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Rotating Rayleigh-Bénard convection

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Rotating Rayleigh–Bénard Convection

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de TechnischeUniversiteit Eindhoven, op gezag van de rector magnificus

prof.dr.ir. F.P.T. Baaijens, voor een commissie aangewezen doorhet College voor Promoties, in het openbaar te verdedigen op

dinsdag 19 december 2017 om 16.00 uur

door

Hadi Rajaei

geboren te Torbateheydarieh, Iran

Dit proefschrift is goedgekeurd door de promotoren en de samenstelling van de promotiecom-missie is als volgt:

voorzitter: prof. dr. ir. G. M. W. Kroesen1e promotor: prof. dr. H. J. H. Clercx2e promotor: prof. dr. F. Toschicopromotor: dr. ir. R. P. J. Kunnenleden: prof. dr. D. Lohse (UT)

prof. K.-Q. Xia (CUHK)prof. dr. ir. A. A. van Steenhovenprof. dr. ir. B. J. Geurts (UT)

Het onderzoek of ontwerp dat in dit proefschrift wordt beschreven is uitgevoerd in overeen-stemming met de TU/e Gedragscode Wetenschapsbeoefening.

Dedicated to my whole life, my beloved wife, Maryam.

This work is part of the research programme ofthe Foundation for Fundamental Research on Matter(FOM), which is part of the Netherlands Organisationfor Scientific Research (NWO).

Copyright © 2017 by Hadi RajaeiCover design by Hadi Rajaei

Cover photo by Hadi Rajaei,

All rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem or transmitted in any form or by any means, electronic, mechanical, photocopying,recording or otherwise, without prior permission of the author.

ISBN 978-90-386-4402-8NUR 926

Printed by Gildeprint - Enschede

List of Abbreviations

Abbreviation Description

BL Boundary LayerCCD Charge-Coupled DeviceCTCs Convective Taylor ColumnsDNS Direct Numerical SimulationFFT Fast Fourier TransformGL Grossmann–LohseHPC High Particle ConcentrationHIT Homogeneous and Isotropic TurbulenceLSC Large Scale CirculationLEDs Light-Emitting DiodesLPC Low Particle ConcentrationPIV Particle Image VelocimetryPDFs Probability Distribution FunctionsRBC Rayleigh–Bénard ConvectionRRBC Rotating Rayleigh–Bénard ConvectionSF Structure Function3D–PTV Three-dimensional Particle Tracking Velocimetry

i

Contents

List of Abbreviations i

1 Introduction and theoretical background 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Geostrophic balance and thermal wind . . . . . . . . . . . . . . . . . . 61.4 Ekman boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Useful definitions in convective turbulence . . . . . . . . . . . . . . . . 81.6 Flow phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . 91.7 Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.8 Guide through this thesis . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Methods and Parameters 152.1 Experimental set-ups . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.1 Convection cell . . . . . . . . . . . . . . . . . . . . . . . . . 152.1.2 Three-dimensional Particle Tracking Velocimetry . . . . . . . . . 172.1.3 Time-resolved particle image velocimetry system . . . . . . . . . 28

2.2 Experimental procedure and flow characteristics . . . . . . . . . . . . . 292.3 Numerical set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 Transitions in turbulent rotating convention 333.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Methods and Parameters . . . . . . . . . . . . . . . . . . . . . . . . 333.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 343.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Flow anisotropy in rotating buoyancy-driven 394.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Experimental parameters . . . . . . . . . . . . . . . . . . . . . . . . 40

iii

iv CONTENTS

4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.3.1 Large-scale (an)isotropy . . . . . . . . . . . . . . . . . . . . . 414.3.2 Small-scale (an)isotropy . . . . . . . . . . . . . . . . . . . . . 44

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5 Geometry of tracer trajectories 515.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2 Experimental parameters . . . . . . . . . . . . . . . . . . . . . . . . 525.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.3.1 Nonrotating RBC . . . . . . . . . . . . . . . . . . . . . . . . 525.3.2 Rotating RBC . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6 Velocity and acceleration statistics 596.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.2 Experimental parameters . . . . . . . . . . . . . . . . . . . . . . . . 606.3 Lagrangian rms velocity . . . . . . . . . . . . . . . . . . . . . . . . . 616.4 Lagrangian velocity autocorrelation . . . . . . . . . . . . . . . . . . . 61

6.4.1 Small Ro: emergence of two time scales . . . . . . . . . . . . . 656.4.2 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . 70

6.5 Lagrangian rms acceleration . . . . . . . . . . . . . . . . . . . . . . . 706.6 Lagrangian acceleration autocorrelation . . . . . . . . . . . . . . . . . 72

6.6.1 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . 786.7 Oscillatory behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 786.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

7 Exploring the Geostrophic regime 857.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857.2 Experimental techniques and parameters . . . . . . . . . . . . . . . . . 88

7.2.1 Experimental parameters . . . . . . . . . . . . . . . . . . . . 887.2.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . 887.2.3 Data validation . . . . . . . . . . . . . . . . . . . . . . . . . 88

7.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 907.3.1 Spatial vorticity autocorrelation at z = 0.8H . . . . . . . . . . 917.3.2 Flow coherence along the rotation axis . . . . . . . . . . . . . . 96

7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

8 Concluding remarks 1018.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1018.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

CONTENTS v

A Temporal vorticity autocorrelation 105

Bibliography 107

Summary 121

Samenvatting 123

Curriculum Vitae 127

List of Publications 129

Acknowledgments 131

Chapter 1

Introduction and theoreticalbackground

1.1 Introduction

Thermally driven turbulent convection has been initially studied by Henri Bénard and LordRayleigh more than a century ago. In 1900, Henri Bénard observed that the flow motionis organized into a regular pattern of hexagonal cells when a thin layer of fluid is heatedfrom below [8, 9, 10]. A few years later, Lord Rayleigh published his work on the theor-etical analysis of the convective instability of a layer of fluid [135]. In honor of Bénard’sand Rayleigh’s pioneering works, the term Rayleigh–Bénard convection is designated to thefluid motion heated from below and cooled from above. Rayleigh–Bénard convection is ofinterest to theoretical, computational and experimental physicist since it is mathematicallywell-defined and its experimental realization is comparatively straightforward.

In this thesis classical Rayleigh–Bénard convection (RBC) is subjected to backgroundrotation about its vertical axis. Buoyancy-driven flows affected by background rotation areomnipresent in nature and technological applications. These flows can be classified intothree important categories; geophysical flows, astrophysical flows and flows in technologicalapplications.

Large-scale flows in Earth’s oceans and atmosphere are primarily driven by temperature-induced buoyancy. These flows typically have such enormous length scales that Earth’s ro-tation affects them. The combined effects of buoyancy and rotation lead to the formationof the so-called Hadley cells in the atmosphere [53]. The generated winds by Hadley cellsnear the surface of the Earth are known as “trade-winds”. At higher latitudes, mid-latitudecells (Ferrel cells) and polar cells drive the large-scale dynamics of the atmosphere, see Fig-ure 1.1(a). In addition to the atmospheric large-scale motions which are driven by combinedeffects of buoyancy and rotation, flow in the Earth’s outer core [20, 44, 65, 139] and ocean[103, 40, 180] dynamics are also governed by the combined effects of buoyancy and rotation.

Similar to Earth’s atmospheric zonal flows, such large-scale zonal flows have also beenobserved on other planets like Jupiter, Saturn, Uranus and Neptune [58, 63], although it is notknown whether these flows are solely driven by convection. Rotating convection also occurs

1

2 CHAPTER 1. INTRODUCTION AND THEORETICAL BACKGROUND

in the outer layer of the Sun [105, 21], the interior of some planets [17, 19], the liquid metalcores of terrestrial planets, and rapidly rotating stars, e.g. see Figure 1.1(b-c).

The interplay between rotation and convection also plays an important role in technolo-gical applications. For example, in convective cooling in rotating turbomachinery blades bothconvection and rotation play important roles [35, 64]. A second example is chemical vapourdeposition on rotating heated substrates [169]: considerable convective heat transfer due tothe heated substrate, combined with rotation. Another example includes efficient separationof carbon dioxide (CO2) from methane or nitrogen gas. In this process, the gaseous mixtureis placed inside a centrifuge: a cylinder rotating at high speed. In a pressurized centrifuge,the enormous centrifugal forces near the cylindrical walls result in condensation of the CO2

into droplets (due to the radial compression) [170]. The condensation of droplets producessignificant heat while the flow is also strongly affected by rotation.

Figure 1.1: (a) Large-scale motions induced by buoyancy and background rotation in theEarth’s atmosphere (Credit: seas.harvard.edu), (b) zonal flows in the atmosphere of Jupiter(Credit: nasa.gov), (c) a cross section of the Sun presenting different regions including theconvective zone in the outer layer of the Sun (Credit: solarscience.msfc.nasa.gov).

The wide range of applicability of rotating thermal convection forms the reason whyit attracts so much attention and has been extensively studied by laboratory experiments

1.2. EQUATIONS OF MOTION 3

e.g. Refs. [136, 12, 193, 91, 138, 177, 79, 194, 115, 82, 73, 182, 81, 36], numerical sim-ulations e.g. Refs. [154, 73, 61, 153, 69, 77, 142, 70] and theoretical analysis e.g. Refs.[72, 22, 28, 23, 34, 176]. The most recent review paper on rotating RBC (RRBC) is Ref.[155]. However, up to now, the effects of background rotation on RBC are mainly char-acterized by global parameters like the overall heat transfer. In this thesis, however, wecharacterize RRBC by examination of the flow field. A combined experimental-numericalapproach is employed to study the Lagrangian and Eulerian statistics of neutrally buoyantimmersed tracers in RRBC. The results are further introduced in Section 1.8. First, a generaltheoretical background is presented to familiarize the reader with the basic concepts involvedin this thesis.

1.2 Equations of motion

Rayleigh–Bénard convection is typically studied in the Oberbeck-Boussinesq approximation[119, 14]. In this approximation, the fluid properties (e.g. viscosity, thermal expansion coeffi-cient, thermal diffusivity) are independent of the temperature and the fluid density is assumedto be linearly dependent on the temperature,

ρ(T ) = ρ(T0) (1− β (T − T0)) , (1.1)

where ρ is the fluid density, T is the fluid temperature, T0 is some reference temperatureand β is the fluid thermal expansion coefficient. These assumptions are reasonably valid forsmall temperature differences (β∆T . 0.2 [25, 116]). Keeping this in mind, we can writethe equations for conservation of mass, momentum and energy for a Newtonian fluid as [23]

∇ · u′ = 0,

∂u′

∂t+ u′ ·∇u′ = −∇p+ ν∇2u′ + βgT z,

∂T

∂t+ u′ ·∇T = κ∇2T,

(1.2)

where u′ is the velocity, t is time, p is the pressure (mean density is incorporated in thepressure term), ν is the kinematic viscosity of the fluid, g is the gravitational acceleration, Tis the temperature relative to a reference temperature, κ is the thermal diffusivity of the fluidand z is the vertical unit vector.

Equations (1.2) are valid for an inertial frame of reference. However, we are interested inthese equations in the rotating frame. The transformation from an inertial to a rotating frameof reference can be found in fluid mechanics textbooks [46, 75]. Let us assume that u′(r′)and u(r) are the velocity fields in the inertial and non-inertial (rotating) frames, respectively.r′ = (x′, y′, z′) and r = (x, y, z) are the corresponding position vectors in those frames.Thus, for the velocity one can write

dr′

dt=dr

dt+ Ω× r, (1.3)

where Ω is a constant rotation vector. See also Figure 1.2 for clarification.


r

Ω×r

Ω

θ

Figure 1.2: Definition sketch of motion in a rotating reference system.

Following the same approach as for velocity, the derivative of velocity with respect totime (i.e. acceleration) is expressed as

d2r′

dt2=

d

dt

(dr

dt+ Ω× r

)+ Ω×

(dr

dt+ Ω× r

)=d2r

dt2+ 2Ω× dr

dt+ Ω×Ω× r,

(1.4)

or in other words we have

a′ = a + 2Ω× u + Ω×Ω× r, (1.5)

where a′ and a are the accelerations in the inertial and rotating frames, respectively. Thesecond and third terms on the right hand side of Equations (1.4) and (1.5) represent the Cori-olis and centrifugal accelerations, respectively. The centrifugal acceleration can be rewrittenas a gradient

Ω×Ω× r = −∇(

1

2(Ω× r) · (Ω× r)

)= −∇

(1

2Ω2r2

⊥

), (1.6)

where r⊥ = |r| sin θ is the distance to the rotation axis (see Figure 1.2) and Ω = |Ω|. There-fore, the centrifugal acceleration can be incorporated into the pressure term in the Oberbeck-Boussinesq equations (Equations (1.2)); ∇P = ∇

(p− 1

2Ω2r2⊥). Thus, the momentum

equation in the rotating frame yields

∂u

∂t+ u ·∇u + 2Ω× u = −∇P + ν∇2u + βgT z. (1.7)

1.2. EQUATIONS OF MOTION 5

We only consider a convection cell rotating about its vertical axis, thus for the remainder ofthis thesis the rotation vector is Ω = Ωz. In order to obtain the important dimensionlessnumbers, the Oberbeck-Boussinesq equations in a rotating frame of reference are nondimen-sionalized by defining the following variables; x = x/H (dimensionless position), u = u/U(dimensionless velocity), t = tU/H (dimensionless time), T = T/∆T (dimensionless tem-perature) and P = P/U2 (dimensionless pressure). Tildes indicate dimensionless variableshere and H,U, and ∆T are the typical length scale (the separation of heating and coolingplates), the typical velocity scale, and the temperature difference between bottom and topplates, respectively. Therefore, the dimensionless equations can be rewritten as

∇ · u = 0,

∂u

∂t+ u · ∇u +

2ΩH

Uz× u = −∇P +

ν

UH∇2u +

gβ∆TH

U2T z,

∂T

∂t+ u · ∇T =

κ

UH∇2T .

(1.8)

In thermal convection, the typical velocity is usually expressed as U =√gβ∆TH , the so-

called free-fall velocity [128]: the maximum buoyancy-generated velocity. Three differentdimensionless numbers can be defined, namely the Rayleigh number, the Prandtl number andthe Rossby number,

Ra =gβ∆TH3

νκ, (1.9)

Pr =ν

κ, (1.10)

Ro =U

2ΩH. (1.11)

In these equations, the Rayleigh number indicates the strength of thermal forcing and is ameasure of the ratio of buoyancy and dissipation, the Prandtl number describes the relativeimportance of momentum diffusivity and thermal diffusivity, and the Rossby number is theratio of the inertial and Coriolis forces. Considering that

ν

UH=

√Pr

Raand

κ

UH=

1√PrRa

,

the Equations (1.8) can be rewritten as

∇ · u = 0,

∂u

∂t+ u · ∇u +

1

Roz× u = −∇P +

√Pr

Ra∇2u + T z,

∂T

∂t+ u · ∇T =

1√PrRa

∇2T .

(1.12)

Apart from the aforementioned dimensionless numbers (Ra,Pr,Ro), one can also introducethe Nusselt number, the Ekman number, the Taylor number, the Froude number and the cell


aspect ratio, defined as

Nu =qH

k∆T, (1.13)

Ek =ν

2ΩH2, (1.14)

Ta =

(2ΩH2

ν

)2

, (1.15)

Fr =Ω2R

g, (1.16)

Γ =D

H, (1.17)

where q is the mean heat-current density, k is the thermal conductivity, R is the cell radiusand D is the cell diameter when considering an upright cylindrical domain. The Nusseltnumber is the ratio of total vertical heat flux and the conductive heat flux. The Ekman numbershows the importance of the viscous compared to Coriolis forces. The Taylor number is theinverse square of the Ekman number. The importance of centrifugal buoyancy (nominallydisregarded in the Oberbeck-Boussinesq equations but unavoidable in experiments) can beassessed by the Froude number. The last number is the aspect ratio; a number representing inthis case the cylindrical geometry.

1.3 Geostrophic balance and thermal windThe complete set of Navier-Stokes equations with the Oberbeck-Boussinesq approximationhas a complicated structure. However, by introducing some relevant assumptions for a certainclass of flows, it is possible to simplify and interpret these equations. Assume that the flow isquasi-steady, the inertial force is negligible compared to the Coriolis force (i.e. Ro 1) andviscous effects are negligible in the bulk (i.e. Ek→ 0), the motion of such a flow is governedby

2Ωz× u = −∇P + gβT z. (1.18)

We can rewrite Equation (1.18) in component form as

−2Ωv = −∂P∂x

,

2Ωu = −∂P∂y

,

0 = −∂P∂z

+ gβT,

(1.19)

with u and v the velocity components in x and y directions, respectively. The horizontalbalance in Equation (1.19) is called the geostrophic balance. Taking the derivatives of thefirst and second terms with respect to y and x, respectively, and subtracting, we have

−2Ω

(∂u

∂x+∂v

∂y

)= 0.

1.4. EKMAN BOUNDARY LAYER 7

Considering the incompressibility condition we arrive at

∂w

∂z= 0, (1.20)

withw the vertical velocity component. If we take the derivative of the horizontal componentsof Equations (1.19) to z, and substitute the vertical pressure gradient by gβT , see the verticalcomponent of Equation (1.19), we arrive at

2Ω∂v

∂z= gβ

∂T

∂x,

2Ω∂u

∂z= −gβ ∂T

∂y.

(1.21)

Equations (1.20) and (1.21) together are called the thermal wind balance [123] and they canbe rewritten as

∂u

∂z=gβ

2Ωz×∇T. (1.22)

This equation states that vertical gradients of the horizontal velocity components depend onlyon the horizontal temperature gradient, while the vertical gradient of the vertical velocitycomponent is always zero.

For a fluid with constant density (i.e. ρ = ρ0, independent of temperature), we arrive atthe well-known Taylor-Proudman theorem [129, 164]

∂u

∂z= 0, (1.23)

stating that the vertical gradient of all the velocity components are zero.

1.4 Ekman boundary layer

As mentioned before, when rotation is dominant and viscous effects are negligible, the hori-zontal flow motion is governed by the geostrophic balance, see the horizontal components inEquation (1.19). The geostrophic balance is only valid away from the boundaries, where theviscosity can be neglected. However, boundary layers at the cylinder sidewall and top andbottom plates are required to satisfy the viscous no-slip conditions; the boundary layers attop and bottom boundaries are the so-called Ekman boundary layer, see e.g. Refs [46, 75].

Starting with the Navier-Stokes equations with relevant assumptions (i.e. neglecting timederivatives and the horizontal derivatives in the viscous term and assuming that the bulk flowis in geostrophic balance) [46, 75], the horizontal velocity components inside the Ekmanboundary layer are expressed as

uE = uI − [uIcos (z/δE) + vIsin (z/δE)]e−z/δE ,

vE = vI + [−uIsin (z/δE)− vIcos (z/δE)]e−z/δE ,(1.24)

where uE and vE are the horizontal velocities inside the Ekman boundary layer, uI and vIare the horizontal velocities in the interior region, z is the distance to the plate and δE is the


Ekman boundary layer thickness: δE =√ν/Ω [46, 75]. Note that δE is independent of the

interior velocity (bulk velocity) and the flow configuration. Another important parameter isthe Ekman time scale, the adjustment time for a fluid when a new rotation rate is introduced,which is defined as τE = H/

√νΩ [46].

1.5 Useful definitions in convective turbulenceThe importance of the study of turbulence is well captured by the famous quote of RichardFeynman as “Turbulence is the most important unsolved problem of classical physics”. Tur-bulent flows are ubiquitous; most of the flows in nature and technological applications areturbulent. The flow transitions from laminar to turbulent when the large-scale Reynolds num-ber goes beyond some threshold, i.e. advection becomes more important than viscous effectsat large scales. Each and every turbulent flow is characterized by some shared characteristicssuch as irregularity, high diffusivity, large Reynolds numbers, high dissipation, a wide rangeof active length scales, etc [165]. A turbulent flow can be driven by different means, herewe focus on thermally-driven turbulence. With this introduction in mind, we briefly discusssome basic concepts in (thermally-driven) turbulence which are used in this thesis. We referthe readers to general turbulence text books for a detailed discussion, e.g. Refs. [165, 126].

Turbulence is a dissipative process: a continuous input of energy at large scales is requiredto keep the turbulence running. In convective turbulence, the buoyant production acts as theenergy input at large scales. In statistically steady turbulence, the input energy is transferredinto the smallest scales (turbulence cascade), where the viscosity plays a major role, anddissipates into heat. In convective turbulence, two dissipation rates are involved: the kineticenergy and thermal variance dissipation rates, ε and εθ, respectively. Therefore, two differentdissipative length scales exist: the Kolmogorov length (η) and the Batchelor length (ηB)[107]. These length scales are given by

η =

(ν3

ε

)1/4

and ηB =

(νκ2

ε

)1/4

, (1.25)

where ε is the kinetic dissipation rate. These two length scales dictate the grid spacing (res-olution) in direct numerical simulation (DNS): the grid spacing should be smaller than theselength scales.

As mentioned, the kinetic energy and thermal variance dissipation rates are involved inthermally driven turbulence. RBC is an inhomogeneous system: local dissipation rates de-pend strongly on the vertical and radial positions; particularly in the vicinity of the boundariesthey may attain considerably larger values than in the center. However, one can derive an ex-act relation for the global (volume-averaged) kinetic and thermal variance dissipation rates in(non)rotating RBC as [148, 149]

〈ε〉 =ν3Ra

Pr2H4(Nu− 1) and 〈εθ〉 =

κ∆T 2

H2Nu, (1.26)

where 〈ε〉 and 〈εθ〉 are the volume-averaged kinetic energy and thermal variance dissipationrates, respectively. The global dissipation rates calculated in DNS give a firm foundation tovalidate the accuracy of the calculated Nu.

1.6. FLOW PHENOMENOLOGY 9

1.6 Flow phenomenology

It is well-known that rotation introduces different regimes in RBC, see e.g. Refs. [13, 82, 84,156]. In this section, we will discuss these regimes (three in total) and their main features.These regimes are reflected in the heat transfer efficiency (Nu) and flow field morphologies,see Figures 1.3 and 1.4. Figure 1.3 shows the normalized heat transfer as a function of Ro on alogarithmic scale. The demarcations between different regimes are given by the vertical dash-dotted lines. Figure 1.4, on the other hand, shows the flow morphologies near the top platefrom top to bottom for Ro =∞, 1 and 0.05, respectively, corresponding to regimes I, II andIII, respectively, to be defined below. The blue curves are example particle trajectories nearthe top plate and the red dots indicate the starting point of each trajectory. In the followingwe explain these regimes and their characterizations.

10−1

100

101

0.7

0.8

0.9

1

1.1

1.2

RegimeI

RegimeII

RegimeIII

Ro

Nu(R

o)/Nu(∞

)

Figure 1.3: The normalized heat transfer as a function of Ro on logarithmic scales. Red dotsare experimental data and open square are numerical data for Ra = 2.73×108 and Pr = 6.26.Data taken from Refs. [159, 195]. The vertical dashed lines represent the demarcation linesbetween different regimes.

• Rotation-unaffected regime (Regime I): The rotation-unaffected regime occurs forRo & 2.5 for a convection cell with Γ = 1 (the transition between rotation-unaffectedregime and rotation-affected regime is aspect ratio dependent) [78, 184, 182, 132].Starting with Figure 1.3, the heat transfer efficiency does not change throughout thisregime; i.e. weak background rotation does not affect Nu. The rotation-unaffectedregime is characterized by a domain-filling Large Scale Circulation (LSC) [4, 78, 184].Figures 1.4(a-b) show some trajectories starting at the same time close to the top platein this regime. As one can partially see from the graph, the fluid parcels move upwardfrom one side and they travel downward at the other side of the cell. The presence


of the LSC dictates vertical mean velocities near the side walls and horizontal meanvelocities near the top and bottom plates, while the flow has zero mean velocity at thecell center. The LSC exists for large values of Ro (Ro & 2.5 for Γ = 1) [78, 184].

• Rotation-affected regime (Regime II): The rotation-affected regime exists for 0.1 .Ro . 2.5 (for Γ = 1 and Pr ≈ 7): The upper bound is aspect ratio dependent whilethe lower bound is Pr dependent [156]. This regime is characterized by a continuousenhancement in the heat flux with increasing background rotation for Pr > 1, seeFigure 1.3. The enhancement does not occur for Pr < 1. In this regime, the LSChas disappeared and is replaced by rotation-aligned vortical plumes, see Figure 1.4(c-d) for Ro = 1. These vortical plumes add swirling motion in the horizontal plane.The vortical plumes spin down (become weaker) as they approach the cell center. Ascan be seen (qualitatively) from the figure, the horizontal and vertical velocities possesapproximately zero mean values in this regime. Quantitative analysis of the presentvelocity statistics confirms the zero mean values for vertical and horizontal velocities.

• Rotation-dominated regime (Regime III): The rotation-dominated regime, alsoknown as the Geostrophic regime, is characterized by a dramatic drop in the heattransfer efficiency with increase in the background rotation, see Figure 1.3. This dropcontinues until the flow motions halt and diffusion becomes the only active way of heattransfer (i.e. Nu = 1). In the geostrophic regime, the flow is principally governed by abalance of the Coriolis and the pressure gradient forces. The rotation-aligned vorticalplumes become more prominent (they can penetrate further into the bulk) and sim-ilar to regime II, the vertical and horizontal mean velocities are approximately zero:it is confirmed by the velocity statistics from the current study, see Figure 1.4(e-f) fora qualitative analysis. There are four distinct flow structures in this regime, namelycellular convection, convective Taylor columns, plumes, and geostrophic-turbulence[151, 117, 153]. Depending on the state, the flow phenomenology is different. Thecellular state occurs just above the onset of convection (i.e. Nu = 1); it is character-ized by densely packed thin hot and cold columns spanning the entire vertical extent ofthe flow domain. Departing from the onset of convection, these cells may develop intowell-separated vertically aligned vortical convective Taylor columns (CTCs), surroun-ded by shields of vorticity of opposite sign. Another mode of convection consists ofplumes with less vertical coherence and no shields. A final state is called “geostrophic-turbulence”, where the vertical coherence is lost almost completely and the interioris fully turbulent. Note that all four convection modes are part of the geostrophic re-gime, which should thus not uniquely be identified with the geostrophic-turbulencestate alone. The occurrence of these four states is strongly dependent on the Prandtlnumber Pr: at lower Pr . 3 no Taylor columns are formed, while the geostrophic-turbulence state remained out of reach for Pr & 7 in the simulations [70]. Consideringour experimental set-up, it is expected that we cover the cellular, CTCs and plumestates while the geostrophic-turbulence state is out of reach.

1.7. GOAL 11

−40−20

020

40

−40−200

2040

160

180

200

xy

z

a)

−40−20

020

40

−40−200

2040

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c)

−40−20

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2040

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e)

−40 −20 0 20 40−40

−20

0

20

40

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b)

−40 −20 0 20 40−40

−20

0

20

40

x

y

d)

−40 −20 0 20 40−40

−20

0

20

40

x

y

f)

Figure 1.4: Particle trajectories from experiments near the top plate for (a) Ro =∞, isometricview (b) Ro = ∞, top view, (c) Ro = 1, isometric view, (d) Ro = 1, top view (e) Ro = 0.05,isometric view and (f) Ro = 0.05, top view. The axes are in mm and the top plate is locatedat z = 200 mm.

1.7 GoalAs mentioned before, rotating thermal convection has been studied by laboratory experi-ments, numerical simulations and theoretical analysis. However, up to now, the effects ofbackground rotation on RBC are mainly characterized by global parameters like the overallheat transfer: less attention has been given to the flow field. Additionally, the available stud-ies on the flow field are performed in the Eulerian frame of reference. Recently, nonrotatingRBC has been studied in the Lagrangian frame of reference numerically [144, 37, 90] andexperimentally [113, 114, 90, 89]. However, to the best of our knowledge, the Lagrangianstudies of the rotating RBC are non-existent.

There are still many open questions with regard to the rotating RBC convection, whichcan be answered by Eulerian and particularly Lagrangian flow field measurements. In this


thesis, we address such questions, of which the main ones are:

• What are the driving mechanisms behind transitions from one regime to another re-gime? How are these transitions reflected into different Lagrangian statistics? Howcan we characterize each regime using Lagrangian statistics? (Chapters 3, 5 and 6)

• What are the effects of background rotation on the large- and small-scale flow field atdifferent heights in a cylindrical convection cell? Do the small-scales remain isotropicor is Kolmogorov’s hypothesis of local isotropy violated? (Chapter 4)

• What are the effects of background rotation on the geometrical aspects of fluid parceltrajectories? Can we recover the power-law scaling, reported earlier for homogeneousand isotropic turbulence (HIT), in our non-HIT thermally-driven turbulence throughoutthe cylindrical domain? If not, how does the power law scaling changes with rotationand position of the measurement volume in the cell? (Chapter 5)

• Is the geostrophic regime accessible with a conventional RBC set-up: a convectioncell with H = 0.2 m and water as working fluid? If so, can we reproduce the basicsignatures of the asymptotic solutions in an experiment? (Chapter 7)

In the following chapters we will address the above questions. Brief answers to these ques-tions are presented in the next section, Section 1.8, to familiarize the reader with the results.

1.8 Guide through this thesisWe start with the description of our experimental and numerical approaches to study RRBCin Chapter 2. Two different approaches are used for flow measurement, namely three-dimen-sional particle tracking velocimetry (3D-PTV) and time-resolved particle image velocimetry(PIV). The experimental data are complemented by direct numerical simulations (DNS) whenit is appropriate and possible.

Chapter 3 concerns the rotation-unaffected and rotation-affected regimes and the trans-ition between them. Using measurements of the Lagrangian acceleration of neutrally buoyantparticles at two different heights (close to the top plate and at the center of the cylindricalconvection cell) and accompanying DNS, we study the role of the boundary layers in thetransition between regimes I and II. We perform an analysis for different cell aspect ratios,supported by DNS simulations, which provides the deeper understanding of the dependenceof the transition on the aspect ratio.

In Chapter 4, we study the effects of background rotation on large- and small-scale iso-tropy in RRBC from both Eulerian and Lagrangian points of view. 3D-PTV and DNS areemployed at three different heights within the cell. The Lagrangian velocity fluctuation andsecond-order Eulerian structure function are utilized to evaluate the large-scale isotropy fordifferent rotation rates. Moreover, we examine the experimental measurements of the Lag-rangian acceleration of neutrally buoyant particles and the second-order Eulerian structurefunction to evaluate the small-scale isotropy as a function of the rotation rate.

In Chapter 5, we perform an experimental investigation on the effects of the backgroundrotation on the geometry of tracer trajectories for RRBC in regimes I and II. We focus on thegeometry of tracer trajectories by computing curvature statistics. Scaling laws for curvaturestatistics have been derived for HIT in previous studies. We always recover this HIT scaling

1.8. GUIDE THROUGH THIS THESIS 13

in the bulk of RRBC. However, in the horizontal boundary layers, we only recover this HITscaling for higher rotation rates. We discuss the physics behind these phenomena in thischapter.

In Chapter 6 the experimental velocity and acceleration statistics from 3D-PTV data forall three regimes are studied. The main focus of the chapter is on the Lagrangian velocity andacceleration autocorrelations and how the transition from one regime to another affects thesestatistics. Different time scales are observed in the velocity autocorrelations in regimes II andIII which are found to be associated with the strong vortical plumes in these regimes.

Chapter 7 of this thesis focuses on the transition to the rotation-dominated regime (regimeIII) and the onset of convection. Approaching the geostrophic regime of rotating convectiontypically requires dedicated set-ups with either extreme dimensions (height of > 1.5 m) [26]or some troublesome working fluids (e.g. cryogenic gases) [36]. In this chapter, we push ourexperiment into a regime where it is possible to compare the experimental data with thoseof the asymptotically reduced equations. These equations, derived from the incompressibleNavier-Stokes equations, are the so-called non-hydrostatic quasi-geostrophic set of equationsin the limit Ro → 0 [68, 151]. We show that it is also possible to enter the geostrophicregime of rotating convection with “classical” experimental tools: a table-top set-up with aconventional convection cell with a height of 0.2 m and water as working fluid. We compareour experimental data with results from simulations of the asymptotically reduced equationsreported in previous studies [117] and reach a very satisfactory agreement.

In the last chapter, Chapter 8, we summarize our main findings and provide an outlook tofurther studies of this interesting flow problem.

Chapter 2

Methods and Parameters

The present study employs a combined numerical-experimental approach to explore rotat-ing Rayleigh–Bénard convection. The main part of this study is based on the experimentaldata obtained from three-dimensional particle tracking velocimetry (3D-PTV). We also em-ploy particle image velocimetry (PIV). The experimental data are complimented by directnumerical simulations (DNS) when required. In this chapter, the experimental and numericalapproaches are explained. We first start with the experimental set-up for RRBC in Section2.1. Next, the experimental procedure and flow characteristics are given in Section 2.2. Fi-nally, the DNS set-up is discussed in Section 2.3.

2.1 Experimental set-upsThe experimental set-up consists of a convection cell and an optical measurement systemmounted on a rotating table. Two different optical systems are used for the flow measurement,namely 3D-PTV and PIV. Both systems use the same convection cell. Schematic views of theexperimental set-ups (consisting of the convection cell and an optical system on the rotatingtable) are presented in Figures 2.1(a-b). Figure 2.1(a) shows the 3D-PTV system combinedwith the convection cell and Figure 2.1(b) displays the PIV system combined with the sameconvection cell. In the following sections, first the convection cell is explained in Section2.1.1. Then, the 3D-PTV and PIV systems and the corresponding post processing proceduresare discussed in Sections 2.1.2 and 2.1.3, respectively.

2.1.1 Convection cellFigure 2.2 shows a schematic view of the convection cell. The convection cell, similar tothe one used in Refs. [82, 81, 76, 78, 79], is composed of three main parts: (i) a heatedplate at the bottom, (ii) a cooling chamber at the top, and (iii) a cylindrical Plexiglas vesselconnecting top and bottom plates, see Figure 2.2. At the bottom, an electrical resistanceheater is attached to a copper plate. The electrical resistance heater is manufactured in theform of a disk, thus it is in contact with a large area of the copper plate. Furthermore, copperhas an excellent thermal conductivity (kcopper = 390 W/(m·K)), guaranteeing a uniformtemperature distribution. There is a layer of insulation beneath the electrical heater to avoid

15

16 CHAPTER 2. METHODS AND PARAMETERS

Figure 2.1: Schematic views of (a) the 3D-PTV system and (b) the PIV system. 3D-PTVuses a volumetric illumination using light-emitting diodes (LEDs) while PIV utilizes a 2Dillumination using a laser sheet.

heat leakage. The copper plate temperature is measured by a thermistor placed inside a holeclose to the wetted surface at the center of the copper plate. The thermistor is connected toa heater controller unit which keeps the temperature of the plate constant by variation of theinput power. The temperature fluctuations do not exceed 0.1 K during operation.

The cooling chamber is placed at the top of the convection cell. A sapphire plate, withksapphire = 33 W/(m·K), is positioned at the bottom of the cooling chamber, guarantee-ing good thermal contact between working fluid and the recirculating fluid in the coolingbath while retaining transparency. Sapphire has a smaller thermal conductivity compared tocopper, however, the thermal conductivity of sapphire is substantially better than other trans-parent materials e.g. ksapphire/kplexiglas ≈ 180. A 12 mm thick Plexiglas plate is locatedon the top of the cooling chamber. The cooling chamber is transparent and as a result itmakes the cell optically accessible from above. The cooling chamber is equipped with eightnozzles; four for coolant inflow and four for coolant outflow. The inlet and outlet nozzles areequipped with fine meshes to avoid formation of large-scale flow structures and to guaran-tee a well mixed flow with a uniform temperature inside the cooling chamber [76]: indeed,the temperature inside the cooling chamber has been found to be homogeneous by measur-ing the temperature at nine different positions inside the cooling chamber [76]. A coolingbath (Haake V26/B refrigerated bath, with Haake DC50 temperature control unit) keeps thecooling chamber temperature constant. A thermistor is located in the cooling chamber andconnected to the cooling bath. During operation, the temperature fluctuations are smallerthan 0.05 K.

The cylindrical Plexiglas vessel (with thickness of 10 mm) with equal inner diameter Dand height H of 200 mm, filled with water, is placed inside a rectangular Plexiglas box witha wall thickness of 15 mm. The space between cylinder and box is filled with water as wellto avoid the distortion of the illumination from the side. The focus of this study is on themeasurements of the flow field rather than heat transfer. Therefore, optical accessibility is

2.1. EXPERIMENTAL SET-UPS 17

Figure 2.2: A sketch of the convection cell and its components (left panel). Different com-ponents are labeled with letters in the figure. A picture of the convection cell (right panel).

required: it is not possible to insulate the convection cell. However, we try to keep the meantemperature close to the ambient temperature to minimize the heat loss to the ambient. Notethat the thick Plexiglas cylinder and box also decrease the heat loss to the ambient.

