Mark Brouns

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    Abstract

    In medical applications it is important that aerosols reach the alveolarzone of the respiratory tract, to be effective. Before this region is reached,the aerosols have to pass the upper airway (UA), starting with the mouthand a 90-degree bend leading into the trachea. The UA geometrys irreg-ularity and constrictions (such as the vocal cords) potentially affect thedeposition of inhaled aerosols.

    The goal of this dissertation was to develop from the available CT-scans,a simplified yet realistic human UA geometry. From this computer gener-ated UA geometry, a suitable physical model was created for Particle ImageVelocimetry (PIV) measurements. Via Reynolds similitude, a seeded water-glycerine mixture, matching the refraction index of the transparent modelwas measured in a central sagittal plane of the model at four flow rates (cor-responding to 10, 15, 30 and 45 L/min air breathing flow rate). These PIVmeasurements were compared with Computational Fluid Dynamics (CFD)simulations of the fluid phase. Of the various available turbulence mod-els that were combined with the Reynolds Averaged Navier-Stokes (RANS)equations to compute the fluid phase in this UA model, the k

    Shear-

    Stress Transport (SST) turbulence model best reproduced the experimentalresults.

    For the simulation of the particle phase, particles with diameter rangingfrom 1 to 20 micrometer were tracked in a Lagrangian frame of referencethrough the obtained converged flow field. Simulations of total depositioncompared well with experimental deposition data, for particles with a valuefor the non-dimensional parameterStk.Re0.37 higher than 0.1(Stokesnumber.Reynoldsnumber0.37 > 0.1). Total deposition of particleswith a smaller value was overpredicted, probably due to exaggerated turbu-lence simulated at low flow rates. Simulations of local particle deposition

    patterns in the UA model were much more realistic than local depositionpatters previously reported in simplified geometries. In particular, simu-lated mouth deposition more closely resembled that obtained experimen-tally in realistic upper airway geometries. The influences of gravity, of car-rier gas, of degree of turbulence at the model entrance, and of consideringnon-steady flow at particle injection were discussed.

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    Finally a clinical problem of tracheal stenosis was tackled by introducingvarious degrees of constriction in the upper third of the trachea in the UAmodel. CFD simulations of pressure drops across the stenosis allowed usto propose a rule of thumb from which pressure drops over the stenosiscan be estimated, simply on the basis of breathing flow and stenosis crosssection. In addition, the best-fit exponent in the power law that relatespressure drop to breathing flow was proposed as a diagnostic tool in thenon-invasive monitoring of tracheal stenosis patients.

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    Acknowledgements

    Doing a Phd is a strenuous and cumbersome work, which I was not ableto finish without the help and support of many people. Therefore I want tothank everyone who contributed in any way to the making of my thesis.

    I wish to thank the head of the research group Fluid Mechanics andThermodynamics, Prof. Dr. Ir. Chris Lacor for giving me the opportunityto work on this very interesting field of research. Im particular grateful

    that he gave me the chance to develop my own ideas, which helped me togrow as a researcher.

    Secondly, I would like to thank my co-promotor, Prof. Dr. Sylvia Ver-banck, who always helped me to focus not only on the computational partof the research but also the physiological side of the research. For not beinga CFD-specialist, she posed many questions, which helped me to look in acritical way to the obtained results.

    Thirdly, I would like to thank my colleagues and former colleagues atthe Fluid Mechanics research group: Kris Van den Abeele, Sergey Smirnov,Patryk Widera, Santosh Jayaraju, Ghader Ghorbaniasl, Matteo Parsani,Mahdi Zakyani Roudsari, Dean Vucinic and former colleagues Jan Ram-

    boer and Tim Broeckhoven. First, Tim and Santhosh thanks a lot for theproofreading of this dissertation. I know you both had a lot work and read-ing someones Phd can be quite strenuous. Tim, also thanks for sharing anoffice during 4 years, we had a lot of fun together. Jan, you always helpedme to put things into perspective. Things are quite different without ouroffice-ninja. Kris, you are now almost 2 years in the department but it looksa lot longer... We had and hopefully will have a lot of fun together. Sergey,Patryk, Ghader, thanks for the nice discussions during the coffee breaks. Iwould express my gratitude to Alain Wery for his unlimited help with allthe computer problems and lab problems, I encountered over the past years

    and also our secretary Jenny Dhaes for her administrative support.I also want to thank my family and friends, who always reminded methat there was more in life than upper human airways.

    Last but certainly not least I want to thank my girlfriend Annemie. Youalways put up with my bad mood after an unsuccessful day and gave me

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    the courage to finish my Phd. Like you supported me during my Phd, I willhelp you to go through the upcoming difficult time.

    Mark Brouns, Vilvoorde,Oktober 2007

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    Contents

    Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

    Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

    1 Introduction 11.1 Asthma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 History of asthma . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Asthma in the world . . . . . . . . . . . . . . . . . . . . . . 2

    2 Anatomy of the Human Respiratory Tract 4

    2.1 The respiratory system . . . . . . . . . . . . . . . . . . . . . 42.1.1 Function of the respiratory system . . . . . . . . . . 42.1.2 The structure of the respiratory system . . . . . . . 6

    3 Theoretical Background of Particle Image Velocimetry (PIV) 123.1 Historical background of fluid measurements . . . . . . . . . 123.2 The principle of particle image velocimetry . . . . . . . . . . 153.3 Mathematical background of PIV evaluation . . . . . . . . . 17

    3.3.1 Auto-correlation . . . . . . . . . . . . . . . . . . . . 193.3.2 Cross-correlation of a pair of two single exposed record-

    ings . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3.3 Optimization of the correlation . . . . . . . . . . . . 22

    3.4 Evaluation of PIV images . . . . . . . . . . . . . . . . . . . 233.4.1 Error estimation . . . . . . . . . . . . . . . . . . . . 263.4.2 Detection of spurious vectors (outliers) . . . . . . . . 26

    3.5 Tracer particles . . . . . . . . . . . . . . . . . . . . . . . . . 29

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    4 Theoretical Background of Computational Fluid Dynamics(CFD): The Particle Phase 324.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.2 Geometric properties of particles . . . . . . . . . . . . . . . 32

    4.2.1 Particle size . . . . . . . . . . . . . . . . . . . . . . . 334.3 Dilute and dense flows . . . . . . . . . . . . . . . . . . . . . 344.4 Phase coupling . . . . . . . . . . . . . . . . . . . . . . . . . 344.5 Modeling two-phase flows . . . . . . . . . . . . . . . . . . . 35

    4.5.1 Eulerian continuum approach . . . . . . . . . . . . . 354.5.2 Lagrangian trajectory approach . . . . . . . . . . . . 36

    4.6 Mass Balance . . . . . . . . . . . . . . . . . . . . . . . . . . 364.7 Momentum Balance . . . . . . . . . . . . . . . . . . . . . . . 37

    4.7.1 Interphase Force . . . . . . . . . . . . . . . . . . . . 374.7.2 Body Force . . . . . . . . . . . . . . . . . . . . . . . 41

    4.8 Stochastic trajectory approach . . . . . . . . . . . . . . . . . 41

    5 Theoretical background of Computational Fluid Dynamics(CFD): The Fluid Phase 435.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.2 History of CFD . . . . . . . . . . . . . . . . . . . . . . . . . 445.3 The Navier-Stokes equations . . . . . . . . . . . . . . . . . . 45

    5.4 The Reynolds Averaging . . . . . . . . . . . . . . . . . . . . 465.5 Turbulence modeling . . . . . . . . . . . . . . . . . . . . . . 485.5.1 k turbulence models . . . . . . . . . . . . . . . . 495.5.2 k turbulence models . . . . . . . . . . . . . . . . 535.5.3 The Reynolds Stress Model (RSM) . . . . . . . . . . 56

    6 State-of-the-Art of the Research in Upper Airway Geome-tries 58

    7 PIV of the Flow in a Model of the Upper Human Airways 697.1 Creation of the phantom . . . . . . . . . . . . . . . . . . . . 69

    7.1.1 Creation of the computer model of the upper humanrespiratory tract . . . . . . . . . . . . . . . . . . . . 697.1.2 Creation of the upper respiratory airways phantom . 72

    7.2 Flow measurements . . . . . . . . . . . . . . . . . . . . . . . 777.2.1 The experimental-set-up . . . . . . . . . . . . . . . . 77

    7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

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    7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

    8 CFD of the Flow in a Model of the Upper Human Airways 938.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

    8.1.1 Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . 938.1.2 Numerical Method . . . . . . . . . . . . . . . . . . . 96

    8.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . 998.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

    9 Numerical Particles Deposition Study in a Model of theUpper Human Airways 115

    9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1159.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1169.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

    9.3.1 Validation and total deposition analysis . . . . . . . . 1189.3.2 Local deposition analysis . . . . . . . . . . . . . . . . 1229.3.3 Influence of gravity . . . . . . . . . . . . . . . . . . . 1249.3.4 Influence of turbulence (Eddy Interaction Model) . . 1269.3.5 Influence of the carrier gas (Heliox vs Air) . . . . . . 1289.3.6 Influence of unsteady flow rate . . . . . . . . . . . . . 131

