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Advisor(s): Prof. Dr. Ir. Chris Lacor

Prof. Dr. Sylvia Verbanck

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Outline

Inhaled pharmaceutical aerosols have been playing a crucial role in thehealth and wellbeing of millions of people throughout the world for severalyears. Since the mid 1950s, aerosol forms of medication have been signif-icant in treating the most common respiratory illnesses such as asthmaand chronic obstructive pulmonary disease (COPD). However, administra-tion of drugs by the pulmonary route is technically challenging and ourunderstanding of the aerosol transport in the lungs is far from complete.The main contributing factors are:

• Variable filtering effects of the upper and central airways before themedication can reach the alveolar region of the lungs where they areeventually taken up by blood.

• Considerable inter-subject variations in the airway morphology.

• Variations in inhalation techniques.

For the above reasons, devising an efficient aerosol delivery system re-quires a systematic understanding of the effect of aforementioned vari-ables on aerosol behavior in the human airways. Indeed, performing suchsystematic in-vivo measurements is not feasible. Alternatively, the use ofComputational Fluid Dynamics (CFD) has emerged as an effective tool formethodical analysis of various parameters affecting the airflow as well asaerosol dynamics in the human airways.

As will be seen in the introduction chapter of this thesis, the human air-way is mainly divided into three regions, namely the extra-thoracic (upperairway), the tracheo-bronchial, and the alveolar region. The present the-sis is focused on the study of air-breathing patterns and medical aerosoltransport-deposition in the upper airways. From CFD simulation point ofview, the upper airway region is the most challenging due to transitionalnature of the airflow. The present thesis is broadly divided into eight chap-ters. The contents and relevant significance of each chapter is briefly de-scribed below.

Chapter 1 gives a brief introduction to the human respiratory system.Physical features of each part of the airway are highlighted. Particularattention is given to the upper airway region due to its relevance in thepresent thesis. A brief introduction is given as to how the aerosols and

the human airways are linked, followed by some details of the factors andmechanisms affecting the aerosol transport-deposition in the upper air-ways.

The literature survey on the existing study of flow patterns and particletransport in the upper airway constitutes Chapter 2. The upper airway ge-ometries with varying degrees of geometrical complexities used in litera-ture are drawn out. The airway geometries are presented in the ascendingorder of their complexity, i.e., from simplest to the more complex. The mod-eling methods used for the CFD study of fluid as well as particle phase arereported. The fluid flow patterns as well as the aerosol deposition charac-teristics observed by various researchers in different upper airway geome-tries are discussed. By combining various in-vivo and in-vitro depositiondata in the human upper airways, several authors have devised relativelysimple mathematical models for predicting the amount of inhaled aerosolthat may deposit in the upper airways. All such correlations pertaining tothe upper airways and their particular limitations are discussed. In addi-tion to this chapter of literature survey which provides a brief summary ofthe most important observations, the introduction section of each chapterin this thesis also discusses the relevant literature pertaining to the con-cerned chapter.

In Chapter 3, the governing equations for the fluid phase, pertaining to themodeling methods employed in the present thesis, are discussed. The mod-eling methods include Reynolds Averaged Navier Stokes (RANS), LargeEddy Simulation (LES) as well as Detached Eddy Simulation (DES). Fi-nally, the feasibility of using each of these methodologies to study the fluid-particle dynamics in the upper airways is discussed.

Chapter 4 describes the governing equations for the particle phase followedby the main aspects of Lagrangian modeling methods employed for han-dling unstructured grids. Even though the concept of unstructured gridsexists since long, the practical applicability is still under budding stage,especially for two-phase simulations. In this view, the modeling methodsdescribed in this chapter can be seen as the first step towards applicabil-ity of unstructured grids for biomedical applications, which is by far theonly option to avoid expensive experiments for realistic geometry configu-rations. The next three chapters, i.e., Chapter 5, 6 and 7 are applications ofRANS methodology to study the fluid flow and particle deposition charac-teristics, convective mixing, and the effect of tracheal stenosis in the upperairways.

ii

In Chapter 5, the fluid flowand particle deposition in a realistic CT-extractedupper airway model is studied. RANS k ! ! turbulence model is usedfor the fluid phase, and the Lagrangian particle solver module developedin the previous chapter is used to study the aerosol deposition. Typicalsteady inhalation modes of slow (15 l/min), normal (30 l/min) and heavy(60 l/min) breathing are simulated. Micro-particle (1-20 µm) transport anddeposition is investigated. Extensive quality control tests are performed tostudy the effect of grid on fluid and particle phase; the effect of number ofaerosols on total upper airway deposition. The sensitivity of flow transi-tion on the airway complexity is demonstrated. The pronounced effect ofsedimentation at very low flow rates on the deposition of heavy particles isobserved. Based on the enhanced oral airway deposition, the need for con-sidering more realistic CT-based geometries is highlighted. The contentsof Chapter 4 and 5 are published in Jayaraju et al. [76].

In Chapter 6, the degree of volumetric dispersion undergone by a boluswhile passing through the upper airway passage is studied, both experi-mentally and numerically. In addition to aerosol bolus deposition study,which was the main focus in the previous chapter, the study of volumet-ric dispersion also offers a sensitive tool to characterize aerosol transport.Whether it is for the study of convective mixing or for medication target-ing, an aerosol bolus inhaled to any given lung depth must transit theupper airway, and it is crucial to quantitatively predict its dispersive ef-fect, which was lacking in the literature. In this chapter, the dispersiveeffects in both inhalation and exhalation modes are studied. Experimentsshowed that the upper airway induces a relatively mild dispersion on thetraversing aerosol bolus and that the dispersive coefficients during inhala-tion and exhalation modes were very similar. The experimental dispersionis found to be only 1/10th of the relatively arbitrary axial dispersion valuethat is currently being used to characterize the upper airway transit forsimulations of aerosol transport in the deeper lung. Hence this quantifica-tion of upper airway dispersion is a considerable advancement in the field.For the CFD simulations, RANS k ! ! methodology is used. The inabilityof this model in accurately predicting the dispersive effects during expi-ratory mode has been observed, highlighting the need to critically assessthe most commonly used RANS k ! ! turbulence methodology. The workcorresponding to Chapter 6 is published in Jayaraju et al. [78].

Chapter 7 deals with the effect of tracheal stenosis on the flow dynamicsas well as the ensuing aerosol dispersion and deposition. The potential ofusing aerosol boluses inhaled at normal breathing of 30 l/min to detect tra-cheal stenosis ranging 50-90% obstruction of tracheal cross sectional area

iii

is investigated. The aerosol bolus deposition efficiency and bolus disper-sion, in terms of bolus half-width (HW) or bolus standard deviation (SD),were numerically simulated as a function of the degree of stenotic obstruc-tion. The effect of aerosol particle size on bolus deposition efficiency wasalso considered. While the particle dispersion is seen to be quite insensi-tive to the stenotic constriction, 5 µm particle deposition seemed to exhibitconsiderable sensitivity, making it a probable non-invasive diagnostic toolfor the detection of tracheal stenosis. Detailed fluid flow characteristics inthe presence of stenosis is published in Brouns et al. [19].

The main objective of Chapter 8 is to test the validity of RANS, LES andDES for the description of fluid/particle behavior in an upper airway model.It was observed in Chapter 5 that the deposition percentage for the medicalaerosols which generally lie in the respirable range (1-5 µm) were consis-tently over-predicted by RANS methodology. In Chapter 6, it was also seenthat RANS was inaccurate in predicting the dispersive effects during expi-ration mode. Based on these observations, the need to switch towards moreadvanced numerical methodologies such as LES and DES is recognized.To validate the fluid phase simulations, we performed PIV measurementsin a central sagittal plane of a simplified upper airway model cast. Thesame airway model and fluid flow conditions are considered for the simu-lations. The transport and deposition of 1-10 µm particles is investigated.Extensive quality control tests have been performed. The superiority ofboth LES and DES when compared to RANS in accurately predicting fluidphase and the deposition of micro-particles pertaining to medical aerosols(1-5 µm) is demonstrated. The work presented in this chapter is publishedin Jayaraju et al. [77].

iv

Acknowledgments

First and foremost, I would like to gratefully acknowledge the enthusias-tic supervision of my promoter Prof. Chris Lacor. I particularity thankhim for our weekly technical discussions, which had a major influence onthis thesis. I am indebted to him for showing great confidence in me andalways pushing me to achieve greater heights. I can say for sure that thepast four years at VUB have been the most productive days ofmy learning.

The present thesis was simply not possible without the consistent guid-ance of my co-promoter Prof. Sylvia Verbanck. She virtually taught me ev-erything; from making me understand the physiological aspects of humanbreathing, to having those perfect final draft of articles we published to-gether. Her leadership, attention to details, urge for perfection and down-to-earth nature have set an example that I would like to match some day.

I warmly thank our system administrator Alain Wery, for his tremendoussupport from my day one at VUB. I am yet to meet someone who is so pa-tient and always ready to help others.

The support of our secretary Jenny D’haes started even before I arrived inBelgium. She was there for me, starting from filling down my admissionforms in Dutch, to organizing my PhD defense. Thanks a lot Jenny, youtruly have been great!

It is my pleasure to acknowledge my seniors, Mark Brouns and GhaderGhorbaniasl, for their valuable guidance through different phases of myPhD. While Mark taught me his tried-and-tested practical ways of ap-proaching a PhD, Ghader was like a walking handbook of Mathematicswhom I referred for various problems of mine.

I am also grateful to Kris Van den Abeele, firstly for guiding me throughvarious teaching assignments we carried out together, and secondly for go-

v

ing through my thesis and giving his valuable inputs. On the same note, Iwould like to thank Patryk Widera for sharing the office space and for thenumerous constructive discussions we had over four years.

I am very pleased to acknowledge my present and former colleagues SergeySmirnov, Matteo Parsani, Mahdi Zakyani, Willem Deconinck, Khairy El-sayed, Dean Vucinic, Nikolay Ivanov, Cristian Dinescu, Jan Ramboer, TimBroeckhoven, and Vijay Kumar Verma, for their constant support and en-couragement in all my professional endeavors.

Lastly, and most importantly, my utmost gratitude is reserved to my par-ents and family members for always being there through my good and badtimes.

vi

Jury Members

President Prof. Johan DECONINCKVrije Universiteit Brussel

Vice-President Prof. Rik PINTELONVrije Universiteit Brussel

Secretary Prof. Patrick KOOLVrije Universiteit Brussel

External Members Prof. Jan VIERENDEELSUniversiteit Gent

Prof. Gerard DEGREZUniversite Libre de Bruxelles

Promoters Prof. Chris LACORVrije Universiteit Brussel

Prof. Sylvia VERBANCKUniversitair Ziekenhuis Brussel

vii

Contents

1 Introduction 11.1 The respiratory system . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Extra-thoracic region . . . . . . . . . . . . . . . . . . . 11.1.2 Tracheo-bronchial region . . . . . . . . . . . . . . . . . 51.1.3 Alveolar region . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Aerosols and the respiratory system . . . . . . . . . . . . . . 51.3 Factors affecting pharmaceutical aerosol targeting . . . . . . 61.4 Mechanisms of particle deposition . . . . . . . . . . . . . . . 91.5 Clinical aerosol measurements (bolus tests) . . . . . . . . . . 11

2 Literature Survey 152.1 Airway geometries . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Modeling methods . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Fluid phase . . . . . . . . . . . . . . . . . . . . . . . . 192.2.2 Particle phase . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Fluid flow characteristics in upper airways . . . . . . . . . . 232.4 Particle deposition characteristics in upper airways . . . . . 272.5 Empirical deposition relationships . . . . . . . . . . . . . . . 28

3 The Fluid Phase 353.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . 37

3.1.1 Reynolds Averaged Navier Stokes (RANS) . . . . . . 403.1.2 Large Eddy Simulation (LES) . . . . . . . . . . . . . . 443.1.3 Detached Eddy Simulation (DES) . . . . . . . . . . . . 49

3.2 RANS, LES or DES ? . . . . . . . . . . . . . . . . . . . . . . . 52

4 The Particle Phase 554.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . 554.2 Modeling the particle phase . . . . . . . . . . . . . . . . . . . 604.3 Stochastic trajectory approach . . . . . . . . . . . . . . . . . . 614.4 Aspects of Lagrangian modeling . . . . . . . . . . . . . . . . . 63

ix

4.4.1 Time integration . . . . . . . . . . . . . . . . . . . . . 644.4.2 Locating particles inside a control volume . . . . . . . 654.4.3 Cell search algorithm . . . . . . . . . . . . . . . . . . . 674.4.4 Interpolation of flow variables . . . . . . . . . . . . . . 68

4.5 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 704.5.1 Inlet boundary condition . . . . . . . . . . . . . . . . . 704.5.2 Wall and Outlet boundary condition . . . . . . . . . . 73

4.6 Uncoupled and coupled calculations . . . . . . . . . . . . . . 734.7 Programming language . . . . . . . . . . . . . . . . . . . . . . 744.8 Data structures . . . . . . . . . . . . . . . . . . . . . . . . . . 754.9 Flow chart of particle solver module . . . . . . . . . . . . . . 794.10 Testcases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.10.1 Analytical solution . . . . . . . . . . . . . . . . . . . . 814.10.2 2D Planar mixing layer . . . . . . . . . . . . . . . . . . 82

5 Application I: Fluid Flow and Particle Deposition in UpperAirways 895.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.2 Model preparation . . . . . . . . . . . . . . . . . . . . . . . . . 915.3 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . 93


5.4 Quality control . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . 97


5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6 Application II: Convective Mixing in Upper Airways 1116.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.2 Materials and methods . . . . . . . . . . . . . . . . . . . . . . 113

6.2.1 Experimental methods . . . . . . . . . . . . . . . . . . 1136.2.2 Numerical methods and quality control . . . . . . . . 116

6.3 Theoretical axial dispersion coefficient . . . . . . . . . . . . . 1176.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.4.1 Experimental results . . . . . . . . . . . . . . . . . . . 1176.4.2 CFD results . . . . . . . . . . . . . . . . . . . . . . . . 120

6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1206.6 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1266.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

x

7 Application III: Tracheal Stenosis 1297.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297.2 Numerical methods & quality control . . . . . . . . . . . . . 1307.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130


7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

8 Fluid Flow and Particle Deposition in Upper Airways: LESand DES 1378.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1378.2 Model preparation & experimental methods . . . . . . . . . . 1398.3 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . 1418.4 Quality control . . . . . . . . . . . . . . . . . . . . . . . . . . . 1438.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147


8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

9 Conclusions and Perspectives 1579.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1579.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

9.2.1 Future CFD developments . . . . . . . . . . . . . . . . 1599.2.2 Future Airway model developments . . . . . . . . . . 1609.2.3 Future applications . . . . . . . . . . . . . . . . . . . . 161

List of Publications 163

Bibliography 165

xi

Nomenclature

Roman Symbols(u ! up) Slip velocity, m/sFd(u ! up) Drag force per unit particle massgx Gravitational acceleration, m/s2

up Velocity of the particle, m/sxp Position of the particleu

!iu

!j Reynolds stress tensor, m2/s2

Ui, Uj Averaged or filtered velocity components, m/sd Length scale in DES model, mA Van Driest constantC Particle number concentrationC! WALE model constantCd Drag coefficientCs Smagorinsky model constantD Dispersion coefficient, cm2/sdh Hydraulic diameter of the geometry, mdae Aerodynamic diameter of the particle, µmdmean Mean diameter of the geometry, mHW Half-width of an aerosol bolus, mlL Length of the geometry, ml" Length scale for the smallest eddy, mle Length scale of an eddy, mLs Length scale for sub-grid scales, mp Static pressure, papin Static pressure at model inlet, papout Static pressure at model exit, paQ Flow rate, l/minsij Instantaneous strain-rate tensorSD Standard deviation of an aerosol bolus, mltc Crossing time of a particle in an eddy, ste Time scale of an eddy, s

xiii

tint Interaction time of a particle with an eddy, su" Velocity scale for the smallest eddy, m/su# Frictional velocity, m/sui, uj Instantaneous velocity vector components, m/su

!

i, u!

j Fluctuating component of the instantaneous velocity, m/sUp Interpolated flow variable at any vertex pUmean Mean velocity of the flow, m/sV Volume of the geometry, m3

Vp Penetration volume of an inhaled bolus, mlGreek Symbols! Grid spacing, m" Deposition efficiencyµ Dynamic viscosity of the fluid, kg/m ! s# Kinematic viscosity of the fluid, m2/s#t Eddy viscosity, m2/s! Specific dissipation, 1/s!i Weighting function" Conservative variable$ Density of the fluid, kg/m3

$p Density of the particle, kg/m3

%" Time scale for smallest eddy, s%p Relaxation time of the particle, s%ij Viscous stresses, N/m2

%sgsij Sub-grid scale stresses& Turbulent kinetic energy dissipation, m2/s3

' Random numberDimensionless NumbersRe = $Umeandmean/µ Reynolds numberRep = $dp| u ! up |/µ Reynolds number of the particleStk = $pd2

aeU/9µdh Stokes numbery+ = u#d/# Non-dimensional distance to wallAcronymsCOPD Chronic Obstructive Pulmonary DiseaseDE Deposition efficiencyDES Detached Eddy SimulationDNS Direct Numerical SimulationLES Large Eddy SimulationLRN Low Reynolds NumberOOP Object Oriented ProgrammingRANS Reynolds Averaged Navier StokesS-A Spalart AllmarasSGS Sub Grid Scale

xiv

SST Shear Stress TransportUAM Upper Airway ModelWALE Wall Adapting Local Eddy Viscosity

xv

Chapter 1

Introduction

Contents1.1 The respiratory system . . . . . . . . . . . . . . . . . . 1

1.1.1 Extra-thoracic region . . . . . . . . . . . . . . . . . 11.1.2 Tracheo-bronchial region . . . . . . . . . . . . . . . 51.1.3 Alveolar region . . . . . . . . . . . . . . . . . . . . . 5

1.2 Aerosols and the respiratory system . . . . . . . . . . 51.3 Factors affecting pharmaceutical aerosol targeting 61.4 Mechanisms of particle deposition . . . . . . . . . . . 91.5 Clinical aerosol measurements (bolus tests) . . . . . 11

1.1 The respiratory system

Breathing is the process by which oxygen in the air is brought into thegas exchanging part of the lungs and into close contact with the blood.Blood absorbs oxygen and simultaneously gives up carbon-dioxide, whichis carried out of the lungs when air is breathed out. Fig. 1.1 shows aschematic representation of the human respiratory system as well as therespiratory pathway. The respiratory system is basically divided into threecategories namely the extra-thoracic region, the tracheo-bronchial regionand the alveolar region.

1.1.1 Extra-thoracic region

Fig. 1.2 shows a schematic representation of the extra-thoracic regionwhich consists of the nasal cavity, the mouth cavity, pharynx, larynx and

1

CHAPTER 1. INTRODUCTION

Air!enters!from!Nose orMouth

Passes!through!Nasopharynx!or!Oropharynx

Through!the!Larynx!(glottis)!

Into!the!Trachea!

Into!the!left!and!right!Bronchi!

Bronchi!further!branches!into!Bronchioles

Extra!thoracic"Region"

Tracheo!bronchial"Region"

Alveolar"RegionTerminates!in!a!cluster of Alveoli

Figure 1.1: Left: Schematic representation of the human respiratory system [1];Right: The respiratory pathway.

trachea. The extra-thoracic region is also most often referred to as theupper airway region or the nose, mouth and throat region.

Nasal cavity

The nose (nasal cavity) is the preferred entrance for outside air into therespiratory system. The nasal passage serves as a moistener, a filter, anda warm up before the air intake. While the hairs in the nostrils act as afilter for the foreign particles, the mucus and cilia (tail like projections onthe surface) collect dust, bacteria and other particles in the air. The mucusmembrane also helps in moistening the air. The blood in the capillariesjust below the mucus membrane help warm up the inspired air.

Mouth cavity

Mouth becomes the preferred method of air intake for the people who havea mouth-breathing habit or whose nasal passage may be temporarily ob-structed, as by a cold or during heavy exercise. Mouth is also the preferredpassage for the intake of aerosol medication in treating lung diseases suchas asthma. The shape of the mouth cavity varies considerably depending

2


Figure 1.2: Schematic representation of the extra-thoracic (upper) airway [46].

on the position of tongue and jaws.

Pharynx

The main function of pharynx (throat) is to collect the air coming from thenose and mouth, and pass it downstream towards trachea. The pharynxis further subdivided into nasopharynx and oropharynx. In subjects withoral allergy syndrome and related allergies, the pharynx is often a reactionsite to allergens, with common symptoms including burning and itching.

Epiglottis

The epiglottis guards the entrance to the trachea by blocking it duringswallowing so that the swallowed material is guided towards the esopha-gus and stomach.

Larynx

The larynx (glottis, voice box) houses the vocal chords and as the air isexpired, the vocal chords vibrate. Humans can control these vibrationswhich enables us to make sound. The vocal folds affect the shape andmagnitude of glottic cross-section depending on the flow rate and is seento be a crucial geometric feature to be considered in the air-flow dynam-ics study [20]. Brancatisano et al. [12] studied the movements of the

3


Figure 1.3: Electron scanning micrograph of cilia and mucus generating cells.

vocal cords during large lung volume changes in 12 normal subjects andreported substantially different glottic width and glottic area between sub-jects. The glottic width and area increased during inhalation to 10.1±5.6mm and 126±8 mm2 respectively, whereas during exhalation the lowestvalues were 5.7±0.5 mm and 70±7 mm2.

Trachea

The trachea is a tube-like structure which acts as a passage from the phar-ynx to the lungs. The trachea is kept open by cartilage rings within itswalls. The presence of cartilage rings can have a considerable effect onthe flow dynamics [122]. Trachea roughly measures 10-14 cm in lengthand 16-20 mm in diameter. Similar to nasal cavity, trachea is covered withciliated mucous membrane which acts as a filter for foreign particles. Fig.1.3 shows the electron scanning micrograph of cilia and mucus generatingcells on the surface of the airway passage. Cilia extends approximately5-10 µm from the airway surface.

4


1.1.2 Tracheo-bronchial region

The trachea further divides into two cartilage-ringed and ciliated tubescalled the main bronchi. The bronchi enter the lungs and spread into atree-like structure by further subdividing itself into the lobar bronchi, seg-mental bronchi and finally ends up becoming tiny terminal bronchioles(approximately 30,000) leading into the gas exchanging (alveolated) zone.

The tracheo-bronchial region is also referred to as the lower airways. Theextra-thoracic and tracheo-bronchial airways taken together are called the’conducting airways’ as they transport air to the gas-exchange region of thelungs. The term central airways is sometimes used to refer to the upperregions of the tracheo-bronchial airways [46].

1.1.3 Alveolar region

Each terminal bronchiole subtends an air chamber that could be likenedto a bunch of grapes. Each chamber contains many cup-shaped cavitiesknown as alveoli. The walls of the alveoli, which are only about one cellthick, are the respiratory surface. They are thin, moist, and are sur-rounded by several numbers of capillaries. The estimation is that lungscontain about 300 million alveoli and that the thin barrier with a largesurface makes it ideal for the exchange of oxygen and carbon dioxide be-tween blood and through these walls. Their total surface area is roughlyabout 70 m2.

1.2 Aerosols and the respiratory system

Aerosol is defined as a suspension of fine solid particles or liquid dropletsin a gas. Examples are smoke, air pollution, smog etc. The aerosols arerelated to human respiratory system in two ways:

Aerosol pollutants

The effects of inhaling particulate matter has been widely studied in hu-mans and animals and include asthma, lung cancer, cardiovascular dis-ease, and premature death. Particulate matter pollution is estimated tocause 22,000 to 52,000 deaths per year in the United States (from 2000)and 200,000 deaths per year in Europe.

The size of the particle is a main determinant of where in the respira-tory tract the particle will deposit when inhaled. Larger particles (> 10

5


µm) are generally filtered in the nose and throat and do not cause adverseproblems, but particulate matter smaller than about 10 µm can settle inthe bronchi and lungs and cause health problems. The 10 µm size does notrepresent a strict boundary between respirable and non-respirable par-ticles, but has been agreed upon for monitoring of airborne particulatematter by most regulatory agencies. Similarly, particles smaller than ap-proximately 2 µm tend to penetrate into the gas-exchange regions of thelung, and very small particles (< 100 nanometers) may pass through thelungs to affect other organs.

Inhaled pharmaceutical aerosols

According to World Health Organization (WHO), around 300 million peo-ple suffer from asthma and around 255,000 people died of asthma in 2005.It is predicted that the asthma deaths will increase by 20% in the coming10 years if proper care is not taken.

Inhaled medication is the preferred method of drug administration to thelung for the first-line therapy of asthma and chronic obstructive pulmonarydiseases. Asthma is a chronic condition in which the airways occasionallyconstrict, become inflamed, and are lined with excessive amounts of mu-cus. Symptomatic control of episodes of wheezing and shortness of breathdue to asthma is generally achieved with fast acting broncho-dilators, whichare basically pocket-sized pharmaceutical aerosol inhalers. While target-ing a delivery dose of inhaled pharmaceutical aerosol to the lower airways,a high percentage of it is lost due to deposition and clearance in the up-per airways. Furthermore, adverse health effects such as cellular damage,inflammation and tumor formation can occur in the upper airways poten-tially as a result of local deposition and absorption patterns [161].

For the above reasons, it is of utmost importance to understand the fluid-particle dynamics in the upper airways. The present thesis is a step forwardin our attempts to better understand this important and very interesting as-pect of biomechanics.

1.3 Factors affecting pharmaceutical aerosoltargeting

The effective targeting of the inhaled pharmaceutical aerosols to the alve-olar regions of the respiratory system depends on the following factors:

6


Geometrical complexity

The inhaled aerosol particles need to negotiate the mouth-throat struc-ture and the branching airway structures before reaching the alveolar lungzone that could benefit from aerosol therapy. The complexity of the extra-thoracic portion of the oral airway, which includes bends and sudden cross-sectional changes potentially induces considerable local medication depo-sition. The angles of branching and the diameter and lengths of differentelements of the tracheo-bronchial region further influences deposition ofthe medical aerosols. Also, the geometry of the respiratory tract is timedependent and varies during the inhalation-exhalation cycle. Consider-able differences in geometrical details exist between individuals.

Particle parameters

To reach the alveolar region of the lungs, the aerosol particles need to be incertain optimal size range called the respirable range (0.5 - 5 µm). Whilethe particles > 5 µm range tend to deposit in the extra-thoracic region, par-ticles < 0.5 µm get inhaled without depositing and exhaled right back out.The principal approach used in the existing pharmaceutical inhalation de-vices, particularly when targeting the alveolar region, is by using particlesizes near 1-5 µm, assuming that the density of the particle is close to thatof water (" 1000 kg/m3) [2].

Fig. 1.4 shows typical deposition patterns of inhaled droplets in differentregions of the respiratory tract. It should however be mentioned that thegraph should only be viewed to get a qualitative picture of the depositionpatterns in different regions of the airway tract. The actual depositionpercentages may vary depending on the inhalation flow rate and the com-plexity of the airway geometry under consideration.

Breathing flow rate

Breathing flow rates directly affect the aerosol transport/deposition. Basedon the activity of an individual, the inhalation flow rates are roughly clas-sified as: slow breathing (15 l/min), normal breathing (30 l/min) and heavybreathing (60 l/min).

Inhalation devices

Based on the working mechanism, there are three types of inhalation de-vices presently available in the market. a) metered dose inhaler, b) drypowder inhaler, and c) nebulizers. Each of these devices have their own

7


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 2 4 6 8 10 12 14 16

diameter (micrometer)

depo

sitio

n pr

obab

ility

extra-thoracic regiontracheo-bronchial regionalveolar region

Figure 1.4: The probability that inhaled droplets of different diameters will de-posit on different regions of respiratory tract as predicted by a two-way coupledhygroscopic model for a Ventolin aerosol with mass median diameter of 4 µm andgeometric standard deviation of 1.7 with 100,000 droplets/cc and room temperatureambient air of 50% RH is shown [2].

Figure 1.5: Different types of inhalers.

8


advantages/disadvantages based on the needs of individual patients.

a) Metered Dose Inhalers: It is also referred to as the pressurized metereddose inhaler (pMDI) or propellant metered dose inhaler. Fig. 1.5(a) showsa typical hand held pMDI. Pressing down the canister releases a mist ofmedicine that is breathed into the lungs. pMDI’s are currently the mostcommonly used delivery device. Like most inhalation devices, only 10%-20% of the nominal per puff dose reaches the targeted airways [109].

