1
A STUDY OF OSCILLATORY FLOW THROUGH
A BRONCHIAL BIFURCATION
by
Darrell L. Jan
B.S., University of California at Berkeley
(1978)
S.M., Massachusetts Institute of Technology
(1980)
SUBMITTED IN PARTIAL FULFILLMENT
OF 'THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
IN MECHANICAL ENGINEERING
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 9,
1986
@ Massachusetts Institute of Technology 1 986
Signature redacted
.........-.
Signature of Author..
.. .
,.
Department of M
--
............
Signature redacted
.
.....
. . .... ... . ...
. .. . . . .. . . . . . . . . . . . .
Roger D. Kamm
Thesis Supervisor
Accepted by....
Signature redacted
Ain . Sonin
Chairman, Department Graduate Committee
VASSACHUSETTS INSTiTUTE
OF TECHNOLOGY
MAR 0 9 1987
LIBRAWES
Archives
.
Certified b y ......
.
anical Engineering
2
A STUDY OF OSCILLATORY FLOW THROUGH
BRONCHIAL BIFURCATION
A
by
Darrell Leslie Jan
Submitted to the Department of Mechanical Engineering in
September, 1986, in partial fulfillment of the requirements
for the degree of Doctor of Philosophy.
ABSTRACT
Air that flows through the human lung passes through a
multi-generation dividing network, a single unit of which is
called a bifurcation.
The geometry of the bifurcation,
which includes changes in cross-sectional shape and flow
around a
bend, causes a complicated three-dimensional
structure to
the flow field.
This flow field has
previously been assumed to be quasi-steady, even under the
conditions of High Frequency Ventilation (HFV, a relatively
new form of respiratory therapy.) For this study, a threedimensional flow visualization
technique was developed to
allow quantitative and
qualitative examination of the flow
field in a model
bronchial bifurcation, in the range a-321.3 and Ax/d=0.75-19.6, conditions which are encountered in
the lung during HFV. Here a -a(w/v)1/2 is the unsteadiness
parameter,
with "a" the
tube
radius, w the natural
frequency, and v the kinematic viscosity,
and Ax/d is the
local stroke length nondimensionalized by the tube diameter
"d". The flows are divided into 4 regimes, each denoted by
its distinguishing characteristic: quasi-reversible, quasisteady, turnaround zone (transitional), and confined vortex.
These regimes are found, by
comparison to curved tube
theory, to represent the dominance
of one or more of
unsteady, viscous, or curvature forces.
The confined vortex
regime exhibits dependence on acceleration, contradicting
assumptions of quasi-steadiness.
The ranges of tidal
volume and frequency for which the various generations in
the lung experience
the
different
flow regimes are
determined. Based on the characteristics of the flow, a
streaming-mixing model for mass transport is suggested.
Thesis supervisor: Dr. Roger D. Kamm
Title: Professor of Mechanical Engineering
3
Acknowledgements
Much appreciation is due to my advisor, Prof. Roger D.
Kamm, my unofficial advisor,
Prof. Ascher H. Shapiro, and
the other members of my thesis committee, whose criticisms
were most highly
valued.
The construction of the
bifurcation is one of many tasks credited to Richard Fenner,
whose sense of humor was equally important.
I also wish to
acknowledge the small army of undergraduates who helped
enter the data,
as well as special effort from Claire
Sasahara. Many members of the Fluids Lab contributed to an
intellectual,
social,
and
supportive
atmosphere.
A
particular nod goes to my hang-gliding partner and to my
current officemate. The contribution of my fiance, Dr. Ann
M. Neville, was undoubtedly the most expansive.
I thank my
parents, Dr. and Mrs. Dewey Jan, for their continued
patience and support.
This work was supported by a grant from NHLBI, number
PO1-HL-33009.
4
TABLE OF CONTEETS
Abstract
2
Acknowledgements
3
List of figures
5
I.
13
Introduction.
II. Experimental apparatus and techinques.
24
III. Background theory.
44
IV. Qualitative results.
62
V. Quantitative results.
70
VI. Discussion.
83
VII. Mass transport implications.
106
VIII. Conclusions.
124
References
126
Appendix
133
Figures
134
5
List of Figures
Figure 1.1
Qualitative picture of inspiratory flow in
a bifurcation with a Poiseuille entry profile. The 2-vortex
secondary flow pattern in the daughter tube is shown in the
lower branch; the upper
of the axial profile.
branch
shows the "horseshoe" shape
Figure 1.2
Qualitative picture of expiratory flow. A
4-vortex pattern can be seen in the parent tube. The shaded
area is a region of higher axial velocity that extends
vertically.
Figure 1.3
Steady inspiratory
flow velocity data
measured in a
daughter
branch, having an additional
bifurcation upstream.
Re=460.
Left: secondary velocity
field.
Right: axial velocity contours.
All velocities
referenced to the local mean velocity.
The outside of the
bend, or inside of the bifurcation, is at the bottom. (from
Isabey and Chang, 1982).
Figure 1.4
Steady expiratory flow in the trachea.
Re=1060.
The figure is oriented with the daughter tubes in
the plane of the L-R axis. Left: secondary velocity field.
Right: axial velocity contours.
All velocities referenced
to the local mean velocity. (Isabey and Chang, 1982).
Figure 2.1
Idealized
bifurcation, after Pedley.
dimensions
of
a
lung
Figure 2.2
Solid diamonds:
The range of o and Ax/d
encountered in the Weibel lung model under HFV conditions,
frequency = 5 hz and tidal volume = 30 ml. The zeroth
generation begins at the right, the highest generations are
at the lower left.
Open squares: The range of parameter
values covered in this study.
6
Figure 2.3
Diagram of experimental apparatus.
The
mirror provides an additional side view image to the camera
above.
Figure 2.4
Coordinate system used in the experiments.
Figure 2.5
Experimental data for oscillatory flow in
a straight tube compared to the theoretical solution of
Uchida. Dots are experimental data at 6 past peak flow,
squares are data at 51 past peak. Uchida solutions: 60--A-Bcurve; 51 0 --D- curve.
Figure 2.6
experiments.
Range
of
parameter
Figure 3.1
Toroidal coordinate
curved tube geometries.
Figure 3.2
curved tube.
Illustration of
Figure 3.3
Flow regimes
flow identified by dominance
Yamane et al, 1985.
Figure 3.4
Plots of
(above) and axial velocity
curve (below) for 3 sets of
1985).
in
of
values
system
pressure
covered
for
in
use with
balance
in a
oscillatory curved tube
various forces.
After
secondary flow streamfunction
profiles in the plane of the
a and Dn values. (Yamane et al,
Figure 3.5
Maximum value
of
the secondary flow
streamfunction, W max, versus a and Dn.
(Yamane et al,
1985).
Figure 3.6
Diagram showing secondary
from vorticity at the inlet to a bend.
flow
arising
7
Figure 3.7
Variation of axial shear at the inner bend
1/2
with axial distance, n=(1-r)(0.5 Dn)/.
Soh and Berger's
result is the solid line; Stewartson et al's boundary layer
calculation is the dashed line.
The axial velocity w is
normalized by the bulk axial velocity. Dn=680.3, Re=900,
a/R=1/7.
(Soh and Berger, 1984).
Figure 3.8
(a)secondary velocity field,
41.50 past
entry.
(b)axial velocity contours,
2670 past entry. Flat
inlet profile, Dn=680.3, Re-900, a/R=1/7.
Velocities are
normalized to the bulk axial velocity. The outside of the
bend is to the right.
(Soh and Berger, 1984).
Figure 4.1
Diagrams of the
4 flow regimes.
(a)
Reversible flow.
(b) Quasi-steady flow. Secondary flow
patterns in the cross-section are shown. (c) Transitional
flow exhibiting turnaround zone. (d) Confined vortex flow.
Figure 4.2
Particle streak
photographs of quasisteady flow. a=2.3, Ax/d-19.5 (Rectangular lines at the flow
divider are due to a seam in the construction.
Some
calibration marks can be seen.)
(a) End expiration. (b) 450
past end expiration. (c) Peak inspiratory flow. (d) 450 past
peak inspiration.
(e) End inspiration.
(f) 450 past end
inspiration. (g) Peak expiratory flow.
(h) 450 past peak
expiration.
Figure 4.3
Transitional flow.
Photographs taken from
videotape data. Half the bifurcation can be seen in a top
view above, a side view image is below. a=6, Ax/d=9.6 (a)
End expiration.
(b) 450 past end expiration.
(c) Peak
inspiratory flow. (d) 450 past peak inspiration. (e) End
inspiration.
(f) 450
past
end inspiration.
(g) Peak
expiratory flow. (h) 450 past peak expiration.
8
Figure 4.4
Confined vortex flow.
Streak photographs
of top view. a-21.3, 6x/d=9.7 (a) End expiration. (b) 450
past end expiration. (c) Peak inspiratory flow. (d) 450 past
peak inspiration.
(e) End inspiration.
(f) 45 past end
inspiration. (g) Peak expiratory flow.
(h) 450 past peak
expiration.
Figure 4.5
Occurrence of
function of a and Ax/d.
the
Figure 4.6
Yamane
et
al's
oscillating curved tube flow plotted
Ax/d.
Figure 4.7
and Dn.
