compo-

nent which satisfies Rayleigh’s inflection-point criterion, resulting in the exis-

tence of an unstable mode. Therefore, the flow is unstable at infinite Reynolds

number. The flow visualization in figure 1.1(a) shows 28–32 stationary vortices

attributed to this inviscid instability mechanism. Three-dimensional boundary

layers that have an inviscid instability are said to have ‘cross-flow instability’.

Since the work by Gregory et al. (1955), the rotating-disk boundary-layer

flow has been used as a simple model for the swept-wing boundary-layer flow

because the velocity profiles are similar so that both flows are susceptible to

1Kohama, Y. 1984 Study on boundary layer transition of a rotating disk. Acta Mech. 50,193-199, figure 2.b with kind permission from Springer Science and Business Media.

1. INTRODUCTION 3

cross-flow instability. Figure 1.1(b) shows the visualization study of the swept-

wing boundary-layer flow by Crawford et al. (2013). However, the e↵ects of

the pressure-gradient parameter and a variable sweep angle are relevant to the

swept-wing flow whereas, the rotating-disk boundary-layer flow is independent

of these parameters, which makes it a much easier configuration for which to

investigate the nature of cross-flow instabilities. Despite the similarity of lam-

inar velocity profiles between the rotating-disk flow and swept-wing flows, the

rotating-disk flow has a periodic condition in the azimuthal direction, making

it a so-called semi-closed system in contrast to swept-wing flows. Further-

more, the early experimental observations showed relatively small variations in

Reynolds number for the onset of transition in di↵erent facilities for rotating-

disk flow. The flow visualizations in figure 1.1 clearly show the di↵erence in

the turbulent breakdown regions between the two flow cases, namely the loca-

tion is azimuthally homogeneous for rotating-disk flow while the location for a

swept-wing boundary-layer flow varies in the spanwise direction, resulting in a

zig-zagged turbulent breakdown region.

Insights into the robust laminar-turbulent transition for the rotating-disk

flow were given by Lingwood (1995a) who found ‘local absolute instability’

(attributed to an inviscid mechanism of rotating-disk flow) and suggested that

the local absolute instability triggers the onset of nonlinearity and transition.

On the other hand, Lingwood (1997c) revealed that the swept-wing flow could

be absolutely unstable in the chordwise direction under certain conditions how-

ever due to the lack of spanwise periodicity, the laminar-turbulent transition

over a swept wing could still be a convective process. The rotating-disk flow

also has Coriolis e↵ects in contrast to swept-wing boundary layers. However,

as explained above, the primary mechanism for laminar-turbulent transition

of the rotating-disk flow is inviscid in nature and, therefore, the Coriolis and

streamline curvature e↵ects are shown not to be of primary importance for the

transition process unless external excitations, e.g. high turbulence levels in the

outer flow, preferentially excite particular modes, which may lead to di↵erent

transition routes.

Studies of the rotating-disk boundary-layer flow are useful not only for

understanding the nature of cross-flow instabilites but also for industrial ap-

plications. Brady (1987) presented results for flows induced by one or more

rotating disks showing that they have constituted a major field in fluid dy-

namics studies since the last century. Indeed, many application fields, such as

rotating machinery, viscometry, computer storage devices and crystal-growth

processes, require study and understanding of rotating flows. Chemical vapour

deposition (CVD) reactors are one of the direct applications of the rotating-

disk boundary-layer flow and are often used in the semiconductor industry

to deposit thin films of electrical and optical materials on substrates, see e.g.

Hussain et al. (2011); Chen & Mortazavi (1986); Vanka et al. (2004). Further

4 1. INTRODUCTION

applications of the rotating-disk flow to rotor-stator systems are discussed in a

recent review paper (Arco et al. 2005).

As mentioned above, understanding of rotating-disk boundary-layer flow

is important from both scientific and industrial points of view. However, the

exact nature of the laminar-turbulent transition process for the rotating-disk

flow is still not fully understood. In particular, the extent to which the abso-

lute instability a↵ects the laminar-turbulent transition process remains to be

determined. This study gives more insights into the various laminar-turbulent

transition routes for the rotating-disk flow.

The thesis is constituted as follows: Part I, chapter 2, will continue describ-

ing the basis of the work, including the governing equations, previous authors’

works and the aims of the study; chapter 3 describes the experimental set-up

and measurement techniques, including the calibrations. Part I ends with a

summary of results and a list of publications as well as describing the author’s

contribution to the papers in chapters 4 and 5, respectively. Part II contains

six papers on various aspects of the rotating-disk flow.

CHAPTER 2

Studies of the rotating-disk flow

2.1. The governing equations

The Navier-Stokes equation (NSE) describes the momentum conservation for

fluid motion. To apply this equation to the rotating-disk flow system (see

figure 2.1), the equation should be described in a cylindrical coordinate system

as an infinite planar disk with a constant angular speed ⌦

⇤. First the position

vector and instantaneous velocity vector are given as r = (r⇤ cos ✓, r⇤ sin ✓, z⇤),v = (u⇤, v⇤, w⇤

), ! = (0, 0,⌦⇤), respectively. Then the continuity equation to

describe mass conservation of the system and NSE are written, in an uniformly

rotating co-ordinate system, as

Figure 2.1. A sketch of the von Karman boundary layer

on a rotating-disk showing the mean velocity profiles (in a

stationary laboratory frame).

5

6 2. STUDIES OF THE ROTATING-DISK FLOW

r · v = 0, (2.1)

@v

@t⇤+ (v ·r)v + 2! ⇥ v + ! ⇥ (! ⇥ r) = � 1

⇢⇤rp⇤ + ⌫⇤r2

v, (2.2)

where 2!⇥ v is the Coriolis acceleration term and !⇥ (!⇥ r) is a centrifugal

acceleration term, p⇤ is an instantaneous pressure and ⇢⇤ and ⌫⇤ are density

and kinematic viscosity of a Newtonian fluid, respectively. r and r2are the

gradient and Laplace operators, respectively, in cylindrical coordinates. The

continuity equation and the NSE can be decomposed into radial, azimuthal

and axial components and the details of the decomposition are described in the

Appendix of Imayama (2012). The decomposed continuity equation and the

NSE are written as

Continuity equation:

@u⇤

@r⇤+

1

r⇤@v⇤

@✓+

@w⇤

@z⇤+

u⇤

r⇤= 0, (2.3)

Radial component of NSE:

@u⇤

@t⇤+

✓u⇤ @u

⇤

@r⇤+

v⇤

r⇤@u⇤

@✓+ w⇤ @u

⇤

@z⇤

◆� v⇤2

r⇤� 2v⇤⌦⇤ � r⇤⌦⇤2

= � 1

⇢⇤@p⇤

@r⇤+ ⌫⇤

✓@2u⇤

@r⇤2+

1

r⇤2@2u⇤

@✓2+

@2u⇤

@z⇤2

◆+

1

r⇤@u⇤

@r⇤� u⇤

r⇤2� 2

r⇤2@v⇤

@✓

�,

(2.4)

Azimuthal component of NSE:

@v⇤

@t⇤+

✓u⇤ @v

⇤

@r⇤+

v⇤

r⇤@v⇤

@✓+ w⇤ @v

⇤

@z⇤

◆+

u⇤v⇤

r⇤+ 2u⇤

⌦

⇤

= � 1

⇢⇤r⇤@p⇤

@✓+ ⌫⇤

✓@2v⇤

@r⇤2+

1

r⇤2@2v⇤

@✓2+

@2v⇤

@z⇤2

◆+

1

r⇤@v⇤

@r⇤� v⇤

r⇤2+

2

r⇤2@u⇤

@✓

�,

(2.5)

Axial component of NSE:

@w⇤

@t⇤+

✓u⇤ @w

⇤

@r⇤+

v⇤

r⇤@w⇤

@✓+ w⇤ @w

⇤

@z⇤

◆

= � 1

⇢⇤@p⇤

@z⇤+ ⌫⇤

✓@2w⇤

@r⇤2+

1

r⇤2@2w⇤

@✓2+

@2w⇤

@z⇤2

◆+

1

r⇤@w⇤

@r⇤

�.

(2.6)

2.1. THE GOVERNING EQUATIONS 7

2.1.1. Mean velocity profile

In this section, mean velocity profiles of the laminar rotating-disk boundary-

layer flow are calculated. Von Karman (1921) derived an exact axi-symmetric

similarity solution of the Navier-Stokes equation for the (time-independent)

base flow (over a disk of infinite radius). Here, the mean velocity profiles and

the mean flow direction profile will be presented.

The instantaneous radial, azimuthal and axial velocities (u⇤, v⇤, w⇤) and

instantaneous pressure (p⇤) are decomposed into mean (time-independent) and

fluctuation (time-dependent) components, i.e. Reynolds decomposition. These

are given as

u⇤= U⇤

+ u⇤,

v⇤ = V ⇤+ v⇤,

w⇤= W ⇤

+ w⇤,

p⇤ = P ⇤+ p⇤,

(2.7)

where U⇤, V ⇤,W ⇤are the mean radial, azimuthal and axial velocities, P ⇤

is

the mean pressure, u⇤, v⇤, w⇤are fluctuating velocities in the radial, azimuthal

and axial directions, and p⇤ is the fluctuating pressure. Then the similarity

variables for the velocities and pressure are defined by

U(z) =U⇤

r⇤⌦⇤ , V (z) =V ⇤

r⇤⌦⇤ , W (z) =W ⇤

(⌫⇤⌦⇤)

1/2, P (z) =

P ⇤

⇢⇤⌫⇤⌦⇤ , (2.8)

where U, V,W, P are nondimensional radial, azimuthal and axial mean velocity

components and mean pressure. z is the wall-normal height from the disk sur-

face normalized by the characteristic length L⇤= (⌫⇤/⌦⇤

)

1/2, namely defined

as

z = z⇤/L⇤. (2.9)

The mean basic flow equations are derived from equations (2.3–2.6) sub-

stituting the similarity variables and taking into account time-independence

and axi-symmetry. Thus, these yield nonlinear ordinary di↵erential equations

written as follows:

2U +W 0= 0, (2.10)

U2 � (V + 1)

2+ U 0W � U 00

= 0, (2.11)

2U(V + 1) + V 0W � V 00= 0, (2.12)

P 0+WW 0 �W 00

= 0, (2.13)

where the prime denotes di↵erentiation with respect to z. The boundary con-

ditions on a rotating-disk flow in the rotating frame are no-slip conditions at

8 2. STUDIES OF THE ROTATING-DISK FLOW

the wall. At z = 1 the nondimensional radial and azimuthal velocities are 0

and -1, respectively. Thus, the boundary conditions are written as

U(0) = 0, V (0) = 0, W (0) = 0,

U(1) = 0, V (1) = �1.(2.14)

The nonlinear ordinary di↵erential equations (2.10–2.13) are written as the

following first-order di↵erential equations,

U 0= g1, (2.15)

V 0= g2, (2.16)

W 0= �2U, (2.17)

g01 = U2 � (V + 1)

2+ g1W, (2.18)

g02 = 2U(V + 1) + g2W, (2.19)

where g1 and g2 are not given by the boundary condition at the wall so an

initial guess has to be provided. Then a fourth-order Runge-Kutta integration

method is applied for the integration of the equations from z = 0 to z = 1,

which was approximated by z = 20 in this calculation. Thus, a Newton-

Raphson searching method was used after each integration pass to adjust the

initial guesses for g1 and g2 until the boundary conditions at z = 20 were

satisfied (see also Appendix A in Lingwood 1995b) thereby determining the

mean velocity profiles. The pressure profile is obtained by an integration of

the equation (2.13) from z = 0 to 1 and with a reference pressure at z = 1,

resulting in

P (z) = �W 0(1)� (W (1)

2 �W (z)2)/2. (2.20)

The solutions of the di↵erential equations in equations (2.10–2.13) are plot-

ted in figure 2.2.