The aforementioned non-idealities are unavoidable in the experimental set-ups with op-tical accessibility [82, 81, 76, 78, 79, 114, 113, 177]. However, the non-idealities in thecurrent convection cell have no significant influences on the measurements as evidenced bythe excellent quantitative agreement of experimental and simulation results achieved in thelater chapters of this thesis.

2.1.2 Three-dimensional Particle Tracking VelocimetryThere are different experimental techniques for particle tracking, e.g. acoustic three dimen-sional particle tracking e.g. Refs. [111, 109, 112, 110, 130], smart particles (non-passive) e.g.[146, 41] and three-dimensional particle tracking velocimetry (3D-PTV) using silicon-stripdetector or high speed cameras, e.g Refs. [98, 101, 31, 106, 191, 85, 179, 178, 108]. Eachmethod has its own pros and cons and has been employed for Lagrangian studies of non-rotating turbulence. We apply optical 3D-PTV based on high speed cameras due to extensiveexpertise in the lab [31, 191, 106]. It is worth mentioning that the Lagrangian exploration ofRayleigh–Bénard convection (RBC) has been done numerically [144, 37] and experimentally[113, 114, 89] (using 3D-PTV based on high speed cameras), but not under rotation. Herewe extend this method to the rotating frame.

Fundamentals of 3D-PTV algorithm

3D-PTV is a non-intrusive optical diagnostic technique in which the fluid motion is measuredthrough tracking small neutrally buoyant seeding particles. The original underlying concepthas been used for visualization in the 1960s [168]. In this system, the tracking has beenperformed manually. The first automated PTV system was realized in 1985 by Chang etal. [24] and later followed by Adamczyk et al. [1] in 1988. However, the PTV algorithmhas been significantly improved since then. A typical 3D-PTV algorithm consists of the


following steps: first the seeding particles are detected in the images, then 3D positions of theparticles in 3D space are calculated, and finally the temporal linking between two subsequent-in-time frames is established needed for evaluation of particle velocity and acceleration. Inthe following, we discuss the aforementioned steps. In the current study, we use a particletracking system based on the system developed at ETH Zürich (Switzerland) [100, 101, 187,186, 98]. We present a brief introduction of the methods applied for each step while mainlyconcentrating on the techniques which are used in the ETH algorithm.

The first step is to detect the seeding particles in an image. The seeding particles typicallyappear as bright spots in the camera image. There are various techniques for determining thecenter of a particle, e.g. peak fitting [2], template matching [162] and weighted mean. TheETH algorithm uses the weighted mean technique as it defines the center of a particle in a 2Dimage by

xc =

∑i xiI(xi, yi)∑i I(xi, yi)

and yc =

∑i yiI(xi, yi)∑i I(xi, yi)

where i is the pixel number, (xc, yc) is the coordinate of the center of the particle and I(xi, yi)is the intensity of the particle at (xi, yi). The weighted mean method is computationally cheapcompared to the other two methods and it provides 0.1 pixel accuracy (sub pixel accuracy)for a particle occupying ∼ 5 pixels [106].

The second step is to find the 3D position of the seeding particles based on their positions(projections) on 2D images. In order to do this, at least two images from two different viewingangles are required. It is possible to reconstruct the 3D position by using a geometricalrelation between the 3D positions and their projections onto the 2D images, known as theepipolar line [99]. The basic concept is plotted in Figure 2.3. In this plot, X is a point in3D space, x1 and x2 are the X projections onto image 1 and 2, respectively, and C1 and C2

are the centers of camera 1 and 2, respectively. Let us assume that the center of the cameras,positioning and orientation of the cameras (obtained from the calibration) and a point inimage 1, e.g. x1, are known. We know that X , x1, x2, C1 and C2 are coplanar, see Figure2.3. Thus, X can be any point on Line 1. The projection of Line 1 onto image 2 is a lineand is called the epipolar line. In theory, if the seeding density is very low, it is most likelythat there is only one bright spot on the epipolar line in image 2. However, in practice, thereare typically multiple bright spots on the epipolar line; another image (camera) is required toresolve these ambiguities. Any arbitrary number of cameras (larger than 1) can be used forPTV. However, more than four cameras was found to be not necessary, impractical and notcost efficient for general applications [100].

Next step is the establishment of the temporal link between subsequent-in-time frames.This step is normally the most challenging step in PTV algorithms. When the seeding densityis low (the inter-particle distance is significantly larger than the particle displacement betweentwo successive frames), the temporal link can readily be established by using a nearest neigh-bor scheme. However, for higher seeding densities, the situation is more complicated. Inthis case, different techniques have been proposed in the literature, e.g. the relation method[7], spring model [120], optical flow scheme [29], multi-frame tracking [56, 101] and streakphotography [33, 60]. The ETH code uses the multi-frame tracking technique introduced byHassan et al. [56] and further modified by Malik et al. [101]. In this method, four consecut-ive frames are considered and based on different criteria, e.g. particle velocity and angle oftravel between successive frames, an acceptable trajectory is established.


Figure 2.3: Schematic representation of the working principle of epipolar line for a twocamera system. C1 and C2 are the camera 1 and 2 centers, respectively. X is a point in spaceand x1 and x2 are its projections onto image 1 and 2, respectively.

As mentioned before, the extrinsic and intrinsic camera parameters are determinedthrough a calibration procedure. The extrinsic parameters are set-up dependent: they de-pend on position and orientation of the cameras with respect to the measurement volume. Onthe other hand, intrinsic parameters are inherent to the camera in use: they are independent ofpositioning and orientation of the cameras. Intrinsic parameters include variables like focallength and lens distortion coefficients. The extrinsic camera parameters are obtained througha calibration procedure by using an optical model based on a pinhole camera model, lensdistortion and different media with different refractive indices [100]. There are two mainapproaches to calibrate a 3D-PTV system: multi-plane calibration (a calibration plate sub-merged inside the measurement volume at different heights) or a 3D staircase body (a 3Dbody consisting of points at different heights). The bottleneck for the multi-plane calibrationprocedure is the precise parallel positioning of the calibration plate at different precise vertic-ally separated heights. Therefore, we use a 3D staircase body, see Figure 2.4. The calibrationis discussed in more detail in the subsequent section.

Figure 2.4: The V-shaped staircase calibration body used in the current study.


Characteristics of the current 3D-PTV set-up

The particle tracking system used in this study consists of four charge-coupled device (CCD)cameras (MegaPlus ES2020, 1600 × 1200 pixels equipped with 50 mm lenses) positionedabove the convection cell, see Figure 2.1(a). The apparatus of the camera is set to f /16 for allexperiments, which provides the maximum depth of field. The 3D-PTV images are recordedat 8-bit dynamic range of grayscales. The active area of the camera sensor is 11.8× 8.9 mm2

resulting in a pixel size of 7.4 × 7.4 µm2. The cameras are located roughly 400 mm awayfrom the measurement volume which results in a field of view of roughly 80× 60 mm2. Thecameras record the flow field at a frequency of either 30 Hz or 15 Hz, depending on the flow.This is adequate to resolve the smallest length and time scales of the flow field. The recordedimages are transferred to a computer simultaneously which amounts to a data transfer rate ofapproximately 240 MB/s (for a frequency of 30 Hz and 8-bit images).

The illumination is provided by four arrays of light-emitting diodes (LEDs). Each arrayconsists of 21 lamps. In order to minimize the heat production of the LEDs, they are operatedin pulsed mode, triggered by an external function generator. To further decrease the effectsof heat disturbances from LEDs on the flow, blue LEDs with a dominant wavelength of 455nm are chosen; the heat absorption coefficient of water at this wavelength is close to itsminimum [125]. A rough estimate (Beer-Lambert Law) of the heat absorbed by the waterfrom the LEDs (QLEDs) confirms the negligible effects of LEDs compared to the mean heatsupplied through the copper plate, QLEDs/QCopper < 1%.

Preliminary experiments reveal that the light from LEDs reflects on the copper plate andtubular walls which deteriorates the capability of the tracking of the particles. To cope withthese reflections, we use high-pass filters in combination with fluorescent particles. Figure2.5 shows the working principle of this combination. The fluorescent Polyethylene particles(supplied by Cospheric Co., USA) have a mean density of 1002 kg/m3. The Polyethyleneparticles have a diameter of 75-90 µm, thermal conductivity of 0.5 W/(m·K) (the same aswater) and heat capacity of 1900 J/(kg·K) (slightly less than half compared to water). Theratio between the particle response time and the Kolmogorov time scale is the Stokes numberSt = τp/τη , where τp = d2

pρp/(18νρf ) is the so-called particle response time with ρf thefluid density. For the present experiments, the Stokes number is small, St ≈ 2 × 10−4; theparticles can be considered tracers. The fluorescent particles emit light at a wavelength of600 nm which provides a sufficient upshift with respect to the illumination wavelength (455nm), resulting in easy separation of direct and fluorescent light. The cameras are equippedwith high-pass filters (OG-570, Schott Glass) which filter out the spurious reflections with awavelength below 570 nm. Furthermore, the quantum efficiency of the cameras is best in therange of 400 to 600 nm and the particle emission wavelength falls into this range as well.

The calibration has been performed by use of a 3D body with a V-shaped staircase, thesame body as used by Ref. [30], see Figure 2.4. In total, 81 dots are manufactured on thetarget. The ETH code calculates the extrinsic and intrinsic parameters based on the calibra-tion images. The calibration is carried out at ambient temperature. However, the refractiveindex of water depends on temperature. The measurement error due to the refractive indexfluctuations is calculated by performing long-time measurements of stationary particles. Asmall plus-shape geometry has been manufactured with five holes each filled with many tracerparticles, see Figure 2.6. The plus-shape geometry is connected to a circular base via a 1 mmbar and placed inside the convection cell. In order to be sure that the LSC does not affect


Flow & particles

High pass filter

Figure 2.5: The working principle of the filter. The blue arrows are the illumination light at awavelength of 455 nm while the red arrows are the emission light of fluorescent particles at awavelength around 600 nm.

the plus-shape geometry, seven bars connect the tubular wall of the vessel to the geometry.The bars are 1 mm in diameter to minimize the interfering with the flow field. The standarddeviation of the position fluctuations in the horizontal and vertical directions are found to beapproximately 6 and 20 µm, respectively, when a typical temperature gradient (∆T = 10K) is imposed. The errors in the velocity and acceleration signals are estimated as follows.First, two sets of random numbers with the same standard deviations as those of position fluc-tuations (6 and 20 µm) are generated. Next, these sets of randomly generated numbers areadded to the position data of a real-case experiment. The new velocities and accelerations,based on the data with added noise in position, are calculated. The new velocities and accel-erations are subtracted from the velocities and accelerations computed based on the originaldata (the data before adding the noise due to the temperature dependence of the refractive in-dex of water). The standard deviations of the acceleration difference are less than 0.1 mm/s2

in xy and 0.18 mm/s2 in z directions. The standard deviations of the velocity difference areless than 0.02 mm/s in xy and 0.05 mm/s in z directions.

All experimental equipment, including the convection cell and the 3D-PTV system, isplaced on the rotating table. The rotating table is controlled from adjacent room for safetyreasons. The rotation rate can vary between 0.01 and 10 rad/s with an accuracy of ±0.005Ω.The accuracy requirements of the table have been checked by Refs. [167, 30]. The maximumvertical misalignment of the surface table is found to be 7.5 µm, measured at the edge ofthe table. The data of the digital water-level measurements confirm that the table can rotateat constant angular velocities with no appreciable angular accelerations [30]. The residualangular acceleration found to be always below 10−3 1/s2. The effects of any possible tablevibration on the PTV system have been measured: these effects are negligible [30]. Furtherdetails of the rotating table can be found in Refs. [30, 167]

Low-pass filtering for particle trajectories

Small errors in the position signal potentially result in considerable errors in the velocityand tremendous errors in the acceleration signal given that they must be reconsidered withdiscrete derivative schemes in post-processing. Different approaches reducing these errors,including Epps’ method [38], Gaussian filters [108] and cubic polynomial fitting [96], havebeen tested. The cubic polynomial fitting described in Ref. [96] and used by e.g. Refs.


Figure 2.6: The plus-shape geometry with five holes on the black plus geometry. A zoomwindow of the plus-shaped geometry with the five holes is shown in the image.

[30, 191] has been adopted for the post-processing of the position, velocity and acceleration.

In this method, a cubic polynomial is fitted for each time step t using N preceding andsubsequent time steps, from t−N∆t to t + N∆t: the filter length is 2N + 1. The first andlast N points of a long trajectory are discarded. Following this approach, it is possible todefine the raw position of a point at time t in the i direction as xi(t) and it is expressed as

xi(t) = ci,0 + ci,1t+ ci,2t2 + ci,3t

3 + e(t), (2.1)

where e(t) is the noise. The constants ci are determined as

ci = (ATxi)T (ATA)−1, (2.2)

where

A =

1 t−N∆t t−N∆t2 t−N∆t3

1 t− (N − 1)∆t t− (N − 1)∆t2 t− (N − 1)∆t3

......

......

1 t+N∆t t+N∆t2 t+N∆t3

,

and

ci =

ci,0ci,1ci,2ci,3

.


Position, velocity and acceleration after filtering are expressed as

xi(t) = ci,0 + ci,1t+ ci,2t2 + ci,3t

3,

ui(t) = ci,1 + 2ci,2t+ 3ci,3t2,

ai(t) = 2ci,2 + 6ci,3t,

(2.3)

respectively. Low-pass filters are used to eliminate the background noise (high frequencysignals). However, depending on the filter length, it might remove too little or too much ofthe high frequency signal. Therefore, the velocity/acceleration signal depends on the filterlength: long filter length (large N ) results in an further smoothing out the signal and under-estimation of the velocity/acceleration signal, while short filter length (small N ) results inan overestimation of the velocity/acceleration signal. In order to check the signal depend-ence on the filter length, we calculate the acceleration variance,

⟨a2⟩

=∑n

1 a2i /n with n the

number of statistics (the mean acceleration is always zero for all measurements), for differentfilter lengths. Figure 2.7 shows the acceleration variance as a function of time period overwhich the filter is applied, nondimensionalized by the local Kolmogorov time scale. The localKolmogorov time scale is defined as τη =

√ν/ε [126] where ε is the local kinetic energy

dissipation rate, calculated from our DNS data. In the figure τf = (2N + 1)/f where f isthe camera frequency. As can be seen from the graph, the acceleration variance always de-pends on the filter length: filtering influences both noise and real signal. For small τf/τη , theacceleration variance decreases dramatically with increase in τf/τη . Beyond some threshold,there exists an exponential decay; the threshold is shown as a red circle and square in Figure2.7 for xy and z directions, respectively. Voth et al. [178] suggested that we can fit a curvethrough the data points of the form

g(τf/τη) = A (τf/τη)B

+ Cexp(Dτf/τη + E (τf/τη)

2), (2.4)

with A, B, C, D and E the fit parameters. Then, an approximation of the acceleration vari-ance can be achieved when τf/τη = 0, i.e.

⟨a2⟩

= g(0) = C. The C values for xy and zdirections are shown in the figure as Cxy and Cz , respectively. The dashed lines in Figure 2.7show the exponential term in Equation (2.4). Voth et al. found that the suggested accelera-tion variance is overestimated by almost 10% compared with the corresponding accelerationvariance from numerical simulations [178]. Ni et al. also followed the same procedure forcalculation of the acceleration variance [114]. Based on our DNS data at the cell center,presented in this thesis, we found that a filter length equal to the filter length where the ex-ponential decay starts is the closest to the DNS data: τf/τη ≈ 0.4 for x and y directionsand τf/τη ≈ 0.65 for the z direction, see the red symbols in Figure 2.7. The differencesbetween this method and Voth’s are negligible; we found an approximately 8% overestima-tion of C compared to our DNS data. However, the proposed approach is computationallyless expensive since the filtering is carried out for only one filter length. The filter length inthe z direction is larger than that in x and y directions because the cameras are located on topof the cell and the z direction is perpendicular to the plane in which the cameras are located.


0 0.2 0.4 0.6 0.8

100

Cz

Cxy

τf/τη

⟨

a2⟩

(mm/s2)2

xyz

Figure 2.7: The acceleration variance for Ro = 2.38 at the cell center for different filterlength nondimensionalized by the Kolmogorov time scale, τη . The dashed lines show onlythe exponential term in Equation (2.4).

Transformation to Eulerian frame of reference

So far, we have discussed the working principles of 3D-PTV, the characteristics of our exper-imental 3D-PTV set-up, and post-processing of the trajectories (low-pass filters). Although3D-PTV naturally provides the data in the Lagrangian frame of reference, it can provide use-ful information in the Eulerian frame as well. The Eulerian data give access to the velocitygradient tensor. Switching from Lagrangian to Eulerian frames of reference is not straightfor-ward and different approaches are proposed in Refs. [97, 30, 98]. These approaches includethe least squares method described in Ref. [98], the finite difference method and differentconvolution methods described in Refs. [97, 30], and interpolation on a regular grid [30]. Aswill be shown in the following, none of the methods is capable of fully retrieving the velocitygradient tensor in the current experimental approach. However, the interpolation on a regulargrid is observed to be the superior approach in terms of data quality and ease of use. We shallonly show the results of this approach.

The number of velocity data points per time step plays a crucial role in achieving accurateresults for all methods. As mentioned before, in the cubic polynomial method N data pointsat the beginning and end of each trajectory are discarded: the beginning and end of eachtrajectory are prone to higher errors. In addition, in the cubic polynomial method, all traject-ories shorter than 2N +1 are ignored. As a result, there are less velocity vectors (data points)at each time step in the cubic polynomial method. Here, however, the number of velocitydata points per time step plays a crucial role. In order to maximize the number of velocitydata points per time step, the velocities of each trajectory are calculated using a second-ordercentral difference method instead of the cubic polynomial method. Note that the second-order central difference method can be performed over short trajectories as well. In addition,


there is no need to discard the beginning and end of each trajectory. Therefore, the number ofvelocity data points per time step increases. The velocities from central difference method areinterpolated on a regular grid. The interpolated velocities on the regular grid are smoothed bya low-pass cubic polynomial filter, the same as the one discussed before, performed on eachfixed grid point over time. For the cubic polynomial filter N = 15 is chosen, with 2N + 1the filter length.

The velocity gradient tensor can be decomposed into the vorticity and strain rate tensorsas

∇u =1

2(∇u + (∇u)T ) +

1

2(∇u− (∇u)T ) = S + Ω, (2.5)

with Ω and S the antisymmetric (vorticity tensor) and symmetric (rate-of-strain) parts ofthe velocity gradient tensor, respectively. The double lines on Ω stand for tensor and thesuperscript T stands for the matrix transpose. The vorticity tensor, given by

Ω =1

2

0 ωz −ωy−ωz 0 ωxωy −ωx 0

,is calculated using Stokes theorem, that gives the relation between vorticity and circulationby

Γ =

∮u · dl =

∫(∇× u) · dA =

∫ω · dA (2.6)

withω = (ωx, ωy, ωz) = ∇×u the vorticity vector, l the path of integration around a surfaceA. For example, for the calculation of ωz from velocity data on a regular grid, we have [131]

ωz =Γi,j

4∆X∆Y(2.7)

with

Γi,j =1

2∆X(ui−1,j−1 + 2ui,j−1 + ui+1,j−1)

+1

2∆Y (vi+1,j−1 + 2vi+1,j + vi+1,j+1)

−1

2∆X(ui+1,j+1 + 2ui,j+1 + ui−1,j+1)

−1

2∆Y (vi−1,j+1 + 2vi−1,j + vi−1,j−1)

(2.8)

the circulation, ∆X and ∆Y the grid spacing in x and y directions, subscripts i and j the gridpoint numbering (see Figure 2.8) and u = (u, v, w) the velocity vector. The rate-of-straintensor is given by

S =

εxx12γxy

12γxz

12γyx εyy

12γyz

12γzx

12γzy εzz

,with εmm = ∂um/∂xm and γmn = γnm = (∂um

∂xn+ ∂un

∂xm), (m,n) ∈ (x, y, z) and m 6= n.


Figure 2.8: Contour for the calculation of Γi,j used in the estimation of the vorticity at point(i, j) [131].

Similar to the vorticity components, the rate-of-strain components can also be readily derived;for instance

γxy =1

8∆X(vi+1,j−1 + 2vi+1,j + vi+1,j+1)

+1

8∆Y(ui−1,j+1 + 2ui,j+1 + ui+1,j+1)

− 1

8∆X(vi−1,j−1 + 2vi−1,j + vi−1,j+1)

− 1

8∆Y(ui−1,j−1 + 2ui,j−1 + ui+1,j−1).

(2.9)

Note that the aforementioned approach is only used for the interpolation on a regular grid.Each method has its own way of retrieving the velocity gradient tensor the details of whichcan be found in the corresponding references.

Two different evaluation criteria are used to check the quality of the velocity gradienttensor:

• the trace of the velocity gradient tensor should be zero (i.e. mass conservation forincompressible fluid is satisfied, ∇ · u = 0),

• the Lagrangian acceleration is equal to the sum of the Eulerian acceleration and theconvective term (i.e. aL = Du

Dt = ∂u∂t + u ·∇u).

Note that these criteria can also depend on the original velocity signal from the Lagrangiandata as well. However, the uncertainty in the Lagrangian velocity measurements are small:0.02 mm/s in the xy direction and 0.05 mm/s in the z direction. Therefore, the main contri-bution in the error comes from the Eulerian-retrieving scheme. The first criterion is examinedby the joint-PDF of ∂u∂x + ∂v

∂y and−∂w∂z for the interpolated velocity on a regular grid, see Fig-ure 2.9(a-b) for Ro =∞ and Ro = 0.1 near the top, respectively. The correlation coefficient


for variables a and b is defined as

C =〈ab〉√〈a2〉〈b2〉

,

which can vary between 0 and 1. The value 0 stands for two uncorrelated variables and 1 is fortwo perfectly correlated variables. The correlation coefficient for a = ∂u

∂x + ∂v∂y and b = −∂w∂z

is found to be 0.9 for Ro =∞ near the top. The color bars show the number of occurrences.The correlation coefficient decreases significantly with an increase in background rotation;the coefficient is 0.2 for Ro = 0.1. The decrease in the correlation coefficient with decreasingRo is most probably due to the decrease in the vertical velocity gradients with decreasing Ro(as a result of Taylor-Proudman theorem, see Equation (1.23)) which leads to a small signalto noise ratio.

dudx

+dvdy

−

dw dz

a)

a)

−0.5 0 0.5−0.5

0

0.5

2000

4000

6000

8000

10000

12000

14000

16000

dudx

+dvdy

−

dw dz

b)

−0.5 0 0.5−0.5

0

0.5

200

400

600

800

1000

1200

1400

1600

1800

2000

Figure 2.9: Joint-PDFs of dudx + dvdy and −dwdz for (a) Ro =∞, and (b) Ro = 0.1 close to the

top plate. The color bars indicate the number of occurrence. The correlation coefficients are0.9 and 0.2 for Ro =∞ and Ro = 0.1, respectively.

The second criterion is evaluated by the joint-PDF of the Lagrangian acceleration and thesum of the Eulerian acceleration and convective term,

aL =Du

Dt=∂u

∂t+ u ·∇u.

The joint-PDF provides a comprehensive check for the Eulerian data as it includes spatialand temporal derivatives. Figure 2.10(a-c) shows the joint-PDFs for x, y and z directionsfor Ro = ∞ near the top. The correlation coefficients are found to be 0.9, 0.9 and 0.6 forx, y and z, respectively. Figure 2.10(d-f) shows the joint-PDFs for x, y and z directionsfor Ro = 0.1 near the top. The correlation coefficients for Ro = 0.1 are found to be 0.9,0.9 and 0.7 for x, y and z, respectively. Contrary to the first criterion (mass conservation),the correlation coefficients are hardly affected by the background rotation. The correlationcoefficient is lower in the z direction due to the camera positioning and associated highererror in the z direction, as mentioned before.

As can be seen from the correlation coefficients, the velocity gradient tensor is accurately


( ∂u∂t + u·∇u)/armsx

(

Du

Dt

)

/arm

sx

a)

−2 0 2−3

−2

−1

0

1

2

3

500

1000

1500

( ∂v∂t + u·∇v )/armsy

(

Dv

Dt

)

/arm

sy

b)

−2 0 2−3

−2

−1

0

1

2

3

500

1000

1500

( ∂w∂t + u·∇w )/armsz

(

Dw

Dt

)

/arm

sz

c)

−2 0 2−3

−2

−1

0

1

2

3

500

1000

1500

( ∂u∂t + u·∇u)/armsx

(

Du

Dt

)

/arm

sx

d)

−2 0 2−3

−2

−1

0

1

2

3

1000

2000

3000

4000

5000

6000

( ∂v∂t + u·∇v )/armsy

(

Dv

Dt

)

/arm

sy

e)

−2 0 2−3

−2

−1

0

1

2

3

1000

2000

3000

4000

( ∂w∂t + u·∇w )/armsz

(

Dw

Dt

)

/arm

sz

f)

−2 0 2−3

−2

−1

0

1

2

3

200

400

600

800

1000

1200

1400

Figure 2.10: Joint-PDFs of the Lagrangian acceleration and Eulerian acceleration plus ad-vective term. Panels (a,b,c) show the joint-PDF in the x, y and z directions, respectively,for a sample experiment at Ro = ∞ close to the top plate (covering a volume betweenz = 0.76H and z = 0.96H). Panels (d,e,f) show the joint-PDF in the x, y and z directions,respectively, for a sample experiment at Ro = 0.1 close to the top plate (same observationvolume as for Ro =∞). The color bars indicate the number of occurrences. The correlationcoefficient are 0.9, 0.9 and 0.6 (0.9, 0.9 and 0.7) for x, y and z directions, respectively, forRo =∞ (Ro = 0.1).

retrieved for large Ro. However, the background rotation introduces difficulties to retrievesome components of the velocity gradient tensor. Therefore, it is not possible to retrieveaccurately all nine components of the velocity gradient tensor for high background rotationrates. However, we will show in Chapters 6 and 7 that the velocity gradient tensor can beutilized, with an acceptable accuracy, to identify the vortex populated regions and to studyautocorrelations of the vertical component of the vorticity.

2.1.3 Time-resolved particle image velocimetry system

In addition to 3D-PTV, two-dimensional time resolved particle image velocimetry (PIV) isused. For PIV measurements the same convection cell, as described above, is used. One CCDcamera, the same camera as the one used in the PTV system, is placed above the convectioncell. The illumination is provided by a dual-head Nd:YAG laser (Quantel CFR400) with awavelength of 532 nm. Negative cylindrical and positive lenses are used to achieve a laserlight sheet with a thickness of less than 1 mm. The camera and the laser are triggered by anexternal function generator at a frequency of 15 Hz.

PIV images are processed with a commercial software package, PIVTech (Göttingen,Germany). The window size is chosen as 32 × 32 pixels with 50% overlap, resulting in99 × 74 vector points in the field. The observation view of the camera is approximately

2.2. EXPERIMENTAL PROCEDURE AND FLOW CHARACTERISTICS 29

100 × 75 mm2; the vector spacing becomes ∆x = ∆y ≈ 1 mm. A two-dimensional fastFourier transform (FFT) is used for cross-correlation of the two corresponding windows fromtwo subsequent images. Spurious vectors are detected and replaced using the universal outlierdetection test described in Ref. [185]. This test uses the velocity of the neighboring points todefine whether the velocity of the desired point is acceptable.

The seeding particles for PIV are Polyamid particles (Dantec Dynamics) with a meandiameter of 50 µm and a density of 1030 kg/m3. For PIV, the camera is placed verticallyabove the cell and the cylindrical sidewalls are not in the observation view of the camera. Asa result, no light reflection is observed. Thus, there is no need for usage of high-pass filtersand fluorescent particles.

Low-pass filtering and vertical vorticity component

The PIV experiments are carried out in a time-resolved fashion. Furthermore, the PIV meas-urements are performed for low Ro numbers: the flow motions are slow and possess longcorrelations. This a priori knowledge can be employed to further decrease the backgroundnoise. Therefore, a low-pass cubic polynomial filter, similar to the one described for 3D-PTVdata in Section 2.1.2, is used. However, this filter only enhances the quality of the PIV dataand does not change any conclusions drawn from these measurements. For the PIV measure-ments, the filter is applied to the velocity vector at each fixed point in space over time. Thehalf-filter length is chosen to be N = 15 for PIV measurements.

The 2D-PIV measurement gives access to the horizontal velocity field and consequentlythe vertical component of the vorticity vector, ωz . We use the same methodology as that of3D-PTV to retrieve the vertical component of the vorticity, see Equations (2.6) and (2.7).

2.2 Experimental procedure and flow characteristicsThe experiments have been performed with water. Water from a boiler (∼ 70C) is keptat ambient temperature for a couple of days to degas. The used fluorescent particles (for3D-PTV experiments) are hydrophobic in nature, therefore they are soaked with a Tween 60solution (Sigma Aldrich) for 24 hours. The statistically steady state in RRBC is assessedby considering two typical time scales: the adjustment time scales when a new temperaturedifference or when a new rotation rate is applied. For each experiment, after setting thetemperatures for top and bottom plates, the system is left undisturbed for at least half an hourto adjust to the new temperature settings. The reported value in the literature is τ ≈ 3× 102

s for a cell of similar dimensions as the current one filled with water [16, 76]. After this time,the table is put in motion and, depending on the rotation rate, different adaptation durationsare considered. The typical time scale for fluid adaptation to the background rotation is givenby the Ekman time scale, τE = H/

√νΩ [46]. The actual time used in the experiments for

adaptation to the background rotation is at least 3τE .The 3D-PTV experiments are performed at the cell center and close to the top plate.

The measurement volume for both the cell center and close to the top plate is approximately80 × 60 × 50 mm3 (x, y, z), see Figure 2.11. The horizontal intersection (in the xy plane)of the measurement volume is rectangular since the camera sensors consist of 1600 × 1200(square) pixels. The measurement volumes at the center and near the top cover the height0.375H < z < 0.625H and 0.75H < z < H , respectively. The center of the


measurement domain is on the cylinder (or z) axis (which coincides with the rotation axis andgravity). The measurement volume near the top extends up to the plate surface. Depending onthe region of interest, e.g. the boundary layer or bulk, the volume can be divided into smallervertically separated subvolumes. Throughout this thesis, the choice of the subvolumes areclearly expressed accordingly. The 2D-PIV experiments, on the other hand, are performedonly at the height of z = 0.8H .

H

z = 0

z = H

Center

Top

g

Ω

Figure 2.11: The RBC cell and the measurement volumes. The blue and red surfaces are thecold and hot plates, respectively. The purple cubes are the measurements volumes with sizeof 80× 60× 50 mm (x, y, z). The direction z is along the gravity, perpendicular to the coldand hot plates.

The 3D-PTV experiments are performed in two different fashions: high particle concen-tration (HPC) and low particle concentration (LPC). Depending on the parameters of interest,either HPC or LPC data sets are used. Note that HPC experiments consist of more trajectoriesat each time step (more data points in space at each time step) but the trajectories are shortercompared to the LPC experiments: the reason behind this has been discussed in Section 2.1.2under Fundamentals of 3D-PTV algorithm. Therefore, for example, for the interpolationon a regular grid HPC measurements are used while for the Lagrangian velocity and acceler-ation autocorrelations LPC measurements are used.

LPC (HPC) experiments are performed for a duration of approximately 300 (30) minutesfor each rotation rate. However, it is not possible to continuously perform PTV measure-ments with high frequency for such a long period due to technical limitations. Therefore, theexperiments with frequency of 30 Hz are divided into segments of approximately 11 minutes(the experiments in the rows 1-12 in Tables 2.1 and 2.2 and rows 18 and 31 in Table 2.2 areperformed at a frequency of 30 Hz). The PIV experiments are performed for a duration of∼ 25 min. In LPC experiments, an average number of ∼ 500 randomly distributed particlesare tracked at each time step. On the other hand, in the HPC experiments, an average numberof ∼ 1600 (∼ 2700) randomly distributed particles are tracked at each time step in the cellcenter (close to the top plate).

Tables 2.1 and 2.2 summarize the parameters for the measurement series in the cell center

2.3. NUMERICAL SET-UP 31

and close to the top, respectively. The check mark sign X indicates that these experimentsare performed. The fluid properties (at the mean temperature) are taken from [88].

Table 2.1: Parameters for the measurement series close to the top plate. The X and× indicatethat the corresponding experiments are performed or are not performed, respectively.

No. Ω Tmax Tmin Raa Roa Ek LPC HPC PIV(rad/s) (C) (C) ×109 (PTV) (PTV)

1 0 27 17 1.3 ∞ ∞ X X ×2 0.035 27 17 1.3 4.78 3.6×10−4 X X ×3 0.058 27 17 1.3 2.88 2.2×10−4 X X ×4 0.07 27 17 1.3 2.38 1.8×10−4 X X ×5 0.175 27 17 1.3 0.95 7.2×10−5 X X ×6 0.35 27 17 1.3 0.48 3.6×10−5 X X ×7 0.875 27 17 1.3 0.19 1.4×10−5 X X ×8 1.65 27 17 1.3 0.10 7.2×10−6 X X ×9 2 27 17 1.3 0.083 6.0×10−6 X × ×10 2.91 27 17 1.3 0.057 4.1×10−6 X × ×11 3.5 27 17 1.3 0.048 3.4×10−6 X × ×12 4.12 27 17 1.3 0.041 2.9×10−6 X × ×

aThe reported Ra and Ro values in Chapters 3 and 4 (the experiments at rows 1-8 of this table are discussedin Chapters 3 and 4) are slightly different from the values reported in this table due to the calculation of the fluidproperties at a too high mean temperature (in Chapters 3 and 4 the fluid properties are taken at 24 C instead of 22C for calculation of Ra and Ro). The errors are not consequential. Thus, we retain the Ra and Ro values as thevalues published in the papers in Chapters 3 and 4.

2.3 Numerical set-upIn parallel to the experiments, direct numerical simulations (DNS) are performed. A cyl-indrical configuration with the same aspect ratio as the experiment (Γ = 1) is realized insimulations, where for convenience the discretization of the equations of motions is done incylindrical coordinates. The governing equations are described in Section 1.2. The boundaryconditions are no-slip at all walls, isothermal plates on top and bottom (bottom plate tem-perature ∆T higher than top), and an adiabatic sidewall. Details of the numerical methodcan be found in Refs. [173, 174, 82]. The number of grid points in radial, azimuthal andvertical directions is 257, 513, and 513, respectively, satisfying the required resolution cri-teria [52, 147, 76]. Grid refinement is applied close to the sidewall, and to the bottom andtop boundaries. The resolution is checked a posteriori by validating that the Kolmogorovscale is resolved and there is an adequate number of grid points within the boundary layers(at least 10 points). Underresolved boundary layers result in too high Nu value, see e.g. Refs.[6, 175, 158]. Apart from simulations with Γ = 1, two other aspect ratios, Γ = 1/2 and 2,are also simulated for a few rotation rates. The radial, azimuthal and vertical resolutions are129× 513× 513 for Γ = 1/2 and 577× 769× 513 for Γ = 2, respectively.


Table 2.2: Parameters for the measurement series close to the top plate. The X and× indicatethat the corresponding experiments are performed or are not performed, respectively.

No. Ω Tmax Tmin Raa Roa Ek LPC HPC PIV(rad/s) (C) (C) ×109 (PTV) (PTV)

1 0 27 17 1.30 ∞ ∞ X X ×2 0.035 27 17 1.30 4.78 3.6×10−4 X X ×3 0.058 27 17 1.30 2.88 2.2×10−4 X X ×4 0.07 27 17 1.30 2.38 1.8×10−4 X X ×5 0.175 27 17 1.30 0.95 7.2×10−5 X X ×6 0.35 27 17 1.30 0.48 3.6×10−5 X X ×7 0.875 27 17 1.30 0.19 1.4×10−5 X X ×8 1.65 27 17 1.30 0.10 7.2×10−6 X X ×9 2 27 17 1.30 0.083 6.0×10−6 X X ×10 2.91 27 17 1.30 0.057 4.1×10−6 X X ×11 3.5 27 17 1.30 0.048 3.4×10−6 X X ×12 4.12 27 17 1.30 0.041 2.9×10−6 X X ×13 4.12 22 18.5 0.41 0.023 3.0×10−6 × X X14 4.12 23.5 18.5 0.61 0.028 3.0×10−6 × X X15 4.12 25.5 18.5 0.91 0.034 2.9×10−6 × X X16 4.12 27.5 18.5 1.25 0.039 2.8×10−6 × X X17 4.12 29.5 18.5 1.63 0.044 2.8×10−6 × X X18 4.12 37.5 18.5 3.49 0.062 2.5×10−6 × X ×19 1 21.6 20 0.19 0.065 1.2×10−5 × × X20 1.25 21.6 20 0.19 0.052 9.5×10−6 × × X21 1.5 21.6 20 0.19 0.043 8.2×10−6 × × X22 1.75 21.6 20 0.19 0.037 7.0×10−6 × × X23 2 21.6 20 0.19 0.032 6.2×10−6 × × X24 2.25 21.6 20 0.19 0.029 5.5×10−6 × × X25 2.75 21.6 20 0.19 0.023 4.5×10−6 × × X26 4.12 18.5 14 0.37 0.023 3.4×10−6 × × X27 4.12 19.5 14 0.48 0.026 3.3×10−6 × × X28 4.12 21 14 0.65 0.030 3.2×10−6 × × X29 4.12 22.5 14 0.81 0.034 3.2×10−6 × × X30 4.12 29 14 1.89 0.049 2.9×10−6 × × X31 1 31 26 0.94 0.13 1.0×10−5 × X ×

aThe reported Ra and Ro values in Chapters 3 and 4 (the experiments at rows 1-8 of this table are discussedin Chapters 3 and 4) are slightly different from the values reported in this table due to the calculation of the fluidproperties at a too high mean temperature (in Chapters 3 and 4 the fluid properties are taken at 24 C instead of 22C for calculation of Ra and Ro). The errors are not consequential. Thus, we retain the Ra and Ro values as thevalues published in the papers in Chapters 3 and 4.

