Effects of Mesh Style and Grid Convergence on Particle Deposition in Bifurcating Airway Models With...

8/10/2019 Effects of Mesh Style and Grid Convergence on Particle Deposition in Bifurcating Airway Models With Comparisons

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Medical Engineering & Physics 29 (2007) 350366

Effects of mesh style and grid convergence on particle deposition inbifurcating airway models with comparisons to experimental data

P. Worth Longest , Samir Vinchurkar

Department of Mechanical Engineering, Virginia Commonwealth University, 601 West Main Street,

P.O. Box 843015, Richmond, VA 23284-3015, United States

Received 10 January 2006; received in revised form 11 May 2006; accepted 16 May 2006

Abstract

A number of research studies have employed a wide variety of mesh styles and levels of grid convergence to assess velocity fields and

particle deposition patterns in models of branching biological systems. Generating structured meshes based on hexahedral elements requires

significant timeand effort; however, these meshes are oftenassociated with high quality solutions. Unstructured meshes that employ tetrahedral

elements can be constructed much faster but may increase levels of numerical diffusion, especially in tubular flow systems with a primary

flow direction. The objective of this study is to better establish the effects of mesh generation techniques and grid convergence on velocity

fields and particle deposition patterns in bifurcating respiratory models. In order to achieve this objective, four widely used mesh styles

including structured hexahedral, unstructured tetrahedral, flow adaptive tetrahedral, and hybrid grids have been considered for two respiratory

airway configurations. Initial particle conditions tested are based on the inlet velocity profile or the local inlet mass flow rate. Accuracy of

the simulations has been assessed by comparisons to experimental in vitrodata available in the literature for the steady-state velocity field in

a single bifurcation model as well as thelocalparticle deposition fraction in a double bifurcation model. Quantitative grid convergence was

assessed based on a grid convergence index (GCI), which accounts for the degree of grid refinement. The hexahedral mesh was observed to

have GCI values that were an order of magnitude below the unstructured tetrahedral mesh values for all resolutions considered. Moreover, the

hexahedral mesh style provided GCI values of approximately 1% and reduced run times by a factor of 3. Based on comparisons to empirical

data, it was shown that inlet particle seedings should be consistent with the local inlet mass flow rate. Furthermore, the mesh style was foundto have an observable effect on cumulative particle depositions with the hexahedral solution most closely matching empirical results. Future

studies are needed to assess other mesh generation options including various forms of the hybrid configuration and unstructured hexahedral

meshes.

2006 IPEM. Published by Elsevier Ltd. All rights reserved.

Keywords: Respiratory particle dynamics; Grid convergence index; Respiratory dosimetry; Bifurcation models; Lagrangian particle tracking

1. Introduction

Accurate assessment of the dose received from respiratory

aerosols is critical in a number of applications including tox-icology analysis of pollutant exposures and pharmacology

analysis of inhaled medications[15].To assess respiratory

aerosol dynamics within sub-regions of the lung, a number

of researchers have employed computational fluid dynamics

(CFD) analysis. Numerical simulations provide a powerful

technique to assess the effects of geometric form and mod-

Corresponding author. Tel.: +1 804 827 7023; fax: +1 804 827 7030.

E-mail address:[email protected](P.W. Longest).

ifications, particles sizes, breathing patterns, and regional

ventilation rateson localparticle transport and depositionpat-

terns both in the lung and in other biological systems. Com-

putational fluid dynamics simulations have been employedto evaluate airflow patterns and the resulting particle depo-

sition profiles in model geometries of the extrathoracic

[611], bronchial [1222] and pulmonary [2325] regions

of the lung. Simulations have been conducted for micron

[10,14,19,2630],submicron[8,10,12,31,32]and nanoscale

[11,15,3335]particle sizes. The influence of physical struc-

tures such as the larynx[36],various shapes of carinal ridges

[37], cartilaginous rings [38] and lung tumors [31,39,40]

on the flow field have been assessed numerically. Compu-

1350-4533/$ see front matter 2006 IPEM. Published by Elsevier Ltd. All rights reserved.

doi:10.1016/j.medengphy.2006.05.012
mailto:[email protected]://localhost/var/www/apps/conversion/tmp/scratch_4/dx.doi.org/10.1016/j.medengphy.2006.05.012http://localhost/var/www/apps/conversion/tmp/scratch_4/dx.doi.org/10.1016/j.medengphy.2006.05.012mailto:[email protected]


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P.W. Longest, S. Vinchurkar / Medical Engineering & Physics 29 (2007) 350366 351

tational simulations have also been employed to consider the

effects of variable outflow division ratios to simulate differ-

ent regional ventilation conditions[21],asymmetric branch-

ing[22],and transient flow[20]. In some cases, flow field

validations of these models have often been performed by

comparisons to in vitro results [16,20]. Validations of particle

depositions are widely based on comparisons to regional orbranch-averaged values[13,29,33,35,4143].Very few stud-

ies have considered quantitative validations in comparison

to more localized particle deposition patterns in single and

double bifurcation models[44].

In addition to providing a better understanding of transport

phenomena associated with respiratory aerosols, regional

CFD models have also been employed to determine branch-

level deposition functions and local deposition enhancement

factors for use in whole-lung dosimetry modeling [11,12,15].

Improved model realism and the effects of upstream flow fea-

tures, which have been shown to be important[45]but not

fully quantified, require increasingly larger and more com-

plexsimulations. As such, numericalefficiency balancedwithacceptable accuracy becomes increasingly more important in

the simulation of regional respiratory aerosol dynamics and

biological flows in general. However, methods that improve

system efficiency while maintaining accuracy are difficult to

identify considering that current CFD studies employ a large

number of solution methods, mesh styles, and initial condi-

tions and that particle deposition has rarely been validated at

the sub-branch level.

When performing a CFD solution, generating the com-

putational mesh often requires a significant amount of the

total time and can affect the solution quality. As such,

faster and automated mesh generation techniques can greatlyenhance the total efficiency of CFD solutions. Tetrahedral

meshes, composed of four-sided triangle-faced elements, can

be largely automated by current mesh generation software

once a clean or seamless surface model is available. These

meshes can be refined in regions of interest or automati-

cally adapted to the flow field to provide increased resolution

and reduced discretizationerrors. However,tetrahedral-based

meshes are generally considered to be less accurate than hex-

ahedral meshes due to a number of factors, including poor

alignment with the primary flow direction and fewer control

volume nodes[46].Recently, hybrid meshes have also been

introduced that are typically a combination of lower order

internal elements surrounded by a layer of higher order pyra-

mid, prism or hexahedral elements on the surface in order to

better resolve the near-wall flow field. To assess the improved

total efficiency in CFD solutions offered by tetrahedral and

hybrid meshes in comparison to the higher accuracy that is

typically associated with hexahedral meshes, comparisonand

validation studies are necessary. This is especially true for

biological systems where the flow is predominantly oriented

in one direction with superimposed secondary velocities aris-

ing from bifurcations and geometric curvature.

