Experimental and analytical study on the stability of a bounded jet at low Reynolds numbers


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Experimental and analytical study on the stability of a bounded jet at low Reynolds numbers
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xvi, 190 leaves : ill. ; 28 cm.
López, Juan Luis, 1941-
Publication Date:


Subjects / Keywords:
Reynolds number   ( lcsh )
Hydrodynamics   ( lcsh )
Jets -- Fluid dynamics   ( lcsh )
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )


Thesis--University of Florida.
Includes bibliographical references (leaves 183-189).
Statement of Responsibility:
by Juan L. López.
General Note:
General Note:

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University of Florida
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All applicable rights reserved by the source institution and holding location.
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aleph - 000185180
notis - AAV1762
oclc - 03330218
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Full Text






To my wife Elsa


I wish to express my sincere appreciation to Dr.

U. H. Kurzweg and Dr. E. R. Lindgren for their research

counseling, teaching,and assistance through my studies

at this University. I also wish to thank Dr. K. T.

Millsaps and Dr. R. L. Fearn for excellent teaching and

assistance and to Dr. R. J. Gordon for serving on my

advisory committee.

A special word of thanks is extended to M. F. Wehling

for his assistance in proofreading the manuscript, to

Mrs. M. Boyce for the drawing of some of the figures, and

to Mrs. S. A. Palkowski for typing the final copy. Finally,

I would like to express appreciation to the United States

Air Force for making it all possible.



ACKNOWLEDGEMENTS ..................................... iii

LIST OF FIGURES...................................... vii

LIST OF SYMBOLS...................................... xi

ABSTRACT............................................ xiv

CHAPTER 1 INTRODUCTION............................. 1

The Problem of Transition to
Turbulence ................................. 1
Previous Investigations.................. 3
Investigations on Jets in the Presence
of Rigid Boundaries........................ 7
Physiological Implications. Blood Flow 10
Stenotic Flow............................. 11


Description of the Experimental Apparatus
and Procedure............................. 16
Preparation and Physical Characteristics
of Bentonite Solutions................... 25
Pressure Drop Measurements............... 28


Flow Visualization Experiments........... 30
Flow Patterns Downstream of the
Constriction.............................. 30
Quantitative Observations................ 32

The Onset of Disturbances............ 32
The Position of the Stationary
Turbulent Region.................... 45
Effects of Changes in Constriction
Geometry on the Onset of Disturb-
ances .............................. 55

Correlation with Previous Experiments.... 58

Introduction......................... 58
Johansen's Experiments with Pipe
Orifices............................. 59



Transition in Models of
Arterial Stenoses.................... 60
In Vivo Experiments on the Onset
of Sound in Arteries................ 62
Turbulence Spectra .................. 66
Onset of Sound in Flexible Tubes.... 71

Pressure Drop Measurements............... 76

Introduction.......................... 76
Experimental Results................. 80
Effects Produced by Changes in
Geometry............................ 86
Correlation with Momentum Balance
Approach............................. 90

Experiments using Polymer Solutions...... 95

Introduction.......................... 95
Polyox Solution Characteristics and
Preparation Techniques.............. 96
Pressure Drop Measurements.......... 98

CHAPTER 4 STABILITY ANALYSIS....................... 103

Introduction......................... 103
Mathematical Formulation in
Rectangular Coordinates............. 105

Mathematical Form of the
Disturbance..................... 107
Inviscid Analysis .............. 108

Experimental Velocity Profiles...... 110
Instability Modes of Propagation.... 113

Symmetric Disturbances......... 113
Antisymmetric Disturbances..... 119
Antisymmetric Disturbances in
the Presence of a Back-Flow
Region......................... 126

The Streamlines of the Perturbation. 130
Mathematical Formulation in
Cylindrical Coordinates............. 134


Axisymmetric Disturbances........ 138
Non-Axisymmetric Disturbances
in an Unbounded Jet.............. 145
Mathematical Formulation......... 149

Correlation with Experimental
Observations.. ................... .... 161
Effects Produced by the Boundaries.... 162

CHAPTER 5 CONCLUSIONS .......................... 166


BIBLIOGRAPHY...................................... 183

BIOGRAPHICAL SKETCH ................................ 190



2.1 Diagram of the experimental apparatus.. 21

2.2a Experimental Apparatus................. 22

2.2b Pressure Drop Measurements.............. 22

2.3a Constriction Models.................... 23

2.3b Model with Rounded Edges................ 23

3.1 Development of Jet Instability
(Do=1.90cm, D1/Do-1/2)................. 35

3.2 Development of Jet Instability
(Do=1.27cm, D1/Do=1/4)................. 37

3.3 Development of Jet Instability
(Do=1.90cm, D1/Do=1/6)................. 38

3.4 Development of Jet Instability
(Do=l.27cm, D1/Do=13/16).............. 39

3.5 Turbulent Region (Do=0.32cm, D1/Do=1/2) 40

3.6 Breakdown of Disturbances into
Turbulence.............................. 40

3.7 Critical flow rate...................... 42

3.8 Critical Reynolds number................ 43

3.9 Critical Reynolds number versus area
ratio.................................. 44

3.10 Transition region....................... 49

3.11a Turbulent region, Do=1.90cm, D1/Do=1/12. 50

3.11b Turbulent region, Do=1.90cm, D1/Do=1/6.. 50

3.11c Turbulent region, Do=1.90cm, D1/Do=1/3.. 51

3.11d Turbulent region, Do=1.90cm, DI/Do=1/2.. 51

3.12a Turbulent region, Do=0.95cm, D1/Do=1/6.. 52




3.12b Turbulent region. Do=0.95cm, DI/Do=1/3. 52

3.13a Turbulent region. Do=0.32cm, Di/Do=1/6. 53

3.13b Turbulent region. Do=0.32cm, D,/Do=1/2. 53

3.14 Length of turbulent region when close
to orifice . .. 54

3.15a Turbulent region for different
geometries. DI/Do=1/8 .. 57

3.15b Turbulent region. Di/Do=1/4. ... 57

3.16 Correlation with Sacks experiments. 65

3.17 Comparison of turbulence spectra. 69

3.18 Turbulence spectra at various probe
locations . ... 70

3.19 Comparison with the onset of sounds in
flexible tubes . ... 75

3.20 Constriction geometry .. .... 79

3.21 Dimensionless pressure drop .... 84

3.22a Pressure distribution. DI/Do=l/4. ... 85

3.22b Pressure distribution. DI/Do=3/8. 85

3.23 Effects produced by geometry change
in pressure drop. . ... 88

3.24 Effects produced by geometry change
in pressure distribution .. 89

3.25 Pressure drop using Polymer solution. 102

4.1 Analytical profiles in rectangular
coordinates . ... .111

4.2 Approximated experimental velocity
profiles .. 112

4.3 Growth factor curves for symmetrical
disturbances . 120




4.4 Growth rate curves for symmetrical
disturbances. . .. 121

4.5 Growth factor curves for antisymmetrical
disturbances. . .. 123

4.6 Growth rate curves for antisymmetrical
disturbances. . .. 124

4.7 Wave phase velocity curves. ... 125

4.8 Stability of a profile with a back-flow
region. ..................131
region. . . 131

4.9 Relative stability of velocity profiles
at different axial positions. ... 132

4.10 The streamlines of the perturbation .. 135

4.11 Velocity profiles in cylindrical
coordinates . 140

4.12 Wave phase velocity (m=o) .. 146

4.13 Growth factor curves (m=o). 146

4.14 Growth rate curves for axisymmetric
disturbances (m=o) . .. 147

4.15a Growth rate curves for non-axisymmetric
disturbances. R=0.10 .. 156

4.15b Growth rate curves for non-axisymmetric
disturbances. R=0.20 . .. 157

4.15c Growth rate curves. R=0.50 ...... 158

4.15d Growth rate curves. R=1.0 158

4.15e Growth rate curves. R=2.5 ...... 159

4.15f Growth rate curves. R=2.75 ...... 159

4.16 Wave phase velocity curves. .. .160



4.17 Growth factor curves .

4.18a Correlation with experiments.
DI/Do=l/6. . .

4.18b Correlation with experiments.
D1/Do=1/2. . .

4.19a Effects of boundaries (rectangular).

4.19b Effects of boundaries (cylindrical).

A.la Bentonite solution viscosity (0.2%).

A.lb Bentonite solution viscosity (0.1%).

A.2 Fluid viscosity at large deformation

A.3 Polyox solution viscosity (100PPM)







S. 179

S. 179




(Arranged in Alphabetical Order)

Ao cross-sectional area (unobstructed)

AI cross-sectional area (obstructed)

c complex axial wave velocity

c dimensionless axial wave velocity

C viscometer calibration constant

d interface distance

Do test section diameter (unobstructed)

D, constriction diameter

e model eccentricity

H step height

Hz Hertz

I modified Bessel function of the first kind
K modified Bessel function of the second kind
k axial wavenumber

k unit vector in the z direction

LD length of the turbulence region when close to the

m azimuthal wavenumber

p total pressure

p(r) amplitude of the pressure fluctuation

p pressure fluctuation

Po mean flow pressure


q total velocity vector

Q flow rate

r radial cylindrical coordinate

R interface distance, R = R2-R,

Ro boundary position

Re Reynolds number based on unobstructed pipe diameter Do

s asymmetric parameter

t time of fall

T temperature in 0C

u longitudinal velocity component

u longitudinal fluctuation velocity

u(r) amplitude of the radial velocity fluctuation

U longitudinal component of the mean flow velocity

Uo mean flow centerline velocity

U mean flow orifice velocity

v transverse component of velocity

v transverse fluctuation velocity

V transverse component of the mean flow velocity

w axial velocity fluctuation

W(r) axial velocity distribution

Wo mean flow centerline velocity in cylindrical

x longitudinal rectangular coordinate

y transverse rectangular coordinate

Zo constriction length



Greek letters

6 velocity profile parameter, 6=dl/(d2-d,)

6 angular cylindrical coordinate

p fluid viscosity

v kinematic viscosity

p fluid density

a disturbance growth rate

T wall shear stress
perturbation amplitude

V Stoke's stream function

w complex amplification factor


i imaginary part

r real part

N normalized variable

Symbols not listed here have all been described in the

text when used for the first time.