4
flow
regimes
as a
flow
regimes
for
on a graph of a and
Flow regimes of this study on a graph of a
Figure 5.1
Top view of streak data. Regime 1, quasisteady and quasi-reversible. a=2.3, Ax/d=0.75
. (a)peak
inspiration. (b) peak expiration.
Figure 5.2
Secondary flow magnitude vs distance. The
dashed line represents the expected noise level. Regime 1.
a=2.3, Ax/d=0.75 (a) peak
inspiratory flow.
(b) peak
expiratory flow.
Figure 5.3
Axial velocity contours in the parent
tube. Velocities are normalized by peak bulk axial velocity.
Regime 1, a=2.3, Ax/d=0.75.
Figure 5.4
Top view of streak
unsteady and quasi-reversible. a=21.3,
inspiration. (b) peak expiration.
Figure 5.5
Secondary
Regime 2, a=21.3, Ax/d=0.75
peak expiratory flow.
.
data.
Regime 2,
Ax/d=0.75 . (a)peak
flow
magnitude vs distance.
(a) peak inspiratory flow. (b)
9
Figure 5.6
Regime 2, a=21.3, Ax/d=0.75
.(a) Axial
velocity contours in the parent tube.
(b)Secondary flow
field. The distance between nodes corresponds to 0.25 times
the peak bulk axial velocity.
Figure 5.7
Top view of streak data.
Regime 3, quasisteady,
curvature
effects.
a=2.3,
Ax/d=19.5
(a)accelerating flow, 450 before peak inspiration. (b) peak
inspiration. (c) 450 past peak inspiration. (d) accelerating
flow, 450 before peak expiration.
(e) peak expiration. (f)
450 past peak expiration.
Figure 5.8
Secondary flow
magnitude vs distance.
Regime 3, a-2.3, Ax/d=19.5
.
(a)accelerating flow, 450
before peak inspiration. (b) peak inspiration. (c) 450 past
peak inspiration.
(d) accelerating flow, 450 before peak
expiration.
(e) peak
expiration.
(f)
450 past peak
expiration.
Figure 5.9
Axial
velocity
contours
(above) and
secondary flow field (below). Parent tube, Regime 3, a=2.3,
Ax/d=19.5
(a)
accelerating
flow,
450
before
peak
inspiration.
(b) peak inspiration.
(c) 450 past peak
inspiration.
Figure 5.10 Axial
velocity
contours
(above) and
secondary flow field (below).
Daughter tube, Regime 3,
a=2.3, Ax/d=19.5.
(a) accelerating flow, 450 before peak
inspiration.
(b) peak inspiration.
(c) 450 past peak
inspiration.
Figure 5.11 Axial
velocity
contours
(above) and
secondary flow field (below). Parent tube, Regime 3, a=2.3,
Ax/d=19.5 (a) accelerating flow, 450 before peak expiration.
(b) peak expiration. (c) 450 past peak expiration.
10
Figure 5.12 Axial
velocity
contours
(above) and
secondary flow field (below).
Daughter tube, Regime 3,
a=2.3, Ax/d=19.5 .
(a) accelerating flow, 450 before peak
expiration.
(b) peak
expiration.
(c)
450 past peak
expiration.
Figure 5.13 Top view of streak data.
Regime 4,
unsteady,
curvature
effects.
a=21.3,
Ax/d=9.7
(a)accelerating flow, 450 before peak inspiration. (b) peak
inspiration. (c) 450 past peak inspiration. (d) accelerating
flow, 450 before peak expiration.
(e) peak expiration. (f)
450 past peak expiration.
Figure 5.14 Secondary flow
magnitude vs distance.
Regime 4,
a=21.3, Ax/d-9.7 . (a)accelerating flow, 450
before peak inspiration. (b) peak inspiration. (c) 450 past
peak inspiration.
(d) accelerating flow, 450 before peak
expiration.
(e) peak
expiration.
(f)
450 past peak
expiration.
Figure 5.15 Axial
velocity
contours
(above) and
secondary flow field (below).
Parent tube, Regime 4,
a=21.3, Ax/d=9.7 (a) accelerating flow, 450 before peak
inspiration.
(b) peak inspiration.
(c) 450 past peak
inspiration.
Figure 5.16 Axial
velocity
contours
(above) and
secondary flow field (below).
Daughter tube, Regime 4,
a=21.3, Ax/d=9.7 .
(a) accelerating flow, 450 before peak
inspiration.
(b) peak inspiration.
(c) 450 past peak
inspiration.
Figure 5.17 Axial
velocity
contours
(above) and
secondary flow field (below).
Parent tube, Regime 4,
a=21.3, Ax/d=9.7 (a) accelerating flow, 450 before peak
expiration.
(b) peak
expiration.
(c)
450 past peak
expiration.
11
Figure 5.18 Axial
velocity
contours
(above) and
secondary flow field (below).
Daughter tube, Regime 4,
a=21.3, Lx/d=9.7 .
(a) accelerating flow, 450 before peak
expiration.
expiration.
(b)
peak
expiration.
(c)
450
past
peak
Figure 5.19 Stereo pairs.
Regime 1, quasi-steady and
quasi-reversible. a=2.3, Ax/d=0.75 .
(a)peak inspiration.
(b) peak expiration.
Figure 5.20 Stereo pairs.
Regime 2, unsteady and
quasi-reversible. a=21.3, Ax/d=0.75 . (a)peak inspiration.
(b) peak expiration.
Figure 5.21 Stereo pairs.
Regime 3, quasi-steady,
curvature effects. a=2.3, Ax/d=19.5 . (a)accelerating flow,
450 before peak inspiration.
(b) peak inspiration. (c) 450
past peak inspiration.
(d) accelerating flow, 450 before
peak expiration.
(e) peak expiration.
(f) 450 past peak
expiration.
Figure 5.22 Stereo pairs.
Regime
4, unsteady,
curvature effects. a=21.3, Ax/d=9.7 . (a)accelerating flow,
450 before peak inspiration.
(b) peak inspiration. (c) 450
past peak inspiration.
(d) accelerating flow, 450 before
peak expiration.
(e) peak expiration.
(f) 450 past peak
expiration.
Figure 6.1
Flow regimes encountered in the Weibel
lung in normal quiet breathing.
(15 breaths/minute, 10
liters/minute). Each symbol represents a Weibel generation,
from the zeroth generation at the right and proceeding into
the lung to the left.
Figure 6.2
Effect of
varying
regimes encountered in the Weibel lung.
frequency
on
flow
12
Figure 6.3
Effect of varying tidal
regimes encountered in the Weibel lung.
volume
on flow
Figure 6.4
Flow regimes encountered in the Weibel
lung for a wide range of HFV conditions. Frequency=5-15 Hz,
tidal volume=10-100ml.
Figure 7.1
Experimentally
determined
normalized
transport coefficient for a branching network, compared to
Watson's theory for a straight tube,
as a function of a.
(Kamm et al, 1984b).
Figure 7.2
Normalized transport coefficient for a
branching network as a function of P2 compared to Watson's
theory.
(Slosberg, 1983).
a
.
Figure 7.3
Normalized transport coefficient for
branching network, as a function of 82 (Paloski, 1986).
Figure 7.4
Normalized transport coefficient for an
anatomical branching network,
as
a
function of
32.
(Keramidas, 1986).
Figure 7.5
Comparison of the normalized transport
coefficient predicted by Watson to that of the streamingmixing model with a blunt velocity profile with approximate
Stokes boundary layers.
13
I. Introduction
The lung is
the
During normal breathing, fresh
drawn through a complex
airways
eventually
air
in
terminate
surface (the alveolar zone)
enters
a
within
large
mass
transfer
which oxygen and carbon
On its journey from
blood.
the
the lung and is
system of airways. These
branching
dioxide are exchanged with
organ of gas exchange.
body's
human
the mouth to this gas
exchange region, gas passes through a
branching
single
system
(a
bifurcation) producing an
terms of the shape
of
unit
of
axial
is
called a
pattern, both in
flow
intricate
the
which
velocity profile and the
creation and changes of various secondary motions.
The conventional
explanation
exchange employs a
bulk
completely ignores
any
field.
The tidal
flow
for
model
effects
,
volume
or
of
normal, healthy gas
(eg,
the
volume
West, 1979) and
detailed velocity
of air inhaled per
breath, is simply larger than the volume of the airways (the
dead space); thus fresh
air
reaches the alveolar zone with
each breath. This approach is
largely justified in the case
of normal tidal
However, these detailed flows
play
an
breathing.
important
role
conditions, including
in
High
gas
transport
under certain
Frequency Ventilation (HFV), a
technique of artificial respiration using higher than normal
breathing
frequency
and
tidal
volumes
too
low
for the
successful gas exchange of this technique to be explained by
the traditional
bulk
model.
The
details
of the velocity
14
field
in
the
lung
are
doubtlessly
also
important
in
determinating the locations of deposition of inhaled aerosal
particles.
In this study we examine the velocity field produced by
purely oscillatory flow through
bifurcation for
various
a
flow
regimes.
relevant not only to pulmonary
flow through arterial
relates
also
to
model
gas
These
results are
flow, but also to blood
bifurcations.
the
of a typical lung
Thus the present study
considerable
body
of
literature
addresing the fluid mechanics at cardiovascular junctures to
the formation of atherosclerotic lesions.