2.1.2. Perturbation equations

The derivation of the perturbation equations for the rotating-disk boundary-

layer flow has been described in paper 4 in Appelquist (2014). Here, only the

final form of the linearized perturbation equations, with the parallel-flow ap-

proximation made and neglecting terms of the order of 1/R2are described.

Therefore, the perturbation continuity equation is given as

@u

@r+

u

R+

1

R

@v

@✓+

@w

@z= 0. (2.21)

The perturbation equations for the rotating-disk boundary-layer flow are ob-

tained as

2.1. THE GOVERNING EQUATIONS 9

!1 !0.8 !0.6 !0.4 !0.2 0 0.20

2

4

6

8

10

z

Figure 2.2. Laminar mean velocity and pressure profiles U(dashed line), V (solid line), W (chain line), and P (thick solid

line), respectively, in a rotating frame.

Radial component:

@u

@t+ U

@u

@r+

uU

R+

V

R

@u

@✓+

W

R

@u

@z+ wU 0 � 2V v

R� 2v

R

= �@P

@r+

1

R

✓@2u

@r2+

1

R2

@2u

@✓2+

@2u

@z2

◆,

(2.22)

Azimuthal component:

@v

@t+ U

@v

@r+

V

R

@v

@✓+

W

R

@v

@z+ wV 0

+

Uv

R+

2uV

R+

2u

R

= � 1

R

@P

@✓+

1

R

✓@2v

@r2+

1

R2

@2v

@✓2+

@2v

@z2

◆,

(2.23)

Axial component:

@w

@t+ U

@w

@r+

V

R

@w

@✓+

W

R

@w

@z+

wW 0

R

= �@P

@z+

1

R

✓@2w

@r2+

1

R2

@2w

@✓2+

@2w

@z2

◆.

(2.24)

By solving these equations considering infinitessimal disturbance ampli-

tudes, it is possible to investigate the linear instability of rotating-disk boundary-

layer flow. This analysis is called local linear stability analysis. In paper 5, some

results from this analysis are compared with experiments and direct numerical

simulations.

10 2. STUDIES OF THE ROTATING-DISK FLOW

2.1.3. Reynolds-averaged equations

The Reynolds-averaged continuity equation and the Navier-Stokes equations

(RANS) are derived to describe the governing equations for the turbulent

boundary-layer flow over the rotating disk. These equations are derived from

equations (2.3–2.6) with some assumptions, see Appendix A in Imayama (2012)

for the detailed derivations. The decomposed velocity and pressure components

are shown in (2.7) and are used to derive the Reynolds-averaged continuity

equation. These are substituted into (2.3) and time averages taken. Deriva-

tives with respect to the ✓ direction are neglected because of the axi-symmetry

of the mean flow. Therefore, the Reynolds-averaged continuity equation is

given as

@U⇤

@r⇤+

@W ⇤

@z⇤+

U⇤

r⇤= 0. (2.25)

By substituting (2.7) into (2.4), (2.5) and (2.6) and taking time averages (de-

noted with an overscore), the RANS equations are derived. Applying the usual

assumptions and assuming axi-symmetry, the RANS equations for the incom-

pressible turbulent rotating-disk boundary-layer flow are obtained as

Radial component:

U⇤ @U⇤

@r⇤+W ⇤ @U

⇤

@z⇤� V ⇤2

r� 2V ⇤

⌦

⇤

= � 1

⇢⇤@P ⇤

@r⇤+ r⇤⌦⇤2

+

1

⇢⇤@

@z⇤

✓µ⇤ @U

⇤

@z⇤� ⇢⇤u⇤w⇤

◆,

(2.26)

Azimuthal component:

U⇤ @V⇤

@r⇤+W ⇤ @V

⇤

@z⇤+

U⇤V ⇤

r⇤+ 2U⇤

⌦

⇤

=

1

⇢⇤@

@z⇤

✓µ⇤ @V

⇤

@z⇤� ⇢⇤v⇤w⇤

◆,

(2.27)

Axial component:

@w⇤w⇤

@z⇤= � 1

⇢⇤@P ⇤

@z⇤. (2.28)

2.2. CONVECTIVE INSTABILITY AND ABSOLUTE INSTABILITY 11

2.2. Convective instability and absolute instability

Lingwood (1995a) showed theoretically that for the rotating-disk boundary-

layer flow, some travelling disturbances change from being locally convectively

unstable to become locally absolutely unstable above RCA

. Briggs (1964) in-

troduced the concept of convective and absolute instabilities in the study of

plasma physics. In this section, the basic concept of local linear convective and

absolute instabilities is shown. More mathematical details to distinguish local

convective and absolute instabilities for rotating-disk boundary-layer flow are

shown in e.g. Lingwood (1997b).

Both convective and absolute instabilities are related to growth of distur-

bances in space and time. These are distinguished by the di↵erent behaviours

of the linear impulse response given at a certain spatial location. Figure 2.3

shows a sketch of three di↵erent impulse responses at nondimensional time

t = t1 for di↵erent instability conditions, where the linear impulse is intro-

duced at a nondimensional position xs

= 0 and at t = 0. Figure 2.3(a) is a

stable condition. In this case, the introduced impulse amplitude decays in time

and at t = t1 the system reverts back to the initial condition. Figure 2.3(b)

0

0

t

0

x

(a)

(b)

(c)

t1

t1

t1

xs

Figure 2.3. The concept of a linear impulse response to

distinguish between convective and absolute instability in the

x�t plane: (a) stable, (b) convectively unstable, (c) absolutely

unstable. The linear impulse is introduced at x = xs

at t = 0

for all cases.

12 2. STUDIES OF THE ROTATING-DISK FLOW

shows the impulse response for a convectively unstable condition. The intro-

duced impulse creates a wave packet that grows as it convects downstream

within the chain lines. Figure 2.3(c) shows the impulse behaviour for an abso-

lutely unstable condition where the introduced impulse creates a wave packet

that stays at the introduced location and grows exponentially within the chain

lines.

2.3. Overview of previous studies and remaining problems

Here, some early studies on the rotating-disk boundary-layer flow are intro-

duced. The following two subsections cover flow instability and transition, and

the turbulent boundary-layer, respectively. Finally, remaining problems on the

rotating-disk flow are discussed.

2.3.1. Instabilities and the laminar-turbulent transition process

The study of the incompressible rotating-disk boundary-layer flow was pro-

moted by von Karman (1921) who derived the exact similarity solution for

the laminar boundary layer with an infinite-radius disk from the Navier-Stokes

equation. Since the flow is induced by rotation of the disk, the rotating-disk

boundary-layer flow is a simple model to understand cross-flow instability for

comparison with swept-wing boundary-layer flows, e.g. Gregory et al. (1955).

An early study of the instability of the rotating-disk flow was performed

by Theodorsen & Regier (1944) using hot-wire measurements of the boundary

layer within the laminar-turbulent transition region, and they captured an in-

stability wave and describe that “a pure tone of a frequency of about 200 cycles

per second was observed in the transition region”. They also measured velocity

profiles from the laminar to turbulent regions showing that the boundary-layer

thickness is constant for laminar profiles and it grows as the flow becomes

turbulent. They mentioned that the transition Reynolds number is R = 557.

Smith (1947) also performed experiments using a hot-wire probe and showed

instability waves at di↵erent Reynolds numbers showing that the amplitude of

sinusoidal wave becomes larger and larger as the Reynolds number increases

and finally the signal becomes chaotic indicating breakdown to turbulent flow.

These indications of instabilities observed experimentally have been investi-

gated using di↵erent approaches. Flow visualization techniques were applied by

Gregory et al. (1955); Fedorov et al. (1976); Clarkson et al. (1980); Kobayashi

et al. (1980); Kohama (1984); Wilkinson et al. (1990); Faller (1991); Kohama

& Suda (1992); Astarita et al. (2002) to capture the structures of the instabil-

ities. These visualization studies (Kobayashi et al. 1980; Astarita et al. 2002)

revealed that the rotating-disk flow typically has around 22–32 primary spiral

vortices (the number increasing with Reynolds number), and Kohama (1984)

showed that the phase velocity of the spiral vortices is zero, which means that

the primary spiral votices observed using these flow-visualization techniques

2.3. OVERVIEW OF PREVIOUS STUDIES AND REMAINING PROBLEMS 13

are stationary with respect to the disk in its rotating frame. The visualiza-

tions also show that breakdown of vortices for rotating-disk flow occurs at a

certain radius independent of azimuthal location, resulting in a well-defined

circular transition front in visualizations, e.g. Gregory et al. (1955); Kobayashi

et al. (1980); Kohama (1984). This behaviour contrasts with the transition to

turbulence of swept-wing boundary-layer flows, for which visualizations (e.g.

Dagenhart & Saric 1999) show that the transition zone zigzags in the spanwise

direction.

The instabilities were also investigated using a theoretical approach us-

ing local linear stability analysis (e.g. Malik et al. 1981; Mack 1985; Itoh &

M. 1982; Watanabe 1985; Malik 1986; Faller 1991; Lingwood 1995a; Hwang

& Lee 2000; Itoh 2001; Hussain et al. 2011). The local linear stability anal-

ysis solves eigenvalue problems of the Navier-Stokes equations with boundary

conditions, neglecting non-parallel, nonlinear and some high-order terms. The

analysis (e.g. Malik et al. 1981) shows that the critical Reynolds number for

the stationary instability (in the rotating frame), so-called Type-I stationary

cross-flow instability, is about R = 290, which agrees well with experimental

observations (e.g. R = 297 by Kobayashi et al. 1980). This Type-I instability

is attributed to an inviscidly unstable mechanism due to an inflection point

of the radial mean velocity profile. Malik et al. (1981) discussed that Coriolis

and streamline curvature e↵ects need to be taken into account for the critical

Reynolds number to agree better with experimental observations. This Type-I

mode is unstable not only for stationary waves but also travelling disturban-

ces that have non-zero phase speed relative to the disk. Hussain et al. (2011)

showed that the mode with maximum spatial growth rate is a Type-I mode that

has large negative phase speed in the rotating frame, so that the disturbance

is travelling significantly slower than the rotating disk. There is another (vis-

cously) unstable mode (Type-II), which was found to have significantly lower

critical Reynolds number at around R = 50�69 (e.g. Malik et al. 1981; Itoh &

M. 1982; Faller 1991) but also lower radial spatial growth rate than the Type-I

mode, as shown by Hussain et al. (2011).