    9.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

    10 Clinical Application: Tracheal Stenoses 13810.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 13810.2 Materials and methods . . . . . . . . . . . . . . . . . . . . . 14010.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14310.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

    11 Conclusions and Future Challenges 153

    Bibliography 170

    A List of publications 171

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    List of Figures

    2.1 representation of the breathing cycle of a human [4] . . . . . 5

    2.2 representation of an alveolus [4] . . . . . . . . . . . . . . . . 62.3 picture from the cilia in the respiratory tract [5] . . . . . . . 72.4 frontal view of the mouth with the different structures [2] . . 82.5 side view of the mouth and pharynx [6] . . . . . . . . . . . . 92.6 detailed front(left) and top (right) view of the larynx [8] . . 10

    3.1 Leonardo da Vinci sketched various flow fields over objectsin a flowing stream . . . . . . . . . . . . . . . . . . . . . . . 13

    3.2 Ludwig Prandtl next to his famous water tunnel . . . . . . . 143.3 Experimental arrangement for PIV in a wind tunnel . . . . . 163.4 The three modes of particle image density: (a) low (PTV),

    (b) medium (PIV) and high image density (LSV) . . . . . . 173.5 Schematic representation of geometric imaging . . . . . . . . 183.6 Example of an intensity field I . . . . . . . . . . . . . . . . . 193.7 Schematic representation of the auto-correlation of the inten-

    sity field I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.8 The intensity fieldIrecorded at timetand the intensity field

    I

    recorded at time t + t . . . . . . . . . . . . . . . . . . . 213.9 Schematic representation of the cross-correlation of the in-

    tensity fields I andI

    . . . . . . . . . . . . . . . . . . . . . . 223.10 Idealized linear digital signal processing describing the func-

    tional relationship between two successively recorded particleimage frames . . . . . . . . . . . . . . . . . . . . . . . . . . 243.11 Measurement uncertainty in digital cross-correlation PIV eval-

    uation with respect to varying particle image diameter . . . 273.12 Arbitrary example of a PIV measurement result containing

    spurious displacement vectors . . . . . . . . . . . . . . . . 28

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    3.13 Light scattering by a (from top to bottom) 1m, 10m and30 m glass particle in water . . . . . . . . . . . . . . . . . . 31

    4.1 Particle size distribution in gas-solid flows (after Soo, 1990) . 334.2 Map for particle-turbulence modulation (after Elghobashi,

    1994) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.3 Drag coefficient for spheres as a function of particle Reynolds

    number (after Schlichting, 1979) . . . . . . . . . . . . . . . . 384.4 Drag coefficient computed with the different formulations for

    spheres as a function of particle Reynolds number; rightpanel shows a zoom . . . . . . . . . . . . . . . . . . . . . . 40

    6.1 geometry developed by Katz and Martonen [68] . . . . . . . 596.2 geometry used by Corcoran and Chigier [27] . . . . . . . . . 606.3 geometry developed by Zhang et al. [145] . . . . . . . . . . . 636.4 geometry developed by Stapleton et al. [115] . . . . . . . . . 65

    7.1 Creation of realistic geometry . . . . . . . . . . . . . . . . . 707.2 Comparison of flow field in realistic (left) and simplified ge-

    ometry (right) . . . . . . . . . . . . . . . . . . . . . . . . . . 717.3 side-view and rear-view of the smoothed 3D model of the up-

    per human airways with cross-sections at different locations

    of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.4 STL-model suspended in plasticine . . . . . . . . . . . . . . 767.5 metal positive placed in perspex box . . . . . . . . . . . . 767.6 Phantom of upper human airway model with the glycer-

    ine/water mixture in the pharynx . . . . . . . . . . . . . . . 797.7 scheme of the experimental set-up . . . . . . . . . . . . . . 807.8 an image pair (a. image 1, b. image 2) with the obtained

    correlation coefficient (c) . . . . . . . . . . . . . . . . . . . . 817.9 comparison of results obtained with evaluating 3000 and 4000

    image pairs; left: normalized magnitude of velocity at 1 tra-

    cheal diameter downstream the glottis for an air flow rateof 45 l/min; right normalized turbulent kinetic energy at 1.5tracheal downstream the glottis for the same flow rate . . . . 82

    7.10 streaklines and contour of normalized magnitude of velocityin the upper human airway model at 15 l/min (a) and 30l/min (b) air flow rate . . . . . . . . . . . . . . . . . . . . . 83

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    7.11 detailed view of the streaklines of velocity in the mouth at 10l/min (a), 15 l/min (b), 30 l/min (c), 45 l/min (d) air flow rate 85

    7.12 detailed view of the streaklines of velocity in the pharynx at10 l/min (a), 15 l/min (b), 30 l/min (c), 45 l/min (d) air flowrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

    7.13 Cartesian plot of the normalized axial velocity in the pharynxfor 10, 15, 30 and 45 l/min air flow rate . . . . . . . . . . . . 87

    7.14 Cartesian plots of the normalized velocity at different loca-tions in the trachea( 0.5, 1, 2 and 3 tracheal diameters down-stream the glottis) for 15 l/min and 30 l/min air flow rate . 88

    7.15 contour plots of the normalized turbulent kinetic energy (k norm)for 10 l/min (a), 15 l/min (b), 30 l/min (c) and 45 l/min (d) 90

    7.16 normalized turbulent kinetic energy at four different locationsin the trachea for all measured flow rates . . . . . . . . . . . 91

    8.1 Three dimensional view of the grid in the central sagittalplane with a detailed view of the boundary layer cells . . . 94

    8.2 Comparison of normalized magnitude of velocity (left) andnormalized turbulent kinetic energy (right) for three gridsized (400 000, 800 000 and 1 500 000 cells) at 5mm above

    the epiglottis (up) and one tracheal diameter downstream theglottis (down) . . . . . . . . . . . . . . . . . . . . . . . . . 95

    8.3 Comparison of normalized turbulent kinetic energy (right)for two different inlet boundary conditions at 5mm abovethe epiglottis (left) and one tracheal diameter downstreamthe glottis (right) at 45 l/min . . . . . . . . . . . . . . . . . 98

    8.4 contour plots of normalized velocity (left) and normalized ki-netic energy (right) of k--sst (a), k--realizable(b), reynoldsstress model (c) and experiments (d) at 15 l/min . . . . . . 100

    8.5 contour plots of normalized velocity (left) and normalized ki-netic energy (right) of k--sst (a), k--realizable(b), reynoldsstress model (c) and experiments (d) at 30 l/min . . . . . . 101

    8.6 zoom of the streaklines in the pharynx at 30 L/min: panela: k--sst, panel b: k--realizable, panel c: reynolds stressmodel and panel b: experiments at 15 l/min . . . . . . . . . 102

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    8.7 zoom of the streaklines in the pharynx at 15 L/min: panela: k--sst, panel b: k--realizable, panel c: reynolds stressmodel and panel b: experiments at 15 l/min . . . . . . . . . 103

    8.8 zoom of the streaklines in the trachea at 15 L/min: panela: k--sst, panel b: k--realizable, panel c: reynolds stressmodel and panel b: experiments at 15 l/min . . . . . . . . . 103

    8.9 comparison of velocity profiles for all tested turbulence mod-els with experiment at 5 mm above the epiglottis (a), one (b)and three (c) tracheal diameter downstream the glottis for15 l/min . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

    8.10 comparison of velocity profiles for all tested turbulence mod-els with experiment at 5 mm above the epiglottis (a), one (b)and three (c) tracheal diameter downstream the glottis for30 l/min; same legend as figure 8.9 . . . . . . . . . . . . . . 108

    8.11 comparison of turbulent kinetic energy profiles for all testedturbulence models with experiment at one (a and c) and two(b and d) tracheal diameters downstream the glottis for 15(a and b) and 30 (c and d) l/min . . . . . . . . . . . . . . . 111

    8.12 cross-sectional view of the streamlines in the mouth (a), phar-ynx (b) and trachea (c) . . . . . . . . . . . . . . . . . . . . . 112

    8.13 a three-dimensional view of the streamlines . . . . . . . . . . 113

    9.1 Inspiratory deposition efficiency: comparison of simulateddeposition with reported experimental data . . . . . . . . . . 119

    9.2 Simulated total deposition and experimental best fit as afunction of Stokes number and Reynolds number as definedin Grgic et al. [47] . . . . . . . . . . . . . . . . . . . . . . . 120

    9.3 sites of deposition: inlet tube (blue), mouth (green), pharynx(red) and larynx + trachea (yellow) . . . . . . . . . . . . . . 122

    9.4 Simulated deposition values (expressed as % of total numberof particles) in three model subparts for four different flowrates (10, 15, 30 and 45 L/min) . . . . . . . . . . . . . . . . 124

    9.5 Two-dimensional representation of individual particle depo-sition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