Several computational studies have considered the effects

of various meshing styles on solution characteristics in bio-

logical systems with respect to in vitro validations, grid

convergence, and other parameters either directly or indi-

rectly. Liu et al.[47]performed CFD simulations of the total

cavopulmonaryconnectionusingstructured and unstructured

meshes. In this study, multi-block hexahedral meshes were

considered structured and tetrahedral meshes were referred

to as unstructured. For this bifurcating geometry, flow fieldswere reported to be similar between the structured and

unstructured meshes; however, the structured model resolved

more secondary vortices. Furthermore,values of the velocity-

derived energy loss were reported to be significantly different

based on the two mesh styles. Validation of the velocity field

with experimental measurements was not reported. Prakash

and Ethier [48] conducted a mesh convergence study of blood

flow in a right coronary artery model using unstructured high-

order tetrahedral meshes and a finite element solution. Mesh-

independent velocity fields for a solution adaptive tetrahedral

mesh were obtained based on a 5% cut-off criterion. How-

ever, mesh convergence of the wall shear stress field was

only observed to within 10% for the upper limit of ele-ments considered. Tambasco and Steinman[49] employed

a Lagrangian-based 3D volumetric residence time (VRT)

model in stenosed carotid artery bifurcations. Theoretically,

the VRT should approach unity as Lagrangian mass con-

servation is achieved. Using uniform and adaptively refined

tetrahedral meshes, Tambasco and Steinman [49] reported

that a uniform VRT could not be achieved for the most

refined tetrahedral mesh considered. Similar to the VRT con-

cept, Longest et al.[50]calculated near-wall residence time

(NWRT) values of platelet-like spheres in an anastomotic

model employing hexahedral elements. In this model, the

Lagrangian-based NWRT values were shown to converge;however, the VRT criterion was not tested.

In addition to the appropriate mesh resolution, an uncer-

tainty associated with Lagrangian particle tracking in bio-

logical flows is the appropriate initial particle density or

seeding. With respect to respiratory dynamics, Comer et

al.[51]and Zhang and Kleinstreuer[19]initiated parabolic

particle seedings that were consistent with the initial veloc-

ity profiles employed in the corresponding in vitro experi-

ments. These simulations show good agreement with branch-

averaged deposition efficiencies reported by Kim and Fisher

[52]. Subsequent simulations employing initially parabolic

particle profiles matched branch averaged deposition results

very well [29,33,42]. However, Kim [53] suggests that a con-

stant initial profile may be most appropriate for respiratory

dynamics simulations in some cases, as was employed for

nanoparticle validation studies by Shi et al. [35]. Longest

and Kleinstreuer[54]employed a parabolic particle profile

multiplied by the local radial coordinate to account for local

inlet mass flow rate in an axisymmetric investigation of blood

particle deposition.

While it is commonly accepted that hexahedral meshes

provide more accurate solutions than tetrahedral meshes at

similar densities, this difference has not been quantified for

bifurcating domains with comparisons to experimental data.


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352 P.W. Longest, S. Vinchurkar / Medical Engineering & Physics 29 (2007) 350366

Quantitatively assessing the accuracy of various mesh styles

in representative geometries is necessary to help modelers

choose the most effective mesh type for a given application.

For instance, if branch-averaged deposition rates are of inter-

est, a rapidly generated tetrahedral mesh may be sufficient.

In contrast, if local particle accumulations are needed, then a

block structured hexahedral mesh may be necessary. Consid-ering the significant differences in time required to generate

these two mesh styles, it is critical to quantify their associated

accuracy in representative bifurcating geometries. Compar-

isons to localized in vitrodata are necessary to make these

assessments. Moreover, quantitative comparisons of numer-

ical results to sub-branch particle deposition data has only

been reported in one previous study by Oldham et al. [44].

Theobjectiveof this work is to betterevaluate theeffectsof

mesh generationtechniquesand grid convergenceon velocity

and local particle deposition fields in representative bifurcat-

ing geometries. In order to achieve this objective, four widely

used mesh styles including structured hexahedral, unstruc-

tured tetrahedral, flow adaptive tetrahedral, and hybrid gridshave been considered for two respiratory airway models. Ini-

tial particle conditions tested are based on the inlet velocity

profile or the inlet mass flow rate profile. Conditions required

for grid convergent results based on a 1% relative error cri-

terion are established. Accuracy of the simulations has been

assessed by comparisons to experimental in vitrodata avail-

able in the literature for the velocity field in a single bifurca-

tion model [55] as well as thelocalparticledeposition fraction

in a double bifurcation model[44].For this study, conditions

of the models have been limited to match the in vitroexperi-

mental systems considered. However, the Reynolds numbers,

Stokes number and particle size implemented are generallyrepresentative of airflow and inhaled micron aerosols in the

central tracheobronchial airways and particle hemodynamics

in large artery bifurcationsfor laminar flow conditions. These

benchmark studies are intended to serve as a basis for future

Table 1

Geometry parameters for the single bifurcation (SB) model

Bifurcation First

Parent diameter (cm) D1= 3.81

Daughter diameter (cm) D2= 2.70

Bifurcation radius of curvature Rb1= 7D2Carinal radius of curvature Rc1= 0

Bifurcation half angle = 35

Branch Reynolds number Re1=518

Re2=367

improvements in accuracy and computational efficiency and

to facilitate modeling broader regions of the respiratory tract.

2. Methods

2.1. Bifurcation models and boundary conditions

Geometric surface models of single and double bifurca-tions of the respiratory tract have been considered to evaluate

the effects of mesh style and grid convergence on veloc-

ity fields and particle deposition locations. The first model

is an idealized single bifurcation (SB) geometry as used in

the velocity field measurement studies of Zhao and Lieber

[55](Fig. 1a). This model was selected due to the availabil-

ity of a quantitative geometric characterization as well as

detailed reports of the velocity field under various conditions

[5557]. Consistent with the experimental model of Zhao

and Lieber[55],characteristics of the geometry are outlined

inTable 1.This model includes a sharp carinal ridge formed

by the intersection of the two daughter branches as well as aconstant cross-sectional area in the region of the bifurcation,

i.e., 2D22/D21 = 1. The intent of this model is to capture key

geometric parameters of the respiratory airways in a manner

that can be quantitatively defined and modified. However, the

Fig. 1. Geometric surface models of the branching respiratory airways. (a) An idealized single bifurcation model used by Zhao and Lieber[55]for velocity

field measurements and (b) the physiologically realistic bifurcation (PRB) model proposed by Heistracher and Hofmann [58]and used in the particle deposition

study of Oldham et al.[44].


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model itself is not representative of any specific respiratory

generation and may not be appropriate for particle deposi-

tion studies due to the inclusion of a sharp carinal ridge. Two

planes of symmetry were included in generating the compu-

tational meshes of this geometry.