Most recommendations suggest that patients should slowly and fully in-hale while firing the pMDI dose. The most common difficulties with pMDI’sare failure to coordinate actuation of the device with inhalation and an in-voluntary cessation of inhalation when cold aerosol particles reach the softpalate. A means to avoid this problem is to place a spacer (ranging 50-750 ml in volume) in between pMDI and patient, such that the aerosol canbe inhaled from the spacer immediately after the pMDI is actuated. Thespacer also filters out large particles that would otherwise stick to the up-per airways.

b) Dry Powder Inhalers: Dry powder inhalers (DPI’s) are breath-actuateddevices which eliminate the co-ordination problem seen with the pMDI’s.Fig. 1.5(b) shows one such DPI. It is very similar to PMDI, except for thereduced surface area of the mouthpiece exit. DPI’s are among the most re-cent delivery devices and generally adult patients find it to be more user-friendly than pMDI’s. With DPI’s, the rate of inspiratory flow is criticaland generally, a forceful and deep breath is required for optimum outputfrom this device. This flow is also necessary to desegregate the drug parti-cles from their carrier (usually lactose).

c) Nebulizers: Nebulizers are among the oldest of inhalation devices. Atypical jet nebulizer is shown in Fig. 1.5(c). High pressure air from thecompressor is passed through nebulizer which houses a nozzle and a baf-fle. While nozzle helps in primary droplet production, the baffle filtersout larger particles before the mist of drug-containing droplets reach themouthpiece for inhalation. Nebulizers are employed chiefly for the deliveryof large bronchodilator doses during acute asthma attacks and for patientsunable to use other inhalation devices [3].

1.4 Mechanisms of particle deposition

The three main transport mechanisms acting on the particles in the res-piratory system are impaction due to inertia of the particles, sedimenta-

9


Figure 1.6: Total and regional (extrathoracic, upper bronchial, lower bronchial,and alveolar) deposition of unit-density spheres in the human respiratory tractpredicted by the semi-empirical model proposed by the International Commissionon Radiological Protection (ICRP) [6]. Density of sphere is 1000 kg/m3 and theflow rate is 18 l/min [65].

tion due to gravitational acceleration, and diffusion due to Brownian mo-tion. Fig. 1.6 illustrates the different mechanisms influencing total as wellas regional deposition of unit density spheres orally inhaled at the meanbreathing pattern of an adult male in the sitting position.

Inertial Impaction: Deposition due to impaction is directly proportionalto the mass of the individual particle, ie., the size and the density of theparticle. The flow velocity also has an influence on the inertial deposition.While deposition occurs throughout the airways, inertial impaction usu-ally occurs in the first few generations of the lung, where the air velocityis high and the airflow is generally turbulent [96].

Sedimentation: Deposition due to sedimentation is mainly due to largeparticle size and relatively long residence times of the particles which is

10


Figure 1.7: Inhaled and exhaled bolus concentration curves for a given penetrationvolume. Typical bolus characteristics such as mode and half-width are shown forboth inhaled and exhaled boluses [31].

usually the case in the last five to six generations of airways (smallerbronchi and bronchioles) and in the alveolated region of the lung, wherethe air velocity is low [90].

Diffusion: The random movement of the particles is represented by diffu-sion. The distance a particle travels by diffusional transport increases withdecreasing particle size and flow rate. The highest probability of aerosoldeposition due to diffusional displacement occurs for very small particlesinhaled into the lung periphery where the airway dimensions are small.

1.5 Clinical aerosolmeasurements (bolus tests)

Bolus dispersion technique is a widely used non-invasive experimental toolto characterize the convective gas transport which can be altered in dis-eased lungs (e.g., [13, 21, 36, 35, 121, 125, 149]). It requires the subject tobe in the inhalation mode, while a small aerosol volume (typically 50 - 70

11


Figure 1.8: Exhaled bolus concentration curves at different penetration volumes.At each penetration volume, there is a comparison between a normal patient anda patient with cystic fibrosis [11].

ml) is inspired at a predetermined moment during inhalation [152] (Fig.1.7). The penetration depth of the bolus, i.e., how deep it is transportedinto the lungs is given by the volume of air following the bolus peak (Fig.1.7). The distribution of the concentrations of aerosol during the subse-quent exhalation is then plotted as a function of penetration volume. Anexample plot of aerosol dispersion at different penetration volumes for anormal subject and a patient with cystic fibrosis is shown in Fig. 1.8. Ascan be seen, the deeper we probe into the lungs, the more broader (dis-perse) the exhaled curve becomes. It is also interesting to see that theexhaled concentration curve for the patient with cystic fibrosis is more dis-persed when compared to the normal patient at all lungs depths. In thisway, aerosol bolus behavior becomes a non-invasive diagnostic tool.

Since we will be dealing only with the extrathoracic portion of the airwayin the present thesis, the aerosol bolus is to be inhaled only to a very smalllung depth (shallow bolus). For the extrathoracic region (oral and laryn-geal part), 49.3 ml was used as the penetration volume for the numericalsimulations of aerosol transport [32].

12


The inhaled and exhaled boluses are characterized by their half-width (H)and deposition efficiency (DE). Half-width is the bolus width at one-halfof the bolus peak. The change in half-width (H) reflects the aerosol dis-persion that has occurred to the bolus transit in the airways. H is definedas,

H =!H2

ex ! H2in

"1/2(1.1)

where Hin and Hex are the inspired and expired half-widths respectively.

The deposition efficiency (DE) is obtained by,

DE =#

1 ! Np,ex

Np,in

$(1.2)

where Np,in and Np,ex are the number of inspired and expired aerosolsrespectively. The ratio Np,ex/Np,in is obtained by comparing the areas ofthe inspired and expired boluses on the plot of aerosol concentration vs.volume.

13


14

Chapter 2

Literature Survey

Contents2.1 Airway geometries . . . . . . . . . . . . . . . . . . . . . 15

2.2 Modeling methods . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Fluid phase . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.2 Particle phase . . . . . . . . . . . . . . . . . . . . . 21

2.3 Fluid flow characteristics in upper airways . . . . . 23

2.4 Particle deposition characteristics in upper airways 27

2.5 Empirical deposition relationships . . . . . . . . . . . 28

2.1 Airway geometries

The overall complexity of the upper pathway between the mouth and thetrachea, with its bends and cross-sectional changes, poses serious chal-lenges for both experimental and computational studies. Previous studieshave used the upper airway models with varying degrees of geometricalcomplexity.

Katz and Martonen [84] created a simple three-dimensional model of thelarynx (Fig. 2.1) based on morphometric measurements of replica humancasts and Weibel morphology of the tracheal dimensions. The larynx wasmodeled as a 6 cm long cylinder with a circular entrance and exit cross-sections. The apertures created by ventricular and vocal cords were mod-eled as ellipse.

15

CHAPTER 2. LITERATURE SURVEY

Figure 2.1: Simplified three-dimensional model of larynx [84].

Figure 2.2: Simplified three-dimensional model of larynx [28].

Corcoran and Chigier [28] measured the axial velocity and turbulence in-tensity, using Phase Doppler Interferometry (PDI) in a cadaver-based sim-ple larynx-trachea model (Fig. 2.2). The model consisted of a polyurethanecasting of the human larynx, connected to a glass tube with an inside di-ameter matching the tracheal diameter of the cadaver.

Even though the models of Katz and Martonen [84] and Corcoran andChigier [28] are able to provide basic information about the flow patterns,they are often too simplistic and not complete, particularly with respect tothe flow inlet conditions.

Zhang et al. [171] simulated air flow and micro-particle transport in a sim-plified, but more complete model of the upper human airways (Fig. 2.3),incorporating a bend between mouth and trachea. The model consists ofa single circular tube with local diameter variations, based on data setsprovided by Cheng et al. [25].

Stapleton et al. [143] created an ’average’ geometrical model (Fig. 2.4) ofthe extra-thoracic airways based on data from computed tomography (CT)

16


Figure 2.3: Simplified three-dimensional mouth-throat model [171].

scans, magnetic resonance imaging (MRI) scans, and direct observationof living subjects. By using these characteristic dimensions, a model ofthe extra-thoracic airways is constructed using simple geometric shapes.One advantage of this approach over using a model based on an airwaycast is that extremely high grid resolutions are not needed at the walls toresolve any small airway irregularity. However, the oral cavity region istoo simplistically modeled as a long tube-like structure with an unrealisti-cally large airspace in the mouth cavity (see also the realistic cavity due totongue position in Fig. 1.2).

Yu et al. [165] constructed an upper airway including nasal, oral, laryn-geal and the first two generations of tracheobronchial airway (Fig. 2.5).The geometry was a simplified teaching model used by the medical schoolstudents. While the nasal airways did not duplicate the exact nasal mor-phology, the oral airway was too simplistically represented. The tracheaand first two bifurcations were modeled as circular tubes.

Recognizing the need for more realistic representation of the oral cavity re-gion, Brouns et al. [19] at the Vrije Universiteit Brussel performed a multi-slice CT imaging study in five otherwise healthy never-smoker male sub-jects and constructed a more realistic representation of the airway model(Fig. 2.6).

Experimental aerosol deposition data obtained by Grgic et al. [58] on var-

17



Figure 2.5: Simplified three-dimensional upper airway model including nasal, oral,laryngeal and first two generations of tracheobronchial airways [165].

18



ious realistic upper airway casts indicate considerable intra- and inter-subject variability of the extra-thoracic airway deposition. The influenceof the inlet of the aerosol device, as well as individual mouth and tracheamorphology on local and total aerosol deposition suggests that individual-ized computation of extra-thoracic deposition would result in more accu-rate estimation of the amount of aerosol available to the deeper lung for agiven patient or for a given aerosol device. Xi and Longest [161] studied themicro-particle deposition in a) CT based realistic mouth-throat model; b)Simplified model based on the CT scans. It was concluded that the realisticgeometry provided the best predictions of regional deposition in compari-son to experimental data and hence are needed for accurate evaluation oflocalized deposition patterns. Recognizing this, Jayaraju et al. [76] at theVrije Universiteit Brussel performed RANS simulations on a truly realis-tic upper airway model (Fig. 2.7) obtained from the CT scan data.

2.2 Modeling methods

2.2.1 Fluid phase

There are rapidly growing works in the literature describing the use ofCFD for studying the fluid flow and particle deposition in the upper air-

19


Figure 2.7: Three-dimensional mouth-throat model as extracted from CT scan data[76].

ways. The numerical methods are broadly classified based on the natureof flow, i.e., laminar or turbulent. When dealing with turbulent flows, thewhole range of turbulence models and their ability in accurately predictingthe physics of turbulence comes into picture.

There is an extensive literature focused on the regions of respiratory tractwhere the flows are predominantly laminar. This is mainly to avoid thecomplications that come with turbulence modeling. Laminar fluid flow andparticle depositions in tubes and bifurcations where the Reynolds numberis only few hundred have been previously studied [69, 27, 93, 88]. Reynoldsnumber is a dimensionless number that gives a measure of the ratio of in-ertial forces to the viscous forces. In case of tubular flows, it is given byud/#, where u is inlet velocity, d is the diameter of the pipe, and # is thekinematic viscosity. At a low flow rate of 15 l/min, Yu et al. [165] studiedthe fluid flow and ultra-fine (0.001-0.01 µm) particle diffusion in the oraland nasal passages followed by larynx, trachea and main bronchi. Goodagreement with experiments were reported. Martonen et al. [100] per-formed laminar flow simulations in a larynx, trachea, and main bronchimodel to provide some insights into the basic flow features. As Stapletonet al. [143] points out, the flows in larynx and trachea are normally turbu-lent or at least transitional and hence the results of Martonen et al. [100]must be interpreted with caution.

20


Two equation RANS models are the most commonly used turbulence mod-els for predicting the fluid-particle dynamics in the upper airways. Katzet al. [83] used the standard k ! & model to understand the effect of flowrate on the flow patterns and the particle trajectories in a geometry basedon laryngeal casts (Fig. 2.1). Stapleton et al. [143] further tested theperformance of standard k ! & model in a human mouth throat geometryat a turbulent flow rate of 28.3 l/min. Significant deviations in simulatedversus experimental pressure drop were reported. Several studies, e.g.,[51, 84, 85, 120] have used standard k ! & to analyze flow patterns in thelarynx.

The poor performance of the k!&model clearly highlighted the need for anexperimentally validated, low-Reynolds-number (LRN) turbulence model,which can represent laminar, transitional and turbulent flows for the com-putational analysis of transport phenomena in upper airways covering re-alistic inhalation rates (ranging 10 and 60 l/min). In an attempt to addressthis problem, Zhang and Kleinstreuer [169] tested the performance of LRNk ! ! model of Wilcox [160] in a test conduit with local constriction, at atransitional Reynolds number of 2000. Good agreement with the experi-ments were seen when comparing velocity and kinetic energy levels acrossthe geometry.

2.2.2 Particle phase

Stochastic modeling of the particle phase involves direct simulation ofparticle motion through a random turbulent flow field. There are sev-eral approaches in developing stochastic models. Few examples in litera-ture include models based on the Langevin equation [97], random Fouriermodes [103] and pdf models [118]. However, one of the most widely usedmodel is the Eddy Interaction Model (EIM) first introduced by Hutchin-son et al. [73] and further developed by Gosman and Ioannides [53]. Eventhough several variants of EIM model have been proposed in the literature[134, 57, 54, 55, 56], the originally proposed EIM of Gosman and Ioannides[53] remains the most widely used, mainly because of its simplicity.

Katz and Martonen [83] were one of the first to study the effect of tur-bulence on particle motion within the human larynx and trachea usingk ! & turbulence model coupled with EIM of Gosman and Ioannides [53].However, no quantitative validations (e.g., comparison with experiments)were provided. Stapleton et al. [143] further tested the performance ofstandard k ! & model coupled with EIM in a human mouth throat geom-

21


etry at a turbulent flow rate of 28.3 l/min. The transport of polydisperseaerosol particles of 4.8 µm were simulated and a huge over-prediction indeposition was seen in the computations as opposed to the experiments.This was mainly attributed to the poor fluid flow predictions by k!&model.Riding on the confidence of good performance by LRN k!!model of Wilcox[160] in a constricted tube at transitional Reynolds number [169], Zhanget al. [171] performed transitional flow simulations (15, 30 and 60 l/min)in a simplified upper airway model (Fig. 2.3) using LRN k ! ! turbulencemodel coupled with EIM of Gosman and Ioannides [53]. Good agreementof the simulated micro-particle (0.001 < Stk < 1) deposition were reportedwhen compared to experimental deposition correlation function. In mostof the works that followed [87, 167, 8, 102, 170, 60, 76, 77], it has become anorm to use LRN k ! ! model in predicting the fluid-particle dynamics inthe upper airways.

Matida et al. [102] studied the deposition of monodisperse particles (1-26 µm) in an idealized mouth-throat geometry (Fig. 2.4) at inhalation flowrates of 30 and 90 l/min. LRN k!! turbulence model along with EIM basedon Gosman and Ioannides [53]was used. Contrary to the good performanceof LRN k ! ! model reported by Zhang et al. [171], the simulations ofMatida et al. [102] showed huge over-prediction in deposition percentages(>50%), even for the lowest Stokes number particles. The reason accord-ing to the authors was the assumption of isotropy in the EIM model andonce the anisotropy effects next to the walls were modeled, a better overalldeposition prediction was obtained. However, recent attempts [161, 76, 77]in predicting the micro-particle deposition in simplified as well as realisticupper airways do not reproduce such behavior. All these recent works usedLRN k ! ! model along with an isotropic EIM model and the results werecomparable to those obtained by Matida et al. [102] using LRN k!! modelalong with an anisotropic EIM model. Whether the near-wall anisotropy istaken care or not, the main observation that can be made in all the abovediscussed works is that there still remained a considerable over-predictionin deposition at smaller Stokes number range, and the particles in theseStokes number range are the ones pertaining to inhaled medication (1-5µm).

Several authors have started to explore the possibility of using large eddysimulation (LES) methods for the study of particle deposition in the up-per airway, by applying LES in relatively simple structures such as a 900

bend [16] or a constricted tube [99]. Together with LES simulations ofdeposition in a simplified mouth cavity model [101], and in a simplifiedupper airway model [79, 77], the potential of more accurately simulating

22


Figure 2.8: Lengthwise velocity vector lines for 15, 30 and 60 l/min respectively[85].

aerosol transport in the upper airways has been recognized. Jayaraju etal. [77] also explored the possibility of using DES instead of LES, as it iscomputationally less expensive.

2.3 Fluid flow characteristics in upper airways

Katz and Martonen [84] were among the first to create a three-dimensionalsimplified larynx model (Fig. 2.1) to understand the basic flow patterns at15, 30 and 60 l/min. Swirling circumferential flows through the larynx dueto changes in cross-sectional area was observed. Katz et al. [85] furtherextended this preliminary work to study the effect of glottal aperture mod-ulation on inhalatory laryngeal fluid dynamics. Similar to their previouswork, three flow rates of 15, 30 and 60 l/min were considered. It was foundthat the complex geometry produces jets, recirculation zones, and circum-ferential flows which may have a profound influence on particle depositionnear the larynx. Fig. 2.8 shows the recirculation zones formed at 15, 30and 60 l/min, due to the presence of laryngeal constriction. It is a knownfact that the glottis has different shapes and cross-sectional areas at dif-ferent moments during the respiratory cycle. In order to access this effect,Renotte et al. [120] have studied the effect of pseudo-time-varying glot-tic aperture on the flow conditions at quiet breathing. In particular, two

23


glottic shapes were considered: one representing the glottal opening dur-ing inhalation and the other during exhalation. The geometry used was avery simplified larynx-trachea model that closely resembled the one usedby Katz and Martonen [84] (Fig. 2.1). Minor differences were outlined be-tween inhalatory and exhalatory flow profiles. This perhaps is due to sim-plistic representation of larynx-trachea (which is basically a constrictionin a straight tube). Brouns et al. [20] studied the influence of a circular,elliptical and triangular shape glottis on the flow dynamics in a simplifiedupper airway (Fig. 2.4) at a quiet breathing rate of 15 l/min. Even thoughconsiderable variations in the flow patterns downstream of glottis wereobserved, the total pressure drop was seen to be more dominated by thecross-sectional area than by its shape.

Corcoran and Chigier [28] used Phase Doppler Interferometry to charac-terize axial velocity and turbulence intensity contours in the tracheal sec-tion of a cadaver-based larynx-trachea model (Fig. 2.2). The flow was char-acterized for steady state flow at three Reynolds numbers (1250, 1700, and2800). Reverse flows with significant velocities were noted in the anteriortrachea within one diameter downstream of the larynx, for all three flowcases. The cross-sectional area of the reverse flow regions was larger forthe lower Reynolds number cases. High levels of axial turbulence inten-sity were noted near the anterior/left tracheal walls within one diameterdownstream of the larynx. Turbulence levels were still significant afterfour downstream diameters, indicating the potential for turbulent deposi-tion at positions further downstream, including the bronchial tree wherepassage diameters are smaller.

Laminar-to-turbulent air flow for typical inhalation modes (15, 30 and 60l/min) in a representative human upper airway model (Fig. 2.3) have beensimulated by Zhang et al. [171]. At normal breathing of 30 l/min, thevelocity profiles (Fig. 2.9) were seen to become skewed in the curved por-tion of the oral cavity and pharynx/larynx due to centrifugal force effects.Flow separation was seen due to abrupt geometric changes. While a uni-form flow was observed in the glottis, an asymmetric laryngeal jet wasseen to be generated after the glottis. The turbulence kinetic energy (Fig.2.9) was seen to become strong after the constriction of soft palate, to riserapidly after the glottis and to eventually decay more slowly, approachingan asymptotic level. It was concluded that the turbulence that occurs afterthe glottic constriction in the upper airways for moderate and high-levelbreathing can enhance particle deposition in the trachea near the larynx.

Johnstone et al. [80] experimentally (hot wire anemometry) studied the

24


Figure 2.9: Left: Mid-plane velocity contours at a flow rate of 30 l/min; Right:Variations of cross-sectional area-averaged turbulence kinetic energy [171].

Figure 2.10: Left: 2-D streaklines from PIV at a mid-plane and a normal inhalationflow rate of 30 l/min [64].

25


mean and RMS axial velocity field in the central sagittal plane of an ide-alized representation of the human extra-thoracic airway (Fig. 2.4) duringsteady inhalation flow rates of 10, 15, 30, 45, 60, 90 and 120 l/min. Regionsof separated and recirculating flow downstream of the mouth inlet, uvula,and larynx were seen. Circumferential secondary flow patterns have beenreported in the oral cavity (Fig. 2.10). Normalized mean hot-wire axial ve-locity profiles across the first five model traverse sections (i.e. from the oralinlet to the oropharynx)were typically found to increase as Reynolds num-ber decreases, demonstrating the presence of stronger viscous effects atlower inhalation flow rates. RMS velocities seemed to be least for the firstthree flow rates, but were found to converge closely onto a similar curve atthe higher flow rates. Both mean and RMS axial velocity profiles distal tothe oropharynx region appear to be primarily shaped by the geometry ofthe airways, since the basic features of these profiles were discovered to beessentially independent of the Reynolds number of the flow.

Yu et al. [165] numerically simulated the human breath patterns at 15l/min, in an idealized model (Fig. 2.5) that also incorporated the nasalairway in additions to the mouth-throat geometry. It was seen that the in-halation patterns (i.e., nasal inhalation, oral inhalation and simultaneousnasal-oral inhalation) had significant effects on the velocity profiles withinlaryngeal airways. The larynx was seen to be the key morphological factoraffecting the character of air stream motion within the upper lung. Dur-ing exhalation, the effects of reversing laryngeal jet were reported to beinsignificant.

The fluid flow patterns in the simplified geometry of Brouns et al. [19] (Fig.2.6) at 15, 30 and 60 l/min are discussed in detail by Brouns [18]. UsingLES methodology, Jayaraju et al. [77] (Chapter 8) also describes the flowpatterns at a normal breathing flow rate of 30 l/min. The flow featuresin the realistic geometry of Jayaraju et al. [76] (Fig. 2.7) at 15, 30 and60 l/min are discussed in Chapter 5. For the sake of completeness, a briefoverview of velocity vector lines in both simplified and realistic geometriesis shown in Fig. 2.11 .

In summary, the consistent observation among different studies was thatthe velocity profiles were seen to become skewed in the curved portions ofthe mouth, pharynx and larynx in general. Also, major amplification inkinetic energy is reported to take place downstream of larynx. However,soon after the larynx, the skewness of the flow profile as well as the sever-ity of the recirculation zone vary from one airway model to the other. Ascan be seen from all the velocity profiles presented before, this mainly de-

26


Figure 2.11: Mid-plane velocity vector lines at a normal breathing rate of 30 l/min,in a simplified model (left) and a realistic model (right) that were created in theframework of the present thesis [77, 76].

pends on the orientation of larynx as well as the upstream flow conditionswhile approaching the larynx. Hence, pharynx as well as larynx should beconsidered as the key morphological factor affecting the flow characteris-tics.

2.4 Particle deposition characteristics in up-per airways

Katz et al. [83] studied the particle deposition in one of the simplestlarynx-trachea model (Fig. 2.1) in order to understand the effect of lar-ynx on the particle deposition. The key quantitative observation was thatthe turbulence can have a profound effect on particle deposition in the lar-ynx and trachea. It was concluded that any calculation for the depositionof inhaled aerosols must consider the turbulence phenomenon.

Zhang et al. [171] studied the micro-particle (0.001 < Stk < 1) trans-port and deposition in a simplified oral airway model (Fig. 2.3) at 15,30 and 60 l/min. The turbulence that occurs after the constriction in the

27


oral airways for 30 and 60 l/min led to enhanced particle deposition in thelarynx-trachea region, and the turbulence enhanced deposition was seento be more profound for smaller particles (Stk < 0.05) in particular. Theparticles that were released around top and bottom part of the inlet planewere seen to deposit more easily on the curved oral airway surfaces. Kle-instreuer and Zhang [87] further extended this work with the same modeland boundary conditions to demonstrate that the particles nicely followthe airflow stream at 15 l/min, whereas the particle motion seemed to bemore random and disperse, i.e., influenced by flow fluctuations in case of60 l/min. The particle size and inhalation flow rate were reported to bethe main factors influencing particle deposition when compared with theturbulent dispersion alone.

In order to understand the effects of particle size, flow rate and flow Reynoldsnumber on the regional particle deposition, Grgic et al. [59] used gammascintigraphy and gravimetry tomeasure the deposition of radioactive monodis-perse sebacate oil particles of diameter 3, 5 and 6.5 µm at two constant flowrates of 30 and 90 l/min. In addition to particle size and flow rate, anotherReynolds number effect on deposition was identified which is related tovarying flow cross-sectional velocity profiles. The aerosols were seen to de-posit mostly in the laryngeal area and the upper part of the trachea. Theregional deposition profiles depended only weakly on different flow andparticle conditions. The pharyngeal and glottal constrictions were seen tobe the key morphological factors affecting downstream aerosol deposition.Most of the above physical phenomena have also been numerically repro-duced in the LES simulations of Jin et al. [79] in the same upper airwaymodel.

In summary, it was seen across different studies that the turbulence in-duced by the larynx has a profound effect on the particle deposition inlarynx/trachea region. The particle size, the fluid flow rate, and the fluidReynolds number were reported to have an influence on the particle depo-sition. The detailed analysis of aerosol transport and deposition patternsin more complex geometries such as the ones shown in Fig. 2.7 and 2.6 arediscussed in Chapter 5 and 8 respectively.

2.5 Empirical deposition relationships

By combining various in-vivo and in-vitro deposition data in the humanupper airways available in the literature, it is possible to establish a rel-atively simple empirical relationship for predicting the amount of inhaled

28


Figure 2.12: Upper airway deposition efficiency in human subjects measured dur-ing mouth breathing shown as a function of impaction parameter d2

aeQ. Solid curveis an empirical fit (Eq. 2.1) to the average of all of the data points while the dashedlines indicate the approximate range of the data from Lippmann [95] and Chanand Lippmann [23]. The above figure is taken from Stahlhofen et al. [142].

aerosol that may deposit in the upper airways. One of the most widelyused empirical relationship was proposed by Stahlhofen et al. [142] (Fig.2.12) by using regional deposition data in oral airway of human volunteerswhich were measured using mono-disperse particles tagged with radio-label [95, 47, 23, 43, 139, 140, 141]. The measured deposition fractionswere a combination of both inhalatory and exhalatory deposition. Stahlhofenet al. [142] defined the following empirical fit,

" = 1 ! 13.5 # 10!8(d2

aeQ)1.7 + 1(2.1)

where dae is the aerodynamic diameter (µm) and Q is the inhalation flowrate (cm3/s). Please note that dae and Q are normalized by unit diame-ter and unit flow rate respectively, so that we have a non-dimensionalizedimpaction parameter on the x-axis and a non-dimensionalized deposition

29


efficiency on the y-axis.

Cheng et al. [25] derived a new empirical fit for the inhalatory depositionefficiency assuming that the inhalatory and exhalatory depositions wereidentical. The empirical fit is given by,

" = 1 ! exp(!ad2aeQ) (2.2)

dae is the aerodynamic diameter (µm) and Q is the inhalation flow rate(l/min). Same as before, both dae and Q are normalized by unit diameterand unit flow rate respectively, to have a non-dimensionalized impactionparameter on the x-axis and a non-dimensionalized deposition efficiencyon the y-axis. a =0.000276±0.000188 (mean±SE) is a best-fitted non-dimensional parameter obtained by using a nonlinear regression program.The empirical fit of Cheng et al. [25] is shown in the Fig. 2.13. The datapoints represented by filled circles in Fig. 2.13 are experimental measure-ments obtained by Cheng et al. [25] at three different flow rates of 15, 30and 60 l/min. In an attempt to account for the geometry specific length andvelocity scales, Cheng et al. [25] also represented the measured depositionefficiency as a function of Stokes number (Fig. 2.14) which was definedhere as Stk = $pd2

aeU/9µdh. U is a measure of the mean velocity definedas Q/A, A is the mean cross-sectional diameter, and dh is the minimumhydraulic diameter. A new empirical fit was derived based on the Stokesnumber which is given by,

" = 1 ! exp(!6.66Stk) (2.3)

It is apparent from Fig. 2.14 that the experimental data tend to fall intoa single curve even though there were three different flow rates consid-ered. This was not the case when the deposition efficiency was simplyrepresented as a function of impaction parameter (Fig. 2.13). A similarapproach was adopted by Grgic et al. [58] who measured regional as wellas total depositions in seven realistic upper airway geometries that spanthe range of key dimensions of a larger set of 80 geometries. Representingthe deposition efficiency as a function of inertial parameter showed largescatter which was attributed to inter-subject variations and different inletdiameter conditions. Representing the deposition efficiency as a functionof Stokes number showed better collapse of data, but significant scatter re-mained due to different geometric configurations downstream of inlet. Inorder to account for this, Grgic et al. [58] replaced the hydraulic diameterdh in the definition of Stokes number with a mean diameter dmean calcu-lated simply by dividing cast volume V by the path length L of the centralsagittal line of the model to obtain a measure of area. A corresponding ve-

30


Figure 2.13: Upper airway deposition efficiency in human subjects measured dur-ing mouth breathing shown as a function of impaction parameter d2

aeQ [25].

Figure 2.14: Experimentally measured deposition efficiency as a function of Stokesnumber [25].

31


!!

0

10

20

30

40

50

60

70

80

90

100

0.01 0.1 1 10

Stk Re0.37

Dep

osit

ion

Eff

icie

ncy, %

S5b

S5a

S4

S3

S2

S1a

S1b

idealized

100-100/(11.5Stk1.912Re

0.707+1)

Figure 2.15: Experimentally measured deposition for 8 different models plotted asa function of Stokes and Reynolds number correlation. Error bars are standarddeviations.

locity scale Umean was calculated from the volume flow rate and the meancross-sectional area. The Stokes number then looks like,

Stk =$pd2

pUmean

18µdmean(2.4)

dmean = 2%

V

(L(2.5)

Umean =QL

V(2.6)

Using this Stokes number, the scatter among the data was markedly re-duced. In their previous deposition tests on an idealized upper airwaygeometry, Grgic et al. [59] had noticed that the deposition experimentswith constant Stokes number showed a Reynolds number dependence, dueto changes in flow field with Reynolds number. Therefore, Grgic et al. [59]proposed the following empirical relationship where the Stokes number ismultiplied by Reynolds number to the power of 0.37.

