A Three-dimensional model of pulsatile flow at an arterial bifurcation

Rochester Institute of Technology

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5-1-1996

A Three-dimensional model of pulsatile flow at an

arterial bifurcation

Andrew Hayes

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Recommended Citation

A Three-Dimensional Model of Pulsatile Flow

at an Arterial Bifurcation

by

Andrew R. Hayes

A Thesis Submitted

m

Partial Fulfillment

of the

Requirements for the Degree of

MASTER OF SCIENCE

IN

tvffiCHANICAL ENGINEERING

Approved by:

Professor s.

Kandlikar

(Thesis Advisor)

Professor

Alan Nye

Professor

P. Marletkar

Professor

Charles Haines

(Department Head)

DEPARTMENT OF tvffiCHANICAL ENGINEERING

COLLEGE OF ENGINEERING

ROCHESTER INSTITUTE OF TECHNOLOGY

Abstract

Atherosclerosis in

known

to

form

at sites ofpredilection.

These

sites are

typically

areas of reduced pressure or shear stress.

In

the

thesis a

link between

hemodynamics

and atherosclerosis

formation

will

be

investigated. In

the

study

three-dimensional

pulsatile

flow

at a simplified carotid

artery bifurcation

has

been

modeled with computational

fluid

dynamics

(CFD).

A

flow

visualization

experiment

has

also

been

performed on a glass replica of

the

bifurcation

to

verify

the

CFD

results.

An

intense literature

review

has

been

included in

the

thesis

to

Title of Thesis:

..A Three-dimensional Model of Pulsatile Flow at an Arterial Bifurcation."

I, Andrew R. Hayes, hereby grant pennission to the Wallace Memorial Library of

Rochester Institute of Technology to reproduce my thesis in whole or in part. Such

reproduction shall not be for commercial use or profit.

Date: May 1996

Andrew R. Hayes

Forward

I

would

like

to

express

my

appreciation and

thanks to the

following

people:

Dr. Kandlikar for

all of

his

help, inspiration,

and motivation

during

this

workand

throughout

my

yearsat

RIT,

Christine for her

support and

caring

when

I

wasunable

to

give

her

the

time,

Preston

Blay

for his

help

and alternativeviews on

things,

and

my

parents,

Paul

and

Karen,

who

have

supported and

helped

me

through

my

college career.

Table

of

Contents

List

of

Figures

vii

List

of

Tables

xi

1

.

Introduction

1

1

.

1

Heart

Disease

1

2.

Literature

Review

2

2. 1

General

2

2.1.1 Pulsatile Flow

7

2.1.2

Anatomy

8

2

.

2 Numerical

Modeling

1

0

2.3 Experimental Investigations

14

2.4

Importance

of

Study

20

2.5

Objectives

21

3. Theoretical

22

3.1

Computational Fluid Dynamics

22

3

.2

Computational

Fluid

Dynamics Model

23

3.3

Assumptions

25

3.4

Boundary

Conditions

28

3.5

Numerical Validation Analysis

29

4. Experimental Investigation

37

4.2

Experimental Procedure

41

5.

Results

and

Discussion

44

5.1 General

44

5.2 Theoretical

Results for

Clean Model

45

5.3 Theoretical

Results

for

Blockage

in

Model

84

5.4 Experimental

Results

for

Clean Model

1 19

5

.

5 Experimental

Results

for

Blockage in Model

1 25

5.6 Comparison

131

6. Conclusion

134

References

136

Appendix

138

A. CFDS Command File

139

B. CFDS FORTRAN Subroutine

142

List

of

Figures:

2.1.2 I

Anatomy

of

the

Large

Arteries

2.3. 1

Bharadvaj

et al.

(1982)

Vortex Flow Pattern

3.2.1 CFD

Grid Distribution

3.5.1

90

Tee-junction

3.5.2

Case

1,

CFD

Results

3.5.3

Case

1,

Work

of

Karino,

et al.

3.5.4

Case

2,

CFD

Results

3.5.5

Case

2,

Work

of

Karino,

et al.

4.1.1

Experimental

Setup

5.2.1 Cardiac Cycle

5.2.2

Centerline

Velocity

Vectors

at

0.149s

5.2.3

Centerline

Velocity

Vectors

at

0.186s

5.2.4

Centerline

Velocity

Vectors

at

0. 189s

5.2.5

Centerline

Velocity

Vectors

at

0.191s

5.2.6

Centerline

Velocity

Vectors

at

0.196s

5.2.7

Centerline

Velocity

Vectors

at

0. 199s

5.2.8

Centerline

Velocity

Vectors

at

0.200s

5.2.9

Centerline

Velocity

Vectors

at

0.203s

5.2. 10

Centerline

Velocity

Vectors

at

0.209s

5.2.

1 1

Centerline

Velocity

Vectors

at

0.212s

5.2.13

Centerline

Velocity

Vectors

at

0.230s

5.2.

14

Centerline

Velocity

Vectors

at

0.255s

5.2. 15

Centerline

Velocity

Vectors

at

0.257s

5.2.16

Centerline

Velocity

Vectors

at

0.263s

5.2.17

Centerline

Velocity

Vectors

at

0.268s

5.2.18

Centerline

Velocity

Vectors

at

0.278s

5.2.19

Centerline

Velocity

Vectors

at

0.281s

5.2.20

Centerline

Velocity

Vectors

at

0.290s

5.2.21

Centerline

Velocity

Vectors

at

0.293s

5.2.22

Centerline

Velocity

Vectors

at

0.302s

5.2.23

Centerline

Velocity

Vectors

at

0.308s

5.2.24

Centerline

Velocity

Vectors

at

0.365s

5.2.25

Centerline

Velocity

Vectors

at

0.383s

5.2.26

Centerline

Velocity

Vectors

at

0.392s

5.2.27

Centerline

Velocity

Vectors

at

0.395s

5.2.28

Centerline

Velocity

Vectors

at

0.410s

5.2.29

Grid

Distribution

with

Sites labeled

5.2.30 Shear

Stress

Plot

at

Site

1

5.2.3 1 Shear

Stress

Plot

at

Site

2

5.3.1 Centerline

Velocity

Vectors

with

Blockage

at

0. 138s

5.3.2 Centerline

Velocity

Vectors

with

Blockage

at

0. 165s

5.3.3 Centerline

Velocity

Vectors

with

Blockage

at

0. 180s

5.3.4

Centerline

Velocity

Vectors

with

Blockage

at

0. 192s

5.3.5

Centerline

Velocity

Vectors

with

Blockage

at

0. 195s

5.3.6

Centerline

Velocity

Vectors

with

Blockage

at

0. 198s

5.3.7

Centerline

Velocity

Vectors

with

Blockage

at

0.201s

5.3.8

Centerline

Velocity

Vectors

with

Blockage

at

0.210s

5.3.9

Centerline

Velocity

Vectors

with

Blockage

at

0.213s

5.3.10

Centerline

Velocity

Vectors

with

Blockage

at

0.216s

5.3. 1 1

Centerline

Velocity

Vectors

with

Blockage

at

0.219s

5.3.12

Centerline

Velocity

Vectors

with

Blockage

at

0.222s

5.3.13

Centerline

Velocity

Vectors

with

Blockage

at

0.225s

5.3.