Abstract of Dissertation Presented to the
Graduate Council of the University of Florida
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy



Juan L. Lopez

June, 1977

Chairman: Dr. Ulrich H. Kurzweg
Major Department: Engineering Science, Mechanics, and
Aerospace Engineering

A series of steady flow experiments is described, in

which important hydrodynamics factors, including onset of

instabilities and transition to turbulence are considered

for the axisymmetric jet formed downstream of symmetric

constrictions in circular pipes. Results were obtained

for several ratios of test section to constriction dia-

meter, within a Reynolds number (Re=UoDo/v) range from

approximately 50 to 2000. The critical Reynolds number

for onset of velocity undulations was evaluated for a

wide range of area ratios using a non-disturbing flow vis-

ualization technique, based on the double refracting

properties of an aqueous colloidal solution of White

Hector bentonite.


It was found that the critical Reynolds number was

independent of the test section diameter and a function

of the dimensionless area ratio. Different types of

instabilities were noticed and the flow characteristics

changed rapidly with increasing flow velocity.

A brief comparison with previous investigations on

the onset of sound showed a definite correlation with the

onset of undulations in the present experiment, but onset

of sound appeared to occur at a more advanced stage of

flow disturbances, beyond the onset of instabilities.

Experimental observations also demonstrated the exist-

ence of different modes of instability propagation. Of

particular interest was some type of helical motion which

usually appeared at some distance from the orifice. At

sufficiently large flow rates the jet broke down into

turbulence, producing a localized turbulent region of

highly irregular motion bounded by laminar disturbed flow

on both sides. The relative position of this region with

respect to the jet orifice was measured for several geo-

metrical conditions. Pressure drop measurements were

taken across the obstructed pipe section using distilled

water and a solution of polyethylene oxide. Results

demonstrated a strong influence of geometry and area ratio

upon pressure drop measurements on flows through pipe con-


Inviscid stability analysis was applied to study the

propagation of symmetrical and non-symmetrical disturb-

ances in rectangular and cylindrical geometry, and the

wave phase velocity and growth rate computed for several

velocity profiles. Numerical calculations showed a

non-vanishing growth rate for the dimensionless wavenumber

range o
Maximum growth rate usually occurred for kd approximately

0.80 and of magnitude very sensitive to the shear-layer


The stability of an unbounded cylindrical jet was

examined and the instability amplification evaluated at

various downstream positions. Results validated some of

the experimental observations by showing the growth rate

to be larger for axisymmetric disturbances near the jet

orifice, but, larger for non-axisymmetric modes at positions

further downstream, where the jet has partially diffused.

Observed instability wave lengths correlated with the

corresponding wavenumber for maximum amplification as

predicted by inviscid analysis.



The Problem of Transition to Turbulence

In 1839 the German engineer Hagen published results

of experiments on the flow of water through cylindrical

glass tubes, reporting observations on the transition

region between steady and irregular flow. This seems

to be one of the first times that the transition phenom-

enon and the existence of turbulence have been expressly

reported in the literature.

In his fundamental investigations 1879-1833, 0.

Reynolds distinguished between "steady" and "sinuous"

fluid motion, and introduced a dimensionless quantity

characterizing homogeneous incompressible viscous flow.

This is Reynolds number, Re=U Do/V (U=mean flow velocity;

Do= tube diameter; v= kinematic viscosity) usually inter-

preted as the ratio of the inertial to the viscous forces

has been used since Reynolds' experiments to indicate the

state of a fluid motion, with transition to irregular

motion occurring at a certain value.

Experimental and analytical studies of the transi-

tion phenomenon have been numerous since the end of the

last century, most of them related to the incidence of

turbulence in relation to flow through straight pipes

and channels. Schiller (1921) reported that if the flow

is highly disturbed at the tube entrance and provided

the entrance length is long enough there occurs an instan-

taneous increase of the flow resistance when a critical

Reynolds number is exceeded. This discontinuity of the

flow resistance was currently interpreted as an instan-

taneous onset of turbulence. Tollmien (1929) computed

the neutral stability curve for the boundary layer on a

flat plate using the theory of instability to small

oscillations. Tatsumi (1952) using Tollmien's method,

obtained a lowest possible critical Reynolds number of

approximately 9700 for Poiseuille flow.

Binnie (1945, 1947) reported the appearance of flow

disturbances by visual observations using the stream

double refraction technique at a Reynolds number close

to 1970, with continuous turbulence established at 2900.

Lindgren (1957) made similar studies and the interpreta-

tion of these observations suggests that the transition

phenomenon in viscous flow does not occur instantaneously

at some specific Reynolds number but rather within a

transitional Reynolds number range.

While there are numerous experiments on transition in

cylindrical test sections, there appears to be no exten-

sive data published concerning the critical Reynolds

number at which an axisymmetric jet becomes unstable, or,

the characteristics and position of the stationary turbu-

lent region formed downstream of a jet orifice.

The present investigation has been directed to

investigate the flow characteristics for a jet in the

presence of rigid boundaries. The jet was formed by

symmetric constriction sections of various diameters

within the test pipe. The method of observation was

based on the streaming double refraction properties of

White Hector Bentonite. An inviscid stability analysis

was then applied to determine the mode of disturbance

propagation that grows most rapidly at large Reynolds

number. The analysis was applied to study the propaga-

tion of symmetrical and non-symmetrical disturbances,

the wave phase velocity, and instabilities growth rate

for the approximated symmetrical velocity profiles.

The Reynolds number at the onset of undulations

was compared with some available date from previous

investigations and with the conditions for the onset of

sound in cylindrical sections with flexible walls.

Previous Investigations

The first major contribution to the study of hydro-

dynamical stability can be found in the theoretical papers

of Helmholtz (1868). Previous to this time, many

scholars had certainly become aware of the question but

their efforts did not progress beyond the stage of

description. After the findings of Helmholtz, the com-

bined efforts of Reynolds (1883), Kelvin (1880) and

Rayleigh (1878) produced a rich harvest of knowledge.

Lord Rayleigh, between 1878 and 1917 published a great

number of papers on this subject and was able to prove

that the existence of a point of inflexion on the velo-

city profile constitutes a necessary condition for the

occurrence of instability in inviscid fluid flow. Much

later Tollmien (1926) succeeded in showing that this

constitutes also a sufficient condition for the amplifi-

cation of disturbances.

The key stability equation was arrived at indepen-

dently by Orr (1907) and by Sommerfeld (1908). At this

stage, it was generally believed that the study of

infinitesimal small pertubations would lead to some

definite answers to the problem of transition from lam-

inar to turbulent flow. Tollmien (1929) using the Orr-

Sommerfeld stability equation calculated the first

neutral eigenvalues for two-dimensional boundary layer

flow. Soon the same road led Schlichting (1932) to

further evaluation of critical Reynolds number and ampli-

fication rates of disturbances.

Later, Lin (1945) reviewed and improved the mathe-

matical procedures and laid the foundations for a

general expansion of the stability analysis.

At this time the stability of plane Poiseuille

flows had become a particularly controversial issue

and Lin put it in order by his new and more general

analysis. Also at this time a new tool, the digital

computer, became available for the analysis of hydro-

dynamic stability. After a great deal of effort it

became apparent that the theory of amplification of

small flow perturbations could not explain the transi-

tion to turbulent flow.

The efflux of a laminar jet from an orifice in the

absence of solid boundaries was treated by Bickley

(1939) applying boundary layer theory. The velocity

profiles determined by Bickley were correlated with

experiments by Andrade (1939) confirming the theoretical

predictions. It was also determined that the jet becomes

unstable at very low orifice Reynolds number.

The laminar circular jet was treated by Schlichting

(1933), and Tollmien (1926) determined the velocity

distribution for the circular turbulent jet using

Prandtl's mixing-length theory, both cases for jets

immersed in a medium with no radial boundaries. Experi-

mental results on circular jets were given by W. Zimm

(1921) and P. Ruden (1933). The immersed free jet, which

occures when a fluid is discharged from a nozzle or

orifice far from any solid boundaries, becomes completely

turbulent a short distance from the point of discharge

producing strong mixing with the surrounding fluid.

A. J. Reynolds (1962) conducted experiments on the

instabilities of a liquid-into-liquid jet directed into

a large tank of water. The experiments were performed

with the purpose of resolving observational discrepan-

cies reported by Schade and Viilu (See Reynolds, 1962).

Schade reported that a long rectilinear jet could be

maintained until the Reynolds number (based on orifice

diameter) was a few hundreds. Viilu (1960) reported,

after careful experiments at low flow rates, a value of

11 for the Reynolds number at breakdown. Reynolds (1962)

in his observations encountered a surprising variety of

flows, among them, "pedal" breakdown, "sinuous"oscilla-

tions, and "shearing" puffs. According to his observa-

tions while it was difficult to obtain steady jets at

relative low Reynolds numbers, progressively longer jets

could be maintained as the Reynolds number increased

toward 150. For Reynolds numbers above 300, jets became

disordered near the nozzle, although not so near as in

the range between 10 and 30.

The stability of the two-dimensional laminar free

jet against infinitesimal antisymmetric disturbances was

treated by Tatsumi and Kakutami (1958), obtaining a

curve of neutral stability separating the regions of

stable and unstable flow in the (k, Re) plane, where k

is the oscillation wave number. Calculation showed that

the free jet becomes unstable at very low Reynolds number

of less than 10.

Batchelor and Gill (1962) made a general mathemati-

cal analysis of the stability of axisymmetric jets for

homogeneous unbounded fluids. A condition similar to

Rayleigh's about the inflexion point was shown to be a

necessary condition for instability, also it was shown

that the propagation of non-axisymmetric mode of oscil-

lations were favored for velocity profiles varying

slowly with respect to the radial coordinate, as observed

previously by Reynolds (1962).