In the remainder of this chapter, we review the prevous
studies of bifurcation
flow
to
provide
a context for the
present work.
A. Steady vs. unsteady conditions in the lung.
In spite of the probable importance of unsteady effects
during many pulmonary
previous work has
maneuvers,
assumed
Consequently, many have
under which the
1977),
using
system developed
by
to
model
Weibel
(1963),
flow
a
in
parameter is defined as
a = a(w/v)1/2;
or quasi-steady flow.
identify the conditions
assumption
symmetric
parameter,
unsteady effects in
steady
sought
quasi-steady
the
value of the a
a
including HFV, nearly all
of
is valid.
Pedley
the lung branching
first
calculated the
measure
of the importance of
a
straight tube. The a
long
15
where a is the tube
oscillation,
and
radius,
v
is the natural frequency of
w
is
the
parameter represents a ratio
kinematic
viscosity.
This
of unsteady inertial forces to
viscous forces, and can also be thought of as a ratio of the
pipe
radius
to
the
Stokes
boundary
layer
thickness,
(v/w)1/2. Under conditions of
normal quiet breathing, about
15 breaths/minute (0.25 Hz),
a<
the second. For
panting
(180
a
faster
1 in all generations past
breathing
breaths/min,
or
generations. However, Pedley
boundary layer develops on
3.0
then
the
Hz),
argued
use
and Sudlow (1969),
u/(v/w)1/2,
=
a>
1
in
many
that since a new
carina of each bifurcation,
the correct parameter to
P
rate corresponding to
is one introduced by Schroter
where
a = the steady
boundary layer thickness. The thickness a is approximated by
(vL/u) 1 /2 , where u is the mean velocity and L is the length
of te hus
tue. 3
of the tube. Thus
(wL/u)
. Pedley actually used
=1/2
6E
2
and found it to be
even during rapid
much less than 1 for all airways
breathing.
therefore seems justified
The quasi-steady assumption
for
these two conditions; others
will be addressed later in this study.
A
time-varying
enhancement to the P criterion was introduced by Chang et al
(1985).
They
claimed
neccessary for a complete
that
the
following
parameter was
description of the relevant fluid
mechanics:
L(dU(t)/dt)
E(t)
2
U(t)
16
where L is the distance
from
the flow divider, and U(t) is
the mean axial velocity. (In Isabey
U(t) is replaced by a
constant
vicinity of flow reversal,
of the relative importance
f
for
parameter e(t) is an estimate
but
with
variation
the cycle can be examined.
<<1
to avoid division by
of unsteady forces to convective
above,
variability so that the
Chang, et al 1986,
representing a value in the
apparently
zero at flow turnaround.) The
forces, as was
and
They
quasi-steadiness.
the
inclusion of time
of this parameter within
reasoned that E(t) must be
For
an
oscillating sinusoidal
flow, E(t) is smaller at peak velocity and largest near zero
flow. Thus it indicates that quasi-steadiness is most likely
to be
found
at
peak
velocity
and
dominant at zero flow. Measurements
that
having
unsteadiness is
an
value up to
16 were made at peak velocity and were shown to compare well
to steady
flow
data
at
the
number. However, it is not
valid measure of
same
shown
to
quasi-steadiness
cycle, and they do not address
instantaneous Reynolds
what degree E(t) is a
at
the
other phases of the
question of whether the
cycle as a whole can be considered primarily quasi-steady or
primarily unsteady.
Also, the character of a truly unsteady
regime in bifurcation flow has not been described, as far as
we know.
Our
study
examines
oscillatory cycle and
tests
for quasi-steadiness based on
shown that at least
two
several
the
a,
phases
during
the
validity of the criterion
#,
parameters
or
E(t).
It will be
are needed to describe
17
the flow completely.
We
will also characterize the nature
of the flow when it cannot be considered quasi-steady.
B. Previous work.
Steady flow experiments. Some
measurements in a bronchial
Schroter and Sudlow,
horizontal (in the
who
bifurcation
used
plane
axial flow profiles.
of
Using
demonstrated the existence
motion in the daughter
cell vortex pattern
in
of the earliest velocity
a
hot
model were made by
wire
probe to find
the bifurcation) and vertical
smoke
of
visualization, they also
the
2-cell vortex secondary
tubes
during inspiration and the 4-
the
parent tube during expiration.
One can form a qualitative description of steady inspiratory
and expiratory flow
based
on
their
results
and those of
Pedley, Schoroter, and Sudlow (1971).
The incoming flow
is
streams, forming developing
wall of each
branch.
split
by
boundary
the
layers
The
flow
entering
around a bend, causing the
flow
to
tube: the faster moving
fluid
moves
divider into two
on the inside
each branch goes
develop as in a curved
toward the outside of
the bend, which is the inside of the bifurcation, and slower
moving fluid moves in
flow pattern forms
the opposite direction. The secondary
two
counter-rotating vortices, as shown
in Fig. 1.1. The axial velocity profile in the daughter tube
takes on a "horseshoe"
near the inside
wall
but
along the wall. (Here
to the plane of the
shape,
with the axial velocity peak
spreading
vertical
up
and down somewhat
is defined as perpendicular
bifurcation.) Along a vertical diameter
18
the velocity profile is
M-shaped,
while along a horizontal
diameter the profile exhibits a peak toward the inside wall.
In expiratory
flow,
seen
again 2 vortices form from
merge in the parent tube
a diameter of the
flow
schematically
in Fig. 1.2,
each daughter tube, but they now
to form a 4-vortex pattern. Within
divider, the axial velocity becomes
greater along a vertical ridge. The velocity profile along a
horizontal diameter
reveals
a
peak
in
the center, while
along a vertical diameter the profile is fairly flat.
For either flow direction,
the axial profile shapes
the qualitative features of
were
found
range of parent tube Reynolds
to
be similar over a
(Re = Ud/v where U is
number
the axial velocity and d is the tube diameter) = 50 to 4500.
Schroter
and
Sudlow
also
used
demonstrate the importance of
of curvature at
the
from
non-anatomically low
value
curvature to
radius),
reversal, while a
higher
bend.
100-4500,
of
to
caused
of
1:1
flow
on
(radius of
separation and
4:1
made
Examining a
they found that a
curvature,
value
measurements were nonetheless
visualization
having the appropriate radius
parent-to-daughter
range of Reynolds number
parent
smoke
did not.
a
Their
model having too
sharp a bend.
Olson
measurements
(1971)
of
made
all
3
through
Because
dificulty
the
measurements for only a
detailed
velocity
inspiratory flow
of
very
an
few
components
anatomically
of
the
and
accurate
in
steady
accurate model.
technique,
he
made
values of the Reynolds number,
19
and made no expiratory
are the same
separation
as
described
for
daughter
above.
tube
Transition to turbulence was
The magnitude of
of
secondary
Schroter
flows
This
and
Schroter
and
number =300-1500.
occur for Re0
was
at
>1300.
most about 0.3
is somewhat lower than the
but
who
did
described
not
them
measure
as
being
helical cycle within 3 diameters
divider.
This
The
Sudlow
to
Sudlow,
sufficient to complete one
times the axial flow.
Reynolds
motion
directly,
downstream of the
He reported seeing no
said
secondary
times the axial velocity.
results
The overall features
measurements.
translates to about 0.5
higher secondary motion seen by
is
attributed
nonanatomical sharpness of
their
by
bend;
Pedley
to
the
Olson's values are
probably the correct ones.
Using
a
model
of
an
arterial
branch
which
was
geometrically quite similar to a lung bifurcation, Brech and
Bellhouse (1973) measured
pulsatile flow.
axial
They visualized the secondary flow with dye
streams, noticing what
the outside wall,
seem
of
the
flow, as described in
to
flow
field
in
Chapter
blood
painstakingly following
Balasubramanian et al
through a model
and dye. Their
the
(1980)
carotid
be embedded vortices along
type
different geometries, Karino
the
profiles during steady and
seen
III.
et
in curved tube entry
In similar studies on
al (1979, 1983) visualized
vessel
T-shaped
movement
junctions
by
of tracer particles.
qualitatively visualized flow
bifurcation using hydrogen bubbles
quantitative
data
came
measurement of axial velocity profiles.
from laser doppler
20
Unsteady flow experiments.
relatively little data
unsteady
on
conditions.
As
the
stated above, there is
velocity
Haselton
and
field under truly
Scherer
(1982, also
Scherer and Haselton, 1982) did not study the velocity field
directly,
but
velocity in
motion can
instead
lung-like
be
examined
the
branching
thought
convective streaming
systems.
of
simply
as
expiratory velocity profile
having
a
the inspiratory profile.
may
be
in
location.
either
This
leads
direction,
the
result
of the
different shape than
to a net motion which
depending
on
the
spatial
Haselton and Scherer performed experiments on two
models, one consisting of a
single bifurcation, the other a
multigeneration network in which all
diameter.
This streaming
They
injected
clouds
tubes were of the same
of particles and measured
the net convected motion
per cycle upstream and downstream.
The
per
streaming
distance
cycle
was
at
most slightly
greater than 0.1 times the stroke length (equal to the tidal
volume divided by the cross-sectional area), and was usually
less.
