The Pennsylvania State University

The Graduate School

College of Engineering

EFFECT OF GRID TOPOLOGY AND

RESOLUTION ON COMPUTATION OF STEADY

AND UNSTEADY INTERNAL FLOWS

A Thesis in

Mechanical Engineering

by

Steven P. McHale

c© 2009 Steven P. McHale

Submitted in Partial Fulllment

of the Requirements

for the Degree of

Master of Science

May 2009

The Thesis of Steven P. McHale has been reviewed and approved* by the follow-

ing:

Eric G. Paterson

Associate Professor of Mechanical Engineering

Thesis Advisor

Laura L. Pauley, P.E.

Arthur L. Glenn Professor of Engineering Education

Professor of Mechanical Engineering

Karen A. Thole

Professor of Mechanical Engineering

Department Head of Mechanical and Nuclear Engineering

*Signatures are on le in the Graduate School

Abstract

Internal ow has been studied under steady and unsteady conditions. The work was motivated

by previous research in canine olfaction which showed that the structure of the dog's nose consists

of a highly complex pipe network in which the unsteady air ow transports odorant molecules

to the sensory region. Due to geometric complexity of the canine turbinates, a body-tted hex-

dominant mesher was used, thus generating the question: What is the impact of grid topology and

resolution on solution accuracy?

The governing equations are dened and reduced to an analytical solution for a circular pipe

with both steady and unsteady components. The numerical methods and solution algorithms are

described. Simple test cases are performed using an open source computational uid dynamics

(CFD) code, OpenFOAM, to understand the underlying physics and computational challenges

inherent in internal pipe ows.

The characteristic steady, Poiseuille ow was found and a grid study was performed over a range

of Womersley numbers (Wo) for the oscillatory, unsteady calculations of the straight pipe. It was

shown that the Womersley number directly aects the amount of near wall resolution needed to

accurately resolve the Stokes layer. The same calculations were also performed for two other grid

topologies to compare the eectiveness of alternate grid types. The nal simulations performed

were steady inhalation/exhalation and unsteady respiration of a straight pipe within an external

environment.

iii

Contents

List of Figures vii

List of Tables ix

Nomenclature x

Acknowledgments xi

1 Introduction 1

2 Literature Review 4

2.1 Motivation of the Current Work: Canine Olfaction CFD . . . . . . . . . . . . . . . 4

2.2 Studies of Non-Bifurcating Pipe Flow . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Studies of Bifurcating Pipe Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Studies of Complex Pipe Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5 Computational Aspects of Internal Flow . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Governing Equations and Numerical Methods 18

3.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 Poiseuille Flow Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Pulsatile Flow Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3.1 Pulsatile Shear Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.4 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.5 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5.1 Spatial Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.5.1.1 Convection term . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

iv

3.5.1.2 Laplacian Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.5.1.3 Source Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.5.2 Temporal Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.5.2.1 Temporal Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.5.2.2 Treatment of Spatial Derivatives in Transient Problems . . . . . . 30

3.6 Solution Algorithm for the Navier-Stokes Equations . . . . . . . . . . . . . . . . . 32

3.6.1 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.6.2 Derivation of the Pressure Equation . . . . . . . . . . . . . . . . . . . . . . 32

3.6.3 Pressure-Velocity Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.6.3.1 PISO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.6.3.2 SIMPLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 OpenFOAM 37

4.1 Using OpenFOAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.1.1 Pre-processing grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.1.2 Building cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1.3 Post-processing data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 OpenFOAM Solvers and Linear Algebraic Methods . . . . . . . . . . . . . . . . . . 39

4.2.1 simpleFoam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2.2 icoFoam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2.3 channelIcoFoam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2.4 Fourier adaptations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2.5 Linear algebraic methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3 OpenFOAM Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3.1 xedValue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3.2 zeroGradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3.3 cyclic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3.4 pressureInletOutletVelocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3.5 timeVaryingUniformFixedValue . . . . . . . . . . . . . . . . . . . . . . . . . 42

5 Internal Flow Computations 43

5.1 Grid and Topology Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.1.1 Steady Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

v

5.1.2 Unsteady Results: Global Error Evaluation . . . . . . . . . . . . . . . . . . 51

5.1.3 Unsteady Results Radial Error Evaluation . . . . . . . . . . . . . . . . . . . 57

5.2 The Plenum Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2.1 Steady Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.2.2 Unsteady Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6 Conclusions and Recommendations for Future Work 73

Bibliography 76

vi

List of Figures

1.1 The internal geometry of the canine nasal passages [4]. . . . . . . . . . . . . . . . . 2

2.1 Reynolds number distribution in the canine nasal airway at peak inspiratory ow

rate during sning [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Axial distribution of the Womersley number in the canine nasal cavity during sning

(f = 5 Hz). For reference, the background shows an appropriately-scaled sagittal

section of the canine nasal airway [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.1 Parabolic velocity prole that is characteristic of Poiseuille ow . . . . . . . . . . . 21

3.2 Velocity proles at Wo = 9 across one period. Stokes layer is evident near the wall

as the overshoot phase lag from the centerline ow. . . . . . . . . . . . . . . . . . . 24

3.3 Fourier decomposition of pulsatile analytical solution into velocity magnitude and

phase lag for Wo = 1, 3, 9, 27. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.1 Comparison of the internal spatial resolution of the (1) coarse, (2) medium, (3) ne,

and (4) nest CFD grids in the maxilloturbinate region. Comparable grid resolution

is found in the nasal vestibule and ethmoidal region [5]. . . . . . . . . . . . . . . . 44

5.2 Coarse, medium, and ne grids for the grid study. . . . . . . . . . . . . . . . . . . 45

5.3 Unstructured, hybrid, and structured grids used for the topology study. . . . . . . 46

5.4 The grid used for the plenum case shown as the view on the inlet face, along the

axial length, and on the plenum face. . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.5 Example of statistically steady for Wo = 9, computation was ran for 15 cycles and

mass ux was observed before the nal Fourier cycle was computed. . . . . . . . . 50

5.6 Steady state results for the grid and topology study: velocity proles compared to

the analytical solution and error from the wall to centerline. . . . . . . . . . . . . . 52

vii

5.7 The maximum value of the Fourier velocity magnitude across a range of Wo for the

grid and topology studies, with the error from the analytical solution. . . . . . . . 58

5.8 The thickness of the Stokes Layer across a range of Wo for the grid and topology

studies, with the error from the analytical solution. . . . . . . . . . . . . . . . . . . 59

5.9 The phase lag of the Fourier velocity at the wall across a range of Wo for the grid

and topology studies, with the error from the analytical solution . . . . . . . . . . 60

5.10 The value of the maximum mass ow rate across a range of Wo for the grid and

topology studies, with the error from the analytical solution. . . . . . . . . . . . . 61

5.11 The phase lag of the mass ux to the pressure oscillation across a range of Wo for

the grid and topology study, with the error from the analytical solution. . . . . . . 62

5.12 Error across the radius to the analytical solution of the Fourier velocity magnitude

and phase lag for Wo = 1, 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.13 Error across the radius to the analytical solution of the Fourier velocity magnitude

and phase lag for Wo = 9, 27 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.14 The velocity distribution for the steady inspiration and expiration for the plenum

case. The values for the inspiration velocity have been changed from negative to

positive for comparison purposes, but in reality have negative value. . . . . . . . . 68

5.15 Fourier velocity magnitude and phase lag data sampled at four locations along the

pipe, Wo = 1. Error is dened as the dierence to the analytical solution. . . . . . 70

5.16 Wall shear stress compared to the analytical shear stress at twenty instances across

one period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.17 Velocity proles at every t = T20 seconds at four locations along the pipe. The

dash-dot-dot line is the earliest prole in the phase, followed by the long-dash line

second, the dash-dot line third, and the solid line is the last prole of the phase

in each gure. The inhalation errors increase as the velocity magnitude does. The

prole at L=100*D is most aected by the proximity to the plenum. . . . . . . . 72

viii

List of Tables

5.2 Parameters of each pipe grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.3 Physical parameters of grid/topology studies and the plenum case. . . . . . . . . . 48

5.4 Unsteady parameters of the grid/topology studies and the plenum case. . . . . . . 49

ix

Nomenclature

CFD Computational Fluid Dynamics

Re Reynolds number

Rec Critical Reynolds number

Wo Womersley number

NMR Nuclear Magnetic Resonance Imaging

GCI Grid Convergence Index

LDV Laser-Doppler Velocimetry

TCI Time Convergence Index

pIOV pressureInletOutletVelocity

tVUFV timeVaryingUniformFixedValue

GAMG Geometric-Algebraic MultiGrid method

PCG Preconditioned Conjugate Gradient method

DIC Diagonal Incomplete-Cholesky preconditioner

PBiCG Preconditioned Bi-Conjugate Gradient method

DILU Diagonal Incomplete Lower Upper preconditioner

x

Acknowledgments

This thesis was supported by the Oce of Naval Research and the Exploratory and Founda-

tional program of Penn State University's Applied Research Laboratory. I would rst and foremost

like to sincerely thank Dr. Eric Paterson for providing me with an opportunity to explore an area

of science that I found so fascinating. I thank you for the knowledge you imparted and the incred-

ible patience with which you directed my eorts. I would like to thank Dr. Pauley for reviewing

my thesis. I would also like to thank Dr. Gary Settles, Dr. Brent Craven, and Mike Lawson

for your assistance to my research. The Computational Mechanics Division at the Water Tun-

nel has my thanks, in particular Dr. Joel Peltier, Dr. John Mahay, Dr. Rob Kunz, and Jack

Poremba. Thanks to Dr. Hrvoje Jasak for the creation of OpenFOAM. I would like to thank John

Pitt for his advice on numerical methods and Cory Smith for his overall support. Thank you to

fellow grad students: Ning Yang, Bob Erney, Bill Moody, Chandan Kumar, Cli Searfass, Ryan

Mosse, and Chris Applegate. Thank you to the Penn State Mechanical Engineering department

for providing such an excellent place to study Mechanical Engineering for these past seven years.

To anyone that read my thesis and provided grammatical advice, I thank you. Lastly, I would

like to thank God, my wonderful family, and exceptional friends for your absolute love and support.

xi

Chapter 1

Introduction

Internal ow is a vital aspect of physics that is intrinsic to all levels of current civilization and

human survival itself. Oil pipelines transport uid energy across the world; clean drinking water

ows into communities while waste ows out, providing a high standard of living and preventing the

spread of disease. Blood courses through the veins of the body providing nutrients to the organs,

while respiration imports oxygen, exports unnecessary air, and transports odorant molecules to

the olfactory region in order to smell. For these reasons and more, scientists have endeavored to

understand the underlying physics that dene this particular branch of uid dynamics.

The gold standard of olfactory acuity lies in the canine's olfactory region [1]. The internal

structure of the dog's nasal cavity is a complex network of scrolling pipes that creates an optimal

situation for odorant transport to the olfactory organs (ethmoidal region) [2]. Understanding

how the aerodynamics of this animal's nasal cavity complements its superior odorant recognition

capabilities can provide insight into bettering the current technology of trace molecular detection

[1].

Though mathematical derivations are possible for very simple instances of internal ow, the

complexity of the dog's nasal cavity does not lend well to that form of exploration. In those

instances, experimental techniques and Computational Fluid Mechanics (CFD) calculations ll the

void of understanding. CFD, in particular, is a powerful asset that can discretize the governing

equations of internal ow, apply the appropriate boundary conditions, and nd a solution with

a high degree of accuracy given proper spatial and temporal resolution. With the computational

power available today, CFD codes can take high delity grids and resolve very intricate ow

1

Figure 1.1: The internal geometry of the canine nasal passages [4].

characteristics, though such computations can be quite costly due to licensing fees of commercial

codes.

As an alternate approach to licensing fees, OpenCFD and Wikki, Ltd. have developed Open-

FOAM, an open source object-oriented CFD toolbox. OpenFOAM contains utilities to create a

mesh or convert some common meshes into the OpenFOAM format; a large library of dierent

solvers and models to simulate many diverse aspects of uid dynamics; and post-processing util-

ities for sampling and conversion to some common graphical formats. In using CFD, accuracy

of the solution must be quantied through verication and validation. Thus, a study of internal

ows physically similar, but geometrically simpler, to the dog's nose was performed using Open-

FOAM in an attempt to understand the characteristics of steady and unsteady ow and to validate

OpenFOAM in this regard.

In uid dynamics, turbulent-ow regimes can be quantied by the Reynolds number (Re), the

ratio between inertial and viscous forces. The transition from laminar to turbulent ow is dened by

the critical Reynolds number. For steady, internal ow in a circular pipe, the value is: Rec ≈ 2300

2

[3]. Therefore, any ow that is less than the critical Reynolds number can be considered laminar

and modeled as such. It will be shown in Chapter 2 that the canine's nasal geometric parameters

in conjunction with its respiration mass ow rate produces laminar ow in all but the entrance

region of the nostrils (nasal vestibule). That being stated, all computations were performed at

750 ≤ Re ≤ 1000 to ensure laminar ow. Respiration through the nasal cavity may be considered

as pulsatile with zero mean ow. As turbulent-ow regimes are quantied by the Reynolds number,

the degree of unsteadiness is quantied by the Womersley number (Wo). The Womersley number

represents the relationship between the ow length scale, pulsatile frequency, and viscous eects.

For a circular pipe, as the Womersley number increases, a Stokes layer develops. The Stokes layer,

which will be discussed in greater detail in Chapter 3, is an out-of-phase velocity phenomenon

that occurs near the wall. Again in Chapter 2, it will be shown that within the regions of interest

in the nasal cavity Wo∼ 1. The computations were performed across a wide range of Wo that

encompasses the physical range of the dog to fully understand the physics and computational needs

of this type of pulsatile ow. The capstone calculation, a straight pipe connected to an ambient

external environment was computed at Wo = 1.

The purpose of this work endeavors to explore four areas: the grid resolution requirements

needed to accurately capture the unique characteristics of pulsatile pipe ow over range of Womer-

sley numbers; the eect of grid topology at the same Womersley range to discover the eectiveness

of alternate grid types; a simplied unsteady situation resembling respiration within an exter-

nal environment; and a validation of OpenFOAM in regards to the ows considered and overall

usability.

This thesis is presented in the following order. First, the motivation of the research as related

to canine olfaction and a review of the literature is detailed. Then, the derivations of analytical

solutions and discussion of physics related to pulsatile ow. Next, the discussion moves onto the

computational aspects of OpenFOAM, and the validation cases that were computed. Lastly, a

summary and discussion of future work is provided.

