Rotating Rayleigh-Bénard convection

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1 Rotating Rayleigh-Bénard convection Rajaei, H. Published: 19/12/2017 Document Version Publisher s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. The final author version and the galley proof are versions of the publication after peer review. The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication Citation for published version (APA): Rajaei, H. (2017). Rotating Rayleigh-Bénard convection Eindhoven: Technische Universiteit Eindhoven General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal? Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 01. Jul. 2018

2 Rotating Rayleigh Bénard Convection PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de rector magnificus prof.dr.ir. F.P.T. Baaijens, voor een commissie aangewezen door het College voor Promoties, in het openbaar te verdedigen op dinsdag 19 december 2017 om uur door Hadi Rajaei geboren te Torbateheydarieh, Iran

3 Dit proefschrift is goedgekeurd door de promotoren en de samenstelling van de promotiecommissie is als volgt: voorzitter: prof. dr. ir. G. M. W. Kroesen 1 e promotor: prof. dr. H. J. H. Clercx 2 e promotor: prof. dr. F. Toschi copromotor: dr. ir. R. P. J. Kunnen leden: prof. dr. D. Lohse (UT) prof. K.-Q. Xia (CUHK) prof. dr. ir. A. A. van Steenhoven prof. dr. ir. B. J. Geurts (UT) Het onderzoek of ontwerp dat in dit proefschrift wordt beschreven is uitgevoerd in overeenstemming met de TU/e Gedragscode Wetenschapsbeoefening.

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

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6 This work is part of the research programme of the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organisation for Scientific Research (NWO). Copyright 2017 by Hadi Rajaei Cover design by Hadi Rajaei Cover photo by Hadi Rajaei, All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission of the author. ISBN NUR 926 Printed by Gildeprint - Enschede

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8 List of Abbreviations Abbreviation BL CCD CTCs DNS FFT GL HPC HIT LSC LEDs LPC PIV PDFs RBC RRBC SF 3D PTV Description Boundary Layer Charge-Coupled Device Convective Taylor Columns Direct Numerical Simulation Fast Fourier Transform Grossmann Lohse High Particle Concentration Homogeneous and Isotropic Turbulence Large Scale Circulation Light-Emitting Diodes Low Particle Concentration Particle Image Velocimetry Probability Distribution Functions Rayleigh Bénard Convection Rotating Rayleigh Bénard Convection Structure Function Three-dimensional Particle Tracking Velocimetry i

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10 Contents List of Abbreviations i 1 Introduction and theoretical background Introduction Equations of motion Geostrophic balance and thermal wind Ekman boundary layer Useful definitions in convective turbulence Flow phenomenology Goal Guide through this thesis Methods and Parameters Experimental set-ups Convection cell Three-dimensional Particle Tracking Velocimetry Time-resolved particle image velocimetry system Experimental procedure and flow characteristics Numerical set-up Transitions in turbulent rotating convention Introduction Methods and Parameters Results and discussion Conclusions Flow anisotropy in rotating buoyancy-driven Introduction Experimental parameters iii

11 iv CONTENTS 4.3 Results Large-scale (an)isotropy Small-scale (an)isotropy Conclusions Geometry of tracer trajectories Introduction Experimental parameters Results Nonrotating RBC Rotating RBC Conclusions Velocity and acceleration statistics Introduction Experimental parameters Lagrangian rms velocity Lagrangian velocity autocorrelation Small Ro: emergence of two time scales Concluding remarks Lagrangian rms acceleration Lagrangian acceleration autocorrelation Concluding remarks Oscillatory behavior Conclusions Exploring the Geostrophic regime Introduction Experimental techniques and parameters Experimental parameters Experimental setup Data validation Results and discussion Spatial vorticity autocorrelation at z = 0.8H Flow coherence along the rotation axis Conclusions Concluding remarks Conclusions Outlook

12 CONTENTS v A Temporal vorticity autocorrelation 105 Bibliography 107 Summary 121 Samenvatting 123 Curriculum Vitae 127 List of Publications 129 Acknowledgments 131

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14 Chapter 1 Introduction and theoretical background 1.1 Introduction Thermally driven turbulent convection has been initially studied by Henri Bénard and Lord Rayleigh more than a century ago. In 1900, Henri Bénard observed that the flow motion is organized into a regular pattern of hexagonal cells when a thin layer of fluid is heated from below [8, 9, 10]. A few years later, Lord Rayleigh published his work on the theoretical analysis of the convective instability of a layer of fluid [135]. In honor of Bénard s and Rayleigh s pioneering works, the term Rayleigh Bénard convection is designated to the fluid motion heated from below and cooled from above. Rayleigh Bénard convection is of interest to theoretical, computational and experimental physicist since it is mathematically well-defined and its experimental realization is comparatively straightforward. In this thesis classical Rayleigh Bénard convection (RBC) is subjected to background rotation about its vertical axis. Buoyancy-driven flows affected by background rotation are omnipresent in nature and technological applications. These flows can be classified into three important categories; geophysical flows, astrophysical flows and flows in technological applications. Large-scale flows in Earth s oceans and atmosphere are primarily driven by temperatureinduced buoyancy. These flows typically have such enormous length scales that Earth s rotation affects them. The combined effects of buoyancy and rotation lead to the formation of the so-called Hadley cells in the atmosphere [53]. The generated winds by Hadley cells near the surface of the Earth are known as trade-winds. At higher latitudes, mid-latitude cells (Ferrel cells) and polar cells drive the large-scale dynamics of the atmosphere, see Figure 1.1(a). In addition to the atmospheric large-scale motions which are driven by combined effects 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 been observed on other planets like Jupiter, Saturn, Uranus and Neptune [58, 63], although it is not known whether these flows are solely driven by convection. Rotating convection also occurs 1

15 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 metal cores 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 technological applications. For example, in convective cooling in rotating turbomachinery blades both convection and rotation play important roles [35, 64]. A second example is chemical vapour deposition on rotating heated substrates [169]: considerable convective heat transfer due to the heated substrate, combined with rotation. Another example includes efficient separation of carbon dioxide (CO 2 ) from methane or nitrogen gas. In this process, the gaseous mixture is 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 CO 2 into droplets (due to the radial compression) [170]. The condensation of droplets produces significant heat while the flow is also strongly affected by rotation. Figure 1.1: (a) Large-scale motions induced by buoyancy and background rotation in the Earth 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 the convective 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 why it attracts so much attention and has been extensively studied by laboratory experiments