32


" = 100 ! 10011.5(Stk Re0.37)1.912 + 1

(2.7)

Re =$Umeandmean

µ=

2$Qµ

%L

(V(2.8)

Stk =$pd2

pUmean

18µdmean=$d2

pQ

36µ

%(L3

V 3(2.9)

An excellent collapse of data measured on various subjects (Fig. 2.15), onto a single curve indicates that the above defined correlation is indeed anadequate tool to compare and validate upper airway deposition values sim-ulated on any given upper airway geometry. In the present thesis, Eq. 2.7has been consistently used to represent the total deposition in the upperairway.

33


34

Chapter 3

The Fluid Phase

Contents3.1 Governing Equations . . . . . . . . . . . . . . . . . . . . 37

3.1.1 Reynolds Averaged Navier Stokes (RANS) . . . . . 40

3.1.2 Large Eddy Simulation (LES) . . . . . . . . . . . . 44

3.1.3 Detached Eddy Simulation (DES) . . . . . . . . . . 49

3.2 RANS, LES or DES ? . . . . . . . . . . . . . . . . . . . . 52

Before writing down the governing equations and different modeling ap-proaches for the fluid phase, it is worth giving an introduction to the mostimportant flow phenomenon called Turbulence.

Turbulence is that state of fluid motion which is characterized by its ran-domness, increased diffusivity, relatively high Reynolds number, three-dimensionality, and dissipativeness. In terms of energy cascade, turbu-lence is considered to be composed of various sizes of eddies. Developing amathematical model to mimic the physics of turbulence requires very goodunderstanding of the roles played by the largest and the smallest scales ofeddies in the transport of properties.

The energy cascade

Fig. 3.1 shows a schematic representation of the energy cascade. The largescales are of the order of the flow geometry. If l and u are the length andvelocity scales of the largest eddy, the time scale is derived as,

% =l

u(3.1)

35

CHAPTER 3. THE FLUID PHASE

Energyfrommean flow

Large scales Small scaleswhere dissipationtakes place

Figure 3.1: Schematic representation of the energy cascade [37].

Figure 3.2: Energy spectrum for a turbulent flow [160].

36


The large energy containing eddies give away their kinetic energy to slightlysmaller scale eddies with which the large scales interact. The process ofkinetic energy transfer continues in the similar fashion until we reach thesmallest scale eddies, where the frictional forces become so large that thekinetic energy is converted into internal energy. This process of energytransfer and dissipation is referred to as the cascade process. The dissi-pation (&) which takes place at the smallest scales, also referred to as theKolmogorov scales, can be estimated from the large scale properties as fol-lows,

& =u2

%=

u3

l(3.2)

Since the process of dissipation in the smallest scales are due to viscousforces, we can estimate the properties of smallest eddies using flow kine-matic viscosity (#) and the dissipation (&) itself. The length, velocity andtime scales are given by:

l" =##3

&

$1/4

(3.3)

u" = (#&)1/4 (3.4)

%" =&#&

'1/2(3.5)

The turbulent length scale l is related to the wavenumber ) as ) = 2(/l.The energy spectrum E()) for a turbulent flow is as shown in Fig. 3.2.From dimensional analysis, the Kolmogorov -5/3 law characterizes the in-ertial subrange which is given by,

E()) = C$&2/3)!5/3 (3.6)

C$ is the Kolmogorov constant.

3.1 Governing Equations

All fluid motions (laminar or turbulent) are governed by a set of dynamicalequations namely the continuity, momentum and the energy equation,

37


*$

*t+*

*xi($ui) = 0 (3.7)

*

*t($ui) +

*

*xj($uiuj) = ! *p

*xi+*%ij*xj

(3.8)

*

*t($E) +

*

*xi($Hui) =

*

*xi(%jiuj ! qi) (3.9)

ui(+x, t) represents the i-th component of the fluid velocity at a point inspace +x and time t.

p(+x, t) is the static pressure.

%ij(+x, t) are the viscous stresses.

$(+x, t) is the fluid density.

E and H are the total energy and total enthalpy per unit mass.

qi in Eq. 3.9 is the heat flux which is proportional to the temperaturegradient.

qi = !) *T*xi

(3.10)

where ) is the thermal conductivity.

The Mach numbers associated with air breathing are very nominal whichallows the flow to be treated as incompressible. Furthermore, the airbreathed in and out behaves as a Newtonian fluid, in which case the vis-cous stresses are related to the incompressible fluid motion using a prop-erty of fluid, viscosity.

%ij = 2µ

#sij !

13skk,ij

$(3.11)

sij is the instantaneous strain rate tensor given by,

sij =12

#*ui

*xj+*uj

*xi

$(3.12)

For incompressible flows, Eqs. 3.7 & 3.8 are simplified to the followingform,

38


*uj

*xj= 0 (3.13)

*ui

*t+ uj

*ui

*xj= !1

$

*p

*xi+ #

*2ui

*xj*xj(3.14)

The inhaled air is heated and humidified from the airway walls, which islargely complete within the first few generations of the conductive airwaysdepending on the inhalation rate as well as on the temperature and hu-midity of the air being inhaled [46]. However, due to the difficulties inmeasuring the actual temperature and humidity in the airways, only fewmathematical modeling is available in the literature. In the present thesis,the temperature effects are ignored and hence Eq. 3.9 is uncoupled fromthe continuity and momentum equations.

The four main numerical procedures for solving the Navier-Stokes areDirect Numerical Simulation (DNS), Large Eddy Simulation (LES), De-tached Eddy Simulation (DES) and ReynoldsAveragedNavier Stokes (RANS)approach. The most accurate approach is DNS where the whole range ofspatial and temporal scales of turbulence are resolved. Since all the spa-tial scales, from the smallest dissipative Kolmogorov scales (l") up to theenergy containing integral scales (l), are needed to be resolved by the com-putational mesh, the number of points required in one direction is of theorder,

N =l

l"(3.15)

The number of points required for a resolved DNS in three dimensions canbe estimated as,

N =#

l

l"

$3

"#

ul

#

$9/4

= Re9/4 (3.16)

The number of grid points required for fully resolved DNS is enormouslylarge, especially for high Reynolds number flows, and hence DNS is re-stricted to relatively low Reynolds number flows. DNS is generally usedas a research tool for analyzing the mechanics of turbulence, such as tur-bulence production, energy cascade, energy dissipation, noise production,drag reduction etc. The next three sections explain the RANS, LES andDES methodologies in detail.

39


3.1.1 Reynolds Averaged Navier Stokes (RANS)

When the flow is turbulent, it is convenient to analyze the flow in twoparts, a mean (time-averaged) component and a fluctuating component,

Ui = U i + u!

i

P = P + p!

Tij = T ij + %!

ij

Overline is a shorthand for the time average and in case of RANS, U i $ Ui

and u!i=0. The above technique of decomposing is referred to as Reynolds

Decomposition. Inserting these decomposition into the instantaneous equa-tions and time averaging results in the Reynolds Averaged Navier Stokesequations.

*U j

*xj= 0 (3.17)

*U i

*t+ U j

*U i

*xj= !1

$

*P

*xi+ #

*2U i

*xj*xj! *

*xj

&u

!iu

!j

'(3.18)

u!iu

!j in the last term of Eq. 3.18 represents the correlation between fluc-

tuating velocities and is called as Reynolds stress tensor. All the effectsof turbulent fluid motion on the mean flow are lumped in to this singleterm by the process of averaging. This will enable great savings in termsof computational requirements. On the other hand, the process of aver-aging generated six new unknown variables. Now, in total we have tenunknowns (3-velocity, 1-pressure, 6-Reynolds stresses) and only four equa-tions (1-continuity, 3 components of momentum equation). Hence we aresix equations too few. This is referred to as the Closure problem.

Based on the way we close the Reynolds stress tensor, there are two maincategories, namely the eddy viscosity models and Reynolds stress models.

The Reynolds stress tensor resulting from time averaging of Navier Stokesis closed by replacing it with an eddy viscosity multiplied by velocity gra-dients. This is referred to as the Boussinesq assumption.

u!iu

!j = !#t

#*U i

*xj+*U j

*xi

$(3.19)

40


In order to make Eq. 3.19 valid upon contraction because of Eq. 3.17, itshould be rewritten as,

u!iu

!j = !#t

#*U i

*xj+*U j

*xi

$+

23$,ijk (3.20)

k is the turbulent kinetic energy given by,

k =12u

!iu

!i (3.21)

The eddy viscosity is treated as a scalar quantity and is determined usinga turbulent velocity scale v and a length scale l, based on the dimensionalanalysis.

#t = vl (3.22)

There are different types of EddyViscosity Models (EVM) based on the waywe close the eddy viscosity. Algebraic or zero equation EVM’s normally usea geometric relation to compute the eddy viscosity. In one equation EVM’swe solve for one turbulence quantity and a second turbulent quantity isobtained from algebraic expression. These two quantities are used to de-scribe the eddy viscosity. In two equation EVM’s the two turbulent quan-tities are solved to describe the eddy viscosity.

In Reynolds Stress Models (RSM) we solve an equation for the Reynoldsstress and one length scale determining equation. Since we solve for Reynoldsstress, we don’t need any model to close it. However RSM’s are computa-tionally much more demanding when compared to EVM’s.

Two-equation k ! ! model

Two equation eddy viscosity models have served as the foundation formuch of turbulence research in the past two decades. The main reason fortheir popularity is that they are complete, i.e., they can predict propertiesof a given turbulent flow with no prior knowledge of turbulent structures.

All two equation eddy viscosity models use turbulent kinetic energy (k)as one of the solved turbulent quantities. Along with the transport equa-tion for k, another transport equation is solved for a second turbulentquantity. The only difference in all two equation models is the choiceof this second quantity we solve for. The two most widely used turbu-lent quantity which is linked to the kinetic energy are the dissipationrate at smallest scales & (as defined in Eq. 3.2) or the specific dissipa-tion !. ! is related to & as ! = &/k . k ! ! turbulence model is the

41


most widely used in simulating the transitional fluid flow in the respi-ratory tract [87, 167, 8, 102, 170, 60, 76, 77]. There are two variants of thek ! ! turbulence model, namely the standard k ! ! model of Wilcox [160](also called the high-Reynolds-number model) and shear-stress-transport(SST) k!! proposed by Menter [105] (also called the low-Reynolds-numbermodel). The major ways in which the SST model differs from the standardmodel are as follows:

• Gradual change from the standard k ! ! model in the inner regionof the boundary layer to a high-Reynolds-number version of the k ! &model in the outer part of the boundary layer. The reason for thisswitch is that the k ! ! model has a very strong sensitivity to the(quite arbitrary) free-stream values of !f outside the boundary layer[105].

• The definition of the eddy viscosity is modified to account for thetransport effects of the principal turbulent shear stress.

The above modifications make the SST k ! ! model more accurate and re-liable when compared to the standard k ! ! model.

The modeled k and ! equations read,

*

*t($k) +

*

*xi($kui) =

*

*xj

##k*k

*xj

$+ Pk ! Dk (3.23)

*

*t($!) +

*

*xi($!ui) =

*

*xj

##!*w

*xj

$+ Pw ! Dw + Cw (3.24)

The effective diffusivity for k and ! are given by,

#k = µ +µt

-k(3.25)

#! = µ +µt

-!(3.26)

The turbulent viscosity is related to magnitude of vorticity in the originalmodel [105]. Menter et al. [106] recently modified the definition of eddyviscosity by relating it to the strain-rate magnitude. It is given by,

µt =$k

!

1

max&

1%" , SF2

a1!

' (3.27)

42


where S is a strain-rate magnitude. - is the turbulence Prandtl numbergiven by,

-k =1

F1/-k,1 + (1 ! F1) /-k,2(3.28)

-! =1

F1/-!,1 + (1 ! F1) /-!,2(3.29)

F1 and F2 are the blending functions. For further details on blending func-tions, the reader is referred to the Fluent manual [4]. ." is the dampingfunction for turbulent viscosity causing a low-Reynolds-number correction.It is given by,

." = ."#

#0.024 + Ret/6

1 + Ret/6

$(3.30)

The production terms for k and ! are given by,

Pk = min!µtS

2, 10$/"k!"

(3.31)

P! =.

#tµtS

2 (3.32)

The dissipation terms for k and ! are given by,

Dk = $/"k! (3.33)

D! = $/!2 (3.34)

The cross-diffusion modification for ! equation is given by,

C! = 2 (1 ! F1) $-!,21!

*k

*xj

*!

*xj(3.35)

The values of constants are,

-k,1 = 1.176 -!,1 = 2.0 -k,2 = 1.0 -!,2 = 1.168

a1 = 0.31 /i,1 = 0.075 /i,2 = 0.0828

43


Advantages of low-Reynolds-number k-! model

The two main attractive features of low-Reynolds-number k-! model are,

1. It possesses a non-trivial solution for ! as k goes to zero. It is thus ex-pected to capture flow characteristics which a low-Reynolds-numberk-& model fails to handle.

2. It uses damping functions that only depend on turbulent Reynoldsnumber and hence it is convenient to apply this model to internalflows with complex geometries, which is the case with the humanrespiratory tract.

3.1.2 Large Eddy Simulation (LES)

In case ofRANS, the most challenging aspect was to understand and modelthe largest eddies, which account for most of the transport properties in aturbulent flow. As we go closer towards the walls, the size of the eddies getssmaller and RANS uses viscous damping functions in order to account forthese small dissipating scales. The RANS models often involve simplifiedassumptions which makes it impossible for a single model to represent allturbulent features. Furthermore, it may also require fine-tuning of modelconstants to obtain better results for a given test-case.

The Large Eddy Simulation technique was developed based on an impli-cation from Kolmogorov’s theory of self similarity that the large eddiesof the flow are dependent on the geometry while the smaller scales aremore universal. Hence, the big three dimensional eddies which are dic-tated by the geometry and boundary conditions of the flow involved aredirectly resolved whereas the small eddies which tend to be more isotropicare modeled. Hence, the performance of LES as opposed to RANS will beless problem dependent. An elaborate explanation on LES can be found inseveral books such as [160, 123, 49, 117].

LES equations

In case of RANS, the instantaneous continuity and momentum equations(Eq. 3.7 & 3.8) were time averaged to obtain steady form of averaged equa-tions (Eq. 3.17 & 3.18).

In case of LES, instead of time-averaging, we filter the instantaneous time-dependent equations. Filtering is a method that separates the resolvablescales from the subgrid scales. Filtering can be performed in either wavenumber space or the physical space. The filter cut-off should lie somewhere

44


in the inertial range of the spectrum (Fig. 3.2).

In finite volume methods, box filters are always used because the finitevolume discretization itself implicitly provides the filtering operation. Oneof the earliest volume average box filters was given by Deardorff [38].

0(X, t) =1

!3

( x!0.5!x

x!0.5!x

( y!0.5!y

y!0.5!y

( z!0.5!z

z!0.5!z0(1, t)d1d"d' (3.36)

0 = 0+ 0s (3.37)

In the above equation, 0 denotes the resolvable scale filtered variable and0s denotes the sub-grid scale fluctuation. ! is the filter width given by! = (!x!y!z)1/3.

Leonard [92] defined a generalized filter as a convolution integral whichis given by,

0(X, t) =( ( (

G(X ! 1; !)0(1, t) d31 (3.38)

G is the filter function that determines the scale of resolved eddies. Thefilter function is normalized by requiring that,

( ( (G(X ! 1; !) d31 = 1 (3.39)

The filter function in terms of the volume average box filter (Eq. 3.36) canbe written as,

G(X ! 1; !) =

)*

+

1/!3, |x ! 1| < !x/2

0, otherwise

Finally, the decomposition of the flow into a filtered part and a sub-gridpart looks like,

Ui = U i + usi

P = P + ps

Tij = T ij + %sij

The overline in the above equations represents the filtering operation asopposed to the time-averaging in case of RANS. Also, contrary to RANS,where the average of fluctuations is zero, in case of box filtering, we have

45


U i %= U i and us %= 0. Further details on the filtering methods can be foundin [117, 160, 37, 91].

Inserting the above decomposition into the instantaneous equations re-sults in the following filtered Navier-Stokes equations,

*U j

*xj= 0 (3.40)

,*U i

*t+ U j

*U i

*xj

-= !1

$

*P

*xi+ #

*2U i

*xj*xj!*%sgs

ij

*xj(3.41)

%sgsij are the sub-grid scale stresses.

SGS modeling

From the energy cascade, explained in the beginning of this chapter, itis apparent that the energy transfer occurs from the bigger scales to thesmaller scales. Hence the main purpose of an SGS model is to representthe energy sink. The representation of the energy cascade is an averageprocess. However, locally and instantaneously the transfer of energy canbe much larger or much smaller than the average [48]. Additionally, thereis also the phenomenon of energy backscatter in the opposite direction[116]. Ideally speaking, SGS models should actually account for all thesephenomena. However, if the grid scale is much finer than the dominantscales of the flow, even a crude SGS model will result in good predictions ofthe behavior of the dominant scales [48]. Having this in mind, certain au-thors such as Tamura et al. [145] and Meinke et al. [104] even performedLES without any explicit SGS model, but having refined grids to minimizethe importance of SGS stresses, and the energy drain was achieved by nu-merical schemes. Although this approach yields promising results in somecases, this kind of modeling can hardly be evaluated or controlled [48].Hence, in LES, central or spectral schemes are used and the SGS stressesare explicitly modeled.

The sub-grid scale stresses %sgsij in Eq. 3.41 are given by,

%sgsij = UiUj ! U iU j (3.42)

By using the definition of filtering as given by Eq. 3.37 we can furtherwork out % sgs

ij as,

46


%sgsij = !

&UiUj ! (U i + us

i )(U j + usj)

'(3.43)

%sgsij = !us

iusj. /0 1

Reynolds

+ (!U iusj ! U jus

i ). /0 1Cross!term

+ U iU j ! U iU j. /0 1Leonard

(3.44)

Leonard [92] shows that the Leonard stresses can significantly drain en-ergy from the resolvable scales and they can be directly computed. On theother hand, Wilcox [160] mentions that Leonard stresses are of the sameorder ofmagnitude as the truncation error when a finite-difference schemeof second-order accuracy is used, and thus it is implicity represented.

The cross-term stresses are dispersive in nature and largely account forthe backscatter effects. Modeling them with a purely dissipative modelsuch as Smagorinsky would be in conflict because of its dispersive nature[91]. In many applications, it is assumed that the Leonard and cross-termstresses can be neglected and only the Reynolds stresses remain to be mod-eled. It is the same case in the present work.

A variety of SGS models have been used by different researchers, such astwo-point closures [89], scale-similar models [10], and one-equation mod-els [129, 72, 22] to name a few among others. Please refer to the book ofSagaut [123] for the detailed review of various SGS models available inliterature.

Smagorinsky model

One of the simplest SGS model is the Smagorinsky model [133]. The un-known subgrid-scale stresses are modeled employing the Boussinesq as-sumption as in the case of RANS. The subgrid-scale stress are related tothe eddy viscosity as follows,

%ij !13%kk,ij = !#t

#*U i

*xj+*U j

*xi

$(3.45)

The eddy-viscosity is modeled as,

#t = L2s

22SijSij (3.46)

Ls is the length-scale for the sub-grid scale and is given by CsV 1/3. V is thecomputational cell volume. It is interesting to note that the length scaleis now the filter width rather than the distance to the closest wall as in

47


RANS. Cs is a constant which is taken to be 0.17. The only disadvantageof the Smagorinsky model is the constant Cs, which is not really a constant,but is flow dependent. It is found to vary between 0.065 [107] and 0.3 [81].In the dynamic version, which was first proposed by Germano et al. [52],Cs is dynamically computed based on the information provided by the re-solved scales of motion. However, the influence of the subgrid scale modelis expected to be small due to the low Reynolds numbers associated withthe upper airway flows we are dealing with in the present thesis [14, 15].

The specification of Ls as CsV 1/3 is not justifiable in the viscous wall re-gion as it incorrectly leads to a non-zero shear-stress at the wall. In orderto rectify this, Moin and Kim [107] use a Van Driest damping function tospecify the length scale as,

Ls = CsV1/3

,1 ! exp

#y+

A+

$-(3.47)

where y+ = u#d/# is the non-dimensional distance from wall. u# is thewall shear stress velocity, d is the distance to the nearest wall and A=25 isthe Van Driest constant.

The above described SGS model is a standard version as defined in Smagorin-sky [133]. The LES simulations in the present thesis are performed em-ploying the Fluent flow solver. The Smagorinsky model implemented inFluent deviates slightly from the standard version in the following ways,

• The length-scale for the sub-grid scale is computed as min()d, CsV 1/3).) is the von Karman constant (typically a value of 0.41 is used), d isthe distance to the closest wall. )d is indeed one of the first mixinglength models in the literature to handle the turbulent viscosity andwas proposed by Prandtl [119]. Van Driest damping is basically animproved version of Prandtl’s mixing length model. Both the Prandtland the Van Driest model are algebraic and from the zero-equationmodels category.

• The constant Cs in Fluent is taken to be 0.1 instead of 0.17 as wasoriginally proposed. The value of 0.17 for Cs was originally derivedfor homogeneous isotropic turbulence in the inertial subrange. How-ever, this value was found to cause excessive damping of large-scalefluctuations in transitional flows near solid boundaries, and has to bereduced in such regions [4]. A Cs value of around 0.1 has been foundto yield the best results for a wide range of flows, and is the defaultvalue in Fluent.

48


Wall-Adapting Local Eddy-Viscosity (WALE) model

The WALE model [110] is specifically designed to return correct wall-asymptoticy3 behavior of the SGS viscosity. The model is based on the square of thevelocity gradient tensor and accounts for the effect of both the strain andthe rotation rate to obtain the local eddy viscosity. The eddy viscosity ismodeled as,

µt = $L2s

(SdijS

dij)3/2

(SijSij)5/2 + (SdijS

dij)5/4

(3.48)

Sdij =

12

!g2

ij + g2ji

"! 1

3,ijg

2kk (3.49)

gij =*U i

*xj(3.50)

Ls is given by C!V 1/3. The main advantage of the WALE model is thatneither a damping function nor a dynamic procedure is required to accountfor the no-slip condition. However, in Fluent, the length scale is still keptto be min()d, C!V 1/3). To the best of the author’s knowledge, there is nospecific advantage of introducing the )d in the length scale definition, asthe no-slip condition is already taken care of. The constant C! in Fluent istaken to be 0.325.

3.1.3 Detached Eddy Simulation (DES)

Detached Eddy Simulation was developed as a response to the computa-tional and physical challenges associated with the reliable prediction ofmassively separated turbulent flows in practical geometries at practicalReynolds numbers. In most external aerodynamic computations such asthe flow around an airplane or an automobile, the boundary layers arethin and populated with small attached eddies whose local length scale, l,is much smaller than the boundary layer thickness, ,. LES, even with thebest wall-region treatment, is very far from affordable in aerodynamic cal-culations, and will be for decades [135]. The non-affordable computationalcosts of LES in the attached boundary layers and the ability of a fine-tunedRANS methodology in predicting the attached boundary layers lead to theDES formulation.

DES was first expounded in 1997 together with its formulation based onthe Spalart-Allmaras turbulence model (S-A model) [138]. By definition,DES is a three-dimensional unsteady numerical solution using a single

49


turbulence model, which functions as a subgrid-scale model in the regionswhere the grid density is fine enough for an LES, and as a RANS modelin regions where it is not. The fine enough grid for LES is one where themaximum spatial step, !, is much smaller than the turbulent length-scale,t. Thus in the LES regions, little control is left to the model and thelarger, most geometry sensitive eddies are directly resolved. DES seemsto be most justified as the model adjusts itself to a lower level of mixing,relative to RANS, in order to unlock the large-scale instabilities of the flowand to let the energy cascade extend to length scales close to the grid spac-ing [144].

The most attractive feature of DES is that it is simply formulated unlikemost of the zonal methods where the RANS and LES regions are explicitlytagged by the user. It provides a continuous velocity and eddy viscosityfield and there is no issue of smoothness between the RANS and LES re-gions.

S-A based DES formulation

The driving length scale in the standard RANS S-A model is the distanceto the closest wall d. This length scale plays an important role in the stan-dard S-A model as it controls the destruction term [137]. The length scalein the DES model is defined as,

d = min(d, Cdes!) (3.51)

d is the distance to closest wall and ! is the grid spacing based on thelargest grid space in the x, y or z directions forming the computationalcell. The empirical constant Cdes has a value of 0.65. The intention is thatin the boundary layers, ! far exceeds d and the standard S-A model rulessince d = d. Away from walls, d = Cdes! and the model turns into a simpleone-equation subgrid-scale model, close to Smagorinsky’s in the sense thatboth make the mixing length proportional to ! [130]. Grids with y+ & 1and a stretching ratio of 1.15 are generally used to resolve the log layer[130, 111].

The S-A model, after inserting the length scale as given in Eq. 3.51 re-duces to,

*

*t($#) +

*

*xi($#U i) =

1-v

3*

*xj

,(µ + $#)

*#

*xj

-+ Cb2$

#*#

*xj

$24

+ P& ! Y&

(3.52)

50


P& and Y& are the production and destruction terms of turbulent viscositythat occurs in the near-wall region due to wall blocking and viscous damp-ing. -& and Cb2 are constants and # is the molecular kinetic viscosity.

The turbulence production P& is calculated as,

P& = Cb1$S# (3.53)

where

S = S +#

)2d2fv2 (3.54)

and

fv2 = 1 ! 2

1 + 2fv1(3.55)

Cb1 and ) are constants. S is a scalar measure of the deformation tensorbased on the magnitude of the vorticity,

S =2

(2$ij$ij) (3.56)

where $ij is the mean rate-of-rotation tensor and is defined by,

$ij =12

#*U i

*xj! *U j

*xi

$(3.57)

The destruction term Y& in Eq. 3.52 is given by,

Y& = c!1$f!

##

d

$2

(3.58)

where

fw = g

,1 + C6

!3

g6 + C6!3

-1/6

(3.59)

g = r + C!2(r6 ! r) (3.60)

r $ #

S)2d2(3.61)

The turbulent viscosity µt is computed from,

µt$#fv1 (3.62)

51


fv1 is the viscous damping function given by,

fv1 =23

23 + C3v1

(3.63)

and

2 =#

#(3.64)

The values of the constants are,

Cb1 = 0.1355 Cb2 = 0.622 -& = 23 Cv1 = 7.1

C!1 = Cb1$2 + (1+Cb2)

'!C!2 = 0.3 C!3 = 2.0 ) = 0.4187

As mentioned before, Eq. 3.52 implies that RANS is employed close to thewall and one will switch to the LES model in the core-region. The switchto LES mode implies a reduction in d. This causes the destruction term Y&to increase, resulting in the decrease of turbulent viscosity #t. The resultin turbulent viscosity reduction causes a drop in production P& , so that#t further decreases [91]. Note that at equilibrium (meaning a balance ofproduction and destruction terms), the S-A model becomes a Smagorniskylike SGS model.

Over the years, there have been several potential improvements suggestedto the original procedure ofDES. One such method is the Delayed Detached-Eddy Simulation (DDES) which detects boundary layers and prolongs thefull RANS mode, even if the wall-parallel grid spacing would normally ac-tivate the DES limiter. This detection device depends on the eddy viscos-ity, so that the limiter now depends on the solution. Very recently [131],further corrections are brought into the DDES method with the aim of re-solving the log-layer mismatch and the models are named Improved DDES(IDDES). The present thesis deals only with the original DES model andfor a detailed overview of the new trends in DES, the readers are referredto Spalart [136].

3.2 RANS, LES or DES ?

Deciding between RANS, LES and DES for simulating the fluid-particledynamics in the respiratory tract needs the basic understanding of howthese models work. A few pro’s and con’s of each of these models are brieflydescribed below.

52


Two equation RANS models are the very commonly used models for mosttypes of engineering problems and have become an industry standard.If one is only interested in the averaged quantities, RANS is the bestchoice. Since we are solving for time-averaged equations, the computa-tional cost to perform a simulation is very reasonable. For example, a typ-ical fluid-particle dynamics simulation (grid size - 550,000, injected par-ticles - 15,000) in a realistic model of the extra-thoracic airway requiresabout 14 hours on an AMD Opteron 2.4 MHz dual-core processor [77].The main disadvantage of a RANS model is the modeling error that comeswhile handling Reynolds stresses. The two-equation turbulence models of-ten involve simplified assumptions which makes it impossible for a singleRANS model to represent all turbulent features. Furthermore, it may alsorequire fine-tuning of model constants to obtain better results for a giventestcase.