3

Chapter 2

Literature Review

2.1 Motivation of the Current Work: Canine Olfaction CFD

The impetus for understanding the aerodynamics of canine olfaction came from eorts to

develop better methods of sampling trace molecules via biomimicry. The principle of biomimicry

is to garner knowledge from nature and apply it to a man-made concept. In the Journal of

Fluids Engineering, Settles describes the dog as the ultimate mobile, instinctive, intelligent sning

platform, and cited the implications for security: trace explosives detection, landmine detection,

and chemical and biological sning. Applying the concept of biomimicry, an appraisal of nature's

accomplishments in olfaction was performed, with emphasis on the external dynamics of canine

olfaction [1]. While this is outside the scope of the present work, it did propel a study of the

internal uid dynamics of the dog's nose.

A fundamental study of the geometry, aerodynamics, and transport phenomenon of canine

olfaction was performed by Craven, et al. [2, 4] that unearthed the internal uid dynamics of

the canine nasal cavity. A general overview of the work consists of the reconstruction of the

nasal passages which was then applied to a morphometric analysis. From that model, CFD was

performed, gleaning insight into the aerodynamics, and a reduced-order numerical model was

developed that captured the essential physics of olfactory mass transport phenomena. A detailed

description of the reconstruction, morphometric analysis, and CFD computations will follow that

provides the physical foundation and computational motivation of the current study.

The canine nasal cavity is divided into two bilaterally symmetric airways, comprised of the

4

nasal vestibule, respiratory, and olfactory regions. Upon inspiration, air enters through the nasal

vestibule and exits to the lungs through the nasopharynx. The ethmoidal region contains the

olfactory epithelium which consists of the odorant receptors that send olfactory signals to the brain.

The maxilloturbinate region is responsible for warming, humidifying, and ltering the inspired air.

In Craven's work, a Labrador's nasal cavity was dissected from its cadaver; a large-breed specimen

was selected to allow higher resolution of the nasal cavity. Nuclear Magnetic Resonance Imaging

(NMR) was used to obtain high-resolution data from the bony structures and soft tissue of the nasal

cavity. Slices with a thickness of 200 µm were taken every 1000 µm along the length of the nasal

cavity and surface reconstruction was then performed. Surface reconstruction entails partitioning

the image into airway or tissue designations, then segmenting the two-dimensional images into a

three-dimensional surface model. The surface is represented as a triangulated surface mesh with

slight surface smoothing to reduce surface staircasing.

In the morphometric analysis the airway perimeter, cross-sectional area, cumulative surface

area, and cumulative internal volume were calculated as functions of axial location. The hydraulic

diameter, Dh, was calculated. The hydraulic diameter (Equation 2.1) is the ratio of cross-sectional

area to the perimeter, which represents the average airway diameter. The Reynolds and Womersley

numbers, in Equations 2.2 and 2.3, were found where Vave is the cross-sectional average velocity,

ν is the kinematic viscosity of air, and f is the sni frequency that was found to be approximately

5 Hz.

Dh =4Ac

P(2.1)

Re =VaveDh

ν(2.2)

Wo =Dh

2

√2πf

ν(2.3)

The largest airway perimeter and maximum cross-sectional area occur in the ethmoidal region. The

hydraulic diameter ranges from 1.5-5 mm. The scrollwork of the ethmoidal conchae dramatically

increases the surface area which facilitates the transfer of mass in the nasal cavity. A substantially

larger portion of the overall surface area is relegated to olfaction purposes than to the other

functions of the nasal cavity. It was theorized, but not conrmed, that the higher surface area in

the olfaction region would be needed because the ethmoidal region is not located along the main

ow path, resulting in smaller velocities and lower transfer rates.

5

Figure 2.1: Reynolds number distribution in the canine nasal airway at peak inspiratory ow rateduring sning [5].

Functionally, the morphological data was used to estimate the Reynolds and Womersley num-

bers in order to obtain an approximation of the ow within the dog's nose. During sning, the

majority of the nasal cavity contains low-Reynolds-number ow (Figure 2.1) with the minimum

on the order of 100. The dead end region far beyond the nasopharynx was not shown because

the velocities within this region are not known. The part of the ethmoidal region in which the

Reynolds number could be calculated shows the Re to be at a minimum. The distribution of the

Womersley number is shown in Figure 2.2. Within the maxilloturbinate and ethmoidal conchae the

Wo is at or near unity. The conclusion was made that the quasi-steady approximation is justied

for the ethmoidal region and the ow is expected to be laminar.

The morphological analysis created the framework for the assumptions needed for the CFD

computations [4, 5]. The internal structures of the nasal airways were assumed to be rigid and the

nostrils were assumed to be undilated. During respiration the nostrils were found to move, but

such motion has not been fully dened and was not expected to aect the general results of the

study. The nasal cavity contains regions of high-Womersley number ow indicating that the ow

during sning is unknown a priori and therefore must be modeled as fully-transient. The ow

in the nasal vestibule was expected to be turbulent at peak inspiration; however, relaminarization

is expected to occur downstream in the low-Reynolds number, low-Womersley number regions,

specically within the ethmoidal and maxilloturbinate regions. Thus, the turbulence of the nasal

vestibule was neglected and laminar ow was calculated throughout the entire nasal cavity.

Due to bilateral symmetry, only the left nasal airway was computed in conjunction with the

6

Figure 2.2: Axial distribution of the Womersley number in the canine nasal cavity during sning(f = 5 Hz). For reference, the background shows an appropriately-scaled sagittal section of thecanine nasal airway [4].

entire external nose contained within a large rectangular box, with fareld atmospheric boundary

conditions applied. The optimal minimum fareld distance was found to be 25 diameters of the

left nostril. The no-slip condition was applied to all solid surfaces and the nasopharynx pressure

boundary condition was specied such that the mass ow rates corresponded to previously found

experimental values. Steady computations held the pressure constant, whereas the unsteady CFD

used a sinusoidal pressure boundary condition with a frequency of 5 Hz.

The grids created for the computations were hexahedral-dominated unstructured grids in which

grid resolution was regionally specied. For example, the fareld had a coarse grid density that

was gradually rened with proximity to the nostril, while the main airway regions required ner

grid resolution. Four grids were generated on the order of 14, 28, 55, and 77 million cells; the

largest of which was limited by practical computational requirements. Though the grid resolution

was regionally specied, proper near-wall resolution was missing throughout the entire grid.

A grid dependence study was performed in which airow impedance curves of overall pressure

drop versus ow rate was measured. This method quanties the sum of frictional pressure drop

and minor losses due to ow separation and mixing, which is directly comparable to experimental

data. A nonlinear pressure drop was observed, in which monotonic convergence occurred with grid

renement. A Richardson extrapolation was performed at four dierent pressure drops (∆P =

50, 500, 2000, 4000 Pa) for steady inhalation and exhalation. At the lowest pressure drop, rapid

grid convergence was observed yielding Grid Convergence Index (GCI) values of approximately

7

4%. The GCI is the measure of percent error that has a high probability of bounding the actual

error of the numerical solution [6]. At larger pressure drops, the order of convergence decreased

yielding GCI values of 15-25%. The observed order of convergence for all cases was greater than

unity. Further, a time step study was also performed using four dierent time steps from 20 to 160

time step per sning cycle, with a renement ratio of 2. The pressure drop used was 2.5 KPa. For

the nest grid, the GCI values was near 1% and the order of accuracy was nearly second-order,

with the conclusion that temporal errors were small and the total numerical error is mainly from

spatial resolution.

The unsteady CFD solutions revealed the aerodynamics of canine sning using physiological

values of sni frequency and peak airow rate. Upon entering the nasal cavity, air becomes well

mixed within the nasal vestibule and the fraction of air that enters the dorsal meatus quickly

advects to the rear of the olfactory region. Then the air lters through the ethmoidal region

towards the nasopharynx or front-most ethmoturbinates for the duration of inspiration. Upon

expiration, little air leaves the olfactory region. In conjunction, one respiratory cycle forces the

air unidirectionally through the olfactory region, which is optimal for odorant separation and

discrimination. The quiescence in the ethmoidal region during expiration provides additional time

for the absorption of odorants. The unsteady results showed that the geometry of the dog's nose

provides the optimal aerodynamics to transport odor particles to the olfactory region and the time

to absorb and process odorants into sensory signals [5].

2.2 Studies of Non-Bifurcating Pipe Flow

The study of internal ow in biological systems began, largely, with Womersley [7], with the

development of an analytical solution for velocity proles of pulsatile ow in a straight, circular

pipe and the subsequent experimental validation. The analytical solution, which is fully derived

in Chapter 3, considers the pressure gradient as periodic in time that yields a closed-form solution

involving zeroth-order Bessel functions and a dimensionless parameter, α (which later became

known as the Womersley number). Taking the real part of the pressure gradient corresponds to

the real part of the solution and is expressed in terms of a modal Fourier magnitude and phase lag.

Analyzing the solution found that increasing α resulted in a departure from the steady Poiseuille

form of the velocity solution. Further, analytical values of mass ow rate and viscous drag at the

surface of the pipe were developed. Applying this solution to an experimentally derived pressure

8

gradient in the femoral artery of a dog (minus the steady component), the mass ow was computed

across one pulsatile cycle for values of α ranging from 0 to 10. It was found that the phase lag

tended to zero with low α and approached its asymptotic value of 90 degrees at high α. The mass

ow at low α was the same given by the Poiseuille formula but decreased as α was increased. The

current work will be shown to verify these results in Section 5.1.2.

For oscillatory ow, it was found that the transition from laminar to turbulence occurred at a

Reynolds number that is lower than the critical value for steady motion. An attempt to quantify

an eective Reynolds number was made by Hale, et al [8]. The femoral artery of a dog was

exposed and injected with dye. The blood ow was lmed and the pressure gradients measured.

Observation showed that the ow was laminar at a Reynolds number range of 600-1000. A similar

technique showed transition to turbulence in a rabbit aorta at a similar Reynolds range. Both

arteries had a similar external diameter of 3mm, but the rabbit's pulse frequency was twice that

of the dog's. The velocity prole was calculated using Womersley's solution across the pulse of

the dog to obtain the value of the maximum shear stress. Equating the maximum pulsatile shear

stress ow condition with the steady ow condition resulting in the same shear stress yielded a

ratio that was applied to the pulsatile Reynolds number. The ratio was found to be close to unity

at α < 2 and to increase linearly as α > 3. Scaling the pulsatile Reynolds number with this

ratio gave the eective Reynolds number which became a function of α. In essence, increasing

the frequency would have the same eect on the Reynolds number as increasing the square of the

radius. When applied to the dog and rabbit, it was found that the eective turbulent Reynolds

number was above the critical Reynolds number for the rabbit, but not the dog. The conclusion

was made that the limit of stability depended not only on velocity and diameter of the tube, but

also on the pulse frequency.

In further study of stability, pulsatile ow in a plane channel was considered [9]. For steady

channel ow, the critical Reynolds number (Rec ∼ 5800) is higher than for a straight pipe. When

Wo is nearly zero, the critical pulsatile Reynolds is nearly that of steady ow, but drops to nearly

a third of the steady Rec at Wo 1. As Wo is increased from unity to innity, Rec for pulsatile ow

increases asymptotically from its minimum value back to the steady value. Increasing frequency

was shown to have a stabilizing inuence on planar ow with Wo > 1, though the critical pulsatile

Reynolds number was less than the steady ow situation and the ow was not purely oscillatory.

A similar evaluation was done using a straight pipe [10]. A similar trend to the channel ow

was observed experimentally, where stability increased with increasing frequency for a given Re.

9

For purely oscillatory ow, a linear t was established between critical Reynolds number and

Womersley number that yielded a constant of 475. This constant was within a range of 250-1000

that had been previously established [11].

To further develop understanding of physiological internal ow, the eect of geometry has been

considered. Berger, et al. derived equations for and studied ow in toroidal pipes [12]. Steady

ow within a toroid with a radius-to-curvature 1 had a velocity solution similar to that of

the Poiseuille ow. For Poiseuille ow, contours of constant axial velocity are concentric circles

that are centered in the precise middle of the pipe. For the toroid, the axial velocity contours

are circles whose centerpoints are located further towards the outer diameter along the plane of

symmetry. A centrifugally-induced pressure gradient drives the slower-moving uid near the wall

inward, while the faster moving uid in the core is swept outward, inducing symmetric-about-

the-plane-of-symmetry secondary ows and creating the velocity perspective just described. The

velocity gradient, and thusly the shear stress, is higher on the outer wall than on the inner wall.

As the ratio of the radius to curvature is increased, the secondary ows become more powerful and

push the location of maximum velocity closer to the outer diameter. Low velocity contours are

circular, but as velocity increases, the shape of the contour on the inner side is deected inward. At

high velocities, the contour is kidney shaped and the shear stress increases on the outer diameter.

With further increase of the ratio of radius to curvature, the locations of maximum circumferential

velocity continues to move toward the outer bend. The secondary ow boundary layers near the

inner bend thicken further, and may separate. At the time this article was written, the existence

of separation was in dispute and the attempt to conrm this was not made. Unsteady laminar ow

was also considered. For purely oscillatory ow at large Wo, the centrifugal forces cause a mean

secondary ow to develop in the Stokes layer at the wall. The eects of viscosity are conned to

thin boundary layers near the walls and along the plane of symmetry.

Further work on this subject was performed by Siggers, et al. [13] in which computational

methods were used. It was found that, at Wo = 10, the solution conformed to the analytical values

that were derived in Berger's work [12], but the onset of instability resulted in asymmetric ows that

were periodic or non-periodic, depending on the curvature-to-radius ratio. A lower-than-critical

ratio would retain periodicity in the ow.

The inuence of geometry was also explored by Nishimura [14], in which oscillatory viscous ow

was studied in symmetric wavy-walled channels. Two wall shapes were considered: sinusoidal

and arc-shaped walls of large amplitude. It was found that the wall geometries have little eect on

10

the fundamental characteristics of oscillatory ow, but the arc-shaped walls became more unstable

more quickly than the sinusoidal walls at higher Womersley numbers. At Wo < 4, the ow was a

two-vortex system, but at Wo > 4 the ow became a four-vortex system. It was concluded that

ow above the transition point were inertially-dominated; which resulted in more complicated

secondary ow features. In all, slight modications to geometry induce secondary motion which

aects the stability. Unsteadiness also reduces stability, yielding a lower critical Reynolds number

than that for steady ow.

Unsteadiness in a straight pipe induces velocities that are dierent from a steady case, which

aects stability such that the onset of turbulence comes at lower Reynolds number. Varying the

geometry further complicates the ow, which results in secondary ows and recirculation that in

turn also aect stability.