The second geometry selected is a double bifurcation

model representative of respiratory generations G3G5 (seg-mental bronchi) and is characterized by improved model

realism in the region of the bifurcations (Fig. 1b). This model

is based on the physiologically realistic bifurcation (PRB)

geometry specified by Heistracher and Hofmann [58]. For

the PRB geometry, Heistracher and Hofmann[58]provide a

complete mathematical description of a single symmetric or

asymmetric bifurcation based on a setof 11 geometric param-

eters and two sigmoid functions. Appropriate anatomic data

from themeasurementstudies of Hammersley and Olson [59]

and Horsfield et al.[60]have been incorporated to define the

appropriate surface curvatures of the model. Branch length

and diameter data is available from a number of sources

for adults [61,62]and children[63]. In comparison to thenarrow and wide geometries of Balashazy and Hofmann

[64,65],the PRB model provides a significant improvement

in realism in the region of the bifurcation[58]and is consis-

tent with the endoscopic carinal ridge images of Martonen et

al.[37].

Specific parameters for the double bifurcation model of

generations G3G5 employed in this study are identical to

the values used in the work of Heistracher and Hofmann

[14]and the localized particle deposition measurements of

Oldham et al. [44], as given in Table 2. Curvatures of the

branches and carinal ridges are based on representative val-

ues of the measurements made by Hammersley and Olson[59]and Horsfield et al. [66].The diameters and lengths of

the G4 airway (middle conduit) are based on the ICRP [61]

model for an adult male. Appropriate diameter and length

scaling parameters are then applied for the third and fifth gen-

eration airways[14].One plane of symmetry was included

in generating the computational meshes of this geometry.

For comparisons to the experimental results considered

in this study, steady inspiratory flow has been assumed for

both the single and double bifurcation models. Validation

of the velocity flow field has been based on comparisons to

the results of Zhao and Lieber [55] for an inlet Reynolds

Table 2

Geometry parameters for the physiologically realistic bifurcation (PRB)

model

Bifurcation First Second

Parent diameter (cm) D1= 0.56 D2= 0.45

Daughter diameter (cm) D2= 0.45 D3= 0.36

Length of branch (cm) L1= 1.1

L3= 0.77L2= 0.92

Outer curvature radius Rb1= 2.5D2 Rb2= 2.5D3Carinal curvature radius Rc1= 0.2D2 Rc2= 0.2D3Bifurcation half angle 1= 35

2= 35

Branch Reynolds number Re1= 1788

Re3= 695Re2= 1113

number of 518. Because this geometry is not intended to be

representative of a specific airway, a corresponding inhalation

flow rate is not available.

The steady inspiratory flow rate employed in the PRB

model results in an inlet Reynolds number of 1788 ( Table 2).

For respiratory generations G3G5, this is consistent with an

inhalation flow rate in the trachea of 60 l/min and representsa state of heavy exertion[61].The resulting flow rate in gen-

eration 3 is 125 ml/s, as specified in the experimental study

of Oldham et al.[44].

Both inlet velocity and particle profiles are expected to

have a significant impact on the flow field and particle depo-

sition locations. For comparisons toin vitrodeposition data,

these profiles may be largely influenced by upstream effects

in the experimental particle generation system. Longest and

Vinchurkar[67],have shown that upstream transition to tur-

bulence results in a relatively blunt initial velocity field and

particle profile at the model inlet. However, the flow within

the PRB model can be approximated as laminar. As such,

fully developed blunt turbulent profiles of velocity and ini-tial particle distributions have been assumed at the model

inlet.Within themodel, laminar flow is assumed. Justification

for these assumptions is based on a high level of agreement

between resulting deposition profiles and experimental data.

Outlet flow is assumed to be evenly divided between the left

and right symmetric branches in both models, i.e., homoge-

neous ventilation. The effects of gravity have been neglected

for the velocity field results in the SB model. However, grav-

ity effects are expected to be significant for the deposition

of the 10m particles considered. Therefore, gravity has

been included in the flow field and particle trajectory cal-

culations of the PRB model with the gravity vector orientedin the positivez-direction, i.e., in the plane of the bifurcation,

to remain consistent with the experiments of Oldham et al.

[44].

2.2. Mesh styles

Four commonly implemented mesh styles have been con-

sidered for the generation of a computational solution in both

the SB and PRB models (Fig. 2).For this study, structured

meshes are defined as having continuous grid lines on all

faces, which require the domain to be subdivided into struc-

tured blocks. Unstructured meshes are defined as having at

least one face on which the grid lines do not remain contin-

uous. The mesh styles considered include a block structured

hexahedral mesh (Fig. 2a), an unstructured tetrahedral mesh

(Fig. 2b), a tetrahedral mesh that is adapted to the steady flow

field to better resolve regions of significant velocity gradients

(Fig. 2c), and a hybrid mesh consisting of internal tetrahe-

dral elements surrounded by a layer of higher order five-sided

pyramid elements (Fig. 2d). All meshes were created using

the integrated solid modeling and meshing program Gambit

(Fluent Inc.).

The structured mesh consists of six-sided hexahedral ele-

ments arranged in a system of interconnected blocks. The


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Fig. 2. Four characteristic meshing styles of the PRB geometry including the (a) hexahedral or six-sided element structured multiblock mesh, (b) tetrahedral

or four-sided element unstructured mesh, (c) flow adaptive tetrahedral mesh to better resolve gradients in the flow field, and (d) hybrid mesh consisting of

tetrahedral elements throughout the interior surrounded by a layer of pyramid or five-sided elements on the surface.

blocks have been arranged in a butterfly blocking design

which minimizes control volume distortions while aligninga high percentage of elements with the local flow direction

(Fig. 2a). Moreover, mesh density is increased near the wall

and near the bifurcation points. This multiblock structure is

difficult to develop because gridlines may be distorted, but

must remain continuous throughout the geometry. That is, the

number of cells on mating block faces must match. Design-

ing a quality block-structured meshing configuration for a

geometry with multiple branches in which hexahedral ele-

ments largely align with the streamlines is a user intensive

nontrivial task. However, the solution quality resulting from

brick-style meshes consisting of six-sided elements is often

considered superior to tetrahedral meshes with many more

control volumes or elements[46]. This is because parts of

the errors made at opposite hexahedral cell faces cancel com-

pletely if the cell faces are parallel. In addition, hexahedral

meshes can be aligned with the predominant direction of the

flow. Additional advantages of hexahedral meshes include

improved interpolations of both surface fluxes and particle

positions within each cell.

The tetrahedral meshing style considered consists of four-

sided elements with increasing mesh density near thewall and

at the bifurcation points. In comparison to block-structured

meshes, generation of tetrahedral meshes can be largely auto-

mated by many commercial grid generation software pack-

ages once a watertight surface model is developed. The

creation of internal blocks to ensure continuous gridlines isnot necessary. However, unstructured tetrahedral meshes are

not capable of aligning with the direction of predominant

flow, thereby increasing numerical diffusion. This effect may

be pronounced in the long and thin conduits that characterize

biological flow systems. Furthermore, tetrahedral elements

provide fewer nodes for the calculation of accurate control

volume surface fluxes and for the interpolation of internal

particle locations. Increasing the number of cells is the tech-

nique typically employed to overcome these disadvantages;

however, significant numerical diffusion errors and inaccu-

racies in near-wall particle interpolations may persist.