Chapter 3

Transitions in turbulent rotatingconvention: A Lagrangianperspective

3.1 Introduction

In this chapter we use Lagrangian acceleration statistics from 3D–PTV and DNS to showthat the transition between regime I (rotation-unaffected) and regime II (rotation-affected) isactually a boundary-layer (BL) transition independent of the flow geometry: while the bulkflow is largely unaffected, the kinetic BLs on the plates transition between Prandtl–Blasius[4] and Ekman type [123, 75]. This sudden BL transition will thus take place independentof geometry and is a generic feature of rotating buoyancy-driven turbulence near horizontalwalls.

The remainder of the chapter is organized as follows. The flow parameters are given inSection 3.2. In Section 3.3, the experimental and numerical results on the particle accelerationand viscous boundary layers are presented and discussed. We summarize our main findingsin Section 3.4.

3.2 Methods and Parameters

Two sets of measurements are performed: (i) in the cell center, (ii) near the top plate. Theobservation volume of the cameras is 80× 60× 50 mm3 (x, y, z) at the cell center and closeto the top plate. In this chapter, the original measurement volume is cropped into smallervolumes: the analysis is performed on a volume of 50× 50× 50 mm3 (x, y, z) at the centerand 50 × 50 × 10 mm3 (x, y, z) near the top plate with their centers on the cylinder axis.

The contents of this chapter have been adopted from H. Rajaei, P. Joshi, K.M.J. Alards, R.P.J. Kunnen, F.Toschi, H.J.H. Clercx, “Transitions in turbulent rotating convention: A Lagrangian perspective”, Phys. Rev. E,93(4):043129, leaving out the introductory paragraphs.

33

34 CHAPTER 3. TRANSITIONS IN TURBULENT ROTATING CONVENTION

Based on the DNS we know that the viscous boundary layer near the top has a thicknessbetween 1.6 and 6 mm, depending on the rotation rate.

In the present work Ra = 1.28 × 109, Pr = 6.7 and Γ = 1 are kept fixed while Ro isvaried between 0.1 and ∞ (Ω between 1.65 and 0 rad/s), see rows 1 to 8 in Tables 2.1 and2.2. The number of data points (for the statistics presented in the succeeding sections) foreach experiment is ∼ 7× 107 at the center and ∼ 3.5× 107 near the top plate.

3.3 Results and discussion

To show the effects of rotation on the convective flow, we consider probability distributionfunctions (PDFs) of acceleration in the center and close to the top plate, separated into hori-zontal and vertical components.

−50 −40 −30 −20 −10 0 10 20 30 40 5010

−8

10−6

10−4

10−2

100

ax/armsx

PDF

c) Ro = inf.Ro = 5Ro = 3Ro = 2.5Ro = 1Ro = 0.5Ro = 0.2Ro = 0.1

−40 −20 0 20 4010

−8

10−6

10−4

10−2

100

az/armsz

PDF

d) Ro = inf.Ro = 5Ro = 3Ro = 2.5Ro = 1Ro = 0.5Ro = 0.2Ro = 0.1

−30 −20 −10 0 10 20 3010

−8

10−6

10−4

10−2

100

ax/armsx

PDF

a) Ro = inf.Ro = 5Ro = 3Ro = 2.5Ro = 1Ro = 0.5Ro = 0.2Ro = 0.1

−30 −20 −10 0 10 20 3010

−8

10−6

10−4

10−2

100

az/armsz

PDF

b) Ro = inf.Ro = 5Ro = 3Ro = 2.5Ro = 1Ro = 0.5Ro = 0.2Ro = 0.1

Figure 3.1: Normalized acceleration PDFs from experiments for Ra = 1.28× 109, Pr = 6.7and Γ = 1: (a) horizontal at the center, (b) vertical at the center, (c) horizontal near top (largerhorizontal axis range compared to other PDFs), and (d) vertical near top.

Figure 3.1(a) shows the normalized experimental horizontal-acceleration PDFs in thecell center. The shapes of the horizontal-acceleration PDFs are largely independent of Ω;they nearly collapse (except for the tails). This is further exemplified by the kurtosis val-ues Ki =

⟨a4i

⟩/⟨a2i

⟩2, plotted in Figure 3.2(a) (cyan triangles). All representative variables

for the horizontal directions are obtained by averaging the respective variables along x and ydirections. The transition point at Roc ∼= 2.5 [184] is indicated by the vertical dash-dottedline. A weak non-monotonic trend for the width of the horizontal-acceleration PDFs at the

3.3. RESULTS AND DISCUSSION 35

center is noticeable as a function of Ro: it has a local minimum at Ro ≈ 0.5. The widerPDF tails at higher and lower Ro indicate a more intermittent distribution, which is due tothe presence of coherent structures. At low enough Ro the vortical plumes can actually enterinto the center, a sign of adaptation to rotation of the bulk flow. However, note that Kcenter

xy

does not show significant changes around Roc. The horizontal-acceleration PDFs in the cen-ter are symmetrical, evidenced by the skewness Si =

⟨a3i

⟩/⟨a2i

⟩3/2being basically zero

for all Ro (Figure 3.2(b)). Finally, we present the rms values of horizontal-acceleration,armsxy = (arms

x + armsy )/2, in the center in Figure 3.3(a). The Lagrangian rms acceleration

shows a broad maximum between 1 . Ro . 2.5, while at lower Ro it is suppressed byrotation.

10−1

100

101

0

10

20

30

40

50

60

Ro

kurtosis

III

≀≀∞

Roc

a)K

xycenter

Kzcenter

Kxytop

Kztop

10−1

100

101

−2

−1.5

−1

−0.5

0

0.5

Ro

skew

ness

III

≀≀∞

Roc

b)

Sxycenter

Szcenter

Sxytop

Sztop

Figure 3.2: (a) Kurtosis and (b) skewness of experimental acceleration PDFs as a function ofRo.

The vertical-acceleration PDFs in the center are depicted in Figure 3.1(b). The PDFs areprogressively narrower for lower Ro, as is also clear from the kurtosis (Figure 3.2(a), bluecircles). The PDFs are symmetric (Figure 3.2(b)). The rms value of vertical-accelerationin the center displays a similar behavior as its horizontal counterpart (Figure 3.3(a)). Thesuppression of high-acceleration events along the rotation axis with increasing rotation hasbeen observed before in experiments on rotating turbulence [31].

The ratio RAA = armsxy /a

rmsz in the center, see Figure 3.3(b) (blue squares), is largely

independent of Ro and unaffected by the transition at Roc. Only at the lowest Ro a slightdownward trend can be observed. However, the downward trend is rather insignificant. Weconclude that the shapes of the acceleration PDFs as well as the isotropy of acceleration inthe cell center are largely unaffected while crossing Roc.

Continuing with the measurements near the top plate, the results change drastically. Thehorizontal-acceleration PDFs in Figure 3.1(c) (note that the horizontal axis range is largercompared to other PDFs) reveal a strong dependence on Ro with very wide tails at Ro ≈ 1and narrower distributions for both higher and lower Ro. This trend is emphasized in thekurtosis plot in Figure 3.2(a) (black diamonds), where a huge jump occurs at Roc. Justbelow Roc there is a significantly enhanced probability of extreme acceleration events. Thisis evidence of the BL transitioning from (non-rotating) Prandtl–Blasius to (rotating) Ekman


10−1

100

101

0.5

1

1.5

2

Ro

arm

s(m

m/s2)

≀≀∞

Roc

a)a

xyrms−center

azrms−center

axyrms−top

azrms−top

10−1

100

101

0

0.5

1

1.5

2

2.5

3

3.5

∞≀≀

Roc

Ro

RAA

=arm

sxy/arm

sz

b)centertop

Figure 3.3: (a) Acceleration rms values and (b) the ratio RAA = armsxy /a

rmsz as a function

of Ro. Purple open symbols represent DNS data for corresponding experiments (markershapes match). Experimental acceleration uncertainty is estimated by adding noise to thedisplacement, with an rms value equal to that of displacement measured for points on a fixedtarget inside the RB cell. The uncertainty in DNS is estimated as the deviation from theexpected flow symmetry.

type. Due to Ekman pumping the converging flow feeding a plume spins up cyclonically [77].This swirling flow greatly enhances horizontal-acceleration as also found in the accelerationrms values in Figure 3.3(a). The PDFs remain symmetric at all Ro (Figure 3.2(b)) given thatwithin the vortical structures both horizontal directions are equivalent.

The PDFs of vertical-acceleration near the top plate in Figure 3.1(d) are negatively skewed(see also Figure 3.2(b), turquoise squares): negative vertical-acceleration events are confinedto small areas but with large magnitudes, a sign of formation of coherent structures near theplates at all Ro. However, as Ro decreases the skewness is reduced. Similar strong localizedevents have been observed for Eulerian velocity as well [77]. Ktop

z steps up (milder com-pared toKtop

xy ) at Roc (Figure 3.2(a)), while getting progressively smaller for lower Ro. armsz

near the top is unaffected by rotation for Ro > Roc, while showing a similar decrease as thecentral rms values below Roc. Both horizontal and vertical acceleration rms values in thecell center, plotted in Figure 3.3(a), increase when Ro approaches Roc from the non-rotatingside, while rms values at the top show no changes.

The ratio RAA near the top plate, as depicted in Figure 3.3(b) (black circles), revealsagain the BL transition at Roc: due to Ekman pumping the acceleration becomes stronglyanisotropic below Roc, while being isotropic above. This is in stark contrast with the centerwhich remains isotropic. Anisotropy of acceleration displays a trend different from that ofEulerian velocity. Eulerian velocity statistics show an enhancement in anisotropy at the centerand a decrease near the top (at z = 0.8H) with increase in Ω [79].

Purple open symbols (with the same shapes) represent DNS a data for corresponding ex-

aTracer particles are incorporated into the code by Kim Alards. The simulations use a spatial grid resolution ofradial, azimuthal and vertical directions as 285, 513, and 513, respectively. To interpolate the fluid velocity from thesurrounding eight grid points around the particle position, we use a tri-linear interpolation scheme and for the timeintegration a second-order Adams–Bashforth scheme is used.


10−1

100

101

10−2

2.284δ

E/L

Ro

δu/L

≀≀∞

Roc, Γ = 1/2 1 2

Present DNS,Γ = 1DNS Ref. [...],Γ = 1Present DNS,Γ = 1/2Present DNS,Γ = 2GL theory,Γ = 1GL theory,Γ = 1/2Ekman BL

Figure 3.4: Normalized BL thickness as a function of Ro. The three vertical lines representtransitions for Γ = 1/2, 1 and 2.

periments in Figure 3.3(a-b). Apart from the lower magnitudes of the experimental data, thetrends for DNS and experiments are similar. The results of DNS and experiments deviate forthe horizontal component near the top due to a reduced likelihood of capturing high accelera-tion events in the experiment. DNS and experiments match at the center since the accelerationrms are generally lower. The DNS data for skewness and kurtosis show similar trends as theexperiments, however, the values are generally higher (by ∼ 20− 100%). Since kurtosis andskewness are highly sensitive to extreme events the error bars are large. Therefore, we aremore interested in the trends rather than the actual values.

For further confirmation of the BL transition, we consider the DNS data for three aspectratios; 1/2, 1, and 2. Following Refs. [159, 82, 73], we define the thickness of the kinematicBL, δu, as the position of the maximum horizontal rms velocity. Figure 7.3.1 shows the δu forthe present DNS data and that of Ref. [82] (at Ra = 1× 109, Pr = 6.4). In addition to DNS,we estimate δu for the non-rotating case employing Grossmann–Lohse (GL) theory [48, 49,50, 51, 157] for Γ = 1 and 1/2: δu = aL/

√Re [48, 50] where a ≈ 0.922 (0.684) for Γ = 1

(1/2) [157]. The blue dashed and magenta solid lines in Figure 7.3.1 show the theoretical(Prandtl–Blasius) BL thickness based on GL theory for Γ = 1 and 1/2, respectively. Theymatch well with the DNS data for corresponding Γ. Close to the transition at Roc thesetheoretical estimates no longer hold, as rotation starts to change the BL dynamics.

We use linear Ekman BL theory to describe the BLs under rotation. The problem of thematching of a geostrophic bulk flow with velocity uI in x direction to a solid boundary canbe easily solved [75]:

uE = uI [1− e−z/δE cos(z/δE)] , (3.1a)

vE = uIe−z/δE sin(z/δE) , (3.1b)


where u and v are the horizontal velocity components in the Ekman BL, δE =√ν/Ω is

the characteristic Ekman BL thickness, and z is the distance to the plate (see also Equations(1.24) when vI = 0). Note that δE is independent of the exterior velocity and the flowgeometry. This generic solution suffices for RRBC simulations where the orientation of theLSC is not fixed to the coordinate system. The BL velocity actually possesses a maximum√u2E + v2

E = 1.069uI at z/δE = 2.284, before asymptotically reducing to uI at larger z.This theoretical prediction based on linear BL theory accurately describes the turbulent sim-ulation data including the prefactor, see the turquoise line in Figure 7.3.1. We conclude thatindeed the BLs change from Prandtl–Blasius to Ekman type at Roc. The applicability of Ek-man theory has been validated in a wide parameter range [84, 82, 80, 83, 153]. There is noprimary dependence on Ra or Pr.

3.4 ConclusionsUsing Lagrangian acceleration statistics and accompanying DNS, we find an abrupt trans-ition in the BL dynamics of RRBC between Prandtl–Blasius type and Ekman type BLs. Thistransition in BLs coincides with the transition reported earlier and predicted surprisingly wellby employing Ginzburg–Landau theory in Ref. [184]. Our analysis contributes to the under-standing of the transition by describing the physics of the BLs which drive this process. TheBLs related to the LSC are of Prandtl–Blasius type; as their characteristic thickness δu be-comes comparable to δE the transition occurs and the vortical plumes aid in the disappearanceof the LSC [78].

We have presented the first measurement of Lagrangian acceleration in rotating Rayleigh–Bénard convection. 3D-PTV and DNS have revealed that the turbulent transition in RRBCis a boundary-layer transition. Post-transition, where the BL is already rotation-dominated,the bulk becomes progressively affected by rotation: fluctuations are reduced and vorticalplumes can enter deeper into the bulk, leading to opposite trends in the kurtosis of verticaland horizontal accelerations. We expect that the subsequent regime change at high rotationhappens when the entire bulk becomes rotation-dominated [67].

Chapter 4

Flow anisotropy in rotatingbuoyancy-driven turbulence

4.1 IntroductionKolmogorov’s hypothesis of local isotropy states that the turbulent flow statistics should beuniversal at sufficiently high Reynolds numbers and small scales; they are independent ofthe large-scale flow structure. According to the hypothesis, the information of the preferreddirection of the large-scale flow is lost during the cascade from large to small scales. Severalnumerical and experimental studies address the effects of large-scale anisotropy on the smallscales for different types of flow. Contrary to Kolmogorov’s hypothesis of local isotropy,small-scale anisotropy is reported in the flow between two counter-rotating disks [122], shearflow [145, 181], channel flow [11], and turbulent shear-less mixing [166]. Since the majorityof simulations and models rely on the assumption of local isotropy and all turbulent flowsin nature and technological applications suffer from large-scale anisotropy, the study of theeffects of large-scale anisotropy on small scales plays a major role in our understanding ofthese flows.

In the present chapter, we focus on large- and small-scale anisotropy induced by back-ground rotation in Rayleigh–Bénard convection. It has been reported earlier in Ref. [79]that the background rotation in RBC amplifies the large-scale anisotropy in the bulk whileit reduces anisotropy closer to the plate at height z = 0.8H (H is the height of the con-vection cell). In the present chapter, we first analyze the effects of background rotation onthe large-scale anisotropy at three different heights in rotating RBC. Then we focus on theeffects of rotation-induced large-scale anisotropy on the small scales. We use a combinedexperimental–numerical approach to investigate the small-scale anisotropy from Eulerian andLagrangian viewpoints.

The remainder of this chapter is organized as follows. We start with the experimentalparameters in Section 4.2. In Section 4.3.1, the results on large-scale (an)isotropy are dis-cussed based on velocity statistics and the inertial range of the second-order Eulerian struc-

The contents of this chapter have been adopted from H. Rajaei, P. Joshi, R.P.J. Kunnen, H.J.H. Clercx, “Flowanisotropy in rotating buoyancy-driven turbulence”, Phys. Rev. Fluids, 1(4):044403.

39

40 CHAPTER 4. FLOW ANISOTROPY IN ROTATING BUOYANCY-DRIVEN

ture function (SF). Next, we discuss the results on small-scale (an)isotropy in Section 4.3.2based on the dissipative range of the second-order Eulerian SF and acceleration statistics. Wesummarize our main findings in Section 4.4.

4.2 Experimental parameters

In the present study Ra = 1.3× 109, Pr = 6.7 and Γ = 1 are the fixed parameters while Rovaries between 0.1 and ∞ (Ω between 1.65 rad/s and 0). These experiments correspond tothe rows 1 to 8 in Tables 2.1 and 2.2. The mean temperature is set to 22C and ∆T = 10C.

In this chapter, the velocity fluctuation in the i direction is defined as

ui =⟨(Ui − 〈Ui〉)2

⟩1/2, (4.1)

where Ui is the ith component of the velocity signal and 〈· · · 〉 indicates the time average.Note that the statistics in x and y directions are almost the same, i.e. ux ∼= uy . Thus,we only present the values in one direction (x direction) and refer to it as the horizontalcomponent, uh. Likewise, the acceleration fluctuation in the i direction is defined as

ai =⟨(Ai − 〈Ai〉)2

⟩1/2, (4.2)

where Ai is the ith component of the acceleration signal and 〈· · · 〉 indicates the time aver-age defined as above. Similar to the velocity fluctuation, the horizontal component of theacceleration fluctuation is represented by ax and we refer to it as ah.

In this chapter, we discuss the measurements in three different regions. As in Chapter 3,the original observation volume of the cameras (80× 60× 50 mm3 (x, y, z)) is cropped intosmaller subvolumes. Region I is a volume of 50 × 50 × 50 mm3 (x, y, z), which is locatedin the center of the cylinder thus covering the height 0.375H < z < 0.625H . Region II isa volume of 50 × 50 × 20 mm3 (x, y, z) placed between z = 0.75H and z = 0.85H withits center on the cylinder axis. Region III is close to the top plate, a volume of 50 × 50 ×10 mm3 (x, y, z) with its center on the cylinder axis covering the height 0.975H < z < H .As mentioned before, the original observation volume of the cameras is 80 × 60 × 50 mm3

(x, y, z). Regions II and III are two subvolumes of the same original observation volume. Forthe remainder of the chapter, we shall refer to these locations by the vertical position of theircenter, i.e. z = 0.5H , z = 0.8H and z = 0.975H refer to regions I, II and III, respectively.The statistics are based on approximately 7× 107 data points at the center and z = 0.8H andapproximately 3.5 × 107 data points for z = 0.975H . Note that the kinetic boundary layerat the top plate has a thickness between 1.6 and 6 mm, depending on Ro [132]. Therefore,between 16 and 60% of the upper measurement volume at z = 0.975H is occupied by thekinetic boundary layer.

4.3 Results

In this section, we first discuss the effects of background rotation on large scales. Later on,the small-scale (an)isotropy is examined.

4.3. RESULTS 41

4.3.1 Large-scale (an)isotropyVelocity statistics

It has been shown in Ref. [79], by using the so-called Lumley map, that rotation enhancesthe large-scale anisotropy at the cell center, while it reduces anisotropy close to the top plate(based on planar stereo particle image velocimetry measurements at z = 0.8H). The velocityfluctuation is the velocity scale typical for the largest turbulent scales in the flow. Therefore,anisotropy of the velocity fluctuation indicates anisotropy of the large scales.

As mentioned in Section 1.6, the LSC is the main feature of the flow for Ro & 2.5(for Γ = 1) [78, 184]. The presence of the LSC leads to a non-zero mean for horizontalvelocity at z = 0.8H and z = 0.975H . Therefore, it is required to subtract the mean ofthe velocity from the velocity signal to obtain the velocity fluctuation. As mentioned earlier,the experiment for each rotation rate consists of multiple segments each approximately 11minutes long. The mean of the velocity is calculated based on the mean of the velocity foreach of these segments. Thus, the time averages in Equations (4.1) and (4.2) are performedon approximately 11 min segments. The mean is calculated for each segment since the LSCorientation changes under rotation: the LSC precesses against the direction of rotation [78,55, 194]. Therefore, it is meaningless to use the mean value of the velocity for the entireduration of the experiments (∼ 300 min). Note that these subtractions are only applied on thehorizontal component of the velocity for Ro ≥ 2.5 at z = 0.8H and z = 0.975H where thehorizontal velocity possesses a non-zero mean. For Ro < 2.5 at z = 0.8H and z = 0.975Hand for all measured Ro at the cell center, the velocity has a zero mean and thus no averagingis used.

It is worth mentioning that the calculated mean velocity value depends on the duration ofthe averaging. A shorter averaging time results in a truncation of the low frequency fluctu-ations and thus an underestimation of the velocity fluctuation. Therefore, the uncertainty inthe velocity fluctuation for Ro ≥ 2.5 at z = 0.975H and z = 0.8H is higher compared toother data points.

Figure 4.1(a) shows the velocity fluctuations as a function of Ro for the three meas-urement sets. The results from the current Lagrangian investigation are in good qualitativeagreement with earlier measurements [82] using stereoscopic particle image velocimetry (anEulerian velocity measurement) in a similar system at comparable Ro, Ra and Pr. As can beseen from the graph, at the center (z = 0.5H) the vertical and horizontal velocity fluctuations(blue squares and circles, respectively) are approximately equal for Ro = ∞ (non-rotatingcase). The vertical component is significantly enhanced under weak rotation (2.5 ≤ Ro ≤ 5).A similar but much weaker trend is observed for its horizontal counterpart. For Ro . 2.5,both horizontal and vertical components decrease with decreasing Ro due to the suppressionof turbulence intensity by the background rotation. The horizontal and vertical velocity fluc-tuations scale as u ∼ Ro0.22 (u can be either horizontal or vertical component of the velocity)at the center for Ro ≤ 2.5; similar scaling has been reported for the Eulerian velocity at com-parable Ra and Pr, u ∼ Ro0.2 [82]. The ratio RU = uz/uh, plotted in Figure 4.1 (b), canbe used as a measure of (an)isotropy of the large-scale velocity fluctuations. The ratio RU atthe cell center, blue squares in Figure 4.1 (b), shows clearly that the large-scale anisotropy inthe velocity field increases with decreasing Ro.

For 2.5 ≤ Ro ≤ 5, measurements of the velocity fluctuation values at z = 0.8H , cyantriangles and stars in Figure 4.1(a), show similar behavior as the measurements at the cell cen-


ter; a significant and a slight increase in vertical and horizontal components, respectively. ForRo < 2.5 both vertical and horizontal velocity components decrease with increasing rotationrate, similar to those from the cell center measurements. However, reduction of the verticalcomponent is higher than that of the horizontal one. This is clear from the anisotropy ratioRU , Figure 4.1(b), which approaches one with decreasing Ro. For Ro > 2.5, the large-scaleanisotropy seems to increase from non-rotating to weakly rotating RBC (but note that there ishigher uncertainty for Ro > 2.5). It can be concluded that large-scale anisotropy decreaseswith increasing rotation rate at z = 0.8H for Ro ≤ 2.5. Our findings for measurements atthe center and at z = 0.8H are consistent with earlier reported results in Ref. [79] based onEulerian velocity measurements.

At z = 0.975H , the vertical and horizontal velocity fluctuations are approximately con-stant for Ro & 2.5. The horizontal and vertical velocity fluctuations decrease with reductionin Ro for Ro < 2.5. The anisotropy ratio RU decreases with decrease in Ro (for Ro ≤ 5)and approaches 0.4 for the lowest Ro (highest rotation rate) in our experiment which indicatesthat large-scale anisotropy increases with decrease in Ro at z = 0.975H .

For the measurements at z = 0.975H (and focusing on Ro . 2.5), the horizontal velocityis larger than the vertical one, which is consistent with the presence of strong vortical plumesthat add swirling motion in the horizontal plane [79]. However, for the measurements at thecell center, the vertical component is larger than the horizontal one which is an indicationthat the vortical plumes spin down (become weaker) as they reach the center. The transitionbetween stronger large-scale horizontal motions at the top and stronger large-scale verticalmotions at the cell center occurs at z ≈ 0.8H .

10−1

100

101

0.5

1

1.5

2

2.5

3

3.5

4

Ro

u(m

m/s)

≀≀∞

a)

10−1

100

101

0.4

0.6

0.8

1

1.2

1.4

1.6

∞≀≀

Ro

RU

=uz/uh

b)Center0.8H0.975H

Figure 4.1: (a) Horizontal and vertical velocity fluctuations from the experiments. Bluecircles and squares are the horizontal and vertical components at the cell center, respectively.Cyan triangles and stars are the horizontal and vertical components at z = 0.8H , respect-ively. Red pluses and diamonds are the horizontal and vertical components at z = 0.975H .(b) The anisotropy ratio RU = uz/uh.

In conclusion, we have shown that the large-scale anisotropy increases with increase ofrotation rate both at the center and near the plate (z = 0.975H). However, the nature of theanisotropy at the center and near the top plate (z = 0.975H) are opposite in the sense that at

4.3. RESULTS 43

the center the vertical large-scale motions are stronger while at z = 0.975H the horizontallarge-scale motions are stronger. At z = 0.8H , the large-scale anisotropy seems to decreasewith increasing rotation.

Second-order Eulerian structure function in the inertial range

It has been shown that the second-order Eulerian structure function (SF), defined as D(r) =⟨(u(x+ r)− u(x)

)2⟩with r the magnitude of separation, can be satisfactorily obtained

from 3D-PTV data [121, 113]. In this case, using 3D-PTV data, D(r) is calculated based onrandomly distributed particles which requires the assumption of isotropy and homogeneity inthe flow field. It might therefore serve as a test how well isotropy is satisfied. The dissipationrate can be calculated in homogeneous and isotropic turbulence (HIT) based on the second-order Eulerian SF in both inertial (η r L) and dissipative (r < η) ranges, with ηthe Kolmogorov length scale and L the integral length scale. In this section, we focus onthe equations valid for the inertial range of HIT to provide evidence of (an)isotropy at thesescales. Anisotropy at inertial scales implies anisotropy at large scales since it is supposed thatthe anisotropy is introduced at the large scales.

Within the inertial range of HIT, turbulent flow characteristics can be defined by the dis-sipation rate alone. The following equations relate the second-order Eulerian SF in the inertialrange to the dissipation rate:

DLL(r) = C2(εr)2/3, (4.3)

DNN (r) =4

3C2(εr)2/3, (4.4)

whereDLL(r) is the second-order Eulerian longitudinal SF, ε is the dissipation rate, C2 is theuniversal Kolmogorov constant with C2 ≈ 2.13 [152], r is the magnitude of separation, andDNN (r) is the second-order Eulerian transversal SF.DLL(r) is calculated as follows; at eachPTV time step,

(u1 − u2

)2is calculated for each pair of randomly distributed particles with

separation r±δr (δr is the bin size used for the calculations), where u1 and u2 are respectivelythe projections of the velocities of the first and second particle along r (r is defined as a unitvector parallel to the line connecting the two particles). DLL(r) is an average over all pairswith the same separation r ± δr. DNN (r) is calculated with the same method where u1 andu2 are the projections of the velocities of particles along a line perpendicular to r and e, withe an arbitrarily chosen unit vector here, defined as e = (1, 1, 1)/

√3. Note that in HIT, u1 and

u2 can be any projection perpendicular to r. Therefore, the direction of unit vector e is notimportant.

Figure 4.2(a-b) shows the compensated structure function, calculated as both(DLL(r)/C2)3/2/r and (3DNN (r)/(4C2))3/2/r withC2 as proposed earlier, based on Equa-tions (4.3) and (4.4), respectively, for Ro =∞ (Figure 4.2(a)) and Ro = 0.1 (Fig 4.2(b)) as afunction of separation distance at the cell center. It is reported in Ref. [43], using this method,that this curve should reach a plateau in the inertial range of HIT at a level corresponding toε. However, as can be seen from Figure 4.2(a-b), no clear plateau is reached in these cases.Figure 4.2(a), for Ro =∞, shows an approximate plateau for DNN . However, the measuredε based on this plateau (∼ 1.4 × 10−7 m2/s3 for ε evaluated from DNN ) is considerablysmaller than that of DNS data (∼ 2.6× 10−7 m2/s3, see Figure 4.4). On the other hand, forRo = 0.1 a steep decay is observed after the maximum value for ε. The lack of a plateau


is due to both the large-scale anisotropy and the comparatively low Taylor-based Reynoldsnumbers (Reλ ∼ 39 for Ro = ∞ and Reλ ∼ 8 for Ro = 0.1) which limits the extent ofthe inertial scaling range. The Taylor-microscale Reynolds number, Reλ, is estimated hereusing the isotropic relation Reλ = u2

√15/εν with u = (ux + uy + uz)/3 (ux, uy and uz

are all three positive numbers, see Sec. 4.2). Let us consider the non-rotating case. The flowfield at the center of RBC for Ro =∞ is reported to be close to homogeneous and isotropicturbulence [161]. However, the measured dissipation rate based on the inertial range of thesecond-order structure function for Ro =∞ does not result in the correct value of dissipationrate. Therefore, it is expected that the limited inertial range is the more direct reason for thenon-rotating case (Figure 4.2(a)). However, Figure 4.2(b) (Ro = 0.1) is different as alsoanisotropy plays a role.

The same analysis is done for the measurements at z = 0.8H and z = 0.975H forRo ≤ 1 when the mean value of the velocity is zero. A rapid decay, similar to the one inFigure 4.2(b), is observed.

0 20 40 60 80 1000

0.5

1

1.5

2x 10

−7

r (mm)

(DLL/C

2)3

/2 /r,

(3D

NN/(4C

2))

3/2 /r

a)(DLL/C2)

3/2/r(3DNN/(4C2))

3/2/r

0 20 40 60 80 1000

0.5

1

1.5

2x 10

−7

r (mm)

(DLL/C

2)3

/2 /r,

(3D

NN/(4C

2))

3/2 /r

b)(DLL/C2)

3/2/r(3DNN/(4C2))

3/2/r

Figure 4.2: Compensated structure functions for (a) Ro = ∞ and (b) Ro = 0.1 based oninertial-range scaling at the cell center from the experiments.

It can be concluded that the inertial range (if present at these values of Reλ) has signsshowing anisotropy at all three measurement locations.

4.3.2 Small-scale (an)isotropySecond-order Eulerian structure function in the dissipative range

Continuing with the analysis of the second-order SF, we test the dissipative range (r < η)to measure the compensated structure function and the dissipation rate. At small scales, HITflow characteristics are determined by the dissipation rate and viscosity. It can be shown[107] that for the dissipative range in HIT the following equations hold:

DLL(r) = εr2/15ν, (4.5)

DNN (r) = 2εr2/15ν. (4.6)

4.3. RESULTS 45

This method has been used before to calculate the dissipation rate at the cell center in non-rotating RBC [113]. Figure 4.3 shows the compensated structure function calculated as both15νDLL(r)/r2 and 15νDNN (r)/(2r2) based on Equations (4.5) and (4.6), respectively, forRo = ∞ and Ro = 0.1 at the cell center. The probability of finding a velocity pair with asmall separation r is considerably lower than at comparatively larger separation. The scatterof the data at small scales is due to the lack of convergence at these scales. However, theplateau is well-defined. The black lines are fits to the data to determine ε. Similar plateausare observed for the measurements at z = 0.8H and z = 0.975H for all Ro as well.

100

101

10−8

10−7

10−6

10−5

r (mm)

DNN(r)×15ν/(2r

2),

DLL(r)×15ν/(r2)

Figure 4.3: Compensated structure functions at the cell center for Ro = ∞ (pink circles,DNN (r) × 15ν/(2r2); black triangles, DLL(r) × 15ν/(r2)) and Ro = 0.1 (red squares,DNN (r)× 15ν/(2r2); blue stars, DLL(r)× 15ν/(r2)) based on the dissipative range fromthe experiments.

The Kolmogorov length scale varies between 0.93 and 1.38 mm in the current experi-ments. The dissipation-range scaling is valid up to ∼ 10η [93]. As can be seen from Figure4.3, the current technique resolves down to 0.4 mm. Therefore, the dissipation range is indeedresolved.

The experimental dissipation rate calculated based on the dissipative range and the dis-sipation rate calculated directly from DNS are plotted in Figure 4.4 for all Ro. The DNSdissipation rate is calculated directly without any assumption of HIT, while the experimentaldissipation rate is calculated based on the assumption of HIT at small scales. The compar-ison shows a very good agreement between DNS and experimental dissipation rate at the cellcenter, see blue filled and open diamonds. At z = 0.8H , the experimental dissipation ratesfollow the trends of the DNS, while they are generally lower than those from the DNS, seered filled and open circles. Close to the top plate at z = 0.975H , neither the trend nor thevalues match with the DNS dissipation rates, see black filled and open squares.

Surprisingly good agreement between DNS and experimental dissipation rate at the cellcenter indicates that the flow field is indeed homogeneous and isotropic at dissipative scales.


At z = 0.8H the velocity fluctuations of the small-scale flow field is not far from isotropyas well, however, at z = 0.975H , the assumption of HIT is no longer supported in thedissipative range since the experimental dissipation values are different from DNS data andshow a different trend with regard to Ro. In general, the small-scale isotropy deterioratesfrom center to top, due to the proximity of the wall.

10−1

100

101

0

0.2

0.4

0.6

0.8

1

1.2

1.4x 10

−6

∞≀≀

Ro

ǫ(m

2/s3)

Figure 4.4: Dissipation rate for experiment and DNS as a function of Ro. Open symbols arefor DNS and filled symbols are for experiments. Blue diamonds, cell center; red circles, atz = 0.8H; black squares, at z = 0.975H .

Acceleration constant

In sufficiently small space-time regions in HIT the variance of the Lagrangian accelerationdepends only on dissipation and viscosity. The following equation can be derived [107]:⟨

a2k

⟩= a0ε

3/2ν−1/2, (4.7)

where ak is the Lagrangian acceleration component in direction k and a0 is a universal con-stant. The above equation was derived by Heisenberg [59] and Yaglom [190], and is usuallyreferred to as the Heisenberg-Yaglom relation. It is known that the viscosity influences onlythe small-scale flow motions for large Re [107]. Furthermore, the acceleration scale of thedisturbances of length scale l increases with decreasing l. Therefore, it can be concluded thatLagrangian acceleration is determined largely by small-scale flow motions (r < η) [107].Thus, we use the acceleration statistics to further evaluate the small-scale isotropy in rotatingRayleigh–Bénard convection.

The acceleration fluctuations are reported in Chapter 3 and Ref. [133]. Here, we focuson the values of the constant a0. Generally, the a0 values are a function of Reλ. We coverdifferent Reλ, which are achieved by changing the rotation rate rather than a variation of Ra

4.3. RESULTS 47

101

102

103

100

101

a0

Reλ

Figure 4.5: The Heisenberg-Yaglom constant a0 versus Reλ. Filled symbols are from thecurrent experiments. Red circles and blue squares are horizontal and vertical component atthe center, respectively. Cyan triangles and green six-point stars are horizontal and verticalcomponents at z = 0.8H , respectively. Yellow pluses and dark red diamonds are the hori-zontal and vertical components at z = 0.975H , respectively. The estimated error for our datais approximately 15 − 25%. Open symbols from previous studies; downward-pointing tri-angles and five-point stars are transverse and axial components, respectively, for experimentson turbulence generated between coaxial counter-rotating disks from [85]; circles for exper-iments on non-rotating Rayleigh–Bénard convection at the cell center from [114], six-pointstars isotropic turbulence DNS data from [45]; and right-pointing triangles isotropic turbu-lence DNS data from [172]. The black line (taken from [114], a0 = 4.6/(1 + 77.5/Reλ))is a fit to all experimental (only axial component from [85]) and numerical data.

or Pr. Table 4.1 shows the relation between Reλ and Ro at the three measurement locationsfor all eight experiments. The estimated values of Reλ for the measurements at z = 0.8Hand z = 0.975H for Ro ≥ 2.5 have higher uncertainty due to the subtraction of mean valuesrelated with the presence of the LSC, as discussed in Sec. 4.3.1. The values of a0 for differentReλ are plotted in Fig 4.5. Filled symbols are data of the current study and open symbolsrepresent the data from previous studies [85, 45, 172, 114]. The black line is a fitted curve tothe data in literature, taken from [114].