2.3 Studies of Bifurcating Pipe Flow

New ow phenomena occurs with the introduction of a bifurcating pipe system, like the vascular

system of mammals. Two dimensional ow characteristics of blood ow through a symmetric

branch, under realistic ow pulsations, was studied by Friedman, et al. [15]. The branch area

ratio and mean ow parameters correspond to the physical aortic bifurcation of a human with a

45-degree branch. Flow reversal was observed on every point of the outer wall at one point during

the cycle, while no ow reversal was found on the inner wall until a point further downstream

from the ow divider. From this model, it was concluded that ow separation in the arterial

tree is a rare phenomenon and may not be factor in the development of atherosclerosis. The

conclusions were later disputed as more sophisticated technology facilitated better computational

models; however, this study did show that a bifurcation under physiological condition will exhibit,

at times, secondary ow features. A similar 2D numerical study modeled unsteady ow at renal

branches [16]. Two branches, half the diameter of the main line, split o orthogonally as an

idealized model of aortic branching. The Reynolds number was varied sinusoidally about a mean

value at 200 and peak value of 400. A dividing streamline was found to always exist that separated

the central outow with the sidearm outow. The intersection of the streamline with the wall was

at or downstream of the downstream branch/main-line corner, except at zero net ux (Re = 0)

when the dividing stream cut o the branch. The ow was completely reversed along the main

wall from the centerline ow. The simulations found localized regions near the bifurcations of high

11

shear stress magnitude.

Three-dimensional models were eventually developed in order to more realistically simulate

arterial ow. The benchmark in human carotid modeling, by Bharadvaj, et al. [17, 18], was a two

part analysis of steady ow: ow visualization and Laser-Doppler velocimetry (LDV). Geometric

and Reynolds number similarity was maintained between the human carotid and laboratory model.

The carotid artery bifurcates from the common carotid into the internal and external carotid - the

internal carotid with a larger diameter than the external. The ow visualization, using Hydrogen

bubbles, found secondary ow in the daughter carotids that moved downstream in a helical path.

The secondary ow, which occured along the outer walls, redistributed the velocity so that the

maximum was closer to the bifurcation apex. The ow was shown to likely separate along the

outer wall, though ow reversal was only seen at lower-Reynolds number ow. The study also

found that the zones with low or reversed axial ow were coincident with locations in the carotid

with a predilection for disease. The ndings were able to disprove previous conclusions of ow

separation and recirculation regions and showed that the 2D unsteady assumptions would not be

representative of 3D unsteady ow.

The second part of the study provided a more quantitative analysis of steady carotid ow. The

ow was fully developed before entering the bifurcation and was studied at mean and peak Re

for a variety of ow divisions ratios that were found to exist during a physiological pulse within

the human. The outer corner of the internal-common junction was subjected to low shear stress

and was postulated that reversed ow zones would be expected there during a pulsatile simulation

yielding oscillating wall shear stresses as well. The outer corners at both splits were regions of

comparatively higher pressure. The apex of the bifurcation was subjected to unidirectional and

moderately high shear stress. The ndings were able to indicate that the ow phenomenon within

the carotid is linked to atherogenesis but no direct mechanism was veried. Future studies relied

heavily on the work of Bharadvaj for validation and geometrical modeling.

One such study, by Rindt, et al. [19], was a numerical analysis of steady ow of a carotid

geometry based upon the one described by Bharadvaj. The inlet boundary condition was Poiseuille

ow at Re = 640 and a ow division of 52/48 in the internal/external carotids. The results

found good agreement with LDV results, except near the ow divider. The grid generation was

problematic in this region, and overall the grid renement was limited by computational power.

The computation was able to reveal the ow reversal that was observed at low Reynolds number

by Bharadvaj [17] even though the ow ratios were dierent.

12

The same human carotid geometry was used for ow visualization and LDV studies of pulsatile

ow by Ku and Giddens [20, 21]. Under physiological pulsatile conditions, similar results were

found from the steady ndings of the previous articles: low shear stress on the outer wall and

relatively higher shear stress on the inner wall. Though, pulsatility was found to generate a

changing region of ow reversal and separation that disappeared during early systole (near peak

mass ux) and redeveloped during late systole into a region that was larger than the steady

separation zones on the outer wall of the internal carotid. Bubbles introduced into the ow were

found to remain in this region for several cycles, indicating an increased uid residence time, while

the LDV conrmed that the outer wall experienced low mean shear stress and oscillatory shear

stress. This recirculation region was found to correspond directly to sites where atherosclerosis is

most prominent initially in atherogenesis.

Numerical analysis of unsteady carotid ow followed the experimental studies. Perktold, et

al. [22, 23], compared the results to the unsteady experiments under dierent bifurcation angles.

The model was based on the geometry from Bharadvaj, but not in great detail. This created

discrepancies with the results quantitatively, but found good agreement in the overall ow patterns.

The inlet velocity was assumed to be theWomersley velocity prole (see Chapter 3). With increased

bifurcation angle, it was found that ow reversal increased and ow recirculation was enhanced.

Decreased angle led to separation further upstream. Yung [25] and Jou [25] performed the same

computations for geometries of dierent angles and contours with similar results, providing solid

data for a broad range of carotid geometry found in the human being.

From the previous studies, a conceptual analysis of more complex geometry can be made,

but no further unless high-delity CFD or experiments can be performed. The introduction of a

bifurcation to a pipe system creates secondary ows, recirculation, and at times fully-reversed ow.

The ow is further altered as the model is upgraded from two-dimensional to three-dimensional or

planar to cylindrical and can be expected to become more complicated as the geometry becomes

more physiological.

2.4 Studies of Complex Pipe Systems

Airow in the human nasal cavity was simulated by Keyhani, et al. [26] from a reconstructed

model in a procedure similar to that of the dog nose and reconstruction compared to an experi-

mental model. Velocity proles were compared and the results between the two methods varied

13

within 20% of each other. Experimental error was subject to high uncertainty due to low velocities

and uncertainty in the position of the sensor. Computational renement was found to have neg-

ligible eect on the results; however, small dierences to the physical model were inevitable due

the discretization of the geometry.

A similar study was performed by Horschler, et al. [27] on the human nasal cavity. The

comparison was of velocity streamlines and found good qualitative agreement. Neither a quantita-

tive error assessment nor an uncertainty analysis (computationally or experimentally) was given.

Croce, et al. [28] compared pressure drop against Reynolds number between CFD and experiment.

Excellent agreement was found with both the standard deviation and mean having less than 2%

dierence. The conclusion was drawn that the gross aerodynamics in the CFD solution were accu-

rate especially in the boundary layers where the pressure drop is most heavily aected, but could

not properly validate the local velocity distribution.

The dierences between the comparisons claries the diculty in validating a computational

result to that of an experiment for such complicated geometry. While gross ow features such

as pressure drop were found to compare well, the ner aerodynamics had large disagreement or

only qualitative agreement. Computationally, the implication is that grid requirements must be

fully understood and independently veried to reduce numerical error, along with sophisticated

and proper boundary condition application.

2.5 Computational Aspects of Internal Flow

In order to have condence in a CFD solution, verication and validation are integral to success.

Roy [29] reviewed the procedure to ensure such condence. Verication is proving the mathematical

correctness of the solution. Code verication is a process designed to nd mistakes in the computer

code. In general, a simplication of the geometry and governing equations will provide an exact or

manufactured solution that is used to pinpoint coding mistakes. Solution verication is concerned

with assessing the numerical errors in the computation. Error sources are round-o error, iterative

convergence error, and discretization error. The error from discretization is the dominant source

of error and the several methods exist to estimate the eect. An extrapolation-based Richardson

method is popular because it is readily applicable to nite-volume, nite-dierence, and nite-

element methods. In cases where an analytical solution of the governing equations is not available,

Richardson extrapolation takes the solution on two dierent meshes and extrapolates to a higher-

14

order accurate solution, which provides an estimation of the observed order of accuracy. When an

analytical solution is known, then a one-to-one error estimation is possible. Roache [6] outlines a

procedure to calculate a Grid Convergence Index (GCI) that provides an estimation for an error

band to predict the overall uncertainty of a computational simulation. Validation of CFD proves

the appropriateness of the equations to provide accurate predictions of reality. This can be done

by comparison to experimental data or analytically-derived solutions, as in the case for oscillatory

ow in a circular straight pipe. Validation shows that the boundary conditions and physical models

are appropriate for the reality that is being simulated.

Gokaltun, et al. performed verication and validation of a pulsating laminar ow in a straight

pipe [30]. The ow conditions were such that Re = 233 and Wo = 10. Velocity values were

extracted across every twelfth of the period and subjected to a Richardson Extrapolation. Using

the steady-state Poiseuille ow solution, the GCI calculation showed the order of convergence to

be 1.999 with an error band of ±0.00049. The nest grid had 24 cells uniformly spaced across

the radius; the domain was treated two-dimensionally due to symmetry. While GCI calculation

was not performed for the unsteady case, a Time Convergence Index (TCI) was calculated. The

nest uniform time step was set such that 244 time steps were calculated per period. The pressure

gradient was the ow function used for verication and showed an order of convergence that varied

between 1.7 and 2.3. For validation of oscillatory laminar ow, the velocity proles were compared

to the analytical solution and experimental data. The spatial and temporal errors were smaller

than precision errors due to round-o.

Creating computational grids of physiological geometries is challenging, more so without the

use of an automatic mesher. Typically, the wall and outlet surfaces are generated and then the

internal volumes are created from this boundary. Grid resolution and quality in the bifurcation

region is especially dicult when the angle is sharp, yet this region has signicant impact on the

ow. Several techniques have been developed to address this problem, most notably in employing

novel block structures for purely hexahedral meshes or using unstructured tet-meshes. Vinchurkar

and Longest [31] evaluated hexahedral, prismatic and hybrid meshes for simulating respiratory

aerosol dynamics. For clarication, Vinchurkar denes a prismatic volume as a pyramid with a tet

or square as the base and a hybrid volume made entirely of tets. However, the work of this thesis

refers to an unstructured mesh as volumes made entirely of tets and a hybrid mesh as a combination

of hexahedral and tet volumes with either square or tet bases. The diculty in quantifying grid

convergence on these meshes is that grid renement does not necessarily double the mesh density.

15

The relative error was found through a GCI of values at specied locations. Linear interpolation

was employed because the locations did not coincide with a cell center. The prismatic mesh was

found to have comparable GCI values to the hexahedral grid with only a moderate increase in grid

volumes. The elds showed good agreement, except in the secondary velocity proles. The hybrid

grid had a signicantly higher GCI than the other meshes, and a signicant increase in cell volumes.

Though, the hybrid grid was the quickest to mesh because it was a fully automated procedure.

The prismatic mesh took longer because a rectangular surface grid was made by hand before it

was lled with tets. The structured hexahedral grid was very time intensive because its inherent

nature allows for little automation. The result of the study found that the prismatic yielded the

best avenue for gridding in that the time of construction was manageable with comparable results

to the hexahedral mesh. However, when a particle deposition model was included on the same

grids [32] , the hexahedral mesh most closely matched the empirical data. The other grids showed

much lower correlation to the empirical data.

Prakash, et al. [33] studied mesh resolution requirements for 3D computational hemodynamics

of the right coronary artery. The purpose was to nd renement levels that would accurately

resolve separation zones and wall shear stress. Interestingly, the standard for resolution was set to

previous studies' approximation of big enough and found that the results yielded only moderate

accuracy. Non-adaptive renement yielded grids that provided mesh-independent solutions but

were computationally unfeasible. Using an adaptive, localized renement procedure, like the dog's

nose grid, the arterial grid remained manageable and achieved mesh-independence. In essence,

localized renement in regions of interest and complex ow elds yielded larger returns with man-

ageable grid size for complicated physiological geometries.

Inclusive to verication and validation is the application of boundary conditions. Taylor, et

al. [34] developed a method for a fully-developed Womersley-type inlet velocity prole for a given

transient mass ow rate. Mass ux data in the circular and respiratory systems is easier to obtain

than a velocity prole. Admittedly, the velocity prole in nature is unlikely to correspond exactly

to the Womersley solution, but the model here is more accurate than a Poiseuille or plug ow

inlet and allows for a reduction in grid size because no development length is necessary. As it

stands, the Poiseuille solution is inherent in this method as it corresponds to the steady part of the

solution. Barring experimentally-derived velocity proles, the inlet boundary condition provides

the closest approximation to the actual ow and has the exibility for any periodic mass ux

pulse. However, for the current work, this boundary condition application was not needed due to

16

the simple geometry and application of a periodic boundary at the inlet and outlet.

In terms of dening the pressure boundary conditions, the simplest method is prescribing

pressure values at the boundaries. For a geometry with one inlet and one outlet, this is often

sucient; however, with multiple outlets, constant pressure application is dicult when only the

mass ow ratios are known, i.e. the carotid artery. Vignon, et al. [35] detailed the deciencies

of such a boundary condition and oer two alternatives: resistance and impedance matching.

The resistance boundary condition derives a relationship between pressure and ow that yields

a resistance constant. This is similar to the voltage-current-resistance relationship in electricity.

Voltage being pressure and current being ow. The impedance boundary is based on a solution of a

linear, damped wave equation in the downstream domain. The resistance boundary condition was

found to yield better results than a constant pressure application, but still had very high numerical

damping. This led to un-physiological mass uxes across a pulse period, especially at peak ows;

similar but muted. The impedance boundary condition improved results over the resistance and

was determined to be the most physically relevant application.

17

Chapter 3

Governing Equations and Numerical

Methods

The governing equations described below are the Navier-Stokes equations. OpenFOAM reduces

those PDE's into a set of linear algebraic equations and solves them. In order to validate a CFD

code, computational results must be compared against some prepossessed knowledge (i.e., previous

experimental data or mathematical formulation). In the instance of a straight, circular pipe, steady

and transient analytical solutions can be derived, providing a foundation for validation. The

following derivation of the governing equations describes the response of a radial velocity prole to

an axial pressure gradient. The steady, Poiseuille ow has a characteristic parabolic velocity prole

induced by a constant pressure gradient. For the pulsatile ow, a complex pressure waveform is

used in the derivation. Taking the imaginary component of the velocity solution corresponds to

the imaginary component of the complex pressure waveform. In this way, a velocity solution can

be found for a sinusoidal pressure wave and is shown below [36].

3.1 Governing Equations

The governing equations used in this study are the full Navier-Stokes equations. Cylindrical co-

ordinates are simpler to use due to the circular bounding geometry of the ows under consideration.

The x, r, and θ coordinates correspond to the u, v, and w velocity components contained in the

governing equations. For an incompressible, Newtonian uid, the equations are the Conservation

18

of Mass (continuity, Equation 3.1) and Conservation of Momentum (Equation 3.2).

∇ · u = 0 (3.1)

∂u∂t

+ u · ∇u = −1ρ∇P + ν∇2u (3.2)

where u = (u, v, w)

The aforementioned analytical solutions can be found by reducing the governing equations

under these assumptions:

1. The ow is fully-developed.

2. The ow is incompressible.

3. The pipe has a rigid, symmetric, circular cross-section.

4. There are no external forces.

5. The time derivative is zero for steady ow (Poiseuille ow).

6. No slip condition applies at the walls.

For fully-developed ow, all derivatives in the x direction are zero.