To overcome the increased level of numerical diffusion

that is typically associated with tetrahedral meshes, mesh

refinements based on gradients in the flow or other parame-

ters are possible. A representative flow-adaptive tetrahedral

mesh is shown in Fig. 2cbased on one iteration of refine-

ment to better resolve velocity gradients in the flow field.

As with the generation of the initial tetrahedral mesh, the

refinement process can be automated, thereby reducing the

time required for grid generation in comparison to hexahe-

dral meshes. However, refinements are generally restricted

to steady flows and may not be sufficient to overcome the

numerical diffusionerrorsassociated with tetrahedral meshes

in long and thin biological conduits. For this study, mesh


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refinement has been based on a velocity gradient adaption

approach[68]that employs hanging nodes to divide a single

tetrahedron into eight tetrahedra.

Recently, a number of commercial mesh generation pack-

ages have incorporated the ability to develop hybrid grids

consisting of multiple styles of elements. For bifurcating

flows, the most practical hybrid mesh style employs tetra-hedral elements in the core of the flow surrounded by a layer

of higher order five- or six-sided pyramid or hexahedral ele-

ments at the wall. This is intended to better resolve surface

gradients, wall shear stress values, and particle depositions.

However, user intensive structuring of the external surface is

required to assure that the surface gridlines remain continu-

ous throughout the bifurcation in order to apply the higher

order elements in this region (Fig. 2d).

2.3. Governing equations

Flow conditions in both the geometries considered are

assumed to be isothermal, incompressible and laminar. Fur-

thermore, in cases where particle trajectories are considered,

the particle concentrations are assumed to be sufficiently

dilute such that momentum coupling effects of the dispersed

phase on the fluid can be neglected, i.e., a one-way coupled

flow. The governing equations for the respiratory airflow of

interest include the conservation of mass

u = 0 (1)

and momentum

u

t

+ (u )u = 1

(p+ ) (2a)

where u is the velocity vector,p the pressure and is the fluiddensity. The shear stress tensor is given by

= [u+ (u)T] (2b)

whereis the absolute viscosity.

Hydrodynamic inlet and boundary conditions, in addi-

tion to the no-slip wall condition, were selected to match the

particular flow system and experimental conditions of inter-

est. To approximate a uniform outflow distribution, equally

divided mass flow was specified. Furthermore, flow field out-

lets were extended far downstream such that the velocity was

normal to the outlet plane, i.e., fully developed flow profiles

with no significant radial velocity component.

One-way coupled trajectories of monodisperse 10m

aerosols have been calculated on a Lagrangian basis by

integration of an appropriate version of the particle tra-

jectory equation within the PRB model for comparison

to the experimental results of Oldham et al. [44]. Char-

acteristics of the 10m aerosols of interest within the

PRB model include a particle density p= 1.06 g/cm3,

a density ratio =/p 103, a Stokes number St=

pd2pCcU/18D 0.25, and a particle Reynolds number

Rep = |u v|dp/ 10.

The appropriate equations for spherical particle motion

under the conditions of interest can be expressed

dvi

dt=

Dui

Dt+ f

p(ui vi) + gi(1 ) + fi,lift (3a)

and

dxidt

= vi(t) (3b)

subject to

xi(t0) = xi,0 (3c)

In the above equations, vi andui are the components of the

particle and local fluid velocity, respectively. The ratioof fluid

to particle density is represented as =/p, andgi denotes

gravity, when warranted. The characteristic time required

for particles to respond to changes in the flow field, or the

momentum response time, is p = pd2p/18. The pressure

gradient or acceleration term is often neglected for aerosols

due to small values of the density ratio. However, it has beenretained here to emphasize the significance of fluid element

acceleration in biological flows [50]. The drag factorf, which

represents the ratio of the drag coefficient to Stokes drag, is

based on the expression of Morsi and Alexander[69]

f=CDRep

24 =

Rep

24

a1 +

a2

Rep+

a3

Re2p

(4)

where ai coefficients are available for multiple particle

Reynolds number ranges [69] expected for the respiratory

aerosols of interest. The influence of the lubrication force,

or near-wall drag, is expected to be reduced for the aerosol

system of interest in comparison to liquid flows due tonon-continuum effects[50,70].As such, this term has been

neglected for the simulations considered here. Saffman style

lift has been considered based on the force term described in

Longest et al.[50].Due to the significant size of the particles

considered and the dilute concentrations, Brownian motion

and particle-particle collision effects have been neglected.

Particles have been assumed to deposit upon initial wall con-

tact.

2.4. Initial particle profiles

Previous studies have shown that the initial particle pro-

file has a significant impact on the resulting branch-averaged

deposition results [19]. Furthermore, a number of studies

have initialized particle profiles to be consistent with the

inlet velocity flow field, taking either a parabolic or a con-

stant shape. This is equivalent to specifying that local particle

mass flow rate mp is proportional to the local velocity field,

or

mp u(r) (5a)

An alternative approach is to specify the initial particle mass

flow rate as proportional to the local fluid mass flow rate m

[71]


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mp pu(r) dA m u(r) dA (5b)

wherepis the bulk density of particles in the flow, typically

assumed to be a constant. In the above expression, local par-

ticle and fluid velocities are assumed to be in equilibrium.

Based on a mass flow rate specification, particles within an

infinitesimal ring of fluid at the cylindrical inlet should then

be specified as proportional to the local mass flow rate withinthe ring

mp,ring mring=

r2r1

u(r)2r dr (6a)

Furthermore, it is often convenient to specify initial particle

locations within discrete or finite rings with a locally constant

velocityu12 and a mean radial distance from the center of

(r1+ r2)/2. The resulting mass flow within the discrete ring

can be written

mp,ring mring= u122r1 + r2

2 r (6b)

If we assume a constant fluid density and rings of constant

widthr, the particle mass flux is

mp,ring u12r1 + r2

2 (6c)

where u12 is the mean velocity within the ring of interest

and (r1+ r2)/2 is the distance to the center of the ring. Eq.

(6c)represents a rectangle rule approximation to the integral

presentedin Eq.(6a), which is convenient for the specification

of initial particle seedings in finite rings. Similar results can

be obtained for the integration of Eq. (6a). In the case of

Eq.(6c), the approximate particle profile that accounts for

the local mass flow rate is consistent with the local velocityof each ring multiplied by the local radial coordinate at the

center of the ring.

To generate random particle seedings consistent with the

two methods described above, corresponding probability

density functions were first defined. These functions were

scaled to range from 0 to 1 in the radial direction and nor-

malized such that R=10

f(r) = 1 (7)

For each density function, random variantsrpwere selected

to correspond to particle positions in the radial direction. This

was accomplished by subdividing the radial direction into 32

concentric rings, or bins, and applying the rejection method

described in Press et al. [72]. An angular or value was

then selected from a second uniform distribution function to

specify the second polar coordinate of each particle location

on a unit circle. Transformations to linear coordinates in the

plane of the inlet were then performed.