The range studied was a=0.3-19 and Re=0.5-2000 in the
single generation model and
a=0.17-11.8 and Re=0.15-1400 in
the multigeneration model.
In the single generation model,
a 2 /Re was kept nearly constant, in the range 0.18-0.52.
dimensionless stroke length
ranged from
greater in
-1.5-3.5.
the
[Ax/d -
The
range
multigeneration
considering each branch
was examined for the
The
(2/r)(Re/a2)] therefore
of
stroke
length was
model, Ax/d=0.24-7.8, when
individually, however the streaming
model
as
a
whole.
Only the single
21
generation model had a
reasonable diameter to length ratio,
and both
made
models
were
with
very
probably accentuated the secondary
sharp
bends which
motion and increased the
likelyhood of separation, for the reasons discussed above.
Isabey and
inspiratory
Chang
and
multigeneration
(1982)
made extensive 3-dimensional
expiratory
velocity
asymmetric
specifications developed
anatomical
by
Horsfield
Most of their measurements were
as mentioned aboved
they
measurements
model,
on
and Cumming (1968).
that their results also
applied to oscillatory flow at peak flow rates.
the same overall
based
a
made using steady flow, but
verified
more complex geometry, many
in
Despite the
of their measurements exhibited
velocity
patterns
single, symmetric bifurcation.
discussed
above for a
An example of their results
for inspiratory flow in a daughter branch is shown in Figure
1.3.
Although the
velocity
field
another bifurcation upstream, it
character
of
a
2-vortex
horseshoe-shaped
axial
expiratory
pattern
flow
is modified somewhat by
still exhibits the general
secondary
contour
in
shown in Figure 1.4.
The
flow
their model covered a
range
flow was studied only
in
pattern
pattern.
coupled
velocity ridge can be seen
flow
with
The
a
vertical
and
a
4-cell
axial
data taken from the trachea,
in
the
secondary flow magnitude reached
at the various stations in
Re - 656-8846.
Expiratory
trachea, having Re=1060. The
an
average of up to 15.7%
during inspiration and 21.4% during expiration.
22
Theoretical analyses.
by
Pedley
describes
problem. Due to the
symmetric
the
very complete review article
difficulty
complicated
bifurcation,
considered out of the
solutions for
A
low
a
complete
high
the
geometry
question.
and
of
theoretical
of even a single
analytical
theory
is
Olson has found limiting
Re
for
bifurcation where the cross-section
the
section
of the
changes from a circular
shape to an elliptical one, before the daughter tubes branch
off. Both
his
solutions,
require a slowly
and
varying
very simple vorticity
the
improvements by Sobey,
ellipticity.
model
for
which
that resembled experimental results.
inviscid curved tube
flow
Winter and others, using
theory
the
parent tube to daughter
tube
Scherer developed a
he found solutions
His
model is based on
developed
by Squire and
fact that flow traveling from
or
vice
versa goes around a
bend.
The only three-dimensional
that of Wille (1984).
The
numerical solution seems be
flow
condition was for Re=1O, a
sufficiently low Re that nearly
found to be parabolic,
all the axial profiles were
and secondary flows were negligible.
Nevertheless, the solution required
making
large
dimensional
however to
numbers
of
calculations
quote
Pedley,
two months of CPU time,
calculations
exist,
since
eg
the three-dimensional flow, it
is
results. Perhaps this judgment is
Liou
such
excludes all the interesting phenomena
uneconomical.
a
et
al
Two
(1980),
model inevitably
and gives no clue to
not worth describing the
a
bit harsh; at any rate
23
with
machines
techniques will
meantime,
becoming
faster
certainly
however,
and
become
cheaper,
more
experimental
numerical
practical.
investigation
In the
is
a
necessity.
C.The goal of this study.
Our goal in the work is to observe and characterize the
flow phenomena that are found
lung-like
observation
bifurcation.
of
many
in oscillating flow through a
This
is
visualized
quantitative measurement
of
done
flow
by
qualitative
conditions
and
by
the three-dimensional velocity
field for a few selected cases. We examine the criterion for
unsteadiness and study the character
of the flow when it is
truly unsteady.
gain
The
intent
is
to
insight into the
unsteady fluid mechanics that occurs, which in turn suggests
modelling techniques for mass transport.
D. Organization of the thesis.
This work is presented in the following manner: Chapter
I introduces the need for
the
study
of unsteady flow in a
bronchial environment. Chapter II describes the experimental
apparatus
and
techniques
used.
fundamental fluid mechanical theory
The fourth chapter presents
number
of
flow
conditions
dynamic phenomena. The
fifth
Chapter
results of a large
suggest
chapter
with representative quantitative results
flow conditions. Chapter VI
reviews the
relevant to this study.
qualitative
which
III
certain
fluid
gives greater detail
from a sampling of
is a theoretical description of
those results and a discussion of various implications.
24
II. Experimental apparatus and technipques.
A. The idealized model.
Experiments
are
anatomically shaped
performed
on
bifurcation.
a
It
was
single,
rigid,
anticipated that
under many conditions the flow field would be quite complex,
therefore using a single
effects of unsteadiness
adjacent junctions and
symmetric bifurcation isolates the
from
has
further complications of
asymmetry
further study). This model
is rigid, dry, and
the
is
(which are candidates for
further idealized in that it
smooth
walls.
Some remarks on the
appropriateness of these assumptions are made here.
During normal breathing, the length and diameter of the
airways varies roughly as the
cube
root of the lung volume
in the small airways, and considerably less than that in the
large ones (Pedley). As the
tidal volume, is
much
normal breathing, it
real airways
under
quite small. Thus
problem.
True
less
is
in
these experiments than for
likely
our
the
simulated volume per breath, or
that
the volume change in
experimental
rigidity
bifurcation
of
walls
conditions
would be
our model poses little
are
lined
with
a thin
coating of mucus, which makes the surface which contacts the
air extremely
smooth.
In
coating is thin, following
normal
the
coating has very little effect
healthy individuals this
curvature of the walls. The
on the wall shear (Chang and
Menon).
B. The bifurcation and apparatus.
25
A transparent plexiglas model was constructed following
Pedley's description of an idealized lung bifurcation, shown
in Figure 2.1 and described further in the next chapter. The
dimensions are: (1) a parent tube
diameter of 2.54 cm and a
daughter tube diameter of
1.91 cm, for a parent-to-daughter
area ratio of 1.125,
a
(2)
branching
angle of 700,
(3) a
parent-to-daughter bend having a radius of curvature of 15.2
cm,
(4) the change
in
shape
described in the previous
and
area
chapter,
of the parent tube
(5) The flow divider is
fairly sharp.
Briefly, the construction
as follows: Three
drilled into a
holes
plexiglas
block was small enough
reached by hand.
shaped
with
of
The
hand
controlled angles. The
the inside could easily be
of
the
The
a
into
at
that
inside
the
appropriate diameter were
block
so
separately by hand from
the block was cut
the
tools.
into a slot between
of the bifurcation proceeded
flow
bifurcation was then
divider
was
carved
sheet
of plexiglas and inserted
daughter
branches. The outside of
a
flat Y-shape. Inside and outside
surfaces were smoothed and polished.
The ends
of
the
bifurcation
30cm), straight,
parent
tubes terminate
smoothly
connects to a smooth area
driven by a scotch yoke
a sinusoidal
oscillatory
and
connect
daughter
into
change
to long (about
tubes. The daughter
reservoirs. The parent tube
which feeds into a piston
apparatus. The scotch yoke provides
motion
speed. Sinusoidal motion was
when
rotated at constant
chosen for its reproducibility
26
and for comparison to other work. The stroke length Ax/d
adjustable on the scotch
yoke
and 19.5.
driven
The
yoke
is
to
is
values of 0.75, 5.0,9.7,
by
a
1-hp variable speed
electric motor. The frequencies used were 0.1 Hz and 1.0 Hz.
The bifurcation region
contains liquid
having
plexiglas. This tank
is
the
surrounded
same
eliminates
index
optical
by a tank which
of
refraction as
distortion due to
curvature of the outside walls of the bifurcation.
C. Flow parameters and desired range.
Figure 2.2 shows the range of
and Ax/d encountered for
typical application of HFV
to
by Weibel. It is desirable
to collect data both within this
range and to some degree
the idealized lung developed
outside it, for comparison of flow
regimes where HFV is effective to
those where it is not. In
order to take data at convenient frequencies of oscillation,
from about 0.1 hz to
1.0
hz,
kinematic viscosities. It is
we use fluids of 2 different
also
necessary to control the
index of refraction and the density of the fluid, the former
to prevent optical
distortion
latter
neutral
to
permit
particles. Not the least
from
wall curvature and the
buoyancy
important
of
flow visualiztion
is that the liquid must
not have a solvent effect on the apparatus.
Choice of experimental fluid
The desired property values
are a density of
1.23
g/cm3,
1.49, and kinematic viscosity
cases) or
-18
(for
high
for the experimental fluid
index of refraction equal to
of
alpha)
either
-2 (for low alpha
centistokes.
After much
27
experimentation, we
over each
of
determined
the
three
desired
basic component consists
of
ammonium
"fine-tuned" by
fluid. The
index
addition
of
index
of
and
1 part saturated
(NH4SCN).