Despite many possible unstable modes in the rotating-disk boundary-layer

flow, the dominant observed instability in most experimental studies is the

Type-I stationary cross-flow instability. This is because randomly distributed

unavoidable surface roughnesses continuously excite stationary disturbance in

the flow field, whereas any sources of travelling disturbances do not continu-

ously and repeatably excite the flow without an artificial source. Mack (1985)

performed temporal and spatial analysis using local linear theory comparing

with Wilkinson & Malik’s (1983) experimental study and concluded that “the

spiral streaks observed in flow visualization experiments are the constant-phase

lines of the merged wave patterns produced by several random sources on the

disk”. To study the Type-II mode due to the viscous streamwise-curvature ef-

fect, Faller (1991) performed theoretical and experimental studies and showed

14 2. STUDIES OF THE ROTATING-DISK FLOW

that if the external flow has su�ciently high turbulence level to excite Type-

II instability then the rotating-disk flow can undergo transition to turbulence

at lower Reynolds number than usual. Furthermore, Garrett et al. (2012)

suggested that Type-II mode could become dominant if the homogeneously

distributed surface roughnesses on the disk are big enough.

To investigate the characteristics of Type-I stationary cross-flow instability,

many measurements have been performed using hot-wire anemometry. Due

to azimuthal periodicity of the configuration, the azimuthal wavenumber, �,is a (real) integer. Spectra of hot-wire time series show that the stationary

cross-flow instability consists of a superposition of multiple wavenumbers (e.g.

Jarre et al. 1996b; Lingwood 1996; Corke & Knasiak 1998; Othman & Corke

2006; Corke et al. 2007). Hot-wire and visualization studies (e.g. Wilkinson &

Malik 1985; Astarita et al. 2002) showed that the number of stationary vortices

increase as a function of Reynolds number in the unstable region. Wilkinson

& Malik (1983) showed the angle, ✏, of the stationary vortices are distributed

between 10

�and 14

�at R = 300 � 500. Kobayashi et al. (1980) also got the

angle of the spiral vortices as approximately 14

�at a later stage of transition,

and found that it decreased to ✏ = 7

�at higher Reynolds number. From linear

stability theory Kobayashi et al. (1980) found a value of around 14

�.

Lingwood (1995a) recognized that the variation in transition Reynolds

number, Rt

, for the rotating-disk boundary-layer flow observed using vari-

ous experimental facilities is relatively small e.g. Malik et al. (1981) reports

Rt

= 513 ± 3%. This characteristic is not observed in purely convectively un-

stable flows. Table 2 summarizes transition Reynolds numbers obtained from

various rotating-disk experiments. For example, the purely convectively un-

stable transition to turbulence of the flat-plate boundary-layer flow leads to a

transition location that is highly dependent on the initial disturbance environ-

ment. Considering an impulsive forcing, Lingwood performed theoretical anal-

ysis using Briggs’ method (Briggs 1964) to distinguish convective and absolute

instabilities. Thus, it was found that some travelling disturbances become lo-

cally absolutely unstable above R > 507.3 (Lingwood 1995a, 1997a), which is a

Reynolds number very close to the experimentally reported transition Reynolds

number. Based on these theoretical results Lingwood (1995a) suggested that

“absolute instability may be a better explanation for transition than the con-

vective radial growth of disturbances, leading to nonlinearity”. Furthermore,

Lingwood (1996) also performed experiments by exciting the flow impulsively

thereby introducing a travelling wave packet into the boundary layer to see the

impulse response. As the travelling wave packet approached RCA

, the trail-

ing edge became fixed (radially) in space, which corroborated the theoretical

finding of local absolute instability.

In more recent years, the global behaviour of rotating-disk flow has been in-

vestigated. The local linear stability analysis neglects streamwise development,

2.3. OVERVIEW OF PREVIOUS STUDIES AND REMAINING PROBLEMS 15

Authors Rt

Method

Theodorsen & Regier (1944) 557 Hot-wire

Gregory, Stuart & Walker (1955) 533 Visual, China-clay

Cobb & Saunders (1956) 490 Heat transfer

Gregory & Walker (1960) 524 Pressure probe

Chin & Litt (1972) 510 Mass transfer

Fedorov et al. (1976) 515 Visual, napthalene

Clarkson, Chin & Shacter (1980) 562 Visual, dye

Kobayashi, Kohama & Takamadate (1980) 566 Hot-wire

Malik, Wilkinson & Orszag (1981) 520 Hot-wire

Wilkinson & Malik (1985) 550 Hot-wire

Lingwood (1996) 508 Hot-wire

Othman & Corke (2006) 539 Hot-wire

Table 2. Experimental Rt

(di↵erently defined) given in pre-

vious studies.

which is included in global stability analyses. Davies & Carpenter (2003) inves-

tigated the linear global behaviour using direct numerical simulations (DNS) of

the linearized Navier-Stokes equations. Thus, it was found that the rotating-

disk flow is linearly globally stable in contrast to Lingwood’s (1996) obser-

vations. Furthermore, Davies & Carpenter (2003) suggested that convective

behaviour is dominant even if the flow is strongly locally absolutely unstable.

To explain this contradiction, Pier (2003) performed nonlinear global stability

analysis and showed that the rotating-disk flow is nonlinearly globally unstable

with su�cient background disturbances, with the nonlinear front located at

the onset of the local absolute instability (via a subcritical mechanism). Pier

(2003) also suggested that the self-sustained finite-amplitude disturbances trig-

ger secondary absolute instability leading to turbulence. Further discussions of

linear global behaviour of the rotating-disk flow was given by Healey (2010),

who suggested, using the linearized complex Ginzburg-Landau equation, that

the rotating-disk flow with a finite disk radius configuration becomes locally

linearly unstable leading directly to nonlinear global instability, via a supercrit-

ical mechanism. Furthermore, adding a nonlinear term in the equations, Healey

described small variations in the experimentally-observed transition Reynolds

number in terms of a nonlinear stabilization e↵ect as the edge Reynolds number

approaches the front. Healey also pointed out that Davies & Carpenter (2003)

regarded upstream propagation in their simulations from the downstream con-

dition as a spurious numerical artifact so that the simulations were stopped

before the upstream-propagating waves reached the domain of interest, which

e↵ectively rendered their disk with an infinite radius.

16 2. STUDIES OF THE ROTATING-DISK FLOW

Previous studies also investigated the late stage of laminar-turbulent tran-

sition of rotating-disk flow. Kobayashi et al.’s (1980) visualization study cap-

tured the secondary instability sitting on the top of primary instability just

before the turbulent breakdown, which they described as “a new striped flow

pattern originating along the axis of a [stationary] spiral vortex”. Kohama

(1984) also observed “ring-like vortices which occur on the surfaces of each spi-

ral vortices (sic)” in his flow visualization. At the final stage of the transition,

hot-wire measurements (Kobayashi et al. 1980 and Wilkinson & Malik 1985)

showed ‘kinked’ velocity time series just before turbulent breakdown. Further-

more, Kohama et al. (1994) fixed a hot-wire probe on the disk surface so that

the probe rotates with the disk. The hot-wire probe captured two di↵erent

travelling frequency components at 150 Hz and 3.5 kHz. Thus, they concluded

that the higher frequency component can be attributed to the secondary in-

stability, which was captured as ring-like structures just before the turbulent

breakdown by the visualization technique. Balachandar et al. (1992) conducted

a theoretical analysis to investigate the secondary instability and found that if

the root-mean-square amplitude of the primary stationary disturbances exceeds

approximately 9% of the local disk velocity at R = 500, then the travelling sec-

ondary instability is triggered that consists a pair of counter-rotating vortices.

Lingwood (1996) performed experiments in a low-disturbance environment, i.e.

the clean-disk condition, and stated that “the stationary disturbances are suf-

ficiently small, even close to the onset of transition, for the boundary-layer

stability to be governed by the mean velocity profiles rather than secondary

instabilities”.

As a summary of previous studies on the laminar-turbulent transition of

the rotating-disk flow, the exact nature of the transition process is not yet fully

understood. In particular, the following points were still unclear and will be

discussed in this study.

• Many experimental studies have been performed and it seems that there

are some links between the onset of the nonlinearity and the onset of

absolute instability. However, prior to this study there was no direct

experimental evidence of the absolute instability except the propagation

of a wave packet in an impulsively-excited rotating-disk boundary-layer

flow (Lingwood 1996).

• It was not clear how absolute instability a↵ects the transition process

and interacts with other instabilities, e.g. Type-I stationary vortices.

• Healey (2010) suggested, using the linearized complex Ginzburg-Landau

equation with an additional nonlinear term, that the nonlinear e↵ect

stabilizes the flow as the edge Reynolds number approaches the critical

Reynolds number for the absolute instability. This indicates that the

2.3. OVERVIEW OF PREVIOUS STUDIES AND REMAINING PROBLEMS 17

edge e↵ects of the disk are somehow important for transition. Experi-

mental investigation of this was required.

• At a final stage of the transition process, some experimental studies

have captured secondary instabilities of the rotating-disk flow. However,

some di↵erent opinions on the secondary instability have been proposed.

Thus, further measurements of the secondary instability were required.

Tables 3 and 4 show lists of past experimental studies of the rotating-disk

flow.

2.3.2. Turbulent boundary-layer flow

In contrast to the many studies of the laminar-turbulent transition process,

as mentioned above, experimental studies of the turbulent boundary-layer flow

are limited despite the industrial applications (e.g. rotor-stator systems, Arco

et al. 2005). Table 5 lists early experimental studies of rotating-disk turbulent-

boundary layer flows. The turbulent flow over a rotating disk is a three-

dimensional boundary layer with an inflection point in the radial velocity

(cross-flow) component. Littell & Eaton (1994) showed that the maximum

of the mean cross-flow component reaches 11% of the local disk velocity.

Goldstein (1935) performed torque measurements on the turbulent rotating-

disk flow and Theodorsen & Regier (1944) measured mean azimuthal velocity

profiles for the turbulent boundary-layer flow up to R = 2646. In this mea-

surement range, the mean velocity profile was in good agreement with the 1/7

power law. Pitot and entrainment measurement techniques were applied to tur-

bulent radial and azimuthal velocity profile measurements by Cham & Head

(1969). They concluded that both the radial and azimuthal velocity profiles

conform well to Mager’s (1952) cross-flow expression and Thompson’s (1965)

two-dimensional family, respectively. Cham & Head (1969) also evaluated the

azimuthal local skin-friction coe�cient using a Clauser (1954) plot. Experimen-

tal turbulent statistics measurements on a rotating-disk flow were performed

by Erian & Tong (1971) and they concluded that “the eddy viscosity in the

turbulent boundary layer generated by the disk rotation is substantially larger

than that of the turbulent boundary layer over a flat plate”. Littell & Eaton

(1994) reported that the azimuthal velocity profile, normalized by wall units

obtained from a conventional two-dimensional law of the wall, showed a lack

of a wake component compared with the two-dimensional turbulent boundary

layer. They commented that the reason for the absence of a wake region was

not understood. A two-dimensional boundary layer has a similar wake profile if

there is a streamwise favourable pressure gradient however this is not the case

for the rotating turbulent boundary-layer flow since there is no azimuthal or

radial pressure gradient. Itoh & Hasegawa (1994) performed hot-wire velocity

18 2. STUDIES OF THE ROTATING-DISK FLOW

profile measurements of both azimuthal and radial components. They evalu-

ated local skin-friction coe�cients by direct measurements of velocity profiles

in the viscous sublayer. The direct measurement of the friction velocity al-

lowed the turbulent statistics to be more accurately evaluated than by classical

empirical methods.