    9.6 Top view of the deposited 10 m particles for a flow rate of15 L/min . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

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    9.7 Total deposition in zero gravity, gravity vector under an angleof 45,gravity vector under an angle of 90, normal gravityvector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

    9.8 Comparison of total deposition between mean flow trackingand Eddy Interaction Model . . . . . . . . . . . . . . . . . . 128

    9.9 Comparison of total deposition between particles suspendedin air and heliox . . . . . . . . . . . . . . . . . . . . . . . . . 131

    9.10 Scheme of the inhalation profile and particle injection forunsteady flow accelerating through 30 L/min for FIR of 2L/s2132

    9.11 Scheme of the inhalation profile and particle injection . . . . 134

    10.1 Side view of the realistic (smoothed) 3D upper and trachealairway model including stenosis with 3D grid refinements inthe stenotic area. Inserts are a zoom of a weblike steno-sis (length 2mm) and an elongated stenosis (length 30mm).Cross-sections A-H refer to different locations along the model141

    10.2 Velocity streaklines in the model for 0%, 50% and 90% steno-sis (panel A, B and C, respectively). Dark grey areas repre-sent regions where the velocity is equal or higher than 80% ofthe peak velocity anywhere in the model. Inserts represent

    3D streamlines in the stenotic area . . . . . . . . . . . . . . 14310.3 CFD simulated pressures along the model with a stenosis of

    50%, 75%, 85% and 90% (weblike stenosis; solid circles) andwith no stenosis (indistinguishable from 50% stenosis). Forthe 90% constriction, an elongated stenosis was also consid-erd (open circles, triangles and squares refer to 10mm, 20mmand 30mm stenosis length); inlet flow is 30 L/min . . . . . . 144

    10.4 Panel A: CFD simulated pressure drops over the stenosis as afunction of degree of stenosis constriction. Open and closedsymbols refer to CFD simulations with air breathing at 15

    and 30 L/min flow rate; crosses refer to Heliox breathing at30 L/min. The line plots are corresponding pressure dropestimates obtained by use of Equation 3 with K=1.2 for 15L/min (solid line) and 30 L/min (dotted lines). Panel B: Kvalues for use in Eq.(10.7), obtained for all simulation condi-tions of panel A (see text for details). . . . . . . . . . . . . 146

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    10.5 CFD simulated pressure drops between model inlet and out-let for different flows up to 60 L/min, in the case of no stenosis(open triangles), of 60% constriction (solid squares) and of85% constriction (solid circles). The line plots are the corre-sponding best-fit power laws, leading to power values of 1.77(no stenosis), 1.92 (60% stenosis), and 2.00 (85% stenosis) . 149

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    List of Tables

    7.1 The measured air flow rates and corresponding Reynolds

    numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789.1 Influence on total injected particles on the total deposition

    percentage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1189.2 Comparison of total deposition between mean flow tracking

    and Eddy Interaction Model . . . . . . . . . . . . . . . . . . 1299.3 Comparison of total deposition for Heliox and air as carrier

    gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1309.4 Comparison of total deposition for steady and unsteady flow

    with a FIR of 2 L/s2) . . . . . . . . . . . . . . . . . . . . . 1339.5 Comparison of total deposition for steady and unsteady flow,

    where the particles are released at the moment the flow ratereaches the maximum value . . . . . . . . . . . . . . . . . . 134

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    List of Symbols

    Latin symbolsXi position vector of particle i at given timex the coordinates in the image plane, positionM magnification factor

    I image intensity field

    V0 transfer function giving the light energy of the particle imagein the correlation volume

    R1 auto-correlation of a single exposure image

    ai interrogation area is separation vector in the correlation planeRC convolution of the mean intensities of I

    RF fluctuating noise component

    RC convolution of the mean intensities of I

    RP self-correlation peakD constant displacement of all particles inside the interrogation

    volume

    d constant displacement of all particles inside the interrogationareaRD cross-correlation of particle images from the first exposure

    with identical particle images of the second exposured(t) distance traveled of particle images within the pulse separa-

    tion time

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    t pulse separation timeU, u (instantaneous)fluid velocity (in PIV it is assumed that fluid

    velocity is equal to the tracer particle velocity)

    dp particle diameter

    Up, up particle velocity

    r residual

    q normalized tracer diameter

    Dp distance between particles

    l resolution length (size of the computational cell)L characteristic length scale of the flow field

    m mass of a particleF aerodynamic interphase force, inviscid fluxGb body force, generation of turbulent kinetic energy due to

    buoyancyFD drag forceA exposed frontal area of the particle

    CD drag coefficients surface area of a sphere, having the same volume as the par-

    ticleS actual surface of the particle, scalar measure of the deforma-

    tion tensor, local cross sectiong gravity vectorrp particle position

    TL lagrangian integral time

    Le eddy length scale

    tcross particle eddy crossing time

    tji viscous stress tensor

    sij strain rate tensor

    p pressure

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    t timek turbulent kinetic energy

    Gk,Gk generation of the turbulent kinetic energy due to mean veloc-ity gradients

    YM contribution of the fluctuating dilitation in compressible tur-bulence

    Sk source term in the turbulent kinetic energy equation

    S source term in the turbulent dissipation rate equation

    S source term in the specific turbulent dissipation rate equation

    S generation of the specific turbulent dissipation

    Yk dissipation of the turbulent kinetic energy

    Y dissipation of the specific turbulent dissipation rate

    D cross diffusion in the specific turbulent dissipation rate equa-tion

    Gij generation of turbulent kinetic energy due to buoyancy

    DT,ij turbulent diffusion in RSM

    DL,ij molecular diffusion in RSM

    Cij convection term in RSM

    Pij stress production term in RSM

    Fij production of system rotation in RSM

    Mt turbulent Mach number

    Q volume flow rate

    y+ dimensional wall distance

    y distance to the nearest wall

    y friction velocity

    DH hydraulic diameter

    f damping function

    V volume, local velocity

    L central sagittal line of the geometry

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    u,v,w three velocity componentsp, P pressureV viscous flux

    E total energy

    Cp constant-pressure specific heat capacity

    T temperature

    Greek symbols state of the ensemble at a given time

    point spread function of the imaging lens

    tot absolute measurement errors

    resid residual errors of the measured image displacements

    sys systematic errors

    U average value of the nearest neighbors of scalar U

    relaxation time

    p particle density

    dynamic fluid viscosity

    kinematic fluid viscosity

    wavelength of the incident light, mean free path of the flowfield

    particle shape factor, pressure strain term in the ReynoldsStress Model

    e characteristic lifetime of an eddy

    ij specific Reynolds stress tensor

    t turbulent viscosity

    turbulent dissipation rate

    specific turbulent dissipation rate

    k turbulent Prandtl number for the turbulent kinetic energy

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    turbulent Prandtl number for the turbulent dissipation rate turbulent Prandtl number for the specific turbulent dissipa-

    tion ratek inverse turbulent Prandtl number for the turbulent kinetic

    energy

    inverse turbulent Prandtl number for the turbulent dissipationrate

    turbulent Prandtl number for the specific dissipation rate

    k effective diffusivity of the turbulent kinetic energy

    Y effective diffusivity of the specific turbulent dissipation rateij turbulent dissipation rate tensor

    Y deposition efficiency

    w wall shear stress

    1 preconditioning matrix

    Subscripts

    air of air

    mixture of the water/glycerine mixture

    inlet at inlet

    norm normalized

    rms root mean square

    ref reference

    loc local

    g gauge

    Other symbols

    . averaged

    . favre averagedxix

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    . fluctuating part

    Abbreviations

    LDV Laser Doppler Velocimetry

    LDA Laser Doppler Anemometry

    PDI Phase Doppler Interferometry

    PIV Particle Image Velocimetry

    PTV Particle Tracking Velocimetry

    LSV Laser Speckle Velocimetry

    FFT Fast Fourier Transform

    CFD Computational Fluid Dynamics

    DNS Direct Numerical Simulation

    LES Large Eddy Simulation

    DES Detached Eddy Simulation

    RANS Reynolds Averaged Navier Stokes

    EIM Eddy Interaction Model

    RNG Renormalization group

    SST Shear Stress Transport

    RSM Reynolds Stress Model

    MRI Magnetic Resonance Imaging

    CT Computer Tomography

    STL stereolithography

    DPI Dry Powder Inhaler

    pMDI pressurized Metered Dose Inhaler

    FIR flow increase rate

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    Dimensionless numbersRe Reynolds number

    Stk Stokes number

    De Dean number

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    Chapter 1

    Introduction

    1.1 Asthma

    Asthma is a chronic disease of the respiratory tract in which the airwayoccasionally constricts, in response to one or more triggers, by things suchas exposure to an environmental stimulant (or allergen), cold air, warm air,moist air, exercise or exertion, or emotional stress. The airway often be-comes inflamed, and is lined with excessive amounts of mucus. In children,the most common triggers are viral illnesses such as the ones that cause

    the common cold [120]. This narrowing causes symptoms such as wheez-ing, shortness of breath, chest tightness, and coughing. Between episodes,most patients feel well but can have mild symptoms and they may remainshort of breath after exercise for longer periods of time than the unaffectedindividual.