∂

∂x= 0 (3.3)

Under assumptions 3, 4, and 5, the angular velocity and all θ derivatives are zero.

∂

∂θ= 0 (3.4)

w = 0 (3.5)

Considering Equations 3.3, 3.4, 3.5, and assumption 6, the Continuity Equation (Eq. 3.1) can

be reduced to:

∂

∂r(rv) = 0 (3.6)

19

rv = constant (3.7)

rwall > 0, vwall = 0 : therefore (3.8)

v = 0 (3.9)

Reducing the Momentum Equations (Eq. 3.2) yields the following:

∂u

∂t= −1

ρ

∂P

∂x+ ν

(∂2u

∂r2+

1r

∂u

∂r

)(3.10)

∂P

∂r= 0 (3.11)

The governing equations, under the six assumptions, have been reduced to Equations 3.10 and

3.11. The axial velocity is only a function of radial distance and time, u = f(r, t). The pressure is

only a function of axial position and time, P = f(x, t). Under the assumption that dPdx = f(t) only,

an analytical solution is obtainable with each component having a steady and oscillatory part.

u(r, t) = us(r) + uφ(r, t)

P (x, t) = Ps(x) + Pφ(x, t) (3.12)

Inserting into the reduced governing equation (Eq. 3.13) and separating the steady (Eq. 3.14) and

unsteady (Eq. 3.15) components yields two separate equations.

∂uφ

∂t+

1ρ

(dPs

dx+

dPφ

dx

)= ν

(∂2us

∂r2+

∂2uφ

∂r2

)+

ν

r

(∂us

∂r+

∂uφ

∂r

)(3.13)

1ρ

dPs

dx= ν

∂2us

∂r2+

ν

r

∂us

∂r(3.14)

∂uφ

∂t+

1ρ

dPφ

dx= ν

∂2uφ

∂r2+

ν

r

∂uφ

∂r(3.15)

3.2 Poiseuille Flow Solution

Poiseuille ow describes a laminar, steady state, internal ow in a circular pipe that is charac-

terized by a parabolic, axial velocity prole in the radial direction and a constant axial pressure

gradient. Poiseuille ow is derived using a no slip condition at the wall and a nite velocity at

20

Figure 3.1: Parabolic velocity prole that is characteristic of Poiseuille ow

the centerline. Considering the steady governing equation, Eq. 3.14, a solution is possible if both

sides of the equations are set to a constant, Ks.

dPs

dx= µ

∂2us

∂r2+

µ

r

∂us

∂r= Ks (3.16)

Integrating the pressure side, and applying P (x = 0) = P0 yields:

Ps(x) = Ks · x + P0 (3.17)

Integrating the velocity and applying some simple algebra yields:

us =Ks

4µr2 + A ln(r) + B (3.18)

Applying the no slip condition and then the nite velocity at the centerline results in the nal

form of the steady velocity equation:

us =−Ks

4µ

(R2 − r2

)(3.19)

Ks =Ps(L)− P (0)

L

The Poiseuille velocity prole is parabolic with the maximum velocity at the centerline, Figure 3.1.

21

3.3 Pulsatile Flow Solution

Pulsatile ow of a straight pipe is characterized by a periodic pressure waveform that induces

an unsteady response to the axial velocity. An analytical solution for the velocity can be found for

a purely oscillatory pressure waveform in such a way that allows, via Fourier series, for a similar

solution involving a pressure waveform that is periodic, but not necessarily sinusoidal. The solution

found below has been derived using an oscillatory, complex pressure gradient with an amplitude

equal to that of the steady case. The pressure is a function of axial position and time, P = f(x, t),

and the axial velocity is a function of radial position and time, u = f(r, t). The same boundary

conditions apply as in the steady case: zero velocity at the wall and nite velocity at the centerline.

dPφ

dx=

Pφ(L, t)− Pφ(0, t)L

= Kφ = Kseiωt (3.20)

∂uφ

∂t+

Ks

ρeiωt = ν

∂2uφ

∂r2+

ν

r

∂uφ

∂r(3.21)

If separation of variables is applied, where uφ = U(r) · eiωt, and inserted into the unsteady

governing equations, the PDE can be reduced to a solvable ODE.

(iωU) · eiωt +Ks

ρ· eiωt = ν

(d2U

dr2+

1r

dU

dr

)· eiωt (3.22)

If the Womersley number is dened as Ω = R√

ων , and inserted into Eq. 3.22 along with some

algebraic manipulation, the ODE takes the form of:

U(r) = AJ0(ζ) + BY0(ζ) + iKsR

2

µΩ2(3.23)

The general solution of which is:

U(r) = AJ0(ζ) + BY0(ζ) + iKsR

2

µΩ2(3.24)

Where A,B are arbitrary constants, and J0, Y0 are zero-order Bessel functions of the rst and

second kind, respectively, satisfying the standard Bessel Equations:

d2J0

dζ2+

1ζ

dJ0

dζ+ J0 = 0 and

d2Y0

dζ2+

1ζ

dY0

dζ+ Y0 = 0 (3.25)

22

The independent variable ζ is dened as ζ(r) = λ rR , where λ is the complex frequency param-

eter dened as λ =(

i−1√2

)Ω. In order to solve for constants A and B, the boundary conditions

must be applied to Eq. 3.24:

Boundary Condition1:r = 0 → u(0, t) = finite, ζ = 0, U(0) = finite

U(0) = iKsR

2

µΩ2+ AJ0(0) + BY0(0)

Y0(0) →∞, B = 0

Boundary Condition 2: r = R → u(R, t) = 0, ζ = λ, U(0) = 0

0 = iKsR

2

µΩ2+ AJ0(λ)

A = iKsR

2

µΩ2· 1J0(λ)

Resulting in the nal form of U(r):

U(r) =(

iKsR2

µΩ2

)(1 +

J0(ζ)J0(λ)

)(3.26)

The oscillatory axial velocity is therefore:

uφ(r, t) =(

iKsR2

µΩ2

)(1 +

J0(ζ)J0(λ)

)(cos(ωt) + i sin(ωt)) (3.27)

Combining the steady and unsteady velocities yields the full analytical solution for the unsteady

pulsatile ow equation:

u(r, t) =−Ks

4µ

(R2 − r2

)+(

iKsR2

µΩ2

)(1 +

J0(ζ)J0(λ)

)(cos(ωt) + i sin(ωt)) (3.28)

The nal solution of the velocity equation is complex, as is the driving pressure waveform. In the

zero-mean computations, the imaginary component of the pressure gradient was used, therefore

the imaginary part of the velocity equation was extracted to validate the solutions.

23

Figure 3.2: Velocity proles at Wo = 9 across one period. Stokes layer is evident near the wall asthe overshoot phase lag from the centerline ow.

∂Pφ

∂x= Ks · sin(ωt) =

(Ks · eiωt

)I, therefore

uCFD = (uφ)I =[(

iKsR2

µΩ2

)(1 +

J0(ζ)J0(λ)

)(cos(ωt) + i sin(ωt))

]I

(3.29)

The shape of the unsteady response varies with the Womersley number. At low Womersley

numbers, near unity, the ow is considered quasi-steady. The shape of the velocity prole has a

similar parabolic prole akin to the steady case. At such a low Womersley number, the velocity

transitions in phase with the transient pressure gradient. As the Womersley number increases,

thereby increasing the frequency, the pressure oscillates quicker than the viscous uid can react.

An out-of-phase velocity prole develops, where the near wall region leads the ow closer to the

centerline. This is described as the Stokes layer. As frequency increases, the centerline velocity

decreases towards zero while the magnitude of the Stokes layer increases, but its radial thickness

24

decreases. At a Womersley number of innity, the velocity prole is radially zero.

3.3.1 Pulsatile Shear Stress

Like the velocity solution, the shear stress will oscillate back and forth throughout the period. The

oscillatory shear stress equation is:

τφ(t) = −µ

(∂uφ(r, t)

∂r

)r=R

(3.30)

For a purely oscillatory ow, there is no steady component. Inserting the analytical velocitysolution into the shear stress equation yields:

τφ(t) = − iKsR2

Ω2

[d

dr

(1− J0(ζ)

J0(λ)

)]r=R

eiωt (3.31)

The nal form of the shear stress equation is:

τφ = −KsR

λ

(J1(λ)J0(λ)

)· eiωt (3.32)

3.4 Fourier Series

A Fourier Series decomposes a periodic function into an innite series of sines and cosines [37].

A periodic function is dened as having the same value at any point across an integer number of

periods. If the length of one period is dened by a length T , then a periodic function would have

the following property and general Fourier denition:

f(t) = f(t + T ) (3.33)

f(t) = A0 +∞∑

n=1

An cos(nωt) + Bn sin(nωt) (3.34)

Where A0 represents the average value across one period, or the steady component of the

function, whereas An and Bn represent the magnitude of the nth mode, or the unsteady component

of that mode. The innite sum of these values will return an exact replica of a given function, yet

a discrete number of modes can result in a high degree of replication, leaving out only the highest

frequency content.

25

A0 =12π

∫ 2π

0

f(t) · dt

An =1π

∫ 2π

0

f(t) cos(nωt) · dt (3.35)

Bn =1π

∫ 2π

0

f(t) sin(nωt) · dt

The constants An and Bn can also be converted into a magnitude, |An|, and phase lag, φ per

mode.

f(t) = A0 +∞∑

n=1

| An | ·sin(nωt + φ) where (3.36)

| An | =√

A2n + B2

n

φ = tan−1

(Bn

An

)

A feature of the Fourier Series is the capability of reducing time dependent information into several

manageable constants. This procedure was used within the unsteady computations (Chapter 5)

to facilitate understanding of the results across a period. The decomposition of the analytical

solutions can be seen in Figure 3.3.

3.5 Numerical Methods

The governing equations for conservation of mass, conservation of momentum, and transport of

scalars can be represented by the generic transport equation for φ

∂ρφ

∂t+∇ · (ρUφ)−∇ · (ρΓφ∇φ) = Sφ(φ) (3.37)

which is comprised of 4 basic terms: temporal acceleration, convective acceleration, diusion, and

a source term. The nite volume method requires that each term in Eq. 3.37 be satised over the

control volume Vp around the point P in the integral form.

26

Figure 3.3: Fourier decomposition of pulsatile analytical solution into velocity magnitude andphase lag for Wo = 1, 3, 9, 27.

∫ t+∆t

t

[∂

∂t

∫VP

ρφdV +∫

VP

∇ · (ρUφ)dV −∫

VP

∇ · (ρΓφ∇φ)dV

]dt

=∫ t+∆t

t

[∫VP

Sφ(φ)dV

]dt (3.38)

The discretization of each term will be discussed in the following sections.

3.5.1 Spatial Discretization

The solution domain is discretized into computational cells, or nite volumes, on which the gov-

erning equations are discretized, reduced into algebraic form, and solved with numerical methods.

OpenFOAM is based upon general polyhedral cells. The cells are contiguous, i.e., they do not

overlap each other and completely ll the domain. Dependent variables and other properties are

stored at the cell centroid P, although they may be stored on faces or vertices. The cell is bounded

by a set of faces, given the generic label f . Since the mesh can be a general polyhedral, there is

no limitation on the number of faces bounding each cell, nor any restriction on the alignment of

each face.

The rst step in spatial discretization is to transform the volume integrals in Eq. 3.38 to surface

integrals by using Gauss's Theorem (also known as the Divergence Theorem). In its most general

27

form, Gauss's Theorem can be written as

∫V

∇ ? φdV =∫

S

dS ? φ (3.39)

where S is the surface area vector, φ represents any tensor eld, and the star operator ? is used to

represent any tensor product, i.e., inner, outer, cross and the respective derivatives (div, grad, curl).

Volume and surface integrals are then linearized using appropriate schemes which are described

for each term.

3.5.1.1 Convection term

The convection terms is integrated over a control volume and linearized as follows:

∫V

∇ · (ρUφ)dV =∫

S

dS · (ρUφ) =∑

f

Sf · (ρU)fφf =∑

f

Fφf (3.40)

where the face eld φf can be evaluated using a variety of schemes including central, upwind,

and blended dierencing. Central (or linear) dierencing, which is second-order accurate, can be

written as:

φf = fxφP + (1− fx)φ (3.41)

fx =fN

PN(3.42)

OpenFOAM also includes two other centered schemes: cubicCorrection and midPoint. Upwind

dierencing for φf , which guarantees boundedness but is rst-order accurate, can be written as

φf =

φp for F ≥ 0

φN for F ≤ 0(3.43)

In addition to the standard upwind scheme, linearUpwind, skewUpwind and Quick schemes are

variants of upwinded schemes which are available in OpenFOAM. Finally, there are a number of

blended schemes which attempt to preserve both boundedness and accuracy of the solution. The

gamma scheme can be written as

28

φf = (1− γ)(φf )UD + γ(φf )CD (3.44)

where the blending factor 0 ≤ γ ≤ 1 determines how much dissipation is introduced.

The mass ux F in Eq. 3.40 is calculated from interpolated values of ρ and U. Similar to

interpolation of φf , F can be evaluated using a variety of schemes including centered, upwinded,

and blended schemes, the latter of which includes a number of schemes (such as limitedLinear,

vanLeer, MUSCL, limitedCubic, SFCD, and Gamma).

3.5.1.2 Laplacian Term

The Laplacian term is integrated over a control volume and linearized as follows:

∫V

∇ · (Γφ∇φ)dV =∫

S

dS · (Γφ∇φ) =∑

f

ΓfSf · (∇φ)f (3.45)

The face gradient discretization is implicit when the length vector d between the center of the cell

of interest P and the center of neighboring cell N is orthogonal to the face plane, i.e., parallel to

Sf :

Sf · (∇φ)f = |Sf |φN − φP

|d|(3.46)

In the case of non-orthogonal meshes, an additional explicit term is introduced which is evaluated

by interpolating cell center gradients, themselves calculated by central dierencing cell center

values.

3.5.1.3 Source Terms

Source terms can be specied in three ways: explicit, implicit, and implicit/explicit. For explicit

source terms, they are incorporated into an equation simply as a eld of values. For example, to

solve Poisson's equation ∇2φ = f ; φ and f would be dened as volScalarField and then do

solve(fvm :: laplacian(phi) == f)

29

In contrast, an implicit source is integrated over a control volume and linearized by

∫V

Sφ(φ)dV = SpVP φP

The Implicit/Explicit approach changes between the two based upon the sign of the source term.