Since the large three-dimensional problem-dependent eddies are directlyresolved in LES, the accuracy of this method will generally be better thanthat of RANS. However, LES still requires substantially finer meshes thanthose typically used for RANS calculations. Also, the LES method is bothspace and time dependent and needs to be run for a sufficiently long flow-time to obtain stable statistics of the flow being modeled. This resultsin huge computational costs, which are typically few orders of magnitudehigher than those of RANS. For example, a typical fluid-particle dynamicssimulation (grid size - 1.9#106, injected particles - 5000) in a realisticmodelof the extra-thoracic airway requires about 60 days on 4 AMD Opteron 2.4MHz dual-core processors [77].

Detached eddy simulation (DES) is a modification of a RANS model inwhich the model switches to a sub-grid scale formulation in regions fineenough for LES calculations. Regions near solid boundaries where the tur-bulent length scale is less than the maximum grid dimension are assignedfor RANS mode of solution. As the turbulent length scale exceeds the griddimension, the regions are solved using the LES mode. Therefore the gridresolution is not as demanding as pure LES, thereby cutting down the costof the computation to some extent. While the reduced mesh decreases com-putational effort, we need to solve the additional S-A equation which addsto the computational cost. Also, due to the switch between RANS and LES,generating an optimal cost-effective grid is more complicated.

In Chapter 8, the performance of RANS, LES as well as DES in simulatingthe fluid-particle dynamics has been studied in detail. Hence, a definitive

53


answer for the preferred method between RANS, LES and DES can befound in Chapter 8.

54

Chapter 4

The Particle Phase

Contents4.1 Governing Equations . . . . . . . . . . . . . . . . . . . . 554.2 Modeling the particle phase . . . . . . . . . . . . . . . 604.3 Stochastic trajectory approach . . . . . . . . . . . . . 614.4 Aspects of Lagrangian modeling . . . . . . . . . . . . . 63

4.4.1 Time integration . . . . . . . . . . . . . . . . . . . . 644.4.2 Locating particles inside a control volume . . . . . 654.4.3 Cell search algorithm . . . . . . . . . . . . . . . . . 674.4.4 Interpolation of flow variables . . . . . . . . . . . . 68

4.5 Boundary conditions . . . . . . . . . . . . . . . . . . . . 704.5.1 Inlet boundary condition . . . . . . . . . . . . . . . 704.5.2 Wall and Outlet boundary condition . . . . . . . . . 73

4.6 Uncoupled and coupled calculations . . . . . . . . . . 734.7 Programming language . . . . . . . . . . . . . . . . . . 744.8 Data structures . . . . . . . . . . . . . . . . . . . . . . . 754.9 Flow chart of particle solver module . . . . . . . . . . 794.10 Testcases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.10.1 Analytical solution . . . . . . . . . . . . . . . . . . . 814.10.2 2D Planar mixing layer . . . . . . . . . . . . . . . . 82

4.1 Governing Equations

Based partly on the physical properties of pharmaceutical aerosols andpartly on the mathematical modeling effort required, there are certain rea-

55

CHAPTER 4. THE PARTICLE PHASE

sonable assumptions made to describe the pharmaceutical aerosol trans-port in a fluid medium. The major simplifying assumptions are as follows,

• The pharmaceutical aerosol is assumed to be spherical: This is a rea-sonable assumption, especially for the liquid based inhaled pharma-ceutical droplets, since the small liquid droplets are spherical. Theaerosols generated from DPI’s and pMDI’s are not exactly spherical,but it is still a reasonable assumption because the drag on these par-ticles is very close to that on a sphere [46].

• The ratio of particle to fluid density is very large: The density of phar-maceutical compounds are much higher when compared to the fluidmedium which is air. Typically, the density of aerosols is close to thatof water, which is "1000 kg/m3 as opposed to " 1.2 kg/m3 for airbreathed in.

• Drag force is the dominant point force: This is a direct result of theprevious assumption. Since the density of the aerosols are muchhigher than the density of the fluid medium, several forces such asthe lift force, Basset force, Magnus force and buoyancy force can bereadily discarded as they have negligible effect on the aerosol trans-port.

• The aerosol phase is dilute: A quantitative way of assessing a diluteversus dense flow is by taking the ratio of the particle relaxation time(tr) to the particle collision time (tc). The particle relaxation time isdefined as the time required by the particle to loose 63% of its initialvelocity. If tc > tr, the particle has enough time to respond to thelocal fluid dynamic forces before collision occurs. Hence, a dilute flowis one in which the particle motion is controlled by fluid forces suchas the drag force. In case of pharmaceutical aerosol transport, thecollision time is much larger than the relaxation time, making thephase dilute.

• One-way coupling: The phenomenon of mutual mass, momentumand energy transfer between the phases is termed as coupling. El-ghobashi [42] proposed a map of regimes of interactions betweenparticles and fluid turbulence as shown in Fig. 4.1. For values ofdispersed-phase volume fraction less than 10!6, particles have neg-ligible effects on turbulence and this is termed as one-way coupling.The volume fraction of pharmaceutical aerosols we are dealing within the present thesis is much less than 10!6 and hence one-way cou-pling is assumed. In the second regime which lies between 10!6 !10!3, the existence of particles can augment the turbulence if the ra-tio of the particle response time to the turnover time of a large eddy

56


Figure 4.1: Map for particle-turbulence modulation [42].

is greater than unity, or can attenuate turbulence if the ratio is lessthan unity. This interaction is called two-way coupling. In the thirdregime where the volume fractions are greater than 10!3, in addi-tion to two-way coupling between particles and turbulence, particlecollisions take place and hence this regime is termed as four-way cou-pling.

Incorporating all the above assumptions, the Lagrangian equations gov-erning the particle motion can be written as,

dxp

dt= up (4.1)

dup

dt= Fd (u ! up) + gx

($p ! $)$p

(4.2)

xp is the particle position, gx is the gravitational force, $ and $p are thedensity of the fluid and the particle respectively.

Generally, the particle moves with a different velocity than the fluid atany given point. The difference in fluid velocity (u) and the particle veloc-ity (up), termed as the slip velocity (u!up), leads to an unbalanced pressure

57


distribution as well as viscous stresses on the particle surface which yieldsa resulting force called drag force. In Eq. 4.2, the term Fd (u ! up) is thedrag force per unit particle mass. Fd is given by,

Fd =1%p

CdRep

24(4.3)

%p is the particle relaxation time given by,

%p =$pd2

p

18µ(4.4)

Laws of drag coefficient

The drag coefficient Cd is a function of particle Reynolds number (Rep).Various experimentally based empirical correlations for the drag coeffi-cient based on Rep are available in the literature. One such correlationof Schiller and Neumann [126], which is employed in the present work, isgiven below.

Cd =24

Rep(1 + 0.15Re0.687

p ) (4.5)

The Reynolds number of the particle is defined as,

Rep = $ dp|u! up|

µ(4.6)

Fluent employs a different form of drag coefficient put forth by Morsi andAlexander [108]. It is given by,

Cd = a1 +a2

Rep+

a3

Re2p

(4.7)

where a1, a2 and a3 are constants that apply to smooth spherical particlesin a stipulated range of Rep as given in the table below,

58


10−1 100 101 102 103 104 105 106

10−1

100

101

102

Rep

C d

Experimental dataSchlichting (1979)Schiller and Neumann (1933)Morsi and Alexander (1972)

Figure 4.2: Drag coefficient for spheres as a function of Rep.

Rep a1 a2 a3

< 0.1 0 24 0

0.1 < 1.0 3.69 22.73 0.0903

1 < 10.0 1.222 29.1667 -3.8889

10.0 < 100.0 0.6167 46.5 -116.67

100.0 < 1000.0 0.3644 98.33 -2778

1000.0 < 5000.0 0.357 148.62 -4.75

5000.0 < 10000.0 0.46 -490.546 57.87

10000.0 < 50000.0 0.5191 -1662.5 5.4167

Fig. 4.2 shows the comparison of the above two formulations for Cd withrespect to the experimental data of Schlichting [127]. As can be seen, bothmodels perform very well up until Rep=1000, after which, the model ofMorsi andAlexander [108] does a better job. However, the Rep encounteredin the present work are well below 1000 and both formulations are equallysuited.

59


4.2 Modeling the particle phase

Based on the recent experimental observations by Grgic et al. [58] oninter-subject variability of the extra-thoracic airway deposition, the needfor considering realistic model for CFD study is recognized and an efficientLagrangian particle tracking module for unstructured grids to handle com-plex geometrical features is developed in the commercial C++ flow solverof NUMECA. Even though the concept of unstructured grids exists sincelong, the practical applicability is still under budding stage, especially fortwo-phase simulations. In this view, the modeling methods described inthis chapter can be seen as the first step towards applicability of unstruc-tured grids for biomedical applications, which is by far the only option toremove expensive experiments for realistic geometry configurations.

Coming to the fundamental mathematical modeling of two-phase flow, thetwo most widely used approaches are the Eulerian continuum approachand the Lagrangian trajectory approach. For a detailed overview of themodeling techniques for inhaled particle deposition, the readers are re-ferred to Hofmann [68].

Eulerian continuum approach

In an Eulerian approach, the particles are treated as a second fluid whichbehaves like a continuum and the equations are developed for averageproperties 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. Eu-lerian approach is more suitable for simulating large-scale particle flowprocesses. However, this approach requires sophisticated modeling in or-der to describe the key effects and phenomena found in industrial pro-cesses [30].

Lagrangian trajectory approach

A Lagrangian approach is useful when the particle phase is so dilute thatthe description of particle behavior by continuum models is not feasible.The motion of a particle is expressed by ordinary differential equationsin Lagrangian coordinates and are directly integrated to obtain individ-ual tracks of particles. To solve the Lagrangian-equation for a particu-lar moving particle, the dynamic behavior of the gas phase (generally ob-tained by an Eulerian approach) and other particles surrounding this mov-ing particle should be pre-determined. Since the particle velocity and the

60


corresponding particle trajectory are calculated for each particle, this ap-proach is more suitable to obtain the discrete nature of motion of particles.However, to obtain statistical averages with reasonable accuracy, a largenumber of particles will have to be tracked. An advantage of using theLagrangian approach is the ability to easily vary the physical propertiesassociated with individual particles such as diameter, density etc. More-over, local physical phenomena related to the particle flow behavior can beeasily probed. Hence, the Lagrangian models can also be used for valida-tion, testing and development of continuum models [30].

The Lagrangian approach is classified in to two types namely, Determinis-tic trajectorymethods and Stochastic trajectorymethods based on the effectof turbulence. In a deterministic method, all the turbulent transport pro-cesses of the particle phase are neglected where as the stochastic methodtakes in to account the effect of fluid turbulence on the particle motion byconsidering instantaneous fluid velocity in the formulation of the equationof particle motion. In the present thesis, the pharmaceutical aerosols aremodeled with a stochastic Lagrangian approach.

4.3 Stochastic trajectory approach

Stochastic modeling involves direct simulation of particle motion througha random turbulent flow field. There are several approaches in develop-ing stochastic models. Few examples in literature include models based onthe Langevin equation [97], random Fourier modes [103] and pdf models[118]. However, one of the most frequently used model is the eddy inter-action model (EIM) first introduced by Hutchinson et al. [73] and furtherdeveloped by Gosman and Ioannides [53].

The instantaneous motion of particles governed by Equations 4.1 and 4.2can be written in a general form as given below,

dx

dt= up (4.8)

dup

dt=

1%p

(u ! up) + g (4.9)

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

u = U + u!

(4.10)

Assuming isotropic turbulence, we have,

61


u!2 = v!2 = w!2 =23k (4.11)

where k is the turbulent kinetic energy which may be determined from thek ! ! model. Furthermore, it is assumed that the local velocity fluctua-tions of the fluid phase obey a Gaussian probability density distribution.Most stochastic models in practical use are derived from the formulationof Gosman and Ioannides [53], which is given by,

u!=

%23k ' ' (4.12)

where ' is a random number drawn from a normal probability distributionwith zero mean and unit standard deviation. The minimal random num-ber generator of Park and Miller with Bays-Durham shuffle [5] is imple-mented. The random number generator returns a uniform random deriva-tive with zero mean and unit standard deviation.

The chosen fluctuation is referred to a turbulent eddy whose size (lengthscale) and life-time (time scale) is known. Sommerfeld et al. [134] proposedthe following relations for eddy parameters,

te = ctk

&(4.13)

le = te

%23k (4.14)

where ct was taken to be 0.3.

Since we are using a k!! turbulence model, the dissipation rate & is takento be & = !k. Fig. 4.3 shows a 2D schematic representation of an eddy in-side a rectangular domain. At any given particle position (xp, yp), the eddyparameters are first evaluated based on the local fluid kinetic energy anddissipation rate. The particle position (xp, yp) is assumed to be located atthe center of this hypothetical eddy. It is accepted that each eddy has itsown fluctuation u

!, which remains constant until the particle leaves this

eddy. The particle leaving an eddy is based on a certain interaction timeof the particle with the eddy. Once this interaction time is reached whiletime integration of particle equations, the particle is assumed to have leftthe present eddy. Now, based on the new position of the particle, new eddyparameters are calculated and a new fluctuation u

!is assigned to this eddy.

This procedure may be repeated for as many interaction times as required

62


xp ,yp( )!

Figure 4.3: 2D illustration of a particle within an eddy.

for the particle to traverse the required distance. If a statistically signif-icant number of particles are tracked in this way, the ensemble averagedbehavior should represent the turbulent dispersion induced by the prevail-ing fluid field [53].

The interaction time is the minimum of two time scales, one being a typicalturbulent eddy lifetime and the other the crossing-time of the particle inthe eddy [53].

tint = min(te, tc) (4.15)

The crossing-time is defined as,

tc = !%p ln

,1 !

#le

%p|u ! up|

$-(4.16)

where %p is the particle relaxation time, le the eddy length scale and |u!up|the magnitude of slip velocity. In circumstances where le/(%p|u ! up|) > 1,Eq. 4.16 has no solution. This can be interpreted as the particle trappedby an eddy, in which case tint = te [53].

4.4 Aspects of Lagrangian modeling

Developing a Lagrangian module on unstructured hexahedral meshes whichis directly applicable for complex engineering configurations requires thefollowing key issues to be addressed,

(a) Time integration.

(b) Efficient search and location of particles on an unstructured grid.

63


(c) Interpolation of flow variables at particle location.

4.4.1 Time integration

Equations 4.8 and 4.9 are numerically integrated to obtain the updatedparticle velocity and position. Two numerical schemes have been imple-mented, namely the trapezoidal discretization and Runge-Kutta scheme.

Trapezoidal discretization

The particle momentum and the displacement equations can be solved us-ing a numerical discretization scheme. In the trapezoidal scheme, the vari-ables up and u in Eq. 4.9 are taken as averages, while acceleration due togravity (g) is kept constant.

un+1p ! un

p

!t=

1%p

!u" ! u"

p

"+ g (4.17)

The averages u" and u"p are given by

u" =12

!un + un+1

"(4.18)

u"p =

12

!un

p + un+1p

"(4.19)

un+1 = un + !tunp ·(un (4.20)

Inserting the above into Eq. 4.17 results in the particle velocity at newtime n + 1 which is,

un+1p =

unp

&1 ! 1

2!t#p

'+ !t

#p

!un + 1

2!tunp ·(un

"+ !tg

1 + 12

!t#p

(4.21)

The new particle location is computed by the trapezoidal discretization ofEq. 4.8.

xn+1p = xn

p +12!t

!un

p + un+1p

"(4.22)

64


Runge-Kutta scheme

The governing equation for a single particle in a carrier flow can be ex-pressed in a general form as,

dY

dt= L(Y, t) (4.23)

Y = f(r, u) is a vector of physical properties which include the particleposition, particle velocity etc. L = f(Fd, g) represents the rate of change ofparticle properties with time dependent source terms like the drag force,gravity etc. The above non-linear differential equation can be numericallyintegrated using a Runge-Kutta method to obtain the solution at the nexttime level through a number of intermediate steps (stages). A K-stageexplicit Runge-Kutta scheme to solve the above equation is given as,

+Y (s) = +Y n + !t /k+L(s!1), s = 2...K (4.24)

+Y (1) = +Y n (4.25)+Y (n+1) = +Y n + !t +L(K) (4.26)

In specific, a 4-stage explicit Runge-Kutta scheme which is second order ac-curate was employed. The / coefficients are /2=0.25, /2=0.33333, /2=0.5.Since the fluid field is calculated with second-order accuracy, it is reason-able to provide the same accuracy level for particle tracking [166]. Be-ing a one-step method, the trapezoidal scheme has the advantage that themovement of particles from one control volume to the next can easily becontrolled as opposed to a multi-stage Runge-Kutta. The details of thecell-crossing of a particle are discussed in detail in the next sections.

The purpose of particle tracking is to obtain detailed information on par-ticle parameters within all passed control volumes. To achieve this, theparticle should ideally make several time-steps within each control vol-ume it passes through. Hence the integration time-step !t should be ofthe order of !l/up, !l being the minimum edge length of the control vol-ume. Please do note that the integration time-step has further constraintsfrom the eddy interaction model as described in the Stochastic modelingapproach.

4.4.2 Locating particles inside a control volume

Locating particles on a structured grid is straight-forward since physicalco-ordinates can be transformed into a uniform computational space. How-ever, this is not the case in unstructured grids. Westermann [158] pro-posed several approaches to locate particles in particle-in-cell codes based

65


a1

a3

a2a4 xp

Figure 4.4: 2D representation of a hexahedral cell and its respective partial areasa1 to a4.

on different requirements. Two such approaches which are suitable for thepresent work are discussed below.

Computing partial volumes

First approach was based on computing partial volumes. The nodes of thecontrol volume are joined to the particle location and the volumes of theresulting sub-volumes are compared with that of total cell volume. If thevolumes are exact, the particle lies in the control volume.

For example, consider a 2D representation of a hexahedral cell as shownin Fig. 4.4. If xp is the position of a particle within the cell, partial areasare created by connecting the nodes of the cell to the particle position. Ifthe summation of a1, a2, a3, and a4 is equal to the total area of the cell,then the particle lies within this cell.

Projection to cell faces

The second approach is to project the particle location onto the control vol-ume faces. If the projection is positive for all faces of the control volume,the particle lies within the control volume. In Fig. 4.5, xp is the particleposition within the cell, A, B, C and D are the faces of the cell, x is theposition of the cell face B, and f is the corresponding face-normal of cellface B. Now, the projection for face B is computed as PB = (xp !x)' f . Thesame procedure is repeated for faces A, C, and D. If P > 0 for all the faces,it implies that the particle is within the cell.

Both these methods were implemented in the code, and it was seen that themethod of computing partial volumes fails dramatically for highly skewed

66


fx

xp

A

BC

D

Figure 4.5: 2D representation of a hexahedral cell showing the position of a particlewithin a cell and a face-normal fi.

p1

2 34

56

78

9

Figure 4.6: Typical 2D arrangement of unstructured hexahedral cells.

meshes due to round-off inaccuracies in computing partial volumes whereas the projection method was found to be accurate even for highly skewedcontrol volumes. Since skewed meshes are an integral part of complexgeometries, the projection method should clearly be the preferred choice.

4.4.3 Cell search algorithm

Efficient searching of the particle when it crosses the present control vol-ume under consideration is the next challenge. The time step can be cho-sen to restrict the particle to remain in one of the neighboring cells of thepresent control volume. One should also note that each control volumemay have a variable number of neighboring cells depending on the geo-metrical details.

67


For example, consider a typical 2D cell arrangement which can be expectedin an unstructured hexahedral grid as shown in Fig. 4.6, with the particleposition to be in cell p. It can be seen that the cell p has 9 neighboringcells. The particle position is projected on to each of the control volumefaces of cell p. If the projection is negative for a face, the neighboring cellsharing the face is chosen as a reference cell. Now the particle position isprojected on to the faces of reference cell. If all the distances are positive,the reference cell is taken as the new cell with updated particle position.However, this procedure works only when the particle moves from cell p tocell 1, 3, 5, 6 or 8. Only in the case when the particle is not found in cell1, 3, 5, 6 or 8 we consider the remaining neighboring cells of p (2, 4, 7 or9), by checking each of the remaining cells in the same way of projecting tofaces.

As mentioned before, the chosen integration time-step !t restricts the par-ticle to remain in one the neighboring cells of the present control volume.Once the projection is negative for a particular face, the neighboring cellto which this face is common is identified using the face-cell connectivity.The particle location is now projected to the faces of neighboring cell andif all are positive, this neighboring cell is taken as the new control volumeinside which the particle lies. The procedure is repeated until the particlereaches a wall or outlet.

4.4.4 Interpolation of flow variables

Since complex geometries with unstructured meshes always include highlyskewed meshes, a robust interpolation scheme is a requirement. The particle-laden flows of Apte et al. [9] and Zaitsev [166] used a linear interpolationscheme, with the flow solver being second-order accurate in space and con-cluded that linear interpolation is robust and also provides adequate accu-racy for interpolating the flow parameters between nodes.

The viscous flux discretization in the flow solver of Numeca requires theknowledge of flow variables at cell vertices. For irregular hexahedronmeshes, a linearity-preserving interpolation based on the formulation ofHolmes and Connell [70] is used to ensure accurate transfer of variablesfrom cell-centers to vertices. The flow variables are computed at individ-ual particle locations within a control volume using the same generalizedlinearity-preserving interpolation scheme which is described below,

To interpolate at any point p surrounded by n number of cells, the value of

68


flow variable U at vertex p is given by,

Up =

n5

i=0

Ui!i

n5

i=0

!i

(4.27)

Ui is the cell center value of a flow variable. !i is the weighting functionwhich is defined as,

!i = 1 + 3x!xi + 3y!yi + 3z!zi (4.28)

where !xi, !yi and !zi are the coordinate differences between cell i andvertex p (!xi = xi ! xp, ...). The weighing coefficients 3x,3y and 3z arecomputed by assuming that Eq. 4.27 should be exact for linear evolutionof U . Hence we can write,

Ui = Up + !xi*Up

*x+ !yi

*Up

*y+ !zi

*Up

*z(4.29)

Rewriting 4.27 in the following form,

n5

i=0

!i (Ui ! Up) = 0 (4.30)

and inserting 4.29 in 4.30 results in,

n5

i=0

!i

#!xi

*Up

*x+ !yi

*Up

*y+ !zi

*Up

*z

$= 0 (4.31)

For linear functions, this relation should remain valid for any value ofderivatives. Since there are three unknowns determining ! i, three partic-ular sets of derivatives were chosen so that we obtain the following threeequations from Eq. 4.31, where !i is replaced by Eq. 4.28.

n5

i=1

!xi + 3x

n5

i=1

!x2i + 3y

n5

i=1

!xi!yi + 3z

n5

i=1

!xi!zi = 0

n5

i=1

!yi + 3x

n5

i=1

!xi!yi + 3y

n5

i=1

!y2i + 3z

n5

i=1

!yi!zi = 0 (4.32)

n5

i=1

!zi + 3x

n5

i=1

!xi!zi + 3y

n5

i=1

!yi!zi + 3z

n5

i=1

!z2i = 0

69


A

B

C

Figure 4.7: One dimensional representations of A) One particle injection from facecenter; B) Multiple particle injections from a single face; C) Multiple particle injec-tions from face-center, but different fluctuations.

This linear algebraic system with three unknowns 3x, 3y and 3z are solvedusing Cramer’s rule.

4.5 Boundary conditions

4.5.1 Inlet boundary condition

The particles are typically injected from the cell faces that mesh the inletsurface. Depending on the number of particle injections required, thereare three different possibilities.

One particle per face

This is the simplest way, where one particle is injected from the face-centerof each cell face that forms the inlet surface. A one dimensional represen-tation of this scenario is shown in Fig. 4.7A. The face-center is computedbased on the available face-vertex data. For example, consider a face inphysical space as shown in Fig. 4.8. The face center is now computed as,

70


xc =14

(x1 + x2 + x3 + x4) (4.33)

yc =14

(y1 + y2 + y3 + y4) (4.34)

Uniform distribution within a face

In Lagrangian modeling, it is required to track a huge number of particlesin order to obtain good statistical averaging. Most often, the number ofcell-faces available at the inlet are very nominal. Hence, we can inject mul-tiple particles from each cell-face by uniformly distributing them withineach face. A one dimensional representation of this scenario is shownin Fig. 4.7B. Uniform particle distribution is accomplished by Laplaciantransformation. The particles will first be distributed in isoparametricspace and then transformed into the physical space as described below,

In the isoparametric space as shown in Fig. 4.8, i = 1 : m, and j = 1 : nand m # n is the number of particles needed inside the face. The divisionin 1 and " directions are given by,

!1 =1max ! 1min

m + 1=

2m + 1

(4.35)

!" ="max ! "min

n + 1=

2n + 1

(4.36)

1 and " are defined as,

1(i) = !1 + i ' !1 (4.37)

"(j) = !1 + j ' !" (4.38)

The transformed locations on the physical space are calculated by,

xp(i, j) = n1(i, j)x1 + n2(i, j)x2 + n3(i, j)x3 + n4(i, j)x4 (4.39)

yp(i, j) = n1(i, j)y1 + n2(i, j)y2 + n3(i, j)y3 + n4(i, j)y4 (4.40)

x1!4 and y1!4 are the vertex co-ordinates in the physical space. n1!4 aredefined as,

71


x

y

)(i!

)( j"

)1,1( ## )1,1( #

)1,1()1,1(#

Laplacian transformation

Isoparametric space Physical space

),( 11 yx),( 22 yx

),( 33 yx),( 44 yx

Figure 4.8: Representation of the isoparametric space and the physical space.

n1(i, j) =(1 ! 1(i)) (1 ! "(j))

4(4.41)

n2(i, j) =(1 + 1(i)) (1 ! "(j))

4(4.42)

n3(i, j) =(1 + 1(i)) (1 + "(j))

4(4.43)

n4(i, j) =(1 ! 1(i)) (1 + "(j))

4(4.44)

One particle per face with random perturbation

In case of RANS simulations, we can take advantage of the way the eddyinteraction model works. We can inject multiple particles from the sameface-center position, but use a different random number ' in Eq. 4.12 dur-ing the first step of particle injection. In the case of unsteady RANS, LESand DES, injecting a set of particles at different time levels automaticallyensures random fluctuations on the particles.

72


pr

pu

Figure 4.9: Representation of the wall deposition of a particle.

Calculate the particle phase by using an eddy interaction model

Perform the fluid flow calculations to obtain a converged mean-flow solution

Figure 4.10: Uncoupled particle phase calculations.

4.5.2 Wall and Outlet boundary condition

The airway passage is normally wet and it is realistic to assume that theparticle sticks to the wall. As can be seen in Fig. 4.9, as soon as thedistance between the particle and the wall becomes equal to the radius ofthe particle, it is taken to be deposited on the wall. The same procedureapplies to the outlet of the domain.

4.6 Uncoupled and coupled calculations

In case of uncoupled calculations, the solution procedure is as shown inFig. 4.10. The time-averaged fluid flow calculations are performed till therequired accuracy is achieved and then the particles are introduced in thismean-flow solution and tracked using the eddy interaction model whichmodels the fluctuating part of the velocity.

73


Perform one fluid flow time-step calculation to obtain instantaneous-flow solution

Update the particle phase for the above time-step, without any EIM

Figure 4.11: Coupled particle phase calculations.

Fig. 4.11 shows the solution procedure for coupled calculations. The in-stantaneous fluid flow solution is calculated for a given physical time-step,and then the particle phase is updated in this instantaneous fluid field forthe same time-step duration. This fluid phase and particle phase calcula-tions are repeated in a loop till the particles deposit on the walls or reachthe outlet of the domain.

4.7 Programming language

The present all-hexahedral unstructured flow solver ofNumeca is based onthe object oriented programming (OOP) language C++ which has severaladvantages when compared to traditional languages such as FORTRANwhich still dominates the scientific computing community. Rather thantrying to fit a problem to the procedural approach of a programming lan-guage, OOP attempts to fit the language to the problem. The basic organi-zation of a C++ program is shown in Fig. 4.12.

Unlike structured programming where a problem is approached by divid-ing it into functions, in OOP, the problem is divided into objects. Thinkingin terms of objects rather than functions makes the designing and main-tenance of a huge industrial code easier. The basic idea is to make thedata and the functions that operate on this data into a single unit. If onewants to modify the data in an object, it can only be achieved by usingthe member functions of that object. No other functions outside this objectcan manipulate this data. This very feature simplifies the programming,debugging, and maintenance of the code. A typical C++ program consistsof several objects which interact with each other by calling one another’smember functions. A collection of objects together represents a class. Aclass basically serves as a blueprint or a plan which holds specific infor-mation about what data and functions will be included in the objects of

74


OBJECT

Data

Member function

Member function

OBJECT

Data

Member function

Member function

OBJECT

Data

Member function

Member function

Figure 4.12: The organization of a C++ program [82].

its class. One of the most attractive feature of OOP is inheritance, whichallows one to build new classes based on the old ones. The new class isreferred to as the derived class, which can inherit the data structures andfunctions of the original class, also referred as the base class. This allowsprogrammers to add new features to the existing ones, but without alter-ing the base class. This indeed confines errors, if any, to the derived class,which makes maintenance and debugging easier.

4.8 Data structures

Data structure is a way of storing data so that it can be used efficiently.A well-designed data structure allows a variety of critical operations to beperformed using less resources, both in terms of computational time andmemory space required.