The
of
amounts of either
small
refraction
measured with a refractometer
to match the
allow control
physical parameters. The
(KSCN)
thiocyanate
the
that
approximately 1 part saturated
aqueous potassium thiocyanate
aqueous
fluids
of
this
solution
is
combination was
(Extech model 2192) and found
refraction
of
plexiglas (1.49) to
better than 1%, which is
probably better than the variation
from sample to sample of
the
acrylic.
The density of the
solution is 1.23 grams/cm3
which
desired value for
particle buoyancy. The viscosity
neutral
of this solution is 2.2
prepared by mixing
is
also within 1% of the
centistokes. A 17.9 os solution was
approximately
part of this basic solution.
2
parts glycerine to one
Again, the solution was "fine-
tuned" to the correct viscosity by addition of small amounts
of either fluid. Pure
glycerine
(nd
=
1.48 and P=1.26) is
already very close to the desired values of refractive index
and density, therefore those
values change very slightly as
the glycerine is added to the thiocyanate solution.
These solutions meet
strict physical specifications at
much lower cost than, for example, silicone fluids. They do,
however,
have
some
undesirable
Thiocyanate compounds are highly
characteristics:
toxic.
(i)
Care must be taken
to avoid skin contact. If contact occurs, the chemicals must
be washed off quickly. (ii) The solution tended to gradually
28
acquire a reddish tint
upon
This coloration became
noticeable
seemed
to
be
associated
exposure to ultraviolet light.
after
with
about an hour and
the
oxidation
of
iron
compounds. The exact reaction was not determined, however it
was easily reversed by the addition of powdered zinc.
D. Flow visualization technique.
The
goal
information
nature.
of
of
this
a
complex,
Qualitative
necessity.
It
study
is
flow
anemometry,
motion. Laser
used
visualization
by
would
might
modification of the apparatus,
reversing
simultaneously
measuring the
components
tedious. The technique
al,
1973)
was
velocity
therefore
the
a
quantitative
have
and
of
given
required a
view
of
the fluid
required very little
it has the advantage of
flows. However, the requirement
that
can
is
one
is
by
flash
fluid
too
restrictive for
measure three velocity
extremely
one
expensive,
would
and
be extremely
photolysis (eg, Sovova et
serious
photochemically sensitive
have
be impossible to employ
the
measurements
this technique: equipment
components
was
would
obscuring
anemometry
of three-dimensional
that
Olson,
it
without
being useable in
obtain
compatible with visualization. Hot
separate apparatus since
this technique
to
time-varying three-dimensional
desirable
measurement technique be
wire
was
consideration.
it
is
Using
a
possible to create
dark spots or lines of
very neutral buoyancy by exposure to
a flash of ultraviolet
light.
to create "spots" with
great enough precision and intensity
for quantitative work.
Unfortunately we were unable
29
It is
realized,
incidentally,
that
any quantitative
technique will be quite tedious in application to a study of
this sort. The
true
three-dimensional
requires that we sample
a
increase in the amount of
study. Many techniques
would
have
to
be
3-D
which
a
volume which is a geometric
could ordinarily be automated
applied
is
of the flow
data required for a 2-dimensional
geometry, or else automated
nature of the flow
nature
manually
due
to
the unusual
at great cost. The time-varying
further complication ( increasing
the dimensionality by one), but
one which is constrained by
our decision to examine only 6 phases of the flow at most.
The method ultimately chosen
visualization by particles. The
are
somewhat
different
from
requirements for this study
two-dimensional
Imaichi and Ohmi (1983). A 2-D
thin slice of light which
was quantitative 3-D flow
studies, eg
flow can be illuminated by a
illuminates the particles but not
the background. In a 3-D flow the particles would in general
have a velocity component
of light, which would
In addition, in
perpendicular
to any given sheet
therefore exhibit misleading results.
order
to
components it is necessary
different angles. This
simultaneously measure all three
to
view simultaneously from two
places
spatial
restrictions on the
placement of the light sources--they must not shine into the
camera.
Techinques
similar
to
these
have
been
used by
others, eg Dijkstra and van Heijst (1983), but not for flows
of the complexity of this study.
30
The
above
are
difficulties
fluorescent particles.
using
The particles used in this study are
and glow bright orange
tiny (200-30Opm), neutrally buoyant,
under ultraviolet
by
overcome
illumination.
The
plexiglas bifurcation
gives off a slight blue glow under the same illumination. By
using an
orange
filter,
dimmed, yielding a high
most
of
contrast
the
background
can be
image of the particles on
high speed black-and-white film. When color is available, as
in video taping
and
alone
good
provides
direct
viewing,
contrast,
the color difference
eliminating
the
need for
filters.
Particle manufacture.
The
particles
are
made
in
polymerization, using a procedure
Webb (1981).
An insoluble
the
lab
by
suspension
modified after Frisch and
resin is polymerized while being
suspended in an aqueous solution
shear the resin into tiny
and continously stirred to
spherical bubbles which harden in
that shape.
The resin used is Castoglas "C", a commercial polyester
casting formula made by
the
Castolite Corporation. To this
resin we mix in an equal volume of Dayglo "Blaze Orange" dye
powder ( the
dye
used
in
devices). The suspension
(PVA),
many fluorescent traffic safety
contains
-1.5g/l sodium chloride
-2 g/1 polyvinyl alcohol
(NaCl), and -1.Bg/l ammonium
thiocynate (NH4SCN) in distilled water,but the exact amounts
are not critical. The
first
bubbles of resin from
coalescing
two chemicals help prevent the
or sticking together; the
NH4SCN helps prevent the formation of an emulsion.
31
suspension solution is kept in
About 1.5 liters of the
of the resin-dye
peroxide
to
mixture
catalyze
the
rotating
add
we
the
mixture is stirred for
or a blender. To 150ml
stirrer
motion either by a magnetic
30
1ml
polymerization
seconds
suspension
reaction.
The
and then is poured into
solution.
within an hour. They range
methyl ethyl ketone
The
particles
harden
size from <100pm up to -1mm.
in
The desired sizes are separated out by sieves.
Recording data.
When photographed in motion, the particles leave streak
images of length proportional
camera's shutter speed, At.
a pair of 100
watt
high
to
their velocity and to the
The ultraviolet light source was
pressure
mercury vapor arc lamps
with uv filters. These lamps are normally powered by 120VAC,
60 hz, which results
intensity.
This
in
a
120
pulsation
is
velocities, however at
show noticeable
gaps.
ballast resistor one of
higher
By
hz
not
incorporating
the
lamps
apparent intensity was reduced,
As particle velocity increases
because although the amount
of
particle during At is constant,
when
the
noticeable
streak
at
low
velocities the streak images
essentially steady intensity which
larger area
pulsation in the light
was
a
high wattage
made to produce an
filled
probably
in the gaps. The
to the rms value.
the streak images are dimmed
light reflected back by the
the
is
light is spread over a
longer.
At
the highest
velocities we use, the streak
images dim to the point where
light intensity can become
critical
a
factor.
This is in
32
spite of the fact that we use high speed film pushed to over
1000 ASA. We boosted the
the rated
value
by
light
intensity by about 20% over
increasing
variable transformer. However,
the
in
input
voltage with a
practice
it was careful
placement of the light sources that was most crucial.
The
timing
of
the
camera's
synchronized to the flow cycle
infrared detector
senses
chosen location of the
a
release
is
in a controllable manner. An
marker
wheel
marker passes through the
shutter
of
which
is
placed at a
the scotch yoke. When the
detector, a solenoid is activated
which operates a cable release attached to the camera. There
is a delay time between
the
release of the shutter
sensing
which
we
of the marker and the
found to be repeatable to
well within the accuracy of the shutter (0.001 seconds).
The camera is
aimed
directly
bifurcation (see Fig.2.3). A
view provides
a
reflected
at
mirror
image
a
top
view of the
placed in the field of
of
the
side view. When
taking quantitative data it is necessary to place markers in
the field
of
view
outside surface of
bifurcation is
for
orientation
the
carved
block
of
provides
and
calibration. The
plexiglas from which the
convenient
locations. The
surface is rectangular, with top and bottom sides orthogonal
to the top view and
near
the side view. Each
of
and far surfaces perpendicular to
these
two dots of fluorescent paint
know
orientation.
Thus,
orientation and known
four surfaces is marked with
at known distance apart and a
each
distances
view
on
contains
a reference
near and far surfaces.
33
The latter are useful
for perspective correction, described
below.
The coordinate system is
is placed
somewhat
yields data on
x-y
arbitrarily,
locations
coordinates in x-z. The
1000 photographs
of
shown
but
and
velocity
these
in Fig 2.4. The origin
so
the
that
side view data are
data comes from well over
simultaneous
significant amount of time
and
a top view
views.
We save a
money by not making prints;
instead the developed negatives are used directly.
Data entry.
Data
entry
is
done
manually
with
the
aid
of the
digitizing tablet (Numonics 2200). Negatives are mounted
a photographic enlarger and
are
The degree of enlargement is
in
projected onto the tablet.
left
to the discretion of the
operator; it is typically about a factor of 1.5 times actual
size. The tablet contains an electromagnetic grid capable of
sensing the location of a cursor on an 1111 inch
within 0.005 inches.