The calculated a0 at the center shows no appreciable dependence on direction, see theblue squares and red circles in Figure 4.5. For Ro = 0.1 the horizontal component is smaller,however, the difference is within the experimental uncertainty (∼ 15−25%). For z = 0.8H , adifference between horizontal and vertical directions is noticeable, see the cyan triangles andgreen six-point stars in Figure 4.5, which is an indication that the small scales are becominganisotropic. The a0 values are still in the range of the predicted values for HIT. For z =


10−1

100

101

0.5

1

1.5

2

a0

Ro∞

≀≀

Figure 4.6: The Heisenberg-Yaglom constant a0 from the experiments versus Ro. Red circlesand blue squares are horizontal and vertical component at the center, respectively. Cyantriangles and green stars are horizontal and vertical components at z = 0.8H , respectively.Yellow pluses and dark red diamonds are the horizontal and vertical components at z =0.975H , respectively.

Table 4.1: Reλ as a function of Ro at the three measurement locations for all eight experi-ments.

RoPosition 0.1 0.2 0.5 1 2.5 3 5 ∞Center 8 10 15 21 38 39 34 39z = 0.8H 10 13 17 21 30a 30a 27a 28a

z = 0.975H 11 14 18 21 25a 24a 24a 27a

ahigher uncertainty due to the presence of the LSC.

0.975H , the values of a0 for horizontal and vertical directions are significantly different forRo . 2.5, see yellow pluses and dark red diamonds in Figure 4.5. Furthermore, a0 for thehorizontal component is much larger than what is expected for HIT, while for the verticaldirection it is smaller than expected. For Ro ≥ 2.5, the values of a0 for horizontal andvertical directions collapse on top of each other and they are close to the predicted value forHIT. However, the uncertainty in the estimation of Reλ is high and should be kept in mind.

Figure 4.6 shows a0 as a function of Ro. The symbols and colors are the same as inFigure 4.5. The values of horizontal and vertical a0 decrease with decreasing Ro for themeasurements at the cell center for Ro < 2.5 (red circles and blue squares in Figure4.6). Atz = 0.8H , the horizontal component of a0 (cyan triangle) does not experience significant

4.4. CONCLUSIONS 49

changes for Ro > 1, whereas it slightly decreases for Ro < 1. However, the vertical com-ponent of a0 at z = 0.8H decreases with decreasing Ro. The horizontal component of a0 atz = 0.975H increases with decreasing Ro due to the formation of vortical plumes near theplate which add a swirling motion in the horizontal plane. However, the vertical componentdecreases with decreasing Ro. The effects of rotation on the a0 values is similar to that ofthe acceleration fluctuations. For a detailed discussion on the effects of the rotation on theacceleration fluctuations we refer to Ref. [132].

It can be concluded that the large-scale anisotropy is washed away in the dissipative rangeat the center of the cell, while at z = 0.975H the small scales remain anisotropic with regardto small-scale velocity fluctuations and acceleration variance for Ro < 2.5. At z = 0.8H ,the small-scale flow seems not far from isotropy.

4.4 Conclusions

In summary, we have studied the large-scale and small-scale anisotropy in rotating Rayleigh–Bénard convection (RBC) at three different heights from Lagrangian and Eulerian pointsof view by laboratory experiments and direct numerical simulation (DNS). We have shownthat rotation enhances the large-scale anisotropy of velocity fluctuations at the cell centerand close to the top plate (z = 0.975H), while at z = 0.8H , this large-scale anisotropy isreduced. The nature of anisotropy at the center and and near the top plate (z = 0.975H)are opposite in the sense that at the center the vertical large-scale motions are stronger whileat z = 0.975H the horizontal large-scale motions are stronger. Moreover, we have shownbased on the inertial range of the second-order structure function that large-sale anisotropyincreases at the cell center. However, this tool cannot help us further for study of large-scaleanisotropy at z = 0.8H and z = 0.975H .

Furthermore, we have looked into the effects of large-scale anisotropy, induced by rota-tion, on the small scales. In order to evaluate the small-scale isotropy, we have used twoapproaches. First, the second-order Eulerian structure function (SF) has been analyzed.More specifically, we have examined the experimental dissipation rate based on equationsvalid for homogeneous and isotropic turbulence (HIT) in the dissipative range of the Eu-lerian SF. At the center of the cell, the experimental dissipation rates for different rotationrates (with assumption of HIT) agree very well with the dissipation rate calculated usingDNS. At z = 0.8H , the experimental dissipation-rate values follow the trend of the DNSdissipation rate, while the experimental values are generally smaller than the DNS values. Atz = 0.975H , the experimental values show different trends compared to the DNS data. Asfor the second approach, we have analyzed the Heisenberg-Yaglom Lagrangian accelerationconstant for different rotation rates. At the cell center, the data shows no dependency on thedirection of the acceleration at the cell center. Furthermore, the values of a0 for differentrotation rates show good agreement with the predicted values for HIT for this range of Reλat the cell center. At z = 0.8H , a difference between horizontal and vertical accelerationconstant is noticeable which might be an indication that the small scales are becoming aniso-tropic. However, the difference is still within the experimental uncertainty. At z = 0.975H ,the values of a0 computed from horizontal and vertical acceleration components are differentand they do not match with the predicted values for HIT. Thus, based on both approaches wecan conclude that the small scales remain isotropic at the cell center. At z = 0.8H , the small-


scale flow seems not far from isotropy. However, small anisotropic behavior is observed. Atz = 0.975H , small scales are significantly anisotropic.

It can be concluded that the large-scale anisotropy induced by background rotation hashardly any effects on the small scales of the flow in the central region of rotating RBC, whilethe small-scale velocity and acceleration fluctuations remain anisotropic near the plates(heremeasured at z = 0.975H).

We hope that this study contributes to the extension of existing theories and argumentsfor HIT towards realistic flows in which anisotropy is omnipresent.

Chapter 5

Geometry of tracer trajectories

5.1 Introduction

It has been shown in the previous chapters that the Lagrangian velocity and accelerationstatistics of the neutrally buoyant particles provide gainful insight into the flow dynamics.In this chapter, we examine the shape of particle trajectories which provides information onthe geometrical aspects of the flow [15, 141, 189]. The shape of a particle trajectory in 3Dspace can be fully described by its curvature and torsion. The presented work focuses on thecurvature statistics.

There have been only few experimental [189] and numerical studies [141, 27] in homo-geneous and isotropic turbulence (HIT) on the geometrical aspects of trajectories (curvatureand torsion). These HIT studies revealed pronounced power laws in the curvature and torsionprobability distribution functions (PDF): the tails of the curvature PDFs scale as κ1 for smallcurvature values and κ−5/2 for large curvature values [15, 189], where κ is the curvature.If we assume that the velocity components are Gaussian random variables, it is possible toderive these power laws (κ1 and κ−5/2) analytically as well [189, 5]. The analytical solutionis valid for HIT, however, the roles of anisotropy, inhomogeneity and non-Gaussian velocitystatistics on the curvature statistics and the aforementioned power laws are not clear yet. Theanswer to this question is important since turbulence models rely on the assumption of HITwhile most of the turbulent flows in the nature and technological applications do not meetthese conditions. In this chapter, we study the curvature of trajectories in rotating Rayleigh–Bénard convection (RRBC): a system which possesses inhomogeneous regions (boundarylayer (BL) versus bulk), different levels of anisotropy depending on the background rotation[133], and both Gaussian and non-Gaussian velocity statistics depending on the backgroundrotation.

As mentioned earlier, the geometry of a particle trajectory in 3D space is fully describedby two geometrical parameters: the curvature, κ, and the torsion, τ . These parameters are

This chapter is another representation of the data published in K.M.J. Alards, H. Rajaei, L. Del Castello, R.P.J.Kunnen, F. Toschi, H.J.H. Clercx, “Geometry of tracer trajectories in rotating turbulent flows”, Phys. Rev. Fluids,2(4):044601, leaving out discussion of torsion, self similarity and electro magnetic rotating turbulence.

51

52 CHAPTER 5. GEOMETRY OF TRACER TRAJECTORIES

described as (see Ref. [15] for the derivation),

κ =|u× a||u|3

, (5.1)

τ =u · (a× a)

(u · u)3κ2, (5.2)

where u is the particle velocity, a is the particle acceleration and a is the derivative of theparticle acceleration with respect to time. As one can see, the derivative of the accelerationwith respect to time is required for the torsion calculation. However, it is not possible toretrieve a with sufficient accuracy from the experiments. Therefore, we only discuss thecurvature results; we refer to Ref. [5] for the analysis of the torsion data from DNS.

The remainder of this chapter is organized as follows. The experimental parameters arepresented in Section 5.2. In Section 5.3.1, the curvature PDFs at the cell center and nearthe top plate for nonrotating RBC are presented and discussed. The discussion over thenonrotating RBC curvature PDFs is extended to the rotating frame in Section 5.3.2. Wesummarize our main findings in Section 5.4.

5.2 Experimental parametersThe data used in this chapter are for Ra = 1.3 × 109, Pr = 6.7 and Γ = 1 while Ek variesbetween 7.2× 10−6 and∞ (Ω between 1.65 rad/s and 0). These experiments correspond tothe rows 1 to 8 in Tables 2.1 and 2.2.

The curvature data presented in this chapter are based on the data collected in a volumeof approximately 80 × 60 × 50 mm3(x, y, z) for both the center and close to the top plate.The measurement at the cell center covers a volume between z = 0.375H and z = 0.625Hwhile the measurement at the top covers a volume between z = 0.75H and z = H . Theexact same measurement volumes are chosen for the numerical simulations a.

5.3 Results

5.3.1 Nonrotating RBCWe start with the curvature data for the nonrotating RBC at the cell center. It is known that theflow field at the cell center of RBC is close to the HIT condition [133, 82, 196]. However, thevertical velocity component shows non-Gaussian distribution for Γ = 1 [82, 71], with Γ beingthe cell aspect ratio. The non-Gaussian distribution of the vertical component of the velocityfor Ek = ∞ is also confirmed by our experimental and numerical data, see Figure 5.1 fromexperiments for nonrotating RBC. Figure 5.2 shows the curvature PDFs from experimentsand DNS at the cell center. The horizontal axis is the curvature normalized by the cell height,H . A good agreement, within the error bars discussed in the figure caption, between DNS

aTracer particles are incorporated into the code by Kim Alards. The simulations use a spatial grid resolution ofradial, azimuthal and vertical directions as 285, 513, and 513, respectively. To interpolate the fluid velocity fromthe surrounding eight grid points around the particle position to the location of the tracer particle, we use a tri-linearinterpolation scheme and for the time integration a second-order Adams–Bashforth scheme is used.

5.3. RESULTS 53

and experiment is observed. Furthermore, despite the non-Gaussian PDFs of the verticalvelocity component, the HIT power laws, represented by the black lines in the figure, arerecovered. Note that the vertical velocity PDF shows wide tails for large velocities while itshows a Gaussian distribution for small velocities. Since the scaling laws are derived in thelimit of small velocity [189, 5], it can explain why they are still valid for nonrotating RBC.

−8 −6 −4 −2 0 2 4 6 810

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

u/urms

PDF

ux

uz

Gauss.

Figure 5.1: Vertical and horizontal velocity PDFs for Ek =∞ at the cell center from experi-ments. The red solid curve is the Gaussian distribution.

The κ−5/2 scaling is fully recovered for the DNS data while it is recovered up to κH ≈103 for the experiments. The deviation from κ−5/2 scaling for extreme curvatures in theexperiments is the results of particle tracking procedure; it is difficult to capture the extremeevents. The relative error, calculated based on the curvature PDFs at two subvolume at thecell center, shows an error of 20% for κH > 103 and 8% for κH < 103, confirming highererror bars for extreme curvature values.

We continue with the curvature data near the top plate, where the flow is anisotropic[133] and the velocity PDFs show non-Gaussian distributions with non-zero mean horizontalvelocity. Figure 5.3 shows the curvature PDFs for three data sets: experiments, DNS withinthe boundary layer (BL) and DNS outside the BL. As can be seen from the figure, the κ−5/2

scaling is not recovered inside the BL for DNS data. The experimental data do not showthe same behaviour as there are fewer particles that penetrate into the BL in the experiments.However, the DNS outside the BL and experiments show a perfect match: revealing the κ−5/2

scaling. Note the flow is anisotropic both inside and outside the boundary layer near top plateand we still recover the the κ−5/2 scaling outside the boundary layer near top plate. However,the scaling is different inside the BL. This suggests that the power law scaling for HIT holdas long as the flow is turbulent (i.e. the viscosity effects are negligible).

In conclusion, we have found a very good agreement between experiments and DNSat the cell center and near the top plate. In spite of anisotropy and non-Gaussian velocityPDFs, the power laws scaling observed for HIT are recovered for both the cell center and topmeasurements.


10−4

10−2

100

102

104

106

10−12

10−10

10−8

10−6

10−4

10−2

100

κ−5/2

κ1

κH

PDF

Exp.DNS

Figure 5.2: Curvature PDFs for nonrotating RBC (Ek = ∞) at the cell center from experi-ments and DNS. Curvature values are nondimensionalized using the cell height,H . The blacklines show the scaling laws: κ1 and κ−5/2. In the cell center, the errors are estimated by de-viation between curvature PDFs calculated in the upper half of the measurement volume andthe ones calculated in the lower half of the measurement volume (disregarding the outer partsof the tails, where the error is larger). The estimated relative error of the experimental datafor small curvature values (κH < 103) is 8%, while for large curvature values (κH > 103)is approximately 20%. For DNS, the estimated relative error is 4%.

5.3.2 Rotating RBC

In this section, the effects of the background rotation on the curvature statistics are discussed.We shall limit our discussion to the experimental data, while the DNS data is only consideredwhen it is required for a complete analysis.

Starting with the data at the cell center, Figure 5.4(a) shows the curvature PDFs fromexperiments for different background rotations. As mentioned before, the κ−5/2 scaling isnot recovered for the extreme curvature values (κH > 103) at the cell center in the experi-mental data. However, for the small curvature values, the κ−5/2 scaling is recovered in theexperiments. Although the κ−5/2 scaling is not clear from experimental data, the DNS data,see Figure 5.4(b), shows the recovery of the κ−5/2 scaling for different Ek. Note that thelarge-scale anisotropy increases with decreasing Ek at the cell center [133]. However, it doesnot affect the scaling laws. As can be seen from both experiments and DNS, the PDFs shifttowards higher curvature values with decreasing Ek. We will come back to this observationin the subsequent paragraphs.

Continuing with the data near the top plate, the curvature PDFs are plotted in Figure 5.5.The DNS data (outside the BL) is not shown here but it collapses on top of the experimentaldata. The curvature PDFs reveal that in spite of the non-Gaussian anisotropic velocity filed[133], the κ−5/2 scaling is recovered for all considered Ek. It can be concluded that aniso-tropy and non-Gaussian distribution of velocity field do not play a role in the power lawsscaling as long as the flow is turbulent (i.e. viscosity effects are negligible compared to ad-vection). Similar to the data at the cell center, the curvature PDFs shift toward the higher

5.3. RESULTS 55

10−4

10−2

100

102

104

106

10−12

10−10

10−8

10−6

10−4

10−2

100

κ−5/2

κ1

κH

PDF

Exp.DNS−nonBLDNS−BL

Figure 5.3: Curvature PDFs for nonrotating RBC (Ek = ∞) near the top plate from exper-iments and DNS. Curvature values are nondimensionalized using the cell height, H . Theblack lines show the scaling laws: κ1 and κ−5/2 [15, 189]. Close to the top plate, the errorsare estimated by deviation between curvature PDFs calculated in two horizontally-dividedsubvolumes (disregarding the outer parts of the tails, where the error is larger). The estimatedrelative error is approximately 5% for experiments near the top plate. The estimated relativeerror is 6% for DNS outside the BL, while the relative error inside the BL is up to 10% forDNS.

10−4

10−2

100

102

104

106

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

κ−5/2

κ1

κH

a)

10−4

10−2

100

102

104

106

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

κ−5/2

κ1

κH

b)

Ek = Inf.

Ek = 3.6 ⋅ 10−4

Ek = 2.2 ⋅ 10−4

Ek = 1.8 ⋅ 10−4

Ek = 7.2 ⋅ 10−5

Ek = 3.6 ⋅ 10−5

Ek = 1.4 ⋅ 10−5

Ek = 7.2 ⋅ 10−6

Ek = Inf.

Ek = 3.6 ⋅ 10−4

Ek = 2.2 ⋅ 10−4

Ek = 1.8 ⋅ 10−4

Ek = 7.2 ⋅ 10−5

Ek = 3.6 ⋅ 10−5

Ek = 1.4 ⋅ 10−5

Ek = 7.2 ⋅ 10−6

Figure 5.4: Curvature PDFs for different Ek, measured the cell center, for (a) experimentsand (b) DNS. Curvature values are nondimensionalized using the cell height, H . The blacklines show the scaling laws, κ1 and κ−5/2 [15, 189].

curvature values with decreasing Ek.In order to evaluate the shift in the PDFs with rotation, the position of the maximum

value (most probable value) of the curvature PDF is considered as a reference point. Thepeak position, κ∗, as a function of Ek is plotted in Figure 5.6(a). As can be seen from thegraph, the peak is constant for Ek > 1.8× 10−4 (rotation-unaffected regime). For this range


10−4

10−2

100

102

104

106

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

κ−5/2

κ1

κH

PDF

Ek = Inf.

Ek = 3.6 ⋅ 10−4

Ek = 2.2 ⋅ 10−4

Ek = 1.8 ⋅ 10−4

Ek = 7.2 ⋅ 10−5

Ek = 3.6 ⋅ 10−5

Ek = 1.4 ⋅ 10−5

Ek = 7.2 ⋅ 10−6

Figure 5.5: Curvature PDFs for different Ek measured near the top plate for experiments.Curvature values are nondimensionalized using the cell height, H . The black lines show thescaling laws, κ1 and κ−5/2 [15, 189].

of Ek, the LSC is the main feature of the flow. For Ek . 1.8 × 10−4 (rotation-affectedregime), the peak starts to grow eventually. In regime I, the LSC is hardly affected by thebackground rotation, therefore, the curvature peak does not change with rotation. In regimeII, however, the vortical plumes are more dynamic. The size of these vortical plumes dependson Ek; it goes as Ek1/3 [23]. Since the curvature is an inverse length scale, one can use theinverse of the typical length scale of the coherent structures to nondimensionalize κH . Figure5.6(b) shows the curvature PDFs in Figure 5.5, but nondimensionalized by Ek−1/3. As canbe seen from the figure, the PDFs collapse on top of each other. The dashed line in Figure5.6(a) is a line with Ek−1/3 and it also confirms the Ek−1/3 scaling.

10−4

10−2

100

102

104

106

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

κHEk1/3

PDF

κ−5/2

κ1b)

10−5

10−4

10−3

100

101

Ek−1/3

Ek

κ∗H

≀≀∞

a)

Ek = 7.2 ⋅ 10−5

Ek = 3.6 ⋅ 10−5

Ek = 1.4 ⋅ 10−5

Ek = 7.2 ⋅ 10−6

CenterTop

Figure 5.6: (a) The position of the peak values, κ∗ from the experiments at the cell centerand near the top plate. The black dashed line is proportional to Ek−1/3. (b) The curvaturePDFs plotted in Figure 5.5 (near top) nondimensionalized by Ek−1/3. The black lines showthe scaling laws, κ1 and κ−5/2 [15, 189].

5.4. CONCLUSIONS 57

5.4 ConclusionsWe have studied the effects of the background rotation on the geometrical aspects of particletrajectories in rotating turbulent convection. A particle trajectory can be fully described bytwo geometrical parameters namely the curvature and the torsion. We have discussed thecurvature statistics in the cell center and near the top plate.

We have found that in spite of the anisotropy and non-Gaussian distribution of the velocitystatistics in the cell center and near the top plate, the previously reported power laws [15, 189]of curvature PDFs for homogeneous and isotropic turbulence (κ1 and κ−5/2) are recoveredfor all considered background rotations. It can be concluded that the power laws are robustand they have some levels of universality.

Furthermore, we have observed that the curvature PDFs shift towards the higher curvaturevalues with decreasing Ek in the rotation-affected regime. We found that the length scale ofthe typical flow structures is connected to the curvature statistics. If the curvature statistics arenormalized by the inverse of typical length scale of the vortical plumes, Ek1/3, they collapse:confirming the connection between the typical length scale of flow structures and curvaturestatistics.

Chapter 6

Velocity and acceleration statisticsin RRBC: Indicators fortransitions

6.1 IntroductionIn the previous chapters, the rotation-unaffected and rotation-affected regimes have beendiscussed from different points of view. It is shown that the transition from the rotation-unaffected to rotation-affected regime is driven by the top and bottom boundary layers (BL),see Chapter 3. However, the driving mechanism behind the transition from rotation-affectedto rotation-dominated regime is not clear yet. So far, two main mechanisms are proposed forthis transition, by King et al. [73, 72] and Julien et al. [67].

King et al. [73, 72] suggested that the transition depends on the relative thicknesses ofthe viscous (δu) and thermal (δT ) boundary layers. They showed that δT < δu in regime II,while δT > δu in regime III, within their range of parameter values (103 . Ra . 1010, 1 .Pr . 100 and 1 ≤ Γ ≤ 4). Therefore, they hypothesized that the transition between regimesII and III occurs when the viscous and thermal BL thicknesses become approximately equal,δu ' δT . However, a similar transition is also observed from numerical simulations withstress-free boundary conditions and no Ekman layers [142]. On the other hand, Julien et al.[67] suggested that the transition occurs when the bulk is also rotation-dominated and thevortical plumes span throughout the entire domain: the vertical motion in the bulk is thensuppressed, resulting in a drop in the heat transfer efficiency. Recently, Kunnen et al. [83]also performed simulations with stress-free and no-slip boundary conditions showing that thetransition does not always coincide with the intersection of the viscous and thermal boundarylayer thicknesses, in contrast with King’s hypothesis. The study of Kunnen et al. supportsmore the point of view by Julien et al. [67] rather than King’s, however, no certain conclusioncan be drawn [83].

This chapter is in preparation for submission to Journal of Fluid Mechanics, H. Rajaei, K.M.J. Alards, R.P.J.Kunnen, H.J.H. Clercx, “Velocity and acceleration statistics in rotating Rayleigh–Bénard convection: Indicators fortransitions”.

59

60 CHAPTER 6. VELOCITY AND ACCELERATION STATISTICS

Up to now, the transition to regime III has been evaluated through various parameters.King et al. focused on the relative thicknesses of the Ekman and thermal BLs and global heattransfer, measured by laboratory experiments and numerical simulations [73, 72]. Julien et al.examined the vertical velocity and temperature fluctuations by simulations of asymptoticallyreduced equations [67]. Kunnen et al. evaluated the thermal and Ekman BLs thicknesses,heat transfer efficiency, distribution of dissipation and mean temperature gradients throughoutthe domain by direct numerical simulations with no-slip and stress-free boundary conditions[83].

In this chapter, we examine various regimes in RRBC and the transitions between them,with emphasis on the transition to the rotation-dominated regime III, from a new perspective.We use the experimental Lagrangian velocity and acceleration fluctuations and autocorrel-ations as indicators for the transition. The experimental measurements of the thermal andviscous BL thicknesses are not yet possible. Therefore, it is not possible to directly evaluateKing’s hypothesis with experiments. However, the presented Lagrangian data allows us toexamine the flow structures in all three regimes. Furthermore, it opens up new understandingof the physics behind all three regimes, in particular on the rotation-dominated regime andthe transition to this regime.

We start with the experimental parameters used in this investigation in Section 6.2. Next,the Lagrangian velocity rms values and autocorrelations are discussed in Sections 6.3 and6.4, respectively. In Sections 6.5 and 6.6, the acceleration rms values and autocorrelationsare presented and discussed, respectively. In Section 6.7, an intriguing oscillatory behavior,observed in the acceleration autocorrelation, is explained. We summarize our main findingsin Section 6.8.

6.2 Experimental parameters

The data used in this chapter are for Ra = 1.3 × 109, Pr = 6.7 and Γ = 1 while Rovaries between 0.041 and ∞ (Ω between 4.12 rad/s and 0). These experiments correspondto the rows 1 to 12 in Tables 2.1 and 2.2. The details of the 3D-PTV set-up can be foundin Chapter 2. As in Chapters 3 and 4, a cubic polynomial filter is used to eliminate thebackground noise.

We analyze the Lagrangian velocity/acceleration root-mean-square (rms) values and auto-correlations. The Lagrangian velocity and acceleration rms values are calculated based onthe same measurement volumes as those in Chapter 4: z = 0.5H covering the height0.375H < z < 0.625H with a volume of 50 × 50 × 50 mm3, z = 0.8H covering theheight 0.75H < z < 0.85H with a volume of 50× 50× 20 mm3 and z = 0.975H coveringthe height 0.95H < z < H with a volume of 50×50×10 mm3. However, the Lagrangian ve-locity and acceleration autocorrelations are calculated based on the data collected in a volumeof approximately 80× 60× 50 mm3(x, y, z) for both the center (covering a volume betweenz = 0.375H and z = 0.625H) and close to the top plate (z = 0.875H; covering a volumebetween z = 0.75H and z = H), the same volumes as the ones used in Chapter 5. Notethat it is not possible to divide the measurement domain into smaller subvolumes for the Lag-rangian autocorrelations since the particles do not reside in a thin layer of fluid for a longtime.

6.3. LAGRANGIAN RMS VELOCITY 61

6.3 Lagrangian rms velocity

We start with the presentation of the Lagrangian velocity fluctuations at the center, at z =0.8H and at z = 0.975H . We focus on different regimes in rotating RBC and how the velo-city fluctuations can contribute to our understanding of these regimes and transitions betweenthem. As mentioned in the previous chapters, close to the top plate, the horizontal velocitysignal has a non-zero mean value for low rotation rates (Ro & 2.4) due to the presence of theLSC. The same treatment as that in Chapter 4 is performed on the horizontal velocity signalclose to the top plate (only for Ro & 2.4).

The (an)isotropy of the velocity fluctuations have been discussed in Chapter 4. Figure6.1(a,b) shows the velocity fluctuations and the inverse of the anisotropy ratio,1/RU = urms

xy /urmsz , as a function of Ro for three measurement sets for regimes II and

III. We plot 1/RU instead of RU as it shows the trends more clearly. Regime II of thesegraphs have been treated in Chapter 4. Here, we limit our discussion mainly to regime III.

The measurements at the cell center, dark blue symbols in Figure 6.1(a,b), show thatthe horizontal and vertical velocity fluctuations continue to decrease in regime III with de-creasing Ro, but at faster decay rates. The large-scale anisotropy becomes even larger withdecreasing Ro in regime III, see blue squares in Figure 6.1(b). At z = 0.8H , cyan symbols,the vertical velocity fluctuation decreases with decreasing Ro as a result of the turbulencesuppression with background rotation [23]. The horizontal component decreases as well, butreduction of the vertical component is stronger than that of the horizontal one. The horizontalvelocity fluctuation is governed by two contributing factors: due to turbulence reduction withbackground rotation and enhancement due to swirling motions, contributed by the vorticalplumes in the horizontal plane. Therefore, the horizontal velocity fluctuation decreases at aslower rate than the vertical counterpart. The horizontal and vertical components intersectin regime III: 1/RU goes from below one to values slightly above one. At z = 0.975H ,both horizontal and vertical velocity fluctuations decrease with decreasing Ro. However, asexplained for z = 0.8H , the vertical velocity fluctuation reduces faster compared to its hori-zontal counterpart due to the enhancement of the horizontal velocities by the vortical plumes.This results in an increase in the ratio 1/RU with decreasing Ro, as is clear from Figure6.1(b).

In conclusion, in regimes II and III, the horizontal velocity fluctuations at z = 0.975Hand z = 0.8H decrease at a slower rate compared to their vertical counter parts due to thevortical plumes. In regime III, a faster decay is observed for the vertical component of thevelocity throughout the convection cell. This observation supports the hypothesis proposedby Julien et al. [67] stating that the strong suppression of the vertical motion results in a dropin heat transfer efficiency in regime III.

6.4 Lagrangian velocity autocorrelation

Lagrangian statistics are of utmost importance in the study of particle dispersion and heat andmass transport processes in turbulence. The Lagrangian velocity autocorrelation is one of theuseful Lagrangian parameters which provides insight into the turbulent diffusivity of a fluidparcel [163]. The Lagrangian velocity autocorrelation shows the time period over which theparticle velocity remains correlated with the velocity at the previous times. It is expected that


10−1

100

0.5

1

1.5

2

2.5

3

Ro

urm

s(m

m/s)

IIIII

a)

10−1

100

100

IIIII

Ro

1/RU

=urm

sxy/urm

sz

b)Centerz=0.8Hz=0.975H

Figure 6.1: (a) Horizontal and vertical velocity fluctuations from experiments. Blue circlesand squares are the horizontal and vertical components at the cell center, respectively. Cyantriangles and stars are the horizontal and vertical components at z = 0.8H , respectively.Red pluses and diamonds are the horizontal and vertical components at z = 0.975H . Theerror bars are the same size as the symbols. (b) The inverse of the anisotropy ratio 1/RU =urmsxy /u

rmsz .

the particle velocity remains correlated for a longer time if the flow is coherent. Therefore,the Lagrangian velocity autocorrelation can be used as a measure of flow coherence as well.The Lagrangian velocity autocorrelation is defined as

RLui(τ) =

〈ui(t)ui(t+ τ)〉〈u2i (t)〉

, (6.1)

with ui the ith component of the velocity signal and τ the time lag. Another useful parameterin evaluation of the coherent structures in turbulent flows is the integral time scale, defined as

TLui=

∫ ∞0

RLui(τ)dτ, (6.2)

which is a measure for the time a fluid parcel velocity remains correlated. Both the Lag-rangian velocity autocorrelation and the integral time scale give insight into the large-scaleflow coherence.

In the present study the residence time of the particles in the observation domain (estim-ated as dobs./u where dobs. is the length/width/depth of the observation volume and u is thevelocity rms) is ∼ 25τη , while the autocorrelation coefficient drops below 0.1 at about 15τη(see Figure 6.3) for most of the rotation rates; indicating that the bias error induced by thelimited volume is small.

For large time lags τ the autocorrelation values are not perfectly converged due to theshortage of long trajectories. Lack of convergence leads to higher errors in the calculationsof the integral time scale. Therefore, we fit an exponential function over the time period forwhich the velocity autocorrelation shows a clear exponential decay, see e.g. Figure 6.2 (thedeparture from the exponential decay at large τ as shown in Figure 6.2 is due to a lack of theaforementioned statistical convergence). Exponential fitting is chosen since in homogeneous

6.4. LAGRANGIAN VELOCITY AUTOCORRELATION 63

isotropic turbulence (HIT) the velocity decorrelation at large τ is expected to be exponen-tial [108, 111, 42, 32]; it plays a crucial role in some dispersion models [140]. The linear-logarithmic plot confirms the exponential decay of the velocity autocorrelation at large τ , seeFigure 6.2 obtained in the central part of our convection cell. Note that at very small (dissip-ative scale) τ , in our experiments τ . 1 s, the velocity autocorrelation shows non-exponentialbehavior (the curvature of the velocity autocovariance at τ = 0 is equal to the accelerationvariance) [178, 110, 140]. The integral time scale is calculated using a combination of theactual velocity autocorrelation curve for small time lags and the exponentially-fitted curve forlarge time lags. The integral time scale can also be estimated through the aforementioned ex-ponential fitted curve e−τ/τ0 , where τ0 is the integral time scale. However, the latter approachneglects the non-exponential decay for small time lags, so small differences are expected. Weshall use the first method (see Equation (6.2)) unless otherwise stated.

0 5 10 15 20 25 3010

−1

100

τ(s)

RL uxy

Exp.Fitted line

Figure 6.2: The velocity autocorrelation in the xy direction for Ro = ∞ at the cell center,confirming an exponential decay until the correlation coefficient drops below 0.2. The redline is the fitted exponential curve. The departure from exponential decay at large τ is due tolack of long trajectories.

We start with the data at the center. Figure 6.3(a,b) shows the autocorrelation of thevelocity in the xy and z directions at the cell center, respectively. The time lag is nondi-mensionalized by the local Kolmogorov time scale, given in Table 6.1 calculated from DNSusing the same parameter settings. Note that the Kolmogorov time scale differs, dependingon the position within the cylinder. The velocity autocorrelations in regime I collapse for bothhorizontal and vertical components. Figure 6.4 displays the Lagrangian integral time scales,nondimensionalized by the local Kolmogorov time scale, at the cell center and z = 0.875H .The horizontal and vertical integral time scales at the cell center, red circles and blue squares,remain almost constant in regime I as well. As mentioned before, the integral time scale is ameasure of the large-scale coherence. Therefore, one expects smaller integral time scales atRo ' 2.4, where the LSC breaks down and the flow lacks a dominant large-scale coherentstructure. However, Figure 6.4 does not show a decrease around Ro ' 2.4. It can be ex-plained by the fact that the LSC is best visible near the top/bottom and side boundaries: thelarge-scale flow at the cell center is close to HIT [161, 133] and there is no clear signature of


the large-scale coherent structure in the cell center. In regime II, the horizontal and verticalvelocity autocorrelations progressively increase with reduction of Ro, see Figure 6.3(a,b).These enhancements are also reflected in the integral time scales, see red circles and bluesquares in Figure 6.4. In regime II, the vortical plumes are present as the large-scale flow fea-tures, resulting in an enhancement of the large-scale coherence. Note that, in contrast to theLSC, the vortical plumes are visible in the large-scale flow in the center [133]. In regime III,the vertical and horizontal velocity autocorrelations show non-monotonic trends but only re-veal little changes in this regime. The horizontal and vertical integral time scales also shownon-monotonic behaviors. However, the non-monotonic trends are within the error bars.

Table 6.1: Kolmogorov time scale, τη (in seconds), as a function of Ro for the measurementsat the cell center and close to the top plate. The dissipation rates are calculated from DNSdata. The dissipation rates from DNS are for slightly different Ro numbers, but we expectthat they are still representative for the current experiments given the weak Ro dependencedisplayed here.

Ro 0.0425 0.05 0.06 0.0875 0.1 0.2 0.5 1 2.5 3 5 ∞Center 1.27 1.21 1.14 1.04 1.10 1.09 1.13 1.17 1.33 1.52 1.81 1.94z = 0.875H 0.89 0.88 0.88 0.86 0.88 0.92 0.96 0.96 1.05 1.07 1.09 1.13

We continue with the data at z = 0.875H . Figure 6.3(c,d) shows the velocity autocor-relations at z = 0.875H for the xy and z directions, respectively. The flow shows stronghorizontal correlation in regime I. The horizontal velocity autocorrelations remain correlatedfor a long time due to the presence of the LSC, see also the black diamonds in Figure 6.4. Asthe LSC vanishes for Ro . 2.4, the horizontal velocity correlation time drops abruptly, seealso the black diamonds in Figure 6.4. As the LSC has vanished, the mean horizontal velo-city becomes zero, thus no contribution from mean velocity in the correlations in regime II.Note that, in contrast to the velocity rms values, the autocorrelation data are calculated forthe total velocity signal for Ro & 2.4 at z = 0.875H . For the rest of the data (data at thecenter and data for Ro . 2.4 at z = 0.875H) the mean velocities are zero. In regime II(2.4 . Ro . 0.1), the horizontal correlation decreases slightly with decreasing Ro. Inregime III, Ro . 0.1, the horizontal correlation decreases even further as Ro goes downand its corresponding integral time scale (black diamonds) becomes as small as two timesthe Kolmogorov time scale (TLuxy

/τη ∼ 2). The decrease in the horizontal autocorrelationthroughout regimes II and III can be explained by knowing that the swirling motion in thehorizontal plane near the top plate results in a continuous change of velocity direction in thisregion. In regime III, an oscillatory behavior is observed in the horizontal component of thevelocity near the top plate. We will treat this intriguing oscillatory behavior in detail in Sec-tion 6.7 and show that not only Earth’s gravity determines the flow dynamics but also Earth’srotation.