Recently the breakup of a free liquid jet was

studied by Taub (1976) using a high resolution optical

probe technique.

Investigations on Jets in the Presence of Rigid Boundaries

Most of the experiments in connection with the flow

of fluids through orifices in bounded regions have been

directed toward the establishment of the orifice as a

flow meter device. The flow of fluids through locally

constricted regions has been extensively investigated

both experimentally and theoretically in recent years

for the purpose of determining characteristics of flow

through constrictions in arteries, commonly referred to

as stenosis.

The characteristics of the flow of fluids through

pipe orifices within the transitional region between

steady laminar and turbulent flow have been described

by Johansen (1929). The flow pattern downstream of the

orifice was studied by means of a dye in distilled water.

Johansen found that the magnitude of the Reynolds number

for a given event or type of flow increased progressively

as the orifice to pipe diameter ratio was increased.

Also it was possible to relate many of the pressure drop

characteristics to features of the flow pattern observed


In the last two decades the flow passing an abrupt

expansion has been studied experimentally by many

researchers. Abbott and Kline (1962) observed the

turbulent flow over single and double steps in rectangu-

lar sections, determining the length of separation.

Meisner and Rushmer (1963) studied the disturbed region

downstream of the expansion using a colloidal suspension

of White Hector Bentonite.

The effects on fluid flow produced by a geometry

change, of a constricted region in a pipe were deter-

mined by Robbins and Bentov (1967) with the purpose of

identifying the most influential factors in stenotic

flow. Back and Roschke (1972) in an experimental inves-

tigation on the flow of water through an abrupt circular-

channel expansion described the position of the

reattachment point as a function of the upstream

Reynolds number within the range of transition to


Experimental studies on instabilities and velocity

distribution downstream of a constricted flow region were

reported by Durst, Melling and Whitelaw (1974), in a

rectangular symmetric expansion, and by Iribarne, Frantisak,

Hummel, and Smith (1972) in a cylindrical symmetric expan-


Pressure drop measurements across constricted regions

simulating stenosis have been made by Young and Tsai (1973)

and by Seeley and Young (1976), while the turbulence

spectra were obtained for flows downstream of the con-

striction by Kim and Corcoran (1974).

Numerical computations and correlation with experi-

ments were reported by Macagno and Hung (1967) for a

circular conduit with an abrupt expansion, and by Lee

and Fung (1970) for local constrictions with smooth

axisymmetric expansions at low Reynolds number flows.

Morgan and Young (1974) using an integral method

approach presented a solution to the problem of incom-

pressible flow through an axisymmetric constriction for

low Reynolds number flows below the onset of turbulence.

The results showed that even a mild constriction can

cause a radical alteration in flow characteristics which

become more drastic with constriction ratio and increas-

ing Reynolds number.

Later Aillo and Trefil (1976) studied the entrance

flow problem by linearization of the convective derivative

to evaluate the establishment of Poiseuille flow in

locally constricted tubes.

Physiological Implications, Blood Flow

While most of the flows we encounter in daily life

are turbulent; many biological flow systems appear

designed to function within the laminar regime. The

possible occurrence of turbulence in hemodynamics has

been with us for a long time. Physiologists, biophysi-

cists and more recently fluid dynamicists have speculated

and performed experiments to determine the flow charac-

teristics of the circulatory system.

A good review of the differences between turbulent

and laminar flow and previous research work in the area

of circulatory blood flow is given by Robertson and

Herrick (1975). The mentioned paper discusses the

conditions under which turbulence may be expected to

occur in flowing blood in vivo, and makes an attempt to

classify the flow as laminar, disturbed or turbulent;

with blood flow being mostly of the dissipating type and

not permanent or self-preserving turbulence.

Considerations of turbulent motion in hemodynamics

are complicated by several factors usually not encoun-

tered in other fluid dynamic studies. In vivo blood

flows are maintained by pulsatile pressure gradients,

and take place in tubes with flexible walls, which are

tapered, and not long enough between bifurcations to

permit establishment of fully developed turbulent flow


At low shear, blood behaves as a non-newtonian

fluid with a yield stress (Charm, 1972). Blood also

contains suspended particles erythrocytess, leucocytes

and platelets) and is not considered a homogeneous medium.

The physiological range of average flow Reynolds

numbers is usually considered to be below 2000, but due

to the pulsating nature of the flow, Reynolds numbers

above this value do occur. McDonald (1960) estimated

peak Reynolds numbers in the aorta of humans and animals

noting values in the range 2400 to 10000. We must remind

ourselves that although Reynolds numbers of this magni-

tude actually occur in the blood circulatory system,

these occurrences will be over very short periods of time

and over short tube lengths, so that conditions for fully

developed turbulence are not satisfied.

Stenotic Flow

The partial occlusion of arteries due to stenotic

obstruction is one of the most frequently occurring abnor-

malities in the circulatory system. The causative factors

for the initial development of arterial lesions, leading

to stenoses, are not well understood, but regardless of

the cause it is clear that once an obstruction has

developed the flow characteristics will change and

hydrodynamic factors will play an important role as

the stenosis continues to develop.

Flow through local constrictions has been widely

studied analytically, numerically and experimentally

using models of different geometries simulating stenoses.

Macagno and Hung (1967) used computer solutions of the

vorticity-transport equation for two-dimensional viscous

flow through a conduit expansion, and Lee and Fung (1970)

made numerical calculations on the problem of flow in

constructed tubes.

Meisner and Rushmer (1963), Back and Roschke (1972),

Durst, Melling and Whitelaw (1974), made experimental

studies on the flow behavior through cylindrical and

rectangular expansions, while Young and Tsai (1973)

measured the pressure loss for flows through tube con-

strictions. These researchers among many others con-

cluded that the most improtant factors having potential

physiological significance in the flow of blood through

a constricted artery are

(a) Resistance of the constriction to the flow

characterized by an increased pressure drop across the

constricted region as determined by Young and Tsai (1973)

and Seely and Young (1976). Consequently the heart

would have to work harder to maintain a required blood

supply to the circulatory system.

(b) Damage to the wall tissues produced by turbu-

lent velocity fluctuations in the flow which are created

downstream from the stenosis. Fry (1968) reported a

possible damage to the endothelium by increased velocity

gradient at the wall in arterial blood flow. Morgan and

Young (1974) conducted theoretical studies in the wall

shear distribution. These results revealed significant

alterations caused by severe contractions producing an

order of magnitude increase in the maximum shearing

stress at low Reynolds number and two orders of magni-

tude in the moderate range, (about 100), with separation

in the downstream diverging section producing a reversal

in the wall shearing stress.

(c) The nearly stagnant conditions in the separated

region may significantly affect vessel-to-blood mass

transfer as reported by Caro (1971) and possibly affect

the tissue properties in that region.

(d) Localized turbulence appearing downstream the

constricted region. Eddy formation and turbulence were

identified by Meisner and Rushmer (1963) using flow

visualization methods. The vibrations transmitted to

the wall were studied by Foreman and Hutchison (1970) and

the fluctuation frequency spectrum measured by Kim and

Corcoran (1974) distal (downstream) to a simulated


(e) Noise production attributed to the appearance

of turbulent motion downstream of a stenosis. This

characteristic can be used as a diagnostic tool (phon-

angiopraphy) for assessing the degree of narrowing of a

blood vessel as suggested by Lees and Dewey (1970).

One of the most dramatic consequences associated

with a stenosis is the commonly observed enlargement

which occurs in the low pressure region distal to a

stenosis. Radiologically this post-stenotic dilation

(PSD) is often easier to identify than the stenosis, and

may be caused by wall damage induced by turbulence

(Foreman, 1970). Roach (1963a) indicated that turbulence

was the causative agent in PSD developed in dog arteries

after artificially producing stenoses in the femoral and

carotid arteries with a nylon band. The alterations to

the arterial wall which eventually becomes more distensi-

ble seems to be related to the flow distortions produced

by the stenosis, affecting in some undetermined way the

elastin and intercollagen links (Roach, 1963b).

While most authors seem to believe that the flow

distortions produced by the stenosis is turbulence, others

have questioned this conclusion. Bruns (1959) suggested

that turbulence is a random phenomenon and cannot create

enough energy to produce sound (murmurs). He proposed that

vortex formation (a Karman Trail) was the most likely cause

of murmurs and post-stenotic dilatations in arteries.

Another group (Robicsek et al., 1958; Rodbard et al.,

1967) feels that cavitation is the most likely cause of

murmurs and hence of PSD.

Our intention in bringing the problem of flow through

an occluded artery was not an attempt to explain and

solve a physiological problem, but to try to relate our

experiments in locally constructed tubes and our analyti

cal results to an interesting problem which is not fully

understood. We feel that the results obtained during

the present investigation on the onset of disturbances,

the position of localized turbulence and the pressure

drop measurements under steady flow conditions can supply

useful information about stenotic flow and provide a good

example to which experimental and analytical engineering

techniques can be applied to other sciences.


Description of the Experimental Apparatus and Procedure

Preliminary investigations were conducted using a

very simple experimental apparatus, consisting mainly of

straight rectangular test sections connected directly to

an electric pump and a storage tank, forming a closed

loop. These transparent rectangular test models were

made of three separated sections of acrylic glass, joined

together, with the rectangular channel and constriction

located in the midsection. Two constrictions were made,

one with sharp edges and the other with rounded edges,

both 3.0 cm long, with a channel to orifice diameter

ratio of 1/4 and a channel width of 0.63 cm. These two-

dimensional constrictions consisted of a slit which

extended to the full depth of the flow section.

The optical properties of a solution of a commercial

dye, Milling Yellow, were used to observe the flow

pattern produced by the presence of the channel constric-

tion. The Milling Yellow solution exhibited excellent

birefringent properties, showing a clearly visible inter-

ference pattern when viewed through crossed polarizers,

but unfortunately the solution also exhibited a strong

non-newtonian behavior, and the viscosity was not only a

function of shear stress but was also sensitive to changes

in temperature.