From previous studies of turbulent boundary-layer flow on a rotating disk,

the following areas were open for further research.

• As mentioned above, there are not many experimental studies of the

rotating-disk turbulent boundary-layer flow. In particular, there were

no reports on the high-order turbulent statistics (e.g. skewness and flat-

ness) and spectra of the velocity component.

• To evaluate accurately turbulent statistics normalized by inner scales

experimental measurements of the variables were required for the tur-

bulent boundary layer of a rotating-disk flow.

2.3. OVERVIEW OF PREVIOUS STUDIES AND REMAINING PROBLEMS 19

Authors

Fluid

Diskmaterial

Diskdia.(mm)

Method

Theodorsen&

Regier(1944)

Air

-305,610

HW

Smith(1947)

Air

Steel

305

HW

Gregoryetal.(1955)

Air

Perspex

305

Visual,yawmeter

Gregory&

Walker(1960)

Air

Slabofdural

914

HF

Chin&

Litt(1972)

Water

Lucite

150

Pointelectrodes

Fedorovetal.(1976)

Air

Steel

100-200

Visual

Clarksonetal.(1980)

Water

Plexiglas

610

Visual

Kobayashietal.(1980)

Air

Aluminium

400

Visual,HW

Maliketal.(1981)

Air

Plexiglas

457

HW

Kohama(1984)

Air

Aluminium

400,600

Visual,HW

Itoh&

M.(1982)

Water

Acrylicglass

150,250

Visual

Wilkinson&

Malik(1985)

Air

Glass

456

HW

Watanabe(1989)

Air

Aluminium

300

HW

Wilkinsonetal.(1990)

Air

Glass

330

Visual

LeGal(1992)

Water

Stainlesssteel

500

HF

Kohama&

Suda(1992)

Air

Aluminium

400,600

Visual

Table 3. Experimental studies of the laminar-turbulent

transition of the rotating-disk boundary-layer flow performed

between 1944 and 1992. HW and HF in the method column

indicate use of a hot-wire probe and hot-film probe, respec-

tively.

20 2. STUDIES OF THE ROTATING-DISK FLOW

Authors

Fluid

Diskmaterial

Diskdia.(mm)

Method

Kohama&

Suzuki(1994)

Air

-400

HW

Aubryetal.(1994)

Water

Stainlesssteel

500

HF

Kohamaetal.(1994)

Air

-400

HW

Lingwood(1996)

Air

Aluminum

alloy

475

HW

Jarreetal.(1996b)

Water

-500

HF

Jarreetal.(1996a)

Water

Stainlesssteel

500

HF

Corke&

Knasiak(1998)

Air

Aluminum

457

HW

Astaritaetal.(2002)

Air

Printedcircuit

450

Visual

Zoueshtiaghetal.(2003)

Water

-388

HF

Othman&

Corke(2006)

Air

Aluminum

457

HW

Corkeetal.(2007)

Air

Aluminum

457

HW

Siddiquietal.(2009)

Air

Glass

500

HW

Imayamaetal.(2012)

Air

Glass

474

HW

Harris&

Thomas(2012)

Water

-400

HF

Garrettetal.(2012)

Water

-400

HF

Imayamaetal.(2013)

Air

Glass

474

HW

Siddiquietal.(2013)

Air

Glass

500

HW

Pier(2013)

Air

Syntheticresin

500

HW

Imayamaetal.(2014a)

Air

Glass

474

HW

Table 4. Experimental studies of the laminar-turbulent

transition of the rotating-disk boundary-layer flow performed

between 1994 and 2014. HW and HF in the method column

indicate use of a hot-wire probe and hot-film probe, respec-

tively.

2.3. OVERVIEW OF PREVIOUS STUDIES AND REMAINING PROBLEMS 21

Authors

Fluid

Diskmaterial

Diskdia.(mm)

Method

Cham

&Head(1969)

Air

Steel

914

Pitot,entrainment

Erian&

Tong(1971)

Air

Aluminum

457

HW

Itoh&

Hasegawa(1994)

Air

Aluminum

1000

HW

Littell&

Eaton(1994)

Air

Aluminum

1000

HW

Imayamaetal.(2014b)

Air

Glass

474

HW

Table 5. Experimental studies of the turbulent boundary-

layer flow of the rotating-disk boundary-layer flow.

CHAPTER 3

Experimental methods

3.1. Experimental set-up of the rotating-disk system

In the following chapter, the experimental apparatus and measurement tech-

niques are explained. First of all, the details of the experimental apparatus

used in this study are described. Secondly the velocity measurement proce-

dures using hot-wire anemometry and some other measurement methods are

discussed.

The experimental set-up is shown in figure 3.1. It was originally manu-

factured and used by Lingwood (1996) in Cambridge, UK. It was transferred

from Cambridge to Stockholm before the author started his doctoral work. Al-

though some of the components are identical to the ones used by Lingwood,

most of them were modified and replaced by the author to obtain high-quality

data needed for the new studies.

3.1.1. Rotating apparatus

In this study, a new float-glass disk was prepared. The disk diameter D⇤is

474 mm and the thickness is 24 mm. The edge of the disk was ground down

with a 45

�angle so that the actual radius was reduced by about 1.5 mm giving

r⇤d

= 235.5 mm. To study the laminar-turbulent transition of the rotating-disk

flow under so-called ‘clean’ conditions, the disk surface must be highly polished

and as flat as possible to minimize the excitation of certain instabilities. It is

especially important to have a controlled disturbance environment if one wants

to enable comparisons with theoretical and numerical studies.

The main purpose of this experimental study was to investigate the mech-

anism of absolute instability, which is considered to be important in the transi-

tion process from a ‘clean’ disk. Some studies e.g. Lingwood (1996) and Healey

(2010), however, suggest that surface roughnesses excite convective stationary

disturbances in the flow field and, if the excited amplitudes are su�ciently

large, the flow may undergo transition to turbulence via a convectively un-

stable transition route instead. For these reasons, the glass disk surface was

polished resulting in a surface roughness of less than 1 µm and an azimuthal

imbalance of less than 10 µm, see figure 3.2. However, stationary disturban-

ces attributed to unavoidable roughnesses on the surface were still observed,

22

3.1. EXPERIMENTAL SET-UP OF THE ROTATING-DISK SYSTEM 23

(A)

(c)(b)

(a)

(d)

(e)

(f)

(g)

(B)

Figure 3.1. (A) The experimental set-up of the rotating disk

with plate edge condition

2. (B) Vertically-separated schematic

of main components of the apparatus: (a) glass disk; (b)

aluminium-alloy disk; (c) clamps; (d) iron disk; (e) brass disk

with slits; (f) air bearing; and (g) main motor.

as with many earlier experimental studies e.g. Wilkinson & Malik (1985) and

Lingwood (1996). In the present study, a convectively unstable transition route

due to stationary disturbances was also investigated putting roughness elements

on the disk surface, giving the so-called ‘rough’ disk condition. The details of

the roughness elements are given later. The disk surface was cleaned carefully

before every set of measurements using non-flammable air spray and acetone.

To avoid breakage of the glass disk under operation, the maximum opera-

tional speed was estimated by the following procedure. The relation between

the failure stress, �⇤f

, and the maximum angular velocity, ⌦

⇤max

, of the rotating

disk (Ashby 2005; Lingwood 1995b) is given as

D⇤

2

⌦

⇤max

=

8�⇤

f

Sf

⇢⇤glass

(3 + ⌫Po

)

!1/2

, (3.1)

2Imayama, S., Alfredsson, P. H. & Lingwood, R. J. 2012 A new way to describe the transitioncharacteristics of a rotating-disk boundary-layer flow. Phys. Fluids 24, 031701, figure 1 with

kind permission from AIP Publishing LLC.

24 3. EXPERIMENTAL METHODS

57.5

115

172.5

230

330

150

300

120

270

90

240

60

210

30

180 0

r* [mm]

!*[o ]

Figure 3.2. The azimuthal imbalance measured by a me-

chanical test indicator up to r⇤ = 230 mm. The colour contour

indicates the surface height variation �I from the reference

position (-2 µm (blue) < �I < +7 µm (red)) with 1 µm steps.

where ⌫Po

is Poisson’s ratio, which has an approximately constant value of 1/3for all solids, and S

f

is an appropriate safety factor; Sf

= 10 was selected in

this study. Table 6 shows the typical values for the float-glass parameters. As

a result the maximum rotational speed of the glass disk is given, using the

parameters in table 6, as ⌦

⇤max

= 2553 rpm.

The glass disk was fixed on the aluminium-alloy disk from the side by eight

stationary clamps, see figure 3.1(B). The aluminium-alloy disk had a 475 mm

diameter and 30 mm thickness. To adjust the weight balance of the disks, one

screw of the eight clamps was made slightly larger than the others. The disks

were connected to a vertical shaft of a d.c. servo motor via an iron disk with a

diameter of 270 mm. The d.c. servo-motor (Mavilor MS6) is controlled the disk

rotation via a main motor control inverter (Infranor SMV 1510). A brass disk,

which had 30 slits equally spaced in the azimuthal direction, was mounted on

the iron disk to measure rotational speed and to determine the angle location

of the disk using an optical sensor. One of the slits was covered by tape

to specify a reference angle position. The output voltage from the optical

sensor was recorded together with the hot-wire signal so that it was possible

to make ensemble-averaged time series from the hot-wire signals. Figure 3.3

3.1. EXPERIMENTAL SET-UP OF THE ROTATING-DISK SYSTEM 25

�⇤f

(MPa) ⌫Po

⇢⇤glass

(kg/m

3)

41 0.23 2.53⇥ 10

3

Table 6. Float-glass parameters, where �⇤f

is the failure

stress, ⌫Po

is the Poisson’s ratio and ⇢⇤glass

is the density of the

glass (Source: http://www.industrialglasstech.com/pdf/soda

limeproperties.pdf).

shows a typical example of a simultaneously measured hot-wire signal and an

optical-sensor output. The first voltage drop after the masked slit, for which

the optical sensor gave a high-voltage reading, was defined as the zero angle.

Rotation of the disk package, consisting of the glass disk, the aluminium-alloy

disk, the iron disk, the brass disk and the vertical shaft was supported by a

high-pressure air bearing attached to the motor to ensure smooth and quiet

operation. Pressurized air of 5.5 bar from a compressor and passed through

an air filter (HPC, DomnickHunter AO-0013G) and air dryer (KAESER KMM

Compressed Air Dryer) was supplied to the bearing.

The base of the apparatus consisted of a black-painted steel box with two

sandbags inside to stabilize the apparatus. The main motor was mounted

on the box. Steel arms for the traversing system were also connected to the

box. The total weight of the apparatus was approximately 250 kg. During the

author’s doctoral work the apparatus was moved twice due to reconstruction

of the laboratory building.