The vertical velocity autocorrelations, Figure 6.3(d), show no appreciable changes withrotation in regime I. As can be seen from the graph, they show persistent negative values.This is in agreement with our picture with an LSC present that a fluid parcel moves upwardon one side and downward on the other side of the cell. Considering the general picture,one expects that the anti-correlation occurs at 0.5lobs./uLSC, where lobs. is the length of theobservation view and uLSC is the fluid velocity in the LSC, calculated based on the LSCReynolds number [150]. The anti-correlation starts at τ/τη ≈ 7.0 (τ ≈ 7.5 s), which is


0 5 10 15 20 25

0

0.2

0.4

0.6

0.8

1

τ/τη

RL uxy

c)

0 5 10 15 20 25−0.5

0

0.5

1

τ/τη

RL uz

d)

0 5 10 15

0

0.2

0.4

0.6

0.8

1

τ/τη

RL uxy

a)

0 5 10 15

0

0.2

0.4

0.6

0.8

1

τ/τη

RL uz

b)

Ro = Inf.Ro = 4.78Ro = 2.88Ro = 2.38Ro = 0.95Ro = 0.48Ro = 0.19Ro = 0.10Ro = 0.083Ro = 0.058Ro = 0.048Ro = 0.041

Figure 6.3: (a) Horizontal and (b) vertical Lagrangian velocity autocorrelations at the centerand (c) horizontal and (d) vertical Lagrangian velocity autocorrelations close to the top as afunction of Ro.

compatible with 0.5lobs./uLSC ≈ 6 s. The vertical autocorrelations collapse in regime I,with a rather large anti-correlation; −0.5 at τ/τη = 25. The LSC vanishes for Ro . 2.4, andthe vertical velocity anti-correlation gradually diminishes. For high rotation rates, Ro . 0.5,the anti-correlation in the vertical velocity autocorrelation disappears.

The vertical integral time scales in regime I and part of regime II at z = 0.875H arenot reported due to the negative values in the vertical velocity autocorrelations. The verticalintegral time scales in regime II and III (purple stars in Figure 6.4) increase with decreasingRo which is an indication of enhancement in the flow coherence, which we expect to be dueto a stronger vertical coherence of the vortical plumes.

6.4.1 Small Ro: emergence of two time scales

As mentioned before, an exponential decay at large time lags in the velocity autocorrela-tion is expected; it plays a crucial role in Sawford’s stochastic dispersion model [140]. Ourexperimental data also confirms the exponential decay, proportional to e−τ/τ0 , in the velo-city autocorrelation, see e.g. Figure 6.2. As mentioned before, the behavior of the velo-city autocorrelation is far from an exponential decay at the dissipative time lags τ/τη . 1


10−1

100

101

0

2

4

6

8

10

12

14

Ro

TL ui/τη

I

IIIII

≀≀∞

TLxy

−center

TLz−center

TLxy

−top

TLz−top

Figure 6.4: Integral time scale as a function of Ro for both measurements at the cell centerand top (z = 0.875H). The error is estimated by calculating the integral time scale fordifferent segments of the experimental data. The error is approximately 15%.

[178, 110, 140]. However, here we focus on the large time lags τ/τη & 1 of the velocityautocorrelations.

For all rotation rates at the cell center and most of the rotation rates at z = 0.875H ,the slope of the exponential decay is constant for τ/τη & 1, i.e. a single decay constant τ0in e−τ/τ0 is observed. However, for the vertical velocity autocorrelation for Ro . 0.19 atz = 0.875H a second time scale emerges. Figure 6.5 shows the vertical velocity autocorrel-ation for Ro = 0.058 on a logarithmic scale. The green and red lines are exponential fits,proportional to e−τ/τi , fitted to the part where the autocorrelation shows a clear exponentialdecay. As can be seen from Figure 6.5, there exist two time scales τi; τ1 for small time lags(1 s . τ . 6 s) and τ2 for large time lags (τ & 6 s). The dash-dotted black line is at τ = 6s, where the change in slope occurs. Note that the Kolmogorov time scale at z = 0.875H isabout 1 s.

We can calculate the two correlation time scales (τ1 and τ2) for the vertical velocityautocorrelations plotted in Figure 6.3(d). The results are plotted in Figure 6.6 (solid symbols)as a function of Ro. The smaller correlation time scales (τ1), present for smaller τ andrepresented by the solid red circles, remain almost unchanged in the regimes II and III whilethe larger correlation time scales (τ2), present for larger τ and represented by solid bluesquares, increase with decreasing Ro. The purple stars are the vertical integral time scales,taken from Figure 6.4. As expected, they fall between the solid squares and circles. The opensymbols will be discussed later.

In order to study the origin of the two correlation time scales, the flow domain is dividedinto plume and non-plume regions. The so-called Q-criterion [62] is a widely used methodfor vortex detection. We apply the Q-criterion for a three-dimensional flow field: a vortex is


0 5 10 15 20 25 30

10−1

100

τ(s)

RL uz

Exp.f1 ∝ exp(−τ/τ

1)

f2 ∝ exp(−τ/τ

2)

Figure 6.5: The vertical velocity autocorrelation for Ro = 0.058 at z = 0.875H . The greenand red lines are the exponential fits, proportional to e−τ/τ0 with τ0 = τ1 for 1 s . τ . 6 sand τ0 = τ2 for τ & 6 s. The dash-dotted black line is located at τ = 6 s.

10−1

4

6

8

10

12

14

16

18

20

Ro

τ/τη

IIIII

τ1

τ2

TLz−top

Figure 6.6: Two correlation time scales as a function of Ro. The small and large time scalesare represented by τ1 and τ2, respectively. The solid squares and circles represent the cor-relation time scales calculated based on the data in Figure 6.3(d). The open symbols arethe correlation time scales calculated based on the Q-criterion method. Purple stars are thevertical integral time scales at z = 0.875H , taken from Figure 6.4, calculated using Equa-tion (6.2).

defined as a spatial region where

Q =1

2

(||Ω||2 − ||S||2

)> 0, (6.3)


where Ω and S are the antisymmetric and symmetric parts of the velocity gradient tensor,respectively (see Equation (2.5)). The operator || . . . || represents the Euclidean norm definedas

||A|| =√

Tr(AAT

). (6.4)

As mentioned in Chapter 2, we cannot accurately retrieve all nine components of the velocitygradient tensor from the interpolated data at high rotation rates. Here, however, the situationis different since we are mainly interested in the sign of the Q value. Therefore, the exact Qvalues are not important: the measurements should be good enough to resolve the sign of theQ values. In order to evaluate whether we can resolve the sign of Q values, we consider theQ isosurface plots for different Ro. Judging from the Q isosurface plots of the flow field fordifferent Ro, see e.g. Figure 6.7, the vortical plumes are clearly distinguishable. Therefore,based on this analysis, the vortex and non-vortex regions can be identified (with sufficientquality required for the purpose of our study).

Figure 6.7: Isosurfaces of Q = 0.5Qrms, where Qrms is the root mean square of Q at thetime of the snapshot, for (a) Ro = 1 and (b) Ro = 0.041. The blue surfaces represent thecold plumes and red are the hot plumes. This threshold (0.5Qrms) is introduced only for thepurpose of better illustration.

Detection of the plume and non-plume regions allows for differentiation between velocityautocorrelations inside and outside the plumes. Figure 6.8 shows the vertical velocity auto-correlations inside and outside the plumes for Ro = 0.048. In this figure, the open squaresrepresent the vertical velocity autocorrelation of particles inside the plumes (Q > 0). Thefilled circles, on the other hand, show the vertical velocity autocorrelation of the particles innon-plume regions (Q ≤ 0). The solid and dashed red lines are the fitted lines (proportionalto e−τ/τ0 ) through the open squares and filled circles, respectively. Clearly, the time scalefor correlation within the plumes (the inverse slope of the solid red line) is larger than thatfor outside the plumes (the inverse slope of the dashed red line), indicating larger verticalflow coherence inside the plumes. The time scale values for different rotation rates computedwith this method are given in Figure 6.6 as open symbols. It is worth mentioning that thesolid symbols in Figure 6.6 are calculated based on the velocity autocorrelation from the lowparticle concentration (LPC) data set (∼ 300 minutes of measurement) while the plume de-tection method and consequently the open symbols are calculated based on the high particle


concentration (HPC) data sets (∼ 20 minutes of measurement). Convergence for the largetime lags of the HPC data set is questionable, thus the open symbols in Figure 6.6 have lar-ger uncertainties. Furthermore, the solid and open symbols are calculated through differentapproaches. Therefore, a one by one comparison might not be legitimate. However, the openand solid circles and squares show similar trends.

It can be concluded that τ1 in the vertical velocity autocorrelation (represented by thegreen line in Figure 6.5), which is the governing time scale at small time lags 1 s . τ . 6 s,is mainly determined by the flow time scale in the non-plume regions. Obviously, there is acontribution from the particles inside the plumes as well, however, this contribution is smallersince the number of particles in the non-plume regions is larger than that inside the plume.As has been shown, the particles in the non-plume regions decorrelate faster than those inthe plume regions, e.g. the autocorrelation coefficient at τ = 6 s is below 0.2 for the non-plume region compared to 0.5 for the plume region in Figure 6.8. Thus, it is expected that thecorrelation at large time lags (τ & 6 s) is mainly determined by the time scale τ2 of the plumeregions; the non-plume regions are already decorrelated for such large time lags. Knowingthat τ1 is a characteristic correlation time of the flow outside the plumes, makes it possibleto comment on the reason why τ1 is constant while τ2 increases with decreasing Ro. As τ1is the correlation time outside the coherent structures, when τ1 is nondimensionalized by τη ,one would expect no significant changes with variation in turbulence intensity (i.e. variationin Ro). On the other hand, τ2 is the correlation time of the plumes which is found to varywith Ro.

0 2 4 6 8 1010

−1

100

τ(s)

RL uz

Figure 6.8: The vertical velocity autocorrelations for particles inside plumes (open squares)and outside plumes (filled circles) for Ro = 0.048. The red solid and dashed lines are theexponential decays fitted through the open squares and filled circles, respectively. The fits areperformed for 1 s < τ < 6 s.


6.4.2 Concluding remarks

In conclusion, although the first transition (between regime I and II) is reflected in the ve-locity autocorrelations and the integral time scale (more profoundly near the top plate), thesecond transition (between regime II and III) is not as clearly reflected. This can be explainedby the fact that the former transition is sudden while the latter transition appears to be moregradual. However, the crossover to regime III coincides with the appearance of an additionalcorrelation time scale in the vertical velocity autocorrelations near the top plate. Although thelonger correlation time τ2 increases with decreasing Ro, the shorter time τ1 remains constant.We have indicated that these two different correlation time scales are associated with the cor-relations inside and outside the plumes. The emergence of two correlation time scales suggestthat the crossover to regime III is caused by the presence of sufficiently strong plumes whichinteract less with their surroundings. In other words, the fluid exchange between plume andnon-plume regions becomes considerably less in regime III. Thus, the plumes are separatedfrom the non-plume regions; each characterized by their own time scale. The flow in regimeIII has therefore two distinct characteristics, one inside and another outside of the plumes.

6.5 Lagrangian rms acceleration

So far, the discussion was on quantities which are largely determined by the large-scale flowfield. In this section, however, we focus on the Lagrangian acceleration rms values which aregoverned by comparatively smaller scales.

In this section, similar to Section 6.3, three different regions are analyzed; cell center,z = 0.8H and z = 0.975H . The Lagrangian acceleration fluctuations and the ratio RAA =armsxy /a

rmsz are plotted in Figure 6.9(a,b). The acceleration fluctuation and its ratio in regimes

I and II have been discussed in Chapter 3. Here, however, we focus on the accelerationfluctuation in regime III and the transition from regime II to III.

The horizontal and vertical components of the acceleration at the cell center, see bluecircles and squares in Figure 6.9(a), continue to decrease as Ro decreases due to the sup-pression of turbulence by the background rotation in regime III. The acceleration fluctuationincreases slightly for the lowest considered Ro, most probably due to the formation of inertialwaves as will be explained in Section 6.7. The ratio RAA, which can also be used as a toolto partially evaluate small-scale isotropy, does not change while transitioning from regime IIto regime III.

The measurements at z = 0.8H show that both horizontal and vertical accelerationsdecrease slightly in regime II with decreasing Ro, resulting in an approximately constantRAA. However, after the transition to regime III, the horizontal component starts to increasewith decreasing Ro due to the formation of strong vortical plumes capable of entrainingfurther into the bulk. The vertical component, on the other hand, keeps decreasing withdecreasing Ro in regime III, similar to regime II. As a result, the ratio RAA at z = 0.8Hincreases with decreasing Ro in regime III: a clear indication that the vortical plumes arepenetrating further into the interior region.

In regime III close to the top plate (z = 0.975H), the horizontal acceleration increaseswith decrease in Ro, similar to that of regime II. The vertical acceleration, on the other hand,continue to decrease in regime III with decreasing Ro, but at a higher decay rate compared

6.5. LAGRANGIAN RMS ACCELERATION 71

10−1

100

101

0.5

1

1.5

2

Ro

arm

s(m

m/s2)

IIIIII

≀≀ ∞

a)

10−1

100

101

0

1

2

3

4

5

6

∞

III

III

≀≀

Ro

RAA

=arm

sxy/a

rms

z

b)Centerz = 0.8Hz=0.975H

Figure 6.9: (a) Horizontal and vertical acceleration rms values from the experiments. Bluecircles and squares are the horizontal and vertical components at the cell center, respectively.Cyan triangles and stars are the horizontal and vertical components at z = 0.8H , respectively.Red plusses and diamonds are the horizontal and vertical components at z = 0.975H . (b)The ratio RAA = arms

xy /armsz as a function of Ro. Experimental acceleration uncertainty is

estimated by adding noise to the displacement, see Section 2.1.2 for details.

to regime II. As a result, the ratio RAA increases in regime III at an even higher rate. Thedemarcation between regimes II and III is very well captured by the ratio RAA.

It is worth mentioning that the decrease in armsxy at the cell center might not directly be

related to whether vortical plumes reach the cell center. Note that the sign of the verticalvorticity will flip as we move down far enough in a vortex column [127, 47]; the vorticalplumes spin down as they approach the center which effectively reduces arms

xy . The changein sign of vertical vorticity can be explained by thermal-wind balance. Together with thedecrease of the vertical vorticity, the plumes are also squeezed due to the presence of theboundaries and widened due to conservation of angular momentum.

In conclusion, we can clearly see that the flow dynamics have changed while crossingto regime III near the top plate. If we assume that the vortical plumes interact considerablyless in regime III (see the concluding remarks in Section 6.4.2), a particle inside a plumestays inside the plume, thus experiencing a strong horizontal centripetal and weak verticalaccelerations. In regime II, however, fluid parcels (particles) are exchanged between plumeand non-plume regions, resulting in a comparatively weaker horizontal centripetal accelera-tion. Therefore, the increase in the ratio RAA in regime III supports the conclusion drawn inSection 6.4.2: there is less exchange between plume and non-plume regions in regime III.

Furthermore, the horizontal component of the rms acceleration at z = 0.8H starts toincrease while crossing the transition point: the vortical plumes penetrate further into thebulk. However, at the cell center the horizontal component of rms acceleration continue todecrease with decreasing Ro in regime III.


6.6 Lagrangian acceleration autocorrelation

The analysis in this section is based on the data collected at the cell center and at z = 0.875H ,similar to Section 6.4. The Lagrangian acceleration autocorrelation is defined in a similarfashion as the Lagrangian velocity autocorrelation as

RLai =〈ai(t)ai(t+ τ)〉〈a2i (t)〉

, (6.5)

where ai is the ith component of the acceleration of a fluid parcel. The residence time ofthe particles in the observation volume is ∼ 25τη which is larger than the desired residencetime ∼ 10τη for the acceleration autocorrelation to decorrelate [31, 114]. In order to have abetter insight into the effects of the background rotation on the acceleration autocorrelation, adifferent coordinate system is employed in parallel. Following Del Castello and Clercx [31],we calculate the autocorrelation of the longitudinal (al), transversal Coriolis (at,Cor), andtransversal remainder (at,rem) components of the acceleration, thus a = (al; at,Cor; at,rem).Here, al is the component along the trajectory, defined as al = a·u where a is the accelerationvector and u is the unit vector in the direction of velocity, u = u/|u|. The transversal Corioliscomponent, at,Cor = a·(u×z), where z is the vertical unit vector. The third component is thetransversal remainder acceleration. Such a decomposition is chosen since the Coriolis accel-eration (−2Ωz× u), which is introduced by the background rotation, influences fluid parcelacceleration directly in the direction perpendicular to both the velocity vector and the rota-tion axis. Therefore, it is expected that the background rotation directly affects only at,Cor(and the two other components indirectly by turbulence modification due to background ro-tation). We refer to the acceleration autocorrelations calculated in the Cartesian coordinatesas Cartesian acceleration autocorrelation and acceleration autocorrelations calculated in thenew coordinates as Coriolis-coordinate acceleration autocorrelation.

For the remainder of this section, we first discuss the Cartesian acceleration autocorrela-tions and their corresponding first zero crossing values at the cell center. Then, the Coriolis-coordinate acceleration autocorrelations at the cell center are discussed. Next the Cartesianacceleration autocorrelations and their corresponding first zero crossing values close to thetop plate (z = 0.875H) are elaborated. Finally, the Coriolis-coordinate acceleration autocor-relations close to the top plate (z = 0.875H) are treated.

Measurement at the cell center, Cartesian acceleration autocorrelations and the cor-responding first zero crossing values (Figures 6.10(a,b) and 6.11): Figure 6.10(a,b) showsthe Cartesian acceleration autocorrelations (axy and az) for the horizontal and vertical direc-tions at the cell center in regimes I, II and III. For clarity of the figure, only eight autocorrela-tions are shown. The vertical and horizontal acceleration autocorrelations for the nonrotatingcase, blue diamonds (known to be close to the HIT condition [161, 133]), agree well with thedata in the literature e.g. Refs. [114, 31, 192]: a fast decorrelation for small time lags due tochange in the acceleration direction and a negative loop for larger time lags. An interestingparameter that can be obtained from Lagrangian acceleration autocorrelations is the first zerocrossing of the Cartesian acceleration autocorrelation which shows how fast the accelerationdecorrelates. Yeung et al. [192] found that the first zero crossing of the acceleration auto-correlation in HIT is almost independent of Reλ, and it is around 2.2τη (for a limited rangeof Reλ, 38 . Reλ . 93). Ni et al. [114] found values between 2.13τη and 2.33τη at the

6.6. LAGRANGIAN ACCELERATION AUTOCORRELATION 73

cell center in RBC. The value we find at the cell center is (2.30± 0.25)τη for the nonrotatingcase.

0 2 4 6 8 10 12−0.5

0

0.5

1

τ/τη

RL axy

a)

Ro = Inf.Ro = 2.38Ro = 0.95Ro = 0.19Ro = 0.10Ro = 0.083Ro = 0.058Ro = 0.041

0 2 4 6 8 10 12−0.5

0

0.5

1

τ/τη

RL az

b)

Figure 6.10: (a) Horizontal and (b) vertical Cartesian acceleration autocorrelations at the cellcenter as a function of Ro.

We continue with the Cartesian acceleration autocorrelations at the cell center for therotating case. Figure 6.10(a,b) shows that a decrease in Ro does not significantly changethe acceleration autocorrelations in regime I for both vertical and horizontal directions: theyapproximately collapse. After crossing to regime II, the negative loop in the accelerationautocorrelation becomes more prominent. The negative loops for both horizontal and verticalacceleration autocorrelations show a non-monotonic behavior in regime II. The crossover toregime III coincides with the oscillatory behavior for the acceleration autocorrelation (forRo . 0.1). The oscillatory behavior is stronger in the z direction at the cell center. Thefrequency of the oscillatory behavior is Ω (within the range of the accuracy). We will treatthese waves in depth in Section 6.7.

The first zero crossing values of the Cartesian acceleration autocorrelations are plottedin Figure 6.11. It shows the first zero crossing values (horizontal and vertical) both at thecell center and at z = 0.875H . Here, we only focus on the vertical and horizontal first zerocrossing values at the cell center, blue circles and red triangles, respectively. The vertical(blue circles) and horizontal (red triangles) first zero crossing values at the cell center inregime I show small differences within the error bars. In regime II, the vertical and horizontalfirst zero crossings (blue circles and red triangles in Figure 6.11, respectively) show an overalldownward trend as Ro decreases. The first zero crossing values in regime III are not reporteddue to the fact that the oscillatory behavior makes the analysis to obtain the first zero crossingdifficult.

Measurement at the cell center, Coriolis-coordinate acceleration autocorrelations(Figure 6.12(a,b,c): Figure 6.12(a,b,c) shows the acceleration autocorrelations at the centerfor the new coordinate system defined as before, with acceleration components al, at,rem,at,Cor. The two components al and at,rem at the cell center do not show significant changesby applying the rotation rate. They show a fast decorrelation similar to the Cartesian accel-eration autocorrelation. at,Cor, on the other hand, remains correlated for a long time. Wecan explain this as follows. As mentioned before, at,Cor is expected to be the componentthat is directly affected by the Coriolis acceleration, −2Ωz × u. The Coriolis accelerationscales with the horizontal velocity. As at,Cor is directly affected by the Coriolis acceleration


10−1

100

101

0

0.5

1

1.5

2

2.5

3

3.5

Ro

τ/τ

η

∞

III

≀≀

τLxy

−center

τLz−center

τLxy

−top

τLz−top

Figure 6.11: The value of the first zero crossing of the Cartesian acceleration autocorrelationfor the center (horizontal, red triangles and vertical, blue circles) and close to the top plate(horizontal, black diamonds and vertical, cyan squares) as a function of Ro.

(horizontal velocity), we expect that RLat,Corshows similar long decorrelation time as RLuxy

.Note that the velocity is a feature of the large-scale flow field, thus at,Cor is also governed bythe large-scale flow field.

In regime I, RLat,Corshows no change with rotation; the LSC is dominant and the large-

scale flow structure is fully determined by the LSC. In regime II, a non-monotonic behaviorat the cell center is observed for RLat,Cor

: First it increases and then it decreases for thetwo highest rotation rates. Such non-monotonic behavior for RLat,Cor

has been observed inmagnetically driven turbulent rotating flow as well [31]. In regime III, the waves appear inthe autocorrelation of at,Cor.

Measurement at z = 0.875H , Cartesian acceleration autocorrelations and the cor-responding first zero crossing values (Figures 6.13(a,b) and 6.11): We continue withthe Cartesian acceleration autocorrelations at z = 0.875H (close to the top plate). Fig-ure 6.13(a,b) shows the Cartesian acceleration autocorrelations at z = 0.875H . Similar tothe measurements at the cell center, the Lagrangian acceleration autocorrelations collapse inregime I for both horizontal and vertical components. The vertical acceleration autocorrel-ations show no negative loop (anti-correlation) in regime I. However, in regime II, it showsan anti-correlation which increases with decreasing Ro. The negative loop becomes moreprominent for the horizontal acceleration autocorrelation as well in regime II. Like the cellcenter measurements, an oscillatory behavior is observed in the acceleration autocorrelationfor Ro . 0.1, regime III. However, here the wave is observed mainly in the horizontal direc-tion.

The horizontal first zero crossing at z = 0.875H is comparatively larger than the meas-urements at the cell center for Ro = ∞, see Figure 6.11. The horizontal first zero crossingvalues close to the top plate are approximately constant in regime I (black diamonds). Thevertical acceleration component does not reach zero, thus no first zero crossing is defined inregime I, see Figure 6.13(b). The horizontal first zero crossing values decrease monotonicallywith decreasing Ro in regime II. The vertical counterparts show similar trend, but they start


0 2 4 6 8 10 12−0.5

0

0.5

1

τ/τη

RL al

a)


0 2 4 6 8 10 12−0.5

0

0.5

1

τ/τη

RL at,rem

b)

0 2 4 6 8 10 12

0

0.2

0.4

0.6

0.8

1

τ/τη

RL at,Cor

c)

Figure 6.12: (a) Longitudinal, (b) transversal remainder and (c) transversal Coriolis acceler-ation autocorrelations at the cell center as a function of Ro.

at Ro ' 0.95.

0 5 10 15 20 25−0.5

0

0.5

1

τ/τη

RL axy

a)


0 5 10 15 20 25−0.5

0

0.5

1

τ/τη

RL az

b)

Figure 6.13: (a) Horizontal and (b) vertical Cartesian acceleration autocorrelations at z =0.875H as a function of Ro.

In order to provide an insight into the physics behind the decrease in the first zero cross-ing value of the horizontal acceleration autocorrelations at z = 0.875H , the Lagrangianacceleration is written as proposed by Yeung and Pope [192];

a(t) = A(t)e(t), (6.6)


where A(t) is the acceleration modulus and e(t) is the direction vector. They stated thatthese two quantities are independent. Based on their simulations, they showed that e hasa short decorrelation time (scales as ∼ τη), while A has a much longer decorrelation time.Experimental studies also confirm this observation, e.g. [31, 178]. Figure 6.14 shows theautocorrelation of the acceleration modulus, the acceleration direction and the accelerationin the x direction for Ro = ∞ at z = 0.875H . Our experiments also confirm the generalpicture that the Lagrangian acceleration is governed by both the dissipative time scale (forthe direction) and the integral time scale (for the amplitude). The acceleration modulus andits x direction as a function of Ro are plotted in Figure 6.15(a,b). As can be seen fromthe graph, the acceleration modulus becomes more correlated with decreasing Ro, while theacceleration direction become less correlated. Returning to Figure 6.11, a strictly decreasingbehavior is observed for the horizontal component of the first zero crossing at z = 0.875Hin regime II. The faster decorrelation in the horizontal direction with decreasing Ro is due tothe direction change of the particles at small time lags due to swirling motion of the vorticalplumes in the horizontal plane at z = 0.875H . In other words, the faster change in theacceleration direction results in a faster decorrelation of the acceleration due to the presenceof the vortical plumes.

0 5 10 15 20 25

−0.2

0

0.2

0.4

0.6

0.8

1

τ (s)

RL

ax

ex

A

Figure 6.14: Lagrangian autocorrelation of the acceleration in the x direction, accelerationdirection and acceleration modulus for Ro =∞ near the top plate.

Measurement at z = 0.875H , Coriolis-coordinate acceleration autocorrelations(Figure 6.16(a,b,c): Figure 6.16(a,b,c) shows correlations of al, at,rem, at,Cor close to thetop. Similar to the center,RLal does not show significant changes with rotation in both regime Iand II. In contrast to the center, RLat,remshows more significant changes while transitioningbetween regimes I and II. In regime III, the oscillatory behavior is observed. As mentionedabove, the decomposition is in such a way that the Coriolis force only directly influencesat,Cor. The effects of the rotation on RLat,Cor

is much more profound compared to RLat,Cor

at the center as can be seen from the graph. The vortical plumes, initiated from the top andbottom boundaries, result in a clear distinction between regimes I and II at z = 0.875H .However, the plumes are comparatively weaker at the cell center. The at,Cor autocorrelationscollapse in regime I where the LSC is dominant. However, in regime II, they progressively in-


0 5 10 15 200

0.2

0.4

0.6

0.8

1

τ/τη

RL A

a)


0 5 10 15 20−1

−0.5

0

0.5

1

τ/τη

RL e x

b)

Figure 6.15: The Lagrangian autocorrelation of (a) acceleration modulus and (b) accelerationx direction as a function of Ro at z = 0.875H .

crease as Ro is reduced. In regime III, the oscillatory behavior due to inertial waves is presentin the at,Cor autocorrelations. Apart from this oscillatory behavior, the autocorrelations inregimes II and III display a very similar decay.

0 5 10 15 20 25−0.5

0

0.5

1

τ/τη

RL al

a)


0 5 10 15 20 25−0.5

0

0.5

1

τ/τη

RL at,rem

b)

0 5 10 15 20 25

0

0.2

0.4

0.6

0.8

1

τ/τη

RL at,Cor

c)

Figure 6.16: (a) Longitudinal, (b) transversal remainder and (c) transversal Coriolis acceler-ation autocorrelations at z = 0.875H as a function of Ro.


6.6.1 Concluding remarksIn conclusion, the Cartesian and Coriolis-coordinate acceleration autocorrelations do not ex-perience significant changes in regime I with decreasing Ro. This confirms that the flowdynamics are governed by the LSC in regime I.

The transition between regimes I and II is well reflected in the Lagrangian accelerationautocorrelations at z = 0.875H , particularly in RLat,Cor

and the first zero crossing values ofthe autocorrelations of horizontal acceleration. However, there is no sudden change in trendsin the data at the cell center while crossing the transition: a hint that the transition initiatesfrom top and bottom boundaries.

In regime II, in contrast to regime I, the Lagrangian acceleration autocorrelations changecontinuously with decreasing Ro. The anti-correlations in the vertical and horizontal accel-eration autocorrelations become stronger with decreasing Ro. The first zero crossing valuesshow stronger trends with decreasing Ro compared to regime I. At z = 0.875H , the hori-zontal first zero crossing value decreases with decrease in Ro which show that the vorticalplumes become stronger with decreasing Ro: the horizontal acceleration of the fluid parcelexperiences a faster decorrelation due to change in the direction.

The crossover to regime III coincides with strong waves which are observed in bothCartesian and Coriolis-coordinate acceleration autocorrelations. These waves hinder the fur-ther evaluation of the first zero crossing in regime III. The waves will be treated in the nextsection.

6.7 Oscillatory behaviorAs mentioned in the previous sections, wavy behaviors are observed in the Lagrangian velo-city and acceleration autocorrelations. It is known that in a rotating frame, the Coriolis forcepromotes waves, so-called inertial waves [46]. One can readily derive the dispersion relationfrom the inviscid linearized Navier-Stokes equations [46]. Imposing boundary conditionsmakes the solutions dependent on the geometry; we focus on the cylindrical geometry. Thereare a variety of studies on inertial waves in a cylindrical geometry under different forcingmechanisms: steady differential rotation [54, 94], forced sidewall oscillations [95], libration[18, 86, 118] and precession [39, 74, 102, 87, 104]. The first three conditions are not relev-ant for our experimental measurement. We consider the effects of precession due to Earth’srotation in our experimental measurements. In precessing flows, a cylinder, filled with fluid,rotates along its axis while it precesses along another axis, see Figure 6.17 for a schematicview of the problem. In this figure ΩP is the precession angular velocity, Ω is the cylinderangular velocity and α is the angle between Ω and ΩP .

The study of precessing flows is motivated by its relevance in the field ofaerospace (propellant fuel in a spinning spacecraft e.g. Refs. [160, 3]) and atmosphericscience (tornadoes and hurricanes [171]). In our experimental set-up, Earth’s rotation canact as the precession angular velocity. The important dimensionless parameters in precessingflows in cylindrical domains are the Ekman number, the Poincaré number and the aspect ratio[87]. The Poincaré number is defined as Po = |ΩP |/|Ω|; in our case |ΩP | = |ΩEarth| =7.292 × 10−5 rad/s and Ω is the rotation rate of the rotating table, Ω = Ωz, resulting in1.8× 10−5 . Po . 4.4× 10−5 in regime III. The Poincaré number indicates the strength ofprecession [188] while the aspect ratio defines the resonance condition [39].

6.7. OSCILLATORY BEHAVIOR 79

Figure 6.17: Sketch of the precessing flow. Ω is the angular frequency of the cylinder andΩP is the angular frequency of the precession.

The momentum equation for precessing thermally-driven flow in cylindrical coordinates(r, φ, z) is given by (see Ref. [87] for momentum equation for non-thermally-driven precess-ing flows)

∂u

∂t+ u ·∇u

+2(

Ωz + |ΩP | (sinα cos (φ+ Ωt)) r− sinα cos (φ+ Ωt) φ+ cosαz)× u

= −∇P + ν∇2u− 2|ΩP |Ωr sinα cos (φ+ Ωt) z + βgT z,

(6.7)

where r, φ and z are the radial, azimuthal and vertical unit vectors, respectively. Followingthe steps in Section 1.2, one can readily write the momentum equation in dimensionlessform. We can further simplify the dimensionless momentum equation when Po 1 (weakprecession), Ro 1 and u < 1 (tilde stands for the dimensionless variable) [87]

∂u

∂t+ u · ∇u +

1

Roz× u = −∇P +

√Pr

Ra∇2u− Po

4Ro2 r sinα cos

(φ+

t

2Ro

)z + T z.

(6.8)

As can be seen from the equation, the third term on the right hand side, the so-called Poincaréforcing term, is the principle forcing term due to precession. To quantify the effects of Poin-caré forcing on the flow field in our current experiments, direct numerical simulations withthis precession forcing are performed at Ra = 1.3 × 109, Pr = 6.7, Ro = 0.058, sinα =0.6232 (Eindhoven latitude = 51.44) and corresponding Po = ΩP /Ω = 2.51× 10−5. As inprevious chapters, tracer particles are tracked and evaluated.

The experimental and corresponding numerical acceleration autocorrelations at the cellcenter and at z = 0.875H are plotted in Figure 6.18. The excellent agreement between exper-


imental and simulation data confirms the role of the Poincaré force emanating from Earth’srotation. The wave amplitudes are slightly different between experiment and numerical sim-ulation, however, one should consider that the equations are simplified in the numerical sim-ulations. The wave frequency is Ω for both methods. For an inviscid weakly precessing fluid,when the frequency of the primary inertial mode is the same as that of the Poincaré force(the frequency of Poincaré force is Ω, see Equation (6.7)), a primary resonance happens. Asshown by Meunier et al. [104] the primary resonance for a cylindrical cell with aspect ratioΓ = 1 happens at a forcing frequency of 0.996Ω. Our forcing frequency is Ω, very close tothe primary resonance for Γ = 1. The primary resonances for Γ = 1/2 and 2 are 0.49Ω and1.57Ω, respectively.

0 5 10 15 20 25 30 35−0.5

0

0.5

1

τ (s)

RL axy

c)Num.Exp.

0 5 10 15 20 25 30 35−0.5

0

0.5

1

τ (s)

RL az

d)Num.Exp.

0 5 10 15 20 25 30 35−0.5

0

0.5

1

τ (s)

RL axy

a)Num.Exp.

0 5 10 15 20 25 30 35−0.5

0

0.5

1

τ (s)

RL az

b)Num.Exp.

Figure 6.18: Comparison of experimental and numerical acceleration autocorrelations. (a)Horizontal and (b) vertical acceleration autocorrelations at the cell center. (c) Horizontal and(d) vertical acceleration autocorrelations at z = 0.875H .

The velocity field belonging to the near-resonant inertial waves due to Poincaré forcingcan be illustrated by plotting the solution of the wave equations. The wave equations andtheir solution are given in Ref. [104]. Figures 6.19(a,b,c) show snapshots of the waves attime t = 0. Panel (a) shows the velocity of the waves at three cross sections perpendicularto the rotation axis. The colors and arrows show the vertical and horizontal velocities, re-spectively. Two vertical cross sections along line C are presented in panels (b) and (c). Thecolors in panels (b) and (c) are the horizontal velocity along line C and the vertical velocity,

6.7. OSCILLATORY BEHAVIOR 81

respectively. See the figure caption for more details. As can be seen from the graph, the hori-zontal component of the wave becomes stronger when approaching the top or bottom platesfrom the center. The vertical component, on the other hand, is strongest at the cell center anddiminishes when departing from the center. The same trend has been observed from exper-iments: strong vertical oscillations at the cell center and strong horizontal oscillations nearthe top plate. This pattern rotates about the cylinder axis in an anticyclonic fashion at a rateΩ. Note that these plots are formally for an inviscid flow; corrections for no-slip conditionsin a realistic situation are restricted to minor boundary layer corrections [104].

C

a) b) uC

r/H

z/H

−0.5 −0.25 0 0.25 0.50

0.25

0.5

0.75

1

c) uz

r/H

z/H

−0.5 −0.25 0 0.25 0.50

0.25

0.5

0.75

1

−1

−0.5

0

0.5

1

Figure 6.19: Illustration of the dominant inertial wave mode in a Γ = 1 cylinder. Panel (a)represents three cross sections at z = 0.85H , z = 0.5H and z = 0.15H from top to bottom,respectively. The colors indicate the vertical velocity (perpendicular to the cross sections)and arrows represent the horizontal velocity. The horizontal velocity at z = 0.5H is zero.Panel (b) shows a vertical cross section along line C, shown in panel (a). The colors indicatethe horizontal velocity along line C, uC . At r/H = 0 the velocity uC is not defined. Panel(c) shows the vertical velocity at vertical cross section along line C, similar to panel (b). Thecolor bar and consequently all the velocities are nondimensionalized by their correspondingmaximum velocities at each panel.

In conclusion, the observed oscillatory behavior is emanated from precession by Earth’srotation: these waves are thus unavoidable. However, one can significantly reduce the amp-litude of these waves by choosing a proper Γ further from a resonant mode [104]. Nonethe-less, it was unfortunate that Γ = 1 has a primary resonance at a driving frequency very closeto Ω. It is worth pointing out that the good agreement between experimental and numericaloverall heat transfer measurements for small Ro for Γ = 1, reported earlier in literature (seee.g. Figure 1.3 [159, 195]), hint that the effects of these inertial waves on the heat transfer


efficiency are negligible. However, one should note that the Poincaré forcing term increaseswith decreasing Ro: the Poincaré force dominates over thermal forcing for small enoughRo depending on α. It is worth pointing out that the Poincaré force is not limited to thethermally-driven motions: it plays a role whenever the Poincaré force becomes comparableto the other means of forcing. Therefore, experiments in rapidly rotating flows should bedone with special care: the geometry and its aspect ratio should be chosen with care.