The test section was connected to the pump and the

storage tank using tygon vinyl tubing. Care was taken

to taper and smooth the inside of the connecting section

between the cylindrical tubing and the rectangular inlet

to the test section, in order to prevent unnecessary

disturbances of the flow patterns at these points.

The experimental observations in the constructed

rectangular channels can be summarized as follows: At

very low flow rates the over-all flow field remained

laminar. This situation was characterized by undisturbed

alternating dark and light lines. As the flow rate was

increased the central core became disturbed and then

small localized eddies appeared. A few diameters down-

stream from the localized disturbed region, the disturb-

ances were damped and the flow again assumed laminar

characteristics. As the flow rate was increased, the

eddies formed closer to the jet orifice and the turbulence

activity became more pronounced. For high flow rate tur-

bulence occurred over a wider region starting at the

orifice, but decaying back to stable flow several dia-

meters downstream from the constriction. No turbulence

was observed upstream of the constricted area.

A stability problem similar to that experienced in

fluidic devices was observed. The jet profile in the

expansion region showed preference to reattach to one of

the walls first, producing asymmetric flow condition.

The observations made in the rectangular cross-

section models using the Milling Yellow solution were

primarily of a qualitative nature and served only as a

preliminary step to more refined experiments using cylin-

drical test sections.

The arrangement of the experimental equipment for

the investigations on flows through cylindrical test sec-

tions is shown in Figures 2.1, 2.2a, b. The liquid flow

system consisted of an overhead tank, whose level was

maintained by continuously pumping the fluid from the

storage tank with 1/35 hp 3000 rpm pump, permitting the

excess fluid to return to the lower tank through an over-

flow pipe. This way a truly constant head pressure main-

tained the flow through the test pipe.

Fluid from the overhead tank flowed through an

entrance tube, 130.2 cm long. before entering the test

section. The function of the entrance section was to

insure the decay of entrance disturbances and the devel-

oping of a parabolic flow profile prior to the constric-

tion entrance. Entrance section and all test sections

were made of commercially smooth extruded acryllic tubes.

Several test section diameters were used during the

experiment as described in Table 1. Their average dia-

meters did not vary more than approximate 1% from the

value specified by the manufacturer, but local variations

(up to 6%) were noticed in some cases. The average

diameter of each tube was determined by filling the tube

with distilled water and measuring the weight of the

fluid. Test sections were joined to the entrance and

return sections using flanges made of plexiglas, the

constriction placed between the entrance and the test

section as indicated in the upper right-hand corner of

Figure 2.1. The return section drained the fluid into

the storage tank forming a re-circulating system.

Flow rate was regulated by a needle valve which

allowed very fine adjustments in the fluid velocity.

Flow rate was measured with a spherical float flowmeter,

manufactured by Gilmont Instruments, Inc., with a water

flow range of 30 1900 ml/min.

The entrance and the test sections were firmly fixed

to a heavy test bench to avoid any external mechanical

vibrations, and was mounted horizontally. Connecting

tubing between different sections were made of 3/4" ID

(1.905 cm) tygon vinyl hose. Also two calibration com-

partments (of known volumes) were included as part of the

storage or collecting tank and, used to calibrate the flow

meter and to measure the flow rate outside the flowmeter


An important point in the arrangement and design of

the experimental apparatus was the avoidance of pressure

fluctuations or mechanical vibrations that could induce

premature fluid oscillations. It was observed that very

small vibrations of the apparatus induced propagation of

instabilities when the fluid was close to the critical

point. For this reason the test bench was mounted on

rubber and rubber strips introduced between the rigid

test tubes and the bench tubes supporter. Measurements

of the critical flow rate (onset of velocity undulations)

were only made when external disturbances were at a


Temperature was measured with a thermometer located

prior to the entrance section, and all readings were

taken at room temperature between 24.00 and 25.00 C.

The apparatus described allowed for a complete re-circu-

lating system, very easy to operate and to maintain. At

the beginning of each experiment the fluid was circulated

for a few minutes through the apparatus to obtain homo-

geneous flow conditions.


A Gravity tank
B Flowmeter
C Thermometer
D Entrance section (130.2 cm)
E Test section (57.1 cm)
F Polarizer-Analyzer
G Needle valve
H Control valve
I Calibration compartments
B J Storage tanks
K Pump
L Constriction

C ----

J I-- ~

Diagram of the experimental apparatus

Figure 2.1

Experimental Apparatus


Figure 2.2b

Apparatus for Pressure
Drop Measurements

F''-,zir 2.2a

Constriction Models

Model with Rounded Edges

Figure 2.3a

Figure 2.3b


Models and Test Sections

Model Pipe Orifice Diameter Area Radio %
No. Do(cm) Dl(cm) Ratio A1 Constricted
DI/Do /Ao










































(*) R: Rounded edges

B: Beveled edges

Cylindrical constriction models were made of cast

acrylic rods, machined to the test section diameter,

and cemented to a plexiglass flange. Different holes

were drilled and polished to the specific values of the

desired area ratios as described in Table I. In order

to study the effect that a different constriction geo-

metry had on the flow characteristics, some of the con-

strictions were made with rounded or beveled corners as

shown in Figures 2.3a and 2.3b; all models were 3 cm


During the flow visualization experiment the flow

was normally illuminated by a 60-watt sodium vapor lamp,

from which light passed through a polarizer analyzer


Preparation and Physical Characteristics of
Bentonite Solutions

An aqueous suspension of White Hector Bentonite,

supplied by Tansul Industries, showed good stream-

double refraction properties even at very low Bentonite

concentration, with a density and viscosity very close

to that of water, as described in Appendix A. A

decision was then made to use Bentonite solutions instead

of Milling Yellow to avoid the non-newtonian behavior of

the latter.

Several stock solutions of Bentonite were prepared

during the course of the experiment, all using the same

procedure, and their density and viscosity determined

over a wide range of shear stress and temperature.

The procedure to prepare a 5% concentration stock

solution was to slowly dissolve the bentonite powder

in distilled water at room temperature, agitating con-

tinuously until all bentonite lumps had been dissolved

and the bentonite particles well dispersed. To prepare

the working solution the procedure was to dilute the 5%

stock solution with distilled water until the desired

0.5% concentration was reached.

After standing for a few days solid material that

did not go into suspension precipitated to the bottom of

the container and the remaining solution was decanted.

To determine the final concentration, a sample was heated

in an oven at 820C for about 20 hours, until all the

liquid evaporated. By proper weighing with a Sartorius

balance (accuracy 0.1 mg) the amount of solid in sus-

pension was determined to be approximately 0.20%.

To stabilize the suspension 0.01% of tetra sodium

pyrophosphate was added. The final solution appeared

clean and almost transparent, but after use for a period

of about 6 weeks, it turned cloudy and showed tendencies

to flocculate when left undisturbed for some time,

although it still maintained birefringent properties.

For a more extensive research on the physical and

optical properties of bentonite solutions of different

solid concentration the reader is referred to E. R.

Lindgren (1957).

The method used to determine the solution viscosity

is described in Appendix A. The solution density was

approximately 0.99 g/cc at room temperature. Photo-

graphs were taken of the flow patterns over a wide range

of flow rate for various area ratios and test section

diameters. The optical components were carefully

aligned to obtain the best dark background possible.

Flow rate and temperature were recorded for each photo-

graph taken. Illumination was provided by a sodium

vapor lamp and the results recorded on ASA 400 film,

using a 35mm Mamiya Sekor DSX-1000 camera with close-up


To freeze the fluid motion in time and observe the

turbulent eddies it was necessary to reduce the exposure

time to 1/250 or 1/500 of a second depending on the flow

conditions, with an aperture setting corresponding to f/2.

Even with the lens wide open the amount of light

was insufficient and the film had to be overdeveloped and

printed on high contrast paper. For future experiments

it is recommended that either a more intense light source

or a more sensitive film be used so the lens aperture can

be smaller, thereby improving the depth of field.

Pressure Drop Measurements

The pressure drop measurements were performed only

on flow through the 1.27 cm test section, using pressure

taps located at different positions upstream and down-

stream from the constricted region. A Robinson-Halpern

low pressure transducer P20 series connected to a

Hewlett-Packard D.C. Vacuum Tube Voltmeter Model 412A

were used as shown in Figure 2.2b to detect and record

the pressure drop across the constricted region in the

cylindrical section.

To cover the pressure drop range encountered during

the experiment two pressure transducers were needed, one

of range 0-0.15 psig and the other from 0-1.0 psig both

calibrated with distilled water.

The Robinson-Halpern P20 series low pressure trans-

ducers are electromechanical devices used to measure

gas, liquid or vapor pressures. The principle of opera-

tion is a pressure sensing element (capsule). The motion

of the capsule is transmitted to the core of a linear

position transducer by means of a stainless steel support

rod. A change in the pressure of the medium being mea-

sured moves the core and produces an AC electrical output

that is directly proportional to the pressure change.

This AC signal output is demodulated filtered, and then

fed into an output network that is matched to the specific

load impedance to be used.

Both pressure transducers were calibrated by adding

distilled water to two vertical manometers tubes connected

to the low and high pressure ports of the transducer.

The distance (cm of water) between the meniscuses formed

on each pressure side was measured with a travelling

microscope. A photographic wetting agent was added to

the water to reduce the surface tension variations. The

calibration curve (pressure in cm of water versus output

voltage) showed a good linear behavior in both cases.

It was of extreme importance to eliminate all the

air trapped in the tubing between the pressure port and

the capsule in order to obtain accuracy and repeatability

in the measurements. This occurred because the output of

the transducer changed due to surface tension of any air

bubbles in the transducer ports. The pressure transducer

and voltmeter were allowed to warm-up for about 30

minutes before taking any readings.