14

Centerline

Velocity

Vectors

with

Blockage

at

0.228s

5.3.15

Centerline

Velocity

Vectors

with

Blockage

at

0.23 Is

5.3. 16

Centerline

Velocity

Vectors

with

Blockage

at

0.234s

5.3. 17

Centerline

Velocity

Vectors

with

Blockage

at

0.239s

5.3.18

Centerline

Velocity

Vectors

with

Blockage

at

0.250s

5.3. 19

Centerline

Velocity

Vectors

with

Blockage

at

0.252s

5.3.20

Centerline

Velocity

Vectors

with

Blockage

at

0.254s

5.3.21

Centerline

Velocity

Vectors

with

Blockage

at

0.258s

5.3.22

Centerline

Velocity

Vectors

with

Blockage

at

0.261s

5.3.23

Centerline

Velocity

Vectors

with

Blockage

at

0.273s

5.3.24

Centerline

Velocity

Vectors

with

Blockage

at

0.279s

5.3.26

Centerline

Velocity

Vectors

with

Blockage

at

0.363s

5.3.27

Shear Stress Plot

with

Occlusion

at

Site 1

5.3.28 Shear

Stress

Plot

with

Occlusion

at

Site 2

5.4.1 Experiment Video

Capture

I

5.4.2 Experiment

Video

Capture

2

5.4.3 Experiment Video

Capture

3

5.4.4 Experiment Video

Capture

4

5.5.1 Experiment Video

Capture,

Blockage 1

5.5.2 Experiment

Video

Capture,

Blockage 2

5.5.3 Experiment Video

Capture,

Blockage 3

List

of

Tables:

I

Introduction

l.l

Heart

Disease

Heart disease

is

the

leading

cause of

death in

the

world today.

In

a resent article

the

Wall

Street Journal

reported

that

150,000 Americans

die

annually

from

stroke,

a

derivative

of

heart disease. Some

75%

of

those

deaths

are related

to

arterial occlusive

disease,

or atherosclerosis.

Atherosclerosis is

a variation of atheroma

in

which

the

arterial

walls thicken and

harden. Atherosclerosis

may

initiate

as

early

as

the

late

teens

or

early

twenties.

The

disease

propagates at

different

rates

in different

people,

depending

on numerous variables and

risk factors.

Some

of

these

risk factors

are

age,

heredity,

occupation,

general

health,

sex,

etc.

All

the

risk

factors

do

not

have

to

be

present

concurrently,

and no one combination of

factors

is

more

detrimental

than another.

Generally,

there

is

no

way

to

detect

the

state of

2

Literature Review

The

purpose of

this

section

is

twofold.

The

first

goal

is

to

develop

an

understanding

of atherosclerosis and the possible

relationship between

disease

initiation

and

hemodynamics. The

second goal ofthe section

is

to review some ofthe

important

published studies

in

this

area.

The

focus

of

this

review will

be

on two-dimensional and

three-dimensional

modeling

studies.

Both

analytical and experimental studies will

be

reviewed.

2.1 General

In

this section,

early

papers on

blood flow

and

its

possible effect on

atherosclerosis will

be

reviewed.

The

following

papers present

this

information in

a

very

detailed

manner and are reviewed

here

in

somewhat greater

detail.

-

Imparato

_{et al.}

(1979)

-Sigmundet.al.

(1964)

-Fry

(1969)

-Texon(1970) -

Nerem

and

Cornhill

(

1 98 1

)

-

Liepsch

and

Moravec

(1984)

-

Harison

and

Marshall

(1983)

Atherosclerosis

is

the

fatty

degeneration

of

the

large

arteries.

At

the

earliest stage

of

the

disease,

fatty

plaques,

or

deposits,

form in

the

vessel wall.

These

plaques

then

progressively

become

more pronounced and

interfere

with

the

blood flow. The inner layer

of

the

artery,

the

endothelial

layer,

is

enamciated or worn

through, followed

by

the

deposition

of cholesterol and glycerides

from

the

blood

stream

to the

exposed area.

As

the

reduced.

The

effects of

the

reduction

in blood

supply

are

dependent

on

the

particular

organ or region of

the

body

the

artery

supplieswith

blood.

Imparato

et. al.

(1979)

reported on

the

locations

and characteristics of plaques

found

in

the

carotid

bifurcation. The

review analyzed symptoms

in

various patients and

lesions

removed

from

their carotid

bifurcations.

The

plaques were

found

to

vary

considerably

from

patient to patient.

However,

two

characteristics were seen

in

all

the

cases.

The

typical

extent of

involvement

of

the

distal

common and proximal

internal

and

external carotid arteries was shared

by

most plaques.

All

the

plaques

had

some amount of

fibrous

tissue

in

their

structure.

Other

than

these two characteristics, the

plaques were

largely

different in

structure, appearance,

and

formation.

Thrombus

was

found

to

occur

in

a

variety

ofconditions and was either

gray, pink,

or red.

Gray

or pink

thrombus

was

found

to

form

while

blood flow

was still rapid and was

initiated

by

the

adherence of platelets

to the

arterialwall.

The

red clot was

found distal

and

proximal

to

areas of marked stenosis or

in

association with

totally

occluded arteries and

was

interpreted

by

the

authors as

having

occurred where

blood flow had

either ceased or

become

extremely

sluggish.

The

author

further discovered

that

a

variety

of

seemingly

unrelated pathologic events

may

contribute

to the

occurrence of stenosis of

the

carotid

arteries with

the

area of

the

carotid

bulb

and

bifurcation

being

most

commonly

involved.

The

link

to the

study

is

the

fibrotic

plaqueat

the

carotid

bifurcation,

which

may

develop

in

response

to

hemodynamics factors.

Over

the

past

25

years,

a

tremendous

amount of work

has been done in

the

area of

atherosclerosisand

the

possible

link between

hemodynamics

and

the

disease.

The

earliest

studies were

distributed into

two

groups.