All computational grids are made up of basic entities that are cells, faces,edges and vertices. The most complete data-structure showing all connec-tivities is shown in Fig. 4.13. The higher the available connectivities, theless searching operations are needed during a computation. However, thisis at the expense of the memory cost required to store the connectivities.

75


Cell Face

EdgeVertex

Figure 4.13: All possible connectivities between cells, faces, edges and vertices[114].

Cell

Vertex

Face

Figure 4.14: Connectivities used by flow solver module.

Flow solver connectivities

Since the implemented Lagrangian module is an extension of Numeca’sflow solver module, understanding the data structure of the flow solver isessential. In terms of connectivities, the flow solver module uses a light,face-based data structure but stores more mesh related information suchas face-normals or cell-volumes in order to avoid expensive re-calculations.Fig. 4.14 shows the connectivities built for the flow-solver module.

The choice of an appropriate data structure is mainly guided by the dis-cretization methods used. The flow solver of Numeca uses a finite-volumecentral scheme with artificial dissipation [75], which requires the knowl-edge of face-cell connectivity. The viscous fluxes, which are purely cen-trally discretized require the knowledge of velocity gradients at cell faces.In order to obtain the velocity gradients at cell faces, the knowledge of flow

76


Face

Cell

Vertex

Figure 4.15: Connectivities used by particle solver module.

variables at the cell vertices is needed. To perform an interpolation fromthe cell centers to the cell vertices, the face-vertex connectivity is needed[114].

Particle solver connectivities

As explained in section 4.4.3, it is impossible to identify the cell to whichthe particle moves when it travels from cell p to cell 2, 4, 7 or 9 unless weknow all the neighboring cells of p in advance. To identify the neighboringcells of each control volume in a domain, we need to know the faces eachcell is associated with. In other words, we need the cell-face connectivity.Once we have built this connectivity, the neighboring cells are identifiedby looping over the faces of each cell and using the face-cell connectivitywhich is built in the flow solver module.

It was concluded in section 4.4.1 that the integral time-step !t should beof the order of !l/up. !l is the shortest edge of a cell in which the particletravels. Each face of a cell is recognized using cell-face connectivity. Short-est edge of each face is constructed using the face-vertex connectivity whichis available from the flow solver module. A structured way of numberingthe face-vertex connectivity enables to construct the shortest edge of eachface. Finally, the shortest edge among all faces is taken as the shortestedge of the cell.

Fig. 4.15 summarizes the connectivities needed for particle solver mod-ule. Please note that the cell-vertex connectivity was also built in orderto perform the cell-search by computing partial volumes. However, thismethod was found to be inaccurate for highly skewed meshes as describedin section 4.4.3, and hence the cell-vertex connectivity can be skipped.

77


Data storage in particle solver module

In unstructured meshes, all mesh-related searching operations demandhuge amounts of computational resources. At the same time, it is not fea-sible to perform all the operations once and store them, due to memorycost. Hence, there should be an ideal combination of what should be storedand what can be computed dynamically during the simulation. All datastored are in vectors or multi-dimensional arrays. Below is the list of datawhich are stored or dynamically computed for the particle phase module.

Cell-face connectivity

The cell-face connectivity is required for the following two reasons,

a) To identify the surrounding cells of each cell.

b) To identify if a particle is inside the cell or not, by using the projec-tion method.

Considering a maximum grid refinement of 4, 24 integers per control vol-ume have to be stored. This connectivity is built at the beginning of parti-cle solver module before the particle injection.

Surrounding cells connectivity

The surrounding cells are required for the following two reasons,

a) For interpolation at any point inside a cell.

b) For cell searching during particle tracking.

Considering a maximum refinement of 4, 56 integers per cell have to bestored. Having such a huge storage in memory for each control volume isnot practical. Hence, the surrounding cells are dynamically calculated asand when the particle travels through that control volume.

Shortest edge of a cell

This is required to define the time step for integrating the particle equa-tions. One real variable has to be stored per cell. This is performed at thebeginning of the particle phase.

78


Boundary cells

The ID’s of boundary cells are also stored in a single array. This is helpfulfor two purposes.

a) For deciding if the particle has crossed a boundary face or not.

b) To consider the boundary values when interpolating the flow parame-ters at any point inside a boundary cell or inside any cell associated withthe boundary cell.

4.9 Flow chart of particle solver module

The flow chart in Fig. 4.16 gives a brief of the steps involved in trackinga single injected particle using the present particle solver module imple-mented in the unstructured all-hexahedral C++ flow solver of Numeca.Once the inlet particle distribution is decided, an array containing the x,y, and z positions of the injections is generated. Now, the particles areinjected one by one by looping over the number of injections. Once the in-jection position is known in a given control volume, the surrounding cellsof this control volume are found using the cell-face and face-cell connec-tivities. The flow variables at the particle position are interpolated usingthe variables available in the present control volume and its surroundingcells. Once the flow variables are known, estimates of eddy length and timescales are made based on the turbulence kinetic energy and its dissipationrate. An estimate of the fluctuating velocity is also calculated based on thekinetic energy level and employing a random number generator. Basedon the prescribed injection properties of the particle such as the density($p), diameter (dp), injection velocity (up), and the interpolated fluid veloc-ity at the injection position (u), the external forces acting on the particleare calculated. As mentioned before, the two main forces considered in thepresent study are the drag force and the gravitational acceleration. Oncethe forces are computed, the physical time-step for integration of the par-ticle equations is determined based on the cell size of the present as wellas the surrounding cells, and the eddy parameters. One time-step is per-formed to obtain the updated particle velocity and position. It is checkedif the new particle position is within the present control volume. If no, thecontrol is passed back to compute the surrounding cells of the new con-trol volume. If yes, it is further checked if the current control volume is aboundary cell. If it is not a boundary cell, the control is passed to inter-polating the flow variables at the new location and continuing the particletracking until the distance between the particle center and the nearest

79


Determine the inject co-ordinates based on the required inlet distribution and inject one particle

Compute the surrounding cells of the current control volume where the particle lies

Interpolate the flow variables at the particle position

Determine the eddy parameters and fluctuating velocity using interpolated flow variables and the

random number generator

Compute the forces acting on the particle

Determine the integration time-step based on the cell size & eddy parameters

Perform a time-step to obtain updated particle position and velocity

If the particle is still in the current cell

Check the distance between particle position & the nearest wall/outlet

Particle is stuck on wall or reached outlet

If the current cell is a boundary cell

If the distance to wall/outlet <= particle radius

YESNO

NOYES

NO YES

Figure 4.16: Flow chart demonstrating the steps involved in tracking one injectedparticle.

80


wall/outlet becomes equal to or less than the radius of the particle.

4.10 Testcases

4.10.1 Analytical solution

The mathematical implementation of the code can be checked by formu-lating a test-case for which the analytical solution exists. Consider thefollowing ordinary differential equation which governs the particle motion,

%d2rp

dt2+

drp

dt= u(rp) (4.45)

t = 0 : rp = rpo;drp

dt= upo

where u(rp) is a linear function describing carrier flow velocity distribu-tion. % is the relaxation time due to drag force.

Considering the carrier flow velocity components, ux = xp, uy = yp, uz = 0in Eq. 4.45 results in the following two independent equations,

%d2xp

dt2+

dxp

dt! xp = 0 (4.46)

t = 0 : xp = xpo;dxp

dt= upo

%d2yp

dt2+

dyp

dt! yp = 0 (4.47)

t = 0 : yp = ypo;dyp

dt= vpo

The analytical solution for particle positions in time is given by,

xp(t) = P1exp(31t) + P2exp(32t) (4.48)

P1 =32xpo ! upo

32 ! 31; P2 =

31xpo ! upo

31 ! 32; 31,2 =

!1 ±)

1 + 4%2%

yp(t) = Q1exp('1t) + Q2exp('2t) (4.49)

Q1 ='2ypo ! vpo

'2 ! '1; Q2 =

'1ypo ! vpo

'1 ! '2; '1,2 =

!1 ±)

1 ! 4%2%

81


0 0.5 1 1.5 2 2.5 3 3.5 4−0.5

0

0.5

1

1.5

2 Tau = 1Tau = 0.1Tau = 0.001Computed trajectory

Figure 4.17: Comparison of simulated particle trajectories with analytical solutionfor different relaxation times.

* : analytical solution for % = 0.001; xo = 0.35! : analytical solution for % = 0.1; xo = 0.25+ : analytical solution for % = 1; xo = 0.15! : computed trajectory of the particles

Fig. 4.17 shows that the computed trajectories of particles are in verygood agreement with the analytical solution for a varied range of particlerelaxation times, which proves that the mathematical part is correctly im-plemented. Please note that drag force is the only considered part of theinterphase forces.

4.10.2 2D Planar mixing layer

Chang et al. [24] noted that very few measurements are available in re-lation to the polydispersed properties of two-phase flows and made it anobjective to provide well-defined benchmark quality data for model valida-tion. The experimental setup is described below.

Measurements were made in a vertical tunnel which was split in to twoseparate flow paths by a central splitting plate. The contraction ratio ofthe tunnel was 16:1 with a 150 # 150 mm2 cross-sectional area at thetest section. The mean velocities at high and low-streams were 10.0 and2.3 m/s respectively. The high speed stream was seeded by polydispersed

82


(a)

Figure 4.18: The computational domain.

water drops with diameters ranging from 3-100 µm. Data was collectedat four stations in the streamwise direction, 5, 20, 40 and 80 mm from theseparation point of the splitting plate. The data obtained at 5 mm providedall the required inlet boundary conditions for the flow simulation, exceptfor the rate of dissipation of turbulent kinetic energy. Additional measure-ments of turbulent shear stress and velocity gradients were performed at5 mm and the rate of dissipation was estimated by using the Boussinesqhypothesis.

The carrier flow field is simulated using Numeca’s flow solver for unstruc-tured hexahedron grids, FINE-Hexa 2.1/. The computational domain isas shown in Fig. 4.18. It varies from 5 mm to 120 mm in the streamwisedirection and -35 mm to +35 mm in the transverse direction. Box adapta-tion was used to refine the mesh in the region of the mixing layer. Velocity,kinetic energy and the dissipation profiles are specified at the inlet, staticpressure at the outlet and the mirror for boundaries.

Fig. 4.19 summarizes the fluid phase results. It can be seen that thestreamwise velocities are very well predicted whereas the transverse ve-locities are not. Increase in mesh size did not show any improvements. Za-itsev [166], who developed the Lagrangian module in the structured codeof Numeca performed the same test case for model validation and also re-ported the under-estimation of the transverse velocity field. It can be seenthat the peak values of kinetic energy move from the upstream side (highvelocity region) towards the downstream side (considerably low velocities).Two k!&models were tested, namely the low-Reynolds number Yang-Shihmodel [164] and an extended wall function model of Hakimi [61]. The LRN

83


model of Yang-Shih provided the best results.

Fig. 4.20 represents the streamwise velocity predictions of the particles.The comparison between the predictions and measurements is favorable.However, it is seen that the drop penetration depth into the low-speedstream side considerably reduces with reduction in particle diameter. Thedispersion of particles is due to the combined effect of fluctuating veloci-ties and inertia of the particle. It is known that the turbulent dispersionof large particles is dominated by their inherent inertia. This explains thedeepest penetration depth predicted for large particles (80 µm). The rea-son for reduction in penetration depth for smaller particles is prehaps dueto the fluctuating velocity. The fluctuating velocity is predicted using arandom number generator with zero mean and standard deviation of one,but theoretically, we should choose a random number generator with zeromean and standard deviation between -, to +,, which is impractical dueto limited computer capabilities.

The spanwise velocity predictions are given in Fig. 4.21. The under-prediction of spanwise velocities for all particle sizes is understandablydue to under-prediction of transverse velocity of the flow. However, the ex-tent of under-prediction reduces with increasing particle diameter, whichagain highlights that the heavier particles are less sensitive to turbulentdispersion effects.

84


Figure 4.19: Flow field velocity and kinetic energy profiles.+ : experimental data [24].! : computed profiles.

85


Figure 4.20: Streamwise velocity profiles of the particles.+ : experimental data [24].! : computed profiles.

86


Figure 4.21: Spanwise velocity profiles of the particles.+ : experimental data [24].! : computed profiles.

87


88

Chapter 5

Application I: Fluid Flowand Particle Deposition inUpper Airways

Contents5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 895.2 Model preparation . . . . . . . . . . . . . . . . . . . . . 915.3 Numerical methods . . . . . . . . . . . . . . . . . . . . . 93

5.3.1 Fluid phase . . . . . . . . . . . . . . . . . . . . . . . 935.3.2 Particle phase . . . . . . . . . . . . . . . . . . . . . 94

5.4 Quality control . . . . . . . . . . . . . . . . . . . . . . . . 945.5 Results and discussion . . . . . . . . . . . . . . . . . . . 97


5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.1 Introduction

Although inhaled medication is the preferred method of drug administra-tion to the lungs, aerosol deposition in the extrathoracic airways consti-tutes a major obstacle to efficient aerosol drug delivery to the intratho-racic airways. The overall geometrical complexity of the extrathoracicpathway, with its bends and sudden cross sectional changes, poses seriouschallenges for both experimental and computational studies. In addition,

89

CHAPTER 5. APPLICATION I: FLUID FLOW AND PARTICLEDEPOSITION IN UPPER AIRWAYS

experimental aerosol deposition data obtained on various realistic upperairway casts indicate considerable intra- and inter-subject variability ofthe extrathoracic airway deposition [58]. The influence of the inlet of theaerosol device, as well as individual mouth and trachea morphology on lo-cal and total aerosol deposition suggests that individualized computationof extrathoracic deposition would result in more accurate estimation of theamount of aerosol available to the deeper lung for a given patient or fora given aerosol device. Assuming that individualized imaging of the ex-trathoracic airway can be obtained, an individualized estimate of aerosoldeposition would require a computational fluid dynamics (CFD) simula-tion that can be performed on an unstructured mesh with an appropriateparticle tracking module. Indeed, the complexity of a realistic structureand the distorted nature of its walls make it difficult to mesh with a struc-tured grid.

Previous studies have used upper airway models with varying degrees ofgeometrical simplification for simulation or experimental studies of deposi-tion [167, 40, 87, 171, 102]. The deposition study undertaken by Grgic et al.[58] in seven casts of realistic extrathoracic models, which were thought tobe representative of magnetic resonance imaging scans obtained in 80 sub-jects, showed pronounced inter-subject deposition variability attributed tothe variations in the morphological dimensions with individuals studied.While acknowledging the necessity for individualized aerosol depositionstudies, these authors also developed an empirical deposition curve simi-lar to the most widely used Stahlhofen curve [142], which could accountfor geometrical differences such that different deposition curves would col-lapse onto one curve. This was done by incorporating a critical lengthscale which is characteristic of the geometry, into the definition of theStokes number. In the case of total deposition, Grgic et al. [58] used a so-called equivalent diameter, computed on basis of the overall geometry vol-ume and its total centerline path length. A supplementary dependence onRe, observed experimentally, was also incorporated and the best fit curveshowed a dependence on the product Stk 1.91Re0.707. A similar approachwas used in the experimental study by DeHaan and Finlay[40] where in-let diameter was the crucial length introduced into the definition of Stokesnumber to compute mouth deposition. In this case, the deposition curveonly contained Stk1.91.

Based on the recent experimental observations on inter-subject variabilityof the extrathoracic airway deposition, the need for considering realisticmodel for CFD study is recognized and an efficient Lagrangian particletracking module for unstructured grids is developed to handle complex ge-

90


ometrical features. The details of Lagrangian modeling are as explained inChapter 4. Even though the concept of unstructured grids exists since long,the practical applicability is still under budding stage, especially for two-phase simulations. In this view, the present work is the first step towardsapplicability of unstructured grids for biomedical applications which is byfar the only option to remove expensive experiments for realistic geome-try configurations. To the best of our knowledge, these are the first CFDsimulations of particle deposition in a truly realistic extrathoracic airwaygeometry.

5.2 Model preparation

Local ethics approval was obtained for a multi-slice CT imaging study infive otherwise healthy never-smoker male subjects who were referred forlung imaging due to possible low concentration occupational asbestos ex-posure. CT images were acquired using a Sensation 16 CT-scanner withfollowing settings: acquisition 16#0.75 mm, reconstruction slice thickness1mm, pitch 1.25, rotation time 0.5 seconds, Care dose, reference mAS 100,KV 120, Medium filter B45f. An approximately oval shaped mouth-piecewas used with major axis 17.36 mm and the minor axis 11.68 mm. Noseclamps were used to prevent nose breathing. Imaging was initiated abovethe nasal cavity and synchronized to start with the subject slowly inhal-ing at about 15-20 l/min to near total lung capacity where a breath holdwas inserted to complete data acquisition just below the carina. An exam-ple of one such CT-scan is as shown in the Fig. 5.1. From the scans ob-tained on the five subjects, one representative image of upper and centralairway anatomy was selected by the pulmonary radiologist and pneumol-ogist. The multi-slice CT images were imported in to a 3D reconstructionsoftware Amira 4.0 and a raw upper and central airway 3D geometry wasconstructed by triangulation.

The highly irregular and complex nature of the model (Fig. 5.1) makesthe creation of a structured grid tedious and work-intensive. For suchcomplex geometries, unstructured meshes are more feasible. Using Nu-meca’s unstructured grid generator - Hexpress (2.2/), an all-hexahedralunstructured mesh is generated.

91


Figure 5.1: A: Sample of a CT-Scan; B: The reconstructed 3D airway geometry withdifferent angle views of mouth cavity.

92


5.3 Numerical methods

5.3.1 Fluid phase

The computational domain is imported intoNumeca (Fine/Hexa 2.3-1) com-pressible Reynolds-Averaged-Navier-Stokes (RANS) solver. The basic RANSequation in a cartesian frame of reference integrated over a control volumeV is expressed as,

( (

V

(*"*t

dV +( (

S

+F · +dS =( (

S

+Q · +dS (5.1)

The conservative variables " being density ($), 3 components of velocity($U, $V, $W ) or energy ($E). F and Q represents inviscid and viscous fluxrespectively. In the low subsonic Mach number regime, time-marchingalgorithms designed for compressible flows show a pronounced lack of effi-ciency due to large disparity of acoustic wave speed u+c compared to shearand entropy waves which are convected at fluid speed u. This problemis overcome by bringing a correction to discretization of the conservationequations called preconditioning. The preconditioning matrix used here isa combination of those suggested by Turkel[147] and Choi and Merkle[26].The central scheme with Jameson type dissipation [75] is used for spatialdiscretization. A fourth order Runge-Kutta scheme is used for time inte-gration. Multigrid V-cycle strategy [67] with three grid levels is used forconvergence acceleration. The flow is assumed to be converged when thedimensionless mass and momentum residuals were less than 10!3.

The flow is modeled using low-Reynolds-number shear-stress-transport k!! turbulence model of Menter [105], with the recent modification broughtto the eddy viscosity by Menter et al. [106]. The value of turbulent kineticenergy at the inlet is prescribed assuming 5% turbulence intensity. Whenthe air is inspired from still atmosphere, the turbulence intensity levelswill generally be very small, however, the mist of aerosols coming from theinhaler may enhance the turbulence intensity at the inlet. Heenan et al.[64] noted that varying turbulence boundary conditions had little effect onthe final results in their simulations on idealized mouth-throat geometrywith similar flow rates as in our simulations. We specifically carried outthree different simulations by plugging 1%, 5% and 10% turbulence inten-sities respectively at the inlet. The maximum difference in total depositionof a mixture of five different particle diameters (2, 4, 6, 8 and 10 µm) wasseen to be less than 1.9%, signifying the insensitivity of particle deposi-tion on the inlet turbulence intensity. This is understandably so because

93


Figure 5.2: Left: Normalized Axial velocity contour; Right: Normalized kineticenergy contour.

the increase in turbulent kinetic energy levels is mainly triggered by thecomplex geometrical features of the mouth-throat geometry.


With a view of efficient particle tracking on unstructured grids, a stochas-tic Lagrangian trajectory model was implemented in the flow solver of Nu-meca. The details of Lagrangian modeling are as described in Chapter4.

5.4 Quality control

The reliability of fluid flow solver using the present LRN SST k ! ! modelto simulate transitional flow is validated by performing a simulation inan axisymmetric tubular flow with stenosis [7] at a transitional Reynoldsnumber of 2000. The normalized axial velocity contours and kinetic energycontours are shown in Fig. 5.2. Two dimensional cross-sectional veloci-ties at different diameters downstream of stenosis (Fig. 5.3) compare well

94


0 0.5 1012345

u/u in

r/R

z = 0d

Exp. data (Ahmed 1983)LRN k−w (Zhang &Kleinstreuer 2002)SST k−w (present)

0 0.5 1012345 z = 1d

0 0.5 1012345 z = 2.5d

0 0.5 1012345 z = 4d

0 0.5 1012345 z = 5d

0 0.5 1012345 z = 6d

Figure 5.3: Comparison of normalized axial velocity (at different sections down-stream of the glottis) for the constricted tube at Re = 2000 with the ex-perimental data of Ahmed and Giddens[7] and the simulations of Zhang andKleinstreuer[168].

95


Figure 5.4: Velocity magnitude at section A-B near glottis for 3 different grid sizesat an inhalation flow rate of 60 l/min.

with the experimental data of Ahmed and Giddens[7] and the LRN k ! !model simulations of Zhang and Kleinstreuer[168] which is suggested tobe the best readily available LRN turbulence model. The presently em-ployed LRN SST model was able to adequately predict the uniform flow atthe glottis and the rapid decrease in velocity at z = 4d due to flow transi-tion. The peak velocities in the center of the domain at z = 2.5d and z = 4dare also comparatively well predicted.

As we are dealing with transitional flow, it is certain that few portionsof the flow will remain locally laminar. We performed both laminar andturbulent simulations for a very low flow rate of 3 l/min which is certainlylaminar and compared the velocity profiles at different sections in the ge-ometry. The present LRN k ! ! model was able to reproduce the laminarflow behavior which gave further confidence in the turbulence model em-ployed.

The adequacy of grid resolution is tested by verifying both fluid and parti-cle results. The fluid results are checked on three different mesh sizes atthe maximum flow rate of 60 l/min. In Fig. 5.4 we see that the velocitymagnitudes taken at a cut-section near the glottis region agrees well for550,000 and 950,000 meshes. Also, the total pressure drop from inlet tooutlet for both meshes varied only by 0.5 Pa.

The effect of grid-resolution on the particle deposition is tested by consid-ering two different mesh sizes of 550,000 and 950,000. Three different

96


particle diameters (2, 8 and 14 µm) and two different flow rates of 30 and60 l/min were considered. The maximum difference in total deposition didnot exceed 4%. Hence the 550,000 mesh suffices for accurate flow and par-ticle simulations.

The effect of total number of injected particles on the deposition percent-age was also tested by considering 1500 and 3500 particle injections. Fourdifferent particle diameters (2, 6, 10 and 20 µm) at flow rates of 30 and 60l/min were tested. The maximum difference in total deposition was seen tobe less than 2% and hence 1500 particles suffice for accurate prediction ofdeposition percentage. This observation is consistent with that of Matidaet al. [102] who stated that their simulated deposition results on an ide-alized geometry didn’t change when the number of particles injected werechanged from 1000 to 10,000.

5.5 Results and discussion

5.5.1 Fluid phase

Fig. 5.5 shows contours of the velocity magnitude for sedentary breathingcondition, i.e., 15 l/min. In addition to velocity magnitude contours, thecross-sectional views also show the secondary velocity vector lines. Thecross-sections D1-D2, E1-E2 and F1-F2 are approximately one, three andsix diameters from C1-C2 which marks the larynx (glottis). We refer theside towards C1 as posterior and the side towards C2 as anterior. Theaxial-velocity profiles are highly skewed with many recirculation zones dueto the complex nature of the domain. The flow entering through mouth-piece impinges on the tongue and accelerates as it moves through the mid-dle region of the mouth due to the reduction in cross-sectional area. Theaxial velocity profile at section A1-A2 shows that the maximum velocityis not at the center as in most of simplified geometries, but is inclinedtowards anterior side. This feature may have considerable effect on theparticles and may result in higher mouth deposition compared to simpli-fied geometries where the mouth region is symmetric. The presence of theuvula just next to the soft palate at the end ofmouth region (Fig. 5.1) formsa huge restriction for the airway passage resulting in the flow entering thepharynx in the form of a jet which is from hereon referred to as oropha-ryngeal jet. The streakline representation in Fig. 5.5 shows recirculationregions in the epiglottis and upper part of pharynx region as a result of theoropharyngeal jet. When the flow bends from the mouth to the pharynx,it experiences complex secondary motions due to the pressure gradient.Section B1-B2 shows three distinct secondary vortices as the flow moves

97


Figure 5.5: Velocity profiles (cm/s) in the oral airway model at 15 l/min. Above:Mid plane velocity contour and 2-D streaklines. Below: Axial velocity contours andsecondary velocity vector lines at six different cross-sections.

98


Figure 5.6: Velocity profiles (cm/s) in the oral airway model at 60 l/min. Above:Mid plane velocity contour and 2-D Streaklines. Below: Axial velocity contoursand secondary velocity vector lines at six different cross-sections.

99


downstream of the pharynx. A sharp step at the end of pharynx on theposterior side results in a laryngeal jet beginning from the glottal regionand developing towards the anterior side of trachea. The secondary mo-tions at the glottis region (C1-C2) are considerably damped out in a veryshort span between B1-B2 and C1-C2. The posterior side peak velocity atC1-C2 develops towards the anterior side as the flow moves downstream.At three diameters downstream, secondary motions again become moreprevalent due to highly skewed laryngeal jet and partly also because ofthe complex nature of trachea. It is interesting to note that the tracheais at an inclination to the mouth inlet face, where as most of the simpli-fied geometries assume a straight tube parallel to inlet face. This indeedmay also increase particle deposition, especially those with higher stokesnumber due to inertial effects. As a result of laryngeal jet, there is a bigflow separation on the posterior side as seen in the streakline representa-tion. Heenan et al. [64] observed that the development of the laryngealjet widely varies across studies and that it is essentially due to the vari-ation in cross-sectional shapes used to model the glottis. Also, Brouns etal. [20] studied the influence of glottic aperture on the tracheal flow inan idealized mouth-throat geometry and showed how the interaction be-tween glottis size and shape on the one hand, and the geometry of themouth-throat model on the other hand, is crucial to the laryngeal jet andthe overall fluid flow field.

The kinetic energy plot in Fig. 5.7 shows amplification and transition toturbulence soon after the glottis region. In the previous simulations onan idealized airway model, Kleinstreuer and Zhang[87] had indicated thatthe turbulence intensity prescribed at inlet is damped out and laminar flowprevails throughout the geometry under such low flow rate condition. Bycontrast, Heenan et al. [64] studied the effect of inlet boundary conditionin an idealized oropharynx model and found transitional flow regimes atsimilar flow rates. Stapleton et al. [143] refers to a previous experimentalstudy where turbulent structures were found in the flow of tracheal castsfor inhalation flow rates above 3 l/min. From these different observations,we conclude that the degree of turbulence critically depends on the degreeof geometrical complexity of the extrathoracic airway model.

At 60 l/min (Fig. 5.6), the flow features in the mouth and the pharynxremain very similar to those obtained for 15 l/min. However, the laryn-geal jet which develops soon after the glottis region, expands more quicklyand transfers more momentum to the flow downstream. Corcoran andChigier[28] observed a similar behavior in their phase doppler study of thelaryngeal jet in a cast of larynx/trachea. The length of the separated flow

100


Figure 5.7: Turbulent kinetic energy profile (m2/s2) in the oral airway model for15 l/min (left) and 60 l/min (right).

zone in the trachea due to the laryngeal jet reduces considerably comparedto 15 l/min due to faster expansion of the jet.

Fig. 5.8 shows the area-averaged pressure drop (p! pin) across the airwaymodel for 15, 30 and 60 l/min. The pressure drop remains very nominalthroughout the airway geometry for 15 l/min. For 30 and 60 l/min, we seethat the major losses are in the mouth and larynx region. The drop in thesecond half of the mouth and the larynx region is a direct result of the flowacceleration.


Total deposition

Since the flow velocities in the extrathoracic airways are relatively highand the residence time of the particles are short, inertial impaction is thedominant mechanism for particle deposition. Hence it is common to repre-sent the extrathoracic deposition as a function of inertial impaction param-eter $pd2

pQ. However, the experimental data available in literature show alot of scatter when plotted with respect to the inertial parameter and this

101


0 50 100 150 200 250 300−70

−60

−50

−40

−30

−20

−10

0

10

Axial Length (mm)

Pres

sure

Dro

p (P

a)

15 l/min30 l/min60 l/min

Mouth Pharynx Larynx Trachea

Figure 5.8: Area average pressure drop (p! pin) (Pa) at every 5mm along the axialdirection.

10−3 10−2 10−1 100 1010

20

40

60

80

100

Stk Re0.37

Depo

sitio

n (%

)

Experimental fit (Grgic 2004)Simulated deposition

Figure 5.9: Simulated total deposition (open symbols) as a function of Stokes num-ber and Reynolds as defined in Grgic et al. [58]. The experimental best fit curve isalso represented (solid line).