3.1.2. Traverse system

A traverse system with two axes was connected to a steel-box base through

aluminium and steel beams (see figure 3.1(A)). One of the traverses moved

in the horizontal (radial) direction, and the other traverse was mounted on

the horizontal traverse at a 45

�inclination so as not to disturb the axial flow,

which approaches the rotating-disk from above. The horizontal and inclined

traverses were made of stainless-steel pipes and lead screws were inserted into

the pipes. They were operated by absolute encoders (AVAGO AEAS-7000 and

Mitsutoyo ID-C125B) and d.c. motors (micro motors E192.14.67 and RH158

510:1), respectively. The encoder (AVAGO AEAS-7000) for the horizontal

traverse was inserted between the d.c. motor and the lead screw. The other

encoder (Mitsutoyo ID-C125B) was mounted on the inclined traverse. The

resolution was 5 µm for the horizontal traverse and 3 µm for the inclined

traverse, respectively. At the edge of the inclined traverse, adapters to connect

a hot-wire probe were attached. The attached hot-wire probe was able to

reach all the required measurement positions, namely from the centre of the

disk to beyond the disk outer edge for the horizontal direction, and from the

26 3. EXPERIMENTAL METHODS

6.4

6.5

6.6

6.7

V*[m

/s]

0 45 90 135 180 225 270 315 360

012345

![°]

E* ta

cho[V

]

(a)

(b)

Figure 3.3. A typical example of a time series for one disk ro-

tation. (a) Azimuthal velocities obtained by a hot-wire probe.

(b) Simultaneously obtained voltage from an optical sensor to

determine the disk speed and angular position.

(a)

100 150 200 250

!40

!30

!20

!10

0

10

r*

ref [mm]

! h

* [

µm

]

(b)

Figure 3.4. (a) The set-up of a vertical alignment measure-

ment of the horizontal traverse with a mechanical indicator

attached to the edge of the inclined traverse. (b) The relative

height variations �h⇤in horizontal traverse movement mea-

sured by the mechanical indicator. r⇤ref

is a relative radius

location.

3.1. EXPERIMENTAL SET-UP OF THE ROTATING-DISK SYSTEM 27

wall to beyond the turbulent boundary-layer thickness. The parallelism of the

horizontal traverse to the surface of the disk was checked using a mechanical

indicator, see figure 3.4(a). Since the movement of the horizontal traverse is

not perfectly straight, some vertical height variations were observed by the

indicator under the assumption that the radial imbalance of the disk was small

compared to the vertical variations, see figure 3.4(b). The obtained data, i.e.

vertical-variations data, as shown in figure 3.4(b), were used to compensate the

vertical positioning of the horizontal traverse. The orthogonality of the inclined

traverse was checked using a machinist square. An in-house manufactured

electronic circuit was developed to operate the d.c. motors. The board and

encoders were connected to a controller board (National Instruments USB-

6216) and this traverse system was operated by a computer using LabVIEW8.6

software.

3.1.3. Edge conditions

In the present study, motivated by Healey (2010), the e↵ects of edge conditions

and edge Reynolds numbers on the laminar-turbulent transition of a rotating-

disk flow have been investigated. Here, the edge Reynolds number, Redge

, is

defined as Redge

= r⇤d

(⌦

⇤/⌫⇤)1/2 where r⇤d

= 235.5 mm is the actual radius of

the disk.

Three di↵erent edge conditions were prepared as shown in figure 3.5, and

they are called ‘open type’, ‘ring type’ and ‘plate type’, respectively. The

steel beams were mounted on the apparatus to hold various edge components

around the disk. The open-type edge condition had no extended plate or cover

around the disk. The ring-type edge condition had a steel-ring cover around

the aluminium-alloy disk that covered the eight clamps that fix the glass disk.

This edge condition was included because the eight clamps create a period-eight

disturbance flow field in the laboratory frame of reference (which is stationary

in the rotating-disk frame) and these disturbances were damped by the ring

cover. The ring cover was fixed to the steel beams so that it did not rotate with

the disk and the horizontal slit width between the ring cover and the glass disk

was adjusted to be less than 1 mm. The edge of the glass disk is still exposed

in a similar way to the open-type edge condition since the top of the ring is

located 11 mm vertically below the disk surface. The plate-type edge condition

had a wooden annular plate with outer diameter of 900 mm fixed on the steel

beams. This extended plate eliminated the e↵ects of the eight aluminium fixing

components and also reduced the e↵ects of noise coming from the air bearing

and d.c.-servo motor. The horizontal gap between the disk and plate was less

than 1 mm and vertically the disk surface and plate are approximately flush.

The edge Reynolds number was varied by changing the rotational speed

since the radius of the disk was fixed. In paper 2, measurements were performed

with various edge Reynolds numbers and the e↵ects on the laminar-turbulent

transition process are discussed.

28 3. EXPERIMENTAL METHODS

(a) open type (b) ring type

(c) plate type

Figure 3.5. Three edge conditions.

3

3.1.4. Roughness elements

Here, the details of the roughness elements are described. The roughness ele-

ments are used to excite Type-I stationary disturbances in the flow field and to

establish transition to turbulence due to the growth of stationary disturbances

before the flow reaches the absolutely unstable region, i.e. via a so-called con-

vective unstable transition route. Direct comparisons between this convectively

unstable transition route and the absolutely unstable transition route, without

surface roughness elements, are made in paper 4.

In this study, dry transfer lettering by Letraset (Letraset Ref. 13045) was

used to create the roughness elements put on the disk surface. Each element

has a circular shape and the diameter is approximately 2 mm. A laser pointer

3Imayama, S., Alfredsson, P. H. & Lingwood, R. J. 2012 An experimental study of edge e↵ectson rotating-disk transition. J. Fluid Mech. 716, 638-657, figure 1 with kind permission from

Cambridge University Press.

3.1. EXPERIMENTAL SET-UP OF THE ROTATING-DISK SYSTEM 29

0 45 90 135 180 225 270 315 3603

4

5

6

7

8

! [°]

h* [

µm

]

(b)

1 8 9 1617

2425

32

Figure 3.6. (a) Roughness height measurement set-up. (b)

Measured heights, h⇤, of roughnesses. The numbers in the

figure indicate the numbering of the roughness elements.

was mounted on the steel beam of the apparatus to indicate where a rough-

ness element should be put on the glass surface. Once a roughness element

was put on the surface, the disk was rotated 11.25� manually, then the loca-

tion indicated by the laser pointer changes gives the next roughness location.

In this way, 32 roughness elements were put at r⇤ = 110 ± 0.5 mm, corre-

sponding to approximately R = 287 in the present study, at angular intervals

of 11.25 ± 0.4�; see figure 3.6(b) and figure 3.7(a). The individual roughness

heights were measured by a laser distance meter (opto NCDT 1700-10, which

has a resolution of 0.5 µm), as shown by the set-up in figure 3.6(a). Since the

laser meter was fixed in the laboratory frame, the roughness height was sam-

pled by rotating the glass disk manually with a sampling frequency of 625 Hz.

A relative di↵erence between the height average of each roughness element and

the neighbouring glass-disk level was evaluated as the height of the roughness

elements. Figure 3.6(b) shows the measured heights of each roughness element.

The averaged height of the 32 roughness elements was 5.4 µm and the nondi-

mensional height normalized by the characteristic length L⇤= (⌫⇤/⌦⇤

)

1/2is

0.014. After measurements were performed with 32 roughnesses, some of the

elements were carefully removed using acetone. Then measurements with 24,

30 3. EXPERIMENTAL METHODS

Figure 3.7. Top view of the glass-disk surface showing each

roughness elements configuration. The roughnesses were put

at r⇤ = 110±0.5 mm, corresponding to approximately R = 287

in this study. (a) 32 roughnesses [1-32]. (b) 24 roughnesses [9-

32]. (c) 16 roughnesses [17-32]. (d) 8 roughnesses [25-32]. (e)

1 roughness [32]. (f) 0 roughnesses (clean-disk condition).

3.2. MEASUREMENT TECHNIQUES 31

16, 8 and 1 roughness elements were taken. Figure 3.7 shows the top view of

the glass disk with each roughness configuration.

3.2. Measurement techniques

3.2.1. Hot-wire anemometry

Several methods to measure fluid velocity have been developed. Typical ex-

amples are Prandtl tube, Pitot tube, hot-wire anemometry, Particle Image Ve-

locimetry (PIV) and Laser Doppler Velocimetry (LDV). In the present study

hot-wire anemometry was used to measure fluid velocity due to the advantage

of high temporal and spatial resolutions. Hot-wire anemometry consists of a

hot-wire probe and an electronic circuit to operate the probe. The hot wire-

probe was operated by a constant temperature anemometer (CTA) and the

resistance of the sensor wire, made of platinum, depends on its temperature.

The wire temperature depends on the surrounding environment, e.g. fluid ve-

locity and fluid temperature. The CTA keeps the wire temperature constant

and gives an output voltage depending on the feedback amount. Thus, if the

fluid temperature was constant, the CTA-output voltage correlated to the fluid

velocity, via its cooling e↵ect, at the wire.

Figure 3.8. A hot-wire probe for laminar-turbulent transi-

tion measurements. (a) Bottom view of the probe. (b) Sensor

part of the probe indicated by red dashed square in (a). (c)

Side view of the probe. The measures in the figures are in

millimeters.

32 3. EXPERIMENTAL METHODS

In the present study, two in-house manufactured hot-wire probes were pre-

pared for the laminar-turbulent transition and turbulence measurements. The

one for the laminar-turbulent transition measurements had straight prongs with

a sensor made of platinum with a diameter of 5 µm and 1 mm length, see fig-

ure 3.8. The other one for the turbulent measurements was designed to capture

as small spatial and temporal turbulent scales as possible. The prongs were

bent and the sensor wire, made of platinum with a diameter of 1.3 µm and

0.3 mm length, was soldered at the tips, and figure 3.9 shows pictures of that

probe. These hot-wire probes were operated by a CTA system (DANTEC

StreamLine) with an overheat ratio (↵R

) of 0.8, defined as

↵R

=

R⇤(T ⇤

h

)�R⇤(T ⇤

ref

)

R⇤(T ⇤

ref

)

, (3.2)

where R⇤(T ⇤

h

) is the resistance of the sensor at the operating temperature of T ⇤h

and R⇤(T ⇤

ref

) is the resistance of the sensor at the reference of T ⇤ref

. The hot-

wire probe was mounted on the traverse system and the sensor wire was aligned

with the radial direction making it mainly sensitive to the azimuthal velocity

component. The hot-wire probe was carefully aligned to be parallel to the

Figure 3.9. Turbulent boundary-layer probe. (a) Bottom

view of the probe. (b) Sensor part of the probe indicated by

the red dashed square in (a). (c) Side view of the probe. The

measures in the figures are in millimeters.

3.2. MEASUREMENT TECHNIQUES 33

wall surface using a micro lens and a camera. To remove unphysical noise, e.g.

electronic noise, a low-pass filter with a cut-o↵ at 30 kHz (for laminar-turbulent

transition measurements) or 100 kHz (for turbulence measurements) is applied

to the CTA circuit. The output voltage from the CTA was digitalized using a

16-bit A/D converter (National Instruments USB-6216) at a specific sampling

rate and sampling time and saved to a hard-disk drive or solid-state drive by

the same computer used in traverse operation using LabVIEW8.6 software.

3.2.2. Hot-wire calibration

Here the hot-wire calibration methods for both laminar-turbulent transition

and turbulence measurements are described.

Calibration of hot-wire probes is usually conducted in the free stream of a

measurement or calibration wind tunnel with a reference velocity meter (e.g.