    1.2 History of asthma

    The word asthma is derived from the Greek aazein, meaning to exhale

    with open mouth, to pant. The expression asthma appeared for the firsttime in the Iliad (written by Homerus), with the meaning of a short-drawnbreath, but the earliest text where the word is found as a medical term isthe Corpus Hippocraticum. However it is difficult to determine whetherin referring to asthma, Hippocrates and his school (460-360 B.C.) meantan autonomous clinical entity or simply a symptom. He thought that the

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    spasms associated with asthma were more likely to occur in tailors, anglers,and metalworkers.The best clinical description of asthma in later antiquity isoffered by the master clinician, Aretaeus of Cappadocia (1st century A.D.).The numerous mentions of asthma in the extensive writings of Galen (130-200 A.D.) appear to be in general agreement with the Hippocratic texts andto some extent with the statements of Aretaeus [80].

    Moses Maimonides, a renowned 12th century rabbi, philosopher, andphysician practiced in the court of Saladin (1137-1193), sultan of Egyptand Syria. He wrote a treatise on asthma for his royal patient, Prince Al-Afdal. He noted that his patients symptoms often began with a commoncold, especially in the rainy season, forcing him to gasp for air until phlegmwas expelled [3].

    Jean Baptiste Van Helmont, a Belgium physician during the 16th cen-tury, wrote that asthma originated in the pipes of the lungs. In the 17thcentury, Bernardino Ramazzini, an Italian physician, noted a connection be-tween asthma and organic dust. During the early 1800s asthma was rarelymentioned in medical literature. At that time 5 patients with asthma con-stituted a case report. Asthma was first described in the medical literaturein the mid-1800s and still considered rare at that time [3].

    The use of bronchodilators started in 1901. Early 20th century studiesfocused on the premise that asthma was a psychosomatic disease, and this

    side-tracked the major advances which loomed on the horizon. Eventuallyresearchers would refute these erroneous psychiatric theories, and provethat asthma was a physical condition. It was not until the 1960s that theinflammatory component of asthma was recognized, and anti-inflammatorymedications were added to the regimen [3].

    1.3 Asthma in the world

    According to World Health Organization (WHO) estimates, 300 million

    people suffer from asthma and 255 000 people died of asthma in 2005.Asthma is the most common chronic disease among children. It is not just apublic health problem for high income countries but it occurs in all countriesregardless of the level of development. Over 80% of asthma deaths occursin low and lower-middle income countries. Asthma deaths will increase byalmost 20% in the next 10 years if urgent action is not taken [9].

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    A recent study of the European Federation of Allergy and Airway (EFA)Diseases Patients Associations, presented in Brussels on March 5 2007 in-volved 1,300 people with severe asthma in five European countries: France,Spain, Germany, Sweden and the UK and was conducted by NOP health-care and coordinated by Asthma UK on behalf of EFA. The study revealedthat 90% of the 6 million people in Europe with severe asthma are not re-ceiving optimum care, leaving 1.5 million of them to live in constant fearthat their next attack could be fatal. All these people are missing at leastone of the five treatment goals recommended by the Global Initiative forAsthma (GINA) [33]. Unfortunately, patients who are inadequately treatedare more likely to suffer frequent attacks and are at greater risk of hospital-ization and in some cases, death [122], [124].

    Asthma affects 30 million people across Europe [39] and costs healthcareservices approximately 17.7 billion euro a year [11], a cost which could besignificantly reduced if access to effective patient centered care was a rulenot a privilege across Europe. In Western Europe one person dies everyhour as a result of severe asthma [95] , but 90% of these deaths could beprevented with effective management of the disease [10].

    Many respondents of the survey are optimistic that new more effectivedrugs will be available in future (71%). Approximately one in three, 29%,say investing in research is the single most useful thing their government

    could do to improve their asthma.With this in mind, this dissertation aims to make a small but hopefully

    a meaningful contribution to the control of this disease.The dissertation is organized as follows: first an explanation is given of al

    essential parts of the upper human airways, followed by a theoretical back-ground of Particle Image Velocimetry, the applied computational methodsof the fluid phase and particle phase. In the next chapter the creation ofthe model of the upper human airways is described and the measurementsof the flow in this model. Followed by chapter 8 where the applied fluidflow simulations are compared with the experiment. The simulation of theaerosol deposition in the developed model is described in chapter 9. To

    conclude, a numerical study, which is carried out on the developed modelwith inclusion of a tracheal stenosis, is discussed.

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    Chapter 2

    Anatomy of the Human

    Respiratory Tract

    This chapter describes the terminology of the respiratory tract

    2.1 The respiratory system

    Each human being breaths about 20000 times every day, this results into

    600 million breaths at the age of seventy. At rest an average adult breathsabout 15 kg of air (10000 to 20000 liters) each day. This section providesa simplified explanation on how breathing works and a description of thelung physiology.

    2.1.1 Function of the respiratory system

    The main function of the respiratory system is to supply the blood (andcells) with oxygen and to remove the carbon-dioxide from the blood (andcells).

    The inhalation process is driven by the diaphragm. When it contracts,the contents of the abdomen is pushed downwards and the thorax or ribcageexpands. This creates a larger thoracic volume and thus an under pressurein the lungs with respect to the atmospheric pressure at the level of themouth and nose. This makes the air, which contains about 21% oxygento travel down into the deeper lung. During forced inhalation, the external

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    intercostal muscles and accessory muscles come into play and further expandthe thoracic volume.

    The exhalation process is passive. The lungs are by nature elastic andbecause of the recoil from the stretch of inhalation, the air is pushed out-wards until the pressure in the thorax reaches equilibrium with the at-mospheric pressure. During forced exhalation the expiratory muscles, theabdominal muscles and internal intercostal muscles force the air to flow outof the lungs. Figure 2.1 shows a representation of the respiration cycle.

    Figure 2.1: representation of the breathing cycle of a human [4]

    As already been mentioned, the primary function of the respiratory sys-tem is to exchange gases. This exchange happens at the level of the alve-oli (figure 2.2), where oxygen attaches to the hemoglobin molecules in theblood. These molecules transport oxygen from regions of supply (the alve-oli) to the region of demand (the blood and cells).

    Carbon dioxide travels from the metabolically active cells into the capil-laries. The concentration of carbon-dioxide in the cells is much greater than

    in the capillaries, this process of movement of materials from a higher to alower concentration is called diffusion. According to Ficks law of diffusiondiffusion, the rate of gas transfer through a sheet of tissue is proportionalto the tissue area and inversely proportional to the sheet thickness. In theblood, water is combined with carbon-dioxide to form bicarbonate. This re-moves the carbon-dioxide from the blood and keeps the concentration levels

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    Figure 2.2: representation of an alveolus [4]

    of carbon-dioxide low in the blood, so that the diffusion process can con-tinue. In the alveolar capillaries, bicarbonate combines with a hydrogen ionto form carbonic acid, which breaks down into water and carbon-dioxide.This carbon-dioxide diffuses into the alveoli.

    2.1.2 The structure of the respiratory system

    In order to reach the alveolar zone of the lungs, where the gas exchangetakes place, the air has to pass several complex structures. In the following

    section a detailed description of the most important structures, which aredealt with in this dissertation, is given.usually, when normal breathing, air enters the respiratory tract through thenostrils or nares and flows through the nose. The open spaces in the noseare celled nasal passageways or nasal cavities, which act as a filter of dustand other foreign material. The nasal cavities are covered with tiny hairs

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    called cilia (figure 2.3).

    Figure 2.3: picture from the cilia in the respiratory tract [5]

    The cilia move back and forth pushing the particles and mucus eithertoward the the pharynx or the nostrils. The nasal cavities also warm upand moisten the incoming air. Going down the nasal cavity the air passes

    through the nasopharynx, which extends to the level of the uvula.The other opening of the respiratory tract, the mouth has the same

    function also warms up and moistens the air but to a lesser degree, becauseair travels much faster through the mouth compared to the nose. At theroof of the mouth,the hard palate, a thin, bony plate of the skull is situated.This is followed by the soft palate or the palatine velum. The uvula is infact a soft process that extends from the posterior edge of this soft palate.

    At the end of the mouth, the air travels through the fauces (Latin pluralfor throat) to the oropharynx, which is also connected to the nose by thenasopharynx. The fauces is the hinder part of the mouth and are regarded

    as the two pillars of mucous membrane. One being anterior, known as thepalatoglossal arch and the second is posterior, the palatopharyngeal arch.Between these two arches is the palatine tonsil, which protects the bodyfrom infection as shown on figure 2.4.

    The oropharynx extends from the uvula to the epiglottis and is linedwith stratified squamous epithelium that protects against abrasion due to

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    Figure 2.4: frontal view of the mouth with the different structures [2]

    the high volume of food intake. Coming from the oropharynx, air movesinto the laryngopharynx, which extends to the opening of the larynx and theesophagus, which leads to the digestive system, as can be seen on figure 2.5.The laryngopharynx is like the oropharynx lined with squamous epithelium.The nasa-, oro- and laryngopharynx form together the pharynx.