In this study, initial particle seedings consistent with fully

developed velocity (Eq. (5a)) and mass flow rate profiles

(Eq.(6c)) have been considered. A normalized representa-

tion of the fully developed turbulent velocity inlet profile is

shown inFig. 3a. The corresponding inlet particle seeding

shows a significant concentration of particles at the inlet cen-

ter (Fig. 3b). In contrast, the normalized inlet mass flow rate

profile is zero at the radial center due to a reduction in cross-

sectional area (Fig. 3c). The corresponding particle seeding

displays a more even distribution of particles arising from the

balance of velocity and area within each ring (Fig. 3d).

2.5. Numerical method

Discretization errors may be defined as any difference

between the exact solution of the governing equations and

the discretized system[73].These errors arise from numeri-

cal algorithms, the mesh style and quality used to discretize

the equations, and boundary conditions[46]. In this study,

errors arising from the mesh style and quality are of inter-

est. To isolate mesh related discretization errors, a common

solution procedure implemented by a commercial code has

been selected and consistently applied to the four mesh styles

considered.

To solve the governing mass and momentum conservationequations in each of the geometries and for each mesh style,

the CFD package Fluent 6 has been employed. This commer-

cial software provides an unstructured control-volume-based

solution method for multiple mesh styles including structured

hexahedral, unstructured tetrahedral, and hybrid meshes.

User-supplied Fortran and C programs have been employed

for the calculation of initial particle profiles, particle depo-

sition locations, grid convergence, and post-processing. All

transport equations were discretized to be at least second

order accurate in space. For the convective terms, a second

order upwind scheme was used to interpolate values from cell

centers to nodes. The diffusion terms were discretized usingcentral differences. To improve the computation of gradients

for the tetrahedral and hybrid meshes, face values were com-

putedas weighted averagesof values at nodes, whichprovides

an improvement to using cell center values for these meshes.

Nodal values for the computation of gradients were con-

structed from the weighted average of the surrounding cells,

following the approach proposed by Rauch et al.[74].A seg-

regated implicit solver was employed to evaluate the resulting

linear system of equations. This solver uses the GaussSeidel

method in conjunction with an algebraic multigrid approach

to solve the linearized equations. The SIMPLEC algorithm

was employed to evaluate pressurevelocity coupling. The

outer iteration procedure was stopped when the global mass

residual had been reduced from its original value by five

orders of magnitude and when the residualreduction rates

for both mass and momentum were sufficiently small. To

ensure that a converged solution had been reached, resid-

ual and reduction-rate factors were decreased by an order

of magnitude and the results were compared. The stricter

convergence criteria produced a negligible effect on both

velocity and particle deposition fields. To improve accuracy,

cgsunitswere employed,and allcalculations were performed

in double precision. To further improve resolution in the par-

ticle deposition studies, geometries were scaled by a factor


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Fig. 3. Two methods for specifying the inlet particle profile. (a) Normalized inlet velocity profile based on upstream conditions and (b) initial particle locations

based on the velocity profile. (c) Normalized inlet mass flow rate profile and (d) initial particle locations based on the mass flow rate profile.

of 10 and the appropriate non-dimensional parameters werematched.

To determine grid convergenceand establishgrid indepen-

dence of the velocity field solutions, successive refinements

of each mesh style havebeen considered. For each refinement,

grid convergence is evaluated using a relative error measure

of velocity magnitude between the coarse and fine solutions,

=

ui,coarse ui,fineui,fine (8)

A vector of relative error values was determined for 1000

consistent points located in the region of the bifurcation. The

root-mean-square of the relative error vector was used to pro-vide an initial scalar measure of grid convergence for the

points considered,

rms=

103i=1

2i

103

1/2(9)

Rigorously, grid convergence measures should be based on

refining the grid by a factor of 2, i.e., grid halving. However,

dividing hexahedral elements by a factor of 2 in three dimen-

sions increases the total number of elements in the model by

a factor of 8. Due to this order of magnitude increase in ele-

ment count for successive grid refinements, it is not alwayspractical to apply true grid halving. However,rmsvalues are

a function of the grid refinement factor, r, as well as the order

of the method employed [75]. That is, a rmsof 1% on a mesh

reduced by a factor ofr= 1.5 is less stringent than a rms of

1% on a mesh reduced by a factor of 2. Assuch, relativeerror

values must be adjusted to account for cases in which grid

reduction factors less thanr= 2 are employed. To extrapolate

rmsvalues to conditions consistent with true grid halving, the

grid convergence index (GCI) has been suggested by Roache

[75].This method is based on Richardson extrapolation and

can be applied for the fine grid solution as

GCI[fine] = Fs rms

rp 1 (10)

In the above equation,rrepresents the grid refinement factor

and p is the order of the discretization method. Based on

second-order discretization of all terms in space,p = 2 for the

systems of interest. Refinement of the meshes was performed

to maintain a constant reduction value in the three coordinate

directions. The associatedr-value has been calculated as the

ratio of control volumes in the fine and coarse meshes

r=

Nfine

Ncoarse

1/3(11)


9/17


10/17


Table 3

Grid convergence measures for the hexahedral mesh scheme

Grid sizes

(103)

r-Value rms (%) GCI (%) Approximate run

for fine grid (min)

6.222 1.52 1.87 4.23 10

2279 1.53 0.855 1.96 14

79377 1.68 0.630a 1.04a 20

a Minimum value for all meshes considered.

Table 4

Grid convergence measures for the tetrahedral mesh scheme

Grid sizes

(103)


for fine grid (min)

2697 1.55 3.83 8.19 70

97354 1.54 3.63 7.93 105

3541092 1.455 4.31 11.55 150

comparison points was positioned to be less than 5% of the

internal radius away from the wall. Selections of other sets of

1000 points as well as doubling the number of points consid-ered had a negligible (i.e., less than 1%) impact on the grid

convergence values reported.

For the hexahedral mesh style, successive grid refine-

ments resulted in a reduction of both the rmsand GCI values

(Table 3).A high quality grid-converged solution is typically

assumed forrmsvalues less than 1%. This criterion is satis-

fied for the 79,000 hexahedral mesh. However, adjusting for a

grid reduction factor ofr= 1.53 results in a GCI value of 2%.

Achieving a GCI of approximately 1% requires considering

377,000 hexahedral control volumes.

For the tetrahedral mesh style, minimum grid convergence

values are observed for the mid-resolution mesh comparison

of 97,000354,000 elements resulting in an rms of 3.63%and a GCI of 7.93% (Table 4).These variations in the veloc-

ity field are relatively high and may significantly impact

particle trajectories, deposition rates, deposition locations,

and other velocity-dependent parameters such as wall shear

stress. However, increasing the mesh resolution to 1,092,000

cells resulted in an increase in grid convergence values. This

increase is attributed to an upper limit on mesh resolution

associated with the scale of the flow and the enhancement of

Table 5

Grid convergence measures for the flow adaptive tetrahedral mesh scheme

Grid sizes

(103)


for fine grid (min)

2873 1.37 3.19 10.84 30

73241 1.48 2.00 5.00 45

241657 1.40 3.97 12.55 60

Table 6

Grid convergence measures for the hybrid mesh scheme

Grid sizes

(103)


for fine grid (min)

1851 1.40 4.07 12.5 30

51139 1.39 3.50 11.19 90

139385 1.40 4.25 13.20 120

round-off errors. To guard against this scenario, the geometry

was scaled by a factor of 10 (D1= 3.81 cm) and cgs units as

well as double precision calculations were employed. Under

these conditions, an upper limit on the number of tetrahedral

cells in the geometry of interest appears to be on the order

of 300,000400,000. Beyond this resolution, further degra-

dation of the flow field is expected.The flow adaptive tetrahedral mesh provides a significant

improvement to the non-uniform tetrahedral mesh for the

mid-resolution scenario (Table 5). For the flow adaptive style

andmeshes of 73,000and 241,000 cells,the grid convergence

values are rms= 2.0% and GCI = 5.0%. These values are sig-

nificantly lower than reported for the non-uniform tetrahedral

mesh and more consistent with the hexahedral mesh results.