Prandtl tube). In the present study, however, that method was not applied,

and instead the azimuthal velocity profile of the laminar boundary layer of the

rotating-disk flow was used for the calibration. This method did not require a

free stream, which the rotating-disk flow does not have in the laboratory frame,

and it removed the risk of breaking the hot-wire probe during the movement

between a calibration wind tunnel and the traverse system of the rotating-disk

apparatus. However, to perform this calibration method the distance of the

hot-wire probe away from the disk surface must be known. The calibration

procedure for the flow over a rotating disk is explained below.

First of all, the wall-normal height of the hot wire was determined by taking

a picture with a precision gauge block with a thickness of 1.000 mm. Figure 3.10

shows a typical set-up for the hot-wire height determination. After the surface

of the disk was cleaned by an air spray and acetone, the gauge block was put

next to the hot-wire probe. The picture was taken from the front using a micro

lens (Nikon Micro-Nikkor AF 200mm f/4 D ED) and a camera (Canon EOS

7D) through a mirror inserted between its optical path. To get a sharp image,

the room light was turned o↵ and the flash light (Canon Speedlite 550EX) was

applied from the front during the shutter exposure. A typical photograph of

the wall-position determination is shown in figure 3.11. In this figure one pixel

of the image is equivalent to 2.4 µm. By this method, the probe distance from

the wall was determined with an accuracy of 10-15 µm. This error was caused

mainly by resolution of the micro lens and quality of the mirror in the path,

which slightly blurred the image.

34 3. EXPERIMENTAL METHODS

Figure 3.10. A typical hot-wire calibration set-up.

Figure 3.11. Photograph showing the hot-wire probe during

the wall-position determination using a precision gauge block

with 1.000 mm thickness. The upper half-plane shows the real

objects and the objects in the lower half-plane are reflections

in the glass surface.

3.2. MEASUREMENT TECHNIQUES 35

The output voltage from the CTA depends on the surrounding environment

of the hot-wire probe, e.g. temperature and fluid velocity. If the ambient

temperature changes during the measurements, then the hot-wire voltage also

varies and gives rise to an error in the fluid velocity measurements. In this

study, a maximum variation of 1

�C in the ambient temperature was observed.

The following equation (e.g. Bruun 1995) has been proposed to compensate

the output voltage from hot-wire anemometry for temperature variations:

E⇤2 �T ⇤ref

�= E⇤2

(T ⇤)

✓1�

T ⇤ � T ⇤ref

↵R

/↵el

◆�1

, (3.3)

where E⇤(T ⇤

ref

) is a corrected output voltage from the CTA, T ⇤ref

is a refer-

ence ambient temperature, namely the ambient temperature at the time of the

hot-wire calibration, E⇤(T ⇤

) is the output voltage in the measurement, T ⇤is

the time-dependent ambient temperature for the measurements and ↵el

is the

temperature coe�cient of resistivity, which is ↵el

= 0.0038K�1for platinum

(Bruun 1995).

3.2.3. Hot-wire calibration for laminar-turbulent transition measurements

After determination of the wall position of the hot-wire probe, the hot-wire

calibration was performed using the known azimuthal laminar velocity profile of

the rotating-disk boundary layer by changing rotational speed, radial position

and axial height. Figure 3.12 shows a typical example of a hot-wire calibration.

The solid line in figure 3.12 shows a by modified King’s law (for better accuracy

at low velocities, see Johansson & Alfredsson 1982) fit to the calibration data

points, given as

V ⇤= k1(E

⇤2 � E⇤20 )

1/n+ k2(E

⇤ � E⇤0 )

1/2, (3.4)

where E⇤and E⇤

0 are the mean anemometer output voltages at mean velocities

V ⇤and zero, respectively, and k1,2 and n are constants to be determined by

a linear least-square fit of the calibration data. Figure 3.13 shows the devia-

tions of the calibrated data points from the fitted equation (3.4). The devia-

tions are within ±1.5% except in the low-speed region (V ⇤ 0.5 m/s). The

contribution of the radial velocity component to the hot-wire measurements

depends on the rotational speed and the wall-normal position, see figure 2.3

in Imayama (2012). However, figure 3.13 indicates that the deviations of the

calibration data points from the fitting curve are small at di↵erent rotational

speeds and di↵erent heights so that the e↵ects were assumed to be negligible for

the laminar-turbulent transition measurements. The axial velocity component

was also negligible.

36 3. EXPERIMENTAL METHODS

1 1.2 1.4 1.6 1.8 20

5

10

15

20

E* [V]

V* [

m/s

]

Figure 3.12. Hot-wire calibration using the laminar velocity

profile varying the rotational speed, the radial position and the

normal height. The symbols indicate ⌦

⇤=0 rpm (?), 300 rpm

(�), 500 rpm (⇤), 600 rpm (⇧), 700 rpm (4), 770 rpm (O),860 rpm (/). The solid line shows the modified King’s law

fitting of the calibration data.

0 1 5 10 15 20!15

!10

!5

!1.50

1.5

5

V* [m/s]

V*/

V*

fit [

%]

Figure 3.13. Deviations of calibration data points from the

modified King’s law fitting (V ⇤fit

). The symbols are the same

as in figure 3.12.

3.2. MEASUREMENT TECHNIQUES 37

3.2.4. Hot-wire calibration for turbulent measurements

The hot-wire calibration for the turbulent boundary-layer flow also uses the az-

imuthal laminar velocity profile, changing the radial position and wall-normal

height of the hot-wire probe as well as rotational speeds. However, for the tur-

bulent boundary-layer measurements, the azimuthal velocity in the near-wall

region exceeded the calibrated velocity range. For this reason, the hot-wire

calibration procedure was modified in the following way. Using the measured

turbulent velocity profile near the wall, the profiles in voltage units were extrap-

olated to predict the wall voltage of the hot-wire probe. Since the wall speed

can be obtained from the disk rotational speed, one extra calibration point

can be added, see figure 3.14. The obtained data were fitted by a fourth-order

polynomial given as

V ⇤= a⇤0 + a⇤1E

⇤+ a⇤2E

⇤2+ a⇤3E

⇤3+ a⇤4E

⇤4, (3.5)

where V ⇤is a mean azimuthal velocity, E⇤

is a mean output voltage from the

anemometer and a⇤0 � a⇤4 are the coe�cients of the polynomial approximation.

Figure 3.14 shows the deviation of the calibration data points from the polyno-

mial fitting and it is less than ±1% except in the low-velocity region (V ⇤ < 0.5m/s).

3.2.5. Rotational speed of the disk

The disk rotational speed, ⌦

⇤, was measured using a photo-micro sensor (EE-

SX 498). A brass disk with 30 slits at regular intervals in the azimuthal direc-

tion was mounted underneath the iron disk, see figure 3.1(B, e). One of the slits

was taped over to di↵erentiate it from the others and to determine the disk’s

absolute angular position. A typical voltage output from the photo-sensor is

shown in figure 3.3(b). The photo-micro sensor measured the passing of the 29

rotational slits (except the taped slit) with a sampling rate of 80 MHz using a

sampling board (National Instruments USB-6216) and the obtained frequencies

were converted into the rotational speed. The sensor was able to measure the

disk rotational speed up to 3000 rpm, and in this study the highest rotational

speed was 1542 rpm. The measured disk speeds at various rotational speed are

shown in figure 3.15. It shows that the disk rotated within ±1.5 rpm in steady

rotational speed in this study.

3.2.6. Ambient temperature and pressure

The ambient temperature was measured using a platinum resistance thermome-

ter (PT100). The temperature sensor was mounted near the edge of the glass

disk. The resistance of the thermometer changed with ambient temperature.

The resistance was measured using a resistance meter (FLUKE45) and read

by a serial cable. The obtained resistance measurements were converted to

38 3. EXPERIMENTAL METHODS

0 0.02 0.04 0.06 0.08 0.10.8

0.82

0.84

0.86

0.88

z* [mm]

E* [

V]

(a)

0.6 0.7 0.8 0.90

5

10

15

20

25

30

E* [V]

V* [

m/s

]

(b)

01 5 10 15 20 25 30 35!5

!3

!1.5

0

1.5

3

5

V* [m/s]

V*/

V*

fit [

%]

(c)

Figure 3.14. The calibration of the turbulent boundary

layer probe. (a) CTA voltage outputs (E⇤) of the turbulent

boundary layer probe as a function of wall-normal height (z⇤)in the turbulent boundary layer. � is measured voltage. The

solid line is a linear fitting of the voltage outputs. ⇤ is the

estimated wall voltage to obtain high-speed calibration da-

tum. (b) Hot-wire calibration using the laminar profile and

estimated wall voltage. The symbols are the same as in fig-

ure 3.12 and ⇤ is the estimated wall voltage. The solid line is

the calibration curve given by a fourth-order polynomial fit-

ting. (c) Deviations of calibration data points from the fourth-

order polynomial fitting (V ⇤fit

). The symbols are the same as

in figure 3.12.

Reproduced from S. Imayama, R.J. Lingwood & P.H. Alfredsson. The turbulent

rotating-disk boundary layer. Eur. J. Mech. B/Fluids 2014; 48: 245–253. Copyright

©2014 Elsevier Masson SAS. All rights reserved.

3.2. MEASUREMENT TECHNIQUES 39

0 200 400 600 800 1000!5

!2.5

!1.5

0

1.5

2.5

5

Rotations

!"

* [

rpm

]

Figure 3.15. Deviations �⌦

⇤rpm of the rotational speeds

from the target rotational speeds in revolutions per minite.

Target rotational speeds are 400 rpm (Blue), 700 rpm (Green),

1000 rpm (Red), 1500 rpm (Black), respectively.

temperature. The accuracy of this sensor was checked using a high resolution

mercury thermometer with 0.01

�C steps, as shown in figure 3.16. The PT100

used in the present study had a deviation of ±0.15�C in the measurement

range. When the apparatus was moved into a di↵erent building due to a reno-

vation, a constant shift in the temperature compared with the high-resolution

mercury thermometer was observed. Thus, a temperature o↵set was applied

to the PT100.

The kinematic viscosity ⌫⇤ of the fluid is given as

⌫⇤ =

µ⇤

⇢⇤, (3.6)

where µ⇤is the viscosity of fluid and ⇢⇤ is the density. Here µ⇤

is calculated

using Sutherland law, which is written as

µ⇤=

1.4578⇥ 10

�6 ⇥ T ⇤3/2

T ⇤+ 110.4

. (3.7)

The density of dry air is calculated using the gas law, given as

⇢⇤ =

P ⇤atm

287.0⇥ T ⇤ , (3.8)

40 3. EXPERIMENTAL METHODS

19 19.5 20 20.5 21!0.3

!0.15

0

0.15

0.3

T* [°C]

T* !

TM

ar

* [

°C

]

Figure 3.16. The temperature di↵erence between calibrated

PT100 temperature (T ⇤) and a precision mercury thermome-

ter temperature (T ⇤Mar

) with 0.01�C steps.

where P ⇤atm

is the atmospheric pressure in Pascal and T ⇤is the absolute tem-

perature in Kelvin. The atmospheric pressure was measured using a precision

barometer.

CHAPTER 4

Main contributions and conclusions

This chapter summarizes the main contributions and conclusions from the pa-

pers constituting Part II of the thesis. For details on the results the reader is

referred to the appended papers.