If the source is positive, it is treated as an implicit source term so that it increases the diagonal

dominance of the matrix. If the source is negative, it is treated as an explicit source term. In

mathematical terms, the mixed source approach can be written as

∫V

Sφ(φ)dV = SuVP + SpVP φP (3.47)

3.5.2 Temporal Discretization

3.5.2.1 Temporal Derivatives

The rst derivative ∂∂t is integrated over a control volume with one of two schemes: 1st-order Euler

implicit, or a 2nd-order backward dierence

∂

∂t

∫V

ρφdV =(ρP φP VP )n − (ρP φP VP )0

∆t(3.48)

∂

∂t

∫V

ρφdV =3(ρP φP VP )n − 4(ρP φP VP )0 + (ρP φP VP )00

2∆t(3.49)

where the new values of φn = φ(t + ∆t), the old values are φ0 = φ(t), and the old-old values are

φ00 = φ(t−∆t).

3.5.2.2 Treatment of Spatial Derivatives in Transient Problems

Reconsider the integral form of the transport equations Eq. 3.38.

30

∫ t+∆t

t

[∂

∂t

∫VP

ρφdV +∫

VP

∇ · (ρUφ)dV −∫

VP

∇ · (ρΓφ∇φ)dV

]dt

=∫ t+∆t

t

[∫VP

Sφ(φ)dV

]dt

Using Eqs. 3.40, 3.45, and 3.47, Eq. 3.38 can be written in a semi-discretized form

∫ t+∆t

t

∂

∂t

∫VP

ρφdV +∑

f

Fφf −∑

f

(ρΓφ)S · (∇φ)f

dt

=∫ t+∆t

t

(SuVP + SpVP φP ) dt (3.50)

For an Euler implicit approach, this equation can be reduced to

(ρP φP VP )n − (ρP φP VP )0

∆t+∑

f

Fφnf −

∑f

(ρΓφ)S · (∇φ)nf

= (SuVP + SpVP φnP ) (3.51)

In contrast, Eq. 3.50 can also be evaluated with a Crank-Nicholson scheme. The result is

(ρP φP VP )n − (ρP φP VP )0

∆t+∑

f

F

(φn

f + φ0f

2

)−∑

f

(ρΓφ)S ·

((∇φ)n

f + (∇φ)0f2

)

= SuVP + SpVP

(φn

P + φ0P

2

)(3.52)

Regardless of the approach, the equations can be reduced to the algebraic system for every control

volume

aP φnP +

∑N

aNφnN = RP (3.53)

where the coecients aP and aN are the diagonal and o-diagonal coecients, respectively, and

RP is the source-term vector.

31

3.6 Solution Algorithm for the Navier-Stokes Equations

Solution of the incompressible Navier-Stokes equations requires that three items be addressed:

derivation of an equation for pressure; linearization of the momentum equations; and implemen-

tation of a pressure-velocity coupling algorithm. These items will be discussed in the following

sections.

3.6.1 Linearization

The non-linear convection terms in the governing equation, Eq. 3.37, are reduced to∑

f Fφnf where F =

S · (U)f . The challenge is that F , and aP and aN from Eq. 3.53, are functions of (U). The impor-

tant issue is that the uxes F should satisfy the Continuity Equation, Eq. 3.1. Linearization of the

convection term means that the existing velocity (or ux) eld that satises continuity will be used

to calculate aP and aN . For strongly nonlinear phenomenon, two approaches can be used to cap-

ture the non-linearity: sub-iteration over the entire algorithm such the the lagged velocities, and

therefore the uxes and coecients, are iteratively updated; or use of small time-steps such that

the error in not updating the uxes and coecients remains small, both impose a computational

cost.

3.6.2 Derivation of the Pressure Equation

A semi-discrete form of the momentum equations is used to derive the pressure equation:

aP UP = H(U)−∇p (3.54)

This equation is clearly and extension of 3.53, however, the pressure term has been broken out, and

H(U) consists of two parts, the transport part which includes the matrix of coecients for all

neighbors multiplied by corresponding velocities, and the source term part which includes part

of the transient term and all other source terms (apart from the pressure gradient). For example,

H(U) for the incompressible Navier-Stokes equations (excluding source term due to gravity and

32

turbulence) using Euler implicit temporal dierencing is

H(U) = −∑N

aNUN +U0

∆t(3.55)

Equation 3.54 can also be solved for the velocity at the cell center by dividing aP

UP =H(U)

aP− 1

aP∇p (3.56)

From this, the velocity at the cell face can be found though interpolation, i.e.,

Uf =(

H(U)aP

)f

−(

1aP

)p

(∇p)f (3.57)

To derive the pressure equation, the discrete incompressible continuity equation is written

∑f

S ·Uf = 0 (3.58)

and Eq. 3.57 is inserted,

∇ ·(

1aP∇p

)=∑

f

S ·(

H(U)ap

)f

(3.59)

The Laplacian operator on the left-hand side can be discretized using the methods described in

Section 3.5.1.2, which results in the nal form of the discretized incompressible Navier-Stokes

equations

aP UP = H(U)−∑

f

S(p)f (3.60)

∑f

S ·

[(1

aP

)f

(∇p)f

]=∑

f

S ·(

H(U)aP

)f

(3.61)

Finally, if the face uxes F are computed using Uf from Eq. 3.57,

F = S ·Uf = S ·

[(H(U)

aP

)f

−(

1aP

)p

(∇p)f

](3.62)

then the uxes are guaranteed to be conservative.

33

3.6.3 Pressure-Velocity Coupling

The discretized form of the Navier-Stokes system in Eqs. 3.60 and 3.61 are coupled in that each

contains velocity and pressure. While there are algorithms for solving the fully-coupled set of

equations, this remains computationally expensive in comparison to the segregated methods for

coupling the pressure and velocity elds. The most common segregated methods are the PISO and

SIMPLE algorithms and their derivatives (e.g., SIMPLE-C, SIMPLER). Both are commonly used

in OpenFOAM; however, SIMPLE and PISO are typically used for steady and transient problems,

respectively. Each is discussed briey in the following sections.

3.6.3.1 PISO

The Pressure-Implicit Split-Operator (PISO) algorithm is a predictor-corrector method approach

for solving transient ow problems. For the PISO algorithm, the momentum equation is solved rst,

however, since the exact pressure gradient source term is not known at this stage, the pressure eld

from the previous time-step is used instead. This stage is called the momentum predictor and gives

an approximation of the new velocity eld. Using the predicted velocities, the H(U) operator can

be assembled and the pressure equation can be formulated. The solution of the pressure equations

gives the rst estimate of the new pressure eld. This step is called the pressure solution.

Next, conservative uxes consistent with the new pressure eld are computed using Eq. 3.62.

The velocity eld should also be corrected as a consequence of the new pressure distribution.

Velocity correction is done in an explicit manner, using Eq. 3.56. This is the explicit velocity

correction stage.

A closer look at Eq. 3.56 reveals that the velocity correction actually consists of two parts:

a correction due to the change in pressure gradient(

1aP∇p)and the transported inuence of

corrections of neighboring velocities(

H(U)aP

). The fact that the velocity correction is explicit means

that the latter part is neglected. It is therefore necessary to correct the H(U) term, formulate

the new pressure equation and repeat the procedure. in other words, the PISO loop consists of

an implicit momentum predictor followed by a series of pressure solution and explicit velocity

corrections. The loop is repeated until a predetermined tolerance is reached.

Another issue is the dependence of H(U) coecients on the ux eld. After each pressure

solution, a new set of conservative uxes is available. It would therefore be possible to recalculate

34

the coecients in H(U). This, however, is not done; it is assumed that the non-linear coupling is

less important than the pressure-velocity coupling, consistent with the linearization of the momen-

tum equation. The coecients in H(U) are therefore kept constant through the whole correction

sequence and will be changed only in the next momentum predictor.

The overall PISO algorithm can be summarized as follows:

1. Set the conditions.

2. Begin the time-marching loop.

3. Assemble and solve the momentum predictor equation with the available face uxes (and

pressure eld).

4. Solve the pressure equation, and explicitly correct the velocity eld. Iterate until the tolerance

for pressure-velocity system is reached. At this stage, pressure and velocity elds for the

current time-step are obtained, as well as the new set of conservative uxes.

5. Using the conservative uxes, solve all other equations in the system.

6. Go to the next time step, unless the nal time has been reached.

3.6.3.2 SIMPLE

The SIMPLE algorithm (Semi-Implicit Method for Pressure-Linked Equations) provides an

iterative method for solving steady-ow problems. The algorithm is comprised of several steps:

1) an approximation of the velocity eld is obtained by solving the momentum equation; 2) the

pressure equation is solved in order to obtain an updated pressure eld; 3) conservative uxes

are computed using Eq. 3.62; 4) velocity eld is explicitly corrected using Eq. 3.56; and 5) all

secondary equations (e.g., turbulent kinetic energy and specic dissipation) are solved. At each

step, under-relaxation is used to stabilize the non-linearity of the solution algorithm.

The overall SIMPLE algorithm can be summarized as follows:

1. Set initial conditions of all the eld values.

2. Begin the global iteration loop.

35

3. Assemble and solve the under-relaxed momentum predictor equation.

4. Solve the pressure equation and calculate the conservative uxes. Update the pressure eld

with an appropriate under-relaxation. Perform the explicit velocity correction using Eq.

3.56.

5. Solve the other equations in the system using the available uxes, and the updated pressure

and velocity elds.

6. Check the convergence criterion for all equations. If the system is not converged, start a new

global iteration.

36

Chapter 4

OpenFOAM

The use of Open-Source Field-Operation and Manipulation (OpenFOAM) at the practical

level is explained in this chapter. The process of grid construction, case assembly, and post-

processing application is discussed with a focus on the particular methods used for the internal

ow computations.

4.1 Using OpenFOAM

4.1.1 Pre-processing grids

A grid can be described as a contiguous computational domain (see Sec. 3.5.1). The programs

used to build the grids were OpenFOAM's blockMesh utility and Gridgen [38]. The blockMesh

utility is a mesher without a graphical interface. The user inputs the spatial values of the bounding

edges of a hexahedral volume. Simple geometric shapes like lines or circular arcs can be used to

describe the volume geometry. The user then inputs the number of cells and cell growth rate on

each edge. While this approach is unt for complicated geometry, which is outside the scope of

this research, it does provide the benet of proper face ordering. Equipping a grid with cyclic

(periodic) boundary conditions can only be successfully done if each corresponding face on the

cyclic boundaries is ordered in the same way.

Gridgen allows for direct manipulation of grids and has the ability of creating wholly structured,

wholly unstructured, or a blend of both grid types. Gridgen can directly export to OpenFOAM's

mesh format, though it could not at the time the current work was performed. Grids were exported

37

as a Fluent grid and converted with an OpenFOAM utility. However, the face ordering was not

always consistent through this method. For example, the unstructured grid used in the topological

study was capable of cyclic boundary conditions, though other attempts at a similar unstructured

grid could not be used. The hybrid and fully structured grids could not be assembled in Gridgen

such that cyclic boundary conditions could be implemented. A utility, couplePatches, attempts to

renumber the boundaries appropriately, but was unsuccessful for the grids.

OpenFOAM has a function to evaluate a mesh called checkMesh. This utility quanties the

geometrical statistics and cell shapes; checks the topology and boundaries; and assesses the delity

of the grid. The parameters of mesh delity are the boundary openness; non-orthogonality of the

cells; orientation and skew of the cell faces; edge lengths; aspect ratios; and face atness. The

mesh check is useful in predicting the success of the grid and the non-orthogonal correctors needed

to resolve the pressure solution per time step. While passing the mesh checks does not guarantee

that the grid will run successfully, failing one or more mesh checks will almost certainly cause the

computation to lose stability.

4.1.2 Building cases

A case is initially composed of three directories: the system, constant, and zero. Solvers require

dierent les within these for a case to start, but only les needed for laminar, incompressible ow

are described within this section. The constant directory contains the polymesh directory and

transportProperties le. The contents within the polymesh directory describe the grid geometry

and connectivity matrix upon which the cell volumes were created. Within the transportProperties

le, the kinematic viscosity constant, ν, must be set to the value that corresponds to the particular

uid. In the incompressible form of the governing equations, the kinematic viscosity is the only

constant needed, but the pressure gradient term must be divided by the density, ρ.

The system directory is used to describe the more administrative aspects of the computation.

The controlDict is used to adjust the time controls, read/write properties, and time step values.

The controlDict can also be used to set a maximum Courant number to ensure stability, as well

as calling the probes function. The probes function allows the user to select locations within

the computational domain and obtain a time history of the variables at those locations. The

fvSchemes le contains the information pertaining to the discretization schemes for the dierent

terms in the Navier Stokes equations. Computations done for the current work used schemes that

were second order accurate (Gauss Linear). The fvSolution le sets the linear algebraic solver and

38

convergence criteria. The sampleDict le contains the sample locations and method for the sample

post-processing utility. The decomposePar le sets the method and manner in which a grid is

decomposed for parallel computations.

The zero directory contains a le for each variable that will be calculated once the computation

begins. Within each le, every boundary is given its proper boundary condition and initialized

with the appropriate initial condition. The internal cells are also initialized. For laminar ow, the

velocity and pressure are the variables upon which this is done. The manner that this is done is

specic to each boundary condition and is described in the following sections.

4.1.3 Post-processing data

OpenFOAM provides many dierent ways to post-process a case. During a run, the code writes

a directory, like the zero directory, for each time specied by the controlDict. Within that folder

the variable les contain the value for every cell volume and boundary face. These times can be

visualized using external software. OpenFOAM contains utilities to convert the data to formats for

programs such as Tecplot, Ensight, and Paraview. Paraview is provided within the OpenFOAM

installer.

Several utilities exist to expedite CFD analysis. For example, the wallGradU and wallS-

hearStress utilities calculate the velocity gradient and shear stress at the wall and inserts that

as a variable into the time folders. The sample function can be used to extract variable data at

designated locations that are specied within the sampleDict. For the current work, the wall-

GradU variable was sampled at four points along the wall circumference. The other variables were

sampled along a line and interpolated between the midpoints and faces of the cell. The foamLog

utility decomposes the log le and provides a time history of the convergence details.

4.2 OpenFOAM Solvers and Linear Algebraic Methods

4.2.1 simpleFoam

The simpleFoam solver is a steady state solver for incompressible ow. While there is no

temporal term in a steady state equation, the solver uses a time step to denote the number of

iterations toward the steady solution. SimpleFoam uses the SIMPLE algorithm (Section 3.6.3.2)

to handle the pressure-velocity coupling and employs an under-relaxation solution method. The

39

user species the number of orthogonal iterations and non-orthogonal sub-iterations on the pressure

side of each time step calculation, as well as the relaxation factors for each variable. SimpleFoam

is an inherently stable solver in that the Courant number (the ratio between time step size, cell

spatial dimensions, and velocity magnitude) is not a factor in its stability.

4.2.2 icoFoam

The icoFoam solver is an unsteady solver for laminar, incompressible ow. IcoFoam uses the

PISO method (Section 3.6.3.1) for the pressure-velocity coupling and has the same iterative cycles

to correct for non-orthogonality as in simpleFoam. The solver is subject to stability requirements

dictated by the Courant number. A function can be applied to the solver that allows for a constant

Courant number and adjusts the time step accordingly.