    Now the air flows into the complex structure of the larynx. It consistof an outer casing of nine cartilages connected to each other by musclesand ligaments. The most well known and also being the largest and mostsuperior cartilage is the thyroid cartilage or the Adams Apple. The most

    inferior is the cricoid cartilage, which forms the base of the larynx. Thealready mentioned epiglottis is also one of the nine cartilages and it preventsmaterial (e.g.food) from entering the larynx by covering its opening. Thesix remaining cartilages are stacked in two pillars between the cricoid andthyroid cartilage. Two pairs of ligaments, known as vestibular or false vocalfolds (the superior pair) and the (true) vocal cords (the inferior pair) are

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    Figure 2.5: side view of the mouth and pharynx [6]

    situated in this casting of cartilage. The function of the vestibular foldsis to prevent air from coming from the lungs and prevent material fromentering the larynx, like the epiglottis. Speech is produced by letting thevocal cords vibrate with moving air. The greater the amplitude of the

    vibration, the louder the sound will be. Male adults usually have longervocal cords and therefor have lower voices. The opening between the vocalcords is called the glottis, which varies within the breathing cycle. Thelarynx is connected to a membraneous tube of approximately 12 cm longand 2 cm wide, called trachea or windpipe. It consists of dens regularconnective and smooth muscle reinforced with 15 to 20 C-shape pieces of

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    cartilage, which form the anterior and lateral side of the trachea. It has aprotective function and maintains an open passageway for air. The posteriorwall contains no cartilage and consists of a ligamentous membrane andsmooth muscle, which can alter the diameter of the trachea. The esophaguslies immediately posterior to the cartilage-free wall of the trachea. Thetrachea leads down the thoracic cavity where it divides into the right andleft bronchus. The right bronchus is shorter, wider and more vertical thanthe left bronchus. This difference is caused by the heart which is situatedmore to the left than the right side of the chest. The subdivision of thebronchi are primary, secondary and tertiary divisions. In all, they divide 16times into even smaller bronchioles. These lead to the respiratory zone ofthe lungs, which consists of respiratory bronchioles, alveolar ducts and thealveoli where finally, the gas exchange finds place. The surface available forgas exchange in an average adult is between 100 to 140 m2.

    Figure 2.6: detailed front(left) and top (right) view of the larynx [8]

    The respiratory tract from nasal cavities to the smallest bronchi is cov-ered by a layer of sticky mucus, secreted by the epithelium and small ductedglands. Foreign particles (e.g. dust) which hit the walls of the tract aretrapped in this mucus. Once the foreign material is stuck in the mucus, ithas to be removed. This is carried out by the cilia on the epithelial cellswhich move continually up and down the tract. The cilia in the trachea

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    and in the bronchi push the mucus, with the particles towards the pharynxwhere it is swallowed.

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    Chapter 3

    Theoretical Background of

    Particle Image Velocimetry(PIV)

    In this chapter the principles and theoretical background of particle imagevelocimetry is described.

    3.1 Historical background of fluid measure-ments

    Since the early ages mankind is interested in the observation of nature.Even now children place obstacles in a flow and observe the most fascinat-ing structures, or by throwing pieces of wood in a river allowing them toroughly estimate the velocity of the river. The well known artist and scien-tist Leonardo Da Vinci made very detailed drawings of the flow structureswithin a water flow by simple observation (figure 3.1).

    Ludwig Prandtl made a great step forward in the observation of flow

    structures. In contrast with Da Vinci, he not only observed the flow pas-sively but he tried to extract information out of well planned experiments.He designed a water tunnel (figure 3.2) and by adding mica particles on thesurface of the water he studied the structures of the flow in steady as well asunsteady flow behind wings and other objects. By changing the model sys-tematically , flow velocity and other parameters of the experimental set-up

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    Figure 3.1: Leonardo da Vinci sketched various flow fields over objects ina flowing stream

    Prandtl gained insight to the basic features of fluid flow. However this was avery interesting experiment which described for the first time methodolog-ically fluid flow, no quantitative measurements were possible at that time.However these efforts only gave a qualitative view of the flow.

    The oldest well-know technique to quantatively measure fluid flow is thepitot tube, named after French Engineer Henry de Pitot(1695-1771). He wasthe first person to measure velocity with an upstream pointed tube, whilethe French engineer Henry Darcy (1803-1858) developed most of the features

    of the instrument we use today. Pitot tubes are used in wind tunnels,airplanes, etc. The major disadvantages of this measurement method are:

    the tube has to be placed into the fluid flow, and thus disturbs theflow

    only one point can be simultaneously measured and thus it can takea lot of time to measure the complete profile of a flow

    In the late 1950s hot-wire anemometers were introduced . As the nameimplies, hot-wire (-film) anemometers uses a very thin wire, which is placed

    into the flow and through convective cooling by the flow of a wire whichis heated by an electric current, the flow velocity can be measured. Mosthot-wires have a diameter of 5m and a length of approximately 1 mm andare made of tungsten and can take thousands of velocity measurements persecond, allowing to study the details of fluctuations in turbulent flow. Themajor drawbacks of this method are:

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    Figure 3.2: Ludwig Prandtl next to his famous water tunnel

    the wire has to be calibrated before each experiment which can becumbersome

    the disturbance of the flow by the probe

    like the pitot tube, only one point can be simultaneously measured

    The temporal resolution is a lot higher than the pitot tube and the distur-bance is smaller.

    In the mid 1960s, the Laser Doppler Velocimetry (LDV), also calledLaser Doppler Anemometry (LDA) was developed . This is an optical tech-nique to measure flow velocity in a desired point without disturbing theflow. The operating principle of LDV is based on sending a highly coherentmonochromatic light beam (laser beam) toward the target, collecting thelight reflected by small particles in the target area, determining the change

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    in frequency of the reflected radiation due to the Doppler effect, and relat-ing this frequency shift to the flow velocity of the fluid in the target area.The major advantage of this method over hot-wire anemometry is that isnon-intrusive. Nowadays systems which can measure the three componentsof velocity at once become more and more available. However this methodalso has its disadvantages:

    the desired target area has to be reachable by the laser beams the major cost of a system

    difficult to measure close to a surface

    only 1 point can be measured at onceTo overcome this last drawback, other measuring methods where de-

    veloped, called particle-imaging techniques( Planar Laser-Induced Fluores-cence, Laser-Speckle Velocimetry, Particle Tracking Velocimetry, MolecularTracking velocimetry and Particle Image Velocimetry). An overview of thesemethods are described in literature ([76], [13], [55], [88] [37], [35]). SinceParticle Image Velocimetry (PIV) is the only applied method in this work,it will be discussed in detail.

    3.2 The principle of particle image velocime-

    try

    The principle of PIV is based on the measurement of the instantaneous ve-locity of tracer-particles which are carried by the fluid flow through thedetection of the particle displacement with a sophisticated stroboscopicmethod. These particles have to be illuminated in a plane of the flow atleast twice within a short time interval (figure 3.3). The light scattered bythe tracer particles is recorded on a photographic negative or on two sep-

    arate frames on a special cross correlation CCD (Charge Coupled Device)camera positioned at right angles to the light sheet.For evaluation the digital recording is divided in small interrogation

    areas. The local displacement vector for the image of the tracer particlesreflection is determined for each interrogation area by means of statisti-cal methods. It is assumed that particles moved homogeneously within an

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    Figure 3.3: Experimental arrangement for PIV in a wind tunnel

    interrogation area. One image recording contains more than thousand in-terrogation area. The projection of the vector of the local flow velocity intothe plane of the light sheet is calculated taking into account the magnifi-

    cation at imaging and the time delay between two illuminations. The timeof the illumination should be very short in order to avoid streaks made bythe reflection of the illuminated particles. The time between two pulsesshould be long enough to have a displacement of the particles between tworecordings but also not too long to avoid out-of-plane displacement of theparticles. As already mentioned, tracer particles have to be added into theflow. Particles should faithfully follow the motion of the fluid and have toscatter the light very effectively. Therefore a high energy light source (laser)has to be used for generation of the light sheet.

    Another important issue is the density of images of tracer particles on

    the PIV recording. Qualitatively three different types of image density canbe distinguished ([13]), which is illustrated in figure 3.4.

    In the case of low image density (figure 3.4a), the images of individualtracer particles can be detected. Low image density requires tracking meth-ods for evaluation. This situation is often referred to as Particle TrackingVelocimetry (PTV). In the case of medium image density (figure 3.4b), it is

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    Figure 3.4: The three modes of particle image density: (a) low (PTV),(b) medium (PIV) and high image density (LSV)

    no longer possible to identify image pairs by visual inspection. However itis possible detect the individual particle images on each recording. Mediumimage density needs statistical methods to evaluate the PIV recordings. Incase of high image density ((figure 3.4c), it is not even possible to detect theindividual particle images as they overlap in most cases and form speckles.Therefore this situation is called laser speckle velocimetry (LSV).