Still, an increase in grid convergence values is observed for

the high-resolution flow adaptive mesh indicating degrada-

tion of solution results associated with round-off errors.

Of the mesh styles considered, the hybrid mesh displayed

the highest grid convergence values and required the longest

run times (Table 6).The minimum GCI for the hybrid meshwas approximately 11%. Refinement of the hybrid mesh was

restricted by the requirement that the pyramid surface mesh

should mate with the internal tetrahedral mesh at the wall

boundary. Moreover, hybrid meshes that avoid element tran-

sitions near critical regions and that allow for better mesh

refinement may provide improved results.

Comparisons of GCIvalues among themeshstyles consid-

eredallows for several general observations.First, GCI values

for the hexahedral mesh are approximately an order of mag-

nitude lower than for the other mesh styles for all resolutions

considered. The only exception to this observation is the mid-

resolution case of the flow adaptive mesh which provides aGCI of 5%. However, this flow adaptive mesh contains three

times more control volumes than the mid-resolution case of

the hexahedral mesh and provides a GCI that is 2.5 times

larger. Furthermore, run times for the hexahedral mesh are

approximately three times faster than for the other meshes

considered.

A comparison of midplane velocity field results in the SB

model to the empirical measurements of Zhao and Lieber

[55]for an inlet Reynolds number of 518 is shown inFig. 5.

Results for each of the four mesh styles are based on the mid-

resolution cases presented above. Visually, midplane flow

fields for all the numerical solutions match the experimental

results very well. However, small discrepancies are observed,

especially for the tetrahedral-based meshes, along the cen-

terline and at the locations of maximum velocity for each

profile shown. Reduced centerline velocity for the unstruc-

tured meshes likely arises from numerical diffusion effects

that are more prevalent with tetrahedral elements. However,

these discrepancies appear minor for the simplified flow field

of the SB model.

For thePRB model,a grid convergencestudywas also con-

ducted. Due to the absence of a sharp carinal ridge, less dense

control volume spacing was required to achieve grid conver-

gence values comparable to those observed for the SB model.


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Fig. 5. Comparison of steady grid-independent velocity field solutions (vectors) withReinlet= 518 for the SB geometry to the experimental results of Zhao and

Lieber[55](symbols) using the (a) multiblock hex mesh, (b) the pure tet mesh, (c) the flow adaptive tet mesh, and (d) the hybrid mesh.

Approximately 450,000 control volumes were required to

achieve a GCI value of approximately 1% for the hex mesh

with one plane of symmetry. Only one plane of symmetry

was included to allow for the simulation of gravity effects on

particle trajectories. GCI values less than 10% were achieved

for the tetrahedral and hybrid meshes with 750,000 control

volumes. For the flow adaptive case, a GCI value of approx-

imately 5% was achieved for 630,000 cells. Further mesh

refinements did not improve these GCI values indicating that

the upper limit of mesh resolution had been reached.

Using the mesh resolutions reported above, velocity vec-

tors and contours of velocity magnitude are presented in

midplane and selected cross-sectional slices for the PRB

model and the four mesh styles considered (Fig. 6).In com-

parison to the SB model, discrepancies are more evident

for this highly complex flow field. Initial midplane veloc-

ity vectors appear largely similar indicating a shift from the

turbulent inlet profile to a more laminar parabolic profile near

the bifurcation. However, differences in the midplane veloc-

ity fields become more apparent in the region of the second

bifurcation. Differences related solely to mesh style are also

evident in the cross-sectional slices. Within these slices, 2D

streamlines have been generated to indicate the direction of

secondary motion and vortical flow. These streamlines are

somewhat similar among all mesh styles for slice 1. Differ-ences in the secondary motion associated with various mesh

styles become more evident for slices 2 and 3. It is expected

that these differences will have an impact on particle deposi-

tion locations.

3.2. Particle trajectories and effects of initial particle

profiles

Representative particle trajectories and sample deposition

locations in the PRB model are presented in Fig. 7. Initial

particle locations were based on the inlet mass flow rate.

As expected, the inertia of the 10 m particles considered

results in a high percentage of impaction driven deposition

at the carinal ridges of the bifurcations. Spiraling particle

motion arising from significant secondary velocities can be

seen to result in particle deposition by inception. For the

high inlet flow rate considered, gravity was found to have

a minor impact on deposition. Near-wall Saffman lift was

found to account for approximately 510% of the total depo-

sition arising from particle speed changes in the region of the

bifurcations.

Cumulative deposition percentages as a function of dis-

tance in they-direction for the two inlet profiles considered

and the hexahedral mesh of the PRB are shown in Fig. 8.As

described, the simulation of 20,000 particle trajectories was

found to result in converged deposition profiles. The mass

flow rate based inlet profile results in a cumulative deposition

pattern that matches the experimental deposition results of

Oldham et al. [44] to a high degree. The numerical simulation

of this scenario reports a final deposition of 79% compared

to a value of 81% reported by Oldham et al. [44].The results

of the velocity based initial profile also match the final depo-

sition result very well with an approximate total deposition

value of 85%. However, the results of the velocity based pro-

file do not match the local deposition values throughout the

geometry. As such, it appears that specifying inlet particleseedings consistent with the inlet mass flow rate is appro-

priate. Moreover, the agreement of the numerical result may

be improved by considered transition to turbulence and finite

particle radius.

3.3. Effects of mesh style on particle deposition

predictions

Deposition locations of 10m particles initialized on a

mass flow rate basis for the four mesh styles considered are

shown inFig. 9.The hexahedral mesh results display well-


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Fig. 6. Velocity vectors, contours of velocity magnitude, and in-plane streamtraces at the midplane and selected slice locations for multiple grid independent

solutions of the PRB including the (a) hexahedral mesh with 285 103 control volumes, (b) tetrahedral mesh with 750 103 control volumes, (c) flow adaptive

mesh with 750 103 control volumes, and (d) hybrid mesh with 640103 control volumes.

defined deposition patterns with clear areas devoid of particle

deposition, especially in the region just upstream of the first

bifurcation. Interestingly, little deposition is observed at the

midplane of the second bifurcation carinal ridge for all mesh

styles. It appears that the unstructured meshes distribute the

particle deposition locations more evenly, which may be an

effect of the enhanced numerical diffusion within these solu-

tions. This phenomenon appears most severe for the hybrid

mesh. However, it is difficult to determine the relative depo-

sition fractions due to the number of particles present in the

figure.