16 1.2. EQUATIONS OF MOTION 3 e.g. Refs. [136, 12, 193, 91, 138, 177, 79, 194, 115, 82, 73, 182, 81, 36], numerical simulations 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 characterized by global parameters like the overall heat transfer. In this thesis, however, we characterize RRBC by examination of the flow field. A combined experimental-numerical approach is employed to study the Lagrangian and Eulerian statistics of neutrally buoyant immersed tracers in RRBC. The results are further introduced in Section 1.8. First, a general theoretical background is presented to familiarize the reader with the basic concepts involved in 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 coefficient, thermal diffusivity) are independent of the temperature and the fluid density is assumed to be linearly dependent on the temperature, ρ(t ) = ρ(t 0 ) (1 β (T T 0 )), (1.1) where ρ is the fluid density, T is the fluid temperature, T 0 is some reference temperature and β is the fluid thermal expansion coefficient. These assumptions are reasonably valid for small temperature differences (β T 0.2 [25, 116]). Keeping this in mind, we can write the equations for conservation of mass, momentum and energy for a Newtonian fluid as [23] u = 0, u t + u u = p + ν 2 u + βgt ẑ, T t + u T = κ 2 T, (1.2) where u is the velocity, t is time, p is the pressure (mean density is incorporated in the pressure term), ν is the kinematic viscosity of the fluid, g is the gravitational acceleration, T is the temperature relative to a reference temperature, κ is the thermal diffusivity of the fluid and ẑ is the vertical unit vector. Equations (1.2) are valid for an inertial frame of reference. However, we are interested in these equations in the rotating frame. The transformation from an inertial to a rotating frame of 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 + Ω r, (1.3) dt where Ω is a constant rotation vector. See also Figure 1.2 for clarification.

17 4 CHAPTER 1. INTRODUCTION AND THEORETICAL BACKGROUND Ω Ω r θ r 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 to time (i.e. acceleration) is expressed as d 2 r dt 2 = d ( ) ( ) dr dr dt dt + Ω r + Ω dt + Ω r (1.4) = d2 r dr + 2Ω dt2 dt + Ω Ω r, 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. The second and third terms on the right hand side of Equations (1.4) and (1.5) represent the Coriolis and centrifugal accelerations, respectively. The centrifugal acceleration can be rewritten as a gradient ( ) ( ) 1 1 Ω Ω r = (Ω r) (Ω r) = 2 2 Ω2 r 2, (1.6) where r = r sin θ is the distance to the rotation axis (see Figure 1.2) and Ω = Ω. Therefore, the centrifugal acceleration can be incorporated into the pressure term in the Oberbeck- Boussinesq equations (Equations (1.2)); P = ( p 1 2 Ω2 r 2 ). Thus, the momentum equation in the rotating frame yields u t + u u + 2Ω u = P + ν 2 u + βgt ẑ. (1.7)

18 1.2. EQUATIONS OF MOTION 5 We only consider a convection cell rotating about its vertical axis, thus for the remainder of this thesis the rotation vector is Ω = Ωẑ. In order to obtain the important dimensionless numbers, the Oberbeck-Boussinesq equations in a rotating frame of reference are nondimensionalized by defining the following variables; x = x/h (dimensionless position), ũ = u/u (dimensionless velocity), t = tu/h (dimensionless time), T = T/ T (dimensionless temperature) and P = P/U 2 (dimensionless pressure). Tildes indicate dimensionless variables here and H, U, and T are the typical length scale (the separation of heating and cooling plates), the typical velocity scale, and the temperature difference between bottom and top plates, respectively. Therefore, the dimensionless equations can be rewritten as ũ = 0, ũ t + ũ ũ + 2ΩH U ẑ ũ = P + ν UH 2 ũ + gβ T H T ẑ, U 2 T t + ũ T = κ UH 2 T. (1.8) In thermal convection, the typical velocity is usually expressed as U = gβ T H, the socalled free-fall velocity [128]: the maximum buoyancy-generated velocity. Three different dimensionless numbers can be defined, namely the Rayleigh number, the Prandtl number and the Rossby number, gβ T H3 Ra =, (1.9) νκ Pr = ν κ, (1.10) Ro = U 2ΩH. (1.11) In these equations, the Rayleigh number indicates the strength of thermal forcing and is a measure of the ratio of buoyancy and dissipation, the Prandtl number describes the relative importance of momentum diffusivity and thermal diffusivity, and the Rossby number is the ratio of the inertial and Coriolis forces. Considering that ν UH = Pr Ra the Equations (1.8) can be rewritten as and κ UH = 1, PrRa ũ = 0, ũ t + ũ ũ + Ro z 1 ũ = P Pr + Ra 2 ũ + T ẑ, T t + ũ T 1 = 2 T. PrRa (1.12) Apart from the aforementioned dimensionless numbers (Ra, Pr, Ro), one can also introduce the Nusselt number, the Ekman number, the Taylor number, the Froude number and the cell

19 6 CHAPTER 1. INTRODUCTION AND THEORETICAL BACKGROUND aspect ratio, defined as Nu = qh k T, (1.13) Ek = ν 2ΩH 2, (1.14) ( ) 2ΩH 2 2 Ta =, (1.15) ν Fr = Ω2 R 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 radius and D is the cell diameter when considering an upright cylindrical domain. The Nusselt number is the ratio of total vertical heat flux and the conductive heat flux. The Ekman number shows the importance of the viscous compared to Coriolis forces. The Taylor number is the inverse square of the Ekman number. The importance of centrifugal buoyancy (nominally disregarded in the Oberbeck-Boussinesq equations but unavoidable in experiments) can be assessed by the Froude number. The last number is the aspect ratio; a number representing in this case the cylindrical geometry. 1.3 Geostrophic balance and thermal wind The complete set of Navier-Stokes equations with the Oberbeck-Boussinesq approximation has a complicated structure. However, by introducing some relevant assumptions for a certain class of flows, it is possible to simplify and interpret these equations. Assume that the flow is quasi-steady, the inertial force is negligible compared to the Coriolis force (i.e. Ro 1) and viscous effects are negligible in the bulk (i.e. Ek 0), the motion of such a flow is governed by 2Ωẑ u = P + gβt ẑ. (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 horizontal balance in Equation (1.19) is called the geostrophic balance. Taking the derivatives of the first and second terms with respect to y and x, respectively, and subtracting, we have ( u 2Ω x + v ) = 0. y

20 1.4. EKMAN BOUNDARY LAYER 7 Considering the incompressibility condition we arrive at w z = 0, (1.20) with w the vertical velocity component. If we take the derivative of the horizontal components of Equations (1.19) to z, and substitute the vertical pressure gradient by gβt, see the vertical component of Equation (1.19), we arrive at 2Ω v T = gβ z x, 2Ω u (1.21) T = gβ z y. Equations (1.20) and (1.21) together are called the thermal wind balance [123] and they can be rewritten as u z = gβ T. (1.22) 2Ωẑ This equation states that vertical gradients of the horizontal velocity components depend only on the horizontal temperature gradient, while the vertical gradient of the vertical velocity component is always zero. For a fluid with constant density (i.e. ρ = ρ 0, independent of temperature), we arrive at the 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 horizontal flow motion is governed by the geostrophic balance, see the horizontal components in Equation (1.19). The geostrophic balance is only valid away from the boundaries, where the viscosity can be neglected. However, boundary layers at the cylinder sidewall and top and bottom plates are required to satisfy the viscous no-slip conditions; the boundary layers at top 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 time derivatives and the horizontal derivatives in the viscous term and assuming that the bulk flow is in geostrophic balance) [46, 75], the horizontal velocity components inside the Ekman boundary layer are expressed as u E = u I [u I cos (z/δ E ) + v I sin (z/δ E )]e z/δ E, v E = v I + [ u I sin (z/δ E ) v I cos (z/δ E )]e z/δ E, (1.24) where u E and v E are the horizontal velocities inside the Ekman boundary layer, u I and v I are the horizontal velocities in the interior region, z is the distance to the plate and δ E is the