Interfaced
serial
minicomputer,
port
transmits
to
the
location of the
a
X-Y
through
coordinates
intersection
the
in
of
surface to
a standard RS-232
digitizing
ASCII
format
tablet
of
the
crosshairs in its cursor.
(X-Y refers to the coordinate system of the tablet.)
The X-
Y locations of the calibration points are entered first, and
these locations are also marked on a sheet of paper taped to
the tablet.
Next,
the
locations
endpoints in both views are
written
to
receive
the
of
the
particle streak
entered. A computer program was
data
and
simultaneously display
34
images of the streaks
on
a
entered. The program also
coordinates
and
transforms
stores
frames are aligned
to
graphics
the
the
terminal as they are
the X-Y data into xyz
results
on
calibration
disk. Successive
marks
made on the
tablet.
Once data
from
both
views
compares the x-locations of
groupings of
side
view
by including
xyz
the
view
length appears shorter
streaks show
an
fall
streaks.
of
into a given
It then has a first
This estimate is refined
perspective:
when
farther
points provide quantitative
which is applied to
which
coordinates.
effect
entered, the program
the endpoints and makes initial
streaks
neighborhood of the top
estimate of the
are
A
given streak
away; the calibration
information on this correction,
the first estimate. Properly correlated
improvement
in
the
agreement
of the x-
location of their endpoints.
Since the experiment
is
is necessary to collect data
flow
field.
In
actual
only
quadrants
of
data
the
is
parent
quadrant-to-quadrant
from one quadrant of the
practice,
collected from the upper and
side view. Some
quadrilaterally symmetric, it
most
of
the
data
is
lower quadrants nearest to the
also
obtained
tube.
There
inconsistencies
from the far side
were
which
no
would
apparent
suggest
asymmetry. Bilateral symmetry
was
checking the daughter
flows with an electromagnetic
flowmeter.
branch
verifed independently by
35
It is necessary
for
low enough so that a
the
given
identified in both views.
particle concentration to be
particle streak can be uniquely
A
satisfactory degree of control
over the concentration is maintained by the simple technique
of injecting or removing
particles with a syringe connected
to a long thin tube. The tube is inserted between tests. The
desired
concentration
varies
concentration increases with the
since
length
the
apparent
of the streaks. We
found empirically that a comfortable range of streak lengths
is attained by using a
speed of At = [8(frequency
shutter
of oscillation)(Ax/d)J-1. It is
this exact
value
since
the
not
always possible to use
camera
shutter
speed can be
varied only in finite increments.
Because the particle concentration
certain bounds to
of
particles
variations.
obtain
may
must be kept within
unambigous data, the distribution
sometimes
exhibit
significant
spatial
This is generally not a problem for those cases
where there is strong mixing
cycle (Ax/d and a
both
for
greater
these cases, the particles
at least some part of the
than 2, approximately). In
redistribute from cycle to cycle
and the necessary volume can
be covered by taking data from
many cycles. For
where
reversible,
it
the
cases
was
necessary
the flow is essentially
to
frequently
stop
the
experiment, mix the particles, and restart the experiment.
We tested the
(including camera
combined
and
fluid), digitizing pad,
error
enlarger
of
the optical system
optics
and index matching
perspective-correcting program, and
36
human
operator
by
streaks made by
photographing
attaching
units.
entering
fluorescent
outside of the bifurcation.
0.5mm in real
and
For
The
a
simulated
fishing line to the
total
error was less than
typical flow condition having
maximum velocities corresponding to
streak lengths of about
3 cm, this would correspond to an uncertainty in velocity of
less than 17% of peak.
We tested the measuring technique on
the known solution of oscillating flow in a straight tube of
circular cross-section. The results
compared to the theoretical
can be
seen
Coupled with
somewhat
that
the
the
0.5mm
higher
solution
error
than
are
is
of Uchida (1956).
probably
uncertainty
in
two-dimensional
technique (eg Imaichi and
Ohmi)
measurements of velocity in
a
shown in Fig. 2.5,
It
closer to 10%.
location, this is
versions
of
this
but is comparable to other
bifurcation (eg Schroter and
Sudlow).
The human
operator
must
determine
the
direction of
flow. For most phases
and
flow conditions it is sufficient
to note the phase
the
cycle,
of
axial direction. However at high
bulk flow turnaround there
upstream while others are
are
which indicates the bulk
alpha and stroke length at
regions
of fluid that move
moving downstream. Furthermore, a
given particle may reverse direction during the time At when
the shutter is open. It is therefore impossible to determine
the flow direction at turnaround by phase information alone.
We attempted
to
obtain
coding the streak pattern.
A
turnaround
rotating
flow direction by
wheel was placed in
37
front of the camera. The wheel rotated at constant speed and
was masked, so that
the
consecutive shutter
openings
effect
between. The intent was
streak could be
technique was
used
sometimes
unsatisfactory
particle
for
would
the
infer
three
the
retrace
At
size
gaps in
resulting pattern on the
direction. While this
it
reasons:
obscuring the pattern;(2)
and
variable
successful,
reverse
are accelerating
with
that
to
was that of having several
ultimately proved
(1)
part
of
Very
its
often
own
the
path,
flow turnaround the particles
decelerating
in
various ways. These
changes in speed can alter the apparent pattern of gaps. (3)
Most importantly, even in
human
operator
must
thoughtful decision
the
absence
examine
about
the
the
of (1) and (2), the
pattern
streak
and
make
a
direction. This can
increase the amount of time per streak entry by nearly order
of magnitude, making a
large
amount of data acquisition in
this manner unfeasible.
The total quantity of
data
over 8000 streaks. Since there
entered
are
2 views and 2 endpoints
per streak, this means entry by hand of
It
was
originally
estimated
entered at the rate of one
practice, the
most
at less than half that
that
32,000 data points.
data
points
could be
point per 1-2 seconds. In actual
competent
rate of about one point per
for this study is
undergraduate
could reach a
10 seconds at best. Most worked
speed. Some factors which contribute
to slower data entry rates include: images of lower quality,
which require more concentration;
lack of experience on the
38
part of the
student
operator,
and
fatigue. All operators
were required to enter practice
data of oscillating flow in
a
test
straight
tube,
the
same
used
to
verify
the
experimental technique. There was
a
learning; some entered reasonable
results on the first try;
others were never able
to
wide range of speed of
succeed even after many attempts
(and therefore did not enter actual bifurcation data.)
Two methods of
further
investigated but deemed
encoding the
image
levels.
and
The
the procedure were
unsatisfactory.
The first involved
with
employed was made by
512x512,
automating
digitizing
Hamamatsu
1024x1024
image
a
and was capable of 256x256,
resolution
data
camera. The camera
was
with
sent
up
through
to
an
256 gray
IEEE
488
interface to a MINC
data acquisition computer which applied
a simple gray scale
cutoff
to
find
problem with this technique was
that the hardware was quite
slow, easily several times slower
trial took place a
these
devices
few
are
years
than entry by hand. (This
ago; more modern versions of
probably
technique was to record the
the streaks. The main
much
faster.)
The
second
flow on high speed video rather
than on still photos, then digitally decode the video signal
by computer analysis. Before much
method, it was found that
this technique without
signal was much too
Smith and Paxson (1983) had tried
success.
They
found that the video
noisy for computer interpretation. They
were eventually able to
extract
by
and
photographing
progress was made on this
them
data from the video images
using
a
digitizing
tablet.
39
Apparently human
operators
are
superior
to
computers at
interpreting noisy images.
Recording of qualitative information.
The configuration for recording qualitative information
is quite similar to that
described above. A somewhat higher
concentration of particles is used. For these experiments is
it instructive to be able to
view as many parts of the flow
field as possible simultaneously, so the only restriction is
that
the
concentration
must
particles in front obscure
true, however, that
not
the
be
ones
so
in
high
that
the
back. It is still
the apparent (subjective) concentration
may vary with the flow condition. The probable cause is that
increased velocity in any
apparent
volume
example,
in
a
that
flow
is
high.
If
magnitude
but
particle
paths
look
now
concentration.
until
at
with
a
a
particle.
axial
component
For
of
are straight parallel lines
the
concentration is fairly
with
three
the
the same velocity
velocity
intersect
creating
More
by
the
flow
all
will
concentration, thus
only
streaks
overlap
we
traversed
with
velocity, all particle
which will not
three dimensions will enlarge the
at
relatively
appearance
cross-sectional
components,
of
locations
low
a higher
will
be
sampled, enhancing the effect.
For flows with a
high
redistribute from cycle
concentration is
higher,
change very little.
to
degree of mixing, the particles
cycle.
the
But
since the particle
visualized velocity patterns
40
Qualitative
was
information
recorded
in
two forms:
still photos, in which the side view was omitted in favor of
a closer view of the
top;
and
video recordings, made by a
standard video camera connected to a video casette recorder.
The video camera had a
frame
rate of 30 frames/second. The
recording can be played back
at normal speed, variable slow
speed, or frame-by-frame. It
is also possible to photograph
the video image as it plays back in real time.
Range of experiments.
The values of alpha and
are shown in Fig
2.2.