4.2.3 channelIcoFoam

The channelIcoFoam solver is adapted from two standard OpenFOAM solvers: icoFoam and

channelFoam. ChannelIcoFoam has the same characteristics as icoFoam mentioned in Sec. 4.2.2

but handles the pressure term in the velocity equation as a known quantity. Because the Navier

Stokes equation is a function of the pressure gradient, not the pressure itself, the velocity solution

needs to approximate that value at each time step. The channelFoam solver is used for ows

in which the pressure gradient is constant and always known. In this case there is no need to

approximate the pressure gradient via a numerical discretization method, the pressure gradient

can simply be given a value and set as a body-force term in the velocity equation. The channelI-

coFoam solver adapts that method, but allows for the pressure gradient to sinusoidally oscillate;

thereby permitting a pressure driven computation to have cyclic inlet and outlet boundary con-

ditions. The frequency, gradPMag, and gradPNormalDirection constants must be set within the

gradPProperties le in the constant folder,

4.2.4 Fourier adaptations

A periodic ow can be quantied using Fourier analysis, via the magnitude and phase lag of

the rst (or higher) mode and the average (zeroth mode). For applicable ow situations, the

icoFourierFoam and channelIcoFourierFoam solvers were developed. They are computationally no

dierent from icoFoam and channelIcoFoam; however, at the end of each time step calculation, the

40

coecients A0, A1, and B1 are calculated using the trapezoidal rule. To obtain valid Fourier coef-

cients, the Fourier solvers must be used on a previous calculation that has reached a statistically

steady state and ran for one period, T . On the last time step (at t = T ), the Fourier coecient

variables are valued as the integration of the velocity components across one period and can be

post-processed. The frequency, f = 1T , is specied in the fourierCoeProperties in the constant

folder.

4.2.5 Linear algebraic methods

Though there are many dierent algebraic solvers available in OpenFOAM, three were utilized

for the computations. The pressure equations were solved using a Geometric-Algebraic Multi-

Grid method (GAMG) and a Preconditioned Conjugate Gradient (PCG) method using a Diagonal

Incomplete-Cholesky (DIC) preconditioner. The velocity equation was solved using a Precondi-

tioned Bi-Conjugate Gradient method (PBiCG) using a Diagonal Incomplete Lower Upper (DILU)

preconditioner. Dierent algebraic solvers were used to dually assess the capability of each method

within OpenFOAM.

4.3 OpenFOAM Boundary Conditions

4.3.1 xedValue

The xedValue boundary condition applies a user-specied constant value along the boundary.

The constant can be applied uniformly or non-uniformly as a scalar, vector, or tensor value.

4.3.2 zeroGradient

The zeroGradient boundary condition applies a zero rst-derivative, or Neumann condition

with a constant of zero, in the normal direction of the local face to the boundary.

4.3.3 cyclic

A cyclic boundary condition enables two patches to be treated as if they are physically connected

and are used for repeated geometries. For the pipe cases, a reference pressure must be set to a

reference volume in order for the pressure loop to solve correctly. From that reference pressure and

41

the pressure gradient specied in the gradPProperties, channelIcoFoam can solve for the pressures

with respect to the reference cell.

4.3.4 pressureInletOutletVelocity

The pressureInletOutletVelocity (pIOV) boundary condition evaluates the velocity via the

pressure-induced volume ux. In order to apply this boundary condition, the pressure must be

well-posed or a solution will not converge.

4.3.5 timeVaryingUniformFixedValue

The timeVaryingUniformFixedValue (tVUFV) boundary condition applies a uniform value to

the boundary that is specied as a function of time in a data le. The value is constant for each face

and is interpolated between the two nearest values in the data le. Like the xedValue boundary,

the value can be a scalar, vector, or tensor.

42

Chapter 5

Internal Flow Computations

The CFD results are separated into two sections: a grid and topological study of a straight

pipe and a straight pipe respiring into an external environment (plenum). The grid study was

computed on three hexahedral structured grids, see Figure 5.2. The coarsest of which has an

approximate number of volumes across the diameter as the nest dog's nose grid (Figure 5.1). With

each grid renement, the number of volumes along a connector was doubled from the previous grid

and the growth rate remained constant; thereby doubling the near wall resolution and quadrupling

the total volumes on each circular face with each renement. The grid parameters can be seen in

Table 5.2. The grid study was performed to understand the eect of increasing the resolution to

capture the Stokes Layer characteristics and overall velocity prole.

The grids used for the topology study (see Figure 5.3) were a fully unstructured grid with no

clustering near the wall; a hybrid grid with a structured wall layer and unstructured inner core;

and the middle grid from the grid study. The grids were created to have approximately the same

volume density on the circular face with no variation in the axial direction, except for number of

cells (see Table 5.2). In other words, any slice normal to the axial direction would be uniform to

the inlet and outlet face. The grids were able to be created this way for two reasons: for the ow

under consideration, the fully developed velocity solution will not change in the axial direction

and the application of the boundary conditions is such that an entry length in the pipe is not

required. The increase in the number of downstream volumes for the unstructured and hybrid grid

is purely to ensure solution convergence. The topological comparison was performed to understand

the eectiveness of alternate grid topologies.

43

Figure 5.1: Comparison of the internal spatial resolution of the (1) coarse, (2) medium, (3) ne,and (4) nest CFD grids in the maxilloturbinate region. Comparable grid resolution is found inthe nasal vestibule and ethmoidal region [5].

44

Figure 5.2: Coarse, medium, and ne grids for the grid study.

45

Figure 5.3: Unstructured, hybrid, and structured grids used for the topology study.

46

Figure 5.4: The grid used for the plenum case shown as the view on the inlet face, along the axiallength, and on the plenum face.

47

Total Cells Axial Cells Faces Density(

facesmm2

)In/Out Boundary

Coarse 245 1 245 311.9 cyclicMiddle 980 1 980 1247.8 cyclicFine 3920 1 3920 4991.1 cyclic

Hybrid 2616 3 872 1110.3 pIOV, tVUFVMiddle 980 1 980 1247.8 cyclic

Unstruct. 5190 5 1038 1321.6 cyclic

Table 5.2: Parameters of each pipe grid.

D (mm) L (mm) Re umax (ms ) 1

ρ∂P∂x

(m2

s2

)Pin Pout

Grid/Topology 10 10 1000 0.2 0.032 0 0.00032Plenum (Pipe) 1 100 750 1.5 0.25 0 n/a

Plenum (Plenum) 50 25 n/a 2.5

Table 5.3: Physical parameters of grid/topology studies and the plenum case.

Initially, steady CFD solutions were obtained for all the grids. The velocity proles were

compared to the analytical solutions. The computations were capable of solving for steady ow.

The computational domains for the grid and topology study were created with an equal diameter

and length, D = L = 0.01 m and computed at Re = 1000. For steady ow, this calculates to an

average velocity, u = 0.1 ms , which corresponds to parabolic prole with umax = 0.2 m

s . These

velocity conditions dene the pressure gradient to a value of ∂P∂x = 32 Pa

m . As a laminar solver was

used, the kinematic viscosity was the only known uid parameter needed (ν = 1×10−6 m2

s ), though

the pressure must be divided by the density (ρ = 1000 kgm3 ) to maintain uniform dimensions.

The inlet and outlet pressure boundary conditions were set to a xedValue of uniform Pin =

3.2 × 10−4 m2

s2 and Pout = 0 m2

s2 respectively across the boundary faces. This ensures laminar

ow at the desired Reynolds number. As the pressure is fully specied, the velocity boundary

conditions were set to cyclic (channelIcoFoam) or pIOV (icoFoam/simpleFoam) in the case of the

hybrid grid (Table 5.2). The wall boundary condition sets the pressure boundary condition to

zeroGradient and the velocity to a xedValue of zero. The physical parameters for all the steady

calculations can be seen in Table 5.3. Once the steady solutions were obtained and veried, the

unsteady solutions were ready to be computed.

The unsteady cases were purely oscillatory ow. This was performed by varying the pressure

gradient sinusoidally about zero, via a time data le (tVUFV, icoFoam) or directly specifying the

gradient and frequency in the constant folder within the case (channelIcoFoam). The magnitude of

48

Wo f ( 1s ) T (s) Cycles to Stat. Steady

27 4.6410 0.21547 2509 0.51565 1.9393 153 0.057296 17.4532 51 0.00636618 157.08 2

1 (Plenum) 0.636618 1.5708 2

Table 5.4: Unsteady parameters of the grid/topology studies and the plenum case.

the pressure gradient corresponds to the steady value found for the appropriate Reynolds number,

depending on the grid: straight pipe or plenum. The frequency was varied such that the Womersley

number had four dierent values: Wo = 1, 3, 9, 27. The numerical details are shown in Table

5.4.

For each Womersley number, the cases were run for the same number of cycles until a sta-

tistically steady situation developed, and then ran for one period using the Fourier solver - a

statistically-steady condition was reached when the ow was the same at corresponding times

across a cycle, meaning that the ow has reached its periodic form. This was determined by moni-

toring the mass ow rate across the cycles and comparing its value at peak mass ux. An example

of this is shown in Figure 5.5. For a Wo = 9 computation, each case was ran for 15 periods before

the Fourier solver was implemented. Increasing the Womersley number increased the number of

cycles needed to reach this condition.

Computationally, only a discrete number of times were calculated across a period. In order to

maintain stability, the time steps were moderated by two criteria: a maximum Courant number

and a maximum time step. During the computations to achieve statistically steady, the maximum

Courant number was set to 0.5. The time step was allowed to vary to expedite the computations.

During the last period, in which the Fourier solver was implemented, the time step was set to a

maximum value of 1 ms and the variable time step option was enabled. This lowered the Courant

number, increased the accuracy, and allowed the time step to automatically reduce itself in order to

output the variables at the designated times. The hybrid grid was computed with a slightly dierent

approach. A constant time step per Wo was used to ensure an integer number of computation

cycles between outputting variables. The time steps varied between 1.08 ms and 0.873 ms. The

number of time steps per cycle was specically set so as to greatly exceed the number of time steps

computed in the pulsatile ow grid study [30] and canine olfaction CFD [5]. This eort allowed

the results to be compared through spatial dierences alone, with the exception of the mass ux

analysis. More detail will be provided in Section 5.1.2.

49

Figure 5.5: Example of statistically steady for Wo = 9, computation was ran for 15 cycles andmass ux was observed before the nal Fourier cycle was computed.

For the grid and topology study, the steady results are presented rst, followed by the unsteady

results. The steady proles are compared to the analytical solution, then the error across a

radius is shown for all grids. The unsteady results were compared to several global, analytical

parameters and also to the analytic rst Fourier mode velocity magnitude and phase lag. The

error of the solution to the analytic value is shown for each comparison. For each parameter under

consideration, the results of the grid study are discussed rst and the topology study second.

The plenum case was performed to simulate a more realistic physical environment similar to

that of the canine olfaction CFD computations by introducing an external environment (plenum)

to the straight pipe. The physical parameters are similar to the dog's nose CFD (laminar and

quasi-steady) and can be seen in Tables 5.3 and 5.4; however, the geometry is much simpler,

though scaled on the order of the nasal cavity. The plenum case includes both steady inspiration

and expiration computations. The boundary conditions of the hybrid grid (pIOV, tVUFV) were

applied to the plenum case, though the plenum was considered as the outlet from the previous

cases. The diameter and length of the pipe have dierent values (D = 0.001 m and L = 0.1 m) and

the Reynolds number was lower (Re = 750, Figure 2.1). Therefore, the magnitude of the pressure

gradient changed, and consequently, the pressure at the inlet and plenum became: Pin = 2.5 m2

s2

and Ppl = 0 m2

s2 for the same uid. The plenum case was only ran at Wo = 1 (Figures 2.2 and

5.4) to reect the quasi-steady environment within the ethmoidal (olfaction) region of the canine's

olfactory region.

50

5.1 Grid and Topology Studies

5.1.1 Steady Solution

Considering the steady solution, all grids were capable of converging to the Poiseuille solution.

The results of the grid study can be seen in Figure 5.6 (a, c). The sample function was used to

extract the velocity values across the radii of all the grids.

In evaluating the eect of grid renement, the grid study shows a monotonic decrease in error

with increasing renement. A Richardson extrapolation exercise was not performed to show this

because an analytical solution exists for comparison. The coarse grid had the highest error, min-

imizing at the centerline at 97.6% of the analytical solution. The error increased to a maximum

of 3.2%. With the rst renement, the middle grid halved the error of the coarse grid almost

uniformly across the radius, about 1% minimum error at the centerline and 2% maximum error

near the wall. The nest grid renement reduced the error again nearly by half, capturing just

under 99.5% of the solution at the centerline and 98.9% at the wall.

The comparison of the steady solution on dierent grid topologies, Figure 5.6 (b, c), provides

insight into the value of structured layers near the wall and gridding exibility of unstructured

meshes. The hybrid grid, with the lowest grid density (Figure 5.2), maintains approximately a

1.4% error in its unstructured core, while the structured grid is closer to 1% in the same region.

The hybrid grid error only slightly decreases in its structured region furthest from the wall. The

concentric nature of the structured layers in the hybrid grid, which blockMesh does not create for

the structured grid, appears to lower the error when compared to the structured grid through this

region. The hybrid grid is also the least dense and local renements specically in the structured

layer closest to the wall, as seen in the grid study, would increase accuracy . The completely

unstructured grid (and also the most dense) averaged under 0.7% error from the centerline to the

near wall region. The lack of a structured wall layer did not signicantly impact the solution in

the area where the velocity gradient was highest.

5.1.2 Unsteady Results: Global Error Evaluation

To quantify the results of the unsteady computations, several global parameters were compared

to the analytical value at each Womersley number. To evaluate the performance of capturing

the dynamics of the Stokes Layer, three parameters were selected. First, the maximum value of

the rst Fourier mode velocity magnitude was found. This value was found at the location of

51

(a) Grid study

(b) Topology study

(c) Error across the radius to the analytical value

Figure 5.6: Steady state results for the grid and topology study: velocity proles compared to theanalytical solution and error from the wall to centerline.

52

the peak overshoot when the Stokes layer was present, or the centerline when the velocity had

no overshoot. As Womersley number increased by a factor of three, the magnitude decreased by

nearly a factor of ten. Then, the radial location of the rst parameter on each grid was compared

to the analytical value. These rst two parameters signify how well the grid under consideration

captures the Stokes Layer. The next parameter shows the phase lag at the wall. Analytically,

this value has the smallest phase lag along the radius to the pressure wave. The nal two global

parameters considered are the peak mass ux magnitude and phase lag to the analytical solution.