21 8 CHAPTER 1. INTRODUCTION AND THEORETICAL BACKGROUND Ekman boundary layer thickness: δ E = ν/ω [46, 75]. Note that δ E is independent of the interior velocity (bulk velocity) and the flow configuration. Another important parameter is the Ekman time scale, the adjustment time for a fluid when a new rotation rate is introduced, which is defined as τ E = H/ νω [46]. 1.5 Useful definitions in convective turbulence The importance of the study of turbulence is well captured by the famous quote of Richard Feynman as Turbulence is the most important unsolved problem of classical physics. Turbulent flows are ubiquitous; most of the flows in nature and technological applications are turbulent. The flow transitions from laminar to turbulent when the large-scale Reynolds number goes beyond some threshold, i.e. advection becomes more important than viscous effects at large scales. Each and every turbulent flow is characterized by some shared characteristics such as irregularity, high diffusivity, large Reynolds numbers, high dissipation, a wide range of active length scales, etc [165]. A turbulent flow can be driven by different means, here we focus on thermally-driven turbulence. With this introduction in mind, we briefly discuss some basic concepts in (thermally-driven) turbulence which are used in this thesis. We refer the 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 required to keep the turbulence running. In convective turbulence, the buoyant production acts as the energy input at large scales. In statistically steady turbulence, the input energy is transferred into the smallest scales (turbulence cascade), where the viscosity plays a major role, and dissipates into heat. In convective turbulence, two dissipation rates are involved: the kinetic energy and thermal variance dissipation rates, ɛ and ɛ θ, respectively. Therefore, two different dissipative length scales exist: the Kolmogorov length (η) and the Batchelor length (η B ) [107]. These length scales are given by ( ) ν 3 1/4 ( ) νκ 2 1/4 η = and η B =, (1.25) ɛ ɛ where ɛ is the kinetic dissipation rate. These two length scales dictate the grid spacing (resolution) in direct numerical simulation (DNS): the grid spacing should be smaller than these length scales. As mentioned, the kinetic energy and thermal variance dissipation rates are involved in thermally driven turbulence. RBC is an inhomogeneous system: local dissipation rates depend strongly on the vertical and radial positions; particularly in the vicinity of the boundaries they may attain considerably larger values than in the center. However, one can derive an exact relation for the global (volume-averaged) kinetic and thermal variance dissipation rates in (non)rotating RBC as [148, 149] ɛ = ν3 Ra Pr 2 H (Nu 1) and ɛ θ = κ T 2 Nu, (1.26) 4 H2 where ɛ and ɛ θ are the volume-averaged kinetic energy and thermal variance dissipation rates, respectively. The global dissipation rates calculated in DNS give a firm foundation to validate the accuracy of the calculated Nu.

22 1.6. FLOW PHENOMENOLOGY 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 a logarithmic scale. The demarcations between different regimes are given by the vertical dashdotted lines. Figure 1.4, on the other hand, shows the flow morphologies near the top plate from top to bottom for Ro =, 1 and 0.05, respectively, corresponding to regimes I, II and III, respectively, to be defined below. The blue curves are example particle trajectories near the top plate and the red dots indicate the starting point of each trajectory. In the following we explain these regimes and their characterizations. 1.2 Nu(Ro)/Nu( ) Regime III Regime II Regime I Ro Figure 1.3: The normalized heat transfer as a function of Ro on logarithmic scales. Red dots are experimental data and open square are numerical data for Ra = and Pr = Data taken from Refs. [159, 195]. The vertical dashed lines represent the demarcation lines between different regimes. Rotation-unaffected regime (Regime I): The rotation-unaffected regime occurs for Ro 2.5 for a convection cell with Γ = 1 (the transition between rotation-unaffected regime 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 this regime; i.e. weak background rotation does not affect Nu. The rotation-unaffected regime 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 plate in this regime. As one can partially see from the graph, the fluid parcels move upward from one side and they travel downward at the other side of the cell. The presence

23 10 CHAPTER 1. INTRODUCTION AND THEORETICAL BACKGROUND of the LSC dictates vertical mean velocities near the side walls and horizontal mean velocities near the top and bottom plates, while the flow has zero mean velocity at the cell 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 while the lower bound is Pr dependent [156]. This regime is characterized by a continuous enhancement in the heat flux with increasing background rotation for Pr > 1, see Figure 1.3. The enhancement does not occur for Pr < 1. In this regime, the LSC has disappeared and is replaced by rotation-aligned vortical plumes, see Figure 1.4(cd) 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. As can be seen (qualitatively) from the figure, the horizontal and vertical velocities posses approximately zero mean values in this regime. Quantitative analysis of the present velocity statistics confirms the zero mean values for vertical and horizontal velocities. Rotation-dominated regime (Regime III): The rotation-dominated regime, also known as the Geostrophic regime, is characterized by a dramatic drop in the heat transfer efficiency with increase in the background rotation, see Figure 1.3. This drop continues until the flow motions halt and diffusion becomes the only active way of heat transfer (i.e. Nu = 1). In the geostrophic regime, the flow is principally governed by a balance of the Coriolis and the pressure gradient forces. The rotation-aligned vortical plumes become more prominent (they can penetrate further into the bulk) and similar 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) for a qualitative analysis. There are four distinct flow structures in this regime, namely cellular convection, convective Taylor columns, plumes, and geostrophic-turbulence [151, 117, 153]. Depending on the state, the flow phenomenology is different. The cellular state occurs just above the onset of convection (i.e. Nu = 1); it is characterized by densely packed thin hot and cold columns spanning the entire vertical extent of the flow domain. Departing from the onset of convection, these cells may develop into well-separated vertically aligned vortical convective Taylor columns (CTCs), surrounded by shields of vorticity of opposite sign. Another mode of convection consists of plumes with less vertical coherence and no shields. A final state is called geostrophicturbulence, where the vertical coherence is lost almost completely and the interior is fully turbulent. Note that all four convection modes are part of the geostrophic regime, which should thus not uniquely be identified with the geostrophic-turbulence state alone. The occurrence of these four states is strongly dependent on the Prandtl number Pr: at lower Pr 3 no Taylor columns are formed, while the geostrophicturbulence state remained out of reach for Pr 7 in the simulations [70]. Considering our experimental set-up, it is expected that we cover the cellular, CTCs and plume states while the geostrophic-turbulence state is out of reach.