    3.3 Mathematical background of PIV evalu-

    ationIn the previous section, the principle of PIV was explained. The next sectionwill go more into detail of the PIV evaluation. As mentioned before, theobtained images are divided into interrogation areas and those areas ofsequential images of PIV recordings are statistically evaluated. A detailedmathematical description of statistical PIV evaluation has been given byAdrian [12], Keane [71] and Westerweel [129]. These interrogation areas arealso called interrogation windows The geometric backprojection of theseareas into the light sheet are referred to as interrogation volumes 3.5.

    A single exposure recording consists of a random distribution ofNtracerparticles:

    =

    X1X2 XN

    withXi=

    XiYi

    Zi

    (3.1)

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    Figure 3.5: Schematic representation of geometric imaging

    describes the state of the ensemble at a given time.

    Xi is the positionvector of particle i at that time.

    x = xy ,Xi= xi

    M,Yi = yi

    M (3.2)

    x refers to the coordinates in the image plane. The particle position andthe image position are related by a constant magnification factor M. Theimage intensity field of a single exposure can be expressed by:

    I=I(x , ) =(x) Ni=1

    V0(Xi)(

    x xi ) (3.3)

    where V0 is the transfer function giving the light energy of the image of an

    individual particle within the interrogation volume, the function repre-sents the point spread function of the imaging lens, it is assumed identicalfor each tracer particle. This can be rewritten as [96]:

    I(x , ) =Ni=1

    V0(Xi)(

    x xi ) (3.4)

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    Figure 3.6 an example of an intensity field I of a single expose image isshown.

    Figure 3.6: Example of an intensity field I

    3.3.1 Auto-correlation

    The auto-correlation of a single exposure image can be defined as follows:

    R1(s , ) = I(x , )I(x + s , ) (3.5)where represents the spatial average over interrogation areaa1. Equation3.5 can be approximated by:

    R1(s , ) = 1

    a1

    Ni=j

    V0(Xi)V0(

    Xj)

    a1

    (x xi )(x xj + s)dx

    + 1

    a1

    Ni=j

    V20(Xi)

    a1

    (x xi )(x xj + s)dx (3.6)

    wheres is the separation vector in the correlation plane. The terms i =jrepresent the correlation of different particle images and therefore randomlydistributed noise in the correlation plane. The terms i = j represent thecorrelation of each particle with itself.

    Adrian [96] proposed the following decomposition:

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    R1(s , ) =RC(s , ) + RF(s , ) + RP(s , ) (3.7)

    whereRC(s , ) is the convolution of the mean intensities ofIand RF(s , )

    is the fluctuating noise component both resulting from the i= j terms.RP(

    s , ) is the self correlation peak located at position (0,0) in the correla-tion plane, resulting from the components that correspond to the correlationof each particle with itself.

    Figure 3.7: Schematic representation of the auto-correlation of the inten-sity field I

    In figure 3.7 the schematic representation of the auto-correlation of theexample intensity field Iis given. Correlation peaks (RP andRF) occur atlocations which are given by the vectorial differences between particle loca-tions. Their strength is proportional to the number of all possible differenceswhich result in that location.

    3.3.2 Cross-correlation of a pair of two single exposed

    recordingsMostly used in PIV, is the so-called cross-correlation technique. Hereby, atracer ensemble consists of two single exposure images (figure 3.8).

    If an identical light sheet and windowing characteristics are considered,the cross-correlation function of two interrogation areas can be written as:

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    Figure 3.8: The intensity field Irecorded at time t and the intensity fieldI

    recorded at time t + t

    R2 = (s , ,D) =

    i=j

    V0(Xi)V0(

    Xj+

    D)R(

    xi xj + s d)

    + R(s d)

    Ni=1

    V0(Xi)V0(

    Xj+

    D ) (3.8)

    where

    R(xi xj + s d) = 1a1a1

    (x xi )(x xj + s d)dx (3.9)

    and

    Xi =

    Xi+

    D =

    Xi+ DXYi+ DY

    Zi+ DZ

    ,d = MDX

    MDY

    (3.10)

    D is the constant displacement of all particles inside the interrogation vol-ume, s represents the separation vector in the correlation plane. The termsi

    =j are the correlation of different randomly distributed particles, consid-

    ered as noise in the correlation plane. On the other hand, the terms i= jcontain the displacement information desired. Equation 3.9 can again bedecomposed into three parts:

    R2(s , ,D) =RC(s , ) + RF(s , ,d) + RD(s , ,d) (3.11)

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    where RD(s , ,d) represents the component of the cross-correlation func-tion that corresponds to the correlation of images of particles obtained fromthe first exposure with images of identical particles obtained from the sec-ond exposure (i = j terms). For a given distribution of particles inside the

    flow, the displacement correlation peak reaches a maximum fors =d.Therefore the location of this maximum yields the average in-plane dis-placement, and thus the U and V components of the velocity. In figure3.9, the schematic of the cross-correlation of the example intensity fields I

    andI

    are given. Correlations ofx

    2 dont appear on figure 3.9 because thisparticle image is located outside the interrogation window.

    Figure 3.9: Schematic representation of the cross-correlation of the inten-sity fields Iand I

    One can also consider the correlation of a double exposed recording,but this is less important than the previous correlation. This correlationcontains more noise than the correlation of the single exposed recording.Also an ambiguity in the displacement peaks appears. The reader is referred

    to [96] for a mathematical description of this technique.

    3.3.3 Optimization of the correlation

    The following equation can define the locally measured velocity:

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    |U| =d(t)

    Mt +

    residMt

    (3.12)

    d(t) is the distance traveled by the particle images within the pulse sep-

    aration time t and resid are the residual errors of the measured imagedisplacements. These errors are not affected by the alteration of the pulseseparation time. Therefore, the second term on the right hand side of equa-tion 3.12 will increase if the pulse separation time decreases:

    limt0

    residMt

    = (3.13)

    On the other hand, the particle image displacement decreases if the pulseseparation time decreases. This leads to:

    limt0

    d(t)

    Mt = |U| (3.14)

    As been shown, the accuracy of PIV measurements can be increasedby increasing pulse separation time between the exposures. But for highvalues of the separation time the measurement noise increases. For verylarge pulse separation time, the particle displacement will exceed the extent

    of the interrogation volume.

    3.4 Evaluation of PIV images

    As already explained, some kind of interrogation scheme is required to ex-tract displacement information from a PIV recording. Initially, this interro-gation was manually performed on selected images with low density seedingwhich allowed the individual tracking of particles ([135], [32]). With com-puters and image processing becoming more and more commonplace, itbecame possible to automate the interrogation process ([34], [46], [44]).On

    images with medium seeding density it is almost impossible to detect match-ing image pairs of particle images, by visual inspection. Hence, statisticalmethods had to be developed.

    Looking from an image perspective view to two successively recordedparticle image frames, the first image can be considered as the input tosystem whose output produces the second image of the pair (3.10). The

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    Figure 3.10: Idealized linear digital signal processing describing the func-tional relationship between two successively recorded particle image frames

    input image I is converted, through the displacement function d and anadditive noise process N to the output image I

    . With both images I andI

    known, the aim is to estimate the displacement d while excluding theeffects of the noise process N. This can be done by finding locally thebest match between the images in a statistical sense, like explained in theprevious section. One way is to directly calculating the cross-correlation,where the template I is linearly shifted around in the sample I

    withoutextending over the edges ofI

    . The template I is smaller than the sampleI

    . Each choice of the sample shift, one correlation value is computed. Forshift values at which the template matches the sample, the highest valueof the correlation will be found and from that the displacement d can be

    found. This method has some drawbacks:

    the number of multiplications per correlation value increases in pro-portion to the interrogation window area.

    no rotations or deformations can be recovered by this method. no efficient method of computing (millions of multiplications)

    Alternative is to take advantage of the correlation theorem which states thatthe cross-correlation of two functions is equivalent to a complex conjugate

    multiplication of their Fourier transforms:

    R2 I I

    (3.15)

    where I and I are the Fourier transforms of the functions I and I

    . Inpractice the Fourier transform is efficiently implemented for discrete data

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    using the Fast Fourier Transform (FFT) which reduces computation fromO[N2] operations to O [NlnN] operations ([18], [94]).

    The Fourier transform is an integral over a domain extending from neg-ative infinity to positive infinity. In practise, the integrals are computedover finite domains which is justified by assuming the data to be periodic,that is, the image sample continually repeats itself in all directions. So, ifthe data of length N contains displacements exceeding half the sample sizeN/2, then the correlation peak is folded back into the correlation plane toappear on the opposite side. In this case the Nyquist theorem is violated.The solution is to reduce the laser pulse delay.