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10−3 10−2 10−10

20

40

60

80

100

Stk

Ora

l Cav

ity D

epos

ition

(%)

Experimental fit (DeHaan 2004)Simulated deposition95% prediction limits

Figure 5.10: Simulated oral cavity deposition (%) as a function of Stokes numberas defined in DeHaan and Finlay[40]. The solid line represents the experimentalbest fit curve.

scatter is generally attributed to inter-subject variations. In order to takethe geometry into account, Grgic et al. [58] proposed to plot the extratho-racic deposition as a function of Stokes number. First, the Stokes numberis defined as $pd2

pU/18µD, where the mean diameter, D, is calculated us-ing the cast volume V and the path length L of the central sagittal line ofthe model (D = 2

6V/(L). The corresponding velocity scale is calculated

as U = QL/V . Finally, a Reynolds number dependence is incorporated byplotting deposition data against Stk1.91Re0.707. Using this representation,our CFD simulated total deposition data are plotted in Fig. 5.9, along withthe experimental based best fit curve of Grgic et al. [58]. While the overalldeposition prediction pattern agreed with the experimental fit, there stillremained considerable over-prediction in the low Stokes-number range.This observation is consistent with that of Matida et al. [102] in their ide-alized model and, Xi and Longest [161] in their CT-based realistic model ofthe upper airways.

For the specific purpose of aerosol deposition in the mouth cavity, we usedthe method of DeHaan and Finlay[40], which in our application, consistedof defining the Stokes number as $pd2

pCc(U + 2k0.5)/18µD, where U is theaverage inlet velocity, D is the inlet diameter, and k is the inlet turbulentkinetic energy. Using this representation, our CFD simulated mouth de-

103


Figure 5.11: Simulated deposition values (expressed as % of particles at modelinlet) in three model subparts.

position data are plotted in Fig. 5.10. Again, the good agreement betweensimulated mouth deposition and the best fit curve lends further support tothe validity of our computational methods.

Particle deposition in model subparts

Fig. 5.11 shows that the total deposition increases with increase in par-ticle diameter, suggesting that inertial impaction is the dominant deposi-tion mechanism. The relative deposition in each model subpart, expressedas a fraction of total aerosol entering the mouth, is largely unaffected bychange in flow rate or particle diameter. Note however that for particlesabove 12 µm in the case of 60 l/min, the mouth deposition becomes so highthat the subsequent pharynx and larynx-trachea deposition reduces withfurther increase in particle diameter. Clearly, the mouth region acts as aneffective filter and is responsible for a major percentage of total deposition.This agrees with the experimental observation of Grgic et al. [58] where

104


the oral cavity accounted for most intense deposition in six out of the sevenrealistic upper airway casts studied. In contrast, studies on an idealizedoral airway model (e.g.,[171, 59]) show considerably smaller mouth depo-sitions, which maybe an indication of the need for realistic models of themouth cavity to reliably estimate mouth deposition.

In addition to the deposition values in the model subparts, as representedin Fig. 5.11, a more detailed description of the deposition patterns of indi-vidual particles is shown in Fig. 5.12. Since compendious representation of3D deposition patters is often difficult, we project the particle coordinatesonto a 2D plane. Fig. 5.12 shows the particle deposition coordinates alongwith a best possible 2D cutting plane that passes through the center of thedomain. Note that the geometry is not symmetric and the cutting planedoes not cover the outermost boundary of the domain, which explains whysome particles appear to be deposited outside this plane.

2 µm: For sedentary breathing, there is almost no deposition in the ovalshaped mouth-piece where as for normal breathing, there is a concentrateddeposition. This can be attributed to increase in the magnitude of turbu-lent fluctuations (which is proportional to the flow kinetic energy) whencompared to sedentary breathing. For heavy breathing condition, in ad-dition to mouth-piece deposition, inertial forces become important and weobserve a concentrated deposition at the beginning of tongue. Depositionin pharynx remains low for all three flow rates showing that most of theparticles which escape deposition in the first half of mouth will reach theoutlet (end of trachea).

6 µm: Relative deposition patterns remain almost the same as of 2 µmwith slightly increased deposition in second half of the mouth and in thepharynx region.

10 µm: For both sedentary and and normal breathing condition, thereare some particles deposited at the end of the tongue region as the flowbends from mouth to pharynx. As the particles move downstream of thepharynx, they experience a sharp step at the end of pharynx (beginningof larynx) resulting in some regional deposition which increases with in-crease in flow rate. In addition to these depositions, for heavy breathing,we observe considerable deposition towards the end of the mouth in theuvula region (Fig. 5.1) because the inertia of particles dominates and theparticles can no more follow the sudden bend in the flow from mouth topharynx. There is also a concentrated deposition on the outer wall of thepharynx which is due to the direct impingement of oropharyngeal jet on to

105


15 l/min

2 µm

30 l/min 60 l/min

6 µm

10 µ

m14

µm

18 µ

m

Figure 5.12: Two-dimensional representation of individual particle deposition.

106


0 10 20 30 40 50 600

2

4

6

8

10

12

14

16

18

20

22

Flow rate (l/min)

Depo

sitio

n (%

)

1 µm3 µm5 µm7 µm

(a)

5 10 15 20 25 300

2

4

6

8

10

12

14

16

18

20

22

Flow rate (l/min)

Tota

l dep

ositio

n (%

)

1 µm3 µm5 µm7 µm

(b)

Figure 5.13: (a) Simulated deposition percentage in the model as a function ofparticle size and inhalatory mass flow. (b) Total deposition in the airway modelwhich is a sum of mouth-throat model deposition (present simulations) and thelung model deposition extending from the trachea up to the segmental bronchi[151]. Please note the different scales on x-axis between figures (a) and (b).

107


the outer surface of pharynx.

14 µm: For sedentary and normal breathing, the relative deposition pat-terns remain same as that of 10 µm. For heavy breathing we see additionaldeposition region at the middle region of the mouth on its outer walls whichis clearly due to increased inertia forces.

18 µm: For sedentary breathing, in addition to previous observations at14 µm, we see a chunk of particles trapped in the cavity region soon afterthe mouth-piece where there is flow recirculation. The number of particlestrapped in this region decreases with increase in flow rate suggesting thatit is mostly the turbulence effect. For normal breathing, we observe fewsmall patches of deposition in the middle region of mouth which were notseen for 14 µm, suggesting the beginning of inertial effect in this region.There is also a considerable concentration on the outer wall of pharynxdue to the oropharyngeal jet. For heavy breathing, the deposition patternsare less scattered and more intensely concentrated at the mouth-piece, be-ginning of tongue and upper wall of the middle mouth region. When wecompare the mouth-piece deposition with that of 2 µm, it is evident thatmost of the particles are deposited on lower half of the mouth-piece, sug-gesting that inertia dominates and the turbulence effects are not at all felt.

Finally, Fig. 5.13(a) shows the simulated deposition as a function of flowrate, for particle diameters between 1 and 7 µm, including also a flow rateas low as 3 l/min, to verify whether such a low flow rate generates greaterdeposition as gravity becomes important. This was clearly the case, and asexpected, this gravitational effect at 3 l/min was amplified for the heavierparticles. For the range of flows between 3 and 30 l/min in Fig. 5.13(a), sim-ulated deposition rates have been reported for a 10-generation bronchialmodel by van Ertbruggen et al. [151]. When combining our own depositionefficiencies obtained in mouth and trachea with those predicted from thebronchial model [151], i.e., as if the latter were directly appended to theend of our trachea, we can obtain a rough estimate of total deposition fordifferent combinations of flow and particle diameter which is shown in Fig.5.13(b). These cumulated deposition data indicate that for normal breath-ing conditions, inhalatory deposition in the upper and central airways forthe particle size 5 µm, which is usually referred to as the upper limit ofthe respirable range for inhalation drugs, does not exceed 12% when theaerosol is at rest before inhalation (e.g., inhaled from a spacer).

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5.6 Conclusions

The original contributions of this work are:

1. The realistic airway flow simulations show that the laminar to tur-bulent transition, especially at low flow rates is sensitive to the com-plexity of the airway model. Flow transition was seen soon after theglottis region for a low flow rate of 15 l/min, which are not reportedin the simplified geometry simulations found in the literature.

2. Mouth region acts as an effective filter and is responsible for majorpercentage of total deposition, which again is not the case with sim-plified models. This indicates the need for more realistic representa-tion of mouth cavity to reliably predict regional mouth deposition insimplified models.

3. The deposition due to sedimentation becomes important for heavierparticles at very low flow rates resulting in higher depositions com-pared to the depositions at a considerably higher flow rate.

4. The cumulative depositions (upper airway + central bronchial tree)indicate that for normal breathing conditions, inhalatory depositionin the upper and central airway tree for the particle size 5 µm, whichis usually referred to as the upper limit of the respirable range forinhalation drugs, does not exceed 12%.

109


110

Chapter 6

Application II: ConvectiveMixing in Upper Airways

Contents6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1116.2 Materials and methods . . . . . . . . . . . . . . . . . . . 113

6.2.1 Experimental methods . . . . . . . . . . . . . . . . 1136.2.2 Numerical methods and quality control . . . . . . . 116

6.3 Theoretical axial dispersion coefficient . . . . . . . . 1176.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.4.1 Experimental results . . . . . . . . . . . . . . . . . 1176.4.2 CFD results . . . . . . . . . . . . . . . . . . . . . . . 120

6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 1206.6 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . 1266.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.1 Introduction

The upper airway geometry with its glottic narrowing embedded in a tor-tuous oropharyngeal pathway, affects aerosol deposition and dispersionon its way to the lungs. Previous experimental and numerical studieshave mainly concentrated on aerosol deposition, using realistic upper air-way models with varying degrees of geometrical simplification [76, 102].Aerosol transport is generally considered under steady flow conditions, ex-cept for a recent work by Grgic et al. [60], where small volumes of aerosols

111

CHAPTER 6. APPLICATION II: CONVECTIVE MIXING IN UPPERAIRWAYS

(aerosol boluses) delivered in different stages of the inhalatory phase, wereused to study the effect of flow accelerations on aerosol deposition in theupper airway. Besides the deposition of an aerosol bolus, its degree of volu-metric dispersion also offers a sensitive tool to characterize aerosol trans-port. Indeed, aerosol boluses delivered to different lung depths are beingused to quantify all aerosol mixing processes, except for Brownian diffu-sion, that are collectively referred to as convective mixing [34]. Severalsources of convective mixing have been identified [32, 121], such as turbu-lent mixing, cardiogenic mixing, and asymmetry between inhalatory andexhalatory flow patterns (e.g., secondary flows, ventilation heterogeneity).These effects are operational to a different extent at different lung depths.They can be quantified experimentally via their effect on aerosol boluses(typically confined to an initial volume of 50-100 ml) inhaled to variouslung depths and recovered at the mouth during exhalation.

The degree to which aerosol boluses disperse in extra- and intrathoracicairways has been the subject of several studies [36, 66, 121, 128, 149, 148].It has been suggested that convective mixing of boluses inhaled to pene-tration volumes between "100 ml (past the carina) and "200 ml (end ofanatomical dead space) can be accounted for by empirical formulas for dis-persion in a network of branching conductive airways, such as those pro-posed by Scherer et al. [125], Ultman [148] or van der Kooij et al. [150].Rosenthal et al. [121] speculated that the larynx may elicit a degree ofbolus dispersion comparable to that in the conductive airways, and con-cluded their study with a call for a dedicated study of bolus dispersion inthe larynx. Simone et al. [132] studied the transport of He and SF6 bo-luses through a larynx cast inserted in a straight tube. They pointed outthat the laryngeal jet would tend to increase dispersion and that the tur-bulence propagated from the high shear boundary of the jet would tendto decrease dispersion. Ultman et al. [149] showed in 5 normal subjects,that effective axial dispersion undergone by He and SF6 boluses in thefirst 80 ml of the upper airway ranged 290-390 cm2/s. This is in contrastwith the effective axial dispersion of 2400 cm2/s attributed to the orophra-ynx in order to match simulated and experimental values of aerosol bolusdispersion for very shallow boluses [33]; the same study obtained excel-lent agreement between simulated and experimental bolus dispersion asa function of volumetric depth in conductive and alveolated airways. Inrecent years, sophisticated 3D computational fluid dynamic (CFD) simu-lation studies in realistic upper airway geometries (e.g., [94]) do providedetailed predictions of flow patterns, including laryngeal jets, vortices andturbulent intensities. However, the accuracy of these CFD simulations re-mains difficult to validate, and a relatively easy way to do so, would be to

112


predict the dispersive effect on an aerosol bolus. This simulated bolus thenrepresents a relevant physiological measure that can be obtained experi-mentally for comparison.

Whether it is for the study of convective mixing or for medication target-ing, an aerosol bolus inhaled to any given lung depth must transit theupper airway, and it is crucial to quantitatively predict its dispersive ef-fect, which is currently lacking. We therefore aimed to quantify bolus dis-persion by carrying out aerosol bolus experiments on a physical cast of arealistic upper airway model [19] and by performing corresponding CFDsimulations in exactly the same 3D geometry. We tested the hypothesesthat bolus dispersion may be different in inhalatory and exhalatory flowdirections, and that bolus dispersion in either flow direction may deviatefrom the one dictated by a simple Gaussian from which a 1D axial dis-persion coefficient is derived. Irrespective of the exact experimental bolusshape, it represents a physiological measure that can be used to test theaccuracy of CFD simulation methods currently employed to study aerosoltransport in the upper airway.

6.2 Materials and methods

6.2.1 Experimental methods

The 3D geometry of the upper airway model (UAM) was retrieved fromprevious work [19] and cast into a silicone hollow model, assembled intwo halves (one half is depicted in Fig. 6.1). Briefly, the UAM geometryhad been previously derived from a smoothened representative upper air-way geometry obtained with multi-slice CT imaging on five healthy never-smoking male subjects (average age 38 years; range 26-52 years). Impor-tantly, CT imaging had been initiated above the nasal cavity and synchro-nized with inhalation in order to obtain a representative glottic area dur-ing slow inhalation. Glottic area was 125 mm2 in this UAM model, totalUAM volume was 90.6 cm3 and the average UAM cross section amountedto 2.78 cm2.

The aerosol generating system consisted of an ACORN II nebulizer (Mar-quest Medical Products, Englewood, CO) from which a diluted suspensionof polystyrene latex 1 µm particles (Duke Scientific, Palo Alto CA) wasaerosolized and passed via a silica-gel drying tunnel, into a the 50 ml tubebetween valves 1 and 2. The aerosol continuously entered the 50 ml tubevia one sliding valve (2) and was evacuated through a filter via anothersliding valve (1) so as to obtain a steady concentration of aerosol in the 50

113


flow meter

2aerosol

1

UAM cast

photometer

syringe

Figure 6.1: Schematic view of the experimental setup used for bolus dispersionmeasurement in a 3D hollow cast of the upper airway model (UAM). A dry 1 µmaerosol contained in a 50 ml tube located between valves 1 and 2 is aspired throughthe UAM, and the photometer beyond it, with a 2 liter syringe.

114


ml tube. In preparation of the 50 ml aerosol delivery to the UAM, valves1 and 2 were then switched to obtain an open passage between the airfrom behind the flow meter and the syringe. By having the syringe as-pire a volume of 2 liters, the 50 ml aerosol was pulled through the UAMand the photometer (PARI,Starnberg, Germany); the syringe was operatedmanually using visual feedback from the flow meter. Throughout the ex-periment, the photometer continuously recorded the particle concentrationas a function of aspired volume; data acquisition was at 100 Hz.

Aerosol bolus tests were performed with the UAM in the inhalatory po-sition (as depicted in Fig. 6.1) and alternating target flows between 250ml/s and 500 ml/s in a series of 12 tests in total. At these two flows,Reynolds numbers were about 1300 and 2600, respectively. A similar testsequence was performed with the UAM in the exhalatory position, i.e., byconnecting the tracheal end to valve 1 and UAM mouthpiece end to thephotometer. In both in- and exhalatory UAM configuration, the aerosolwas made to enter the UAM via a 90o bend beyond valve 1. Using a 90o

bend has been shown to render the incoming aerosol profile flatter thanwhen delivered in a straight line [152], enabling a better comparison witha CFD simulated uniform injection of particles at the model inlet. Also,based on our previous experience with a similar setup where boluses couldbe monitored during both inhalation and exhalation by placing the pho-tometer in between valve 1 and an actively breathing subject [154, 155],the bolus was seen to remain well contained within a 50 ml volume uponentering the subject. Preliminary tests on the present setup confirmedthat this was also the case for a bolus entering the UAM.

Bolus dispersion was quantified in terms of its halfwidth, i.e., the volumet-ric width between concentrations at half peak height, and of its standarddeviation, i.e., the second moment of the volume dependent bolus concen-tration curve [13]. Halfwidths and standard deviations were indicated byHWin and SDin for inhalation, and by HWex and SDex for exhalation; allare obtained by a simple formula [121] for each flow direction. For in-stance, HWin is computed as,

HWin =2

HW 2end!trachea ! HW 2

mouth (6.1)

with UAM in inhalatory configuration (i.e., bolus injected at the mouth andrecovered at end of trachea), and HWex is computed as,

HWex =2

HW 2mouth ! HW 2

end!trachea (6.2)

with UAM in exhalatory configuration (i.e., bolus injected at the end of the

115


trachea and recovered at the mouth). The same type of formulas are usedto obtain SDin and SDex.

6.2.2 Numerical methods and quality control

Fluid phase

Using the existing UAM 3D geometrical boundaries of Fig. 6.1, a prelimi-nary grid resolution study was conducted. Comparing flow fields obtainedwith unstructured hexahedral meshes of either 800,000 or 2,170,000 cells(Hexpress, Numeca, Brussels, Belgium) had indicated that the differencesin velocity magnitude in a cross-section downstream of the larynx werenegligible. Therefore, the UAM with the hexahedral mesh containing 800,000cells was used here. All flow field computations were performed using acommercial CFD software package (Fluent 6.3, ANSYS, Canonsburg, PA).An incompressible Reynolds Averaged Navier Stokes (RANS) solver wasemployed to simulate the fluid flow, and a two equation shear stress trans-port k ! ! model was used to model the turbulence. The k ! ! model hasbeen previously proposed to be the most adequate turbulence model forsimulating transitional flows in the upper airways [102, 150]. For the spa-tial discretization, a second-order upwind scheme was used. A third-orderMUSCL scheme was used for the momentum and the k!! equation respec-tively. The SIMPLE algorithm was used for pressure-velocity coupling. Atypical simulation of the flow field to obtain a convergence level of threeorders of magnitude took approximately 11-12 hours on a 2.4 MHz dualcore processor (AMD Opteron, Sunnyvale, CA).

Particle phase

For the particle phase, we used an Eddy Interaction Model (EIM) wherethe fluctuating part of the instantaneous velocity is modeled assumingisotropic turbulence and assigned to an eddy with known life time andlength scale. More details on the EIM is described in Chapter 4. One mi-cron particles were injected homogeneously at the UAM inlet cross sectionso as to obtain a uniform particle number per surface area distribution overthe entire cross-section. A preliminary particle number study injecting ei-ther 10,000, 15,000 or 30,000 particles had indicated that the halfwidthsof particle concentration profiles obtained at the geometry outlet differedby 0.7% between 10,000 and 30,000 particles and by 0.6% between 15,000and 30,000 particles. For the present simulations, 15,000 particles wereconsidered and trapezoidal scheme for particle tracking was employed. Ap-proximately 5 hours were required to track 15,000 particles.

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6.3 Theoretical axial dispersion coefficient

In an attempt to find a one dimensional dispersion coefficient D to charac-terize the axial dispersion undergone by an aerosol bolus in the UAM, theparticle profile recovered at the model outlet was compared with the an-alytical solution of the 1D diffusion equation corresponding to an aerosolprofile initially confined between axial locations x = !h and x = +h [29]

C(x, t) =12

#erf

,h ! x + u0t

2)

Dt

-+ erf

,h + x ! u0t

2)

Dt

-$(6.3)

where C is particle number concentration as a function of space and time,x is axial pathway length; D is the axial dispersion coefficient and u0 isan average velocity defined as the ratio of flow rate (250 ml/s or 500 ml/s)over average cross section (2.78 cm2). With a homogeneous injection of the15,000 particles at the UAM inlet surface, the injected aerosol bolus in theCFD simulations effectively occupied an initial volume of 0.125 cm3, cor-responding to an initial aerosol bolus slab thickness of 0.045 cm (= 2.h) atthe initial bolus position (x = 0). When considering such a bolus passing afixed location, for instance at x = 32.6 cm (i.e., UAM length), the transfor-mation of the time-dependent concentration trace into a volume-dependentone via flow rate, shows a halfwidth which increases as a function of D upto approximately 2500 cm2/s and then levels off (Fig. 6.2A). By contrast,when considering the spatial dispersion of typical concentration curves ob-tained with Eq. 6.3 at a fixed point in time (t = 0.05 s here), and varyingD, a steady increase of spatial dispersion with D is observed (Fig. 6.2B).In this case, the flow rate does not interfere with halfwidth, since it merelytranslates the entire diffusing bolus along the x-direction at a faster pace.

6.4 Results

6.4.1 Experimental results

Actual flows corresponding to bolus tests with the UAM in the inhala-tory and exhalatory configuration and with a target flow of 250 ml/s were260±12(SD) ml/s and 262±14(SD) ml/s respectively. For the UAM testswith a target flow of 500 ml/s, actual inhalatory and exhalatory flows were497±28(SD) ml/s and 490±38(SD) ml/s respectively. Six typical photome-ter traces corresponding to a bolus test with the UAM in the inhalatoryconfiguration are presented in Fig. 6.3A. In Fig. 6.3B, the representa-tive traces of Fig. 6.3A are normalized to peak height, and compared tothe corresponding analytical solution of Eq. 6.3 for D = 200 cm2/s andD = 250 cm2/s. While neither option perfectly captures the entire bolus

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Figure 6.2: Panel A: Halfwidth computed from the 1D theoretical time dependentconcentration of a bolus recovered at x = 32.6 cm (i.e., the axial pathway lengthbetween UAM in- and outlet), after being initially contained within a 0.045 cmslab at x = 0 and transported towards x = 32.6 cm by diffusion and convective flow(the latter is indicated by the block arrow). For each axial dispersion coefficientD, the solution of Eq. 6.3 at x = 32.6 cm is transformed from a time-dependentto a volume-dependent bolus (via flow rate), such that the volume difference athalf bolus peak height, i.e., the halfwidth, can be determined (circles:250 ml/s vs.triangles:500 ml/s). Panel B: Halfwidth computed from the 1D theoretical spatialconcentration of a bolus at t = 0.05 s, after being initially contained within a 0.045cm slab at x = 0 and transported by diffusion and convective flow (the latter isindicated by the block arrow, but only introduces a spatial translation and doesnot affect spatial halfwidth). For each axial dispersion coefficient D, halfwidth isobtained as the x-difference at half peak height (in cm) of the solution of Eq. 6.3 fora fixed t = 0.05 s (solid circles); also represented is the axial bolus halfwidth givenby !x =

"2D!t with !t = 0.05 s (dotted line) .

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Figure 6.3: Panel A: Synchronized photometer traces with the UAM in the inhala-tory configuration and a target flow of 250 or 500 ml/s (6 tests each); synchroniza-tion was arbitrarily set to have all bolus fronts aligned to 100 ml for clarity of rep-resentation. From each set of 6 curves, one representative photometer trace (solidcircles) is derived from which halfwidths are computed as indicated by the dottedlines (HW = 89 ml and 84 ml for respective target flows of 250 or 500 ml/s). PanelB: Combined representative photometer traces of Panel A for 250 and 500 ml/s(solid circles), normalized to their peak bolus value, and set against correspondingtheoretical solutions of Eq. 6.3, considering an initial bolus volume of 50 ml (as inexperiments) with a effective axial dispersion coefficient D set to either 200 cm2/s(left) or 250 cm2/s (right); dotted and dashed lines refer to the theoretical tracesfor 250 and 500 ml/s respectively.

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curve, particularly in its tail portion, the analytical solution correspondingto D = 200 cm2/s best matches the initial part of the bolus, including itshalfwidth, while the D = 250 cm2/s solution somewhat better captures thebolus tail. This is particularly true for D = 250 cm2/s at a flow of 250 ml/s(dotted line in the right panel of Fig. 6.3B), yet, at the expense of an over-estimated bolus halfwidth.

Fig. 6.4 summarizes the average (±SE) values of the experimental bolusdispersion indices. Overall, there were small decreases in HWin, HWex,SDin and SDex with increasing flow rate, but none of the post-hoc pair-wisecomparisons reached significance. There were no statistically significantdifferences between HWin and HWex or between SDin and SDex.

6.4.2 CFD results

CFD simulated inhalatory and exhalatory aerosol deposition in the UAMwas 3% and 4%, respectively for 250 ml/s and 8% and 9%, respectively for500 ml/s. Fig. 6.5 shows the CFD simulated particle concentration tracesat the model outlet, which were normalized to their respective bolus peaks,for 250 ml/s (thick solid lines) and 500 ml/s (normal solid lines). The bo-lus halfwidths corresponding to the CFD simulated particle concentrationcurves were much smaller for exhalation than for inhalation at both 250ml/s (HWin = 69 ml; HWex = 20 ml) and 500 ml/s (HWin = 49 ml; HWex

= 20 ml). Also superimposed on the inhalatory traces of Fig. 6.5A are thecorresponding analytical solutions from Eq. 6.3 with D = 200 cm2/s, for250 ml/s (dotted lines) and 500 ml/s (dashed lines). In Fig. 6.5B, the an-alytical solution with D = 25 cm2/s is also displayed, merely to illustratethe degree of underestimation of axial bolus dispersion for the exhalatoryUAM configuration. The marked difference in halfwidth between the CFDgenerated boluses in panels A and B of Fig. 6.5 is in contrast to the verysimilar halfwidths obtained in bolus experiments with the UAM in inhala-tory and exhalatory configuration (Fig. 6.4).

6.5 Discussion

With respect to the primary aim of this study, we have found that thedispersion of an experimental aerosol bolus transiting a realistic model ofthe upper airway including the trachea can be reasonably approximatedby a Gaussian fit (Eq. 6.3). The remaining discrepancy was in the bolustail, where the experimental bolus showed a slightly greater skew thanthe Gaussian fit, especially at flow rates exceeding quiet breathing (> 250

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Figure 6.4: Panel A: Experimental bolus halfwidth (mean±SE) for target flowsof 250 and 500 ml/s with the UAM in the inhalatory (open circles) and exhala-tory (solid circles) configuration. Panel B: Experimental bolus standard deviation(mean±SE) for target flows of 250 and 500 ml/s with the UAM in the inhalatory(open circles) and exhalatory (solid circles) configuration.

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Figure 6.5: Panel A: CFD simulated outlet profiles with the UAM geometry in in-halatory configuration for flows of 250 ml/s (thick solid lines) and 500 ml/s (normalsolid lines) and corresponding theoretical solutions of Eq. 6.3, using D = 200 cm 2/sfor both flows. Panel B: CFD simulated outlet profiles with the UAM geometryin exhalatory configuration for flows of 250 ml/s (thick solid lines) and 500 ml/s(normal solid lines) and corresponding theoretical solutions of Eq. 6.3, using D =25 cm2/s for both flows.

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ml/s). Depending on the physiological application, and whether it requiresto better capture the bolus peak and halfwidth or its tail, an effective ax-ial dispersion of 200-250 cm2/s adequately characterizes the dispersionprocess in this segment of the airway tree. Inhalatory and exhalatoryboluses showed roughly the same bolus dispersion, and bolus dispersiononly slightly decreased by increasing flow from 250 to 500 ml/s. The CFDnumerical simulations reproduced experimental results for inhalation butnot for exhalation, warranting further scrutiny on the part of the numer-ical methods (turbulence models and parameters) used to describe transi-tional flows in structures such as the UAM used here.

Our experiments show that the upper airway geometry leads to a 80-90ml wide bolus at the model outlet, for a 50 ml bolus at model inlet un-der quiet breathing conditions (250-500 ml/s) and in either inhalatory orexhalatory flow direction. When correcting an average 85 ml bolus atUAM outlet for the non-zero initial aerosol bolus according to Eqs. 6.1and 6.2, the net dispersion halfwidth HWin or HWex amounts to 69 ml(=

6(85ml)2 ! (50ml)2). For medical aerosols, this implies that a typ-

ical aerosol from a pressurized metered dose inhaler, which is typicallyfired into a 250 ml holding chamber prior to inhalation, will undergo a netdispersion in the upper airway (including the trachea) such that its vol-umetric dispersion beyond that point becomes no more than 260 ml (i.e.,(=

6(260ml)2 ! (250ml)2) to obtain 69 ml net dispersion). With respect to

the target lung volume for aerosol medication, this degree of volumetricdispersion induced by the upper airway is negligible from a clinical stand-point.