24 1.7. GOAL b) a) z y y x c) z y y x x 40 f) e) z y x 40 d) y x x Figure 1.4: Particle trajectories from experiments near the top plate for (a) Ro =, isometric view (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 located at z = 200 mm. 1.7 Goal As mentioned before, rotating thermal convection has been studied by laboratory experiments, numerical simulations and theoretical analysis. However, up to now, the effects of background rotation on RBC are mainly characterized by global parameters like the overall heat transfer: less attention has been given to the flow field. Additionally, the available studies on the flow field are performed in the Eulerian frame of reference. Recently, nonrotating RBC has been studied in the Lagrangian frame of reference numerically [144, 37, 90] and experimentally [113, 114, 90, 89]. However, to the best of our knowledge, the Lagrangian studies of the rotating RBC are non-existent. There are still many open questions with regard to the rotating RBC convection, which can be answered by Eulerian and particularly Lagrangian flow field measurements. In this

25 12 CHAPTER 1. INTRODUCTION AND THEORETICAL BACKGROUND thesis, we address such questions, of which the main ones are: What are the driving mechanisms behind transitions from one regime to another regime? How are these transitions reflected into different Lagrangian statistics? How can 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 at different heights in a cylindrical convection cell? Do the small-scales remain isotropic or is Kolmogorov s hypothesis of local isotropy violated? (Chapter 4) What are the effects of background rotation on the geometrical aspects of fluid parcel trajectories? Can we recover the power-law scaling, reported earlier for homogeneous and isotropic turbulence (HIT), in our non-hit thermally-driven turbulence throughout the cylindrical domain? If not, how does the power law scaling changes with rotation and position of the measurement volume in the cell? (Chapter 5) Is the geostrophic regime accessible with a conventional RBC set-up: a convection cell with H = 0.2 m and water as working fluid? If so, can we reproduce the basic signatures of the asymptotic solutions in an experiment? (Chapter 7) In the following chapters we will address the above questions. Brief answers to these questions are presented in the next section, Section 1.8, to familiarize the reader with the results. 1.8 Guide through this thesis We start with the description of our experimental and numerical approaches to study RRBC in Chapter 2. Two different approaches are used for flow measurement, namely three-dimensional particle tracking velocimetry (3D-PTV) and time-resolved particle image velocimetry (PIV). The experimental data are complemented by direct numerical simulations (DNS) when it is appropriate and possible. Chapter 3 concerns the rotation-unaffected and rotation-affected regimes and the transition between them. Using measurements of the Lagrangian acceleration of neutrally buoyant particles at two different heights (close to the top plate and at the center of the cylindrical convection cell) and accompanying DNS, we study the role of the boundary layers in the transition 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 dependence of the transition on the aspect ratio. In Chapter 4, we study the effects of background rotation on large- and small-scale isotropy in RRBC from both Eulerian and Lagrangian points of view. 3D-PTV and DNS are employed at three different heights within the cell. The Lagrangian velocity fluctuation and second-order Eulerian structure function are utilized to evaluate the large-scale isotropy for different rotation rates. Moreover, we examine the experimental measurements of the Lagrangian acceleration of neutrally buoyant particles and the second-order Eulerian structure function 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 background rotation on the geometry of tracer trajectories for RRBC in regimes I and II. We focus on the geometry of tracer trajectories by computing curvature statistics. Scaling laws for curvature statistics have been derived for HIT in previous studies. We always recover this HIT scaling

26 1.8. GUIDE THROUGH THIS THESIS 13 in the bulk of RRBC. However, in the horizontal boundary layers, we only recover this HIT scaling for higher rotation rates. We discuss the physics behind these phenomena in this chapter. In Chapter 6 the experimental velocity and acceleration statistics from 3D-PTV data for all three regimes are studied. The main focus of the chapter is on the Lagrangian velocity and acceleration autocorrelations and how the transition from one regime to another affects these statistics. Different time scales are observed in the velocity autocorrelations in regimes II and III 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 (regime III) and the onset of convection. Approaching the geostrophic regime of rotating convection typically 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 our experiment into a regime where it is possible to compare the experimental data with those of the asymptotically reduced equations. These equations, derived from the incompressible Navier-Stokes equations, are the so-called non-hydrostatic quasi-geostrophic set of equations in the limit Ro 0 [68, 151]. We show that it is also possible to enter the geostrophic regime of rotating convection with classical experimental tools: a table-top set-up with a conventional convection cell with a height of 0.2 m and water as working fluid. We compare our experimental data with results from simulations of the asymptotically reduced equations reported 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 to further studies of this interesting flow problem.

27 14 CHAPTER 1. INTRODUCTION AND THEORETICAL BACKGROUND

28 Chapter 2 Methods and Parameters The present study employs a combined numerical-experimental approach to explore rotating Rayleigh Bénard convection. The main part of this study is based on the experimental data obtained from three-dimensional particle tracking velocimetry (3D-PTV). We also employ particle image velocimetry (PIV). The experimental data are complimented by direct numerical simulations (DNS) when required. In this chapter, the experimental and numerical approaches are explained. We first start with the experimental set-up for RRBC in Section 2.1. Next, the experimental procedure and flow characteristics are given in Section 2.2. Finally, the DNS set-up is discussed in Section Experimental set-ups The experimental set-up consists of a convection cell and an optical measurement system mounted 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 the experimental set-ups (consisting of the convection cell and an optical system on the rotating table) are presented in Figures 2.1(a-b). Figure 2.1(a) shows the 3D-PTV system combined with the convection cell and Figure 2.1(b) displays the PIV system combined with the same convection cell. In the following sections, first the convection cell is explained in Section Then, the 3D-PTV and PIV systems and the corresponding post processing procedures are discussed in Sections and 2.1.3, respectively Convection cell Figure 2.2 shows a schematic view of the convection cell. The convection cell, similar to the one used in Refs. [82, 81, 76, 78, 79], is composed of three main parts: (i) a heated plate at the bottom, (ii) a cooling chamber at the top, and (iii) a cylindrical Plexiglas vessel connecting top and bottom plates, see Figure 2.2. At the bottom, an electrical resistance heater is attached to a copper plate. The electrical resistance heater is manufactured in the form of a disk, thus it is in contact with a large area of the copper plate. Furthermore, copper has an excellent thermal conductivity (k copper = 390 W/(m K)), guaranteeing a uniform temperature distribution. There is a layer of insulation beneath the electrical heater to avoid 15