Flow Visualization Experiments

The flow visualization experiments were conducted

with the dual purpose of obtaining experimental informa-

tion on the flow characteristics close to the transition

region, and, determining the peculiarities and location

of the stationary turbulent region as a function of the

flow rate for different area ratios and pipe diameters.

It should be emphasized that the stream double-

refraction technique does not provide streamline visual-

ization but rather visualizes shear stresses present

at the various locations. Regions of no shearing forces

appear black under crossed polarizers, while increasing

rates of shear cause increase of the illumination in

that flow region, until saturation effects appear.

Flow Patterns Downstream of the Constriction

The fluid patterns downstream of the constricted

region according to experimental observations can be des-

cribed in general terms as follows. Undisturbed laminar

flow when viewed through the analyzer (crossed at right

angles with the polarizer) appears as an illumination of

the outer regions with a smooth transition to a darker

region at the center of the test section, where the shear-

ing forces are smaller. For purely laminar flow this dark

region remains steady (see Figure 3.la) with a width

depending on the dimensionless area ratio; smaller area

ratios having very narrow dark regions, and with a ten-

dency to contract with increasing flow rate, when the

flow is still laminar.

As the needle valve was adjusted increasing the

flow velocity, the dark region was observed to perform

some oscillatory motion (see Figure 3.1b). This

oscillatory motion was interpreted as indicating the

initial propagation of instabilities in the disturbed

laminar flow. These distortions or undulations in the

flow which appeared periodically, were not stationary

but propagated downstream, tending to decay rapidly.

This behavior signifies the onset of a different flow

regime where shear-layer waves and their stability play

the dominant role.

A very small increase in the flow rate caused the

disturbance waves to occur more frequently and with a

higher intensity. Increasing the flow velocity, these

undulations began to lose definition and to form eddies

of more random behavior. It is important to indicate

at this point that this disturbance appeared, to the

naked eye, to start propagating not immediately down-

stream from the constriction but a few pipe diameters

away, and that the flow variations necessary to change

the flow behavior were extremely small, indicating a

very high flow sensitivity close to the onset of undulations.

A still further increase in the flow velocity caused

a stationary disturbed region (localized eddies) to dev-

elop downstream in the test section (see Figure 3.1d).

This region appeared to move closer to the constriction

as the flow velocity was increased, until turbulence

appeared to develop right at the jet exit location. The

extension of this stationary turbulent region was rela-

tively small, with all turbulent activity decaying

gradually to laminar flow as the fluid moved downstream

the test section. Figure 3.la through Figure 3.6 repre-

sent this sequence of events as the flow rate was in-

creased for the specified area ratio and pipe diameter.

Quantitative Observations

The Onset of Disturbances

The first quantitative observations were to determine

the flow rates at the onset of disturbances, for the var-

ious area ratios and test sections. Starting with a purely

laminar flow, the flow rate was increased in the smallest

incremental steps allowed by the needle valve, until the

oscillations appeared almost continuously. The flow rate

was then measured. By closing the needle valve the dis-

turbances were almost made to disappear, indicating again

a laminar flow condition, the flow rate was then measured

and averaged with the first reading. This determined the

flow rate at the onset of disturbances, or the critical

flow rate.

The definition of this critical point in the fluid

presents some uncertainty since there is a small flow

rate range over which the flow shows similar behavior,

and only the frequency and intensity of the disturbances

seems to change gradually, always increasing with increas-

ing flow rate. The onset of disturbances was defined when

disturbances appeared almost continuously to insure that

they were produced by the jet instability mechanism and

not by any external mechanical vibrations.

Next step was to introduce different constriction

models to determine the dependency of the onset parameters

on the dimensionless area ratio, and to experiment with

different test section diameter. The results of these

experiments are shown in Figures 3.7, 3.8, and 3.9. The

magnitude of the critical flow rate was found to increase

progressively as the area ratio was increased as shown

in Figure 3.7. The critical Reynolds number based on

the test tube diameter Do, is plotted against the ratio

between Do and the constriction diameter D1 in Figure

3.8. This relation seems to scale the phenomenon of

onset of flow undulations.

Increments in the area ratio from small numbers to

almost unity produced three distinguishable flow regions,

each one possessing its own characteristic behavior. A

first region corresponding to the smaller area ratios was

highly unstable, with oscillations of high frequency

and intensity and with very precise boundaries between

the laminar region and the region of localized turbulence.

Figures 3.3a, b, c are representative examples of this

region which extended to approximately an area ratio of


Similar flow phenomena occurred in all test sections

as observed visually. However, as the diameter ratio,

D1/Do became larger, the disturbances appeared less

violent, of lower frequency and intensity. The boundar-

ies of the locally disturbed region were not as precise

as in the first region, especially the downstream bound-

ary for the larger area ratio. In this range the critical

Reynolds number showed a linear dependency on the area

ratio A1/Ao, and the linear relation still existed when

the test tube diameter was varied. This linear region

extended approximately between an area ratio of 0.065 and

0.30. For area ratios larger than 0.30 the slope of the

critical Reynolds number versus area ratio curve began

to gradually change, indicating a trend to a more stable

configuration. Flow sensitivity to changes in flow

rate decreased, and the stationary turbulent region as

perceived by the eye composed of larger eddies, which

(a) Re = 311

(b) Re = 507

(c) Re = 570

Figure 3.1. Devevlopmelnto; ol J.et I nst abili(.y
(Do =1.90 cm, DI /Do = 1/2)

(d) Re=685

(e) Re=980

Sf) ReP -1418

Figure 3.1 Continued

(a) Re = 187

(b) Re = 301

(c) Re = 426

(d) Re = 605

Figure 3.2 Development of Jet Instability
(Do =1.27 cm, DI/Do = 1/4)

(a) Re = 108

(b) Re = 185

(c) Re -- 323

Figure 3.3 Development of Jet Instability
(Do = 1.90 cm, Di/Do = 1/6)

(a) Re = 1678

(b) Re = 1920

(c) Re = 2335

Figure 3.4 Development of Jet Instability

(Do = 1.27 cm, Di/Do = 13/16)

Figure 3.5 Stationary Turbulent Region
(Di/Do = 1/2, Re = 692)

Figure 3.6 Breakdown of Disturbances Into
Turbulence (Di/Do = 1/6, Re = 150)

diffused slowly into the laminar region making the down-

stream boundary of this region hard to determine.

Near an area ratio of 0.60 the slope of the curve

increased even more toward stability and what appeared to

be disturbances of a different origin began to propagate.

As the area ratio approached unity the curve leveled

off, showing a different type of breakdown mechanism

which overlapped with the one described before, but taking

priority as the area ratio increased. The onset of dis-

turbances in this region was characterized first by large

wavelength undulations of the dark region in the middle

of the flow, moving occasionally to an unsymmetric posi-

tion in the tube. Disturbances then appeared over large

sections of the tube (see Figure 3.4) and propagated

downstream more in the form of slugs. It was not possi-

ble for these large area ratio to obtain a stationary

disturbed region with laminar flow on both sides, but

rather large sections of disturbed flow which overlapped

as the flow rate was increased, filling the test section


Due to the high degree of difficulties in making

constrictions models of large area ratio, only a few

experimental observations were made in this range, mostly

in the 1.27 cm pipe diameter. Observations made on the

flow in the upstream side of the constriction showed this

region to be laminar and free of disturbances over most

o 1.90

A 1.27

v 0.95

o 0.32

A /Ao

Figure 3.7 Critical flow rate







Do (cm)
o 1.90

A 1.27
o 0.95

v 0.63

O 0.32

* V







D /Do

Critical Reynolds number

2000 -

1600 i




Figure 3.8



Q, *i





0Y .-t0


0 D C

o 0 c O ,-
\ \

; : \ "^
^ \ <
o \ o
"* f \ COo

of the experiment Reynolds number range, even when the

downstream portion was highly disturbed and a localized

turbulent region had developed. As the transition

Reynolds number approached 1700-1900 disturbances were

also present in the upstream side. These oscillations

were probably produced by entrance disturbances propa-

gating in the entrance section that had not decayed

(due to the high flow velocity) before entering the

tube constriction.

The Position of the Stationary Turbulent Region

As described in previous sections, increasing the

flow rate beyond the onset of disturbances produced an

unstable stationary region of highly irregular motion.

The transition phenomenon from a steady laminar jet to

the appearance of disturbances and then to the final

breakdown into turbulence was not instantaneous but

occurred over a small range of Reynolds number. The

transition region as a function of the geometrical

diameter ratio is shown in Figure 3.10. This region

is bounded by the onset of disturbances curve and the

approximated boundary at which the stationary turbulent

region first appeared. Flow corresponding to the area

to the right of these boundaries was considered stable,

while flow corresponding to the area to the left showed

signs of instability.

It is important to notice that the turbulence

generated was of very weak nature, promptly decaying through

viscous dissipation to laminar flow.

The next set of experiments were directed to measure the

location of the stationary turbulent region with respect

to the jet orifice for various test section diameters as

a function of the area ratio. The characteristics of the

flow and the position of the disturbed region have strong

physiological implications during the formation of a post-

stenotic dilatation in arteries. Fluid pressure fluctu-

ations generated by turbulence are transmitted to the

bounding walls and seem to affect the tissue elastic

properties in localized areas downstream of the stenosis.

The boundaries of the disturbed region were defined

at the point of attachment and detachment from the walls.

These boundaries were easier to evaluate in the large

diameter test sections when conducting experiments with

the small areas ratio models. As the area ratio increased

the position of the downstream boundary was progressively

more difficult to locate. In general these boundaries

oscillated slightly around a mean position, which was

determined by taking several readings at a constant flow

rate. As an area ratio close to one was approached, the

turbulent stationary region ceased to exist, and the

fluid started showing characteristics common to the flow

of fluids in uniform cylindrical test sections in the

transition region.