The separation of independent variables intrinsic to the analytical solution can also be expressed

in the analysis of the computations. The constant shape of the velocity magnitude curve is modied

by the temporal term, creating a time sensitive velocity prole. Similarly, the analysis of the Fourier

velocity magnitude and phase lag allows the computational results to be dissected separately.

Considering rst the rst Fourier mode maximum velocity magnitude, the grid study (Figure 5.7

a, c) shows an expected decrease in error as the grid is rened, though the error for coarse grid is

small (error < 1.5%). The middle grid, for all Wo, has less than 1% error and in all cases has less

than 0.5% dierence to the ne grid. This suggests that the middle grid's level of renement is

adequate for a reliable computation with a lower computational cost than that of the ne grid. As

to the diering topologies, the unstructured grid has the lowest error (error < 0.4%) except for Wo

= 27 (error > 1%) though the slightly higher grid density does provide an accuracy advantage. For

the hybrid grid, the error is higher than the structured and unstructured grid when the value was

found in the unstructured core. The performance increases for Wo = 9, 27 when the maximum

magnitude was found in the structured near wall region, though at Wo = 9 the unstructured grid

still has a higher accuracy. The middle grid maintains a steady error of less than 1% whereas the

hybrid grid exceeds 1% for the lowest Wo and the unstructured grid does so for the highest Wo.

While the error was low for the grid study in resolving the value of the overshoot in the Stokes

Layer, the need for higher resolution near the wall as Wo increases is shown in Figure 5.8 (a, c). One

point of clarity needs to be made, the Stokes Layer is dened in this results section as the distance

from the wall to the location of maximum Fourier velocity magnitude. Technically, by denition

the Stokes Layer is also out-of-phase with the ow in the centerline of the pipe due to viscous

interaction with the wall. In eect, the Stokes Layer does not exist for the lower Wo because the

ow is largely in-phase with itself with no overshoot, but this section considers the Stokes Layer to

be the radial point of maximum Fourier velocity magnitude from the wall. Referring back to Figure

3.3 shows that Wo = 1, 3 has no overshoot and therefore no Stokes Layer; however, for ease of

53

discussion in this section the entire radial distance is considered to be the Stokes Layer. The error

in Figure 5.8 (c) was calculated by nding the radial value at the cell center that corresponded to

the value found in the maximum Fourier magnitude in the rst global parameter discussed.

The grid study shows that for the two lowest Womersley numbers, each grid accurately resolved

the thickness to be the entire radius with no error. As the shape of the velocity distribution in

these two cases are axisymmetrically parabolic, these results are expected due to the cell center

located exactly in the center of the circle. As the Womersley number increased, inducing a true

Stokes Layer, the local renement became a factor. The error became more disparate at Wo = 9,

though maintained a higher accuracy with each renement. Interestingly, at Wo = 27 the coarse

grid had an error of 5% and the ne grid had an error of 4.3%, but the middle grid had an error

of 11.5%. The middle grid, in this case, had a cell center that was much further to the analytical

solution than the other grids which resulted in a greater error by the way the error was calculated.

While the exact Stokes Layer thickness was known a priori, no adjustments were made during the

gridding process to make the performance monotonically distributed because for more complex

geometries (i.e. the canine nasal passage) such knowledge is not known a priori.

The introduction of unstructured mesh types (Figure 5.8 b, c) shows that at Wo = 1, 3, the

error is below 2% for the hybrid grid and below 3% for the unstructured grid. This is a disadvantage

when compared to the symmetric middle grid that had no error. At Wo = 9, the Stokes Layer

thickness is within the near-wall-structured portion of the hybrid and middle grids. The middle

grid has the highest error at 6.5%, followed by the hybrid grid at 4.5% and the unstructured grid

at 1%. At the highest Wo, the middle grid again had the highest error. The hybrid grid, with an

error of 2.5%, was better than the unstructured error of 4%. Statistically, the error should increase

as Wo increases because the analytical Stokes Layer thickness, which is the denominator of the

error equation, decreases. The unstructured grid exemplies this increase in error from Wo = 9 to

Wo = 27 because there was not adjustment to the cell sizes near the wall, though Gridgen does

have the capability to do that. Decreasing cell size near the wall through adjusting cell expansion

rates can slightly moderate the error for a structured grid. This is the case for the hybrid grid,

though not for the middle grid which was discussed in the previous paragraph.

The last global parameter concerning the Stokes Layer is the rst Fourier mode velocity phase

lag at the wall. Whereas the rst parameters described the accuracy of the computations in terms

of magnitude, the results of studying the phase lag at the wall imparts information on the transient

part of the calculations. On the surface of the wall, no phase lag occurs because the velocity is

54

zero; though slightly above the wall within the uid, a phase lag to the pressure sinusoid exists.

The data was sampled at the closest cell center to the wall surface. Ostensibly, this location

could have been selected at any radial point. Analytically, the radial phase lag can be evaluated

as the dierence from any radial point and compared as such; however, the location just away

from the wall is within the Stokes Layer and has the smallest phase lag to the pressure waveform.

Furthermore, the gradient of the analytical solution for the Fourier velocity phase lag is sharpest

per Wo at the wall and would be expected to be more dicult for a grid to resolve.

Considering the grid study in Figure 5.9 (a, c), the error increases as the Wo is increased for

all structured grids. Each grid had error of less that 0.05% accuracy at Wo = 1 and ranged from

0.12% to 0.65% error for Wo = 3. The coarse grid increased from 1.3% to 1.5% error from Wo

= 9 to Wo = 27. Similarly, the middle grid increased from 0.6% to 1% error and the ne grid

increased from 0.2% to 0.75% error. The increase in error from the previous Wo for all three grids

is indicative of the diculty in resolving the Stokes layer at higher frequencies.

As to the topology study, it is very apparent that a structured near wall region is benecial in

resolving the phase lag. The unstructured grid has the highest errors, most signicantly at Wo =

27. The hybrid and structured grids have nearly identical errors at or below 1%. The disparity in

the grid densities between the unstructured (highest) and hybrid (lowest) grid further shows that

the phase lag for the higher two Wo is heavily aected by near wall resolution.

The last two global parameters concerns the mass ux. The rst of which is the comparison

to the analytical mass ow rate at maximum ux, or peak expiration. As the mass ux is the

integration of the velocity ux normal to a plane, this parameter provides an indication of how well

each grid resolves the entire ow eld at peak ow. A function inherent in OpenFOAM was applied

to the solvers to calculate the volume ux at all boundaries per time step. As incompressible solvers

were used, the value was scaled by the constant density, ρ, and compared to the analytical mass ow

rate. The wall boundary had no mass ux, which ensured the no slip condition worked properly.

The inlet and outlet boundaries had negligible dierence between the two that was consistent with

the numerical error inherent in CFD.

Unlike the previous parameters discussing the Stokes Layer, the mass ow phase lag must be

analyzed in conjunction with the maximum mass ow magnitude. Analytically, this phase lag

is the dierence between the time of peak pressure and the time of peak mass ux. The phase

lag increases with increasing Womersley number. To reiterate, at Wo = 1 the ow is considered

quasi-steady and the ow oscillates nearly in-phase with the pressure. As the Womersley number

55

increases, the Stokes Layer develops and the near wall viscous eects reduce the overall magnitude

of the mass ux while increasing the response time of the ow. In parallel with the discrete cell

centers and analytical Stokes Layer thickness comparison, the discrete time steps were not chosen

to fall directly on the time of maximum mass ow. At the time when the mass ux peaked in the

computation, the ux value and corresponding time were recorded and compared to the analytical

values. Therefore, any discrepancy between the computed and analytical time induces extra error

into the mass ux comparison because the velocity prole at the computational moment will be

dierent from the velocity prole at peak analytical mass ow. A higher phase lag error would add

to the numerical error of the peak mass ux, but subjectively, a similar phase lag error between

grids would enable fair comparison of the peak ux. Also, each grid had a slightly dierent time

of peak mass ux due to the variable time step discussed previously.

Comparing the mass ux at three levels of renement, Figures 5.10 (a, c) and 5.11 (a, c),

provides a general comparison to the solution as a whole. At Wo = 1, the phase lag error decreased

from 1.45% to 0.59% to 0.22% with each level of renement. The phase lag error corresponded

to the following maximum mass ow error from coarsest to nest: 1%, 0.5%, and 0.25%. The

largest dierence in phase lag error occurs at this Womersley number, skewing the peak mass ux

results the most out of any Womersley number. Though, the peak mass ux error was halved with

each successive renement. The error would likely decrease by less than 50% with each renement

when the two results are combined and adjusted. For Wo = 3, the phase lag error appeared to

have no eect on the mass ow magnitude. The phase lag error decreases from 0.7% to 0.3% from

the coarse to middle grid and then to 0.2% for the ne grid. Though the mass ow magnitude

hovers near 0.1% for all three grids. This does not necessarily mean that all three grids uniformly

computed the solution accurately because any overshoot of the solution would compensate for an

undershoot, thereby decreasing the overall error of the integration. For the last two Womersley

numbers, the phase lag is consistent for all three grids at a value of 0.2%. This provides the

opportunity to compare the mass ux error independently of phase lag. At Wo = 9, the error for

the coarse, middle, and ne grid is 0.3%, 0.14%, and 0.05% respectively. At the last Womersley

number, the error reduced from 2.8% to 2.5% and then to 2.3%. While lower error cannot equate

greater success in resolving the ow eld, higher error suggests an overall diculty in achieving

the analytical velocity prole. The consistency of the error at Wo = 27 for all the grids further

suggests such diculty because upon renement the results did not tend to converge towards zero

error.

56

Comparing the mass ux errors for the three dierent topologies, Figures 5.10 (b, c) and 5.11 (b,

c), also gives an overall comparison to the velocity solution. At Wo = 1, the hybrid and structured

grid have a phase lag error of 0.6% and the unstructured grid has a lower error of 0.25%. The

hybrid grid had the highest mass ow magnitude error at 1.2%, followed by the middle grid with

0.5%, and the unstructured grid with 0.2% error. With no dierence in phase lag error between

the hybrid and middle grid, the indication is that the unstructured core does not resolve the ow

as well as the completely structured grid. At Wo = 3, the hybrid grid again has the highest error

to the mass ow magnitude (0.4%) when compared to the other two grids (0.1% error). The phase

lag error is near 0.3% for all three grids: hybrid (0.33%), middle (0.3%) and unstructured (0.27%).

The results suggest that grid density is a large factor in resolving velocity proles with a parabolic

shape. With the onset of the Stokes Layer at Wo = 9, 27, the phase lag error is consistently

0.2% (except that the hybrid grid had an error near zero at Wo = 9). Compared subjectively,

the completely structured grid computed the mass ux magnitude with highest accuracy (0.12%)

at Wo = 9 while the hybrid and unstructured grid matched errors at 0.5%. The hybrid grid

signicantly resolved the velocity with the highest accuracy at Wo = 27, while the unstructured

grid had the highest error. Section 5.1.3 further validates the global parameters by evaluating the

error across the entire radius.

5.1.3 Unsteady Results Radial Error Evaluation

To further quantify the unsteady results, the magnitude and phase lag error to the analytical rst

Fourier velocity mode was calculated across the radius for each Womersley number. Whereas the

global parameters detailed specic characteristics of the ow, the radial error provides information

on the accuracy of resolving the entire ow eld. While the peak mass ux error could only

suggest high accuracy in solving for the velocity prole, the radial error evaluation can complete

the analysis. For each Womersley number, the grid study results will be discussed followed by

the topological study. A smoothing algorithm was applied to the error curves, so there is a slight

discrepancy in the radial phase lag error at the wall to the global error evaluation.

In Figure 5.12 (a, b) at Wo = 1, the general shape of the error curve remains the same but

the error decreases upon grid renement for both parameters. The Fourier magnitude is most

successfully resolved near the centerline but the error grows in the near wall region. The opposite

is true for the Fourier phase lag, though the error is small for all three grids; the highest error is less

than 0.2% for the coarsest grid. The phase lag error here was the lowest among all the Womersley

57

(a) Grid study

(b) Topology study

(c) Error to the analytical value

Figure 5.7: The maximum value of the Fourier velocity magnitude across a range of Wo for thegrid and topology studies, with the error from the analytical solution.

58

(a) Grid study

(b) Topology study

(c) Error to the analytical value

Figure 5.8: The thickness of the Stokes Layer across a range of Wo for the grid and topologystudies, with the error from the analytical solution.

59

(a) Grid study

(b) Topology study

(c) Error to the analytical value

Figure 5.9: The phase lag of the Fourier velocity at the wall across a range of Wo for the grid andtopology studies, with the error from the analytical solution

60

(a) Grid study

(b) Topology study

(c) Error to the analytical

Figure 5.10: The value of the maximum mass ow rate across a range of Wo for the grid andtopology studies, with the error from the analytical solution.

61

(a) Grid study

(b) Topology study

(c) Error to the analytical value

Figure 5.11: The phase lag of the mass ux to the pressure oscillation across a range of Wo for thegrid and topology study, with the error from the analytical solution.

62

numbers considered, the closest to the pressure wave, and had the least dierence between the wall

and the centerline value. Monotonic convergence was observed.

As the gridding technique was altered for the topology study, the results show that the unstruc-

tured grid has the highest overall accuracy in resolving the Fourier magnitude. The unstructured

grid has more cells on its face than the other two grids, though none of its cells are clustered near

the wall. For a parabolic velocity prole, this suggests that grid density is more important to

solution accuracy than near wall resolution. However, if structured near wall resolution is used, an

axisymmetrical approach generally yields better results; the hybrid grid has a lower error in most

of the structured region.

As the Womersley number is tripled (Wo = 3), the analytical Fourier magnitude stays parabolic,

but with a blunter prole than the previous Wo. The analytical phase lag values increases, as well

as the dierence between wall and centerline phase lag. The grid study again shows that the error

decreases proportionately with grid renement for both magnitude and phase lag in Figure 5.12

(c, d). The Fourier magnitude error near the centerline is less than the previous Wo and the error

near the wall is relatively unchanged. The phase lag error as a whole is much greater than at Wo =

1 and the error between the wall and the midpoint of the radius is more pronounced. The Fourier

phase lag error still remains less than 1%, but it is evident that the ow is more dicult to resolve

with the increase in Womersley number.

In regards to the topology study, the unstructured grid again has the lowest error for the Fourier

magnitude and also for the phase lag. Within the structured region, the hybrid grid again has

lower error than the middle grid for both Fourier components; however, the middle grid has lower

error near the centerline than the hybrid grid.

At Wo = 9, the manipulation of the analytical solution into its Fourier components reveals

a Stokes Layer. The combination of the overshoot near the wall of the Fourier magnitude and

the dierence in phase lag between the wall and the centerline ow creates this unsteady ow

characteristic, while the ow in the centerline begins to move with a uniform plug ow. In

Figure 5.13 (a, b), The grid study shows that Fourier magnitude error becomes negligible in the

plug ow region for even the coarsest grid; however, the phase lag error continues to increase in

the Stokes Layer region near the wall. For a ow that contains a Stokes Layer, this suggests that

less renement is needed in the plug ow region and higher renement is needed to capture the

overshoot phase lag, which will also decrease the Fourier magnitude error near the wall.