29 16 CHAPTER 2. METHODS AND PARAMETERS Figure 2.1: Schematic views of (a) the 3D-PTV system and (b) the PIV system. 3D-PTV uses a volumetric illumination using light-emitting diodes (LEDs) while PIV utilizes a 2D illumination using a laser sheet. heat leakage. The copper plate temperature is measured by a thermistor placed inside a hole close to the wetted surface at the center of the copper plate. The thermistor is connected to a heater controller unit which keeps the temperature of the plate constant by variation of the input 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, with k sapphire = 33 W/(m K), is positioned at the bottom of the cooling chamber, guaranteeing good thermal contact between working fluid and the recirculating fluid in the cooling bath while retaining transparency. Sapphire has a smaller thermal conductivity compared to copper, however, the thermal conductivity of sapphire is substantially better than other transparent materials e.g. k sapphire /k plexiglas 180. A 12 mm thick Plexiglas plate is located on the top of the cooling chamber. The cooling chamber is transparent and as a result it makes the cell optically accessible from above. The cooling chamber is equipped with eight nozzles; four for coolant inflow and four for coolant outflow. The inlet and outlet nozzles are equipped with fine meshes to avoid formation of large-scale flow structures and to guarantee 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 measuring the temperature at nine different positions inside the cooling chamber [76]. A cooling bath (Haake V26/B refrigerated bath, with Haake DC50 temperature control unit) keeps the cooling chamber temperature constant. A thermistor is located in the cooling chamber and connected to the cooling bath. During operation, the temperature fluctuations are smaller than 0.05 K. The cylindrical Plexiglas vessel (with thickness of 10 mm) with equal inner diameter D and height H of 200 mm, filled with water, is placed inside a rectangular Plexiglas box with a wall thickness of 15 mm. The space between cylinder and box is filled with water as well to avoid the distortion of the illumination from the side. The focus of this study is on the measurements of the flow field rather than heat transfer. Therefore, optical accessibility is

30 2.1. EXPERIMENTAL SET-UPS 17 Figure 2.2: A sketch of the convection cell and its components (left panel). Different components 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 mean temperature close to the ambient temperature to minimize the heat loss to the ambient. Note that 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 optical accessibility [82, 81, 76, 78, 79, 114, 113, 177]. However, the non-idealities in the current convection cell have no significant influences on the measurements as evidenced by the excellent quantitative agreement of experimental and simulation results achieved in the later chapters of this thesis Three-dimensional Particle Tracking Velocimetry There are different experimental techniques for particle tracking, e.g. acoustic three dimensional 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-strip detector or high speed cameras, e.g Refs. [98, 101, 31, 106, 191, 85, 179, 178, 108]. Each method has its own pros and cons and has been employed for Lagrangian studies of nonrotating turbulence. We apply optical 3D-PTV based on high speed cameras due to extensive expertise in the lab [31, 191, 106]. It is worth mentioning that the Lagrangian exploration of Rayleigh 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. Here we 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 measured through tracking small neutrally buoyant seeding particles. The original underlying concept has been used for visualization in the 1960s [168]. In this system, the tracking has been performed manually. The first automated PTV system was realized in 1985 by Chang et al. [24] and later followed by Adamczyk et al. [1] in However, the PTV algorithm has been significantly improved since then. A typical 3D-PTV algorithm consists of the

31 18 CHAPTER 2. METHODS AND PARAMETERS following steps: first the seeding particles are detected in the images, then 3D positions of the particles in 3D space are calculated, and finally the temporal linking between two subsequentin-time frames is established needed for evaluation of particle velocity and acceleration. In the following, we discuss the aforementioned steps. In the current study, we use a particle tracking 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 mainly concentrating 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 typically appear as bright spots in the camera image. There are various techniques for determining the center of a particle, e.g. peak fitting [2], template matching [162] and weighted mean. The ETH algorithm uses the weighted mean technique as it defines the center of a particle in a 2D image by x c = i x ii(x i, y i ) i I(x i, y i ) and y c = i y ii(x i, y i ) i I(x i, y i ) where i is the pixel number, (x c, y c ) is the coordinate of the center of the particle and I(x i, y i ) is the intensity of the particle at (x i, y i ). The weighted mean method is computationally cheap compared 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 viewing angles are required. It is possible to reconstruct the 3D position by using a geometrical relation between the 3D positions and their projections onto the 2D images, known as the epipolar line [99]. The basic concept is plotted in Figure 2.3. In this plot, X is a point in 3D space, x 1 and x 2 are the X projections onto image 1 and 2, respectively, and C 1 and C 2 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 in image 1, e.g. x 1, are known. We know that X, x 1, x 2, C 1 and C 2 are coplanar, see Figure 2.3. Thus, X can be any point on Line 1. The projection of Line 1 onto image 2 is a line and is called the epipolar line. In theory, if the seeding density is very low, it is most likely that there is only one bright spot on the epipolar line in image 2. However, in practice, there are typically multiple bright spots on the epipolar line; another image (camera) is required to resolve these ambiguities. Any arbitrary number of cameras (larger than 1) can be used for PTV. However, more than four cameras was found to be not necessary, impractical and not cost 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 density is low (the inter-particle distance is significantly larger than the particle displacement between two successive frames), the temporal link can readily be established by using a nearest neighbor scheme. However, for higher seeding densities, the situation is more complicated. In this 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 streak photography [33, 60]. The ETH code uses the multi-frame tracking technique introduced by Hassan et al. [56] and further modified by Malik et al. [101]. In this method, four consecutive frames are considered and based on different criteria, e.g. particle velocity and angle of travel between successive frames, an acceptable trajectory is established.

32 2.1. EXPERIMENTAL SET-UPS 19 Figure 2.3: Schematic representation of the working principle of epipolar line for a two camera system. C1 and C2 are the camera 1 and 2 centers, respectively. X is a point in space and x1 and x2 are its projections onto image 1 and 2, respectively. As mentioned before, the extrinsic and intrinsic camera parameters are determined through a calibration procedure. The extrinsic parameters are set-up dependent: they depend on position and orientation of the cameras with respect to the measurement volume. On the other hand, intrinsic parameters are inherent to the camera in use: they are independent of positioning and orientation of the cameras. Intrinsic parameters include variables like focal length and lens distortion coefficients. The extrinsic camera parameters are obtained through a calibration procedure by using an optical model based on a pinhole camera model, lens distortion and different media with different refractive indices [100]. There are two main approaches to calibrate a 3D-PTV system: multi-plane calibration (a calibration plate submerged inside the measurement volume at different heights) or a 3D staircase body (a 3D body consisting of points at different heights). The bottleneck for the multi-plane calibration procedure is the precise parallel positioning of the calibration plate at different precise vertically separated heights. Therefore, we use a 3D staircase body, see Figure 2.4. The calibration is discussed in more detail in the subsequent section. Figure 2.4: The V-shaped staircase calibration body used in the current study.