    The highest value in the correlation plane can be computed and thispeak permits the displacement to be determined with an uncertainty of 0.5pixel. But the structure of the correlation peak also contains information.Therefore peak fitting functions for sub-pixel displacement estimates wereintroduced. The three most frequently reported peak interpolation or fittingschemes are the centroid, parabolic and Gaussian (based on a three-by-threepixels kernel). The accuracy associated with those schemes has been widelystudied and detailed results from numerical and theoretical investigationswere given by several authors ([129], [91], [96]). The Gaussian scheme showsthe best performance, confirmed by several authors. If no peak fitting func-tion is used or the estimators are misused, the so-called peak-locking effect

    can occur. The displacement values are locked at integer values. There areseveral reason for this phenomena but the most important one is the par-ticle image being smaller than 1.5 pixel. Also a reduced fill factor of theimage can lead to a peak locking. The interrogation method discussed inthis section is the classical scheme suggested by Willert and Gharib ([136]).More recently advanced algorithms, like multi-pass and multi-grid were de-veloped. In the multi-pass interrogation technique ,the interrogation of theimage is repeated at least once more. In the following passes the imagesample positions are offset by the integer shift determined in the precedingpass. Once the residual shifts are less than one pixel, a re-evaluation of therespective point is no longer necessary (e.g. convergence).

    In combination with this multi-pass interrogation scheme a multi-gridscheme can be applied. With this a pyramid approach is used by starting offwith larger interrogation windows on a coarse grid and refining the windowsand grid with each pass. This is especially useful in PIV recordings withboth a high image density and a high dynamic range in the displacements.

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    In such cases standard evaluation schemes cannot use small interrogationareas without losing the correlation signal due to the larger displacements.Also sub-pixel based image shifting (image deformation) on multiple-pass-or multi-grid interrogation can be applied. This technique is a second or-der method and involves a complete deformation of the image data usingthe displacement data of the previous interrogation passes. More detailedinformation on these methods can be found in the literature ([101], [102],[103], [131]).

    3.4.1 Error estimation

    An important aspect of measurements is the estimation of the errors, whichalways arise within an experiment. The absolute measurement error, tot,can be decomposed into a group of systematic errors, sys, and a group ofresidual errors, resid:

    tot =sys+ resid (3.16)

    Systematic errors are those which arise due to the inadequacy of the statis-tical method of cross-correlation in the evaluation of a PIV record, such asthe use of an inappropriate sub-pixel peak estimator. These errors follow aconsistent trend which makes them predictable. The second type of errorsremain in a form of a measurement uncertainty, even when all systematic

    errors are removed. A very popular approach to estimate these errors isbased on numerical simulation (Monte Carlo simulations) ([96], [129], [70],[71], [72], [73]). Artificial particle images with known content is generatedand by varying only one parameter at a time the influence of that param-eter can be evaluated. In literature detailed information can be found foroptimizing a number of experimental parameters.([96], [101]). An exampleof such a result is shown in figure 3.11. It is shown that the optimal particleimage diameter is slightly higher than 2 pixel.

    3.4.2 Detection of spurious vectors (outliers)

    After automatic evaluation of the PIV recordings, often a number of spu-riousvectors are found back on processed images (figure 3.12).

    These vectors deviate unphysically in magnitude and direction fromnearby valid vectors and often appear at the edges of the data field.(near the surface of the model, at edges of drop-out areas, at the edges of

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    Figure 3.11: Measurement uncertainty in digital cross-correlation PIVevaluation with respect to varying particle image diameter

    the illuminated area). Mostly, they appear as a single incorrect vector. Thiscan be caused by insufficient particle images in the interrogation area, noiseor artifacts not due to the correlation of matched image pairs. Most of thespurious vectors can easily be detected by visual inspection, but when deal-ing with thousands of recorded images it is not obvious to deal with theseoutliers in an interactive way. These erroneous results have to be removed,certainly when wanting to calculate for instance, vorticity. Because of the

    great amount of data involved, this can only be done by an automatic algo-rithm. This algorithm must ensure with a high level of confidence that noquestionable data is stored in the data set. Several algorithms are describedin literature but only the two most used will be explained.

    The dynamic mean value operator

    This algorithms checks each velocity vector individually by comparing itsmagnitude with the average value over its nearest (mostly 8) neighbors.The velocity vector will be rejected if the absolute difference between itsmagnitude and the average over its neighbors is above a certain threshold.

    |U U0| < threshold (3.17)with|U0| the magnitude of the considered vector, Ubeing the average

    value of the nearest neighbors. Problems will arise at the edges of the datafield when there are less than eight neighbors available for comparison.

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    Figure 3.12: Arbitrary example of a PIV measurement result containingspurious displacement vectors

    The (normalized) median test

    The local median test proposed by Westerweel ([130]) showed superior per-formance in detection of outliers in case of a homogeneous and orthogonalrandom field with an single step evaluation procedure. Median filtering sim-ply speaking means that all neighboring velocity vectors are ordered linearlywith respect to the magnitude of their components and a velocity vector isvalid if the difference between the central value of this order (the median)is smaller than a threshold value:

    |U(median) U0| < threshold (3.18)

    Westerweel ([132]) proposed an adaptation to this test, that normalizes themedian residual with respect to a robust estimate of the local variationof the velocity. This method was verified on a large variety of flows anda single threshold value can be applied to effectively detect the outliers.Consider a displacement vector denoted by U0 and its eight neighbors by{U1, U2, U8} and U(median) as the median of this set (without U0). The

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    residualri is defined as:

    ri= |Ui U(median)| (3.19)This residual is defined for each vector{Ui|i = 1, , 8}. The median ofthe residuals r(median) is used to normalize the residual ofU0:

    r0=U0 U(median)

    r(median) + (3.20)

    with being a minimum normalization level. Two seemed to be theoptimal threshold value for r0 . Smaller values lead to a more stringent

    where larger values lead to less stringent outlier detection.

    3.5 Tracer particles

    As mentioned before, tracer particles must be added to the flow and theseparticles must rigorously follow the fluid flow. The quantity which describesthe time required for a particle to adjust or relax its velocity to a newcondition of forces is called the relaxation time of the particle and is givenby:

    =pd

    2p

    18 (3.21)

    where p is the density of the particle, dp is the particle diameter and is the dynamic viscosity of the fluid. The particle relaxation time, as de-fined in 3.21 is restricted to particle motion in the Stokes region, Red

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    must scatter the light sufficient enough, to obtain large enough particle im-ages. It is often more effective and economical to use particles with betterscattering behavior than to increase laser power, to increase image inten-sity. In general, light scattered by small particles is a function of the ratioof the refractive index of the particles to that of the surrounding medium,the shape, size and orientation of the particle. Also polarization and obser-vation angle have an influence on the scattered light. The Mie scatteringtheory can be applied to spherical particles with larger diameters than thewavelength of the incident light. The Mie scattering can be characterizedby the normalized diameter, q, defined by:

    q= dp

    (3.23)

    with is the wavelength of the incident light. If q is greater than one,approximatelyq local maxima appear in the angular distribution over therange from 0 to 180.

    In general, the averaged intensity increases with q2 and the scatteringefficiency depends on the ratio of the refractive index of the particles to thatof the fluid. Therefore the scattering of particles in air will be at least oneorder of magnitude larger than that of the same particles in water, becausethe refractive index of water is considerably larger than the index of air. As

    can be seen on figure 3.13, the light intensity is not blocked by the particlesbut spread in all directions and the highest intensity can be found back inthe forward scattered position. Unfortunately, this direction cannot be usedin PIV in contrast to LDA. In case of heavily seeded flows, a certain amountof the light scattered by the particles is not only due to direct scatteringbut also due to scattered light from other particles. This implies that notonly the diameter of the particles are important but also the density ofthe seeding plays an important role in the scattering efficiency. But withdenser seeded flows, the background noise and thus the noise on recordingswill increase.

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    Figure 3.13: Light scattering by a (from top to bottom) 1 m, 10m and30 m glass particle in water

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    Chapter 4

    Theoretical Background of

    Computational Fluid Dynamics(CFD): The Particle Phase

    In this chapter the principles and theoretical background of the computa-tional method of simulating particle trajectories in combination with RANS,is described

    4.1 Introduction

    Particle Laden flows forms a major class of two-phase flows in which the con-tinuous fluid (gas or liquid) and discrete particles (solid, liquid or gaseous)are treated as two different phases. Laden flows find numerous biologicaland industrial applications such as hemodynamics, flow dynamics in humanairways, biological and chemical reactors, sedimentation, filtration etc. Thepresent chapter describes the governing equation of the particle phase.

    4.2 Geometric properties of particles

    The geometric properties of particles such as size and distribution havedirect influence on the dynamic behavior of gas-solid flows.

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    Figure 4.1: Particle size distribution in gas-solid flows (after Soo, 1990)

    4.2.1 Particle size

    Each particle is implicitly supposed to be made up of large number ofmolecules (minimum of 103 104) which is a pre-requisite to define theirphysical properties. The particle size is assumed to be much higher whencompared to mean free path (average distance a molecule travels in a gas

    between collisions) of continuous phase. Quantitatively, a particle size of1m corresponds to a gas flow with particle size about 10 times more thanthe mean free path.