The position of complete recovery to laminar undis-

turbed flow was not measured since it usually occurred

outside the test section dimensions. The measured loca-

tion of the stationary turbulent region for a pipe diam-

eter of 1.90 cm is shown in Figures 3.11a, b, c, d, for

several diameter ratios. The upstream and downstream

boundaries are described as a function of the average jet

velocity at the orifice. The tail shown in the right-hand

side corresponds to the transition region where the flow

showed extreme sensitivity to flow rate. Increasing the

flow rate caused the disturbed region to move upstream

(i.e., closer to the orifice). Once in close proximity

to the orifice an increase in flow rate produced a small

increase in the region dimensions. Further increasing

the flow velocity produced violent turbulent motion,

filling large portions of the test section. Under these

circumstances the downstream boundary was not clearly


The location of the stationary turbulent region for

the 0.95 and the 0.32 cm ID test sections are shown in

Figures 3,12a, b, and 3.13a, b. The general character-

istics were the same. Of especial interest was the

smallest test section of 1.32 cm ID, since this dimen-

sion is representative of some arteries in humans and

animals. The maximum length of the disturbed region for

this test section was approximately 1.0 to 1.5 cm,

depending on the area ratio. The most common types of

PSD seem to be developed immediately distal to the

stenosis and of dimensions comparable to our experimental


An attempt was made to non-dimensionalize the prob-

lem by plotting the Reynolds number versus the dimension-

less axial position for a constant area ratio and differ-

ent pipe diameters. The results were only fair, showing

some spread in the data, probably caused by the diffi-

culties in determining the boundaries of the turbulent

region. The dimensionless length of the turbulent region

when it extended to the orifice is shown in Figure 3.14 as

a function of the diameter ratio, for various test

sections. By examining these results, three regions were

again noticed, depending on the diameter ratio. Up to a

ratio of approximate 0.20 the length increased, between

0.20 and approximately 0.40, it remained fairly constant,

increasing again for a diameter ratio above 0.40. The

smallest test section of 0.32 cm ID consistently produced

smaller values for the length (LD) of the turbulent









Transition Region






Figure 3. 10






Figure 3.11a




turbulent region

X cm

Turbulent region DO=1.90cm, Di

= 1/12

Turbulent region

4 8 12 16

X cm

Figure 3.11b

Turbulent region Do=1.90cm, D/ D= 1/6

100 -

Turbulent region

o, oo

0,o ---

)4 8 12 16

x cm

Figure 3.11c

Turbulent Region.

Do=l.90cm, D1/ 1/3

I 4 I 12 16 I
S4 8 12 16

x cm

Figure 3.11d

Turbulent Region. Do=1.90cm,D1

= 1/2



30 1




4 8 12 16

Figure 3.12a

x cm
Turbulent Region. Do=0.95cm, D1/Do=1/6


12 16

Figure 3.12b

x cm
Turbulent Region. Do=0.95cm,D1/Do=l/3











Figure 3.1



1 2 3 4 x cm


Turbulent Region. Do=0.32cm, DI/Do=l/6



x cm

Turbulent Region. Do=0.32cm, Di/Do=1/2

Figure 3.13b

Do (cm)
o 1.90

A 1.27

o 0.95

v 0.63

o 0.32

S -o- --



Figure 3.14


Length of turbulent region when
close to orifice




- -- -0-

- -0-

Effects of Changes in Constriction Geometry on the Onset
of Disturbances

The purpose of the following experiments was to

determine if changes in the geometry of the constriction

were an important factor with regard to the critical flow

rate for the onset of disturbances and for the flow top-

ology downstream of the constricted region. Several

models were constructed similar to the square edge models

but with beveled or rounded edges as shown in Figure 2.3b.

The dimensions of all models were 3 cm long and made to

fit in the 1.27 cm pipe. The location of the turbulent

region was determined and critical flow rates for various

orifice sizes. The fluid flow conditions for the differ-

ent geometries at the onset of instabilities are described

in Table II. In all models tested the effects produced

by a geometry change were not significant and the onset

of disturbances appeared to be determined primarily by a

critical jet velocity and the shear stress layer associ-

ated with it. Figures 3.15a, b show the location of the

turbulent region as a function of the average jet velocity

for a diameter ratio of 1/8 and 1/4. The position of the

turbulent region is closer to the constriction for the

rounded geometry than for the straight one, at the same

flow velocity. It seems to be shifted in position by a

distance which is very close to half the constriction

length. This is an indication that the disturbances will

break down into random turbulent motion at a certain

distance from some location at which they originate (in

this case 1.5 cm), depending on their growth rate. For

the sharp edge geometry (900 corners) the high shear

stress layer originates at the end of the constriction,

while for the rounded geometry it is closer to the con-

striction throat.

Geometry Effects

Model No. Do

7 1/8


on the Critical

Edges Geometry



Reynolds Number




Model No.


Model No.


















203 812





523 1026








0I 4 8 1
0 4 8 12

Figure 3.15a

x cm
Turbulent region for different
geometries. D /Do=1/8

o Rounded edges

a Beveled edges

4 8

12 x cm

Figure 3.15b Turbulent region. DI/Do=1/4





Correlation with Previous Experiments


In this section attempt is made to correlate the

experimental observations with some of the previous

investigations on the properties of cylindrical jets in

bounded regions, and, to explain these results in the

light of present observations.

Experiments on fluid flow across locally constricted

test sections can be found in the literature since the

beginning of the century, such as the Johansen's experi-

ments (1929) on orifice plates. Since then, a large

number of experiments have been conducted on orifices and

venturi shape constrictions for a variety of engineering

applications, such as the rate of flow meter. More

recently the problem has been studied from the physiolog-

ical point of view in order to associate hydrodynamic

factors to the development and progression of arterial


The results on the onset of disturbances as determined

from our experiments are correlated with Johansen's obser-

vations and Young and Tsai's (1973) measurements on the

transition Reynolds number. Sound onset as measured by

Sacks et al., (1971) (in vivo) and by Gonzalez (1974) (in

vitro) are related to the onset of disturbances and tur-

bulence to determine the relation between these phenomena.

Johansen's Experiments with Pipe Orifices

The characteristics of the flow through pipe orifices

at low Reynolds number were investigated by Johansen in

1929. Using a dye (methylene blue), Johansen made

observations on the flow pattern downstream from a

diaphragm orifice of 1.34 cm diameter in a cylindrical
tube of diameter 2.68 cm (D- 1/2). These observations
were mainly concerned with jet transition charactersitics

from laminar to turbulent, and the relation between the

discharge coefficient and the differential pressure head

across the orifice.

According to his observations, at a pipe Reynolds

number of about 300 (orifice Reynolds number of 600)

vortex ripples appeared close to the orifice and moved

downstream, losing translational velocity and definition.

Close to a Reynolds number of 500, transition of the jet

occurred, resulting in the formation of a train of vortex

rings which gradually loose identity downstream.

Comparing Johansen's flow photographs (Figures 7 and

8 in Johansen's report) at pipe Reynolds numbers of 300

and 500 with our experimental results using the Bentonite

Solution, it appears that the onset of disturbances in

the present experiments corresponds to Johansen's descrip-

tion of the flow behavior close to or above a Reynolds

number of 300.

As shown earlier in Figure 3.8, for a diameter ratio

of 1/2, onset of undulations occurred at a Reynolds number

of about 470, which is higher than Johansen's value for

flows through an orifice.

At this point it is appropriate to mention a possi-

ble source of disagreement between experimental data

taken for orifice flows as compared with similar data on

extended constricted region. The flow velocity profile

coming out of an orifice is probably almost flat, while

the velocity profile at the jet exit position of an

extended constricted region should be curved. Another

source of discrepancies in the correlation of similar

experiments is the technique used to obtain the informa-

tion. The flow visualization technique based on the

optical properties of Bentonite Solutions is a non-

disturbing method while techniques using velocity probes,

dye injections, etc., are likely to induce flow disturb-

ances. More important still is that the dye follows the

fluid motion, while, streaming birefringence does not.

Transition in Models of Arterial Stenoses

The critical Reynolds number and flow characteristics

in a cylindrical tube with rounded axisymmetric and non-

symmetric constrictions, were partially investigated by

Young and Tsai (1973), using the hot-film probe technique.

By connecting the anemometer to an oscilloscope or strip-

chart recorder, critical flow conditions were determined

for different probe locations. The initiation of the

transition regime was identified by the relative low

frequency oscillations in the hot-film signal. The

nature of the fluctuations varied considerably with the

position of the probe along the axis of the test section.

The experimental critical Reynolds numbers obtained

for a probe position of 15.2 cm with respect to the con-

striction center are shown in Table III, for two models

with different diameter ratio, but with the same length

to diameter ratio (Zo/Do = 2.0). Also included in the

Table are the present experimental results at the onset

of disturbances for models with 900 edges and a ratio

Zo/Do equal to 1.57. Both sets of experiments were

conducted on the 1.90 cm pipe.


Correlation with Young's Experiment

Model Model Experimental
Geometry Length Diameter ratio Critical Technique

Zo/Do D /Do Re

Rounded Axisymmetric Hot-film
(Young's) 2.0 1/3 325+20 Probes

2.0 2/3 800+50
900 Edges 1.57 1/3 2-2 Flow Visuali-
Axisymmetric zation
Present In-
vestigation) 1.57 2/3 872

The correlation of these data appear to be good consid-

ering that the models dimension and geometry are diff-

erent. These results validate our previous conclusion

that the length and geometry of the model play a

secondary role in the transition phenomenon.

In Vivo Experiments on the Onset of Sound in Arteries

Sacks, Tickner and MacDonald (1971) in their investi-

gations on the onset of vascular murmurs conducted a series

of in vivo experiments to determine the minimum flow

Reynolds number necessary to generate sounds in artificially

produced stenosis in the descending aorta of anesthetized


The experimental procedure performed on forty-nine

adult mongrel dogs was to expose the aorta; after applying

tourniquets above and below, the aorta was severed to

allow insertion and fixations of a brass cylinder into

which machined orifice plates could subsequently be placed.