The

division between

these

groups arouse

from

their

respective views on

the

mechanism ofplaque

deposition

and

location.

One group

the regions ofthe aorta where shear stress was reduced.

The

other

group focused

on

high

shear regions.

Fry

(1968)

demonstrated

that

high

shear stress would

damage

the

endothelium.

Sigmund

et al.

(1964)

investigated

the

link

between

turbulence and the

distribution

of atherosclerotic

lesions.

They

found

turbulence

atall

the

sites ofpredilection

for

atherosclerosis.

The determination

of sites of predilection was made through arterial

imaging

studies of

disease

victims.

Two

possible mechanisms

for

the

location

of

atherosclerotic

lesions

to

be determined

by

the

presence of

turbulence

were

introduced.

The

first

related

to

high-frequency

vibration

possibly

having

a

destructive

effect on

the

arterial wall.

This

kind

of

injury

was suggested

to

lead

to

a

type

of

healing

in

the

arterial

tissue

similar

to

atherosclerotic

lesion. The

second mechanism was

the

concentration of

increased

lateral

wall pressure

in

certain areas caused

by

the

presence of

turbulence.

In

the

areas of

increased

pressure atherogenesis

is known

to

be

accelerated.

These

areas

may

create a critical

injury

or retard

the

secretionof

lipids

synthesized

in

the

arterial wall.

The

effects of

the

injury

may

result

in

a

local

accumulation of

lipids,

described

as

the

first

phase

of

the

pathogeneses of atherosclerosis.

The

relations

between

turbulence

and

atherosclerosis were

found

to

be

the

strongest of

any

of

the

earler

hemodynamics factors

investigated.

Fry

(1969)

carried out experiments in-vivo

to

determine

the

effects of

high

shear

stress on

the

endothelial

layer. The

endothelial cells were seen

to

realign

themselves

in

the

direction

of

flow

after

the

cell orientation was altered.

At higher levels

of shear stress

the

cells

may

erode,

whereas at

lower

levels

the

permeability

of

the

endothelium

increased.

Fry

suggests a critical yield stress value

to

quantify

these

effects

(see

figure).

Below

the

critical yield stress value

the

endothelialcells remain

histologically

normal.

At

orabove

the

shear,

Fry

saw an enhancement of protein transport with

increasing

shear stress.

As

the

shear stress was

further

increased,

the erosion

level

is

reached and

denudation

of the

endothelium

invasion

and

deposition

of

lipid

material,

adherence of cellular

elements,

and

deposition

of

fibrin

occur.

Fry

quotesa value

for

the

critical yield stress of

400

dynes/cm2

and a value of

1000

dynes/cm2

for

the

erosion stress.

These

values are

only

representative

due

to the

numerous

factors

controlling

the

chemical and physical environments of the

endothelium.

Texon

(1983)

discussed

the

possible mechanisms of atherogenesis.

Texon

suggested a pressure

differential

leading

to

deposition.

The

area of

localized decreased

static pressure creates a

local

suction on

the

artery

wall.

The

suction results

from

an

imbalance between

the

constant outside pressure and

the

decreasing

internal

pressure of

the

wall.

The

intima

of

the

vessel

is lifted

or raised

due

to the

flowing

blood

upon

the

endothelium.

This

area represents

the

area of

disease impetus.

Texon

calls

the

response a "

biologic

change,

a reparative process or reactive

thickening

due

to the

proliferation of

endothelial cellsand

fibroblasts

from

the

subjacentlayers."

Nerem

and

Cornhill

(1980)

reviewed

the

interaction

of

blood flow

with

the

arterial

wall.

They

describe

the

normal pressure

force acting

on

the

wall as

playing

a

double

role.

The

first

role arises

from

the

hydrostatic

pressure

being

higher

within

the

lumen

of an

artery

than

at

its

outer surface.

There

exists a potential

to

drive

a

bulk flow

across

the

endotheliumand

through the

wall.

This

transfer

will

be

determined

by

the

magnitude of

the

pressure

difference

and

the

resistance offered

by

the

arterial wall.

The

second role

Nerem

and

Cornhill

discussed

was

the

pressure reaction with

the

arterial wall

producing

distension

or

stretching

of

the

wall.

This

will produce stresses within

the

wall

that

may

At

the time

oftheir

work,

Nerem

and

Cornhill

(1980)

concluded that

there

was an

insufficient

amount of evidence

to

link hemodynamics

and

localization

of

lesions.

They

did

highlight

the

fact

that

in

all

the

investigations

regions ofarterial

branching

were

identified

as areas where

early

lesions

occur.

However,

the

exact

location

of

the

initial lesion

development

within

that

region remained _unresolved, with

contradictory

results

from

many

studies.

The

unknown

fluid dynamics

of

blood flow in

these

areas

is believed

to

be

partially

responsible

for

the

incomplete findings.

Another

complication

to

the

problem

is

brought

about

by

the

extremevariation

in

the

geometry

of

the

artery

and

the

bifurcation

region

from

person

to

person.

The flow

pattern

in

one example can not

be

assumed

representative of

the

flow

pattern

in

multiple cases.

Liepsch

and

Moravec

(1984)

found both high

and

low

shear stresses

to

be

responsible

for

the

adhesion and

deposition

ofplatelets and

lipids. The

findings illustrated

the

process of

initial blood

cells

damage

or surface alteration

in

a

high

shear

field.

The

particles

then

have increased

adhesionwith

the

arterial wall and

begin

to

stick

to

the wall

and

form deposits in low

shear regions.

Presently,

low

and

high

shear stresses are proven

to

play

a role

in

atherosclerosis.

High

shear stress

has been

proven

to

play

a role

in

initial depostion.

In

the

areas of

high

shear

stress,

plaque

formations have been discovered

at

the

early

stagesof

arterial

deposition

(Nerem

et.

al.,

1993).

The

body

removes

these

plaques

through

natural

processes similar

to

mechanicalcleaning.

At

a

latter

stage

in

the

disease,

with

the

high

shear regions

clean,

deposits form in

the

low

shear stress regions.

Cell

damage

from

the

high

shear regioncreates an

increase in

the

adhesive properties of

the

cell.

These low

2.1.1 Pulsatile Flow

Atherosclerosis is

a

disease

of

the

large

arteriesproximal

to the

heart.

Blood flow

in

the

large

arteries can not

be described

as

laminar

with a parabolic

velocity

profile.

Instead,

the

flow is

characterized

by

a

variety

of complexities.

These

complexities

include

the

presence of asymmetric

velocity

patterns

due

to

vessel

geometries,

turbulent or

transitional

flow,

secondary

flow

motions,

and

flow

separation.