Paper 1. A new way to describe the transition characteristics of

a rotating-disk boundary-layer flow

• To investigate laminar-turbulent transition of a rotating-disk boundary-

layer flow, a new way to visualize the process has been proposed using

the probability density function (PDF) of azimuthal fluctuating velocity.

The PDF map measured at z = 1.3 over a range of Reynolds numbers

from laminar to turbulent flow dramatically shows the change of the

distribution at R = 550 from exponential growth to a strongly skewed

distribution. This change in PDF corresponds to the change of slope in

vrms

, where vrms

is the disturbance amplitude of the azimuthal velocity

fluctuations. At around R = 600, the skewed PDF starts disappearing

and the positive deviation of azimuthal fluctuation velocity, v, has its

maximum. Above R = 650 the shape of the PDF seems to be symmetric

indicating that the flow has reached a fully-developed turbulent state.

These changes of the flow characteristics are not obvious in the spectral

distributions.

• The application of PDF maps to azimuthal fluctuation velocity-profile

measurements shows the structure normal to the wall. In particular, at

R = 570 peaks in the PDF may be associated with a secondary insta-

bility.

• Measured azimuthal mean velocity profiles are in good agreement with

the theoretical laminar profile at R < 510. The onset of the nonlinearity

has been observed at R = 510 in the spectrum of azimuthal fluctuation

velocity time series which is consistent with Lingwood’s (1995a) sugges-

tion that local absolute instability appears above RCA

= 507.3 and it

triggers the onset of transition (nonlinearity).

41

42 4. MAIN CONTRIBUTIONS AND CONCLUSIONS

• The growth of vrms

shows exponential growth up to around R = 580.

The slope of the exponential growth for 475 < R < 530 corresponds ap-

proximately to the maximum spatial growth rate for stationary linear

disturbances, see e.g. figure 6a in Hussain et al. (2011).

Paper 2. An experimental study of edge e↵ects on rotating-disk

transition

• Healey (2010) showed that, taking into account a finite radius of a disk,

the rotating-disk flow can be linearly globally unstable using the lin-

earized Ginzburg-Landau equation, which is in contrast with the results

of Davies & Carpenter (2003). Furthermore, adding a nonlinear term

into the equation, it was shown that there is a weak nonlinear stabiliza-

tion e↵ect as the edge Reynolds number, Redge

, approaches RCA

. He

compared his suggested stabilization e↵ect with previous experimental

results, which seemed to confirm his hypothesis. In the present study,

laminar-turbulent transition with three di↵erent edge conditions and

various edge Reynolds numbers has been investigated. However, no ob-

vious di↵erence has been observed in the present measurement range for

the di↵erent edge conditions and edge Reynolds numbers.

• High repeatability of the onset of nonlinearity has been observed for

di↵erent edge configurations, edge Reynolds numbers and even di↵erent

background noise levels due to di↵erent edge configurations. This result

corresponds to the observation by Lingwood (1996) that the transition

location of absolutely-unstable flows are likely to be less sensitive to

the precise nature of external disturbances. These results also support

the hypothesis of Lingwood (1995a) that the local absolute instability

triggers the onset of nonlinearity and transition. Furthermore, this is

also in agreement with the suggestion by Healey (2010) that the finite

disk leads the local absolute instability to a (supercritical) linear global

mode, and then to a nonlinear steep-fronted global mode.

• It was found that the variations of transition Reynolds number provided

by previous experimentalists for clean-disk flows and used by Healey

(2010) to support his nonlinear stabilization hypothesis were in fact due

to the di↵erent definitions used. By, as far as possible, applying the

same definition to determine the onset of nonlinearity, the scatter of the

transition Reynolds number was reduced more than recognized previ-

ously. Some early experimental studies are categorized as rough-disk

cases that are likely to favour a convective route to transition and in

these cases the onset of nonlinearity appears at much lower R than RCA

,

4. MAIN CONTRIBUTIONS AND CONCLUSIONS 43

presumably due to the growth of convective instabilities.

Paper 3. On the laminar-turbulent transition of the rotating-

disk flow: the role of absolute instability

• Further studies have been performed to investigate the laminar-turbulent

transition of the rotating-disk flow. The surface of the disk is smooth

enough so that the disk is categorized as a clean disk. The azimuthal

fluctuation velocity time series and the vrms

have been decomposed

to stationary and unsteady components. The growth rate of the dis-

turbance amplitude in both components consistently decreases beyond

R ⇡ 507 with the onset of the nonlinearities.

• Spectra of instantaneous azimuthal fluctuation velocity time series cap-

ture two distinct characteristics; one has spiky peaks located at inte-

ger values of the normalized frequencies and the other one has smooth

peaks. The spiky peaks are shown to be due to stationary disturbances

and smooth peaks are attributed to travelling disturbances.

• The emergence of travelling disturbances and their growth have been

observed at !⇤/⌦⇤ ⇡ 40, where !⇤/⌦⇤is the normalized frequency,

are clearly observed. Although the absolute/global frequency predicted

from theoretical analyses is around !⇤/⌦⇤= 50.3, it is possible that the

experimentally observed travelling disturbances could correspond to the

global mode realized in the physical flow, which is more complex than

the base flow in the theoretical analysis.

• The growth of individual stationary vortices has been investigated. It

is shown that the amplitude of each individual vortex grows exponen-

tially and the breakdown locations at R = 570�580 are well determined

by the Reynolds number rather than their individual amplitude. The

change of wave angle of stationary vortices is observed at R = 540 in-

dependent from their amplitude distributions.

• Travelling secondary instabilities characterized as kinked azimuthal fluc-

tuation velocity are captured in the unsteady time series at R = 570.

The kinked azimuthal fluctuation velocity is not clear at z = 1.3 how-

ever this feature appears above and below this wall-normal height.

• Power spectra show the first harmonics at around R = 510 indicating

the onset of nonlinearity and the harmonics of the primary instabili-

ties grow with increasing Reynolds number. At R = 565 � 590 the

44 4. MAIN CONTRIBUTIONS AND CONCLUSIONS

power spectra captured a jump in disturbance energy at high frequen-

cies. These results correspond to the observations by Viaud et al. (2011)

that showed transition to turbulence through a steep-nonlinear global

mode with a secondary global mode leading to turbulence in direct nu-

merical simulations of an open rotating-cavity flow.

Paper 4. Experimental study of the rotating-disk boundary-

layer flow with surface roughness

• The transition to turbulence of the rotating-disk flow with surface rough-

nesses has been investigated to explore the possibility of a convectively-

unstable route, i.e. the so-called rough-disk condition. 32 roughnesses

were positioned at equal azimuthal spacing on the disk surface at R =

287. The average roughness height was 5.4 µm. The transition to tur-

bulence with 32 roughnesses occurred at earlier Reynolds number than

for a clean disk such that the transition process proceeds before the

flow becomes absolutely unstable. The disturbance profile of the ex-

cited stationary disturbances is in good agreement with the correspond-

ing theoretical eigenfunction. Removing some roughnesses, results in a

contour map of ensemble-averaged azimuthal fluctuation velocity time

series, showing the amplitude development of stationary vortices and

clearly demonstrating the convective nature of the stationary vortices.

• Comparisons of spectra for di↵erent roughness configurations (32, 8 and

0 roughnesses) have been performed. At lower Reynolds numbers, spec-

tra of both instantaneous and ensemble-averaged time series show spiky

peaks at around !⇤/⌦⇤= 30 due to the growth of stationary vortices

excited by surface roughnesses. Even for a clean disk, unavoidable sur-

face roughnesses excite the stationary disturbances in the flow field.

With increasing Reynolds number, spectra from three di↵erent rough-

ness configurations show harmonic peaks indicating the onset of nonlin-

earity. The 32- and 8-roughnesses cases show only spiky peaks in the

spectra implying that the nonlinearities are triggered by the growth of

stationary vortices. However, for the clean disk at R = 510 where the

onset of nonlinearity is shown, the spectra show peaks due to station-

ary disturbances but also due to travelling disturbances as also reported

in Imayama et al. (2014a) and both types grow beyond that Reynolds

number.

• To characterize the observed travelling disturbances at R = 510, band-

pass filtering is applied to the instantaneous time series at the frequency

range for travelling disturbances to extract the disturbance amplitudes,

and the disturbance profile is calculated. The stationary disturbance

4. MAIN CONTRIBUTIONS AND CONCLUSIONS 45

profile is also evaluated from ensemble-averaged time series within the

stationary disturbances’ frequency range. These disturbance profiles

are compared with eigenfunctions from the local stability analysis. It

is found that the travelling disturbance profile is in excellent agreement

with the eigenfunction of local absolute instability at the critical Rey-

nolds number, which is distinct from the stationary disturbance profile.

The stationary disturbance profile is also in good agreement with the

eigenfunction for the stationary mode.

• The results from the clean-disk condition are briefly compared to the-

oretical studies of global behaviour in spatially-developing flows. It is

found that the observed emergence of travelling disturbances and the

onset of nonlinearity at the boundary between convective and absolute

instabilities agrees qualitatively with the theories on front dynamics.

• The development of each stationary disturbance has been investigated

for the 32 roughness cases. In contrast to the stationary disturbances for

the clean disk as shown in Imayama et al. (2014a), all of the stationary-

vortex amplitudes reach the same amplitude at the nonlinear saturation

region and break down showing that the transition behaviour is convec-

tively unstable due to the growth of stationary vortices. The angles

of each stationary vortex are evaluated and in good agreement with

theoretical prediction by the local stability analysis in the linear region.

Paper 5. Linear disturbances in the rotating-disk flow: a

comparison between results from simulations, experiments

and theory

• Comparisons of local linear stability analysis, experiments and simu-

lations are performed for Type-I convectively stationary disturbances.

The local linear stability analysis is performed using a shooting method

and Chebyshev polynomial method and they are found to be in good

agreement with each other. Linear direct numerical simulation (DNS)

is performed with some di↵erent surface roughness distribution cases.

Furthermore, experiments for two di↵erent roughness configurations are

presented.

• The comparisons between the three di↵erent approaches are performed

in the linear region at a fixed azimuthal wavenumber. It is found that

the eigenfunction, growth rate and angle distributions are in good agree-

ment with each other.

46 4. MAIN CONTRIBUTIONS AND CONCLUSIONS

• Comparisons between random and fixed roughness distributions are

made. The di↵erences between the two distributions in growth rate

and numbers of stationary vortices are discussed.

Paper 6. The turbulent rotating-disk boundary layer

• The turbulent boundary layer on the rotating disk has been investigated.

The main purpose of this study is to provide experimental data due to a

lack of previous studies and to compare with two-dimensional boundary-

layer flow. The azimuthal velocity profile measurement was performed

at two di↵erent Reynolds number using a single hot-wire probe with the

length of 0.3 mm and the diameter of 1.3 µm.

• A new calibration technique for the hot-wire probe over the rotating

disk was developed to provide a higher velocity calibration point in

addition to calibration points using the laminar velocity profile. The

voltage outputs from the constant temperature anemometry (CTA) of

the near-wall region of the turbulent boundary layer profile are fitted

and extrapolated to the wall and the output voltage at the wall is eval-

uated. Since the disk wall speed is measured by an optical tachometer,

it is possible to provide this value as a velocity calibration point.