Cumulative deposition patterns in the PRB model for

10m particles and the four mesh styles considered are

shown inFig. 10.Increasing the number of control volumes

beyond the values specified for the PRB geometry had a neg-

ligible impact on the cumulative deposition results. These

results are based on the mass flow rate initial particle pro-

file. The hexahedral result was shown previously in Fig. 8

and agrees very well with the experimental data (within

23%). The tetrahedral and adaptive tetrahedral mesh solu-

tions are largely similar to the hexahedral mesh results and

the empirical data of Oldham et al.[44].Variations from the

hexahedral results are on the order of 23%. The hybrid mesh

results vary more significantly from the hexahedral mesh

results and are on the order of 510% below the experimental

values.


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Fig. 7. Particle trajectories in the multiblock mesh illustrating a high per-

centage of deposition at the bifurcations (carinal ridges). Consistent particle

initial locations result in similar trajectory profiles for the various meshes

considered.

4. Discussion

In this study, the effects of mesh style and grid conver-

gence on flow field and particle deposition variables have

been assessed for SB and PRB models of the respiratory

Fig. 8. Cumulative deposition fraction for the PRB model vs. linear dis-

tance in they-direction (cf.Fig. 1b)compared to the experimental results of

Oldham et al.[44]for the same geometry and flow conditions. Both initial

particle specifications match the final deposition percentage (approximately

81%); however, specifyingan initial particle profile consistent with the mass

flow rate results in the best fit to the local deposition pattern throughout the

geometry.

tract. For the relatively simple flow field of the SB model, themesh style had little visible impact on the flow field. How-

ever, the mesh style did significantly impact the midplane

and secondary flow features of the PRB model. Consider-

ing quantitative grid convergence, the hexahedral mesh was

observed to have GCI values that were an order of magnitude

below the unstructured tetrahedral meshes for all resolutions

Fig. 9. Particle deposition locations for 20 103 initial particles based on converged cumulative deposition profiles for the (a) hex mesh, (b) tet mesh, (c) flow

adaptive tet mesh, and (d) hybrid mesh.


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Fig. 10. Cumulative deposition fraction vs. linear distance in they-direction

for the four meshes considered and an initial particle profile based on the

inlet mass flux.

considered. Moreover, the hexahedral mesh style provided

GCI values of approximately 1% and reduced run times by

a factor of 3. Based on comparisons to empirical data, it was

shown that inlet particle seedings should be consistent with

the local inlet mass flow rate. Furthermore, the mesh stylewas found to have an observable effect on cumulative par-

ticle depositions with the hexahedral solution most closely

matching empirical results.

Grid convergence results in this study are consistent with

observations in other branching biofluid flow systems. As

reported by Liu et al. [47] for a branching hemodynamic con-

figuration, flow fields resulting from structured and unstruc-

tured mesh styles are similar. However, the hexahedral mesh

appears to resolve more secondary flow features such as

the number of vortices observed at cross-sectional slices.

For a hexahedral mesh of a branching hemodynamic sys-

tem,Longest and Kleinstreuer [77] reported grid convergencerates that translate to a GCI of approximately 1%. For a flow

adaptive tetrahedral mesh, Prakash and Ethier [48] reported a

5% relative error based on convergence of the velocity field.

This result is consistent with the 5% GCI observed in this

study for a similar tetrahedral mesh. Other grid convergence

results that report a GCI value of less than approximately

5% for a tetrahedral-based mesh and branching biofluid flow

could not be found. As such, it appears that the GCI limit for

unstructured meshes of branching biofluid systems is on the

order of 510% compared to an approximate value of 1% for

well-resolved hexahedral meshes.

In general, results of this study are consistent with find-

ings for other non-biological systems. Considering aerospace

applications, tetrahedral meshes have been shown to be less

accurate than hexahedral grids at similar resolutions[78,79].

However, very few studies have quantitatively evaluated the

effects of mesh style and grid convergence on velocity fields

and particle trajectories. Other studies have recently consid-

ered the effects of the mesh on turbulence models, including

large eddy simulations (LES) [80], and the evaluation of

polyhedral control volumes. For low Reynolds number two-

equation turbulence models, elements on the surface of the

wall must give very high resolution in the wall-normal direc-

tion. Under this condition, the useof long and thin hexahedral

elements results in a significant reduction in the number of

control volumes. For large eddy simulations, numerical dif-

fusion on non-hexahedral grids may significantly damp tran-

sient terms. Appropriate energy conservation grid styles for

LES are discussed in Moulinec et al. [81]. Recently, Peric has

evaluated the performance of polyhedral elements in compar-

ison to other mesh types[82].Several numerical factors may be responsible for the

improved grid convergence of hexahedral meshes. Hexahe-

dral elements can be aligned with the predominate direc-

tion of flow thereby reducing numerical diffusion errors.

Furthermore, discretization errors partially cancel on oppo-

site hexahedral faces. In contrast, unstructured tetrahedral

meshes cannot be aligned with the direction of predominate

flow, thereby increasing the potential for numerical diffusion.

Therefore, numerical diffusion errors associated with ran-

domly oriented tetrahedral faces are the likely cause of the

higher grid convergence values observed for these meshes.

The occurrence of these errors is enhanced in the predomi-

nately unidirectional flow systems considered.Results of this study show the significant impact of the

initial particle profile on deposition locations and cumulative

deposition results. Similarly, Zhang and Kleinstreuer [19]

showed significant variations in particle deposition locations

associated with deterministic, random, parabolic, and con-

stant inlet profiles. As an extension of previous findings,

results of this study highlight the need to specify inlet par-

ticle distributions consistent with the local inlet mass flow

rate, which accounts for local flow area, and not the inlet

velocityprofile alone.It was also observed that while both ini-

tial particle specifications considered resulted in similar total

deposition percentages, local cumulative deposition valuesvaried significantly by as much as 40%.

The GCI provides a quantitative measure of grid conver-

gence that scales results for cases in which true grid halving is

not applied. That is, the GCI is a measure of grid convergence

which is independent of the grid reduction factor and order

of the computational method employed[75]. However, the

GCI is an extrapolation method that is associated with some

degree of error. The GCI is expected to be less accurate for

values ofrclose to unity. To minimize extrapolation errors

in this study,rvalues have been selected to be approximately

1.5 or greater. Another potential problem with the GCI is

that once a converged solution is reached, the relative error

or rms value will not significantly change for further mesh

refinements. The GCI then becomes a function of only the

r-value selected. However, the rmsvalues were not observed

to become constant in this study.

Selection of the most appropriate mesh style for a specific

task depends on the time available for mesh construction and

the required accuracy of the results. Despite poor grid conver-

gence of the tetrahedral meshes, all mesh styles considered

provided reasonably accurate cumulative particle deposition

profiles. In this study, the block structured hexahedral mesh

required the longest time to create, but provided the best

results. The flow adaptive tetrahedral mesh resulted in a rea-


15/17


sonable grid convergence value of 5%. However, the flow

field adaptive routine was based on steady-state conditions.