In

the

large

arteriesthe

flow

of

blood is

pulsatile,

or

cyclic,

due

to

the

function

of

the

heart.

As

the

left

ventricle of

the

heart

contracts

it

ejects

blood

at amaximum

velocity

into

the

aorta.

At

the

end of

the

blood

ejection phase

the

ventricle pressure

is

reduced.

The

aortic pressure

is

then

greaterthan

the

ventricle

pressure,

allowing blood

in

the

aorta

to

flow

back

into

the

left

ventricle.

As

the

pressure

in

the

ventricle

increases from

the

influx

of

blood,

the

valve

between

the

ventricle and

the

aorta closes.

The

valve

closing

creates a

higher

pressure at

the

base

of

the aorta,

pumping

more

blood

away from

the

heart

and

into

the

systemic circulation system.

As

the

pressure

decreases

at

the

base

of

the

aorta, there

is

anotherperiod of reverse

flow in

the

system.

Finally,

as

the

pressure reaches

2.1.2

Anatomy

In

the

thesis,

blood

flow

through

the

carotid

artery bifurcation

is

analyzed.

The

carotid arteries are

the

major suppliers of oxygenated

blood

to the

brain

and

head.

There

are

two

carotid arteries

in

the

body,

located

symmetrically

about the

neck,

Figure

2.1.2.1.

The

carotid arteries

begin

at

roughly

the

same

location

on the

aorta,

continue

to

the

middle of

the neck,

and split

into

the

internal

and external carotids at

the

carotid

bifurcation.

The

internal

carotid

artery

supplies

blood

to the

brain

while

the

external

carotid

artery

supplies

blood

to the

face

and scalp.

An

occlusion

in

either carotid

artery

is

commonly

known

as

apoplexy,

or stroke.

In

a stroke

the

supply

of oxygenated

blood is

reduced or

halted,

resulting

in

a sudden

loss

of

consciousness,

sensation and

voluntary

movement.

The

severity

of

the

stroke

depends

on

the

period of

time

an organ

is deprived

of oxygen.

In

a stroke of

the

external carotid

artery,

depending

on

the severity, the

victim

may

suffer minimal side

effects,

partial

paralysis of

the

face,

or,

in

the

extreme

case,

mortality.

An

occlusion

in

the

internal

carotid

artery

decreases

or eliminates

blood

supply

to the

brain,

resulting in

a range of side effects

from

partialparalysis ofthe

body

to

,

in

the

most extreme

case,

mortality.

Harrison

and

Marshall

(1983)

performed

autopsy

studies

to

determine

an average

geometry

of

the

carotid arteries and

bifurcation.

The

common carotid

artery

was

determined

to

be

7.6

mm

(1.64 mm) inner

diameter. The

internal

carotid

artery

was

determined

to

be

5. 1

mm

(1.1 mm) inner diameter

with

the

sinus

being

8.3

mm

(

1

.95

mm) inner

diameter. The

angle of

bifurcation,

or

the

angle

dividing

the

internal

and

external

carotids,

was

found

to

be

36.4

_

Internal

Carotid

External

Carotid

Common Carotid

Artery

Aorta

2.2

Numerical

Modeling

In

thissectionthe

following

papers onnumerical

modeling

oftwo-dimensional and

three-dimensional

bifurcations

will

be

reviewed:

-

Fernandez

et al.

(1976)

-Chang and

Tarbell

(1985)

-Rindtetal.

(1987)

-Rindtetal.

(1990)

-Nazemietal.

(1990)

-

Baaijens

_{et al.}

(1993)

Two-dimensional

and

three-dimensional

models

have been

investigated.

The

two-dimensional

models

have

given

limited

insight into

the

problem

because

of

their

simplifications of

the

flow field. At branches

or curves

in

the

flow

field,

secondary

flow is

created.

These secondary

flows

are not present

in

a

two-dimensional

model.

In

a

three-dimensional

model

the

flow field is

better

simulated,

with

the

entire phenomena present.

Fernandez

et al.

(1976)

studied

the

pulsatile

flow

of a

Newtonian,

incompressible

fluid

through

a

two-dimensional

symmetric

bifurcation. The

authors utilized

the

Marker

and

Cell

numerical

technique to

solve

the

equations of

continuity

and motion.

The

results of the

study

show recirculation at

the

nondivider wall

in

the

daughter

branches

and

high

shear stress values on

the

divider

walls.

The

position of

the

high

shear stress values was not changed with

the

pulsatile

flow

pattern.

Due

to the

two-dimensional simplification of

the study,

no

secondary flow

was

determined,

and

thus

no zones ofstagnationwerecalculated.

The

authors

determined

the

presence and

disappearance

of separation at

the

nondivider

wall as

the

pressure

forcing

function

goes through

the

cycle.

As

the

Reynolds

number

increases,

the recirculation area

increases

in

size.

As

the

Reynolds

number

decreases,

the

recirculation zone

decreases.

The

pressure values around

the

stagnation point on

the

nondivider wall were

determined

to

be

1.7

times the

pressure values at

the

boundary. The

authors note that this

pressure

difference

could

be

of

importance in

evaluating

the

impaction

mechanism

for

wall

damage.

Variable

shear stress values were also calculated

for

this

area of

the

daughter

branch,

with

high

shear stress

found

on

the

divider

wall.

To

further

understand

the

flow field development in

the

carotid

artery bifurcation

region,

a

review on pulsatile

flow

through

curved

tubes

is included.

Chang

and

Tarbell

(1985)

employed

numerical methods

to

model pulsatile

flow

through

a curved tube.

The

authors

focused

on

the

secondary flow

field

at

the

curvature section.

The

results of

the

study

show

the

presence of

up

to

seven vortices

in

the

secondary

flow

structure at

the

curve.

These

vortex patterns show vortices

within

vortices,

representing

a

highly

complex

flow field

at

the

curved section of

the

tube.

In

the

secondary

flow

areas

the

authors calculated

the

shear stress values

to

be

equal or

greater

than the

shear stress values

for

the

axial component of

velocity

for

the

flow field.

In

pulsatile

flow

the

peakwall shear stress

is

at

the

inside

wall of

the curve,

with

the

highest

shear

stress

occurring

at

the

outside wall

for

steady

flow.

In

the

pulsatile

flow

study

reverse

flow

was

calculated at

the

inside

wall of

the curve,

implying

an area of recirculation

is

present

for

some

duration

of

the

cycle.

Rindt

et al.

(1987)

studied a

two-dimensional,

rigid

model of

the

carotid

bifurcation.

In

the

study

the

authors performedexperimentalanalysis aswell as numerical.