6.8 ConclusionsIn this chapter, we have discussed different Lagrangian parameters in regimes I, II and III. Inregime I, the presented results confirm the general picture: the flow dynamics are governedby the LSC. The changes in the velocity and acceleration statistics (rms and autocorrelations)both at the cell center and near the top plate are relatively small in this regime.

According to the heat transfer efficiency, the transition from regime I to regime II isknown to be sudden: The LSC is replaced by vortical plumes. The rms velocity and acceler-ation and Lagrangian autocorrelations near the top plate experience a sudden change in trendwhile crossing this transition. On the other hand, the data at the center show a gradual changebetween regime I and II: indicating that the transition starts near the plates, as has been shownearlier in Ref. [132].

The formation of the vortical plumes in regimes II and III is reflected in the Lagrangianvelocity and acceleration statistics. The vertical velocity fluctuations decay faster than theirhorizontal counterparts at z = 0.8H and z = 0.975H . The integral time scales show anincrease as Ro decreases at the cell center and near the top plate, except for the horizontalcomponent near the top plate. The horizontal acceleration fluctuation near the top increaseswhile the other components decrease with decreasing Ro. All these observations are due tothe formation of the vortical plumes close to the top plate.

The transition between regimes II and III is comparatively gradual: the Lagrangian velo-city and acceleration statistics show gradual changes. The crossover to regime III coincideswith the following phenomena and observations:

• The vertical motions are suppressed: The vertical component of the rms velocity andacceleration values are suppressed at a higher rate in regime III compared to regimeII with decreasing Ro. The suppression of the vertical motions result in a decrease inheat transfer efficiency.

• The vortical plumes penetrate further into the bulk: The horizontal component ofthe rms acceleration continue to increases with decreasing Ro at z = 0.975H . Atz = 0.8H , the horizontal component of the rms acceleration start to increase withdecreasing Ro while transitioning to regime III: indicating that plumes have reachedthe interior region in regime III. At the cell center, no enhancement in the horizontalcomponent of the rms acceleration is observed. One should note that acceleration andvelocity might not be perfect criteria for evaluating whether plumes reach the cell cen-ter because it is known that as we go down inside a plume the sign of vorticity changes,e.g. a cold plume has a cyclonic vorticity near the top while it possesses an anticyclonicvorticity near the bottom plate. Therefore, the vorticity field is weaker in the cell centerresulting in a weaker swirling motion in the horizontal plane. This vertical change of

6.8. CONCLUSIONS 83

vorticity can be explained with the thermal-wind balance. As a result, if the horizontalacceleration and velocity statistics at the center do not follow their counterparts nearthe top, it does not directly imply that the plumes have not reached the cell center.

• The vortical plumes interact less with their surroundings: Two correlation time scalesare present for plume and non-plume regions in regime III which indicates that theflow dynamic has distinct character inside and outside the plumes: the fluid exchangebetween plume and non-plume regions decreases in regime III. In addition, a suddenincrease in the ratio RAA = arms

xy /armsz at z = 0.975H , observed in regime III, sup-

ports the same conclusion as the particles inside a plume remain inside the plume andexperience a strong horizontal centripetal acceleration.

The first and second observations support the description of the transition to regime IIIby Julien et al. [67]. The last observation has not been reported earlier: it might be a keyparameter to the transition. However, further experimental and numerical studies are certainlyrequired to evaluate the different mechanisms.

Chapter 7

Exploring the Geostrophic regime

7.1 IntroductionIn this chapter, we focus on regime III. In order to model various geophysical and astrophys-ical flows, e.g. liquid metal cores of terrestrial planets, rapidly rotating stars, and Earth’socean currents [19, 17, 103] with RRBC two conditions must be satisfied: RRBC should behighly turbulent and at the same time dominated by the Coriolis force. Maintaining highlevels of turbulence while the flow is rotationally constrained imposes severe difficultiesin both experiments and simulations. In experiments it is exceedingly hard to maintaina rotation-dominated flow without enacting significant centrifugal buoyancy. The limita-tions on the simulations arise from the enormous range of (physical and temporal) scalesto be resolved: Ekman-type boundary layers near no-slip plates in a rotating fluid are typ-ically very thin and waves in such rotating fluids require small time steps in simulations.These limitations to the numerical solution of the full incompressible Navier–Stokes equa-tions (in the Boussinesq approximation) in the rapidly rotating limit have led Julien andcoworkers to formulate a set of asymptotically reduced equations in the limit Ro → 0[68, 151]. While these asymptotically reduced equations neglect the Ekman boundary layerand fast inertial waves, the gain is huge. Therefore, these equations have been widely usedto characterize the flow morphologies from the onset of convection to geostrophic turbulence[151, 70, 67, 117, 153, 137]. Very recently, Ekman pumping effects are also incorporated inthese equations, see Refs. [66, 124].

At small enough Ro, experiments and direct numerical simulations (DNS) of the fullNavier–Stokes equations (in the Boussinesq approximation) should give similar results assimulations of the asymptotically reduced equations. Up to now, most of the experiments[136, 193, 91, 73, 92, 195, 184, 194, 84, 182, 183] and DNSs [73, 195, 142, 143, 154, 61]did not reach deep into the rapidly rotating convection regime. The few experimental andnumerical studies that entered decisively into this regime [153, 36, 26, 83] primarily use theoverall heat transfer to characterize rapidly rotating Rayleigh–Bénard convection and identifytransitions between flow regimes from changes in the heat-flux scaling. These findings sug-gest a phase diagram as sketched in Figure 7.1, with the Ekman number Ek indicating the

The contents of this chapter have been adopted from H. Rajaei, R.P.J. Kunnen, H.J.H. Clercx, “Exploring thegeostrophic regime of rapidly rotating convection with experiments”, Physics of Fluids 29(4):045105.

85

86 CHAPTER 7. EXPLORING THE GEOSTROPHIC REGIME

ratio of viscous forces over Coriolis forces and the strength of thermal forcing is quantifiedby the Rayleigh number Ra and normalized by its critical value Racr for onset of convectivemotion. Racr grows as the rotation rate is increased [23]. Different turbulent regimes canbe identified: (I) the rotation-unaffected regime where large scale circulation is the main fea-ture of the flow and the heat flux remains constant; (II) the rotation-affected regime whererotation-aligned vortical plumes characterize the flow and the heat flux increases with in-creasing background rotation (for Pr > 1 where Pr is the Prandtl number; it specifies thediffusive properties of the fluid); and (III) the geostrophic regime where the flow is princip-ally governed by a balance of the Coriolis force and the pressure gradient force and the heatflux drops dramatically with increase in the background rotation. In Figure 7.1, the dashedand solid black lines indicate the transitions to the rotation-affected and the geostrophic re-gimes, respectively, for Pr = 0.7, see Ref. [36]. The two gray dashed lines indicate theproposed alternative predictions for the transition to the geostrophic regime suggested byRef. [73] (upper gray dashed line, for Pr ≈ 7) and Ref. [72] (lower gray dashed line, forPr ≈ 4.5−11). The dotted line is the lower bound of the geostrophic regime (Ra/Racr ' 3,see Ref. [36]), below which linear or chaotic convection is expected. The filled blue and openred symbols correspond to the current experimental data and the data from the literature, re-spectively [73, 92, 194, 36, 26]. Red symbols are further explained in the caption of Figure7.1. Note that the transition between the rotation-unaffected and rotation-affected regimes inFigure 7.1 (dashed black line) is plotted for Pr = 0.7. This transition strongly depends on Prand the corresponding transition for Pr ' 7 (similar to Pr number in our experimental data)is vertically above the axis limits of Figure 7.1. On the other hand, the transition between therotation-affected and the geostrophic regimes, which is of interest in this study, is reported tobe less affected by Pr, see Ref. [36].

The experiments that conclusively reached the geostrophic regime are highly dedicatedsetups with either lack of optical access [36] (red open squares in Figure 7.1) or extremedimensions [26] (red open stars in Figure 7.1). In Ref. [36] cryogenic helium gas near itscritical point at 5.2 K is used as working fluid and in Ref. [26] a 1.6 m tall convection cellis used compared to commonly used 0.2-0.4 m tall convection cells. Only qualitative flowvisualizations have been performed in Ref. [26]. The commonly used convection cells androtating tables can reach Ekman values down to approximately 3×10−6, allowing for a smallpart of the geostrophic regime to be accessible. In this chapter we want to push our setup tothe limit of rapid rotation at correspondingly chosen Rayleigh numbers to assess whetherwe can explore the geostrophic regime, and use optical flow diagnostics to obtain statisticsof the flow features to compare to the results from the asymptotic simulations. A favorablecomparison is certainly not guaranteed, given the difference in boundary conditions (no-slipfor experiments and stress-free for the asymptotic simulations) as well as the unavoidablecentrifugal effect in experiments that prevents the application of very large rotation rates.Nevertheless, we achieve a surprisingly favorable agreement.

The simulations of the asymptotic equations have revealed four distinct flow structures,namely cellular convection, convective Taylor columns, plumes, and geostrophic-turbulence[151, 117, 153]. The cellular state occurs just above the onset of convection; it is character-ized by densely packed thin hot and cold columns spanning the entire vertical extent of theflow domain. Departing from the onset of convection (Ra increases), these cells may developinto well-separated vertically aligned vortical convective Taylor columns (CTCs), surroundedby shields of vorticity of opposite sign. Another mode of convection consists of plumes with

7.1. INTRODUCTION 87

10−8

10−7

10−6

10−5

100

101

102

103

Ek

Ra/Racr

rotation−unaffected

rotation−affected

geostrophic

not turbulent

Figure 7.1: Phase diagram of rotating convection as proposed by Ref. [36]. The transitionbetween rotation-unaffected regime and rotation-affected regime (for Pr = 0.7, obtainedfrom heat-transfer measurements in cryogenic helium and given by Ra = 0.086Ek−2) isindicated by the dashed black line [36]. The aforementioned transition for Pr ≈ 6 is outof the graph, given by Ra = 24Ek−2, see Ref. [36]. The solid black line, given by Ra =0.25Ek−1.8, is the proposed crossover to the geostrophic regime for Pr = 0.7. The twogray dashed lines are the alternative predictions of the transition to the geostrophic regimesuggested by Refs. [73, 72]. These lines are given by Ra = 1.4Ek−7/4 (upper dashed line,for Pr ≈ 7) and Ra ≈ 10Ek−3/2 (lower dashed line, for Pr ≈ 4.5 − 11), respectively.The dotted black line indicates the transition to a regime of linear or chaotic convectionat Ra/Racr ' 3 as proposed by Ref. [36]. The filled blue circles are for the currentexperiments with Pr ≈ 6 − 8. The open red symbols are the literature data; diamonds fromRef. [73], circles from Ref. [92], upright triangles from Ref. [194], squares from Ref. [36],and stars from Ref. [26].

less vertical coherence and no shields. A final state is called “geostrophic-turbulence", wherethe vertical coherence is lost almost completely and the interior is fully turbulent. Note thatall four convection modes are part of the geostrophic regime, which should thus not uniquelybe identified with the geostrophic-turbulence state alone. The occurrence of these four statesis strongly dependent on the Prandtl number Pr: at lower Pr . 3 no Taylor columns areformed, while the geostrophic-turbulence state remained out of reach for Pr & 7 in the sim-ulations [70].

Nieves et al. [117] have reported earlier that the spatial autocorrelations of temperature,calculated from simulations of the asymptotically reduced equations, can be employed todifferentiate between these different states. They also mention that for that purpose one couldalso use vertical vorticity or vertical velocity; the results should be almost identical (see alsoRef. [151]). We shall use vertical vorticity as a proxy for temperature.

The remainder of the chapter is organized as follows. The experimental parameters aregiven in Section 7.2.1. The experimental set-up and the measurement techniques are dis-


cussed in Section 7.2.2. In Section 7.3.1, the results of the spatial autocorrelation of the ver-tical vorticity are discussed. Next, the flow coherence along the axis of rotation is discussedin Section 7.3.2. We discuss our main findings in Section 7.4.

7.2 Experimental techniques and parameters

7.2.1 Experimental parametersChandrasekhar [23] used linear stability analysis to demarcate the onset of convection and hefound Racr ' 8.6956Ek−4/3 for small values of Ek, where Racr is the Rayleigh numberat the onset of convection, below which the flow halts and diffusion is the only active wayof heat transfer. The value RaEk4/3 is thus a similar parameter to Ra/Racr that is used tospecify the distance to Racr. We shall use RaEk4/3 to indicate the supercriticality, keepingin mind that RaEk4/3 ' 8.6956 marks the onset of convection.

Approaching the onset of convection can be achieved either by variation of the back-ground rotation (Ω) or by variation of the temperature difference (∆T ). In the present study,both approaches have been examined and compared.

7.2.2 Experimental setupTwo different experimental techniques are used; time-resolved two-dimensional particle im-age velocimetry (PIV) and three-dimensional particle tracking velocimetry (3D–PTV). The3D–PTV data provides information at different vertically distributed horizontal planes alongthe axis of rotation, while PIV data provides information only at one horizontal plane but athigher spatial resolution and an extended planar view.

The PIV experiments are performed in a horizontal planar cross-section at z = 0.8H ,where the bottom plate is at z = 0. The volumetric PTV experiments cover a volumebetween z = 0.76H and z = 0.96H . For both PIV and 3D–PTV measurements, the centerof the measurement domain coincides with the axis of rotation. The details of the experi-mental parameters, conducted by PIV and 3D–PTV are summarized in Tables 7.1 and 7.2,respectively. These experiments correspond to the row 8 to 31 in Table 2.2.

7.2.3 Data validation3D–PTV naturally provides the data in the Lagrangian frame of reference. However, here weare interested in variables in the Eulerian frame of reference. Therefore, interpolation on aregular grid is inevitable. In the current experimental 3D–PTV data an average number of2700 randomly distributed particles are tracked at each time step. The particle velocity alonga trajectory is calculated using a second-order finite difference method. Then, the velocities ofthese randomly distributed particles are interpolated (triangulation-based linear interpolation)on a regular grid with 3 mm grid spacing. Interpolation acts as a filter, consequently some(high-frequency, small-scales) information might not pass through the interpolation. We ex-amine the effects of the interpolation on our 3D–PTV data by comparison of the interpolated3D–PTV data and the spatially-resolved PIV data as an a posteriori check. A comprehensivediscussion of the physical interpretation of the spatial vorticity autocorrelations is given inSection 7.3.1, however, here we are only focused on the (dis)similarities between PIV and

7.2. EXPERIMENTAL TECHNIQUES AND PARAMETERS 89

Table 7.1: Details of the PIV data sets. In the table, Tmax and Tmin are the hot and cold platetemperatures, respectively, Lc is the critical length (defined as Lc = 4.8154Ek1/3, see Ref.[23]), Tb is the buoyancy time scale defined as Tb = Tv((Ra− Racr)Ek4/3/Pr)−1/2 whereTv is the viscous time scale, Tv = (Ek1/3H)2/ν. The definitions of Tb and Tv are takenfrom Ref. [117]. Viscosity and thermal diffusivity are evaluated at the mean temperature;different mean temperatures thus lead to minor variations in the parameter values such as, theEkman number. TC and RC are the acronyms for temperature change and rotation change,respectively.

RaEk43 Ω Tmax Tmin Ro Pr Ek Racr Ra Fr Lc Tb

(rad/s) (C) (C) ×106 ×10−8 ×10−8 (mm) (s)

TC

18 4.12 22 18.5 0.023 7.01 3.02 1.99 4.1 0.17 13.93 7.3819 4.12 18.5 14 0.023 7.85 3.35 1.74 3.74 0.17 14.44 7.1824 4.12 19.5 14 0.026 7.74 3.31 1.76 4.78 0.17 14.34 5.8826 4.12 23.5 18.5 0.028 6.88 2.97 2.03 6.1 0.17 13.85 5.3132 4.12 21 14 0.03 7.58 3.24 1.81 6.50 0.17 14.25 4.7639 4.12 25.5 18.5 0.034 6.71 2.91 2.10 9.1 0.17 13.74 4.0940 4.12 22.5 14 0.034 7.43 3.18 1.85 8.41 0.17 14.17 4.050 4.12 27.5 18.5 0.039 6.53 2.84 2.16 12.5 0.17 13.64 3.4063 4.12 29.5 18.5 0.044 6.36 2.77 2.23 16.3 0.17 13.53 2.9580 4.12 29 14 0.049 6.79 2.94 2.07 18.9 0.17 13.80 2.62

RC

14 2.75 21.6 20 0.023 6.91 4.47 1.18 1.92 0.08 15.86 12.3719 2.25 21.6 20 0.029 6.91 5.47 0.90 1.92 0.05 16.98 10.5622 2 21.6 20 0.032 6.91 6.15 0.77 1.92 0.04 17.64 9.9527 1.75 21.6 20 0.037 6.91 7.03 0.65 1.92 0.03 18.45 9.4532 1.5 21.6 20 0.043 6.91 8.20 0.53 1.92 0.02 19.42 9.0341 1.25 21.6 20 0.052 6.91 9.48 0.41 1.92 0.02 20.64 8.6855 1 21.6 20 0.065 6.91 12.30 0.31 1.92 0.01 22.23 8.4

Table 7.2: Details of the 3D–PTV data sets, see the caption of Table 7.1 for more details.

RaEk43 Ω Tmax Tmin Ro Pr Ek Racr Ra Fr Lc Tb

(rad/s) (C) (C) ×106 ×10−8 ×10−8 (mm) (s)

TC

18 4.12 22 18.5 0.023 7.01 3.02 1.99 4.1 0.17 13.93 7.3826 4.12 23.5 18.5 0.028 6.88 2.97 2.03 6.1 0.17 13.85 5.3139 4.12 25.5 18.5 0.034 6.71 2.91 2.10 9.1 0.17 13.74 4.0950 4.12 27.5 18.5 0.039 6.53 2.84 2.16 12.5 0.17 13.64 3.4063 4.12 29.5 18.5 0.044 6.36 2.77 2.23 16.3 0.17 13.53 2.95120 4.12 37.5 18.5 0.062 5.76 2.53 2.52 34.9 0.17 13.13 2.02

RC

54 4.12 27 17 0.041 6.7 2.9 2.10 13.0 0.17 13.74 3.2767 3.5 27 17 0.048 6.7 3.42 1.69 13.0 0.13 14.51 3.2185 2.91 27 17 0.057 6.7 4.11 1.31 13.0 0.09 15.43 3.16142 2 27 17 0.0083 6.7 5.98 0.80 13.0 0.04 17.49 3.01183 1.65 27 17 0.1 6.7 7.25 0.32 13.0 0.03 18.64 3.07212 1 31 26 0.13 5.68 10.30 0.38 9.43 0.01 20.97 3.84

3D–PTV data. Figures 7.2(a-b) show the spatial vorticity autocorrelations from PIV and 3D–PTV for two experiments at RaEk4/3 = 18 and 63, respectively. Each point in the plot isequidistant from the neighboring points and its distance represents the grid spacing (spatialresolution) of each technique. It is also clear from Figures 7.2(a-b) that PIV has a consider-ably higher spatial resolution compared to 3D–PTV and thus it is considered as a benchmarkto validate the 3D–PTV data. The horizontal bars indicate the uncertainty as a function ofthe separation distance, d. Note that the uncertainty grows with d due to the accumulationof errors. The deviation for RaEk4/3 = 18 is larger than that of RaEk4/3 = 63. Notethat the flow motion is considerably slower (more stable) for RaEk4/3 = 18 compared to


RaEk4/3 = 63. Hence, minor imperfections (e.g. the differences in the ambient temperaturefor two different sets of the experiment) have a greater impact on the reproducibility of thedata for RaEk4/3 = 18, thus the deviations between 3D–PTV data and PIV data might comefrom the reproducibility of the data. However, as can be seen from the graph, the deviationsare well within the uncertainty range for both RaEk4/3 = 18 and 63. It is concluded that theinterpolated 3D–PTV data has sufficient quality for the purpose of this study.

0 10 20 30 40 50−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

d (mm)

R

b)

0 10 20 30 40 50−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

d (mm)

R

a)

3D−PTVPIV

3D−PTVPIV

Figure 7.2: The spatial vertical vorticity autocorrelation from PIV and 3D–PTV data for (a)RaEk4/3 = 18 and (b) RaEk4/3 = 63. The points on each curve are equidistant from theneighbouring points and their distances represent the spatial resolution of each technique.The horizontal bars indicate the uncertainty as a function of the separation distance, d. Theuncertainty grows with d due to accumulation of the errors.

x (mm)

y(m

m)

a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)

20 40 60 80 100

20

40

60

−0.05

0

0.05

x (mm)

y(m

m)

b)b)b)b)b)b)b)b)b)b)b)b)b)b)b)b)b)b)b)b)b)b)b)b)b)b)b)b)

20 40 60 80 100

20

40

60

−2

−1

0

1

2

x (mm)

y(m

m)

c)

20 40 60 80 100

20

40

60

−2

−1

0

1

2

Figure 7.3: The vorticity field from PIV measurements for (a) RaEk4/3 = 18, (b) RaEk4/3 =50 and (c) RaEk4/3 = 80. The vertical and horizontal axes are in mm and the color bar is inunits of 1/s.

7.3 Results and discussionThroughout this paper, the terminology “vorticity" is used to refer to the “vertical componentof the vorticity vector". We start with a qualitative analysis of the flow field. Figures 7.3(a-c)show snapshots of the vorticity field from PIV measurements at RaEk4/3 = 18, 50 and 80(note the differences in the color bar ranges). Figure 7.3(a) can be identified as the cellular


state; it is very similar to figure 1(a) in Ref. [117]. Note that the figure in Ref. [117] is a planarrendering of the temperature with an area of size 20Lc × 20Lc, while Figure 7.3(a) in thispaper shows a planar view of the vorticity in a much smaller region (∼ 7Lc × 6Lc). Judgingfrom these plots, the transition from the cellular state to the plume state is a gradual one.The columnar state, which is represented by well separated convective Taylor columns, is notfully developed in Figure 7.3(b). However, the plume state is apparent for RaEk4/3 = 80, seeFigure 7.3(c) (see also figure 1(c) in Ref. [117]). In the subsequent sections, a quantitativeanalysis based on the vorticity autocorrelation is given.

7.3.1 Spatial vorticity autocorrelation at z = 0.8H

The spatial vorticity autocorrelation provides information on the size of the coherent struc-tures. The spatial autocorrelation of the vorticity in the x direction is given by

Rxω(d) =〈ω(x, y, z)ω(x+ d, y, z)〉

〈ω2(x, y, z)〉, (7.1)

where ω is the vorticity and d is the separation distance. Due to the symmetry imposedby the cylindrical cavity, we define the horizontal autocorrelation as the average of the spa-tial autocorrelations in the x and y directions, thus Rxyω (d) = 1

2

(Rxω(d) + Ryω(d)

). Since

Rxω(d) ' Ryω(d) the averaging only results in a better convergence of the data while it doesnot affect any of the conclusions.

Figure 7.4(a) shows the spatial autocorrelations of the vorticity for some of the exper-iments at z = 0.8H , see Table 7.1 for more details. The spatial vorticity autocorrelationis found to decorrelate with itself within d ' 0.8Lc (Lc = 4.8154Ek1/3) and it shows apersistent negative loop (anti-correlation). The negative loop is stronger for smaller valuesof RaEk4/3 (closer to the onset of convection). The negative loop is an indication that eachvorticity patch is shielded by fluid with an opposite signed vorticity acting like a blanket atsmall RaEk4/3 values. The shields prevent vortex-vortex interactions [151].

As mentioned before the autocorrelations of the vertical vorticity, vertical velocity andtemperature are almost identical, see Ref. [151, 117]. Therefore, we compare vorticity auto-correlations from the experiments with temperature autocorrelations from the simulations.Figure 7.4(b) shows the spatial temperature autocorrelation, taken from Ref. [117]. Experi-ments and simulations show a wavy autocorrelation for small RaEk4/3 values. The wavy be-havior is strongly suppressed for larger values of RaEk4/3. The strength of the negative loopdecreases with increase in RaEk4/3 which indicates that the shield becomes weaker with in-creasing RaEk4/3 in both experiments and simulations. However, in the simulations the neg-ative loop disappears at RaEk4/3 = 140 (geostrophic-turbulence state). The experimentaldata covers a wide range of RaEk4/3 between 14 and 212, however, the anti-correlation isalways present in the experimental data. This dissimilarity between experiments and simu-lations is owing to the fact that the simulation at RaEk4/3 = 140 (geostrophic-turbulencestate) is performed for Pr = 1, while in the experiments Pr is approximately 6. Julien etal. reported that larger values of Pr postpone the transition to the geostrophic-turbulencestate [70]. In fact, these authors found a plume state for RaEk4/3 = 160 at Pr = 7. There-fore, it is expected that the experiments are still in the plume state even for the largest valueRaEk4/3 = 212 applied here.


0 0.5 1 1.5 2 2.5 3−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

d/Lc

Rxy

ω(d)

a)Ra Ek4/3 = 18

RaEk4/3 = 26

RaEk4/3 = 39

RaEk4/3 = 50

RaEk4/3 = 63

RaEk4/3 = 212

0 0.5 1 1.5 2 2.5 3−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

d/Lc

Rxy

T(d)

b)RaEk4/3 = 10

RaEk4/3 = 50

RaEk4/3 = 80

RaEk4/3 = 140

Figure 7.4: (a) The spatial autocorrelation of the vorticity for different RaEk4/3 from theexperiments and (b) the spatial autocorrelation of temperature for different RaEk4/3, takenfrom Ref. [117]. The horizontal axis is nondimensionalized by the critical length scale,Lc = 4.8154Ek1/3.

A more quantitative comparison between experiments and simulations is drawn by ex-amining the half-width, the first zero crossing, the position of the minimum, the second zerocrossing and the position of the local maximum of the spatial vorticity autocorrelation. Theexperimental values are plotted in Figure 7.5 as a function of RaEk4/3. The filled and opensymbols correspond to the PIV and the 3D–PTV measurements, respectively. It is observedthat the spatial vorticity autocorrelation becomes wider with increasing RaEk4/3, e.g. the rateat which the half-width increases with RaEk4/3 is less than that of the second zero crossing.The same result for the spatial temperature autocorrelation is observed from simulations inRef. [117]. Our experimental dataset does not contain the second zero crossing and the localmaximum for RaEk4/3 & 85 due to the fact that the structures become wider and wider withincrease in RaEk4/3 and at some point they become so large that the second zero crossingis beyond the observation view of the cameras. The extrapolation of the data reveals that thesecond zero crossing should keep increasing with RaEk4/3 (e.g. the estimated second zerocrossing value for RaEk4/3 = 212 is at d/Lc ∼ 2.1).

The half-width (circles), the first zero crossing (upright triangles), the minimum (squares)and the second zero crossing (stars) from the vorticity autocorrelations are plotted again inFigures 7.6(a-d), complimented by the same quantities obtained from temperature autocor-relations from simulations of the asymptotically reduced equations (the black solid curves).The simulation data are smaller than the experimental data. However, it should be emphas-ized that a one by one comparison between experiments and simulations is not the purposeand probably not even legitimate since simulations are performed for stress-free boundaryconditions. Additionally, they are formally only valid for the limiting case Ro → 0, thoughStellmach et al. [153] found that the results of simulations of the asymptotic equations arestill consistent with results from full Navier–Stokes DNS for Ek . 10−7. Furthermore, theexperimental data show the spatial vorticity autocorrelation while the simulations are for thespatial temperature autocorrelation. Therefore, we are more interested in the general trendsrather than the actual values. From Figures 7.6(a-d), it can be seen that the half-width, the


first zero crossing, the position of the minimum, and the second zero crossing show similartrends as those of the simulations, but at generally larger values of d/Lc.

50 100 150 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

RaEk4/3

d/Lc

Figure 7.5: The half-width (circles), the first zero crossing (upright triangles), the positionof the minimum (squares), the second zero crossing (stars) and the position of the localmaximum (right-pointing triangles) of the spatial vorticity autocorrelations as a function ofRaEk4/3. The open symbols are the data from 3D–PTV and filled symbols are the data fromPIV. The red and blue symbols represent the experiments with variation in the temperaturedifference and variation in the background rotation, respectively.

Continuing with the spatial vorticity autocorrelation, the autocorrelation coefficient val-ues of the minimum (Rxyω,min) and first local maximum (Rxyω,max) as a function of RaEk4/3

are presented in Figures 7.7(a-b). The solid black lines in Figures 7.7(a-b) are guides to theeye through the data points for RaEk4/3 . 70 and RaEk4/3 & 70. The dashed black linesindicate RaEk4/3 = 70. The minimum values, see Figure 7.7(a), approach to zero with in-creasing RaEk4/3. The rate at which the minimum values change is larger for RaEk4/3 . 70

than that for RaEk4/3 & 70. The data from simulations are also plotted in the figure (blackstars) with a good quantitative agreement. Note that it is expected that the anti-correlationdisappears for the geostrophic-turbulence state thus the minimum value is non-existent inthis state. Therefore, there is no value reported for RaEk4/3 = 140 (geostrophic-turbulencestate) from simulations. As discussed before, our experimental measurements do not coverthe geostrophic-turbulence state. The local maximum values of the spatial autocorrelations,see Figure 7.7(b), show a decrease with increasing RaEk4/3 and they are practically zerofor RaEk4/3 & 70. The simulated values for the temperature autocorrelations are plotted asblack stars and they show a reasonable agreement.

In conclusion, the spatial vorticity autocorrelations show similar trends as those of thespatial temperature autocorrelation calculated based on the asymptotically reduced equations.In addition, the spatial vorticity autocorrelation shows a change in the slope in the vicinityof RaEk4/3 ' 70 which is most probably due to the transition from the cellular-columnarstate to the plume state. The results of the temporal vorticity autocorrelations are discussedin Appendix A.


0 50 100 150 2000.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4

RaEk4/3

d/Lc

a)a)

0 50 100 150 2000.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

RaEk4/3

d/Lc

b)b)

0 50 100 150 2000.6

0.7

0.8

0.9

1

1.1

1.2

RaEk4/3

d/Lc

c)c)

0 50 100 150 2000.8

1

1.2

1.4

1.6

1.8

RaEk4/3

d/Lc

d)d)

PIV−Temp.PIV−Rot.3D−PTV−Temp.3D−PTV−Rot.




Figure 7.6: (a) The half-width, (b) the first zero crossing, (c) the minimum and (d) the secondzero crossing of the spatial vorticity autocorrelation as a function of RaEk4/3. The blackcurves are the corresponding values for the autocorrelations of temperature from simulationsof the asymptotic equations.

Effects of Ekman number on the correlations

The experimental data, see Figures 7.6(a-d), show that the blue symbols (experiments withvariation in the background rotation) are generally lower than the red symbols (experimentswith variation in the temperature difference) for similar RaEk4/3 values. Exploring theTables 7.1 and 7.2 reveals that the blue data points possess relatively higher Ek. In otherwords, for two experimental data sets with equal RaEk4/3 value, the dimensionless spatialvorticity autocorrelation is wider for the data set with smaller Ek number. The measurementfor RaEk4/3 = 55 clearly manifest the effects of large Ek since it possesses the largest Ekin our data sets.

We examine four different points from both the PIV and the 3D–PTV measurements,viz. those with RaEk4/3 = 27, 32, 55 and 67; see Tables 7.1 and 7.2 for details. A truecomparison should be drawn between these points and their limiting case counterparts where


50 100 150 200−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

RaEk4/3

Rxy

ω,m

in

a)

50 100 150 2000

0.05

0.1

0.15

0.2

0.25

RaEk4/3

Rxy

ω,m

ax

b)

PIV−Temp.PIV−Rot.3D−PTV−Temp.3D−PTV−Rot.Asymptotic equations


Figure 7.7: (a) The minimum (Rxyω,min) and (b) first local maximum (Rxyω,max) of the spatialvorticity autocorrelation as a function of RaEk4/3. The values for the asymptotic equationsare taken from the spatial temperature autocorrelations. The dashed black lines are located atRaEk4/3 = 70 and the solid black lines are guides to the eye through the experimental datapoints for RaEk4/3 . 70 and RaEk4/3 & 70.

Ek → 0. However, here the limiting case data are not available, therefore, the points withlarge Ek are compared with the data points with smaller Ek (based on the availability of theexperimental data). The experimental data points with smaller Ek are only available for dif-ferent RaEk4/3 values and thus an interpolation or extrapolation is required. For example,the autocorrelation values for the measurement at RaEk4/3 = 55 (Ek = 12.3 × 10−6-largeEk) are compared with the interpolated values (linear interpolation) based on the measure-ments at RaEk4/3 = 50 (Ek = 2.84 × 10−6) and RaEk4/3 = 63 (Ek = 2.77 × 10−6).Therefore, we have two experimental data points with the same RaEk4/3 but different Ek;for the aforementioned example, we have RaEk4/3 = 55 and Ek = 12.3× 10−6 for Eklarge

and RaEk4/3 = 55 and Ek = 2.8 × 10−6 for Eksmall. Table 7.3 presents the Eklarge andEksmall data for all considered cases.

Table 7.3: Eklarge and Eksmall data for all considered cases. Eksmall represents the inter-polated data.

Eklarge/Eksmall 1.2 2.3 2.7 4.4RaEk4/3 67 27 32 55

Eklarge × 106 3.42 7.03 8.20 12.30Eksmall × 106 2.83 3.10 3.07 2.80

As discussed before, Lc (which is a function of Ek) is used to nondimensionalize theseparation distance in the spatial vorticity autocorrelations (d/Lc). Here, we compare thevalues of d/Lc for Eklarge with those of Eksmall for the half-width and first zero crossing.The ratio RA is defined as

RA =d/Lc(Eksmall)

d/Lc(Eklarge),


where d/Lc(Eklarge) and d/Lc(Eksmall) are the data for the measurements with large andsmall Ek, respectively, and the same RaEk4/3. Figure 7.8 shows the ratio RA at the half-width and first zero crossing as a function of Eklarge/Eksmall. The points in the plot fromthe smallest Eklarge/Eksmall to the largest Eklarge/Eksmall (left to right) correspond toRaEk4/3 = 67, 27, 32 and 55, respectively, see Table 7.3 as well. Note that d/Lc is dimen-sionless and it does not explicitly give information on the physical width of the structures.The ratio RA shows an increase with Eklarge/Eksmall indicating that dimensionless struc-tures become smaller with increasing Ek when RaEk4/3 is maintained constant. While theasymptotic equations do not explicitly depend on Ek and RaEk4/3 is used as a parameter fordescription of the flow dynamics, we thus observed a dependency of the flow dynamics onthe Ekman number for a constant RaEk4/3. However, it should be noted that this depend-ency is small; e.g. if the Ekman number increases by a factor of 5, the typical dimensionlesscoherency length scale decreases by approximately 13%.

The physical width of the structure shows the opposite behavior and it increases with Ek(as expected). Based on the definition of Lc, it increases as Lc ∝ Ek1/3. The physical widthof the autocorrelation increases as well but at a smaller rate. Therefore, the ratio RA showsan increase with increasing Ek.

1 2 3 4 50.95

1

1.05

1.1

1.15

Eklarge/Eksmall

RA

half−widthfirst zero

Figure 7.8: The ratio RA for different Eklarge/Eksmall, indicating the effects of Ek on thelength scales inferred from the correlations. The blue circles and red triangles show the ratioRA calculated for half-width and first zero crossing of the spatial vorticity autocorrelations,respectively.

7.3.2 Flow coherence along the rotation axisUp to now, we examined the spatial vorticity autocorrelations at z = 0.8H for the entire planeconsisting of the plumes (columns) and the intervening regions. However, in this section thecoherent structures (plumes or columns) are examined separately from the regions betweenthe columns or plumes. For this purpose, we are first required to detect the region populated


by the plumes (columns). A widely used method for the vortex detection is the so-calledQ-criterion method [62]. For a three-dimensional flow field a vortex is defined as a spatialregion where

Q =1

2

(||Ω||2 − ||S||2

)> 0, (7.2)

with Ω and S the antisymmetric and symmetric parts of the velocity gradient tensor, respect-ively. In this equation, the operator ||A|| is the Euclidean norm

||A|| =√

Tr(AAT

). (7.3)

Figure 7.9 shows snapshots of the isosurfaces of Q = 0.5Qrms, where Qrms is the rootmean square of Q at the time of the snapshot. This threshold is introduced only for thepurpose of better illustration. The isosurfaces are plotted for three different cases: weaklyrotating Rayleigh–Bénard turbulence, plume state and cellular state. Figure 7.9(a) shows theisosurface of Q for a weakly rotating Rayleigh–Bénard turbulence case (RaEk4/3 = 3644and Pr = 6.7). As can be seen from the figure, there are no clear vertical correlations for thiscase. In Figure 7.9(b), the vertically aligned plumes become apparent for RaEk4/3 = 120(plume state). For small values of RaEk4/3, see Figure 7.9(c) for RaEk4/3 = 18, the cellsare the main features of the flow. The blue and red colored surfaces represent the down-going(cold) and up-going (hot) plumes (columns), respectively.