33 20 CHAPTER 2. METHODS AND PARAMETERS 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, pixels equipped with 50 mm lenses) positioned above the convection cell, see Figure 2.1(a). The apparatus of the camera is set to f /16 for all experiments, which provides the maximum depth of field. The 3D-PTV images are recorded at 8-bit dynamic range of grayscales. The active area of the camera sensor is mm 2 resulting in a pixel size of µm 2. The cameras are located roughly 400 mm away from the measurement volume which results in a field of view of roughly mm 2. The cameras 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 recorded images are transferred to a computer simultaneously which amounts to a data transfer rate of approximately 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 array consists of 21 lamps. In order to minimize the heat production of the LEDs, they are operated in pulsed mode, triggered by an external function generator. To further decrease the effects of heat disturbances from LEDs on the flow, blue LEDs with a dominant wavelength of 455 nm are chosen; the heat absorption coefficient of water at this wavelength is close to its minimum [125]. A rough estimate (Beer-Lambert Law) of the heat absorbed by the water from the LEDs (Q LEDs ) confirms the negligible effects of LEDs compared to the mean heat supplied through the copper plate, Q LEDs /Q Copper < 1%. Preliminary experiments reveal that the light from LEDs reflects on the copper plate and tubular walls which deteriorates the capability of the tracking of the particles. To cope with these reflections, we use high-pass filters in combination with fluorescent particles. Figure 2.5 shows the working principle of this combination. The fluorescent Polyethylene particles (supplied by Cospheric Co., USA) have a mean density of 1002 kg/m 3. The Polyethylene particles have a diameter of µm, thermal conductivity of 0.5 W/(m K) (the same as water) and heat capacity of 1900 J/(kg K) (slightly less than half compared to water). The ratio between the particle response time and the Kolmogorov time scale is the Stokes number St = τ p /τ η, where τ p = d 2 pρ p /(18νρ f ) is the so-called particle response time with ρ f the fluid density. For the present experiments, the Stokes number is small, St ; the particles can be considered tracers. The fluorescent particles emit light at a wavelength of 600 nm which provides a sufficient upshift with respect to the illumination wavelength (455 nm), resulting in easy separation of direct and fluorescent light. The cameras are equipped with high-pass filters (OG-570, Schott Glass) which filter out the spurious reflections with a wavelength below 570 nm. Furthermore, the quantum efficiency of the cameras is best in the range 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, the same body as used by Ref. [30], see Figure 2.4. In total, 81 dots are manufactured on the target. The ETH code calculates the extrinsic and intrinsic parameters based on the calibration images. The calibration is carried out at ambient temperature. However, the refractive index of water depends on temperature. The measurement error due to the refractive index fluctuations is calculated by performing long-time measurements of stationary particles. A small plus-shape geometry has been manufactured with five holes each filled with many tracer particles, see Figure 2.6. The plus-shape geometry is connected to a circular base via a 1 mm bar and placed inside the convection cell. In order to be sure that the LSC does not affect

34 2.1. EXPERIMENTAL SET-UPS 21 High pass filter Flow & particles Figure 2.5: The working principle of the filter. The blue arrows are the illumination light at a wavelength of 455 nm while the red arrows are the emission light of fluorescent particles at a wavelength 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 standard deviation of the position fluctuations in the horizontal and vertical directions are found to be approximately 6 and 20 µm, respectively, when a typical temperature gradient ( T = 10 K) 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 fluctuations (6 and 20 µm) are generated. Next, these sets of randomly generated numbers are added 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 accelerations are subtracted from the velocities and accelerations computed based on the original data (the data before adding the noise due to the temperature dependence of the refractive index of water). The standard deviations of the acceleration difference are less than 0.1 mm/s 2 in xy and 0.18 mm/s 2 in z directions. The standard deviations of the velocity difference are less 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, is placed on the rotating table. The rotating table is controlled from adjacent room for safety reasons. 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 maximum vertical misalignment of the surface table is found to be 7.5 µm, measured at the edge of the table. The data of the digital water-level measurements confirm that the table can rotate at constant angular velocities with no appreciable angular accelerations [30]. The residual angular acceleration found to be always below /s 2. The effects of any possible table vibration on the PTV system have been measured: these effects are negligible [30]. Further details 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 velocity and tremendous errors in the acceleration signal given that they must be reconsidered with discrete derivative schemes in post-processing. Different approaches reducing these errors, including Epps method [38], Gaussian filters [108] and cubic polynomial fitting [96], have been tested. The cubic polynomial fitting described in Ref. [96] and used by e.g. Refs.

35 22 CHAPTER 2. METHODS AND PARAMETERS Figure 2.6: The plus-shape geometry with five holes on the black plus geometry. A zoom window 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 and subsequent time steps, from t N t to t + N t: the filter length is 2N + 1. The first and last N points of a long trajectory are discarded. Following this approach, it is possible to define the raw position of a point at time t in the i direction as x i (t) and it is expressed as x i (t) = c i,0 + c i,1 t + c i,2 t 2 + c i,3 t 3 + e(t), (2.1) where e(t) is the noise. The constants c i are determined as c i = (A T x i ) T (A T A) 1, (2.2) where and 1 t N t t N t 2 t N t 3 1 t (N 1) t t (N 1) t 2 t (N 1) t 3 A =...., 1 t + N t t + N t 2 t + N t 3 c i = c i,0 c i,1 c i,2 c i,3.

36 2.1. EXPERIMENTAL SET-UPS 23 Position, velocity and acceleration after filtering are expressed as x i (t) = c i,0 + c i,1 t + c i,2 t 2 + c i,3 t 3, u i (t) = c i,1 + 2c i,2 t + 3c i,3 t 2, a i (t) = 2c i,2 + 6c i,3 t, (2.3) respectively. Low-pass filters are used to eliminate the background noise (high frequency signals). However, depending on the filter length, it might remove too little or too much of the high frequency signal. Therefore, the velocity/acceleration signal depends on the filter length: long filter length (large N) results in an further smoothing out the signal and underestimation of the velocity/acceleration signal, while short filter length (small N) results in an overestimation of the velocity/acceleration signal. In order to check the signal dependence on the filter length, we calculate the acceleration variance, a 2 = n 1 a2 i /n with n the number of statistics (the mean acceleration is always zero for all measurements), for different filter lengths. Figure 2.7 shows the acceleration variance as a function of time period over which the filter is applied, nondimensionalized by the local Kolmogorov time scale. The local Kolmogorov time scale is defined as τ η = ν/ɛ [126] where ɛ is the local kinetic energy dissipation rate, calculated from our DNS data. In the figure τ f = (2N + 1)/f where f is the camera frequency. As can be seen from the graph, the acceleration variance always depends on the filter length: filtering influences both noise and real signal. For small τ f /τ η, the acceleration variance decreases dramatically with increase in τ f /τ η. Beyond some threshold, there exists an exponential decay; the threshold is shown as a red circle and square in Figure 2.7 for xy and z directions, respectively. Voth et al. [178] suggested that we can fit a curve through the data points of the form ( g(τ f /τ η ) = A (τ f /τ η ) B + Cexp Dτ f /τ η + E (τ f /τ η ) 2), (2.4) with A, B, C, D and E the fit parameters. Then, an approximation of the acceleration variance can be achieved when τ f /τ η = 0, i.e. a 2 = g(0) = C. The C values for xy and z directions are shown in the figure as C xy and C z, respectively. The dashed lines in Figure 2.7 show the exponential term in Equation (2.4). Voth et al. found that the suggested acceleration variance is overestimated by almost 10% compared with the corresponding acceleration variance from numerical simulations [178]. Ni et al. also followed the same procedure for calculation 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 exponential decay starts is the closest to the DNS data: τ f /τ η 0.4 for x and y directions and τ f /τ η 0.65 for the z direction, see the red symbols in Figure 2.7. The differences between this method and Voth s are negligible; we found an approximately 8% overestimation of C compared to our DNS data. However, the proposed approach is computationally less expensive since the filtering is carried out for only one filter length. The filter length in the z direction is larger than that in x and y directions because the cameras are located on top of the cell and the z direction is perpendicular to the plane in which the cameras are located.