A

numerical method

base

on

Galerkin's finite

element method was utilized

to

solve

the

continuity

and

Navier-Stokes

equation.

In

the

study

the

authorsused a sinusoidal pulse

to

model

the

pulsatile

inlet

flow,

where

the

Reynolds

number varied

from

250

to

770.

The

authors

determined

high

velocity

gradients

were present

along

the

divider

wallsofthe

daughter

branches

and

large

zonesof reverse

flow

near

the

nondivider walls of

these

branches. The

extent of

the

reverse

flow

region varied

strongly

with

time.

There

was

high

shear stress at

the

divider

wall

throughout the

entire

cycle,

with

low

and

fluctuating

shear stress on

the

nondividerwall

in

the

carotid sinus.

Nazemi

et al.

(1990)

solved

the

Navier-Stokes

equations

for

pulsatile

laminar flow in

a

two-dimensional

rigid bifurcation.

The

authors

included

the

carotid sinus

in

their carotid

bifurcation

model.

The

numerical model was able

to

create

deposits in

the

area where

the

local

shear stress values

fell

within a predetermined critical range.

The

results of

the

study

show

recirculation zones at

the

divider

walls,

in

the

early

stages of

the cycle,

and at

the

nondivider walls

at

the

latter

stages.

In

the

region of

the

carotid sinus

the

authors

find

separation

in

the

acceleration

phase of

the cycle,

and recirculationwith

the

deceleration

of

the

inlet

velocity.

The

plaque

deposition

model

in

the

study

found

3

possible sites

for

deposition,

the

sinus at

the

upper

wall,

on

the

lower

nondivider

wall,

and

the

upper

divider

wallnear

the

end of

the

sinus.

The

authors compared

their

numerical

deposition

regions with actual

human

angiograms and

found

good agreement

between

the two.

Baaijens,

et al.

(1993)

performed a numerical analysis on

the

steady

flow

of

Newtonian

and non-Newtonian

fluid in

a two-dimensional model of

the

carotid

bifurcation.

In

the

study

the

authors employed a

finite

element model

to

solve

the

Navier-Stokes

equations.

The Reynolds

number was set equal

to

300,

the

typical value at

the

end of

the

diastolic

phase of

the

cardiac

cycle.

With

a

Newtonian

fluid

as

the

working

medium

the

authors

found

the

velocity

pattern

in

authors

determined

the

presence ofreverse

flow.

The secondary

velocities

in

the

region are an

order of magnitude greater than

the

secondary

velocities

in

the

main

branch. When

the authors

repeated

the

calculations with

the

non-Newtonian

fluid

criteria

the

same

velocity

pattern was

determined

in

the model,

but

the

magnitudes of

the

Velocity,

secondary

velocities, and

the

shear

stresses were

5

to

10

%

less,

indicating

the

Newtonian

assumption

introduces

on minimal error

into

the

calculation.

In

the

external carotid

branch,

branch

without

sinus, the

authors

found

no reversal

in

flow.

Negative secondary

velocities were

found

in

the

region.

Low

shear stresses were also calculated

in

the

external carotid.

For

the

calculations with non-Newtonian

fluid,

the

velocity

values were

again

determined

to

be

slightly

lower

than the

Newtonian

model,

with

the

pressure values

being

slightly higher

in

the

non-Newtonian model.

The

authors

determined for large

valuesof

the

length

divided

by

radius ratio

the

differences

between

the two

models

decrease.

The

authors summarize

the

study

by

stating

that the

Newtonian

fluid

model gives good

results.

The

velocity

magnitudes are

slightly higher

than

that

for

non-Newtonian

fluid,

but

the

velocity

patterns are similar.

The Newtonian

model also yields

slightly

larger

recirculation zones

than

those

for

non-Newtonian models.

These

statements are validated

by

comparison with

previously

published experimental studies.

2.3

Experimental

Investigations

In

this

section,

early

papers on experimental

investigations

of

blood

flow

will

be

reviewed.

The

following

papers represent an overview oftheprevious work performed and are reviewed

in

detail.

-

Fukushima

and

Azuma

(1982)

-

Motomiya

and

Karino(1984)

-Bharadvaj

et al.

(1982)

-

Fukushima

et

al.(

1987)

-

Singh

etal.

(1990)

-Liepsh

and

Moravec

(1984)

-Rindtetal.

(1988)

-

Walburn

_and

Stein

(1982)

-Rindtetal.

(1988)

Fukushima

and

Azuma

(1982)

investigated

the

secondary flow fields

generated

in

models

of

stenosis,

bifurcation,

and

branchings due

to

a

steady flow.

The

authors

found

that

fluid

particles

flowing

in

the

center of

the

main

tube

near

the

wall

turn

in

the

vicinity

of

the

apex of

the

bifurcation

and

travel

into

the

branches

swirling

spirally.

The

highest

pressure

is found

at

the

apex

of

the

bifurcation.

The resulting

pressure

differential

in

the

region around

the

apex produces a

vortex

flow

much

like

the

horseshoe

vortex

found in flow

over a cylinder.

The

vortex extends

to

the

outer walls of

the

bifurcation

which are

low

pressure regions and generates

swirling flow in

each

daughter branch. The

authors

found

similar vortices

in

flow

through

a

90

degree

Tee-junction.

Motomiya

and

Karino(1984)

investigated

the

flow

patterns and shear rate

distributions

in

branching

arteries with

rigid

walls and compared

the

results

to

clinical studies on atherosclerosis

and

thrombosis.

Due

to

a new method of

blood

vessel preparation which allowed vessel

Exact

flow

patterns and

velocity distributions

were viewed with

the

aid of

flow

particles

flowing

through transparent

blood

vessels.

Steady

flow

wasanalyzed at various

Reynolds

numbers and

flow

rate ratios.

The

authors

found

a recirculation zone at

the

internal

carotid

artery

sinus.

The

formation

and size ofthe zone

was

largely

dependent

on

the

flow

rate

ratio,

with minimal

dependence

on

Reynolds

number.

At

a

flow

rate ratio of

Q\IQq

=

0.65

(Qj

is

the

external carotid

flow

rate,

Qq

the

internal)

a

recirculation zone was

determined

in

the

internal

carotid sinus at

ReQ

=

170.

The

recirculation

zone grew

in

sizewith

increasing

Reg,

but

remained

in

the

sinus.

At

Rec

=

800,

the

recirculation

zone

decreased in

size as a

counter-rotating helical flow developed

downstream

from

the

stagnation point.

In

the

external carotid recirculation was

only

found

when

the

external was

extremely

occluded

(Q\IQq

=>

0.8).