From a physiological standpoint, the degree of dispersion induced by theupper airway concerns its contribution to the overall convective mixingprocess at different lung penetration volumes (Vp) proximal to the gas ex-changing zone. Only some laboratories have actually measured the halfwidthof exhaled aerosol boluses after inhalation to very shallow penetration vol-umes (Vp < 100ml) [13, 149]. For instance, in 79 normal subjects, Brandet al. [13] have measured an average HW (corrected for the 20 ml inhaledbolus width) of 70 ml and 120 ml for aerosol boluses inhaled to a Vp of 20ml and 50 ml, respectively (respiratory flow was 250 ml/s). Assuming, onbasis of our experiments, that net UAM dispersion in each flow directionamounts to 69 ml, the combined UAM dispersion of a 20 ml bolus duringinhalatory and exhalatory phases can be predicted according to Eqs. 6.1and 6.2 as follows. At end of inhalation, a 20 ml bolus gets dispersed over71.5 ml ( to obtain 69 ml net dispersion) and at end of exhalation, this 71.5ml bolus gets dispersed over 99 ml (to obtain 69 ml net dispersion). Fi-

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nally, when correcting this 99 ml bolus halfwidth at end of exhalation forthe initial 20 ml inhalatory bolus, the predicted cumulative HW for the in-and exhalatory cycle amounts to 97 ml, falling in the 70-120 ml halfwidthrange obtained by Brand et al. [13] for Vp ranging between 20 and 70 mlin human subjects.

In five normal subjects, Ultman et al. [149] had previously obtained anaverage SD value of 55 ml for SF6 gas boluses inhaled to a penetrationvolume corresponding to the UAM volume (91 ml). When cumulating SDvalues from our UAM experiments in either flow direction according toEqs. 6.1 and 6.2, the corresponding SD value we obtain is 59 ml. Thisexcellent agreement can provide an answer to the open suggestion put for-ward by Ultman [148] that the dispersion obtained from a full in- andexhalatory cycle, on basis of experiments which study inhalatory and ex-halatory dispersive effects separately, may represent an overestimation ofthe real cumulative dispersion. The comparison of our cumulative UAMdata to this relatively limited set of experimental data on human subjectsdoes suggest that in the upper airway segment, dispersion effects occur-ring during in- and exhalatory phases are approximately additive.

The present in vitro study shows that the impact of the upper airwayon aerosol dispersion is relatively small. A previous in vitro study by Si-mone et al. [132] had been concerned with the impact of the larynx onmixing in various segments of a 3-generation branching model. These au-thors studied the standard deviation of 5 ml SF6 gas boluses in a broadrange of Reynolds numbers and they normalized the SD2 values they mea-sured in various model segments by the corresponding segment volumesquared in order to obtain a dimensionless number for comparison withother studies. Considering Reynolds number between 1,000 and 2,000,their (SD/volume)2 ratio for the upper airway segment including a some-what simplified larynx, amounted to 0.14 (= (20ml)2/(53ml)2); our corre-sponding value is 0.22 (= (43ml)2/(91ml)2). Like Simone et al. [132], wealso found a decreasing dispersion with increasing flow, but the magnitudeof flow dependent changes seen here is much smaller than the 20% SD de-crease found by Simone et al. [132] between Reynolds number 1,000 and2,000. We can only speculate that this is due to geometrical differences,in particular, the fact that the larynx cast in Simone et al. [132] was em-bedded in a straight tube directly leading to the tracheobronchial model.With the upper airway structure studied here, a transiting aerosol is sub-jected to complex changes in both cross section and angle, all of which mayhave an effect on the flow dependence of bolus dispersion. The presentstudy suggests that a realistic upper airway indeed induces poorly flow-

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dependent bolus dispersion in the 250-500 ml/s flow range.

Another interesting finding by Simone et al. [132] in their in vitro branch-ing plus larynx model was that the largest degree of dispersion originatedat the first bifurcation. Taken together with our finding of a relativelylimited dispersion induced by a more realistic model of the upper airwayand trachea, this could explain the broad range of halfwidths (70-150 ml)obtained in human subjects for shallow volumetric depths (Vp < 100 ml)across different studies. Indeed, the exact penetration volume, i.e., the vol-ume of air following the inhaled aerosol in any experimental setup, maybe subjected to some variability regarding the exact upper airway struc-tures that have been crossed by the aerosol bolus under study. Therefore,the outcome HW value in each experimental setup possibly depends onwhether a shallow bolus has actually passed the carina and whether amarkedly asymmetrical recombination of boluses at the first bifurcationtook place or not. Rosenthal et al. [121] had speculated that there maybe a faster increase of dispersion with Vp in the upper airway structurethan in the conductive airways. The present study suggests that at leastthe upper airway including the trachea has a relatively limited dispersiveeffect.

The CFD simulations only partly confirmed the experimental observationsin that the HWin with UAM in the inhalatory configuration averaged 59ml for flow rates 250 ml/s and 500 ml/s, but exhalatory HWex was just 20ml in this flow range (versus 69 ml in experiments). Like the UAM exper-iments, the corresponding CFD simulations also produced slightly tighterboluses at higher flows. However, the CFD simulations also predicted amarkedly different shape for exhalatory and inhalatory boluses, which wasnot observed experimentally. In particular, the CFD simulated exhalatorybolus peak was almost impossible to associate with a proper Gaussian-derived solution (Eq. 6.3; Fig. 6.5B), given the sharp initial peak and thelong tail.

The above discrepancy between CFD simulations and experiments pointto a problem with the CFD methodology using RANS with a k ! ! turbu-lence model. RANS k!! simulations have been widely used in recent CFDstudies of gas and aerosol transport in the upper airway by us and others[20, 76, 102, 171], and generally tend to overestimate aerosol deposition.A recent comparison of experimental particle image velocimetry and sim-ulated flow patterns of the carrier gas [77] has indicated that large eddysimulation (LES) better capture the experimental patterns of turbulentkinetic energy than RANS k ! !. Given that turbulence is crucial to both

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aerosol deposition and dispersion in the upper airway, it may be worth-while pursuing the LES approach in the future, despite being far moretime-consuming than RANS. Given that CFD tools are currently findingsuch a widespread use in the prediction of the fate of aerosols in the lungs,and that the transitional laminar-turbulent flow regime in the upper air-way poses a particular challenge, it is recommended that the bolus dis-persion be used as a sensitive tool to validate emerging CFD approachessuch as LES. Indeed, it has been observed before [33] that total depositionis a relatively crude measure of aerosol behavior. However, bolus disper-sion may be a more adequate tool to validate CFD simulations of aerosoltransport in the human lung, and in particular, in the upper airway.

6.6 Limitations

In all its simplicity, the aerosol bolus dispersion experiment does presentsome pitfalls and limitations. Firstly, the equipment used for bolus exper-iments monitors aerosol concentrations that are averaged over part or theentirety of the tube cross section, thereby neglecting any non-uniformitythat may potentially develop within a given cross-section. Secondly, anyattempt to a simple quantification of aerosol dispersion usually relies ona 1D Gaussian approach (Eq. 6.3) to extract one axial dispersion coeffi-cient, which is ideally suited for describing concentrations of a dispers-ing gas by molecular diffusion. Since convective mixing of aerosol in theUAM may be more complex, it is not surprising that a Gaussian does notfully mimic the bolus shape. Indeed, the experimental bolus tail cannotbe fully captured by Eq. 6.3, and depending on the exact choice of fittingcriteria (either fitting the entire bolus curve or fitting its halfwidth), thecorresponding dispersion coefficient will slightly vary. We should bear inmind that physiological bolus dispersion studies either consider bolus halfwidth (i.e., ignoring the bolus tail altogether) or exclude all bolus concen-trations below 15% of the bolus peak value when computing bolus SD orbolus skewness (i.e., effectively ignoring part of the bolus tail) [13]. Hence,comparison between physiological bolus experiments will not suffer muchfrom the degree of discrepancy with the Gaussian characterization that weobserve here in the bolus tail. Thirdly, there is a limitation of using bolustraces at the model outlet (Fig. 6.2A) to estimate spatial dispersion actu-ally undergone by a bolus inside the model (Fig. 6.2B). For instance, a time(or volume) dependent concentration trace at the outlet of a 32.6 cm tubein a perfect 1D case of axial dispersion given by Eq. 6.3, shows a levelingoff of bolus halfwidth somewhere between D = 2500 and 5000 cm2/s (Fig.6.2A), which in addition, partly depends on the flow rate. However, for Dvalues below 1000 cm2/s, there is a monotonic increase of halfwidth with

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D and a relatively limited dependence on flow rate. Given that the physi-ologically relevant D values indicated by the present study are well below1000 cm2/s, bolus halfwidth can indeed be considered a suitable param-eter to quantitatively study aerosol transport in the upper airway undernormal breathing conditions.

On basis of experiments [132] or simulations [94], several authors haveduly argued that aerosol transport in the upper airway can have an impacton airways downstream from it. Conversely, the presence of a bifurcationat the tracheal end may affect bolus dispersion inside the trachea. This isa limitation of studying any partial model of the respiratory system, as isthe case here. Yet, it must be recognized that realistic 3D experiments andsimulations in the entire lung are simply not feasible, and in some cases,they are also not necessary. For instance, to test the effect on aerosol bo-lus dispersion or deposition of glottic area, it would suffice to consider onlythis segment and compare numerical data with bolus experiments in ex-actly the same 3D geometry under exactly the same flow conditions, as wasdone here. Also, CFD studies of aerosol transport in the alveolar space(e.g., [62]) do not need more than a reasonable estimate of axial dispersionof a bolus transiting the extra- and intrathoracic airways compartments,for comparison with bolus experiments performed by human subjects. Forthe conductive airways compartment, satisfactory empirical formula of ax-ial dispersion already existed in the literature [125, 148, 150] and for theoropharynx, an axial dispersion value of 2400 cm2/s was adopted [33]. Thepresent experiments provide a direct measure of axial dispersion in theupper airway compartment, comprising oropharynx and trachea, whichranges 200-250 cm2/s. Some variations on this range may exist, depend-ing on, for instance, inter-subject glottic aperture or intra-subject glotticarea variations during respiration [12]. However, we suspect these to havea minor effect on bolus dispersion, for two reasons. First, there is a smallinter-subject variability of bolus dispersion of shallow boluses and absenceof correlation of any bolus parameter with gender [13]. Secondly, a numer-ical study in a laryngeal channel with pseudo-time varying glottis between112 mm2 (peak inhalation) and 66 mm2 (peak inhalation), where Renotteet al. [120] found minor differences in velocity profiles between in andexhalation.

6.7 Conclusions


1. We have found experimentally that a realistic geometry of the upper

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airway between mouth and end of the trachea induces a relativelymild dispersion on a traversing aerosol bolus. For those studies ofaerosol bolus behavior in airways peripheral to the trachea, requiringan estimate of bolus dispersion while trespassing the upper airway,an axial dispersion coefficient of 200-250 cm2/s can be adopted undernormal breathing conditions.

2. Using a realistic UAM leads to unique dispersion curve patterns thatcannot be accurately captured by a simple 1D Gaussian, which isusually employed to extract the axial dispersion coefficient.

3. For both inhalation and exhalation, the values of dispersion coeffi-cient were found to be quite insensitive to the two flow rates consid-ered.

4. The dispersion effects occurring during in- and exhalatory phaseswere seen to be approximately additive.

5. CFD simulations matched the experimental results for inhalation,but not for exhalation, indicating that the turbulence models shouldbe further scrutinized to adequately simulate all aspects of aerosoltransport in the upper airway.

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Chapter 7

Application III: TrachealStenosis

Contents7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1297.2 Numerical methods & quality control . . . . . . . . . 1307.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130


7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7.1 Introduction

Air entering the mouth, passes through pharynx and flows into the tra-chea via the glottal region. As we saw in Chapter 5, flow in these re-gions is highly irregular owing to the complexity of the airway structure.The presence of an obstructive lesion such as tracheal stenosis adds re-sistance to the flow due to changes in pressure and eventually results inbreathing problems. A common form of tracheal stenosis is the so-calledweb-like stenosis, i.e., a marked tracheal narrowing that spans only a fewmillimeters. Patients with tracheal airway stenosis often do not show anybreathing problems even when 50% of the airway lumen cross-section is ob-structed, and then report a relatively sudden appearance of breathing im-pairment when over 75% obstruction is reached [19]. Considering 7 differ-ent stenotic constriction percentages ranging between 50 to 90%, we per-formed CFD simulations at a normal breathing flow rate of 30 l/min. The

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CHAPTER 7. APPLICATION III: TRACHEAL STENOSIS

aim here was to investigate the potential of using the total fluid flow pres-sure drop (i.e., airway resistance), aerosol bolus deposition, and aerosolbolus dispersion to detect tracheal stenosis at the earliest possible stage ofconstriction.

Most of the CFD studies on the effect of local obstructions are focused onthe tracheobronchial region [44, 163, 98, 162, 172]. To the best of the au-thor’s knowledge, our recent work (Brouns et al. [19]) is the first attemptto focus on the fluid flow dynamics in the presence of a web-like trachealstenosis. Pressure drops during normal breathing were analyzed and arule of thumb was derived from which pressure drops over the stenosis canbe estimated on the basis of breathing flow and stenosis cross-section. Ex-perimental studies of stenosis include Wassermann et al. [157] who useda newly developed bronchoscopic technique for the assessment of intra-tracheal pressures as a way to quantify tracheal resistance for use in di-agnosis of patients with tracheal stenosis. Fasano et al. [45] used a non-invasive technique to propose specific airway resistance as a diagnostictool.

7.2 Numerical methods & quality control

The numerical methods and quality control for the fluid and particle phaseare identical to the RANS methodologies described in Chapter 6.

7.3 Results

7.3.1 Fluid phase

In chapter 5, we analyzed the flow profiles in a healthy mouth-throat ge-ometry. In this chapter, we will concentrate on the effect of stenosis on theflow patterns. Fig. 7.1 shows the velocity contours and the velocity vectorlines in the central sagittal plane at a normal breathing flow rate of 30l/min. We focus on the evolution of the flow downstream of the stenosis fortwo different degrees of stenotic constriction. The flow coming from phar-ynx experiences a sharp step at the end of the pharynx which results in thelaryngeal jet developing towards the anterior side of the trachea. A littledownstream, the laryngeal jet which was developing towards the anteriorside encounters the tracheal stenosis which results in second jet like struc-ture developing downstream of stenosis. This jet which has the tendencyof developing towards the posterior side is from now on referred to as thestenotic jet. Due to the stenotic jet, there is a recirculation zone set on the

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Figure 7.1: Mid-plane velocity contours and the velocity vector lines at a normalbreathing flow rate of 30 l/min. 50% and 75% constrictions are shown.

anterior side. As the stenotic constriction increases (from 50% to 75%), theintensity of stenotic jet increases and results in a bigger recirculation zoneon the anterior side. This trend remains consistent for all constrictionsabove 75%. The increase in absolute velocity and the recirculation zonedue to the jet will indeed have considerable effect on the particle deposi-tion downstream of the stenosis.

Fig. 7.2 shows the area-averaged normalized velocity magnitude at every2 mm after the stenotic constriction. Steady levels of velocity are seen upto one diameters downstream of stenosis, after which there is a steep fallbetween one and three diameters downstream.

Fig. 7.3 shows the area-averaged normalized kinetic energy at every 2 mmafter the stenotic constriction. For 50 and 60% constriction, the amplifica-tion levels are very small. For rest of the constrictions, there is a steep in-crease in the kinetic levels up to one, one and half diameters downstream,after which a steady decrease is seen. It is interesting to note that themaximum kinetic energy level at 90% stenosis is twice as much as the lev-els seen for 85% constriction. The higher levels of kinetic energy generally

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Figure 7.2: Area-averaged normalized velocity magnitude (u/uin) at every 2 mil-limeters after the stenotic constriction.

Figure 7.3: Area-averaged normalized kinetic energy (k/u2in) at every 2 millimeters

after the stenotic constriction.

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0

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enhances the turbulent mixing and this perhaps explains the faster fall ofvelocity magnitude (Fig. 7.2) in case of 90% stenosis as opposed to the 85%stenosis.

Fig. 7.4 shows the total pressure drop between inlet and outlet as a func-tion of constriction percentage. We see a modest increase in pressure dropup to about 75% stenotic constriction, beyond which it steeply rises. Such apattern agrees with the appearance of breathing symptoms with patientswho already show a very marked stenosis and explains why patients usu-ally do not experience a major breathing impairment, or associated needfor a stenting procedure (mechanical dilation of the airway), until the con-striction is well above 50%.


The aerosol bolus deposition efficiency and bolus dispersion, in terms ofbolus half-width (HW) and bolus standard deviation (SD), were simulatedas a function of the degree of stenotic obstruction (Table 7.1). It is interest-ing to note that the dispersion and deposition of 1 µm particles with 50%stenosis (40 ml dispersion and 4% deposition) are smaller when comparedto the case with no stenosis (49 ml dispersion and 8% deposition, as seen

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Table 7.1: Dependence of dispersion and deposition on the percentage of stenoticconstriction.

Stenosis Dispersion Deposition(%) (ml) (%)

1 µm 1 µm 5 µm 10 µmHW SD

50 40 22 4 10 62

60 45 25 4 11 64

70 50 28 5 12 67

80 60 31 6 20 70

85 65 37 8 28 78

90 80 39 15 51 86

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in Chapter 6). This indeed is due to the presence of stenotic constrictionwhich restricts the free development of the laryngeal jet which was notseen in the no-stenosis case (Fig. 7.1).

The CFD results show that the upper airway geometry with a maximumstenotic constriction of 90% leads to a 80 ml wide bolus at the trachealoutlet, for a 0.125 ml bolus at the model inlet under normal breathing con-dition of 500 ml/s during inhalation. Since the bolus volume at the modelinlet is very small, the non-zero initial bolus correction is not required. Fordiagnostic aerosol boluses, this implies that a typical 50 ml aerosol boluswill undergo a net dispersion in the upper airway (including the trachea)such that its volumetric dispersion beyond that point becomes no morethan 94.3 ml (i.e., =

6(94.3ml)2 ! (50ml)2 to obtain 80 ml net dispersion).

Such a mild dispersion increase for such a dramatic increase in constric-tion indicates that the bolus dispersion is not an adequate parameter todetect tracheal stenosis.

The effect of airway stenosis on bolus deposition efficiency for differentparticle sizes was also studied. Simulations predicted that shallow bolusesinhaled by subjects with a stenosis of 80% or more, would be affected mostin terms of bolus deposition of a 5 µm aerosol, showing a high relativechange (a 2-3 fold deposition efficiency from 50% to 80-85% obstruction)and a measurable absolute change (10-18% greater deposition for 80-85%stenosis). Smaller (1µm) and larger (10µm) particles would require pickingup smaller absolute differences in bolus deposition efficiency, which couldbe difficult to achieve experimentally. We can therefore conclude that shal-low boluses of 5 µm aerosols, and in particular their deposition efficiency,could be a useful non-invasive diagnostic tool for the detection of trachealstenosis. It should however be noted that only inhalatory phase is consid-ered in the present study due to the fact that CFD has been shown to beunreliable for exhalatory bolus simulation (see previous chapter). In clin-ical setting, the cumulative dispersion during inhalation and exhalationis considered, since the inhaled aerosol bolus is measured at the subject’smouth upon exhalation. Hence bolus dispersion and deposition during ex-halation will be needed to provide a definitive answer as to the potentialdiagnostic tool.

7.4 Conclusions


1. The simulation results have shown a modest increase in pressure

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drop up to 75% stenotic constriction, beyond which it steeply rises.Such a pattern agrees with the appearance of breathing symptomswith patients who already show a very marked stenosis and explainswhy patients usually do not experience a major breathing impair-ment, or associated need for a stenting procedure (mechanical dila-tion of the airway), until the constriction is well above 50%.

2. Particle dispersion was found to be quite insensitive to the stenoticconstriction percentage during inhalatory phase and hence falls shortof being a sensitive tool to detect stenosis.

3. Deposition for 5 µm particles was seen to exhibit a measurable sen-sitivity in relative deposition, making it a potential non-invasive di-agnostic tool for the detection of tracheal stenosis.

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Chapter 8

Fluid Flow and ParticleDeposition in UpperAirways: LES and DES

Contents8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1378.2 Model preparation & experimental methods . . . . . 1398.3 Numerical methods . . . . . . . . . . . . . . . . . . . . . 1418.4 Quality control . . . . . . . . . . . . . . . . . . . . . . . . 1438.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147


8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 154

8.1 Introduction

Inhaled medication is a preferred method of drug administration to thelung for the first-line therapy of asthma and chronic obstructive pulmonarydiseases. The inhaled aerosol particles need to negotiate the mouth-throatstructure in order to reach the smaller airways and the alveolar lung zonethat could benefit from aerosol therapy. The complexity of the extratho-racic portion of the oral airway, which includes bends and sudden cross-sectional changes, potentially induces considerable local medication depo-sition before actually reaching the lungs. A quantitative study of aerosol

137

CHAPTER 8. FLUID FLOW AND PARTICLE DEPOSITION IN UPPERAIRWAYS: LES AND DES

transport and deposition in a realistic model of the mouth-throat posesmany challenges, both from an experimental and simulation standpoint.

Most previous experimental studies in mouth-throat geometries have fo-cused on the characterization of fluid flow fields, with the ultimate aim ofpredicting aerosol deposition patterns. For instance, Corcoran and Chigier[28] used phase doppler interferometry to study the axial flow field in alarynx/trachea cast, assessing the effects of the laryngeal jet on inhala-tion flow patterns. Gemci et al. [50] used laser doppler velocimetry tocharacterize axial velocity fields and turbulent intensity levels in a simplethroat model (essentially a constricted tube). Heenan et al. [64] used en-doscopic particle image velocimetry (PIV) to visualize the velocity fields indistinct parts of the central sagittal plane of an idealized model of the hu-man oropharynx. In specific locations of the same geometry, Johnstone etal. [80] measured mean and RMS axial velocity using x-hot-wire anemom-etry. Heenan et al. [63] combined PIV and scintigraphic aerosol depositionmeasurements in an attempt to relate time-averaged flow fields in the cen-tral sagittal plane to local aerosol deposition in two realistic extrathoracicairway geometries. Finally, DeHaan and Finlay [39] employed ultravioletspectrophotometric assay to measure depositions in a mouth throat geom-etry for different inhalation devices, and DeHaan and Finlay [41] collectedaerosols from dry powder inhalers on disposable filters to determine oralcavity deposition.

Most previous numerical studies have simulated the fluid/particle behav-ior in mouth-throat geometries by means of RANS simulations, mainlyusing two equation turbulence models like k ! & and k ! ! for the fluidphase, sometimes coupled with Eddy Interaction Model (EIM) for the par-ticle phase. Stapleton et al. [143] observed that even in a simplifiedmouth-throat model, k ! & turbulence model was not suitable for the ac-curate prediction of particle deposition. Matida et al. [102] obtained betterresults with standard k ! ! model compared to standard k ! &, yet, forthe particles in low Stokes number range (pertinent to medical aerosols),the simulated total mouth-throat deposition was "50% in contrast to themuch lower depositions encountered experimentally (i.e., less than 5%).When anisotropy effects close to the walls were taken into account in theEIM, simulated deposition came down to "15%. On the other hand, Xi andLongest [161] employed a low Reynolds number k ! ! model to assess theeffects of geometry simplifications on aerosol deposition, reporting deposi-tion values of less than 20% (even without considering anisotropy near thewalls). The same behavior was also seen in our simulations presented inChapter 5. RANS models, which have been basically developed for fully

138


turbulent flows, may be inappropriate for the prediction of particle deposi-tion in the mouth-throat, where flow is in fact transitional. Realizing thisproblem, several authors have started to explore the possibility of usinglarge eddy simulation (LES) methods for the study of particle depositionin the mouth-throat, by applying LES in relatively simple structures suchas turbulent duct flow [146], a 900 bend [16] or a constricted tube [99]. LESsimulations of deposition in a simplified mouth cavity model [101, 17, 74]and in a simplified mouth-throat model [79] have indicated the potentialof LES to more accurately simulate aerosol transport in the extrathoracicairways.

The aim of the present work was to test the validity of the most commonlyused modeling methods, namely the RANS, LES and detached eddy sim-ulation (DES) for the description of fluid/particle behavior in the mouth-throat model. The existing experimental data in literature either resultedfrom intrusive measurement methods, or only provided partial data setson existing model geometries, making the comparison with present sim-ulations difficult. Therefore, we performed PIV measurements in a 3Dmouth-throat cast and used exactly the same mouth-throat geometry forCFD simulations, enabling direct comparison between simulations and ex-periments. On the part of CFD simulations, RANS employed SST k ! !turbulence model, DES was based on the Spalart-Allmaras model for thenear-wall region, and LES was based on two subgrid scale models, namelythe Smagorinsky and the WALE model. Alternatively, we considered thefrozen LES method proposed by Matida et al. [101] where the particlesare released in a frozen (static) instantaneous velocity field and tracked insteady mode, without any EIM.

8.2 Model preparation & experimental meth-ods

The mouth-throat geometry used for PIV measurements and CFD simu-lations is shown in Fig. 8.1. This geometry was previously used for theCFD study of tracheal stenosis [19]. From the mouth-throat geometry, amale model was generated by means of stereolithography, after which aone block transparent female model of the mouth-throat was obtained byfollowing the procedures which are discussed in detail by Hopkins et al.[71]. No geometric re-scaling was necessary for the present PIV measure-ments. A water-glycerin mixture was used with a viscosity of 5.88 # 10!6

m2/s as determined with the AVS300 viscosimeter from Schott Gerateat 25.2oC. Dynamic similarity was achieved for the liquid-based exper-

139


Figure 8.1: Simplified geometry reconstructed from CT-scan data showing differentcross-sections of the geometry.

140


iments by matching the Reynolds number. The PIV measurements wereperformed in a central sagittal plane at a volumetric flow rate of 11.2 l/min,corresponding to a Re number of 2527 (based on inlet diameter).

The water-glycerin mixture was pumped in a reservoir placed "1.5 me-ter above the model in order to create a developed velocity profile. Thereservoir had an inlet which was connected to the pump, an outlet con-necting the model and an overflow exit which guaranteed a constant levelin the reservoir. The outlet of the reservoir was separated from the in-let and overflow exit by a fine maze to stabilize the level and to removeany fluctuations caused by the pump. The flow rate was regulated by avalve which was placed behind the flow meter (KOBOLD InstrumentationNV/SA with accuracy of 4% f.s.).

A New Wave MinilaseII Nd-Yag laser (532 nm wavelength, 100mJ/pulse)was synchronized with a pulse separation, depending on the flow rate. Thepulse separation was chosen in such a way that the reflection of the tracerparticles (10 µm hollow glass spheres) shift 5 pixels between an image pair.The laser beams were combined and formed into a sheet with cylindricaloptics. This pulsed sheet was passed through the model, parallel to theflow, and the light scattered from the particles was recorded with a PCOsensicam QE 5 Hz camera.

Approximately 4000 image pairs were recorded. The images were analyzedusing PIVview 2C software (PIVTEC GmbH, Germany). The vector fieldswere generated using cross-correlation fast Fourier transform (FFT) witha multi-grid procedure combined with a sub-pixel based image shifting orimage deformation with a third order interpolation scheme [124]. The finalinterrogation region was 32#32 pixels with an overlap of 50%. Spuriousvectors were detected by using the normalized median test, which elimi-nates a dependence of the detection criterion on the interrogation domainsize [159]. Only 0.1% of the vectors had to be interpolated. A least squares3-point Gauss fit algorithm was used to recover the sub-pixel displacementof the correlation.

8.3 Numerical methods

The fluid and particle phase were solved employing the incompressiblesolver of FLUENT 6.3.

141


Fluid phase

RANS : The time averaged Navier Stokes equations are modeled employ-ing low Reynolds number variant of SST k!! model [105], which requiresresolving the near-wall region with a fine mesh. This model has been se-lected based on its ability to accurately predict the particle depositions inthe models of mouth-throat geometries [87, 102, 161]. Second-order up-wind scheme for momentum equation and third-order MUSCL scheme fork ! ! equation were employed for spatial discretization. SIMPLE algo-rithm was used for pressure-velocity coupling.

DES : Detached Eddy Simulation is most often referred to as the hybridRANS/LES model where unsteady RANS is employed to model the near-wall region and LES in the core turbulent region. The present DES modelis based on the standard one equation Spalart-Allmaras model. Secondorder implicit formulation for temporal discretization and central differ-encing for spatial discretization of momentum as well as the turbulent vis-cosity equation were employed. DES is computationally more expensivethan RANS but less expensive than LES since the near wall is modeledusing RANS approach. However, DES involves solving of an additionalturbulent viscosity equation.

LES : In the Large Eddy Simulations, the big three dimensional eddieswhich are dictated by the geometry and boundary conditions of the flowinvolved are directly resolved whereas the small eddies which tend to bemore isotropic and less dependent on the geometry are modeled. Two con-stant sub-grid scale models, namely the Smagorinsky model and the WallAdapting Local Eddy Viscosity (WALE)model were tested. Similar to DES,second order implicit formulation is used for temporal discretization andcentral differencing for spatial discretization of momentum equation.The governing equations for RANS, DES and LES methodologies are asdescribed in Chapter 3.

Particle phase

RANS : The governing equations for the Lagrangian modeling of particlephase are as described in Chapter 4. A trapezoidal scheme is used to up-date the particle position and particle velocity.

DES and LES : In the unsteady mode, each fluid phase iteration is fol-lowed by a particle phase iteration and the particles are tracked in realtime. Hence, the effect of resolved large-scale instantaneous velocity onthe particles are accounted for and there is no need for an eddy interaction

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Table 8.1: Dependence of deposition percentage on the inlet condition.