The orifice plate was located at the distal end of the

brass cylinder, thus allowing the vascular wall to parti-

cipate normally in the production of sound associated

with the flow instabilities generated downstream from the

constricted region. Blood flow rate was measured by an

untrasonic flowmeter mounted proximal to the brass cylinder,

and just distal to the aortic arch. A catheter-tip micro-

phone pressure transducer with a frequency response of 0 to

1000 Hz was threaded into the femoral (thigh) artery up

to various locations relative to the orifice in the

thoracic aorta, and its output recorded simultaneously

with the flow rate and pressure.

Critical flow rate at the onset of sound was calcu-

lated from the oscillograph record of the flow rate

during occlusion. The Reynolds number at the instantane-

ous value was based on the internal vessel diameter and

flow velocity proximal to the orifice. The blood

viscosity was calculated from the non-newtonian viscous

parameters described by Hershey and Smolin (1967) using

the rectal temperature.

Using the described experimental apparatus Sacks and

collaborators generated a general sound boundary curve

which established the critical flow Reynolds number at

onset of sound as a function of the constriction to artery

diameter ratio.

The experimental results were approximated by a

least square parabola fitted to the experimental data.

The spread in the experimental data points for the

tethered and untethered artery at a constant diameter

ratio were significant, especially for the large diameter

ratios. According to Sacks the Reynolds number limit

curve for the onset of sound is given by

Reo = 2384 D1 2 (3.1)
and illustrated in Figure 3.16. Flow conditions specified

by Reynolds number and diameter ratio to the left of the

curve in Figure 3.16, representing Sacks data, will pro-

duce sound, whereas those below and to the right will not.

Sacks and collaborators also concluded that this sound

has a broad frequency band, with intensity dropping off

as frequency increases, and is strongly suggestive of

free turbulence or jet instability rather than periodic

vortex shedding.

In Figure 2.16, equation (3.1) representing the "in

vivo" experimental results on sound onset in arteries are

compared with visual observations of this study on the

onset of disturbances in circular pipes of different dia-

meters. We must keep in mind that the in vivo experiments

were conducted on arteries probably of variable internal

diameter, and distensible walls, under pulsating flow con-

ditions through an orifice plate, while the "in vitro"

investigations were performed on rigid tubes and steady

flow conditions through a constricted region 3.0 cm long.

At a diameter ratio larger than 0.45 the data repre-

senting equation (3.1) deviates considerably from our

visual observations which indicate less stable flow

conditions. Near a diameter ratio close to unity the

critical Reynolds number in both experiments is close to


The parabolic curve described by (3.1) falls mostly

in between the curves in Figure 3.16, representing the

onset of disturbances, and the appearance of a locally

disturbed flow region. This result seems to indicate that

sound may be generated at the very early stages of flow

instabilities, at least for severely constricted arteries.

Whether or not the pressure fluctuations produced by

these initial instabilities are strong enough to make the

walls vibrate, so that sound can be detected by an external

microphone, is something still to be discussed.

--Re=2384 (D)2

1200 /

Re Localized
80 turbulence


/ .--Onset of Instabilities

0 0.2 0.4 0.6 0.8


Correlation with Sacks experiments

Figure 3. 16

Turbulence Spectra

Turbulence spectra downstream from a constricted

region in a cylindrical test section were measured by Kim

and Corcoran (1974). The inside diameter of the tube was

0.95 cm and the constriction models made of plastic cylin-

ders with various orifice sizes. Measurements were made

for water with flow rates giving upstream Reynolds numbers

in the range of 800 to 2000. Turbulence spectra were

determined for various orifice sizes at several axial

positions, using a spectrum analyzer with the probe loca-

ted at the center of the tube. Signals were then pro-

cessed with a digital computer which utilized the fast

Fourier processor. The comparison of turbulence spectra

for different orifice diameters is shown in Figures 3.17a,

b, c, d, for upstream Reynolds numbers of 800, 1200, 1600,

and 2000 for measurements taken 2.54 cm downstream from

the orifice. According to Kim and Corcoran no flow dis-

turbances were observed for a Reynolds number of 800 when

the orifice diameter was 0.79 cm (Dl = 5/6). The "onset
of disturbance" curve in Figure 3.8 shows that onset of

disturbances for this diameter ratio does not occur until

a Reynolds number of approximately 1800. Figures 3.17c,

d show that turbulent velocity fluctuations for a 0.79 cm

orifice diameter appeared somewhere between a Reynolds

number of 1600 and 2000.

Sinusoidal fluctuations of low intensity with a fre-

quency of 14 Hz (see square in Figure 3.17a) was observed

at a Reynolds number of 800 for an orifice diameter of

0.635 cm (Dl = 2/3). For this diameter ratio our obser-
vation showed that onset of disturbance occurred at a

Reynolds number close to 840. The correlation between

some of Kim and Corcoran measurements and our visual

observation appears to be good. For smaller orifice

diameter vigorous fluctuations occurred with increasing

turbulence intensity and frequency range, as observed

during the present investigation.

It is apparent by looking at Figures 3.17a, b, c,

and d that the turbulence spectra are different for diff-

erent orifice diameters for small upstream Reynolds

numbers, and somehow this must be related to the differ-

ent distinguishable regions for the onset of disturbances

curve. For large orifices the flow is either laminar or

laminar-disturbed showing low intensity fluctuations and

a low frequency range. As the diameter of the orifice

becomes smaller the flow becomes fully turbulent, with an

increase in turbulence intensity and range of frequency

as interpreted previously from the visual observations.

An increase in flow rate maintaining the same orifice

diameter indicates an increase in turbulence intensity

and the frequency range.

The comparison of the turbulence spectra for the ori-

fice diameter of 3/16 (Dl= 1/2) at different probe locations
is shown in Figure 3.18 for a Reynolds number of 1600. In order

to show the difference in the shape of the spectra clearly,

the spectra were nomalized by dividing by the area under

the curve. It is shown that as the flow moves down-

stream, the eddies with high frequencies disappear, while

there is large drop in turbulence intensity, indicating that

laminar flow is gradually developing.

The turbulence spectra as described in Figure 3.18 for

different axial positions of the probe can be correlated

with Figure 3.11 d describing the position of the highly

disturbed region for a constriction diameter ratio of 1/2

in the 1.90 cm pipe diameter. At a Reynolds number of

1600 the turbulent region begins at about 1.8 cm from the

constriction edge and its length is about 8.5 to 9 cm. In

this case the probe located 2.54 cm from the constriction

is inside the highly active region, and turbulence inten-

sity and range of frequency are high. The probe located

at 10.2 cm from the constriction edge is closer to the

downstream boundary of the locally turbulent region where

turbulent activity begins to decay. The probe located at

about 20 cm is outside the turbulent region, where the flow

is laminar-disturbed and laminar flow is gradually developing.

Kim and Corcoran also conducted experiments trying to

understand the contribution of flow turbulence and the wall

to the spectrum of sound. Preliminary experiments on a

simulated constriction in a latex tube using a contact type

microphone demonstrated that the sound spectra and the




10 100





o 1TT

S 3/8

a 1/2

* 2/3

< 5/6





101 >


10 100





Figure 3.17 Comparison of Turbulence Spectra




Axial Position
o 2.5 cm

>1 10 20.3
,H 0



10 100 1000

Re = 1600

Figure 3.18 Turbulence spectra at various
probe locations

turbulence spectra were quite different for the same flow

rate and orifice diameter. Peak frequencies appeared in

the sound spectra which were not detected in the turbulence

spectra, indicating a strong dependency of the sound spectra

on the vibrations mode of the test section determined by the

geometry and physical properties of the tube.

Onset of Sound in Flexible Tubes

Gonzalez (1974), in his investigations on the origin of

Kortkoff sounds, studied the onset of sound in flexible

tubes locally constricted to simulate a pressure cuff applied

to an artery. The experimental test sections were made of

clear vinyl tubing constricted at the midpoint by squeezing

the sections between two square plexiglass plates held

together by screws at the corners. The edges of the plates

pressing against the tubing were rounded off so that the

contour of the ends of the constriction was smooth and


Steady flow of water at constant pressure head was pro-

vided by an elevated reservoir tank. Flow rate was regulated

by adjusting the exit height, in relation to the reservoir

tank, and measured by collecting the flow in a graduated

cylinder during a known time interval.

Sounds were detected externally via a stethoscope rest-

ing against the tube walls downstream of the constricted

region. The length of the constriction was 4.0 cm and test

sections of various diameters. The method of pressing

flat plates against a round tube produced a non-circular

constriction in a circular tube, and the cross-sectional

area had to be calculated prior to each experiment from

measurement of the vertical clearance under the constric-


According to Gonzalez, resting the stethoscope on

the tube walls just distal to the constriction, flow

noise having a murmur-like character could be heard

only under certain flow conditions. The sound intensity

could be increased by increasing the flow rate or by

decreasing the constriction cross section at different

positions along the tube, and it always decreased in

intensity as the distance from the constriction was

increased. Onset of sound was determined by adjusting

the constriction size and increasing the flow rate until

sound was noticed downstream. Then starting with the

maximum flow, the rate was decreased until the sound dis-

appeared. The average of the two measured flow rates was

recorded as the critical flow rate for the onset of sound.

A linear relationship between critical flow rate and area

ratio was obtained for all tests with sound disappearing

at area ratios between 0.6 and 0.70.

In Figure 3.16 the onset of flow disturbances, as

obtained by our study, was compared with Sacks' sound

onset curve in arteries using an internally placed micro-

phone pressure transducer. In Figure 3.19 the critical

flow rate at the onset of disturbances is compared with

the critical flow rate for the onset of sound (Gonzalez's

experiment) as detected bya stethoscope resting against

the tube walls. Also shown in the same figure are the

approximated magnitude of the Reynolds number when turbu-

lence appeared to be generated right after the constric-

tion exit.

The measurements taken on the 0.67 cm ID tube and

the measurements taken on the 1.27 cm ID pipe are both

shown. Both presented the same general behavior, showing

that the onset of sound as detected by the stethoscope is

well above the critical flow rate at the onset of disturb-

ances as obtained from visual observations.