• The azimuthal wall shear stress was evaluated. For hot-wires used in

the rotating-disk boundary-layer flow, the heat conduction to the wall

from the probe becomes relatively smaller than heat convection so that

it gives possibility to measure the velocity distribution in the viscous

sublayer. The friction velocity is then directly determined from the ve-

locity gradient. This advantage also makes accurate turbulent statistics

in the near-wall region possible. Furthermore, using the similarity of

the cumulative distribution function (CDF) of the velocity in the near-

wall region, the validation of the above velocity calibration method, a

determination of the wall position of the hot-wire probe and evaluation

of heat-transfer e↵ects have been performed.

• The shape factor of the rotating-disk turbulent boundary-layer flow is

smaller than the corresponding two-dimensional turbulent boundary-

layer flow.

• The turbulence statistics close to the wall and in the logarithmic region

of rotating-disk boundary-layer flow are in good agreement with that of

the two-dimensional turbulent boundary-layer flow.

4. MAIN CONTRIBUTIONS AND CONCLUSIONS 47

• The pre-multiplied spectral maps of the rotating-disk turbulent boundary-

layer flow and the two-dimensional turbulent boundary-layer flow are

di↵erent. In particular, in the near-wall region, the maximum energy

content is found at smaller length scales for the rotating-disk case.

CHAPTER 5

Papers and the author’s contributions

Paper 1

A new way to describe the transition characteristics of a rotating-disk boundary-

layer flow

Shintaro Imayama (SI), P. Henrik Alfredsson (HAL) & R. J. Lingwood (RL).

Phys. Fluids 24, 031701.

The laminar-turbulent transition of the rotating-disk flow has been investi-

gated. The original apparatus was borrowed from the University of Cambridge

Department of Engineering and was modified and put into operation by SI.

The experimental investigations were performed by SI under the supervision of

HAL and RL, and the writing was jointly done by SI, HAL and RL.

Parts of these results have been presented at EUROMECH Colloquium

525 Instabilities and transition in three-dimensional flows with rotation, 21 –

23 June 2011, Lyon, France.

Paper 2

An experimental study of edge e↵ects on rotating-disk transition

Shintaro Imayama (SI), P. Henrik Alfredsson (HAL) & R. J. Lingwood (RL).

J. Fluid Mech. 716, 638–657.

The e↵ects of the finite radius of the disk on the laminar-turbulent transition of

the rotating-disk flow have been investigated experimentally. The experimen-

tal investigations were performed by SI using the same facility used in Paper

1 under supervision of HAL and RL, and the writing was jointly done by SI,

HAL and RL.

Some of these results have been presented at the Annual Meeting of the

American Physical Society’s Division of Fluid Dynamics, 20 – 22 November

2011, Baltimore, Maryland, USA.

48

5. PAPERS AND THE AUTHOR’S CONTRIBUTIONS 49

Paper 3

On the laminar-turbulent transition of the rotating-disk flow: the role of abso-

lute instability

Shintaro Imayama (SI), P. Henrik Alfredsson (HAL) & R. J. Lingwood (RL).

J. Fluid Mech. 745, 132–163.

The mechanism of absolute instability in laminar-turbulent transition of the

rotating-disk flow has been investigated. The experimental investigations were

performed by SI using the same facility used in Paper 1 under supervision of

HAL and RL, and writing was jointly done by SI, HAL and RL.

Some of these results have been presented at Svenska Mekanikdagar, 12 –

14 June 2013, Lund, Sweden and 10th ERCOFTAC SIG 33 Workshop, 29 – 31

May 2013, Sandhamn, Sweden.

Paper 4

Experimental study of the rotating-disk boundary-layer flow with surface rough-

ness Shintaro Imayama (SI), P. Henrik Alfredsson (HAL) & R. J. Lingwood

(RL).

The laminar-turbulent transition of the rotating-disk flow with surface rough-

nesses has been investigated and compared with the case without roughnesses.

The experiments were performed by SI using the same facility used in Paper 1

under the supervision of RL and HAL, and the writing was jointly done by SI,

RL and HAL.

Some of these results have been presented at the First Madingley Work-

shop in Fluid Mechanics: an interdisciplinary approach, 23 – 25 July 2014,

Cambridge, UK and at the Annual Meeting of the American Physical Society’s

Division of Fluid Dynamics, 23 - 25 November 2014, San Francisco, California,

USA.

50 5. PAPERS AND THE AUTHOR’S CONTRIBUTIONS

Paper 5

A comparison between the simulated, experimental and theoretical rotating-disk

boundary-layer flow Ellinor Appelquist (EA), Shintaro Imayama (SI), P. Hen-

rik Alfredsson (HAL), Philipp Schlatter (PS) & R. J. Lingwood (RL).

Local linear stability analysis has been compared with experiments and simu-

lations. The experiments were performed by SI using the same facility used in

Paper 1 under the supervision of RL and HAL, and the numerical simulations

were performed by EA under the supervision of RL, HAL and PS. The writing

was jointly done by EA, SI, HAL, PS and RL.

Paper 6

The turbulent rotating-disk boundary layer Shintaro Imayama (SI), R. J. Ling-

wood (RL) & P. Henrik Alfredsson (HAL). Euro. J. Mech. B/Fluids 48,

245–253.

The turbulent boundary layer on the rotating disk flow has been investigated.

The azimuthal friction velocity was determined using hot-wire measurement

directly and turbulence statistics and spectra normalized by the inner scales

are presented. The experiments were performed by SI using the same facility

used in Paper 1 under the supervision of RL and HAL, and the writing was

jointly done by SI, RL and HAL.

Some of these results have been presented at EUROMECH Colloquium

525 Instabilities and transition in three-dimensional flows with rotation, 21 –

23 June 2011, Lyon, France. Furthermore, some of these results have been

presented and also published as a proceeding in Progress in Turbulence V,

Springer Proceedings in Physics, Proceedings of the iTi Conference in Turbu-

lence 2012, 149, 173–176. Parts of the results are also published in Alfredsson

et al. (2013).

51

Acknowledgements

This study has been supported by the Swedish Research Council (VR)

and Linne FLOW Centre at KTH. The University of Cambridge Department

of Engineering provided the main rotating apparatus on long-term loan. The

Marcus Wallenberg Laboratory for sound and vibration research (MWL), De-

partment of Aeronautical and Vehicle Engineering at KTH provided support

through use of its flow instruments (constant temperature anemometry (CTA)

and vibration sensors).

I think it is impossible to describe everything here to thank people who

helped me during my doctoral work. Without support from them, I could not

have succeeded with my doctoral work and life in Sweden. Therefore, even

if you do not find your name in my acknowledgements, you should not be

disappointed. I really appreciate the support of everyone I have met during

this period.

First of all, I would like to thank my main supervisor Prof. Henrik Alfreds-

son. I met you for the first time when I was visiting KTH for the Jamboree

project and then you accepted me to become a doctoral student at KTH. I was

able to do my research in a very nice environment and atmosphere during this

doctoral work. You also taught me how a researcher should be in many di↵er-

ent aspects. Therefore, this five years of life at KTH was extremely worthwhile

and makes a huge impact on my future.

Many thanks also to my co-supervisor Prof. Rebecca Lingwood not only

for discussions of instabilities of a rotating-disk boundary-layer flow but also

showing me how to be a scientist. In particular, I am really grateful to her

for teaching me how a paper should be written and how to discuss with other

researchers.

Many thanks to Ellinor Appelquist for great collaborations on a rotating-

disk boundary-layer flow. It was really enjoyable to discuss the rotating-disk

flow both experimentally and numerically, which led to a much greater under-

standing of the mechanisms. And also thanks to Assoc. Prof. Philipp Schlatter

for the collaborations from numerical point of view.

52

ACKNOWLEDGEMENTS 53

Thanks also to Dr Ramis

¨

Orlu for always helping me and providing a lot

of valuable advice.

I very much appreciate the help of Dr Antonio Segalini gave me to develop a

code for conducting linear stability analysis of the rotating-disk boundary-layer

flow.

I also thank the late Dr Tim Nickels who made the internal arrangements

for the loan of the rotating apparatus from the University of Cambridge De-

partment of Engineering to KTH. And many thanks to Dr Nils Tillmark and

Joakim Karlstrom who packed and transferred the apparatus to Stockholm be-

fore I came to KTH as a doctoral student. This enabled me to start assembling

and modifying immediately when I started my doctoral work. It was very nice

start for me! Furthermore, as great technicians, I appreciate Joakim Karlstrom

(again), Goran Radberg, Rune Lindfors and Jonas Vikstrom for making and

assembling the rotating-disk apparatus. Without their supports, the appara-

tus would never have worked properly. I also thank Dr Markus Pastuho↵ for

teaching me how to make an electric circuit board for the rotating apparatus

and technical advice about electric noise.

During this doctoral work, I was fortunate to have many chances to visit

institutes and other universities’ laboratories, which were great experiences for

me. Here, I would like to thank the following for kind arrangements: Prof.

Bjorn Hof at the Institute of Science and Technology Austria (for visiting the

Max Planck Institute); Dr Junji Shinjo at JAXA Chofu Aerospace Center; Dr

Stefan Hein at DLR Gottingen; and Dr David Ashpis at NASA Glenn Research

Center. I also thank the following for arranging visits to their university lab-

oratories: Prof. Yu Fukunishi and Dr Yu Nishio at Tohoku Univeristy; Prof.

Masahito Asai at Tokyo Metropolitan University; Prof. Koji Fukagata at Keio

University; and Prof. Thomas Corke at the University of Notre Dame. And

also special thanks to my former supervisor, Prof. Yoshiyuki Tsuji, for a lot

of support and advice during my doctoral work and for arranging a visit to a

laboratory in Nagoya University.

I am very grateful to people in the Department of Mechanics, KTH. They

provided a very nice atmosphere and I was able to study with a lot of fun.

Thanks to Julie for sharing an o�ce with me for such a long time. I appreciate

you sharing your time not only in the o�ce but also in the RECEPT project and

in the USA. I also thank people who went for lunch and discussed many things

with me, especially, Mattias, Karl, Tomas, Takuya and Yukinori. Thanks

also to everybody else in the fluid physics laboratory, especially: Fredrik(L),

Jens, Daniel, Bengt, Olle, Malte, Fredrik(H), Thomas, Shahab, Johan, Ylva,

Sissy, Andreas, Marco, Sohrab, Renzo, Bertrand, Elias, Lim, Martin, Robert,

Sembian, Ramin, Nicolas, Jordan, Marcus and Alexandre. I would also like

to thank people at OB18, especially: Alexandra, Emma, Carolina, Malin and

Lailai. I am also grateful to visiting researchers at the Department who shared

54 ACKNOWLEDGEMENTS

fun time with me, especially: Yusuke, Yoshiyuki, Shusaku, Masato, Makoto

and Tetsuya.

I thank many friends I have met during life in Sweden who have supported

me outside of the research world and who have shared fun time, drinking, trav-

eling and many experiences with me. Especially, I would like to acknowledge

my friends in the KTH language cafe, in the Karolinska Institute and in Touhou

Sweden. You have made my life more meaningful, thanks a lot!!

Finally, I appreciate my sister, grandparents and, in particular, my parents

for supporting me and permitting me to choose my own path. Thanks to you,

I was able to see a lot of di↵erent worlds and have great experiences and now

I can convincingly go back to Japan.

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Part II

Papers