Adapting the mesh to a transient flow field is difficult and

computationally expensive, which diminishes the intended

advantage of this mesh style. The hybrid mesh style consid-

ered resulted in the highest GCI values, longest run times

and largest differences from experimental results. The pyra-mid surface elements were selected to improve accuracy in

the near-wall region, which is significant for particle depo-

sition and other surface effects. While this approach was

not successful, many other potential multi-element hybrid

mesh styles are possible. Hybrid mesh performance may be

improved with meshes that allow for better control of ele-

ment refinements and that do not transition between mesh

styles directly on the wall. Unstructured hexahedral mesh

styles may also reduce mesh generation time and provide

for a highly accurate solution. Further studies are neces-

sary to determine if other variations of the hybrid mesh or

unstructured hexahedral meshes can provide results that are

similar to the structured hexahedral mesh considered in thisstudy.

The agreement between the cumulative deposition results

for the hexahedral mesh and the experimental data of Old-

ham et al. [44] can potentially be improved by considera-

tion of several factors. Accounting for finite particle radius

and near-wall drag force modifications may enhance depo-

sition near the first bifurcation and generate a better fit to

the experimental data. Longest and Vinchurkar have shown

that accounting for upstream turbulence in the flow field and

particle trajectory calculations by means of a low Reynolds

numberk turbulence model can moderately improve the

agreement with experimental results. However, turbulencemodeling introduces a significant amount of complexity into

the system which is not necessary to demonstrate the effects

of mesh style and grid convergence as considered in this

study.

In conclusion, the mesh style selected for branching bio-

logical flow systems was shown to impact grid convergence,

velocity fields and cumulative deposition patterns of discrete

elements. The GCI, which is well developed [75]but rarely

applied, was shown to be an effective measure of grid con-

vergence in cases where true grid halving is not practical. A

structured hexahedral mesh style was shown to provide the

best computational solution based on the lowest GCI values

and a high level of agreement with flow field and particle

deposition experimental data. Other mesh styles considered

resulted in much higherGCI values, andsignificant variations

in secondary velocity patterns, but still provide a reasonable

agreement with experimentally observed deposition data on

a local basis. Furthermore, it was shown that the initial parti-

cle profile should be based on the inlet mass flow rate, which

accounts for local flow area, and not the inlet velocity pro-

file. Further numerical simulations are necessary to better

assess the effects of other particle sizes, near-wall particle

effects, other hybrid mesh options, and unstructured hexahe-

dral meshes.

References

[1] Frederick CB, Bush ML, Lomax LG, Black KA, Finch L, Kimbell JS,

et al. Application of a hybrid computational fluid dynamics and phys-

iologically based inhalation model for interspecies dosimetry extrapo-

lation of acidic vapors in the upper airways. Toxicol Appl Pharmacol

1998;152:21131.

[2] Martonen TB, Musante CJ, Segal RA, Schroeter JD, Hwang D,

Dolovich MA, et al. Lung models: strengths and limitations. Respir

Care 2000;45:71236.

[3] Kimbell JS, Overton JH, Subramaniam RP, Schlosser PM, Morgan KT,

Conolly RB, et al. Dosimetry modeling of inhaled formaldehyde: bin-

ning nasal flux predictions for quantitative risk assessments. Toxicol

Sci 2001;64:11121.

[4] Jarabek AM. The application of dosimetry models to identify

key processes and parameters for default doseresponse assessment

approaches. Toxicol Lett 1995;79:17184.

[5] Cohen Hubal EA, Fedkiw PS, Kimbell JS. Mass-transport models to

predict toxicity of inhaled gases in the upper respiratory tract. J Appl

Physiol 1996;80:141527.

[6] Cohen Hubal EA, Kimbell JS, Fedkiw PS. Incorporationof nasal-lining

mass-transfer resistance into a CFD model for prediction of ozone

dosimetry in the upper respiratory tract. Inhal Toxicol 1996;8:83157.

[7] Kimbell JS, Subramaniam RP. Use of computational fluid dynamics

models for dosimetry of inhaled gases in the nasal passages. Inhal Tox-

icol 2001;13:32534.

[8] Martonen TB, Zhang ZQ, Yue G, Musante CJ. Fine particle depo-

sition within human nasal airways. Inhal Toxicol 2003;15:283

303.

[9] Schroeter JD, Musante CJ, Hwang DM, Burton R, Guilmette R,

Martonen TB. Hygroscopic growth and deposition of inhaled sec-

ondary cigarette smoke in human nasal pathways. Aerosol Sci Technol

2001;34:13743.

[10] Zhang Z, Kleinstreuer C, Kim CS, Cheng YS. Vaporizingmicrodroplet

inhalation, transport, and deposition in a human upper airway model.

Aerosol Sci Technol 2004;38:3649.

[11] Zhang Z, Kleinstreuer C, Donohue JF, Kim CS. Comparison of micro-

and nano-size particle depositions in a human upper airway model. J

Aerosol Sci 2005;36:21133.

[12] Balashazy I, Hofmann W, Heistracher T. Local particle deposition pat-

terns may play a key role in the development of lung cancer. Transl

Physiol 2003;94:171925.

[13] Comer JK, Kleinstreuer C, Hyun S, Kim CS. Aerosol transport

and deposition in sequentially bifurcating airways. J Biomech Eng

2000;122:1528.

[14] Heistracher T, Hofmann W. Flow and deposition patterns in successive

airway bifurcations. Ann Occup Hygene 1997;41:53742.

[15] Hofmann W, Golser R, Balashazy I. Inspiratory deposition efficiency

of ultrafine particles in a human airway bifurcation model. Aerosol Sci

Technol 2003;37:98894.

[16] Martonen TB, Guan X, Schreck RM. Fluid dynamics in airway bifur-

cations. I. Primary flows. Inhal Toxicol 2001;13:26179.

[17] Martonen TB, Guan X, Schreck RM. Fluid dynamics in airway bifur-cations. II. Secondary currents. Inhal Toxicol 2001;13:2819.

[18] MartonenTB, Guan X, Schreck RM. Fluid dynamicsin airwaybifurca-

tions. III. Localized flow conditions. Inhal Toxicol 2001;13:291305.

[19] Zhang Z, Kleinstreuer C. Effect of particle inlet distributions on depo-

sition in a triple bifurcation lung airway model. J Aerosol Med: Depos

Clearance Effects Lung 2001;14:1329.

[20] Zhang Z, Kleinstreuer C. Transient airflow structures and particle

transport in a sequentially branching lung airway model. Phys Fluids

2002;14:86280.

[21] Zhang Z, Kleinstreuer C, Kim CS. Effects of asymmetric branch flow

rates on aerosol deposition in bifurcating airways. J Med Eng Technol

2000;24:192202.

[22] Balashazy I, Hofmann W. Deposition of aerosols in asymmetric airway

bifurcations. J Aerosol Sci 1995;26:27392.


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