For the small area ratios (high degrees of contraction)

the flow rate at the appearance of localized turbulence is

close to the critical flow rate at the onset of sound,

indicating that for the highly constricted tubes the

turbulence pressure fluctuations are of sufficient initial

intensity and frequency range to produce wall vibrations

that can be detected by the stethoscope. This was pre-

viously confirmed by our visual observations and Kim's

experiments on rigid tubes. As the area ratio increases

maintaining the same flow rate, turbulence intensity and

frequency decreases (see Figure 3.17) and no sound can be

detected any longer.

As expected, when the area ratio was increased higher

flow rates were needed to produce sound that could be de-

tected by the stethoscope, but the sound boundary curve

shows a steeper slope and for an area ratio larger than 0.20

the flow rate was above the curve describing the flow rate

when the turbulent region developed right at the jet exit

location. Similar behavior was observed in the larger dia-

meter pipe as shown in Figure 3.19.

The critical flow rate at the onset of sound, deter-

mined by means of an internal probe in arteries, showed fair

correlation with the critical flow rate at the onset of

disturbances in this study, while the critical flow rate

at the onset of sound determined externally with a contact

microphone in vinyl test sections deviated considerably,

always occurring at relative large flow rates.

The external contact microphone actually detects the

walls vibrations produced by pressure oscillations in the

jet turbulence. Thus, the sound spectrum in this case will

depend strongly on the vibration mode of the test section,

determined by the geometry and the wall elastic properties,

as observed previously by Kim and Corcoran.

The absence of sound (as detected by a contact micro-

phone) for large area ratios could be explained by looking

at Figures 3.17c, d. For large area ratios the sound in-

tensity and frequency range is greatly reudced even at the

large Reynolds numbers. In some cases the sound intensity

is not only very small but also the frequency range is

below the audible threshold (about 20 Hertz).

- Turbulence at
the Orifice



Flexible Tubes
o Do=1/2"

o Do=1/4"

d Present Investigation
a Do=1/2

v vDo = 1/4

Onset of disturbances





Figure 3.19 Comparison with the onset of sounds in
flexible tubes.








Pressure Drop Measurements


These experiments were performed as a direct contin-

uation of the flow visualization experiments. The main

purpose of these investigations was to determine (a) the

pressure drop along a locally constricted region within a

tube of circular cross section, and the dependency of the

pressure drop on the constriction geometry and the Reynolds

number, (b) the pressure recovery region downstream from

the constricted region, and (c) the correlation (if any)

of pressure drop characteristics with features of the

flow pattern observed visually. The experiments were

carried out using the same apparatus used in the flow

visualization experiments. The fluid used in all tests

was distilled water.

Investigations on the flow behavior and the pressure

drop across locally constricted cylindrical test sections

is an old problem in fluid dynamics, mostly directly to

the establishment of the orifice as a flow meter. The

present investigation is more oriented towards physiology,

so the flow Reynolds number has been maintained inside the

arterial blood rate of flow range. Thus, the range of

Reynolds number considered was approximately 200-2500.

Since the flow of blood in arteries is unsteady,

arterial walls are distensible, and blood is a suspension,

studies based in the steady flow of a Newtonian fluids

(water) in rigid tubes must be considered, at best, a

first approximation to the biological problem.

The pressure drop across a constricted region in a

circular section has strong physiological implications.

One of the most serious consequences of an arterial steno-

sis is the large pressure loss which may develop across a

severe stenosis. The reduced pressure distal to the sten-

osis significantly alters the blood flow to the peripheral

beds supplied by the artery. The pressure loss is primar-

ily dependent on the flow rate and the geometry of the

stenosis since the fluid properties of density and apparent

viscosity are relatively constant. Geometrical factors

like the area ratio (degree of stenosis); the length of the

stenosis; and some measure of the eccentricity could dras-

tically change the pressure drop across the region.

Although arterial blood flow is unsteady, the effect

of the unsteadiness on pressure drop seems to be small for

a severe stenosis, i.e. one for which the reduction in

area is greater than ca, 80% (Young, 1973). It is well

established that under normal physiologic conditions the

effect of the stenosis on flow and pressure is small unless

the stenosis is severe. Thus the pressure drop experiments

with models 8 and 9 (Table I) represent essentially obser-

vations on the moderate to severe stenosis.

The influence of the geometric characteristics upon

the pressure drop, along the constricted region were

studied by a series of experiments on the flow of water

through the 1.27 cm test section. A dimensional analysis

shows that the pressure drop due to a single constriction

in a conduit can be studied in terms of the dimensionless


Ai Do Do Do ( )

Where Ap is the pressure drop over the length L

p fluid density

U mean velocity in the unobstructed tube

Re Reynolds number, DoU/v where v is the

fluid kinematic viscosity

Ao the unobstructed area

Ai constriction flow cross section

Zo length of the constriction

e eccentricity

The length of all the models used was constant and

equal to 3 cm, and the hole drilled along the plub center-

line (zero eccentricity). The pressure drop was measured

over a constant pipe length of 28.4 cm. This reduced the

number of non-dimensional variables to two, the Reynolds

number and the non-dimensional area ratio. The area ratio

was changed by trying different models on the 1.27 cm pipe.

Reynolds number was varied by adjusting the needle valve

controlling the rate of flow.

Pressure readings were taken at different positions

located upstream and downstream of the constricted region

by means of pressure taps separated at a distance in

multiples of the pipe diameter from the constriction edges.

The pressure tap locations are shown in Figure 3.20.

Pressure taps were also made of acrylic glass and machined

to fit the pipe circumference. The top holes in the tube

walls were approximately 0.5 mm in diameter.


-10 -3 -1 0 0 1 2 3 5 7 10

3 cm

Constriction geometry

Figure 3.20

Experimental Results

The pressure drop across the constricted region was

measured using the pressure taps located at +10 Do from

the constriction edges. Since the diameter of the pipe

was 1.27 cm ID and the length of the constriction 3 cm

this resulted in a magnitude for o of approximate 22.3.

This insured that the pressure drop was measured over a

sufficient length for pressure recovery downstream from

the constriction.

For a given constriction, the dimensionless pressure

drop is a function only of the Reynolds number. The

experimental results obtained using models 8, 9, and 10

are shown in Figure 3.21. The general behavior of the

dimensionless pressure drop curves was found to be the

same for all models. At low Reynolds number the pressure

drop is due primarily to viscous effects and the dimension-

less pressure drop decreases with increasing Reynolds

number. Laminar separation may take place at low Reynolds

numbers. With an increase in Reynolds number the flow

becomes locally unstable, and disturbances begin to propa-

gate in the flow.

At higher Reynolds number turbulence develops and

the dimensionless pressure loss along the constriction

is due primarily to turbulence. Thus at high Reynolds

numbers, turbulent losses are dominant and the dimension-

less pressure drop coefficient becomes independent of the

Reynolds number. The pressure drop coefficient illus-

trated in Figure 3.21 includes also the measured pressure

loss in the unobstructed section (Poiseuille pressure


Assuming laminar flow up to a Reynolds number of

2000, the flow rate Q is given by

Q = (Ap/L) (3.3)

Where Ro is the radius of the unobstructed pipe.

Expressing the pressure drop in dimensionless form

and replacing the flow rate by the equivalent Reynolds

number in the experimental pipe, we obtain

L 1
S= 32 (D ) Re) (3.4)

for this case (L/Do) is 22.3 and the dimensionless laminar

pressure drop coefficient given by

S= 713.6 (1) (3.5)

The strong influence of the area ratio upon the

pressure drop is clearly shown through a comparison of

the data for different constrictions in Figure 3.21. For

example at a Reynolds number of 1000 the dimensionless

pressure droD for the heavily constricted model (8) is

aDvroximately 20 times the corresponding value for the

moderately constricted model (10).

No information about the effects of the constriction

length upon the pressure loss can be obtained from these

set of experiments, since this length was the same for all

constrictions. Experiments conducted by Young and Tsai

(1973) on cylindrical test sections demonstrated that the

effect of this parameter is small in the range of Reynold

number studies (100-5000), for rather constricted tubes

having the same area ratios (89%) but different length

to pipe diameter ratio, (Z0/Do = 1,2).

However, experiments conducted by Robbins and Bentov

(1967) on the effect of length of stenosis on flow volume

(using cylindrical models in a 3.3 mm pipe diameter) in-

dicated that the length of the constricted region has a

rapidly increasing influence with increasing severity of

stenosis (small area ratios). When severe levels of

stenosis were reached, such as 90 per cent, increasing

the length of the stenosis from 0.5 to 2.0 cm results in

a decrease of the flow rate by 50%.

The dimensionless axial pressure distributions down-

stream of the constricted region are shown in Figures

3.22a, b, for different area ratios and Reynolds numbers.

Pressure readings were taken at several locations along

the tube as shown in Figure 3.20. The closest pressure

taps were located at a distance of +1 pipe diameter from

the constriction edges; no readings were taken inside the

constriction models. It is noted that as the fluid

approaches the contraction, it accelerates which results

in a rapid pressure drop.

In the downstream side the pressure variations are

smoother and recovery takes place over a greater tube

length. The minimum pressure occurs inside the constricted

flow region. An important point is, that most of the

pressure loss occurs downstream from the abrupt change in

corss-sectional area, where the jet formed by the sudden

contraction attains a minimum area.

The minimum jet cross section area, the so-called

"vena contracta" is located about one constriction diameter

downstream of the constriction entrance (Whitaker, 1968).


D /Do

o 1/4

A 3/8

o 1/2




96 0o o

0 0 0 0 0 0 0

A& A '4 A
AA A 1a A A. A AAA ^A A A
a Lb *p a dl) P dd, a ~a 6

o 00 0 0

0 01 o0 ono on o 0

400 800 1200 1600 2000


Dimensionless pressure drop

0o 0


Figure 3.21