    Fig. 4.1 shows the distribution of particle sizes encountered in variousmultiphase systems as given by Soo [111]. It can be seen that the typicalrange of particle sizes of interest to gas-solid flows is approximately between1 m - 10 cm.

    Particle distribution

    The volume fractions of dispersed phase is relatively low and for gas-solidflows it can be roughly considered to be in the range of 102103 and thespacing between the particles to be about 10 times more than the particlesize. The specific length scales of particle laden flows should satisfy thefollowing relationship,

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    Figure 4.2: Map for particle-turbulence modulation (after Elghobashi,1994)

    particle collisions take place and hence this regime is termed as four-waycoupling.

    4.5 Modeling two-phase flows

    The two most widely used approaches for mathematical modeling of two-phase flows are Eulerian continuum approach and Lagrangian trajectoryapproach.

    4.5.1 Eulerian continuum approach

    In the Eulerian approach, the particles are treated as a second fluid whichbehaves like a continuum and the equations are developed for average prop-erties of the particles. For example, the particle velocity is the averagevelocity over an averaging volume. This approach is most suitable whenone requires a macroscopic field description of dispersed phase propertiessuch as pressure, mass flux, concentration, velocity and temperature.

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    4.5.2 Lagrangian trajectory approachThe Lagrangian approach is useful when the particle phase is so dilute thatthe description of particle behavior by continuum models is not feasible. Themotion of a particle is expressed by ordinary differential equations in La-grangian co-ordinates and are directly integrated to obtain individual tracksof particles. To solve the Lagrangian-equation for a particular moving par-ticle, the dynamic behavior of the gas phase (generally obtained by Eulerianapproach) and other particles surrounding this moving particle should bepre-determined. Since the particle velocity and the corresponding particletrajectory are calculated for each particle, this approach is more suitable to

    obtain discrete nature of motion of particles. However, to obtain statisticalaverages with reasonable accuracy, huge number of particles will have tobe tracked. Ideally, one would like to track each and every particle whichis not computationally feasible. Hence a smaller number of computationalparticles are chosen which represent the actual particles. Each computa-tional particle is regarded as a group of particles which move in the fluidphase with the same physical properties.

    The Lagrangian approach is classified into two types namely, Determin-istic trajectory methodandStochastic trajectory methodbased on the effectof turbulence. In the deterministic method, all the turbulent transport pro-cesses of the particle phase are neglected, whereas the stochastic methodtakes into account the effect of gas turbulence on the particle motion byconsidering instantaneous gas velocity in the formulation of the equation ofparticle motion.

    The flow patterns of both continuous and dispersed phase depend onthe mechanisms of mass, momentum and energy coupling. Below are thegoverning equations for a single particle with no short range interactionssuch as van der Waals force and collision forces. The momentum balance isconsidered in detail as it is of primary importance in the present study.

    4.6 Mass BalanceThe mass transfer of a particle is given by,

    dm

    dt = m (4.1)

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    Change of mass with respect to time is equal to the mass gained/lost by theparticle due to mass flux over the surface. In the present work, the particlesare not considered to gain or loose mass. So equation 4.1 becomes:

    dm

    dt = 0 (4.2)

    4.7 Momentum Balance

    The momentum transfer of a particle is given by,

    d(mup)dt

    =F + m

    Gb+ (m

    up) (4.3)

    Time rate of change of momentum of a particle is equal to the sum of

    external forces acting on it.

    F is the aerodynamic interphase force, mGb

    is the body force and mup is the momentum gained/lost by particle due tomass transfer.

    4.7.1 Interphase Force

    For an idealized sample, the aerodynamic interphase force can be consideredas the sum of Drag force, Basset force, Saffman force and Magnus force,where each force represents some feature of particle fluid interaction. Formost of the practical applications, drag force is the most significant andoften the only considered part of the interface force.

    Generally the particle moves with a different velocity than the fluidat any given point. The difference in fluid velocity (u) and the particlevelocity (up), termed as the slip velocity (u up), leads to unbalancedpressure distribution as well as the viscous stresses on the particle surfacewhich yields a resulting force called drag force. For a rigid sphere the drag

    force is given by,

    FD =CD A

    2|u up| (u up) (4.4)

    where A is the exposed frontal area of the particle to the direction of the

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    Figure 4.3: Drag coefficient for spheres as a function of particle Reynoldsnumber (after Schlichting, 1979)

    incoming flow. For a rigid sphere,

    A= d2p/4 (4.5)

    CD is the coefficient of drag which is a function of the particle Reynoldsnumber and the local turbulent intensity of the fluid. The particle Reynoldsnumber is given by,

    Rep = dp|u up|

    (4.6)

    Laws for Drag Coefficients

    Various experimental data on the drag coefficient for a single sphere atvariousRep were compiled into a single standard curve by Schlichting [105]as shown in Fig. 4.3.

    At high Reynolds number of about 3 105 the wake structure behindthe particle changes which changes the surface pressure distribution aroundthe particle resulting in steep reduction of drag coefficient and transition

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    of flow from laminar to turbulent in the boundary layer of particle surface.Mathematically, the curve can be expressed as follows,

    For Rep >1, but within the limits of transition regime, the flow is gov-erned by inertia effects.

    CD

    = 18.5

    Re0.6p2< Re

    p< 500 (4.8)

    CD = 0.44 500< Rep < 2 105 (4.9)

    Jayaraju et al. [64] used a relatively simple formulation of Schraiber etal. [106]. This formulation is valid for high Reynolds number up to 6 103. It is given by,

    CD = 24

    Rep

    1 + 0.179

    Rep+ 0.013 Rep

    0.5< Rep < 6 103 (4.10)

    Substituting 4.5, 4.6 and 4.10 in 4.4 results in the following final form

    of drag force,

    FD = 3 dp (

    u up)

    1 + 0.179

    Rep+ 0.013 Rep

    (4.11)

    In Fluent several possible laws for the drag coefficient are available. Thedrag coefficient can either be taken from

    CD = a1+ a2Rep

    + a3Rep

    2

    (4.12)

    where a1,a2, anda3 are constants that apply to smooth spherical particlesover several ranges of Re given by Morsi and Alexander [89], or

    CD = 24

    Rep(1 + b1Re

    b2p +

    b3Repb4+ Rep

    ) (4.13)

    where

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    Figure 4.4: Drag coefficient computed with the different formulations forspheres as a function of particle Reynolds number; right panel shows azoom

    b1 = exp(2.3288 6.4581 + 2.44862)b2 = 0.0964 + 0.5565

    b3 = exp(4.905 13.8944 + 18.42222

    10.25993

    )b4 = exp(1.4681 + 12.2584 20.73222 + 15.88553) (4.14)

    which is taken from Haider and Levenspiel [52]. The shape factor, , isdefined as

    = s

    S (4.15)

    where s is the surface area of a sphere having the same volume as theparticle, and S is the actual surface area of the particle. The Reynoldsnumber Rep is computed with the diameter of a sphere having the samevolume.

    Figure 4.4 show the results for the drag coefficient computed with thedifferent formulations. As can be seen, the results of the different formula-tions differ not much.

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    4.7.2 Body ForceThe body force does not depend on carrier phase. It is usually due to gravityand/or the reference system rotation and is given by,

    Gb=

    g ( rp ) 2 up (4.16)where rp is the particle position. In the present work only the gravita-

    tional force acts on the particle. So, that equation 4.16 can be simplifiedto:

    Gb=g (4.17)

    In conclusion, its worth writing down the final equations used to deter-mine the particle position and the velocity,

    drpdt

    = up (4.18)

    mdupdt

    = (FD+

    Gb) (4.19)

    4.8 Stochastic trajectory approach

    Stochastic modeling involves direct simulation of particle motion through

    a random turbulent flow field. Most stochastic approaches use the tech-nique of generating turbulent-like carrier flow field using the variables ofturbulence model which is used for carrier flow simulation.

    The instantaneous motion of particle is governed by,

    d(mup)dt

    = 3 dp (u up)

    1 + 0.179

    Rep+ 0.013 Rep

    + m

    Gb (4.20)

    The instantaneous fluid velocity u in the above equation is representedas the sum of the mean and fluctuating velocity,

    u= u+ u

    (4.21)

    The stochastic model in Fluent is based on the Eddy Interaction Model ofGosman and Ioannides [45]. Assuming isotropic turbulence, the fluctuatingvelocity is given by,

    u

    =

    2

    3k N (4.22)

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    where N is a random number drawn from a normal probability distribu-tion with zero mean and unit standard deviation. k is the turbulent kineticenergy of the flow. The chosen fluctuation is referred to a turbulent eddywhose size (length scale) and life-time(time scale) is known. The Lagrangianintegral time, TL can be approximated by

    TL = CLk

    (4.23)

    where the coefficientCLis to be determined as it is not well known. For thek model and its variants a value of 0.15 is chosen for CL. In equation4.23 =

    k can be substituted. The characteristic lifetime of an eddy isdefined either as a constant:e= 2TL (4.24)

    or as a random variation about TL:

    e = TLlog(r)