Thus,

a continuous recirculation zone exists

in

the

internal

carotid sinus under normal physiological

flow

conditions,

or

Re0

=

600.

At

the

time,

the

existing

data

onatheroscleroticplaque

formation indicated

the

high

occurrence of

formation in

the

internal

carotid

artery

at

the

sinus.

Further study

reveled

the

plaques

localized

on

the

lateral

and outer

walls of

the

internal

carotid

artery

in

the

region of

flow

separation, reversal,

and

low

wall shear

stress,

emphasizing

the

importance

of

local flow

patterns

in

the

carotid sinus

in

atherogenesis.

Bharadvaj

et. al.

(1982)

studied

steady

flow

through

a generalized carotid

bifurcation.

The

working

model

bifurcation

was

fabricated from

glass with

the

geometry

determined

by

statistical

analysis of angiograms.

Reynolds

numbers of

400,

800, 1200,

and

1400

in

the

common carotid

where studied with

varying flow

rate ratios through

the

daughter branches. The

working

fluid

studied was

water,

with

hydrogen bubbles

utilized

for flow

visualization.

At

a given

flow

rate,

the

authors

determined

the

location

of

the

separation zone moves

upstream with

decreasing

flow

through the

internal

carotid

artery,

while at a

fixed

ratio of

flow

rate

it

moves upstream with

increasing

Reynolds

number.

The

flow from

the

main

tube

impinges

on

the

divider

creating

an area of

high

shear stress.

On

the

nondivider walls

the

flow

decelerates,

as

in flow in

a curved

tube,

and creates

low

shear stress and

secondary

flow.

The

Secondary

motion was

detected in

the

area of a three-dimensional stagnation

line. This secondary

motion,

along

with the axial

flow,

leads

to

helical

paths

for

the streamlines ofthe

flow

field,

see

Figure

2.3.1.

Fukushima

et. al.

(1987)

performed an

in depth

study

onthe

secondary flow in steady

and

pulsatile

flow

through a symmetrical

bifurcation.

The

model

bifurcation

was a

rigid

_walled,

symmetrical glass

Y.

The working fluid for

the

investigation

was water and a water-glycerin

mixture.

To

minimize

the

optical

distortion

of

refraction,

the model was placed

in

a transparent

chamber

filled

withthesame

working

fluid. Dye-injection

and aluminum

particles,

20

micrometers

is

diameter,

were utilized

for

visualization ofthe

flow

patterns.

The

characteristics of

the

steady flow in

the

model

bifurcation

were

found

to

be dependent

on

Reynolds

number and

flow

ratio

between

parent and

daughter

vessels.

The secondary

flow

pattern was seen at

Re0

above

500. Initial

flow

separation occurred at

Re0

of about

1490

when

the

flow

rates

in

both daughter branches

was equal.

With

uneven

flow

rates

in

the

daughter

branches,

separationoccurred at

lower

Re0

Increasing

the

Re0

with uneven

flow

rates made

the

swirling

secondary flow

unstableand produce an almost

turbulent

flow

effect

in

the

model.

Mean

Reynolds

and

Womersley

numbers of

1890

and

20,

305

and

8.9,

and

133

and

5.4

were studied

in

the

pulsatile

flow

analysis.

The

Womersley

number

is defined

by

tube

radius,

fundamental

angular

frequency

of

the pulse,

and

kinematic

viscosity.

The

secondary flow

was

mostly

observed

in

the

decelerating

phase and not

during

acceleration.

The

secondary flow

produced a net reversal

flow from

the

apex

to

the

parent vessel on

the

nondivider wall sides.

A

stagnation point

is

produced when

the

back flow

meets

the

forward

flow in

the

parent tube.

The

aluminum

dust in

the

working

fluid deposited

on

the

wall surface

by

gravitational

force. The

deposited

particles moved

slowly

on the wall with

the

flow

and accumulated on an area of

low

shear.

These

areas are coincidental with

the

sitesofatherosclerotic

lesions

in humans. The

author

suggest particles such as

platelets,

leukocytes,

and erythrocytes

play

an

important

role

in

atherosclerosis.

Singh

et. al.

(1990)

performed a multiangle visualization of

flow

patterns

in

arterial

secondly flow

- _waH

layar

ZZZ

frea

stream

Figure 2.3. 1

Bharadvaj

et al.

(1982)

Vortex

Flow

Pattern

diameter,

0.8

cm

daughter

branches,

and angle of

bifurcation

equal to

60 degrees.

Visualization

was performed with an

ink

injection

technique while

rotating

the model to photograph a

three-dimensional

flow field. The Reynolds

number

for

the

flow

was equal

to

297,

andequal

flow

ratios

were allowedthrough

the

daughter

tubes.

The

authors'

results support

the

findings

of

Fukushima

et. al.

(1987)

and

Walburn

and

Stein

(1982).

The

flow

pattern shows

the

horseshoe

vortex as

described. The

dye

injection

method allowed

for secondary

motions visualization

along

the

outer wall ofthe

bifurcation,

also

as

previously

seen.

Rindt

et al.

(1988)

investigated

the

flow

field in

steady

flow

with equal

flow

rate ratios

through

a carotid

bifurcation. This

model

is

a more accurate representation of

the

actual

geometry

than that

of symmetrical

bifurcations,

due

to the

inclusion

of

the

carotid sinus.

The

sinus

is

a

bulb

at

the

beginning

of

the

internal

carotid

artery,

just

after

the

bifurcation

region.

The

authors studied

the

steady

flow,

Reynolds

number equal

to

480,

through the

bifurcation. The

parabolic

velocity

profile

from

the

main

branch

continues about

three-quarters

of

the

way

in

the

bifurcation

region.

Then

the

velocity

increases

at

the

divider,

and

decreases in

the

area of

the

nondivider walls.

In

the

sinus

region,

reverse

flow in

seen onthe nondiver wall side of

the

bifurcation. The

reverse

flow

continues until

the

diameter

of

the

sinus

begins

to

decrease

and

form

the

internal

carotid artery.

As

the

diameter

decreases,

the

nondivider wall

flow begins

to

accelerate

forming

a region of equal

flow

on

both

the

divider

and nondivider sides of

the

artery.

No

recirculation

is

seen on

the

nondivider wall side of

the

external carotid artery.

In

this region, the

flow does

not

decelerate

enough

to

create recirculation.

The

maximum

flow is

on

the

divider

wall side of

the

artery,

as

in

the

internal

carotid artery.

The

authors

determined

the

secondary

flow in

the

common carotid

artery

to

be low

until

the

area of

the

bifurcation

where

the

diameter

begins

to

increase. As

the

artery

widens

the

secondary flow increases

towards the

external carotid.