Ax/d
for which data was taken
Qualitative
videotape data was take
for 18 flow conditions representing
and a range in Ax/d -
a range in a
0.75-20. The case
=
2.3-21.3
a=21.3 and Ax/d=20
is not included since for that case it was apparent that end
effects
from
the
pistontoparent
tube
area
change
were
becoming important. Qualitative
still photos taken directly
of the flow
7
are
a
subset
of
of
those 18 conditions.
Quantitative data is taken primarily of the cases of highest
stroke
length
and
minimum
and
quantitative data is also taken
maximum
frequency.
Some
of the lowest stroke length
cases, also at high and low a.
E. Data analysis
The data are presented
data, as streaks plotted
angles,
of
in
in
the following forms:
(i)raw
3-D space, viewed from various
(ii) stereo views of raw streak data, (iii) displays
secondary
velocity
(iv)contours of axial
magnitude
velocity
vs.
axial
distance,
for a given cross-sectional
41
slice, (v)vector plots of the
secondary velocity at a given
cross sectional slice. All graphs are plotted using software
routines from the
National
The nature of the
data
Center for Atmosperic Research.
acquisition results in an irregular
distribution of the data which requires special attention.
The varying concentration of
data can cause difficulty
in interpretation: too dense a concentration can obscure the
view,
while
addition,
a
sparse
graphics
isocontours
density
may
plotting
generally
routines
require
regularly distributed. It
is
mislead
that
which
the
therefore
the
input
eye. In
produce
data
be
necessary employ a
special procedure to smooth
the data. In general, different
disciplines have
techniques
for
case,
smoothing
DISSPLA
supplied
different
manual).
with
In
the
our
NCAR
the
package,
smoothing (ref
designed
for
routines
plotting
geographic contours for pressure or elevation, were found to
be unsatisfactory for
that the
routines
our
results.
expect
gradients
Cartesian manner,
whereas
section,
gradients
exhibit
orientation.
We
technique based
our
therefore
on
The apparent reason is
local
to
data,
in
used
a
be
for
more
oriented in a
a
given cross-
polar
coordinate
a smoothing/interpolating
averaging
(adopted from DISPPLA
documentation).
The technique is as
follows:
The circular data region
is divided into a mesh with the distance between mesh points
being a
radial
neighborhood
is
length
Ar,
specified,
angular
in
width
which
AO.
the
A local
data
which
42
contribute
to
neighborhood
the
is
meshpoint
defined
in
multiple values of Ar and
any
given
meshpoint
value
the
A.
i
is
r
are
included.
and
0
This
directions by
Then the averaged value at
calculated
by
the
following
formula:
n
z
F(r,6)(5-wt + 6r-wt)
J-i
Fi -
(2.1)
n
Z (60-wt + br-wt)
Equation
(2.1)
summation
is
is
a
taken
weighted
over
neighborhood, where br
averaging
the
and
60
distances from the meshpoint
data
are
to
and wt are weighing terms for
n
formula.
points
in
The
the
the radial and angular
datapoint
j. The terms wr
r and 0 respectively. If this
formula were applied in a single dimension, say Sr, and wr
2, then the geometric interpretation
the importance of a
inverse of its
datapoint
distance,
normally chose wr=0.5
of (2.1) would be that
is weighted according to the
squared,
and
=
from
the meshpoint. We
wt=1.0, giving more significance
to radial gradients than to angular ones.
For the secondary flow
required a
rectangular
vector
plots, the NCAR routine
distribution.
This is satisfactory
and desirable since, for the secondary velocity field, it is
no longer true that radial
gradients are greater in general
43
than
angular
rectangular
gradients.
form
distance, (6 y2 +
of
In
(2.1),
this
using
case
the
we
employed
the
of
the
inverse
5x2)-1/2 as the weighing function.
44
III. Background theory.
The geometrical complexity of
even
the
steady
analytical
solution
flow
solution.
to
the
assumed steady
the bifurcation has made
problem
There
is
resistant
only
one
three-dimensional
flow
at
a
very
to
known numerical
problem
low
complete
[Wille];
Reynolds
it
number and
consumed about 2 months of processor time. In the absence of
a
practical
complete
3-D
portions of the types of
solution,
we
instead
examine
flows seen in the bifurcation, and
compare these to previous studies
This chapter reviews the
of a more general nature.
literature
on flows which exhibit
some similarities to bifurcation flow.
The
geometry
of
the
appropriate, simpler fluid
described
this
bifurcation
mechanical
geometry
for
bifurcation as follows (see
the
problems. Pedley has
an
Chapter
suggests
idealized
bronchial
II's Figure 2.1):
(i)
the area ratio of both daughter tubes to the parent is about
1.2,
(ii) the
angle
divider is sharp,
of
(iv) the
wall is very variable,
times the
branching
parent
in the parent tube,
a
radius,
section can be described
as
(ii) the flow
typical
(v)
the
value of 5 to 10
change in cross-
starting from a circular shape
proceeding
area, followed by a change
70o,
radius of curvature of the outer
having
tube
is
to
to
an
ellipse of the same
a dumbell shape and increase
in area, (vi)the daughter tubes, of constant area, initially
curve, straightening when
the
branching
(vii)the airway walls are generally smooth.
angle is reached,
45
The following are fundamental fluid mechanical problems
which are relevant to this
geometry and which have received
considerable attention:
1. fully developed steady flow in a straight tube.
2. fully developed oscillatory flow in a straight tube.
3. fully developed steady flow in a curved tube.
4. fully developed oscillatory flow in a curved tube.
5. steady entry flow into a curved tube.
The
problem
of
flow
through
the
complicated shape
change is also of interest, but it is probably also the most
difficult and
has
consequently
Some analysis has been done
ellipse portion
of
the
received little attention.
by Sobey on only the circle-to-
shape
change,
Bertelsen (1986) has demonstrated
able to
describe
require a slow
the
change
entire
in
a
and
technique which may be
bifurcation.
shape
more recently
with
Both, however,
distance.
For this
study we will note only that the effects of the shape change
seem to be less important than the effect of curvature.
In this chapter the above
flows (1-5) are reviewed. In
addition, some remarks will be
made on results on pulsating
flow in curved tubes, a related subject.
1. Straight tube flow
46
The solution to fully-developed flow in a straight tube
can be found in any textbook on fluid mechanics. The NavierStokes equation in this case is
ap
--
du
= A -2
ax
dr
which has the well-known parabolic profile solution
a2
u - - -
(-)
4A
where u
In
this
[1
ax
-
(-)
I
a
velocity in the direction x, parallel to
the tube axis
r
a
r 2
ap
radial variable
=
tube radius
case
"fully-developed"
boundary layer has diffused out
means
as
that
the
viscous
far as it can and fills
the cross-section.
2. Oscillating straight tube flow.
In the oscillatory case viscous effects are confined to
a thin Stokes' layer of
natural frequency
of
thickness
oscillation
viscosity. The parameter
which
bu-v/w
and
v
,
where w is the
is the kinematic
indicates the importance of
unsteady effects was introduced in Chapter I:
47
a = a/6u = avU/v
One of the earlier solutions to this problem was obtained by
Uchida. The result for
purely
oscillatory flow driven by a
sinusoidal pressure gradient is
u(r)
a2 /8v
1 ap
8B
8(1-A)
= --{ -- cos wt +
sin wt}
a2
p ax
a2
ber(ka) ber(kr) + bei(ka) bei(kr)
where
A =
ber 2 (ka) + bei 2 (ka)
bei(ka) ber(kr) - ber(ka) bei(kr)
ber 2(ka) + bei 2 (ka)
For low values of a there is sufficient time for the viscous
boundary layer to diffuse to the
center of the tube and the
velocity profile is parabolic, as
increases, the
viscous
layer
can
in
the steady case. As a
only
diffuse a shorter
48
distance
into
the
progressively more
center of
the
tube,
blunt.
tube,
and
An
layer. The core, being
dominated
behind the oscillating
pressure
wall, where viscous
phase with the
reaches
effects
velocities are found
by
core
the
becomes
forms in the
unsteady viscous
by inertial effects, lags
gradient.
In the limit of
90 degrees.
Adjacent to the
dominate,
pressure.
profile
inviscid
surrounded
high a this phase lag
the
the
fluid moves in
Interestingly, the highest peak
in
the
transitional zone between the
core and the boundary layer.
Hino et
al
(1976)
onset of turbulence in
found that the onset of
parameter, Re/a, which
thickness
of
the
have
purely
is
a
occur
examined the
oscillatory pipe flow.
turbulence
unsteady
transition was found to
experimentally
They
is governed by a single
Reynolds
number based on the
Stokes
layer.
Turbulent
at approximately Re/a = 400,
using a Reynolds number based on the peak velocity.
3. Steady fully developed flow in a curved tube.
In studies of curved tubes it is convenient to employ a
toroidal coordinate system, shown
the distance from
the
center
of
in
Fig.
3.1.