Figure 7.9: Isosurfaces of Q for (a) RaEk4/3 = 3644 (weakly rotating Rayleigh–Bénardturbulence, Ek = 6.8 × 10−5, Ra = 1.3 × 109 and Pr = 6.7), (b) RaEk4/3 = 120 and (c)RaEk4/3 = 18. The blue surfaces represent the cold plumes and red are the hot plumes.

The first parameter we examine is the variation of the vorticity along the axis of rota-tion. Figure 7.10 shows the color plot of the vorticity autocorrelation along the rotation axis.The starting point of the autocorrelation is located at z = 0.96H and the final point is atz = 0.762H . The autocorrelation is calculated only in the points where the conditionQ > 0 at z = 0.96H is satisfied. As can be seen from the graph, a gradual decrease in the co-herence is observable. The simulations show similar trends [70]. However, they found morecoherence for small RaEk4/3 values and less coherence for large RaEk4/3 values comparedto the experiments. Note that the sign of vorticity will flip as we move down far enough in acolumn [127, 47]; the vortical plumes spin down as they approach the center. This property


dictates a gradual decrease in the vorticity value (decorrelation) regardless of the RaEk4/3

values. Hence, another criterion which does not depend on the height is more appropriate.Thus, a binary function A is introduced as

A =

1 if Q > 0

0 otherwise.(7.4)

The gradual decrease in the vertical vorticity value is not reflected in the binary function A.Thus, A is a more suitable representative of how far the plumes (columns) extend into thebulk.

z/H

RaEk4/3

0.96 0.92 0.88 0.84 0.8

50

100

150

200

0.4

0.5

0.6

0.7

0.8

0.9

Figure 7.10: Color plot of the autocorrelation of the vorticity as a function of height fordifferent RaEk4/3 values. The “spike" in the plot at RaEk4/3 = 63 depicts a strongercorrelation than the adjacent values, but it is still within the accuracy range.

The autocorrelation of the binary function A is plotted in Figure 7.11. The vertical axisrepresents the autocorrelation coefficient and the horizontal axis is z/H , where z/H = 1 is atthe top plate. The horizontal axis starts at z = 0.96H since the autocorrelation is calculatedonly on the points starting at z = 0.96H . It can be seen that there is a rather abrupt dropdown to z = 0.92H and then the autocorrelation shows an approximately linear decay. Thefirst abrupt drop is a direct consequence of the measurement noise. In other words, the Q-criterion approach identifies some points as a vortex (the binary number for a vortex regionis 1) which become uncorrelated (the binary number for a non-vortex region is 0) at shortdistances. The effect of the noise vanishes for larger distances.

It is possible to fit a line RzA = α zH + γ to the part where the autocorrelation shows a

linear decay and take the slope α as the decay rate at which the plumes become uncorrel-ated. Figure 7.12 shows α as a function of RaEk4/3. Note that α = 0 indicates a fullyvertically correlated flow field. The α values in Figure 7.12 are compatible with the trans-ition observed at RaEk4/3 ' 70; the decay rate of the slope as a function of RaEk4/3 islarger for RaEk4/3 . 70 compared to that for RaEk4/3 & 70. Note that α decreases with


0.96 0.91 0.86 0.81 0.760.4

0.5

0.6

0.7

0.8

0.9

1

z/H

Rz A

RaEk4/3=18

RaEk4/3=212

Figure 7.11: Autocorrelation of the binary function A for RaEk4/3 = 18 and 212.

increasing RaEk4/3; indicating that the flow coherence along the axis of rotation decreaseswith increasing RaEk4/3.

0 50 100 150 200−2

−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

RaEk4/3

α

Figure 7.12: The slopes α of the linear fit RzA = α zH + γ to the autocorrelation of A as

a function of RaEk4/3. The dashed black line is located at RaEk4/3 = 70 and the solidblack lines are guides to the eye through the experimental data points for RaEk4/3 . 70 andRaEk4/3 & 70.


7.4 ConclusionsWe have explored the geostrophic regime of rapidly rotating turbulent convection with labor-atory experiments based on classical tools: a table-top convection cell with height of 0.2 mfilled with water as working fluid. We use two different optical flow diagnostic techniques:3D–PTV and time-resolved PIV.

We pushed the boundaries of our experimental setting to determine if and how far we canenter the geostrophic regime of rapidly rotating convention using a conventional Rayleigh–Bénard convection cell. To achieve this goal, we compare our experimental data of the spatialvorticity autocorrelations with the spatial temperature autocorrelations from the simulationsof the asymptotic equations, taken from Ref. [117]. Surprisingly, most of the reported obser-vations are well reflected in our experimental dataset. This is not guaranteed a priori as wedeal with an experiment with boundary layers on top, bottom and sidewalls, and inevitablynonideality (e.g. heat loss, centrifugal buoyancy, etc.), in comparison to simulations withstress-free conditions at the plates and no sidewall.

The dependence of the spatial autocorrelations on Ek is investigated. It is observed thatthe spatial vorticity autocorrelations are different for two data sets with equal RaEk4/3 butdifferent Ek. Our findings show that the width of the (dimensionless) spatial vorticity auto-correlation becomes smaller with increasing Ek. It is worth pointing out that the physical(dimensional) width of the autocorrelation actually increases with Ek. Finally, the autocor-relations of the vorticity (and a binary function based on the Q-criterion) along the axis ofrotation are examined. The results show an increase in the vertical coherence with decreasein RaEk4/3.

We show that we can indeed reach the geostrophic regime of rapidly rotating convectionusing conventional convection cells. Furthermore, these experimental data provide supportfor the asymptotic approach. We find a good agreement between experimental and simulationdata by examining different parameters and we show that it is possible to reproduce the basicsignatures of the asymptotic solutions in an experiment. This proves that we can definitelylearn more from the asymptotically reduced equations and further attention to this theory inthe rapidly rotating convection community is encouraged.

Currently, a rotating four-meter-tall convection cell combined with stereoscopic PIV sys-tem is being developed which enables further exploration of the geostrophic regime and willallow for more extensive comparisons to the asymptotic simulations.

Chapter 8

Concluding remarks

The main goal of the present thesis is to study the effects of background rotation on thermally-driven turbulence. A combined experimental-numerical approach is employed in a cylindricaldomain and a wide range of different Lagrangian and Eulerian parameters are examined. Inthis chapter, we first summarize our main findings. Next, possible future research directionsare discussed.

8.1 ConclusionsThe presented flow field measurement data in this thesis confirms that the background rotationaffects various parameters in thermally-driven turbulence: the flow phenomenology dependson rotation. Nonrotating and weakly rotating (regime I) Rayleigh–Bénard convection (RBC)is characterized by a domain-filling large-scale circulation. Moderate background rotation(regime II) replaces the LSC with vertically aligned vortical plumes in RBC. Under strongrotation (regime III), the vortical plumes span throughout the entire domain. These regimeshave been studied from different perspectives in this thesis. The most important results of thisthesis are summarized here by answering the questions that were already posed in Chapter 1.

• What are the driving mechanisms behind transitions from one regime to another re-gime? How are these transitions reflected into different Lagrangian statistics? Howcan we characterize each regime using Lagrangian statistics? (Chapters 3, 5 and 6)

The Lagrangian acceleration probability distribution function (PDF) and root-mean-square(rms) values from experiments, the viscous boundary layer (BL) thickness from direct nu-merical simulations and theoretical BL analysis show that the top and bottom boundary lay-ers are of Prandtl-Blasius type for nonrotating RBC and remain as of this type under weakrotation (throughout regime I). However, in regime II the linear Ekman BL theory accuratelydescribes the BLs: the BLs are of Ekman type in regime II. We have shown that the transitionoccurs when the Ekman BL thickness becomes thinner than the Prandtl-Blasius BL thick-ness. It can be concluded that the transition between regimes I and II is driven by the BLs: atransition from Prandtl-Blasius type in regime I to Ekman type in regime II.

While the transition between regimes I and II is sudden, the transition between regimes IIand III is comparatively gradual. The Lagrangian velocity and acceleration rms and autocor-

101

102 CHAPTER 8. CONCLUDING REMARKS

relations show that the transition to regime III coincides with three main observations. First,the vertical motions are strongly suppressed with decreasing Ro; the vertical acceleration andvelocity rms values decrease with decreasing Ro. Second, the vortical plumes penetrate fur-ther into the interior; the ratio arms

xy /armsz start to increase with decreasing Ro at z = 0.8H .

Finally, the exchange of fluid between the plume and non-plume regions diminishes; plumeand non-plume regions have distinct time scales and the ratio arms

xy /armsxy at z = 0.975H sud-

denly increases with decrease in Ro indicating strong horizontal centripetal accelerations ofparticles remaining inside plumes. The first and second observations support the descriptionof the transition to regime III by Julien et al. [67]: the transition occurs when the entire bulkbecomes rotationally dominated and the vertical motions are suppressed. The last observationis new and it opens up a new insight into the transition.

• What are the effects of background rotation on the large- and small-scale flow field atdifferent heights in a cylindrical convection cell? Do the small-scales remain isotropicor is Kolmogorov’s hypothesis of local isotropy violated? (Chapter 4)

Chapter 4 is devoted to the effects of background rotation on small- and large-scale isotropyat three different heights in the convection cell in regimes I and II. It has been shown, us-ing velocity fluctuations and the inertial range of the second-order structure function, that thelarge-scale anisotropy decreases at z = 0.8H , while it increases at the cell center and close tothe top plate (z = 0.975H) with decreasing Ro. Despite enhancement of the large-scale an-isotropy with decreasing Ro at the cell center, the small scales remain isotropic at this height.Small-scale isotropy is evaluated based on the dissipative range of the second-order structurefunction and the Heisenberg-Yaglom Lagrangian acceleration constant. At z = 0.8H , theflow is not far from small-scale isotropy for all considered Ro in regime I and II. However,small anisotropic behavior is observed. At z = 0.975H , the small-scale flow is strongly an-isotropic throughout regime I and II. To summarize, in spite of large-scale anisotropy inducedby background rotation the small-scale flow remains isotropic in the central region of rotat-ing RBC. However, near the plates (here measured at z = 0.975H) the large-scale anisotropypenetrates into the small-scale flow as well.

• What are the effects of background rotation on the geometrical aspects of fluid parceltrajectories? Can we recover the power-law scaling, reported earlier for homogeneousand isotropic turbulence (HIT), in our non-HIT thermally-driven turbulence throughoutthe cylindrical domain? If not, how does the power law scaling changes with rotationand position of the measurement volume in the cell? (Chapter 5)

The geometrical aspects of a particle trajectory are (partially) represented by its curvature.We have considered the curvature statistics at the cell center and close to the top plate (z =0.875H). At the cell center and near the top plate, despite the anisotropy and non-Gaussiandistribution of the velocity statistics, the curvature PDF show the same power-law scalingsas those for HIT [15, 189]. It can be concluded that the power laws are robust and they havesome levels of universality. We have also studied the position of the peak of the PDFs as afunction of Ek. It is observed that the curvature PDFs do not move in regime I, while theyshift toward higher curvature values with decreasing Ek in regime II. We have found that, ifwe normalize the curvature PDFs by the inverse of the typical length scale of flow structure,Ek1/3 (in regime II) [23], they collapse. Therefore, it can be concluded that there is a clearconnection between the typical length scale of the flow structures and curvature statistics.

8.2. OUTLOOK 103

• Is the geostrophic regime accessible with a conventional RBC set-up: a convectioncell with H = 0.2 m and water as working fluid? If so, can we reproduce the basicsignatures of the asymptotic solutions in an experiment? (Chapter 7)

In order to evaluate whether the geostrophic regime is accessible with our experimental set-up, the spatial vorticity autocorrelations from our experimental data are compared with thespatial temperature autocorrelations from the simulations of the asymptotically reduced equa-tions, taken from Ref. [117]. Most of the observations in Ref. [117] are reflected in our exper-imental data. The vertical vorticity analysis and comparisons show that it is indeed possibleto enter the geostrophic regime of rapidly rotating convection using conventional convectioncells: a table-top RBC set-up with a height of 0.2 m and water as working fluid. Further-more, the good agreement between experiments and simulations indicates that in spite of alldifferences between experiments and simulations of the asymptotic equations (i.e. differentboundary conditions and non-asymptotic conditions in the experiment), the basic signaturesof the asymptotic solutions can be reproduced in an experiment.

8.2 OutlookUltimate turbulence and low Pr regimes: Rotating Rayleigh–Bénard convection (RRBC)is a rich system with several unresolved questions. Our understanding of RBC for Ra ofO(106 − 1011) and Pr of O(1) is quite well-established. However, the regime of Ra >O(1013) and/or Pr of O(10−2) are not well-studied. It is of great importance, consideringthe fact that most of the astrophysical and geophysical flows have extreme parameter values:the Rayleigh number in the convective layer of the Sun is expected to be Ra ∼ 1023 and theEarth’s core is made of liquid metal with Pr of O(10−2). The regime of high Ra might bethe so-called ultimate turbulent state: the boundary layers become turbulent in this state [57].It has been recently reported that the ultimate turbulence state is present for Ra > 5 × 1014

in nonrotating RBC with Pr = 0.8 [57]. However, the presence of ultimate turbulence inrapidly rotating convection is still unknown. Approaching the ultimate regime in rapidlyrotating convection is by a great deal more difficult than the nonrotating case due to thestabilizing effects of background rotation in convective flows [23]. The ultimate regime inrapidly rotating convection (if it exists) is a completely untouched terrain; due to its relevancefor astrophysical and geophysical applications further attention to this regime is a sine quanon. Apart from reaching high levels of Ra, the role of very small Pr should be studied totruly model the astrophysical and geophysical flows as well. Current experiments at highestRa are performed at Pr of O(1), experiments at a Pr of O(10−2) can only achieve moderateRa: the effects of extreme changes in the parameter range are not known.Geostrophic regime: Another important unresolved issue in RRBC is the geostrophic re-gime. As discussed in this thesis, four different states with distinct flow morphologies areexpected within the geostrophic regime of rapidly rotating convective turbulence [151, 117,153]. Two of these states are recovered experimentally in this thesis [134]. However, higherlevels of turbulence (i.e. Ra & O(1011)) are required to be able to study all four differ-ent flow structures. Approaching high Ra can be achieved in two ways: either increase theheight of the cell (Ra ∝ H3) or use a working fluid with a large ratio between thermal expan-sion coefficient and thermal and viscous diffusivity (Ra ∝ β/νκ), e.g. cryogenic helium atlow temperature. The latter approach is limited to non-transparent (no optical accessibility)

104 CHAPTER 8. CONCLUDING REMARKS

convection cells for now. Thus, for optical flow measurements we are limited to the formerapproach. Once such a set-up is available, one can probe all four different states which wouldallow for the delineation of state boundaries between the various flow morphologies to con-nect to trends in heat-transfer scaling.Possible issues for rotating experiments: Apart from the aforementioned unresolved mainissues in RRBC, two other questions, stem from this thesis, can be investigated by DNS.Although all rapidly rotating convective turbulence experiments are affected by centrifugalbuoyancy, there is no concrete study on its effects on heat transfer and flow field. The effectsof the centrifugal buoyancy on the heat transfer efficiency and flow field can be considerablefor large Fr. Therefore, further understanding of these effects on RRBC is required for a cor-rect approval of experimental results. In addition, we also found that the Poincaré force dueto precession can be important in rapidly rotating experiments. The effects of the Poincaréforce on the heat transfer efficiency should be better understood too, since it affects all rapidlyrotating experimental measurements, in particular at aspect ratio Γ = 1.

Appendix A

Temporal vorticity autocorrelation

As discussed in Sec. 7.3.1, the spatial vorticity autocorrelation can be employed to studythe geostrophic regime. Therefore, it is expected that the temporal vorticity autocorrelationalso provides insight into the geostrophic regime. The temporal vorticity autocorrelation isdefined as

Rτω(τ) =〈ω(x, y, z, t)ω(x, y, z, t+ τ)〉

〈ω2(x, y, z, t)〉, (A.1)

where τ is the time lag. The temporal vorticity autocorrelation is plotted in Figure A.1(a) asa function of τ and RaEk4/3. The horizontal axis is nondimensionalized by the buoyancytime scale Tb, see Tables 7.1 and 7.2. The temporal vorticity autocorrelations show a fasterdecorrelation with increasing RaEk4/3. Similar behavior is observed in Ref. [117] for tem-perature autocorrelations. Figure A.1(b) shows the dimensionless values of the half-widthof the temporal vorticity autocorrelations (Thw) as a function of RaEk4/3. The data fromthe simulations show noticeably larger Thw (the simulation data are not shown here, see Ref.[117] for more details), approximately six times larger than those of the experiments; thereason behind this discrepancy is not known. However, simulations show similar trends forthe temporal autocorrelation as those of the experiments. Similar to the spatial autocorrela-tion, a change in the slope occurs in the vicinity of RaEk4/3 ' 70.

Nieves et al. use the temporal autocorrelation as an indication of how long it takes fora coherent structure to pass through a fixed point in space i.e. the temporal autocorrelationcan also be expressed as the velocity of a coherent structure (Ucoh) divided by its width.Therefore, the velocity of a coherent structure can be written as the half-width of the spatialvorticity autocorrelation (Lhw) divided by the half-width of the temporal vorticity autocorrel-ation [117]; Ucoh = Lhw/Thw. Figure A.2 shows Ucoh/urms where urms is the root-mean-

square of the horizontal velocity fluctuations, uh =√u2x + u2

y with ux and uy the velocitycomponents in the x and y directions, respectively. As can be seen from the graph, the ratioUcoh/urms becomes larger with increase in RaEk4/3. The simulations are plotted as blackstars in Figure A.2 and they show a decent agreement for RaEk4/3 . 100. Note that thenumerical data for RaEk4/3 = 140 from the simulations is for the geostrophic-turbulencestate (Pr = 1), so no quantitative agreement is to be expected.

To summarize, although the experimental data show a faster temporal decorrelation (faster

105

106 APPENDIX A. TEMPORAL VORTICITY AUTOCORRELATION

τ/Tb

RaEk4/3

a)

0 10 20 30

50

100

150

200

0

0.2

0.4

0.6

0.8

1

0 50 100 150 2000

2

4

6

8

10

12

RaEk4/3

τ/Tb

b)PIV−Temp.PIV−Rot.3D−PTV−Temp.3D−PTV−Rot.

Figure A.1: (a) Color plot of the temporal vorticity autocorrelation from the 3D–PTV data asa function of RaEk4/3 and (b) the half width of the temporal vorticity autocorrelation from3D–PTV and PIV divided by the buoyancy time scale Tb (see Table 7.1) as a function ofRaEk4/3. The dashed black line is located at RaEk4/3 = 70 and the solid black lines areguides to the eye through the experimental data points for RaEk4/3 . 70 and RaEk4/3 &70.

flow motions) than simulations, the ratio of Ucoh/urms shows a good match with the simu-lation data.

0 50 100 150 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

RaEk4/3

Ucoh/urm

s


Figure A.2: Ucoh/urms as a function of RaEk4/3. The dashed black line is located atRaEk4/3 = 70 and the solid black lines are guides to the eye through the experimentaldata points for RaEk4/3 . 70 and RaEk4/3 & 70.

Bibliography

[1] A. A. Adamczyk and L. Rimai. 2-Dimensional particle tracking velocimetry (PTV):technique and image processing algorithms. Exp. Fluids, 6:373–380, 1988.

[2] R. J. Adrian and C.-S. Yao. Pulsed laser technique application to liquid and gaseousflows and the scattering power of seed materials. Appl. Opt., 24:44–52, 1985.

[3] B. N. Agrawal. Dynamic characteristics of liquid motion in partially filled tanks of aspinning spacecraft. J. Guid. Control Dynam., 16:636–640, 1993.

[4] G. Ahlers, S. Grossmann, and D. Lohse. Heat transfer and large scale dynamics inturbulent Rayleigh-Bénard convection. Rev. Mod. Phys., 81:503, 2009.

[5] K. M. J. Alards, H. Rajaei, L. Del Castello, R. P. J. Kunnen, F. Toschi, and H. J. H.Clercx. Geometry of tracer trajectories in rotating turbulent flows. Phys. Rev. Fluids,2:044601, 2017.

[6] G. Amati, K. Koal, F. Massaioli, K. R. Sreenivasan, and R. Verzicco. Turbulent thermalconvection at high Rayleigh numbers for a Boussinesq fluid of constant Prandtl num-ber. Phys. Fluids, 17:121701, 2005.

[7] S. J. Baek and S. J. Lee. A new two-frame particle tracking algorithm using matchprobability. Exp. Fluids, 22:23–32, 1996.

[8] H. Bénard. Les tourbillons cellulaires dans une nappe liquide. Rev. Gen. Sci. PureAppl., 11:1309–1328, 1900.

[9] H. Bénard. Les tourbillons cellulaires dans une nappe liquide. Rev. Gen. Sci. PureAppl., 11:1261–1271, 1900.

[10] H. Bénard. Les tourbillons cellulaires dans une nappe liquide transportant de la chaleurpar convection en regime permanent. Annu. Chim. Phys., 23:62–144, 1901.

[11] L. Biferale and M. Vergassola. Isotropy vs anisotropy in small-scale turbulence. Phys.Fluids, 13:2139–2141, 2001.

[12] B. M. Boubnov and G. S. Golitsyn. Experimental study of convective structures inrotating fluids. J. Fluid Mech., 167:503–531, 1986.

[13] B. M. Boubnov and G. S. Golitsyn. Temperature and velocity field regimes of con-vective motions in a rotating plane fluid layer. J. Fluid Mech., 219:215–239, 1990.

107

108 BIBLIOGRAPHY

[14] J. Boussinesq. Théorie Analytique de la Chaleur, volume 2. Gauthier-Villars, 1903.

[15] W. Braun, F. De Lillo, and B. Eckhardt. Geometry of particle paths in turbulent flows.J. Turbul., 7:N62, 2006.

[16] E. Brown, A. Nikolaenko, D. Funfschilling, and G. Ahlers. Heat transport in turbu-lent Rayleigh-Bénard convection: Effect of finite top-and bottom-plate conductivities.Phys. Fluids, 17:075108, 2005.

[17] F. H. Busse. Convection driven zonal flows and vortices in the major planets. Chaos,4:123–134, 1994.

[18] F. H. Busse. Mean zonal flows generated by librations of a rotating spherical cavity. J.Fluid Mech., 650:505–512, 2010.

[19] F. H. Busse and C. R. Carrigan. Laboratory simulation of thermal convection in rotat-ing planets and stars. Science, 191:81–83, 1976.

[20] P. Cardin and P. Olson. Chaotic thermal convection in a rapidly rotating spherical shell:consequences for flow in the outer core. Phys. Earth Planet. In., 82:235–259, 1994.

[21] F. Cattaneo, T. Emonet, and N. Weiss. On the interaction between convection andmagnetic field. Astrophys. J., 588:1183–1198, 2003.

[22] S.-K. Chan. Investigation of turbulent convection under a rotational constraint. J. FluidMech., 64:477–506, 1974.

[23] S. Chandrasekhar. Hydrodynamic and Hydromagnetic Stability. Oxford UniversityPress, 1961.

[24] T. P. K. Chang, A. T. Watson, and G. B. Tatterson. Image processing of tracer particlemotions as applied to mixing and turbulent flow—I. The technique. Chem. Eng. Sci.,40:269–275, 1985.

[25] X. Chavanne, F. Chillà, B. Chabaud, B. Castaing, and B. Hébral. Turbulent Rayleigh–Bénard convection in gaseous and liquid He. Phys. Fluids, 13:1300–1320, 2001.

[26] J. S. Cheng, S. Stellmach, A. Ribeiro, A. Grannan, E. M. King, and J. M. Aurnou.Laboratory-numerical models of rapidly rotating convection in planetary cores. Geo-phys. J. Int., 201:1–17, 2015.

[27] Y. Choi, Y. Park, and C. Lee. Helicity and geometric nature of particle trajectories inhomogeneous isotropic turbulence. Int. J. Heat Fluid Flow, 31:482–487, 2010.

[28] P. Constantin, C. Hallstrom, and V. Putkaradze. Heat transport in rotating convection.Physica D, 125:275–284, 1999.

[29] T. Corpetti, D. Heitz, G. Arroyo, E. Memin, and A. Santa-Cruz. Fluid experimentalflow estimation based on an optical-flow scheme. Exp. Fluids, 40:80–97, 2006.

[30] L. Del Castello. Table-top rotating turbulence: an experimental insight throughParticle Tracking. PhD thesis, Eindhoven University of Technology, 2010.

BIBLIOGRAPHY 109

[31] L. Del Castello and H. J. H. Clercx. Lagrangian acceleration of passive tracers instatistically steady rotating turbulence. Phys. Rev. Lett., 107:214502, 2011.

[32] L. Del Castello and H. J. H. Clercx. Lagrangian velocity autocorrelations in statistic-ally steady rotating turbulence. Phys. Rev. E, 83:056316, 2011.

[33] P. E. Dimotakis, F. D. Debussy, and M. M. Koochesfahani. Particle streak velocity fieldmeasurements in a two-dimensional mixing layer. Phys. Fluids, 24:995–999, 1981.

[34] C. R. Doering and P. Constantin. On upper bounds for infinite Prandtl number convec-tion with or without rotation. J. Math. Phys., 42:784–795, 2001.

[35] M. G. Dunn. Convective heat transfer and aerodynamics in axial flow turbines. J.Turbomach., 123:637–686, 2001.

[36] R. E. Ecke and J. J. Niemela. Heat transport in the geostrophic regime of rotatingRayleigh–Bénard convection. Phys. Rev. Lett., 113:114301, 2014.

[37] M. S. Emran and J. Schumacher. Lagrangian tracer dynamics in a closed cylindricalturbulent convection cell. Phys. Rev. E, 82:016303, 2010.

[38] B. P. Epps. An impulse framework for hydrodynamic force analysis: fish propulsion,water of spheres, and marine propellers. PhD thesis, Massachusetts Institute of Tech-nology, 2010.

[39] R. F. Gans. On the precession of a resonant cylinder. J. Fluid Mech., 41:865–872,1970.

[40] J. Gascard, A. J. Watson, M. J. Messias, K. A. Olsson, T. Johannessen, and K. Simon-sen. Long-lived vortices as a mode of deep ventilation in the Greenland Sea. Nature,416:525–527, 2002.

[41] Y. Gasteuil, W. L. Shew, M. Gibert, F. Chilla, B. Castaing, and J.-F. Pinton. Lagrangiantemperature, velocity, and local heat flux measurement in Rayleigh-Bénard convection.Phys. Rev. Lett., 99:234302, 2007.

[42] P. Gervais, C. Baudet, and Y. Gagne. Acoustic Lagrangian velocity measurement in aturbulent air jet. Exp. Fluids, 42:371–384, 2007.

[43] M. Gibert, H. Xu, and E. Bodenschatz. Inertial effects on two-particle relative disper-sion in turbulent flows. Europhys. Lett., 90:64005, 2010.

[44] G. A. Glatzmaier, R. S. Coe, L. Hongre, and P. H. Roberts. The role of the Earth’smantle in controlling the frequency of geomagnetic reversals. Nature, 401:885–890,1999.

[45] T. Gotoh and D. Fukayama. Pressure spectrum in homogeneous turbulence. Phys. Rev.Lett., 86:3775, 2001.

[46] H. P. Greenspan. The Theory of Rotating Fluids. Cambridge University Press, 1968.

110 BIBLIOGRAPHY

[47] I. Grooms, K. Julien, J. B. Weiss, and E. Knobloch. Model of convective Taylorcolumns in rotating Rayleigh-Bénard convection. Phys. Rev. Lett., 104:224501, 2010.

[48] S. Grossmann and D. Lohse. Scaling in thermal convection: a unifying theory. J. FluidMech., 407:27–56, 2000.

[49] S. Grossmann and D. Lohse. Thermal convection for large Prandtl numbers. Phys.Rev. Lett., 86:3316, 2001.

[50] S. Grossmann and D. Lohse. Prandtl and Rayleigh number dependence of the Reyn-olds number in turbulent thermal convection. Phys. Rev. E, 66:016305, 2002.

[51] S. Grossmann and D. Lohse. Fluctuations in turbulent Rayleigh–Bénard convection:the role of plumes. Phys. Fluids, 16:4462–4472, 2004.

[52] G. Grötzbach. Spatial resolution requirements for direct numerical simulation of theRayleigh-Bénard convection. J. Comput. Phys., 49:241–264, 1983.

[53] G. Hadley. Concerning the cause of the general trade-winds. Phil. Trans., 39:58–62,1735.

[54] J. E. Hart and S. Kittelman. Instabilities of the sidewall boundary layer in a differen-tially driven rotating cylinder. Phys. Fluids, 8:692–696, 1996.

[55] J. E. Hart, S. Kittelman, and D. R. Ohlsen. Mean flow precession and temperatureprobability density functions in turbulent rotating convection. Phys. Fluids, 14:955–962, 2002.

[56] Y. A. Hassan and R. E. Canaan. Full-field bubbly flow velocity measurements using amultiframe particle tracking technique. Exp. Fluids, 12:49–60, 1991.

[57] X. He, D. Funfschilling, H. Nobach, E. Bodenschatz, and G. Ahlers. Transition to theultimate state of turbulent Rayleigh-Bénard convection. Phys. Rev. Lett., 108:024502,2012.

[58] M. Heimpel and J. Aurnou. Turbulent convection in rapidly rotating spherical shells: Amodel for equatorial and high latitude jets on Jupiter and Saturn. Icarus, 187:540–557,2007.

[59] W. Heisenberg. Zur statistischen Theorie der Turbulenz. Zeitschrift für Phys.,124:628–657, 1948.

[60] L. Hesselink. Digital image processing in flow visualization. Annu. Rev. Fluid Mech.,20:421–486, 1988.

[61] S. Horn and O. Shishkina. Toroidal and poloidal energy in rotating Rayleigh–Bénardconvection. J. Fluid Mech., 762:232–255, 2015.

[62] J. C. R. Hunt, A. A. Wray, and P. Moin. Eddies, streams, and convergence zones inturbulent flows. Report CTR-S88, Center for Turbulence Research, 1988.

BIBLIOGRAPHY 111

[63] A. P. Ingersoll. Atmospheric dynamics of the outer planets. Science, 248:308–316,1990.

[64] J. P. Johnston. Effects of system rotation on turbulence structure: a review relevant toturbomachinery flows. Int. J. Rot. Mach., 4:97–112, 1998.

[65] C. A. Jones. Convection-driven geodynamo models. Phil. Trans. R. Soc. Lond,A358:873–897, 2000.

[66] K. Julien, J. M. Aurnou, M. A. Calkins, E. Knobloch, P. Marti, S. Stellmach, andG. M. Vasil. A nonlinear model for rotationally constrained convection with Ekmanpumping. J. Fluid Mech., 798:50–87, 2016.

[67] K. Julien, E. Knobloch, A. M. Rubio, and G. M. Vasil. Heat transport in low-Rossby-number Rayleigh-Bénard convection. Phys. Rev. Lett., 109:254503, 2012.

[68] K. Julien, E. Knobloch, and J. Werne. A new class of equations for rotationally con-strained flows. Theor. Comp. Fluid Dyn., 11:251–261, 1998.

[69] K. Julien, S. Legg, J. McWilliams, and J. Werne. Rapidly rotating turbulent Rayleigh–Bénard convection. J. Fluid Mech., 322:243–273, 1996.

[70] K. Julien, A. M. Rubio, I. Grooms, and E. Knobloch. Statistical and physical balancesin low Rossby number Rayleigh–Bénard convection. Geophys. Astrophys. Fluid Dyn.,106:392–428, 2012.

[71] M. Kaczorowski, K.-L. Chong, and K.-Q. Xia. Turbulent flow in the bulk of Rayleigh–Bénard convection: aspect-ratio dependence of the small-scale properties. J. FluidMech., 747:73–102, 2014.

[72] E. M. King, S. Stellmach, and J. M. Aurnou. Heat transfer by rapidly rotatingRayleigh–Bénard convection. J. Fluid Mech., 691:568–582, 2012.

[73] E. M. King, S. Stellmach, J. Noir, U. Hansen, and J. M. Aurnou. Boundary layercontrol of rotating convection systems. Nature, 457:301–304, 2009.

[74] J. J. Kobine. Inertial wave dynamics in a rotating and precessing cylinder. J. FluidMech., 303:233–252, 1995.

[75] P. K. Kundu and I. M. Cohen. Fluid Mechanics. Academic Press, second edition,2002.

[76] R. P. J. Kunnen. Turbulent rotating convection. PhD thesis, Eindhoven University ofTechnology, 2008.

[77] R. P. J. Kunnen, H. J. H. Clercx, and B. J. Geurts. Heat flux intensification by vorticalflow localization in rotating convection. Phys. Rev. E, 74:056306, 2006.

[78] R. P. J. Kunnen, H. J. H. Clercx, and B. J. Geurts. Breakdown of large-scale circulationin turbulent rotating convection. Europhys. Lett., 84:24001, 2008.

112 BIBLIOGRAPHY

[79] R. P. J. Kunnen, H. J. H. Clercx, and B. J. Geurts. Enhanced vertical inhomogeneityin turbulent rotating convection. Phys. Rev. Lett., 101:174501, 2008.

[80] R. P. J. Kunnen, H. J. H. Clercx, and G. J. F. van Heijst. The structure of sidewallboundary layers in confined rotating Rayleigh–Bénard convection. J. Fluid Mech.,727:509–532, 2013.

[81] R. P. J. Kunnen, Y. Corre, and H. J. H. Clercx. Vortex plume distribution in confinedturbulent rotating convection. Europhys. Lett., 104:54002, 2014.

[82] R. P. J. Kunnen, B. J. Geurts, and H. J. H. Clercx. Experimental and numerical invest-igation of turbulent convection in a rotating cylinder. J. Fluid Mech., 642:445–476,2010.

[83] R. P. J. Kunnen, R. Ostilla-Mónico, E. P. van der Poel, R. Verzicco, and D. Lohse.Transition to geostrophic convection: the role of the boundary conditions. J. FluidMech., 799:413–432, 2016.

[84] R. P. J. Kunnen, R. J. A. M. Stevens, J. Overkamp, C. Sun, G. J. F. van Heijst,and H. J. H. Clercx. The role of Stewartson and Ekman layers in turbulent rotatingRayleigh–Bénard convection. J. Fluid Mech., 688:422–442, 2011.

[85] A. La Porta, G. A. Voth, A. M. Crawford, J. Alexander, and E. Bodenschatz. Fluidparticle accelerations in fully developed turbulence. Nature, 409:1017–1019, 2001.

[86] M. Le Bars, D. Cébron, and P. Le Gal. Flows driven by libration, precession, and tides.Annu. Rev. Fluid Mech., 47:163–193, 2015.

[87] X. Liao and K. Zhang. On flow in weakly precessing cylinders: the general asymptoticsolution. J. Fluid Mech., 709:610–621, 2012.

[88] D. R. Lide (Editor). CRC Handbook of Chemistry and Physics. CRC Press/Taylor andFrancis, eighty-eighth edition, 2007-2008.

[89] O. Liot, A. Gay, J. Salort, M. Bourgoin, and F. Chillà. Inhomogeneity and Lagrangianunsteadiness in turbulent thermal convection. Phys. Rev. Fluids, 1:064406, 2016.

[90] O. Liot, F. Seychelles, F. Zonta, S. Chibbaro, T. Coudarchet, Y. Gasteuil, J. F. Pinton,J. Salort, and F. Chillà. Simultaneous temperature and velocity Lagrangian measure-ments in turbulent thermal convection. J. Fluid Mech., 794:655–675, 2016.

[91] Y. Liu and R. E. Ecke. Heat transport scaling in turbulent Rayleigh–Bénard convection:effects of rotation and Prandtl number. Phys. Rev. Lett., 79:2257, 1997.

[92] Y. Liu and R. E. Ecke. Heat transport measurements in turbulent rotating Rayleigh–Bénard convection. Phys. Rev. E, 80:036314, 2009.

[93] D. Lohse and K.-Q. Xia. Small-scale properties of turbulent Rayleigh–Bénard convec-tion. Annu. Rev. Fluid Mech., 42:335–364, 2010.

[94] J. M. Lopez and F. Marques. Sidewall boundary layer instabilities in a rapidly rotatingcylinder driven by a differentially corotating lid. Phys. Fluids, 22:114109, 2010.

BIBLIOGRAPHY 113

[95] J. M. Lopez and F. Marques. Rapidly rotating cylinder flow with an oscillating side-wall. Phys. Rev. E, 89:013013, 2014.