In

the

internal

carotid

artery

the

secondary

velocity

goes

towards the

nondivider side.

In

the

region of

the

sinus closest

to the

bifurcation,

velocities.

Further downstream in

the sinus,

secondary

flow

velocities

decrease

near

the

divider

wall and

increase

near the nondivider and adjacent walls.

Secondary

flow in

this region

is

caused

mainly

by

curvature effects and shows avortex with

its

center shiftedtowards

the

nondivider wall.

About half

way

into

the

sinus a stagnation point

is

observed near

the

divider

wall.

The

secondary

flow is

a maximum at

the

adjacent wall near next

to

the

stagnationpoint

As

the

sinus

tapers

into

the

internal

carotid

artery, the

secondary

flow increases in

the

direction

of

the

center of the

branch.

At

the

beginning

of

the

external

branch

the

secondary

flow forms

avortex.

Fluid flows

from

the

nondividerwall

to the

divider

wall andthenreturns

along

theouter radius ofthe artery.

The

changes

in

the

secondary flow further down

the

external carotid are

limited,

and no

stagnation

is detected.

The

secondary

flow

was seen

to

affect

the

flow

through

the

sinus.

In

a

two-dimensional

study, the

secondary flow is

not present.

Thus,

any

effects

from

secondary

flow

on

the

main

flow

field

are

lost

in

two-dimensional

analysis,

limiting

the

usefulness of

two-dimensional

studies.

2.4

Importance

of

Study

Atheroslerotic lesions form

at sites of predilection

in

the

large

arteries.

These

sites

are

typically

areasof

branching,

bifurcation,

and

taper.

At

thesis sitesthe

lateral

pressure

is

decreased

due

to

a combination of

the

geometry

of

the

vessel and

the

blood flow

characteristics.

By

analyzing

the

normal

flow

pattern

through

an

artery,

information

can

be

gained as

to

the

normal,

or

healthy,

pattern of

blood flow in

the

region.

Then

by

examining

the

flow

pattern

in

an

artery

ofan

unhealthy

individual,

the

detection

of

any disturbances

in

the

flow field

may

lead

to

earlier

diagnosis

andtreatment.

2.5

Objectives

The

hypothesis

of

the

hemodynamic influence

on atherosclerosis

is

detailed in

the

literature

review.

The

shear ratevariation and particle stagnation

is

believed

to

be

one of

the

major

factors in

the

role of atherosclerosis

initiation.

The

objects of

the

present work

are as

follows:

1

To

model

the

90tee-junction

flow

visualization

study

of

Katrino, Kwang,

and

Goldsmith

using

a

commercially

available computational

fluid dynamics

software

code,

CFDS-FLOW3D,

asverificationof

the

code.

2.

To

develop

a

transient

fluid flow

model

to

simulate

the

flow

of

blood

at

the

carotid

artery

bifurcation.

3.

To

generate results

showing

the

flow

field,

including

flow

reversal and

recirculation,

~and

the

shear stress variation

during

onecomplete cardiac-cycle.

4.

To study

the

flow in

the

carotid

artery

bifurcation

after a

blockage

has

formed

upstrearrtof

the

bifurcation

5.

Develop

an experimental

flow

loop

to

model

the

pulsatile^low of water

through

a

simplified carotid

artery

bifurcation.

6.

Capture

the

flow field from

the

experiment with

high

speed video

to

determine

the

flow

recirculations and reversals.

7.

Conduct

the

experimental

flow

study

witha

blockage

in

the

main

branch

of

the

model

upstreamof

the

bifurcation,

and capture

the

results with

high

speed video.

3 Theoretical

3.1

Computational Fluid Dynamics

The

thrust

of

the

study

was performed with

Computational

fluid

dynamics (CFD).

All CFD

analysis

in

the thesis

was performed

utilizing

the

commercial software package

CFDS-FLOW3D.

This

software package

is

a

finite

control volume

fluid dynamics

solver.

CFD

analysis

involves

discretization

ofcomplex geometries

into

small quadrilaterals.

The

software allows

the

user

to

set

the

boundary

conditions

for

the

new geometrieswith either

constant values or

FORTRAN

subroutine.

After

the

boundary

conditions are

set, the

relevant equations are solved

for

each new geometry.

The

equations used

in

the

analysis are

the

continuity

equation

^-+V-(p[/)=0,

(3.1)

and

the

momentumequation

^-+V-(pUU)=B+V-a,

(3.2)

at

where

B

is

the

body

force

and a

is

the

stresstensor

a =-pd +n(yU-KVUf).

(3.3)

The CFD

code

has

the

option

to

solve

turbulence

via

the

k-E

model or variations

of

the

Reynolds

stress model

for

turbulence.

In

this

study

the

average

Reynolds

number

is

The

software results can

be

output

in

the

form

of vector _plots, concentration

gradients,

or numerical

tabulation.

In

the

results_sections, vector plotsand numerical tables

are presented.

3.2

CFD Model

A

simplified carotid

artery

bifurcation

was studied with

the

previously

stated

assumptions.

The daughter

branches have

equal

inner

diameters,

branching

off the

centerline of

the

main

tube.

The

main

tube

expands

into

the

daughter

branches in

a smooth

transition.

This

geometry

is

symmetric about

the

centerline.

The

flow

of

blood

at

the

bifurcation is

transient

with

secondary

flow

playing

a

key

role

in flow

field

development.

It

is

this

secondary

flow

that

limits

the

accuracy

of

two-dimensional,

or symmetrical analysis.

The

model studied wasthree-dimensional and symmetric about

the

centeraxis.

The bifurcation

was constructed of

8

blocks,

or

large

quadrilaterals.

The

actual

bifurcation

region

is

comprised of

3

blocks.

Two

blocks

open

into

the two

daughter

branches,

and

the third

forming

the

divider

wall.

Each

daughter

branch

was modeledwith

1

block, beginning

at

the

specific

inner bifurcation block

opening

and

branching

off at

25

degrees

to the

centerline.

The

main

tube

consisted of

3

blocks. These

blocks

all

join into

the

3

bifurcation

blocks. The

main

tube

had

a sufficient

length

as

to

allow

for

fully

developed flow

at

the

bifurcation.

The

model

bifurcation

was created with

the

CFDS-FLOW3D

Preprocessor,

called

Sophia. The first

step

in

creating

the

bifurcation

model

in Sophia

was

to

input

geometrical

points

that

form

the

corners of

the

blocks

of

the

model.

These

points were

then

connected

with arcs

forming

faces.

Two

faces

are connected with

four

lines,

thus

creating

a

quadrilateral,

or

block.