37 24 CHAPTER 2. METHODS AND PARAMETERS xy z a 2 (mm/s 2 ) C z C xy τ f /τ η Figure 2.7: The acceleration variance for Ro = 2.38 at the cell center for different filter length nondimensionalized by the Kolmogorov time scale, τ η. The dashed lines show only the 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 experimental 3D-PTV set-up, and post-processing of the trajectories (low-pass filters). Although 3D-PTV naturally provides the data in the Lagrangian frame of reference, it can provide useful information in the Eulerian frame as well. The Eulerian data give access to the velocity gradient tensor. Switching from Lagrangian to Eulerian frames of reference is not straightforward and different approaches are proposed in Refs. [97, 30, 98]. These approaches include the least squares method described in Ref. [98], the finite difference method and different convolution methods described in Refs. [97, 30], and interpolation on a regular grid [30]. As will be shown in the following, none of the methods is capable of fully retrieving the velocity gradient tensor in the current experimental approach. However, the interpolation on a regular grid is observed to be the superior approach in terms of data quality and ease of use. We shall only show the results of this approach. The number of velocity data points per time step plays a crucial role in achieving accurate results for all methods. As mentioned before, in the cubic polynomial method N data points at the beginning and end of each trajectory are discarded: the beginning and end of each trajectory are prone to higher errors. In addition, in the cubic polynomial method, all trajectories 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 velocity data points per time step plays a crucial role. In order to maximize the number of velocity data points per time step, the velocities of each trajectory are calculated using a second-order central difference method instead of the cubic polynomial method. Note that the secondorder central difference method can be performed over short trajectories as well. In addition,

38 2.1. EXPERIMENTAL SET-UPS 25 there is no need to discard the beginning and end of each trajectory. Therefore, the number of velocity data points per time step increases. The velocities from central difference method are interpolated on a regular grid. The interpolated velocities on the regular grid are smoothed by a low-pass cubic polynomial filter, the same as the one discussed before, performed on each fixed grid point over time. For the cubic polynomial filter N = 15 is chosen, with 2N + 1 the filter length. as The velocity gradient tensor can be decomposed into the vorticity and strain rate tensors u = 1 2 ( u + ( u)t ) ( u ( u)t ) = S + Ω, (2.5) with Ω and S the antisymmetric (vorticity tensor) and symmetric (rate-of-strain) parts of the velocity gradient tensor, respectively. The double lines on Ω stand for tensor and the superscript T stands for the matrix transpose. The vorticity tensor, given by Ω = 1 0 ω z ω y ω z 0 ω x, 2 ω y ω x 0 is calculated using Stokes theorem, that gives the relation between vorticity and circulation by Γ = u dl = ( u) da = ω da (2.6) with ω = (ω x, ω y, ω z ) = u the vorticity vector, l the path of integration around a surface A. For example, for the calculation of ω z from velocity data on a regular grid, we have [131] with ω z = Γ i,j 4 X Y Γ i,j = 1 2 X(u i 1,j 1 + 2u i,j 1 + u i+1,j 1 ) (2.7) Y (v i+1,j 1 + 2v i+1,j + v i+1,j+1 ) 1 2 X(u i+1,j+1 + 2u i,j+1 + u i 1,j+1 ) 1 2 Y (v i 1,j+1 + 2v i 1,j + v i 1,j 1 ) (2.8) the circulation, X and Y the grid spacing in x and y directions, subscripts i and j the grid point numbering (see Figure 2.8) and u = (u, v, w) the velocity vector. The rate-of-strain tensor is given by 1 ɛ xx 2 γ xy 1 2 γ xz S = 1 2 γ 1 yx ɛ yy 2 γ yz 1 2 γ 1 zx 2 γ zy ɛ zz with ɛ mm = u m / x m and γ mn = γ nm = ( um x n, + un x m ), (m, n) (x, y, z) and m n.

39 26 CHAPTER 2. METHODS AND PARAMETERS Figure 2.8: Contour for the calculation of Γ i,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 γ xy = 1 8 X (v i+1,j 1 + 2v i+1,j + v i+1,j+1 ) Y (u i 1,j+1 + 2u i,j+1 + u i+1,j+1 ) 1 8 X (v i 1,j 1 + 2v i 1,j + v i 1,j+1 ) 1 8 Y (u i 1,j 1 + 2u i,j 1 + u i+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 which can be found in the corresponding references. Two different evaluation criteria are used to check the quality of the velocity gradient tensor: the trace of the velocity gradient tensor should be zero (i.e. mass conservation for incompressible fluid is satisfied, u = 0), the Lagrangian acceleration is equal to the sum of the Eulerian acceleration and the convective term (i.e. a L = Du Dt = u t + u u). Note that these criteria can also depend on the original velocity signal from the Lagrangian data 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 contribution in the error comes from the Eulerian-retrieving scheme. The first criterion is examined by the joint-pdf of u x + v w y and z for the interpolated velocity on a regular grid, see Figure 2.9(a-b) for Ro = and Ro = 0.1 near the top, respectively. The correlation coefficient

40 2.1. EXPERIMENTAL SET-UPS 27 for variables a and b is defined as C = ab a2 b 2, which can vary between 0 and 1. The value 0 stands for two uncorrelated variables and 1 is for two perfectly correlated variables. The correlation coefficient for a = u x + v w y and b = 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 decreasing Ro 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 signal to noise ratio. a) b) dw dz dw dz du dx + dv dy du dx + dv dy 200 Figure 2.9: Joint-PDFs of du dx + dv dw dy and dz for (a) Ro =, and (b) Ro = 0.1 close to the top plate. The color bars indicate the number of occurrence. The correlation coefficients are 0.9 and 0.2 for Ro = and Ro = 0.1, respectively. The second criterion is evaluated by the joint-pdf of the Lagrangian acceleration and the sum of the Eulerian acceleration and convective term, a L = Du Dt = u t + u u. The joint-pdf provides a comprehensive check for the Eulerian data as it includes spatial and temporal derivatives. Figure 2.10(a-c) shows the joint-pdfs for x, y and z directions for Ro = near the top. The correlation coefficients are found to be 0.9, 0.9 and 0.6 for x, y and z, respectively. Figure 2.10(d-f) shows the joint-pdfs for x, y and z directions for 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 correlation coefficient is lower in the z direction due to the camera positioning and associated higher error in the z direction, as mentioned before. As can be seen from the correlation coefficients, the velocity gradient tensor is accurately