Inlet 2 µm 4 µm 6 µm 8 µm 10 µmtype Dep. (%) Dep.(%) Dep. (%) Dep. (%) Dep. (%)

Profile 8.96 13.51 22.52 41.11 64.64

Blunt 8.39 12.95 20.61 39.21 63.12

model as in RANS. In case of frozen LES, the fluid flow is simulated forcertain number of through flow cycles followed by the injection of particlesin a stagnant (frozen) instantaneous velocity field and tracked as in RANS,but without the EIM.

8.4 Quality control

The air flow in the mouth-throat geometry is investigated at a normalbreathing rate of 30 l/min. A steady top hat velocity profile along with5% turbulence intensity at the inlet and static pressure at the outlet wereimposed. No-slip boundary condition was used at the walls.

We investigated Lagrangian particles with density $p = 912 kg/m3 anddiameters 2, 4, 6, 8 and 10 µm. Particular care was taken to obtain a uni-form surface area distribution of particles at the model inlet. The particleswere injected with their initial velocity set equal to that of inlet fluid ve-locity. As we are dealing with dilute particulate flow, one-way coupling asdescribed by Elghobashi [42] is assumed. Since the airway passage is nor-mally wet, it is assumed that a particle is taken to be deposited on the wallas soon as it touches the wall.

The effect on inlet condition on the fluid flow and particle deposition wasinvestigated. Applying either a blunt inlet profile or a parabolic inlet pro-file showed negligible effect on the flow while comparing different cross-sectional velocities in the central sagittal plane of the model (Fig. 8.2). Atotal deposition variation of less than 2% was observed for all particle di-ameters considered in the present study (Table 8.1).

143


0 0.5 10

0.5

1

1.5

2

x/d

u/u in

A

Profile inletBlunt inlet

0 0.5 10

0.5

1

1.5

2B

0 0.5 10

0.5

1

1.5

2C

0 0.5 10

0.5

1

1.5

2D

Figure 8.2: Comparison of normalized 3 component velocity magnitude for differentinlet conditions. (A) Five millimeters above epiglottis; (B,C,D) One, two and threetracheal diameters downstream of larynx respectively.

144


RANS : A recent study by Vinchurkar and Longest [156] has shown clearadvantages of using hexahedral meshes in respiratory aerosol transportand deposition. The present mouth-throat model was meshed into 800,000hexahedral cells with clustering in the vicinity of the wall and a stretchingratio of 1.2. The y+ of the first layer of cells next to the wall was about1. For each particle diameter, 5,000 particles were injected at the modelinlet. Meshes of either 800,000 or 2,170,000 cells (essentially obtained bydoubling the number of grid points in each direction) had shown negligiblevariation of cross-sectional velocity magnitude in the larynx region and adeposition variation of less than 2% for all particle diameters. The differ-ence in percentage deposition by injecting 5,000 or 15,000 particles at themodel inlet had been found to be less than 1%. Matida et al. [102] had pre-viously reported that their simulated deposition results were unaffectedby increasing the number of injected particles from 1000 to 10,000. Forthe particle transport, high order trapezoidal scheme was employed. Thetypical simulation time of the flow field to obtain a convergence level ofthree orders of magnitude and to subsequently track 5,000 particles took"14 hours on an AMD Opteron 2.4 MHz dual-core processor.

LES and DES : In case of LES, the mouth-throat model was meshed into1.9 # 106 hexahedral cells with clustering towards the wall. The first celllayer next to the wall had a y+ "0.2 with a stretching ratio of 1.05, whichwas sufficient to resolve the viscous sublayer. For the DES mesh, 1.2 # 106

hexahedral cells were employed with a y+ "1 and a stretching ratio of 1.2.

The converged steady state solution based on SST k ! ! model was per-turbed by adding random fluctuations and was used as an initial solutionfor faster convergence of LES/DES computations. To get rid of any possibleinitial condition effects, 3 through flow cycles were performed before start-ing the time-averaging. A through flow cycle is defined as the ratio of thelength of airway model to the average cross-sectional velocity in the cen-tral sagittal plane. For 30 l/min, it is "0.1 seconds. A through flow cycleis much smaller compared to a typical inhalation period which is "2 sec-onds [93]. The time-step to advance the flow was chosen such that the CFLnumber in the entire domain was less than 1. The typical time step was1 # 10!5 sec when using 1.9 # 106 mesh points. To test mesh independenceof the solution, an additional simulation was performed for LES (WALEmodel) with the same time step but on 2.9 # 106 cells. Essentially, no dif-ferences were found in the average velocity magnitude except for a slightoffset in the location of velocity profile on the anterior side at 5 mm aboveepiglottis (Fig. 8.3(a)). For all LES and DES simulations, obtaining a time

145


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x/d

u/u in

A

LES − 1.9*106

LES − 2.9*106

DES − 1.2*106

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2B

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5C

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4D

Figure 8.3: Comparison of normalized 2 component (ux and uz) velocity magnitudecorresponding to the central sagittal plane. (A) Five millimeters above epiglottis(corresponds to section C in Fig. 8.1); (B,C,D) One, two and three tracheal diame-ters downstream of larynx respectively (corresponds to section H, I, J in Fig. 8.1);x/d = 1 corresponds to the anterior airway wall.

146


independent averaged solution took approximately 4 through flow cycles.Each through flow cycle for LES and DES took approximately 160 hoursand 110 hours respectively, while running parallel on four AMD Opteron2.4 MHz dual-core processors.

Once the time-independent average solution was obtained, Lagrangianparticles were introduced into the domain by injecting 5,000 particles foreach particle diameter uniformly over a time span of one through flow cy-cle. Previous LES studies [79, 101] had indicated that "3500 particleswere sufficient. Approximately 5 through flow cycles were required to getall the particles to either deposit on the wall or reach the outlet. Oncethe unsteady tracking was finished, the flow solution was also used forthe frozen LES simulations where the particles were injected in the frozeninstantaneous velocity field and tracked as in RANS, but without the EIM.

8.5 Results

8.5.1 Fluid phase

The flow patterns obtained with LES, DES and RANS are assessed bycomparing the two-component normalized velocity magnitude profiles withexperimental ones at various model cross-sections (Fig. 8.4). Both LESsubgrid scale models (Smagorinsky and WALE) perform well in all fourcross-sections, with a slightly better prediction of the velocity profile bythe Smagorinsky model in the pharynx region on the anterior side (closeto x/d = 1 in Fig. 8.4(a)). In all four cross-sections, DES and LES pre-dictions of velocity profiles are very similar. By contrast, the widely usedk !! RANS model systematically overestimates velocity near the anteriorairway wall, particularly in the tracheal region (Fig. 8.4(b-d)).

In order to compare kinetic energy across the computations and experi-mental data, we first considered the 2 component kinetic-energy as mea-sured by PIV. The experimental data were compared with the correspond-ing 2 component kinetic-energy obtained with LES and DES (left panels ofFig. 8.5), showing a good agreement between simulations and experimentsand negligible variation across the LES/DES models . Since the kinetic en-ergy in RANS implicitly comprises of all 3 components, direct comparisonwith experimentally measured 2 component kinetic-energy (u

!

x

2and u

!

z

2)

is not possible. Considering the similarity of 2 component kinetic energyprofiles between LES/DES and the experiments, the corresponding 3 com-ponent kinetic energy profiles for LES Smagorinsky was compared withRANS (right panels in Fig. 8.5). It can be observed that the kinetic energy

147


0 0.5 10

0.5

1

1.5

2

x/d

u/u in

A

ExperimentLES − SmagorinskyLES − WaleDESRANS

0 0.5 10

0.5

1

1.5

2B

0 0.5 10

0.5

1

1.5

2C

0 0.5 10

0.5

1

1.5

2D

Figure 8.4: Comparison of normalized 2 component (ux and uz) velocity magnitudecorresponding to the central sagittal plane. (A) Five millimeters above epiglottis;(B,C,D) One, two and three tracheal diameters downstream of larynx respectively.

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0 0.5 10

0.1

0.2

0.3

x/d

k/u2 in

A

ExperimentLES − SmagLES − WaleDES

0 0.5 10

0.1

0.2

0.3

LES − Smag (3 comp)RANS

0 0.5 10

0.1

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0.3

B

0 0.5 10

0.1

0.2

0.3

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C

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0.1

0.2

0.3

0 0.5 10

0.1

0.2

0.3

D

0 0.5 10

0.1

0.2

0.3

Figure 8.5: Left: Comparison of normalized 2 component (u!x

2and u

!z

2) kinetic en-

ergy corresponding to the central sagittal plane between Experiments, LES andDES; Right: Comparison of normalized 3 component kinetic energy between LESand RANS; (A) Five millimeters above epiglottis, (B,C,D) One, two and three tra-cheal diameters downstream of larynx respectively.

149


profile obtained with the k ! ! model is very different from that of LES,with both under- and overestimation of local kinetic energy depending onthe cross-section under study.

The time-averaged flow patterns and the secondary flow structures arepresented in Fig. 8.6 for the LES Smagorinsky model. The six cross-sectional views show velocity magnitudes that are greater than 50% of themaximum velocity in the airway model (i.e., 2.6 m/s and above). The flowentering through the mouth piece impinges on the tongue and takes a bendupwards. As it continues to move forward, it accelerates in the middle partof the mouth due to reduction in cross-sectional area. As can be seen in sec-tion A1-A2, velocity in most of this cross-section is above 2.6 m/s. Also, twodistinct recirculation zones are seen. Towards the end of mouth region,the flow takes a downward turn and enters the pharynx in the form of ajet which undergoes an expansion due to increase in cross-sectional area.Consequently, the velocity is reduced and complex secondary motions areset as shown in slice B1-B2. Just beyond the epiglottis region, the flowagain accelerates due to reduction in cross-sectional area and a clear highvelocity zone develops on the posterior side of section C1-C2. At the endof the pharynx, a step on the posterior side guides the flow towards theanterior side of the trachea in the form of a laryngeal jet. As a result ofthis laryngeal jet, two distinct recirculation zones originate at the poste-rior side at section E1-E2 and move towards the center as the flow movesfurther downstream (section F1-F2).

The normalized velocity magnitude contours in Fig. 8.7 illustrate thatthe shape of laryngeal jet for LES is much closer to what is observed inexperiments, and RANS predicts a more pronounced and longer laryngealjet compared to LES. It also shows that RANS leads to greater velocities atthe anterior tracheal wall, as previously observed at discrete model cross-sections (Fig. 8.4(b-d)).


Fig. 8.8 summarizes the simulated total deposition percentages for dif-ferent particle diameters along with the experimental curve fit of Grgicet al. [58], obtained from deposition measurements in 7 different modelcasts representative of over 80 image-based mouth-throat structures. Forthe 2 and 4 µm particles, LES and DES show particle depositions thatare much closer to the experimental curve than those obtained with RANSk ! !, while for the 8 and 10 µm particles, RANS, LES and DES performequally well. Alternatively, RANS k!! with mean flow tracking, i.e. with-

150


Figure 8.6: Above: Time averaged central sagittal plane velocity magnitude andcorresponding streamlines; Below: Time averaged velocity magnitude (above 50%of maximum velocity in the airway model) and corresponding secondary velocityvector lines at six different cross-sections.

151


Figure 8.7: Time averaged 2 component normalized velocity magnitude contour (ux

and uz) in (a) experiments; (b) LES; (c) RANS.

out EIM, consistently underestimates deposition for all particle diametersgreater than 2 µm. The same is true for the frozen LES method, possi-bly because the under-relaxation parameter settings, which yielded goodresults for LES (Fig. 8.8), are too dissipative for the frozen LES method.Considering that 5 µm is generally referred to as the upper limit of the res-pirable range for inhalation drugs (represented by the dash-dotted line inFig. 8.8), our findings suggest that in the mouth-throat geometry, the pre-diction of medication aerosol deposition inhaled at normal flow rates weremore accurate for LES and DES than for the RANS k ! ! model. At a firstglance, DES can then be seen as the preferred method over LES due toreduced computational requirements. To be certain regarding DES beingbetter than LES on 1.2 # 106 mesh points, an additional LES simulationwas performed on the DES mesh. It is observed that LES did as good asDES for both fluid and particle phase. This clearly means that LES wouldremain the preferred method among the models tested, as it obviates solv-ing an additional equation for #t required for DES. For the description ofparticle transport with diameters above 5 µm (e.g., in the upper range ofair pollutant particle distributions) or for small diameters but inhaled atgreater inhalatory flows (e.g., dry powder inhalers), RANS with its vastly

152


10−2 10−1 100 1010

10

20

30

40

50

60

70

80

90

100

Stk.Re0.37

Depo

sitio

n Ef

ficie

ncy

(%)

Experimental fit(Grgic 2004)RANS−EIMRANS−MEANLESFrozen LESDES

Figure 8.8: Simulated total deposition (expressed as % of particles at model inlet)as a function of Stokes and Reynolds number as defined by Grgic et al. [58]. Thesolid line represents the experimental best fit curve. The dash-dotted line corre-sponds to a 5 µm particle at 30 l/min. In case of RANS, ’+’ represents turbulenttracking i.e. considering EIM; ’#’ represents mean flow tracking i.e. without EIM.

lower computational requirements suffices to adequately predict aerosoldeposition.

In order to better understand the discrepancy in particle deposition be-tween LES (Smagorinsky) and RANS (k!!), deposition in the three modelsub-parts are shown for the five particle diameters (Fig. 8.9). It is seen thatRANS overestimates larynx/trachea deposition, showing relatively greaterdiscrepancy with LES for the smaller particles. This can be explained bythe profound and longer laryngeal jet in case of RANS compared to LES(Fig. 8.7) and the velocity overestimation near the anterior airway wall(Fig. 8.4(b-d)).

Even though LES and DES are more accurate among the models consid-ered, they pose a limitation when attempting to simulate transient inhala-tion/exhalation cycles, partly because of the small sampling time require-ments and partly due to relatively high time-cost associated with typical

153


0

10

20

30

40

50

60

70

Diameter (µm)

Depo

sitio

n Ef

ficie

ncy

(%)

trachea + larynxpharynxmouth

2 4 6 8 10

LES

RANS

Figure 8.9: Simulated deposition values (expressed as % of particles at model inlet)in three model subparts; Mouth deposition excludes deposition in the inlet tube.

transient waveforms. For example, to perform LES/DES simulation of a2 second transient inhalation waveform as considered by Li et al. [93]would require "5 times more computational time compared to the steadyinhalation case simulated in the present work. Alternative RANS mod-els may therefore be worth considering. For instance, Matida et al. [102]applied the near-wall anisotropic corrections to a RANS k ! ! model andshowed considerable improvement in deposition over the one obtained withisotropic assumption. However, the fact that the basic fluid flow field isnot accurately predicted by RANS (Fig. 8.4, 8.5, 8.7) is a problem. Anotherpossibility is to consider a Reynolds Stress Model (RSM) which implicitlyaccounts for near-wall anisotropy, and has been reported to outperformRANS k!&model when predicting nano and micro-particle deposition in aduct flow [146]. However, as pointed out by these authors, the performanceof the different modalities of the Reynolds stress model in a more complexgeometry warrants further investigation.

8.6 Conclusions


1. RANS, LES and DES simulations showed that for the fluid phase,

154


LES/DES lead to similar velocity and kinetic energy profiles and allof which compare well with experimental data, as opposed to RANS.

2. 5 µm is generally referred to as the upper limit of respirable rangefor inhalation drugs. For the micro-particles below 5 µm consideredin the present study, LES and DES more closely match experimentalaerosol deposition than RANS (without near-wall correction). Simu-lating LES on the DES mesh showed that LES would be the preferredmethod among the models tested.

3. For the simulation of particles above 5 µm at normal breathing con-dition, the similar performance of RANS and LES/DES makes RANSthe preferred method.

155


156

Chapter 9

Conclusions andPerspectives

9.1 Conclusions

In order to account for inter-subject variability, the need for consideringrealistic CT based airway models has been recognized and an efficient La-grangian particle tracking module for unstructured grids to handle com-plex geometrical features has been implemented in the commercial C++flow solver of NUMECA. The RANS methodology was used in the follow-ing three applications:

Application 1: Fluid flow and particle deposition in a CT based realisticextra-thoracic airway model was studied. Inhalation flow rates of 15, 30and 60 l/min were considered with particle diameters ranging between 2and 20 µm. The complex flow patterns with skewed velocity profiles andflow separations are discussed. While most idealized mouth-throat modelspredict no transition to turbulence for a flow rate of 15 l/min, we observeincrease in kinetic energy levels soon after glottis, indicating the sensi-tivity to geometrical complexity at low flow rates. Mouth cavity acts asan effective filter and is responsible for the major deposition in the upperairway for all three flow rates. Combining our model depositions with pre-vious simulations in central bronchial tree depositions indicates that fornormal breathing conditions, inhalatory deposition for the particle size 5µm, which is usually referred to as the upper limit of the respirable rangefor inhalation drugs, does not exceed 12%.

Application 2: The axial dispersive effect of the upper airway structure

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CHAPTER 9. CONCLUSIONS AND PERSPECTIVES

(comprising mouth cavity, oropharynx and trachea) on a traversing aerosolbolus was investigated. This was done by means of aerosol bolus experi-ments on a hollow cast of a realistic upper airway model (UAM) and 3Dcomputational fluid dynamics (CFD) simulations in the same UAM geom-etry. The experiments showed that 50 ml boluses injected into the UAM,dispersed to boluses with a half-width ranging 80-90 ml at the UAM exit,across both flow rates (250, 500 ml/s) and both flow directions (inhalation,exhalation). These experimental results imply that the net half-width in-duced by the UAM typically was 69 ml. Comparison of experimental bolustraces with a 1D Gaussian derived analytical solution resulted in an axialdispersion coefficient of 200-250 cm2/s, depending on whether the boluspeak and its half-width, or the bolus tail needed to be fully accounted for.CFD simulations agreed well with experimental results for inhalatory bo-luses, and were compatible with an axial dispersion of 200 cm2/s. How-ever, for exhalatory boluses, the CFD simulations showed a very tight bo-lus peak followed by an elongated tail, in sharp contrast to the exhalatorybolus experiments. This indicates that CFD methods, which are widelyused to predict the fate of aerosols in the human upper airway, where flowis transitional, need to be critically assessed, possibly via aerosol bolussimulations. It is concluded that, with all its geometric complexity, the up-per airway introduces a relatively mild dispersion on a traversing aerosolbolus for normal breathing flow rates in inhalatory and exhalatory flowdirections.

Application 3: The potential of using aerosol boluses inhaled at 30 l/minto detect tracheal stenosis ranging 50-90% obstruction of tracheal crosssectional area was investigated. Computational fluid dynamic simulationswere performed in a realistic upper airway model of the mouth and tra-chea, using a two-equation RANS (with a k ! ! turbulence model). Theaerosol bolus deposition efficiency and bolus dispersion, in terms of bolushalf-width (HW) or bolus standard deviation (SD), were simulated as afunction of the degree of stenotic obstruction. The effect of aerosol particlesize on bolus deposition efficiency was also considered. Simulations pre-dicted that shallow boluses inhaled by subjects with a stenosis of 80% ormore, would be affected most in terms of bolus deposition of a 5 µm aerosol,showing a high relative change (a 2-3 fold deposition efficiency from 50%to 80-85% obstruction) and a measurable absolute change (10-18% greaterdeposition for 80-85% stenosis). Smaller (1 µm) and larger (10 µm) parti-cles would require picking up smaller absolute differences in bolus depo-sition efficiency, which could be difficult to achieve experimentally. It isconcluded that shallow boluses of 5 µm aerosols, and in particular theirdeposition efficiency, could be a useful non-invasive diagnostic tool for the

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detection of tracheal stenosis.

Finally, fluid flow was simulated in a simplified human mouth-throatmodelunder normal breathing conditions (30 l/min) alternatively employing RANS,DES and LES methods. To test the validity of the fluid phase simulations,PIV measurements were carried out in a central sagittal plane of an iden-tical model cast. Velocity and kinetic energy profiles showed good quan-titative agreement of experiments with LES/DES, and less so with RANSk ! !. Mouth-throat deposition was simulated for particle diameters 2, 4,6, 8 and 10 µm. By comparison with existing experimental data, LES/DESshowed considerable improvement over the RANS k ! ! model in predict-ing deposition for particle sizes below 5 µm. For the bigger particles, RANSk!! and LES/DES methods produced similarly good predictions. It is con-cluded that for the simulation of medication aerosols inhaled at a steadyflow rate of 30 l/min, LES and DES provide more accurate results than theRANS k ! ! model.

9.2 Perspectives

9.2.1 Future CFD developments

In case of LES/DES, concurrent simulation of fluid flowand particle motionis indeed a lengthy computational process and one needs to look towardsother cost-effective ways to model the particle motion. Frozen LES as de-scribed and tested in Chapter 8 was already the first step, however, it wasseen to be inaccurate. In future, the following potential ways of particlemodeling should also be tested.

1. Instead of performing frozen LES simulation at just one instanta-neous flow field, it could be useful to perform few such frozen LEScalculations separated by an integral time scale (Tint = length of themodel/average cross-sectional velocity). For example, say we perform25 independent frozen LES calculations, each separated by Tint intime, and finally obtain the particle deposition characteristics by av-eraging the results of the 25 frozen LES calculations. Since we areaveraging the deposition results of 25 different realizations, the errorencountered is expected to be less when compared to performing justone realization as in Chapter 8.

2. Store the time-dependent flow field in terms of three instantaneousvelocity components over a length of one integral time scale (T int).Perform particle tracking by repeatedly playing this stored flow field.

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To avoid discontinuities in the flow field, the stored flow field shouldbe played in a periodic fashion. For example, if 100 instantaneousflow fields are stored, we start the particle tracking using flow fields1 to 100, and then back from 100 to 1.

3. Store the time-averaged three-dimensional flow field in terms of threeaveraged velocity components, the turbulence kinetic energy (whichis the sum of resolved and modeled part), and the energy dissipationrate. By using the time-averaged LES solution, perform the parti-cle tracking as in case of RANS (using an eddy interaction model).The time-averaged solution obtained from LES is indeed more accu-rate when compared to RANS fluid-flow solution. Also, the effect ofanisotropy can be considered in the eddy interaction model as theReynolds stresses are directly available from LES. Hence, the resultsobtained from particle tracking are expected to be more accurate asopposed to RANS.

In Chapter 6, it was seen that CFD simulations did not reproduce the exha-latory bolus measurement data. To resolve this issue, PIV measurementsof both fluid flow and bolus dispersion should be performed on the upperairway model during the exhalation mode. The data should then be usedto validate/improve the turbulence models in CFD.

9.2.2 Future Airway model developments

Airway model extension

In the present thesis, we have been dealing with the upper airway model,i.e., from mouth till end of trachea. However, the inhaled medical aerosolshave still a long way to go before reaching the alveolar regions of the lungs.The present upper airway model can be extended to accommodate the tra-cheobronchial region. The first few generations of the tracheobronchialregion are still a challenging part in terms of CFD simulations. Sincethe kinetic energy levels in trachea are generally high, it is expected thatthe kinetic energy prevails even downstream of it, at least till first two orthree bifurcations. Most of the studies on the fluid flow and particle depo-sition in the tracheobronchial region assume the flow to be laminar. Thisis generally not true, especially for normal (30 l/min) and heavy (60 l/min)breathing conditions. Hence, performing turbulent simulations in the tra-cheobronchial region is required. Also, the flow entering into the tracheo-broncial region is quite distorted due to the presence of larynx upstream.This upstream effect should also be considered in future simulations.

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Moving larynx

The larynx (glottis) area is seen to considerably vary during respiratorycycle, even during quiet breathing. Renotte et al. [120] mentioned thatthe glottis dimension may evolve between 5.7±0.5 and 10.1±0.5 mm dur-ing the respiratory cycle. Considering the dynamically varying geometriesduring a flow simulation is completely a new and unexplored aspect, espe-cially with applications to upper airways, and hence a preliminary study inthis direction is needed to provide insights towards the moving boundaryeffects and the numerical complexities associated with it.

9.2.3 Future applications

Airway resistance measurement

In chapter 7, the potential of using the non-invasive techniques of disper-sion/deposition in order to detect the tracheal stenosis was explored. Apossible additional non-invasive technique is to measure the airway resis-tance and relate the undue increase in resistance to the airway abnormal-ity. There are several possible ways of measuring the airway resistance,however, the most promising method, which is sensitive to the upper air-way component is the forced oscillation technique. Recently, Verbanck etal. [153] at VUB medical school has shown a promising use of forced oscil-lation technique in detecting fixed upper airway obstruction. This poten-tial method should further be explored.

Ultrafine particles in healthy and diseased lungs

Submicron particles, i.e. particles with diameters smaller than 1 µm, fallinto the category of fine respiratory aerosols (diameter ranging from 100nm to 1 µm) or the category of ultrafine particles (diameter <100 nm, alsoreferred as nanoparticles). In the ambient air, the ultrafine particles (<100nm in diameter) are much greater in number than coarser particles. Aconsiderable fraction is emitted from combustion sources such as dieselengines, where particle size ranges from 5 to 500 nm. Other sources aretobacco smoke (18 nm to 1.6 µm), viruses such as Avian flu and SARSwith sizes typically from 20 to 200 nm, and bacteria. Measured concen-trations on roads in Minnesota are as high as 107 particles/cm3 [86]. An-imal studies show that exposure to high doses of such particles can causelung injuries [112, 113]. Also, recent epidemiological studies have indi-cated that ultrafine particles may have greater adverse respiratory effectsthan fine or coarse particles in urban air [115]. Due to all the above rea-sons, studying the deposition characteristics of ultrafine particles in the

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tracheobronchial regions is important.

In a study of the World Health organization (2004), it is stated that themost common cause of death in men and women is cancer-related. Lungcancer is responsible for 1.3 million deaths worldwide annually. It is hy-pothesized that cancerous lesions may arise from a continuous insult fromtoxic particles, creating hot spots in particular locations of the tracheo-bronchial tree. Hence, it is interesting to predict tumor location in anygiven patient on basis of aerosol deposition hot spots simulated for this pa-tients particular traceobronchial structure. This type of study has neverbeen done before. Obviously, such a study necessitates that a CT render-ing of the lung structure of each individual patient, and the localization ofthe tumor with respect to this patients particular structure. This indeedrequires an optimal coordination of medical interventions (examinations,biopsies, CT) with the engineering staff, in order to be able to use thesemedical data in a quantitative way (i.e., use the data for CFD computa-tions).

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List of Publications

Journal articles

1. S T Jayaraju, M Brouns, C Lacor, B Belkassem, S Verbanck. Largeeddy and detached eddy simulations of fluid flow and particle depo-sition in a human mouth-throat, Journal of Aerosol Science, 39:862-875, 2008.

2. S T Jayaraju, M Paiva, M Brouns, C Lacor, S Verbanck. The con-tribution of upper airway geometry to convective mixing, Journal ofApplied Physiology, 105:1733-1740, 2008.

3. S T Jayaraju, M Brouns, S Verbanck, C Lacor. Fluid flow and parti-cle deposition analysis in a realistic extrathoracic airway model usingunstructured grids, Journal of Aerosol Science, 30:494-508, 2007.

4. M Brouns, S T Jayaraju, C Lacor, J De Mey, M Noppen, W Vincken,S Verbanck. Tracheal stenosis: A flow dynamics study, Journal ofApplied Physiology, 102:1178-1184, 2007.

Conference proceedings

1. S T Jayaraju, M Brouns, S Verbanck, C Lacor. Fluid-particle dy-namics in human mouth-throat geometry, International Conferenceof Turbulence and Interaction, Sainte-Luce, Martinique, 2009.

2. S T Jayaraju, M Paiva, C Lacor, W Vincken, S Verbanck. Shallowaerosol bolus tests for detection of tracheal stenosis, American Tho-racic Society Conference, San Diego, USA, 2009.

3. S T Jayaraju, M Brouns, S Verbanck, C Lacor. LES and DES studyof fluid-particle dynamics in human upper respiratory pathway, Eu-ropean Congress on Computational Methods in Applied Sciences andEngineering, Venice, Italy, 2008.

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4. S T Jayaraju, M Brouns, S Verbanck, C Lacor. Lagrangian particletracking on unstructured meshes in a CT based mouth-throat geome-try, International Conference on Multiphase Flow, Leipzig, Germany,2007.

5. S T Jayaraju, M Brouns, S Verbanck, C Lacor. Modeling particleladen flows with applications to clinical aerosols, ERCOFTAC day,Toulouse, France, 2007.

6. C Lacor, S T Jayaraju, M Brouns, S Verbanck. Simulation of the air-flow in the upper airways with applications to aerosols, InternationalWorkshop on Coupled Methods in Numerical Dynamics, Dubrovnik,Croatia, 2007.

7. M Brouns, S T Jayaraju, C Lacor, J De Mey, M Noppen, W Vincken,S Verbanck. Effect of tracheal stenosis on local pressure drop, Amer-ican Thoracic Society Conference, San Francisco, USA, 2007.

8. S T Jayaraju, M Brouns, S Verbanck, C Lacor. Effects of trachealstenosis on flow dynamics in upper human airways, European Con-ference on Computational Fluid Dynamics, Egmond aan Zee, TheNetherlands, 2006.

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