Sophia

creates

blocks

having

straight edges

To

create rounded

edges

the

square edges are projected onto

the

previously

created arcs

by

the

process of

mapping.

After

mapping

of

the

blocks

was complete

the

block

was

in

its finished

form.

Four

more points were

input into Sophia

and

the

process was repeated

to

form

an adjacent

block

to the

one

already

created.

This

process was repeated

for

the

3

blocks

forming

the

main

branch

of

the

bifurcation.

The

process

involved in

building

the

3

blocks

of

the

bifurcation

regionwas much

more complicated.

The

3

faces from

the

ends of

the

main

branch

blocks

were used as

starting

points

in

the

construction.

To

create a

face

joining

the

bifurcation

region with a

daughter

branch

the

points must

be determined

that

will create

the

25

degree departure

angle of

the

daughter branch.

Arcs

are

then

drawn

with

these

points

completing

the

face

joining

the

bifurcation

regionwith

the

daughter branch. Five

points werethen

determined

to

create

the

decrease

in

inner

diameter

of

the

main

tube

at

the

bifurcation

region.

These

points were

then

joined

with splines.

The

straight edges of

the

square

block

representing

the

bifurcation

region were then mapped

to

the

splines.

Creating

the

smooth

transformation of main

branch

to

daughter

branch.

This

process was repeated

for

the

remaining

junction

of main

branch

to

daughter

branch.

The

middle section of

the

bifurcation

region ends with an arced surface

opening

away

from

the

main

branch.

The

arc was constructed

to

create a smooth

transition to the

The

bifurcation geometry

was

broken down

into

8,

000

cells

for

the analysis

The

grid

distribution is

shown

is

Figure 3.2.1. The

cell

distribution for

the main

branch

is

coarse at

the

inlet,

becoming

finer

approaching

the

bifurcation.

In

the

region of the

bifurcation

the

cell

distribution is

very

fine.

The

flow field in

this

area

is

critical,

so the grid

distribution is

very

fine for increased

accuracy

and

to

eliminate

any

grid

dependence

on

the

flow field.

In

the

daughter

branches,

the

cell

distribution

is

a geometric progression.

The

cell

distribution is finer

at

the

base

of

the

branch

were

accuracy

is

critical.

Further down

the

daughter

branch,

near

the outlet, the

cell

distribution is

more sparse

in

an effort

to

save

computational

time

incurred

by

unnecessary

computation.

3.3 Assumptions

Due

to the

difficulty

in

handling

blood

outside

the

body,

certainassumptions

have

been

made

in

the

study.

The first

assumption

is

to

consider

blood in

the

large

arteries a

continuous

fluid.

Blood

is

composed of

plasma, platelets,

and red and white cells.

The

red

blood

cells are

the

most abundant cells

in

the

blood.

However,

the

red cells are

typically

only 8

micrometers

in diameter. There

are

fewer

white

blood

cells

in

the

blood,

but

are

the

largest

of

the

solid

particles,

with a

diameter

of

between

10

to

20

micrometers.

The

commoncarotid

artery

typically

has

a

diameter

of about

8

millimeters,

about

200

times the

diameter

of a white

blood

cell.

Therefore,

the

continuous

fluid

assumption

is

acceptable

in

blood

in

a

large

artery.

Blood

is known

to

behave

as a non-Newtonian

fluid.

The study

focuses

on

large

arteries

only,

in

whichthenon-Newtonian effects

play

a minimal role

in

the

fluid

dynamics

Therefore,

blood

may

be

assumedto

behave

as a

Newtonian

fluid in

the study.

Blood

plasma

is

the

major component of whole

blood.

Plasma

consists of almost

entirely

water.

Therefore,

with

the

above

assumptions, the

flow

ofwater

is

modeled to

representthe

flow blood in

large

arteries.

The

walls of

the

arteries are

distensible,

or compliant.

The

distensibility

of the

artery

wallsacts as a

damper

to the

pulsatile

flow. The

wall

damping

acts

to

eliminate

the

pulsatile nature of

blood

flow,

slowly changing it

to

steady

flow

as

it

goes through the

systemiccirculation.

In

the

arteries

furthest from

the

heart

the

oncepulsatile

flow

of

blood

is

steady.

The

veins are return vessels

in

the

circulation system.

The flow

of

blood is

steady in

the veins,

with valves

strategically

formed

throughout the

vessels

to

eliminate

any

reversal of

flow.

Atherosclerosis

is

not

found

the

veins,

thus

supporting

the

hemodynamic

effects

hypothesis.

In

the

thesis,

the

walls of

the

arteries are assumed

rigid,

forgoing

the

distensibility

characteristics of

artery

walls.

This

assumption will

have

an

effect on

the

flow

pattern,

but it is

unknown

if

the

added

complexity

from

the

distensibility

factor

willamount

to

a comparable

increase in

accuracy.

Finally,

the

geometry

of

the

bifurcation

has been

assumed

symmetric,

both

daughter branches

being

equal

in

diameter. The

bifurcation

angle

is

50,

a

typical

value.

The

main

branch

of

the

bifurcation

is

8

mm

in

diameter,

and

the

daughter

branches

are

5

mm

in diameter.

Throughout

the

study

only

equal

flow

rates

through the

daughter

branches

are allowed.

This

assumption allows

the

flow

pattern

to

develop

in

the

bifurcation

without the

influence

of a

dominant flow

branch.

Thus,

allowing

a clear

depiction

ofthe

flow

field,

eliminating

a

boundary

condition

dependence.

3.4

Boundary

Conditions

The

CFD

program allows

the

user

to

set

boundary

conditions

by

calling

subroutines at

the

appropriate section of

the

code.

The

inlet

condition onthe

bifurcation

is

the transient

output of

the

heart,

the

cardiac cycle.

The

cycle was modeled

by

a

FORTRAN

subroutine

(

see appendix

B ). In

the

subroutine

the

period of

the

cycle

is

set

at

0.8

seconds.

The

heart

output as a

function

of

time

is

approximated over

this

cycle.

As

the

CFD

software solves

the transient simulation, the

subroutine

for

the

velocity

at the

inlet is

called

to

determine

the

inlet

boundary

condition.

Only

equal

flow

ratesthrough

the

bifurcation

are studied

in

the

analysis.

The

outlet

boundary

conditions on

the

daughter

branches

are set

to

equal pressure

boundaries

to

create

the

equal

flow

rates.

All

other

boundaries

are solid

walls,

in

conjunction with

the

rigid

artery

wallassumption stated earlier.

In

the

second section of

the

analysis a

25%

occlusion

is

set

in

the

main

branch

of

the