41 28 CHAPTER 2. METHODS AND PARAMETERS ) /a rms a) x ) /a rms b) y ) /a rms c) z ( Du Dt ( Dv Dt ( Dw Dt ) /a rms d) x ( Du Dt ( u t +u u)/arms x ( u t +u u)/arms x ) /a rms e) y ( Dv Dt ( v t +u v)/arms y ( v t +u v)/arms y ) /a rms f) z ( Dw Dt ( w t +u w)/arms z ( w t +u w)/arms z Figure 2.10: Joint-PDFs of the Lagrangian acceleration and Eulerian acceleration plus advective 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 between z = 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 observation volume as for Ro = ). The color bars indicate the number of occurrences. The correlation coefficient are 0.9, 0.9 and 0.6 (0.9, 0.9 and 0.7) for x, y and z directions, respectively, for Ro = (Ro = 0.1). retrieved for large Ro. However, the background rotation introduces difficulties to retrieve some components of the velocity gradient tensor. Therefore, it is not possible to retrieve accurately all nine components of the velocity gradient tensor for high background rotation rates. However, we will show in Chapters 6 and 7 that the velocity gradient tensor can be utilized, with an acceptable accuracy, to identify the vortex populated regions and to study autocorrelations of the vertical component of the vorticity Time-resolved particle image velocimetry system In addition to 3D-PTV, two-dimensional time resolved particle image velocimetry (PIV) is used. For PIV measurements the same convection cell, as described above, is used. One CCD camera, the same camera as the one used in the PTV system, is placed above the convection cell. The illumination is provided by a dual-head Nd:YAG laser (Quantel CFR400) with a wavelength of 532 nm. Negative cylindrical and positive lenses are used to achieve a laser light sheet with a thickness of less than 1 mm. The camera and the laser are triggered by an external 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 pixels with 50% overlap, resulting in vector points in the field. The observation view of the camera is approximately

42 2.2. EXPERIMENTAL PROCEDURE AND FLOW CHARACTERISTICS mm 2 ; the vector spacing becomes x = y 1 mm. A two-dimensional fast Fourier transform (FFT) is used for cross-correlation of the two corresponding windows from two subsequent images. Spurious vectors are detected and replaced using the universal outlier detection test described in Ref. [185]. This test uses the velocity of the neighboring points to define whether the velocity of the desired point is acceptable. The seeding particles for PIV are Polyamid particles (Dantec Dynamics) with a mean diameter of 50 µm and a density of 1030 kg/m 3. For PIV, the camera is placed vertically above the cell and the cylindrical sidewalls are not in the observation view of the camera. As a result, no light reflection is observed. Thus, there is no need for usage of high-pass filters and fluorescent particles. Low-pass filtering and vertical vorticity component The PIV experiments are carried out in a time-resolved fashion. Furthermore, the PIV measurements are performed for low Ro numbers: the flow motions are slow and possess long correlations. This a priori knowledge can be employed to further decrease the background noise. Therefore, a low-pass cubic polynomial filter, similar to the one described for 3D-PTV data in Section 2.1.2, is used. However, this filter only enhances the quality of the PIV data and does not change any conclusions drawn from these measurements. For the PIV measurements, the filter is applied to the velocity vector at each fixed point in space over time. The half-filter length is chosen to be N = 15 for PIV measurements. The 2D-PIV measurement gives access to the horizontal velocity field and consequently the vertical component of the vorticity vector, ω z. We use the same methodology as that of 3D-PTV to retrieve the vertical component of the vorticity, see Equations (2.6) and (2.7). 2.2 Experimental procedure and flow characteristics The experiments have been performed with water. Water from a boiler ( 70 C) is kept at ambient temperature for a couple of days to degas. The used fluorescent particles (for 3D-PTV experiments) are hydrophobic in nature, therefore they are soaked with a Tween 60 solution (Sigma Aldrich) for 24 hours. The statistically steady state in RRBC is assessed by considering two typical time scales: the adjustment time scales when a new temperature difference or when a new rotation rate is applied. For each experiment, after setting the temperatures for top and bottom plates, the system is left undisturbed for at least half an hour to adjust to the new temperature settings. The reported value in the literature is τ 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 durations are considered. The typical time scale for fluid adaptation to the background rotation is given by the Ekman time scale, τ E = H/ νω [46]. The actual time used in the experiments for adaptation to the background rotation is at least 3τ E. 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 approximately mm 3 (x, y, z), see Figure The horizontal intersection (in the xy plane) of the measurement volume is rectangular since the camera sensors consist of (square) pixels. The measurement volumes at the center and near the top cover the height 0.375H < z < 0.625H and 0.75H < z < H, respectively. The center of the

43 30 CHAPTER 2. METHODS AND PARAMETERS measurement domain is on the cylinder (or z) axis (which coincides with the rotation axis and gravity). The measurement volume near the top extends up to the plate surface. Depending on the region of interest, e.g. the boundary layer or bulk, the volume can be divided into smaller vertically separated subvolumes. Throughout this thesis, the choice of the subvolumes are clearly expressed accordingly. The 2D-PIV experiments, on the other hand, are performed only at the height of z = 0.8H. z = H Ω Top Center g H z = 0 Figure 2.11: The RBC cell and the measurement volumes. The blue and red surfaces are the cold and hot plates, respectively. The purple cubes are the measurements volumes with size of mm (x, y, z). The direction z is along the gravity, perpendicular to the cold and hot plates. The 3D-PTV experiments are performed in two different fashions: high particle concentration (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 trajectories at each time step (more data points in space at each time step) but the trajectories are shorter compared to the LPC experiments: the reason behind this has been discussed in Section under Fundamentals of 3D-PTV algorithm. Therefore, for example, for the interpolation on a regular grid HPC measurements are used while for the Lagrangian velocity and acceleration autocorrelations LPC measurements are used. LPC (HPC) experiments are performed for a duration of approximately 300 (30) minutes for each rotation rate. However, it is not possible to continuously perform PTV measurements with high frequency for such a long period due to technical limitations. Therefore, the experiments 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 are performed 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 particles are tracked at each time step. On the other hand, in the HPC experiments, an average number of 1600 ( 2700) randomly distributed particles are tracked at each time step in the cell center (close to the top plate). Tables 2.1 and 2.2 summarize the parameters for the measurement series in the cell center