[96] B. Lüthi. Some aspects of strain, vorticity and material element dynamics as measuredwith 3D Particle Tracking Velocimetry in a turbulent flow. PhD thesis, Swiss FederalInstitute of Technology, Zürich, 2002.

[97] B. Lüthi, S. Ott, J. Berg, and J. Mann. Lagrangian multi-particle statistics. J. Turbul.,8:45, 2007.

[98] B. Lüthi, A. Tsinober, and W. Kinzelbach. Lagrangian measurement of vorticity dy-namics in turbulent flow. J. Fluid Mech., 528:87–118, 2005.

[99] H.-G. Maas. Digitale Photogrammetrie in der dreidimensionalen Strömungsmesstech-nik. PhD thesis, Swiss Federal Institute of Technology, Zürich, 1992.

[100] H. G. Maas, A. Gruen, and D. Papantoniou. Particle tracking velocimetry in three-dimensional flows- Part I: Photogrammetric determination of particle coordinates. Exp.Fluids, 15:133–146, 1993.

[101] N. A. Malik, T. Dracos, and D. A. Papantoniou. Particle tracking velocimetry in three-dimensional flows, Part II: Particle tracking. Exp. Fluids, 15:279–294, 1993.

[102] R. Manasseh. Breakdown regimes of inertia waves in a precessing cylinder. J. FluidMech., 243:261–296, 1992.

[103] J. Marshall and F. Schott. Open-ocean convection : Observation, theory, and models.Rev. Geophys., 37:1–64, 1999.

[104] P. Meunier, C. Eloy, R. Lagrange, and F. Nadal. A rotating fluid cylinder subject toweak precession. J. Fluid Mech., 599:405–440, 2008.

[105] M. S. Miesch. The coupling of solar convection and rotation. Solar Phys., 192:59–89,2000.

[106] N. R. Moharana. Three-dimensional Lagrangian transport near the surface of a mov-ing sphere. PhD thesis, Eindhoven University of Technology, 2016.

[107] A. S. Monin and A. M. Yaglom. Statistical Fluid Mechanics: Mechanics of Turbu-lence, volume 2. MIT Press, 1975.

[108] N. Mordant, A. M. Crawford, and E. Bodenschatz. Experimental Lagrangian acceler-ation probability density function measurement. Physica D, 193:245–251, 2004.

[109] N. Mordant, J. Delour, E. Lévêque, A. Arnéodo, and J. F. Pinton. Long time correla-tions in Lagrangian dynamics: a key to intermittency in turbulence. Phys. Rev. Lett.,89:254502, 2002.

[110] N. Mordant, E. Lévêque, and J.-F. Pinton. Experimental and numerical study of theLagrangian dynamics of high Reynolds turbulence. New J. Phys., 6:116, 2004.

114 BIBLIOGRAPHY

[111] N. Mordant, P. Metz, O. Michel, and J.-F. Pinton. Measurement of Lagrangian velocityin fully developed turbulence. Phys. Rev. Lett., 87:214501, 2001.

[112] N. Mordant, P. Metz, J.-F. Pinton, and O. Michel. Acoustical technique for Lagrangianvelocity measurement. Rev. Sci. Instrum., 76:025105, 2005.

[113] R. Ni, S.-D. Huang, and K.-Q. Xia. Local energy dissipation rate balances local heatflux in the center of turbulent thermal convection. Phys. Rev. Lett., 107:174503, 2011.

[114] R. Ni, S.-D. Huang, and K.-Q. Xia. Lagrangian acceleration measurements in con-vective thermal turbulence. J. Fluid Mech., 692:395–419, 2012.

[115] J. J. Niemela, S. Babuin, and K. R. Sreenivasan. Turbulent rotating convection at highRayleigh and Taylor numbers. J. Fluid Mech., 649:509–522, 2010.

[116] J. J. Niemela, L. Skrbek, K. R. Sreenivasan, and R. J. Donnelly. Turbulent convectionat very high Rayleigh numbers. Nature, 404:837–840, 2000.

[117] D. Nieves, A. M. Rubio, and K. Julien. Statistical classification of flow morphology inrapidly rotating Rayleigh-Bénard convection. Phys. Fluids, 26:086602, 2014.

[118] J. Noir, M. A. Calkins, M. Lasbleis, J. Cantwell, and J. M. Aurnou. Experimentalstudy of libration-driven zonal flows in a straight cylinder. Phys. Earth Planet. Inter.,182:98–106, 2010.

[119] A. Oberbeck. Über die Wärmeleitung der Flüssigkeiten bei Berücksichtigung derStrömungen infolge von Temperaturdifferenzen. Annalen der Physik, 243:271–292,1879.

[120] K. Okamoto, Y. A. Hassan, and W. D. Schmidl. New tracking algorithm for particleimage velocimetry. Exp. Fluids, 19:342–347, 1995.

[121] N. T. Ouellette, H. Xu, and E. Bodenschatz. Bulk turbulence in dilute polymer solu-tions. J. Fluid Mech., 629:375–385, 2009.

[122] N. T. Ouellette, H. Xu, M. Bourgoin, and E. Bodenschatz. Small-scale anisotropy inLagrangian turbulence. New J. Phys., 8:102, 2006.

[123] J. Pedlosky. Geophysical Fluid Dynamics. Springer, second edition, 1979.

[124] M. Plumley, K. Julien, P. Marti, and S. Stellmach. The effects of Ekman pumping onquasi-geostrophic Rayleigh–Bénard convection. J. Fluid Mech., 803:51–71, 2016.

[125] R. M. Pope and E. S. Fry. Absorption spectrum (380–700 nm) of pure water. II.Integrating cavity measurements. Appl. Opt., 36:8710–8723, 1997.

[126] S. B. Pope. Turbulent Flows. Cambridge University Press, 2001.

[127] J. W. Portegies, R. P. J. Kunnen, G. J. F. van Heijst, and J. Molenaar. A model forvortical plumes in rotating convection. Phys. Fluids, 20:066602, 2008.

BIBLIOGRAPHY 115

[128] L. Prandtl. Meteorologische Anwendung der Strömungslehre. Beitr. Phys. Atmos.,19:188–202, 1932.

[129] J. Proudman. On the motion of solids in a liquid possessing vorticity. Proc. R. Soc.Lond. Series A, 92:408–424, 1916.

[130] N. M. Qureshi, M. Bourgoin, C. Baudet, A. Cartellier, and Y. Gagne. Turbulent trans-port of material particles: an experimental study of finite size effects. Phys. Rev. Lett.,99:184502, 2007.

[131] M. Raffel, C. E. Willert, S. Wereley, and J. Kompenhans. Particle Image Velocimetry:a Practical Guide. Springer, 2013.

[132] H. Rajaei, P. Joshi, K. M. J. Alards, R. P. J. Kunnen, F. Toschi, and H. J. H. Clercx.Transitions in turbulent rotating convection: A Lagrangian perspective. Phys. Rev. E,93:043129, 2016.

[133] H. Rajaei, P. Joshi, R. P. J. Kunnen, and H. J. H. Clercx. Flow anisotropy in rotatingbuoyancy-driven turbulence. Phys. Rev. Fluids, 1:044403, 2016.

[134] H. Rajaei, R. P. J. Kunnen, and H. J. H. Clercx. Exploring the geostrophic regime ofrapidly rotating convection with experiments. Phys. Fluids, 29:045105, 2017.

[135] Lord Rayleigh. On convection currents in a horizontal layer of fluid, when the highertemperature is on the under side. Phil. Mag., 32:529–546, 1916.

[136] H. T. Rossby. A study of Bénard convection with and without rotation. J. Fluid Mech.,36:309–335, 1969.

[137] A. M. Rubio, K. Julien, E. Knobloch, and J. B. Weiss. Upscale energy transfer inthree-dimensional rapidly rotating turbulent convection. Phys. Rev. Lett., 112:144501,2014.

[138] S. Sakai. The horizontal scale of rotating convection in the geostrophic regime. J.Fluid Mech., 333:85–95, 1997.

[139] G. R. Sarson. Reversal models from dynamo calculations. Phil. Trans. R. Soc. Lond.,358:921–942, 2000.

[140] B. L. Sawford. Reynolds number effects in Lagrangian stochastic models of turbulentdispersion. Phys. Fluids A, 3:1577–1586, 1991.

[141] A. Scagliarini. Geometric properties of particle trajectories in turbulent flows. J.Turbul., 12:N25, 2011.

[142] S. Schmitz and A. Tilgner. Heat transport in rotating convection without Ekman layers.Phys. Rev. E, 80:015305, 2009.

[143] S. Schmitz and A. Tilgner. Transitions in turbulent rotating Rayleigh-Bénard convec-tion. Geophys. Astrophys. Fluid Dyn., 104:481–489, 2010.

116 BIBLIOGRAPHY

[144] J. Schumacher. Lagrangian studies in convective turbulence. Phys. Rev. E, 79:056301,2009.

[145] X. Shen and Z. Warhaft. The anisotropy of the small scale structure in high Reynoldsnumber (Rλ ∼= 1000) turbulent shear flow. Phys. Fluids, 12:2976–2989, 2000.

[146] W. L. Shew, Y. Gasteuil, M. Gibert, P. Metz, and J.-F. Pinton. Instrumented tracerfor Lagrangian measurements in Rayleigh-Bénard convection. Rev. Sci. Instrum.,78:065105, 2007.

[147] O. Shishkina, R. J. A. M. Stevens, S. Grossmann, and D. Lohse. Boundary layer struc-ture in turbulent thermal convection and its consequences for the required numericalresolution. New J. Phys., 12:075022, 2010.

[148] B. I. Shraiman and E. D. Siggia. Heat transport in high-Rayleigh-number convection.Phys. Rev. A, 42:3650, 1990.

[149] E. D. Siggia. High Rayleigh number convection. Annu. Rev. Fluid Mech., 26:137–168,1994.

[150] H. Song and P. Tong. Scaling laws in turbulent Rayleigh–Bénard convection underdifferent geometry. Europhys. Lett., 90:44001, 2010.

[151] M. Sprague, K. Julien, E. Knobloch, and J. Werne. Numerical simulation of anasymptotically reduced system for rotationally constrained convection. J. Fluid Mech.,551:141–174, 2006.

[152] K. R. Sreenivasan. On the universality of the Kolmogorov constant. Phys. Fluids,7:2778–2784, 1995.

[153] S. Stellmach, M. Lischper, K. Julien, G. Vasil, J. S. Cheng, A. Ribeiro, E. M. King,and J. M. Aurnou. Approaching the asymptotic regime of rapidly rotating convection:Boundary layers versus interior dynamics. Phys. Rev. Lett., 113:254501, 2014.

[154] R. J. A. M. Stevens, H. J. H. Clercx, and D. Lohse. Optimal Prandtl number for heattransfer in rotating Rayleigh–Bénard convection. New J. Phys., 12:075005, 2010.

[155] R. J. A. M. Stevens, H. J. H. Clercx, and D. Lohse. Heat transport and flow structurein rotating Rayleigh–Bénard convection. Eur. J. Mech. B/Fluids, 40:41–49, 2013.

[156] R. J. A. M. Stevens, J. Overkamp, D. Lohse, and H. J. H. Clercx. Effect of aspect ratioon vortex distribution and heat transfer in rotating Rayleigh-Bénard convection. Phys.Rev. E, 84:056313, 2011.

[157] R. J. A. M. Stevens, E. P. van der Poel, S. Grossmann, and D. Lohse. The unify-ing theory of scaling in thermal convection: the updated prefactors. J. Fluid Mech.,730:295–308, 2013.

[158] R. J. A. M. Stevens, R. Verzicco, and D. Lohse. Radial boundary layer structure andNusselt number in Rayleigh–Bénard convection. J. Fluid Mech., 643:495–507, 2010.

BIBLIOGRAPHY 117

[159] R. J. A. M. Stevens, J.-Q. Zhong, H. J. H. Clercx, G. Ahlers, and D. Lohse. Transitionsbetween turbulent states in rotating Rayleigh–Bénard convection. Phys. Rev. Lett.,103:024503, 2009.

[160] K. Stewartson. On the stability of a spinning top containing liquid. J. Fluid Mech.,5:577–592, 1959.

[161] C. Sun, Q. Zhou, and K.-Q. Xia. Cascades of velocity and temperature fluctuations inbuoyancy-driven thermal turbulence. Phys. Rev. Lett., 97:144504, 2006.

[162] K. Takehara and T. Etoh. A study on particle identification in PTV particle maskcorrelation method. J. Visual., 1:313–323, 1998.

[163] G. I. Taylor. Diffusion by continuous movements. Proc. London Math. Soc., A20:196–211, 1921.

[164] G. I. Taylor. Experiments on the motion of solid bodies in rotating fluids. 104:213–218, 1923.

[165] H. Tennekes and J. L. Lumley. A First Course in Turbulence. MIT press, 1972.

[166] D. Tordella and M. Iovieno. Small-scale anisotropy in turbulent shearless mixing.Phys. Rev. Lett., 107:194501, 2011.

[167] L. J. A. van Bokhoven. Experiments on rapidly rotating turbulent flows. PhD thesis,Eindhoven University of Technology, 2007.

[168] M. van Dyke. An Album of Fluid Motion. Parabolic Press, 1982.

[169] H. van Santen, C. R. Kleijn, and H. E. A. van Den Akker. On turbulent flows incold-wall CVD reactors. J. Cryst. Growth., 212:299–310, 2000.

[170] R. van Wissen, M. Golombok, and J. J. H. Brouwers. Gas centrifugation with wallcondensation. AIChE J., 52:1271–1274, 2006.

[171] J. P. Vanyo. Rotating Fluids in Engineering and Science. Elsevier, 2015.

[172] P. Vedula and P. K. Yeung. Similarity scaling of acceleration and pressure statistics innumerical simulations of isotropic turbulence. Phys. Fluids, 11:1208–1220, 1999.

[173] R. Verzicco and R. Camussi. Numerical experiments on strongly turbulent thermalconvection in a slender cylindrical cell. J. Fluid Mech., 477:19–49, 2003.

[174] R. Verzicco and P. Orlandi. A finite-difference scheme for three-dimensional incom-pressible flows in cylindrical coordinates. J. Comput. Phys., 123:402–414, 1996.

[175] R. Verzicco and K. R. Sreenivasan. A comparison of turbulent thermal convectionbetween conditions of constant temperature and constant heat flux. J. Fluid Mech.,595:203, 2008.

[176] N. K. Vitanov. Convective heat transport in a rotating fluid layer of infinite Prandtlnumber: Optimum fields and upper bounds on Nusselt number. Phys. Rev. E,67:026322, 2003.

118 BIBLIOGRAPHY

[177] P. Vorobieff and R. E. Ecke. Turbulent rotating convection: an experimental study. J.Fluid Mech., 458:191–218, 2002.

[178] G. A. Voth, A. La Porta, A. M. Crawford, J. Alexander, and E. Bodenschatz. Measure-ment of particle accelerations in fully developed turbulence. J. Fluid Mech., 469:121–160, 2002.

[179] G. A. Voth, A. La Porta, A. M. Crawford, E. Bodenschatz, C. Ward, and J. Alexander.A silicon strip detector system for high resolution particle tracking in turbulence. Rev.Sci. Instrum., 72:4348–4353, 2001.

[180] P. Wadhams, J. Holfort, E. Hansen, and J. P. Wilkinson. A deep convective chimneyin the winter Greenland Sea. Geophys. Res. Lett., 29:1434, 2002.

[181] Z. Warhaft and X. Shen. On the higher order mixed structure functions in laboratoryshear flow. Phys. Fluids, 14:2432–2438, 2002.

[182] S. Weiss and G. Ahlers. Heat transport by turbulent rotating Rayleigh–Bénard convec-tion and its dependence on the aspect ratio. J. Fluid Mech., 684:407–426, 2011.

[183] S. Weiss and G. Ahlers. The large-scale flow structure in turbulent rotating Rayleigh–Bénard convection. J. Fluid Mech., 688:461–492, 2011.

[184] S. Weiss, R. J. A. M. Stevens, J.-Q. Zhong, H. J. H. Clercx, D. Lohse, and G. Ahlers.Finite-size effects lead to supercritical bifurcations in turbulent rotating Rayleigh–Bénard convection. Phys. Rev. Lett., 105:224501, 2010.

[185] J. Westerweel and F. Scarano. Universal outlier detection for PIV data. Exp. Fluids,39:1096–1100, 2005.

[186] J. Willneff. 3D particle tracking velocimetry based on image and object space inform-ation. Int. Arch. Photogramm. Rem. Sens. Spatial Inform. Sci., 34:601–606, 2002.

[187] J. Willneff. A spatio-temporal matching algorithm for 3D particle tracking veloci-metry. PhD thesis, Swiss Federal Institute of Technology, Zürich, 2003.

[188] C.-C. Wu and P. H. Roberts. On a dynamo driven by topographic precession. Geophys.Astrophys. Fluid Dyn., 103:467–501, 2009.

[189] H. Xu, N. T. Ouellette, and E. Bodenschatz. Curvature of Lagrangian trajectories inturbulence. Phys. Rev. Lett., 98:050201, 2007.

[190] A. M. Yaglom. On the acceleration field in a turbulent flow. C. R. Akad. URSS,67:795–798, 1949.

[191] M. A. Yavuz. Experimental investigation of spatial clustering and velocities of inertialdroplets in turbulence. PhD thesis, Eindhoven University of Technology, 2016.

[192] P. K. Yeung and S. B. Pope. Lagrangian statistics from direct numerical simulationsof isotropic turbulence. J. Fluid Mech., 207:531–586, 1989.

BIBLIOGRAPHY 119

[193] F. Zhong, R. E. Ecke, and V. Steinberg. Rotating Rayleigh–Bénard convection: asym-metric modes and vortex states. J. Fluid Mech., 249:135–159, 1993.

[194] J.-Q. Zhong and G. Ahlers. Heat transport and the large-scale circulation in rotatingturbulent Rayleigh–Bénard convection. J. Fluid Mech., 665:300–333, 2010.

[195] J.-Q. Zhong, R. J. A. M. Stevens, H. J. H. Clercx, R. Verzicco, D. Lohse, and G. Ahlers.Prandtl-, Rayleigh-, and Rossby-number dependence of heat transport in turbulent ro-tating Rayleigh–Bénard convection. Phys. Rev. Lett., 102:044502, 2009.

[196] Q. Zhou, C. Sun, and K.-Q. Xia. Experimental investigation of homogeneity, isotropy,and circulation of the velocity field in buoyancy-driven turbulence. J. Fluid Mech.,598:361–372, 2008.

120 BIBLIOGRAPHY

Summary

Rotating Rayleigh–Bénard Convection

Rotating Rayleigh-Bénard convection (RRBC), a layer of fluid heated from below and cooledfrom above while rotating about a vertical axis perpendicular to the bottom and top plates, isof practical relevance to various flows in nature and industry. Key examples include oceanicand atmospheric currents, the convective outer layer of the Sun, the interior of giant gasplanets as well as convective cooling in turbomachinery and chemical vapor deposition onrotating substrates.

It is well-known that the application of background rotation leads to a significant flow de-parture from its nonrotating state. Three different regimes can be considered; (I) the rotation-unaffected regime (the flow is characterized by a so-called large scale circulation (LSC) andthe heat flux remains constant), (II) the rotation-affected regime (rotation-aligned vorticalplumes are the main features of the flow and the heat flux increases with increasing back-ground rotation), and (III) the rotation-dominated regime (or geostrophic regime, the flow isturbulent but at the same time dominated by the Coriolis force; the heat flux drops dramatic-ally with increasing background rotation). We refer to these regimes as regimes I, II and III,respectively. In this study, we realize laboratory and numerical set-ups to perform a funda-mental investigation on these regimes and provide insight into the driving forces behind thetransitions between these regimes. Two different approaches are used for flow measurements,namely three-dimensional particle tracking velocimetry (3D-PTV) and time-resolved particleimage velocimetry (PIV). The experimental data are complemented by direct numerical sim-ulations (DNS) when it is appropriate and possible.

The first part of this thesis concerns the rotation-unaffected and rotation-affected regimesand the transition between them. Using the Lagrangian acceleration of tracer particles fromexperiments, simulations of the boundary layer thickness, and theoretical analysis of theboundary layer using Grossmann-Lohse theory, we have shown that the transition betweenregimes I and II is a boundary layer transition between Prandtl-Blasius type (typical of nonro-tating convection) and Ekman type (the boundary layer of many rotating flows). The analysishas been carried out for three different cell aspect ratios, providing a deeper understanding ofthe dependence of the transition on the aspect ratio.

Furthermore, we study the effects of background rotation on large- and small-scale iso-tropy in regimes I and II. The large-scale isotropy for different rotation rates is evaluatedthrough the Lagrangian velocity fluctuation and second-order Eulerian structure function.Concurrently, the small-scale isotropy is examined through the Lagrangian acceleration oftracer particles and the second-order Eulerian structure function. It is found that background

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rotation enhances the large-scale anisotropy at the cell center and close to the top plate, whilereducing it at intermediate height. The large-scale anisotropy, induced by rotation, has negli-gible effect on the small scales at the cell center, whereas the small scales remain anisotropicclose to the top plate.

We also investigate the effects of background rotation on the geometry of tracer traject-ories in regimes I and II. Curvature statistics are used as the representative of the geomet-rical aspects of the particles trajectories. We compare our curvature probability distributionfunctions (PDFs) and the scaling laws from curvature PDFs in regimes I and II with the pre-viously reported curvature PDFs for homogeneous and isotropic turbulence (HIT). Despitethe non-HIT conditions (anisotropy and non-Gaussian distribution of the velocity statistics),we always recover the HIT scaling in the bulk and near the top plate (outside the boundarylayer) of RRBC, indicating that these scaling laws are robust and they have some level ofuniversality. Furthermore, we have found that the length scale of the typical flow structures(LSC in regime I and vortical plumes in regime II) is connected to the curvature statistics.

The second part of this thesis focuses on the rotation-dominated regime (regime III) andthe transition to this regime. There are two main hypotheses proposed for the driving mech-anisms of the transition to regime III: the transition occurs (i) when the relative thicknesses ofthe viscous and thermal boundary layers become comparable or (ii) when the vortical plumespenetrate into the bulk and span throughout the entire domain. These hypotheses are usuallyexamined through different parameters such as viscous and thermal boundary layers thick-nesses and heat transfer efficiency. We have studied all three regimes, in particular regime IIIand the aforementioned hypotheses, from a new perspective: Lagrangian velocity and accel-eration fluctuations and autocorrelations of tracer particles from experiments. We have foundthat the transition to regime III coincides with three main phenomena: strongly suppressedvertical motions, strong penetration of vortical plumes into the bulk and reduced interaction(exchange of fluid) of vortical plumes with their surroundings. The Lagrangian velocity andacceleration fluctuations and autocorrelations of tracers allow us primarily to evaluate thetransition to regime III and the transition hypotheses and secondarily learn more about allthree regimes, in particular regime III.

In the last part of this thesis, we pushed the limitations of our experimental set-up interms of rotation rate to enter the geostrophic regime (regime III) of rotating convection andapproach the limit of this regime; the onset of convection as described by Chandrasekhar.Experiments deep in the geostrophic regime require dedicated set-ups: either large set-upsor set-ups working with fluids with low Prandtl number. We use a table-top convection cellwith height of 0.2 m filled with water as working fluid to evaluate how far we can enter thegeostrophic regime of rapidly rotating convention. A comparison between our experimentalspatial vorticity autocorrelations with the statistics from simulations of geostrophic convec-tion reported earlier (with different boundary conditions) reveals that we can indeed accessthe geostrophic convection regime and can observe the signatures of the typical flow featuresreported in the simulations with a conventional set-up.

The results and the analysis presented in this thesis contributes to the understanding ofvarious flow regimes and the transitions between them in rotating turbulent convection. Inaddition to the description of the transitions and regimes in RRBC, the results contribute tothe extension of existing theories and arguments for HIT towards realistic non-HIT flowswhich are omnipresent in nature and technological applications.

Samenvatting

Roterende Rayleigh–Bénard convectieRoterende Rayleigh–Bénard convectie (RRBC) is de stroming in een gas- of vloeistoflaagverwarmd via de benedenwand en gekoeld van boven die ronddraait om een verticale asloodrecht op de wanden. Het kent praktische relevantie voor diverse stromingen in de natuuren in de industrie. Belangrijke voorbeelden hiervan zijn de stromingen in de oceanen ende atmosfeer, de convectieve buitenlaag van de zon, het binnenste van de grote gasplaneten,maar ook convectieve koeling in turbomachines en het opdampen van laagjes op roterendesubstraten.

Het is bekend dat het toevoegen van rotatie deze stroming significant beïnvloedt ten op-zichte van de statische situatie. Drie verschillende stromingsregimes kunnen worden onder-scheiden: (I) het regime waarin rotatie geen invloed heeft (de stroming is goed beschrevenals een grootschalige circulatiecel en het warmtetransport blijft onveranderd); (II) het regimewaarin rotatie de stroming modificeert (wervelpluimen die zich richten langs de rotatieas zijnde dominante stromingsstructuren en het convectief warmtetransport kan toenemen ten op-zichte van de statische situatie); en (III) het rotatie-gedomineerde regime (ook wel gestroferegime genoemd, de stroming is turbulent maar tegelijkertijd gedomineerd door de Cori-oliskracht, de warmteflux daalt gestaag bij toenemende draaisnelheid). Deze regimes zullenrespectievelijk benoemd worden als I, II en III. In deze studie zijn experimenten en numeriekesimulaties uitgevoerd om een fundamentele analyse uit te voeren van deze stromingsregimesen om inzicht te krijgen in welke factoren de transities tussen de regimes bepalen. Twee ver-schillende experimentele meettechnieken zijn toegepast, namelijk driedimensionale particletracking velocimetry (3D-PTV) en particle image velocimetry (PIV). De resultaten van deexperimenten zijn, waar het mogelijk en bevorderlijk voor het begrip is, aangevuld met nu-merieke simulaties waarin alle schalen zijn opgelost (direct numerical simulation; DNS).

Het eerste deel van dit proefschrift behandelt de regimes waarin rotatie geen rol speelten waarin rotatie de stroming modificeert, alsook de transitie ertussen. Met behulp van Lag-rangiaanse versnellingsstatistiek van tracerdeeltjes uit experimenten, simulaties voor de diktevan de grenslaag, alsmede theoretische analyse van de verwachte grenslaagstructuur zoalsgebruikt in de Grossmann–Lohse theorie is aangetoond dat de transitie tussen regimes I en IIeen transitie van de grenslagen is, en wel tussen Prandtl–Blasius- (voor niet-roterende con-vectie) en Ekman-type grenslagen (de grenslaagstructuur die in veel roterende stromingenwordt gevonden). Deze analyse is uitgevoerd voor verschillende lengte-breedteverhoudingenvan het domein, waarmee deze afhankelijkheid voor de transitie nu beter begrepen is.

Verder zijn de effecten van rotatie op de anisotropie op kleine en grote schaal in regimes

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I en II bestudeerd. The anisotropie op grote schaal is gekwantificeerd als functie van rotatievia de Lagrangiaanse snelheidsfluctuaties en de tweede-orde Euleriaanse structuurfunctie.Daarnaast is de anisotropie op kleine schaal bepaald met de Lagrangiaanse versnellingsstat-istiek van tracerdeeltjes alsmede de tweede-orde Euleriaanse structuurfunctie. De conclusieis dat rotatie zorgt voor toegenomen anisotropie in het centrum van het domein en dicht bijde bovenwand, terwijl de anisotropie juist afneemt op posities ertussen. De anisotropie opde grote schalen, door rotatie veroorzaakt, heeft nauwelijks effect op de kleine schalen in hetcentrum (die blijven isotroop), terwijl de kleine schalen nabij de bovenwand juist anisotroopblijven.

Voorts is het effect van rotatie op de geometrie van trajectorieën van tracerdeeltjes inregimes I en II geanalyseerd. De meetkundige kromming is gekozen als beschrijvende para-meter voor de geometrie van de deeltjesbanen. De gemeten kansverdelingsfunctie van krom-ming en de resulterende schalingswetten zijn vergeleken met resultaten voor homogene enisotrope turbulentie (HIT) uit de literatuur. Ondanks de kenmerken van de huidige strom-ing die duidelijk ongelijk aan HIT zijn (anisotropie en niet-Gaussische snelheidsstatistiek) issteeds dezelfde kansverdeling voor kromming als in HIT gevonden, zowel in het centrum alsdicht bij de bovenwand (maar buiten de grenslagen). Dit maakt duidelijk dat de statistiek vankromming robuust en in grote mate universeel is, ongeacht de specifieke kenmerken van deturbulentie.

Het tweede deel van dit proefschrift is gericht op het door rotatie gedomineerde regimeIII en de transitie naar dit regime. Twee hypothesen zijn gesteld in de literatuur aangaandedeze transitie: (i) de transitie vindt plaats wanneer de thermische en kinetische grenslagen vangelijke dikte zijn (voor beide afhankelijk van rotatie); (ii) de transitie vindt plaats wanneer dewervelpluimen zich over de gehele hoogte van het domein uitstrekken en boven- en onder-kant verbinden. Deze hypothesen zijn tot nu toe getoetst door het verglijken van de dikten vanthermische en kinetische grenslagen en door vergelijking met het convectief warmtetrans-port. Hier is het nieuwe perspectief van Lagrangiaanse snelheids- en versnellingsstatistiekmet bijbehorende autocorrelaties gebruik om de drie regimes te bestuderen, met extra aan-dacht voor regime III en de twee hypothesen voor de transitie. Het blijkt dat de transitienaar regime III samenvalt met drie geobserveerde veranderingen: sterke demping van ver-ticale beweging, sterk doordringen van wervelpluimen in de bulk en verminderde interactievan de wervelpluimen met de omgeving (uitwisseling van vloeistof). De Lagrangiaanse stat-istiek maakt het mogelijk om de transitie naar regime III in detail te beschrijven en om dehypothesen te toetsen. Daarnaast helpt deze ook voor de karakterisering van alle drie destromingsregimes.

In het laatste deel van dit proefschrift zijn de limieten van het experiment opgezocht in ter-men van draaisnelheid, om het geostrofe regime III binnen te gaan en om de uiterste limiet tenaderen: het kritische punt waarbij de convectieve instabiliteit plaatsvindt, zoals beschrevendoor Chandrasekhar. Experimenten die ver in het gestrofe regime kunnen doordringen zijnspeciaal voor dit doel ontworpen: zeer grote opstellingen of opstellingen die vloeistoffen meteen laag Prandtlgetal gebruiken. Hier wordt een conventionele “table-top" convectiecel van0.2 m hoog gevuld met water gebruikt om te zien hoe ver het gestrofe regime kan wordenverkend. Door vergelijking van ruimtelijke correlaties van de verticale component van devorticiteit, gemeten in het experiment, met eerder gepubliceerde resultaten van simulatiesvan gestrofe convectie (weliswaar met andere randvoorwaarden dan in het experiment) isaangetoond dat het geostrofe regime inderdaad is bereikt. De eerder gerapporteerde signatuur

SAMENVATTING 125

van de karakteristieke stromingsstructuren zijn ook in een conventionele experimentele op-stelling waarneembaar.

De resultaten en de analyses die in dit proefschrift zijn gepresenteerd dragen bij aan hetbegrip van de diverse stromingsregimes in roterende turbulente convectie en de bijbehorendetransities. Daarnaast dragen ze bij aan het uitbreiden van bestaande argumentaties en theor-ieën over HIT naar realistischer niet-HIT stromingen die veelvuldig voorkomen in de natuuren in technische applicaties.

126 SAMENVATTING

Curriculum Vitae

Hadi Rajaei was born on March 22, 1988 in Torbateheydarieh, Iran.

Education09/2013-Present Ph.D. in Applied Physics.

Section : Turbulence and Vortex Dynamics.Eindhoven University of Technology (TU/e),Eindhoven, The Netherlands.

09/2011-08/2013 Master of Science in Mechanical Engineering.Section : Energy Technology.Eindhoven University of Technology (TU/e),Eindhoven, The Netherlands.

09/2006-06/2011 Bachelor of Science in Mechanical Engineering.Iran University of Science and Technology (IUST),Tehran, Iran.

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List of Publications

Published journal papers

• H. Rajaei, P. Joshi, K. Alards, R. P. J. Kunnen, F. Toschi, H. J. H. Clercx,Transitions in rotating convection; A Lagrangian perspective,Physical Review E, 93(4):043129(2016).

• H. Rajaei, P. Joshi, R. P. J. Kunnen, H. J. H. ClercxFlow anisotropy in rotating buoyancy-driven turbulence,Physical Review Fluids, 1(4):044403(2016).

• H. Rajaei, R. P. J. Kunnen, H. J. H. ClercxExploring the geostrophic regime of rapidly rotating convection with experiments,Physics of Fluids, 29(4):045105(2017).

• P. Joshi, H. Rajaei, R. P. J. Kunnen, H. J. H. ClercxEffect of particle injection on heat transfer in rotating Rayleigh–Bénard convection,Physical Review Fluids, 1(8):084301(2016).

• K. M. J. Alards, H. Rajaei, L. DelCastelo, R. P. J. Kunnen, F. Toschi, H. J. H. ClercxGeometry of tracer trajectories in rotating turbulent flows,Physical Review Fluids, 2(4):044601(2017)

• Ö. Baskan, H. Rajaei, M. J. M. Speetjens, H. J. H. ClercxScalar transport in inline mixers with spatially-periodic flows,Physics of Fluids, 29(1):013601(2017).

• P. Joshi, H. Rajaei, R. P. J. Kunnen, H. J. H. Clercx,Heat transfer in rotating Rayleigh–Bénard convection with rough plates,Journal of Fluid Mechanics, 830:R3 (2017)

Publications in preparation

• H. Rajaei, K. M. J. Alards, R. P. J. Kunnen, H. J. H. Clercx,Velocity and acceleration correlations in rotating Rayleigh–Bénard convection: Indic-ators for transitions, To be submitted to Journal of Fluid Mechanics.

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130 CURRICULUM VITAE

• K. M. J. Alards, H. Rajaei, R. P. J. Kunnen, F. Toschi, H. J. H. Clercx,Directional change of tracer trajectories in rotating Rayleigh–Bénard convection, un-der preparation.

Acknowledgments

This thesis is just the final step of a four years journey of joy. Before closing this final step,I would like to thank people who helped me in this journey and contribute to this thesis.Without the help of these people, this thesis would not have been possible.

First and foremost, I would like to thank Herman Clercx, Rudie Kunnen and FedericoToschi who gave me the opportunity to work on this interesting project. I would like to thankHerman for all of our discussions, his encouragements and supports. I would like to expressmy deepest appreciation to Rudie who helped me on a daily basis. I have been extremelylucky to have a supervisor who was always available for discussion. I don’t know how I canthank you for your supports during these four years.

I would like to thank Marjan Roedenburg for her priceless efforts for arranging all form-alities. She is always very helpful in every single problem.

I would like to express my sincere gratitudes to the WDY technicians, Ad, Gerald, andFreek. Whenever I needed help, there was always someone who offered it to me. I literallyenjoyed working with you and I learned a lot from you. Thanks a lot guys! The life ofan experimentalist would be tough without your helps. I am in a great debt to all of you,particularly Ad who helped me a lot with the problems ranging from a broken hard disk tosetting up experiments.

I would like to thank my friends and colleagues in WDY. They made these four yearsone the best four years in my life. I would like to thank Pranav and Kim for our insightfuldiscussions. My gratitude also goes to my (ex-)officemates; Pranav, Josje, Wolfram and Jan,who made the work fun. I would like to thank other friends and colleagues at the cascadebuilding, Vitor, Abhineet, Steven, Gianluca, Altug, Ozge, Francesca, Alessandro, Neehar,Dario, Michel, Ivo, Riccardo, Saskia, Matteo (Lulli and Madonia), Dennis, Sudhir, Samuel,Marlies, Matias, Bijan, Jonathan, Pinaki, Abheeti, Sten, Xiao, Chungmin, Andrés, Haijing,Felix, Andrei, GertJan, Leon, Willem, Bas, Maarten, Florian, Dennis, Hadi, Hossein, Saeedand many more who I might have missed to refer. I will remember our fussball matches latein the evening, the unbeatable team of 2.05, our barbecues in the university and in Den Bosch,Happy Italy on Thursdays and our unique camping trips in Germany and Belgium. I enjoyedeach and every of these great activities and thank you so much for making it all happen. Imade some friends for lifetime during these four years.

Finally, I would express my gratitudes to my parents (in-law), sisters and brother. It isdifficult to live 5000 km apart, but they were always supporting me throughout these yearsliving in the Netherlands. Madar o pedare azizam be khatere hameye poshtibanihaton dar inshesh sal doori sepasgozaram.

I opened this thesis by dedication to the love of my life, and I would also like to close

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132 ACKNOWLEDGMENTS

this thesis by thanking her for her support, patience, encouragement and unwavering love.Without your support, I could not be who I am or where I am now.

Hadi Rajaei

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