The unstructured grid still had the least overall error for the Fourier magnitude and phase

63

lag; however, the results begin to deteriorate near the wall, especially in resolving the phase lag.

Unlike the other two grids with structured components, the unstructured grid has seen an increase

in overall error for both Fourier components with an increase in Womersley number. The higher

grid density has been a factor in the the lower errors when compared to the other grids, though this

trend suggests that the lack of localized cells is becoming a factor that is detrimental in resolving

the Stokes Layer. The hybrid and middle have negligible error outside the Stokes Layer. Near the

wall, the hybrid grid has less error in resolving the Fourier phase lag, yet the opposite is true of

the Fourier magnitude. The high error near the wall is very detrimental to the hybrid grid, but

the low grid density increases the overall error that mitigates the results somewhat.

At Wo = 27, the Stokes Layer thickness decreases and the ow outside of it becomes even

more like plug ow. In Figure 5.13 (c, d), the structured grids shows almost no error outside of

the Stokes Layer for both parameters, yet the errors within are higher than for the previous Wo

with a Stokes Layer. This suggests that a very ne degree of resolution within the Stokes Layer

is imperative to accurate solution convergence for high frequencies. Though much relaxation is

possible within the plug ow region, as the coarse grid performs as well as the ne grid.

While the unstructured grid performed well for the previous Womersley numbers, it had the

worst results for both Fourier phase lag and signicantly higher error near the wall for the velocity

magnitude. The phase lag error is more than four times higher within the Stokes Layer than the

other grids. The hybrid grid solves for the centerline ow with more accuracy than the middle and

unstructured grid, but the error in the Stokes Later is higher than the middle grid for both Fourier

components. Computationally, this Wo is the hardest velocity eld to resolve with accuracy.

5.2 The Plenum Study

The purpose of the plenum computation was to increase the complexity of the physical model

to simulate a more realistic computation of respiration. This was done with the addition of an

ambient external environment (plenum) to one end of the straight pipe. The geometry was scaled

on the order of the canine nasal cavity: the diameter of the pipe equals one hydraulic diameter of

the dog's nose and the length equals one hundred diameters. The plenum was extended twenty-ve

diameters axially and radially from the outlet of the pipe. The grid built on the geometry was

fully structured with no near-wall renement (Figure 5.4) and approximately the same number of

64

(a) Fourier velocity magnitude error, Wo = 1 (b) Fourier velocity phase lag error, Wo = 1

(c) Fourier velocity magnitude error, Wo = 3 (d) Fourier velocity phase lag, Wo = 3

Figure 5.12: Error across the radius to the analytical solution of the Fourier velocity magnitudeand phase lag for Wo = 1, 3

65

(a) Fourier velocity magnitude error, Wo = 9 (b) Fourier velocity phase lag error, Wo = 9

(c) Fourier velocity magnitude error, Wo = 27 (d) Fourier velocity phase lag error, Wo = 27

Figure 5.13: Error across the radius to the analytical solution of the Fourier velocity magnitudeand phase lag for Wo = 9, 27

66

cells across the diameter as the nest dog's nose grid (Figure 5.1) .

Initially, computations were performed to model the uid ow of steady exhalation and inhala-

tion at Re = 750 to reect the laminar ow situation within the olfactory region of the dog's nose

(Figure 2.1). The physical and geometrical parameters can be seen in Table 5.3. The exhalation

computation was not able to converge when the steady solver, simpleFoam, was used. The recourse

was to use icoFoam with boundary conditions set such that the steady situation could develop.

The steady inhalation computation was performed using simpleFoam and the absolute value of the

results was used to facilitate comparison. In the frame of reference used, the values are actually

negative. The computational data was sampled at seventy-ve diameters from the outlet of the

pipe, or 25% of the total length from the inlet, and compared to the analytical solution.

To model the unsteady ow, the pressure at the inlet (x = 0) was varied sinusoidally with

a magnitude to that of the steady ow (2.5 KPa) while the plenum boundary pressure was held

at zero (Figure 5.4). The frequency was set so that the Womersley number equaled one, which

reected the value found in the ethmoidal region of the dog's nose in Figure 2.2. After two cycles,

the ow reached a statistically steady state and the Fourier solver was implemented for a third

period (Table 5.4). At the end of the last period, the rst Fourier mode coecients were calculated

and manipulated into magnitude and phase lag. The values and error of both parameters are shown

as a function of radial position. These were sampled at every fourth-length of the pipe 0.25*L,

0.50*L, 0.75*L, 1.00*L where 1.00*L was the interface of the pipe and the plenum. These locations

were selected to evaluate the eect of entry length ow characteristics. The velocity gradient was

also calculated on the wall using the wallGradU function, scaled with the viscosity, and compared

to the analytical wall shear stress at twenty equally spaced times throughout the period. Finally,

radial velocity proles at the four locations are shown at the same twenty times of the shear stress

evaluation.

5.2.1 Steady Results

The analytical solution for steady pipe ow yields the parabolic Poiseuille ow solution. At Re =

750, this yields a maximum velocity of 1.5 m/s at the centerline of the pipe. The errors discussed

are a result of comparing the computed solution to the analytical values. Referencing Figure 5.14,

the expiration simulation had a maximum value at the centerline of 1.458 m/s, which is an error of

2.8% to the analytical solution. Considering the coarseness of the grid, this is an acceptable error.

67

(a) Velocity prole through the diameter at x = 25*D

Figure 5.14: The velocity distribution for the steady inspiration and expiration for the plenumcase. The values for the inspiration velocity have been changed from negative to positive forcomparison purposes, but in reality have negative value.

As the uid was drawn into the pipe during the inspiration simulation, the velocity only peaked at

1.205 m/s (19.7% error). Whereas the expulsion of air into the ambient surroundings had negligible

eect on the ow, the air entering the pipe from the plenum encountered the sharp corners of the

inlet and incurred signicant losses. As the solution is still parabolic for the inhalation, it was

determined that the lower peak velocity was due to losses at the inlet and not a computational

discrepancy.

5.2.2 Unsteady Results

As the unsteady ow in this computation is at Wo = 1, the ow can be considered quasi-steady.

The results of the steady computation can provide insight into the uid dynamics of the unsteady

case. The addition of the plenum to the straight pipe altered the ow characteristics of the steady

computation because the ow was no longer independent of axial position. Proximity to the

68

interface between the pipe and the plenum aected the ow due to losses at the inlet and entry

length eects.

The results of the unsteady computation conrms the insights gained from the steady compu-

tation. Examination of the rst Fourier velocity mode in Figure 5.15 shows that the magnitude

retains a parabolic shape, yet the maximum value does not reach the analytical value. This is the

case because the inspiration prole, while maintaining the parabolic shape, does not attain the

analytical value along most of the radius. The phase lag is a full degree lower than the analytical

value as well. The ow is fully developed for most of the pipe. The dierence in error between

the data sampled at 0.25*L, 0.50*L, and 0.75*L is small: a maximum dierence of 2% for the

magnitude and 0.35% for the phase lag. The velocity prole is heavily aected at the interface

between the pipe and plenum. The shape of the Fourier magnitude curve suggests an unsteady ve-

locity prole that is blunter than the analytical solution, which is consistent with a still-developing

velocity prole. Accepting that the analytical value is not achievable for this type of computation,

the phase lag at the interface is consistent with the fully-developed phase lag, except near the wall.

The sharp corner of the interface has a greater eect closer to the wall than at the centerline,

skewing both the magnitude and phase lag comparison to the ow further along the pipe. The

grid is also coarse which induced higher numerical error into the computation than was seen in the

grid study computations.

The wallGradU function was used to calculate the velocity gradient and mathematically ma-

nipulated into the wall shear stress. Figure 5.16 further conrms that the exhalation part of the

period is very close to the analytical solution, but the inhalation shear stress falls short. As the

values conform to each other, except at the interface, the conclusion can be reached that the lower

inhalation velocities are not a function of distance to the plenum, but in losses to the ow from

the interface edge.

The nal gures of the plenum computation (Figure 5.17) capture the velocity proles at

twenty discrete times during the period. Each graph shows the velocity at the four data locations

and the analytical solution at 90 degrees to each other. This was done to further elucidate the

characteristics of the ow; specically, the lower velocities during the inhalation and the leading

edge eects at the interface. While the coarseness of the grid cannot be neglected, the gures

conrm that the addition of the plenum changes the overall uid dynamics of the system and the

axial dependence of the ow must be taken into account.

69

(a) Fourier velocity magnitude (b) Fourier velocity magnitude error

(c) Fourier velocity phase lag (d) Fourier velocity phase lag error

Figure 5.15: Fourier velocity magnitude and phase lag data sampled at four locations along thepipe, Wo = 1. Error is dened as the dierence to the analytical solution.

70

Figure 5.16: Wall shear stress compared to the analytical shear stress at twenty instances acrossone period.

71

Figure 5.17: Velocity proles at every t = T20 seconds at four locations along the pipe. The dash-

dot-dot line is the earliest prole in the phase, followed by the long-dash line second, the dash-dotline third, and the solid line is the last prole of the phase in each gure. The inhalation errorsincrease as the velocity magnitude does. The prole at L=100*D is most aected by the proximityto the plenum.

72

Chapter 6

Conclusions and Recommendations

for Future Work

The purpose of the current work was to explore steady and pulsatile laminar ow CFD. A grid

study discerned the resolution requirements needed to accurately capture the unique characteristics

of pulsatile pipe ow over a range of Womersley numbers. A topology study evaluated the eect of

alternate grid structures at the same Womersley range to discover the eectiveness of alternate grid

types. A simplied unsteady situation resembling respiration within a straight pipe/plenum system

was modeled to understand some of the underlying physics and computational challenges inherent

in canine olfaction CFD analysis. Validation of OpenFOAM in regards to the ows considered and

overall usability was adjunctly done.

The results of the grid study showed that accuracy increased upon grid renement. At a low

Womersley number, when the ow was quasi-steady and velocity parabolic, each grid computed

the velocity with high accuracy and localized cell distribution near the wall did not signicantly

impact the solution. As the pulsatile frequency increased, a Stokes Layer was induced that altered

the overall dynamics of the system: the overshoot Fourier magnitude descended towards the wall

and grew in phase lag dierence to the centerline ow, which became more like plug ow. Local

renement in the Stokes Layer region increased the accuracy of the computation but the same

renement in the centerline ow had negligible impact on error and increased the computational

cost.

The comparison of dierent grid topologies (fully structured hex mesh, fully unstructured tet

73

mesh, and combination hybrid mesh) provided similar insight into the unsteady ow computations.

Each was successfully able to model the unsteady ow. Neither hex-dominant nor tet-dominant

grids possessed a clear advantage in that regard, but the unstructured grids were less successful in

resolving the location of maximum Fourier magnitude when it was within an unstructured region.

For the ows without a Stokes Layer, the unstructured grid consistently had higher accuracy in the

global and radial parameters studied. The general dierence between the unstructured grid and

the other grids was the lack of cell clustering near the wall and the highest overall grid density. So

gridding eorts for computations of ows at Wo = 1, 3 can yield the highest accuracy with a focus

on overall grid density and can disregard near wall localization. However, this was computed on a

straight pipe and may not hold valid on geometries that induce secondary ows, which was outside

of the scope of the current work. At ows with a Stokes Layer, Wo = 9, 27, the unstructured

grid did not perform with the highest accuracy even with the highest grid density. The structured

and hybrid grid had structured near wall resolution which proved to enhance accuracy in resolving

the Stokes Layer ow. The hybrid grid had higher accuracy in resolving the dynamics of the Stokes

Layer which suggests that an axisymmetrical approach to near wall resolution will result in better

performance, all else being equal. There was negligible dierence in accuracy in the centerline

ows, for all Wo, so grid type can be preferential away from the wall. Unstructured gridding does

provide the advantage of coarsening the inner core of a pipe without sacricing circumferential

grid resolution at the wall, leading to lower computational cost with comparable accuracy. While

the unsuccessful application of the cyclic boundary conditions to the hybrid grid did not aect the

computational cost, it can prove a detriment to ows where cyclic boundaries can simplify a grid.

The plenum computation showed how the ow was altered with the addition of an ambient

external environment to the straight pipe. The model was geometrically simpler than the dog's

nose CFD but modeled the underlying aerodynamics of canine olfaction. The computation showed

that the velocity could achieve the analytical magnitude on the exhalation phase of the period,

but the sharp inlet at the interface created losses that decreased the overall magnitude but still

maintained the parabolic shape of the analytical solution. The ow in the pipe had a developing

length of several diameters from the interface in which the velocity proles diered from further

downstream. At 25 diameters the ow was fully developed. The unsteadiness of the ow at the

interface occurred during both the inspiration and expiration phases of the respiration cycle. The

grid was also very coarse and so the results, especially at the interface, have considerable numerical

error.

74

The nal goal was to validate OpenFOAM in regards to the steady and unsteady ows under

consideration and evaluate its usability from pre-processing grids to post-processing the computed

data. As an open source code, it has no commercial licensing fees and the code itself is transparent

and adaptable, evident by the creation of channelIcoFoam and the Fourier implementation to the

unsteady solvers. OpenFOAM had the necessary boundary conditions available and proved capable

of solving the computations. The gridding utility, blockMesh, is crude but capable for simple

geometries. The diculty in adapting outside grids to implement the cyclic boundary condition

is a detraction, though if cyclic boundaries are not needed then it can adapt a grid from many

dierent commercial CFD formats. The broad range of solvers provides very diverse capabilities

in solving a myriad of CFD problems, though only steady and unsteady laminar solvers have

been validated in this study. The post-processing utilities are competent, from data manipulation

(wallGradU) to format conversion (i.e. foamToTecplot, foamToEnsight). The sampling utility

proved to expedite much of the post-processing time.

There are several areas of the current work where future exploration can expand on the knowl-

edge gained. The same computations can be performed on an unstructured grid that has near wall

renement and a grid created with Harpoon, which was the gridding software used for the canine

olfaction CFD. The usability of OpenFOAM-1.5's automatic mesh generator, snappyHexMesh, can

be evaluated. Pulsatile ow CFD can be performed on a bifurcating pipe to study the aerodynam-

ics of more complicated geometry. A grid study of the plenum computation should be performed

as the plenum grid was very coarse. A method to implement the cyclic boundary condition on

any grid could be developed. The dog's nose grid could be implemented into OpenFOAM and

compared to the previous CFD calculations. A 3D methodology could be developed from Craven's

1D scalar transport study [4] and implemented within OpenFOAM to study vapor transport in

the dog's nose.

75

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