Here r is
the cross-section of the
tube, 4 is angle between the radius vector and the radius of
curvature, and 0 is the angular, or axial, distance into the
tube. The velocity components
in the (r,0,0) directions are
(u,v,w).
equations
system are:
The
Navier-Stokes
in this coordinate
49
2
u -
v au
v
+ --
ar
w
- -
r dG
sinG
-
au
2
r
R + r sinO
a
[ 1
cosG
-
( - ) - v(- 9r
p
r 8a
=
av
+
v
ldul
)(-
+ - --)I
ar r r a4
R + r sinO
2
v av
dr
-
r
+ -
ao
r
-
p
v aw
u -
+ -
ar
-
r ae
a
(-
uw sinO
v
9v
R + r sinO
ar
+ -
ar
r
au
-)
r ae
vw cosO
++
R + r sinO
1
-
sinG
(-) + v(- +
r ae
aw
R + r sin
ap
1
cosO
w
-
R + r sin
a
p
--
(-)
aq
aw
1 a
+ -r ao
p
(
-+
r ao
R + r
sinO
[
+
1
(-
L ar
dw
+
+ -)
r
w cosO
r)e
R + r sinO
w sinO
ar
1
)
+ -
u -
uv
-
dv
R + r sinO
50
assuming a gradual
curvature,
where R is the
S0 a/R <<1,
first solution of these equations
radius of curvature. The
is due to Dean
(1927,
1928),
perturbation expansion in the
Dn = Re 1/2
/.
Dean's
who
solved the problem as a
Dean
number, defined here as
valid
results,
low
for
Dn,
are as
follows:
2 2
2
u/Wo= a Re sin 0(1-r' ) (4-r' )/288R
2
a Re cos 6(1-r'
)(4-23r'
4
+7r' )/288R
2
w/Wo- (1-r'
2
)
[1
- (3/4)(r/R) sin 0 +(Re
4
2
(19
where Wo
is the
maximum
/11520)(r/R)
- 21r'
+ 9r'
6
- r'
)
v/Wo-
2
axial velocity in a straight pipe
of the same radius and pressure gradient.
(In the Dean's solution given,
u,
v, w are pertubations to
the straight pupe solution.)
The physics of the flow
order-of-magnitude estimate of
3.2.
The axial
velocity
W
can
the
sets
be understood by a simple
flow
up
diagrammed in Fig
a an inward pressure
51
gradient
-
-pW 2 /R.
Slower
moving
insufficient axial velocity
to
fluid near the wall has
balance this gradient, thus
it is driven inward by a velocity which we can estimate as
2
W
vsec
R
a
or
vsec Wa a
~-
v R
W
which gives us the
fluid along the
Re
-
-
-
a
R
same
upper
fluid moves outward,
scaling
and
lower
forming
a
as Dean's solution. Thus,
walls moves inward, core
twin vortex secondary flow
pattern.
A number of authors
have found numerical solutions for
higher Dean numbers. One
of
the
most complete studies was
due to Collins and Dennis (1975),
who covered a range of Dn
= 96 to 5000.
results: At
nearly
The
very
parabolic.
following
low
Dean
The
number,
secondary
symmetric about a plane that
halves of the tube. As Dn
shifted outward into a
description is based on their
the
flow
axial profile is
pattern
is nearly
divides the inside and outside
increases, the axial flow peak is
horseshoe-like shape and the centers
of the secondary flow vortices move inward.
52
Nandakumar and Masliyah (1979) and Dennis and Ng (1982)
report numerical solutions
for
flow for Dean number >1000 or
pair of vortices
near
the
so. The pattern shows a small
wall
in addition to the
to Benjamin's (1978ab) study of
how
a
have more than one solution,
his experiments in
4-cell pattern in steady
outer
usual pair. They both refer
Taylor vortices, showing
a
ie
cylinders.
single
2
or 4 Taylor vortices in
Austin
calculated flow for Dn>1000, but
flow condition can
& Seader (1973) also
did not report whether the
flow was 2 or 4 cell.
4. Oscillating flow in a curved tube.
Lyne
(1970)
matching solution
obtained
for
a
high
perturbation
frquency
and
boundary layer
small curvature
which exhibited a secondary flow having 4 vortices, two each
in the inviscid core
and
the
was verified by visualization
fact that for large a,
boundary layer. This pattern
and
is apparently due to the
the highest time mean axial velocity
occurs not at the center
of
the tube but at the transition
region between the core and the boundary layer.
we can do a simple
order
of magnitude estimate to find the
scaling of the secondary flow.
as the steady case except
is that
of
the
Stokes
Once again,
The derivation is the same
that the boundary layer thickness
layer,
secondary flow scales as
vsec/W = W/(R* w).
V/w.
We
find that the
53
complete
curved tube
flow
numerical
problem
study
was
of
conducted
(1985). They obtained numerical results
0-30 and Dn
-5-200
and
categories shown in
represent
divided
Figure
dominance
of
3.3.
by
for
their
Yamane et al
a range of a
results into the 7
Three of these categories
viscous,
forces; the other 4 are
the oscillating
-
A very
unsteady,
or convective
transitional. Two lines of constant
a 2 /Dn are major boundaries of these flow regimes. The region
above the upper line
has minimal curvature effects; below
the lower line the
flow
is quasi-steady. It is interesting
a2 /Dn is directly proportional to
to note that the grouping
a product of the inverse of the stroke length Ax/d times the
square root of the curvature ratio.
The results for the 3 non-transitional regimes resemble
the results from the
above-mentioned studies. Of particular
interest are the two new regimes (VI and VII), in which both
unsteady
and
convective
forces
are
important.
These
transitional flows posess the following features:
(1) Unsteadiness is present
layer, but also as a
the cross-section
likely that this
axial velocity in
axial component so
only as a thin Stokes
larger region extending from the inner
bend which oscillates
flow. Yamane et al
not
out
of
describe
of
the
that
this
peak
region,
high
phase
which
Dn
it
with
the rest of the
as an oscillation across
axial
is
steady
takes
velocity.
known
flow,
on
It seems
to have reduced
has a low enough
the character of an
54
unsteady boundary layer; for
this
reason it oscillates out
of phase with the "inviscid" core.
(2) During most of the cycle (except for peak flow) the
secondary flow pattern is shifted toward the inner bend, and
a stagnant region appears at the outer wall.
Yamane's results for
3
Figure 3.4. All have a Dean
2.8, 7.9 and 30.2.
although some
distortion
The
the
low
and
are shown in
a case is nearly quasi-steady,
appears
horizontal
evident at 0 degrees
conditions
number of 150; the a values are
unsteadiness
in
flow
180
as
plane
a
near sinusoidal
axial
profile, most
degrees. At the intermediate
value, which falls into their regime VII, the secondary flow
patterns and axial profiles are extremely different from the
low a case.
The unsteady
region
at the inside of the bend
extends nearly 2/3 of the way across the diameter, as can be
seen from
the
axial
profiles.
layer at the outside
wall
which
layer thickness predicted by
except a peak flow,
There
is
a-7.9.
is smaller unsteady
close to the boundary
For all phases of flow
the
secondary flow vortex is displaced
toward the inside of the
bend, leaving a stagnant region at
the outside. Once a
III is entered,
is
increased
where
curvature
to 30.2, however, regime
effects
are dominated by
unsteadiness. There is a thin boundary layer that is roughly
the same thickness
all
around
velocity is nearly flat.
like that of straight
Thus,
tube
the circumference. The core
the
axial velocity is much
oscillatory
however, the thin region between
flow. In this case
the boundary layer and the
55
core which moves at the
highest time mean speed also drives
a weak secondary flow pattern of
2 vortices in each half of
the cross-section.
Yamane et al also made experimental measurements of the
phase lag of the pressure gradient
and
Dn
5-200.
=
They
theoretical results.
A
prediction of straight
et
good
deviation
tube
agreement
from
theory
with their
the pressure drop
indicated the onset of
This deviation occurred at a 2 /Dn ~ 1.31.
curvature effects.
Yamane
found
for a range of a - 0-7.9
al's
results
for
maximum
secondary flow
stream function (Wmax) are
shown
in
rapid drop in Wmax as
the
value
a 2 /Dn = 1.31 is exceeded.
Since this same value
indicates the threshold for curvature
effects influencing
pressure
Fig
3.5.
There is a
drop, a relationship between
secondary flow and pressure drop might be expected.
5. Steady entrance flow into a bend
This problem
was
approached
Hawthorne (1951, 1965) and
by
this case,
flow
the
secondary
velocity distribution
Fig.3.5.
The
at
incoming
the vertical plane,
to
fluid has
the
the
bend
an
inviscid
one by
Squire and Winter (1951). In
arises
from
entrance,
velocity
therefore
vorticity parallel
entered
the
as
it
as
a nonuniform
diagrammed in
profile contains shear in
also
has a component of
radius
of
an
angular
curvature. When the
distance 0, this
vorticity vector is, to a first approximation, rotated by an
amount 2
relative to
(o due to the shorter
the
local axial coordinate direction
convection
distance on the inside of
56
+
the bend
the
into
entry
to
due
0
tube). Squire and
Winter's result is:
= -2
q alU/az cos 0
axial vorticity and aU/az is the
is the increase in
where
gradient
of
the
perpendicular to
principle was
axial
entering
the
applied
with
the
plane
the
bend.
The same basic
certain
other
refinements by
of
plane
in
velocity
Hawthorne and more recently
by Rowe. These theories neglect
viscous effects
except
the
gradient at the
entrance
to
arises
extent
that
the velocity
from viscosity. All found
satisfactory agreement with
experimental
of total pressure contours.
(Incidentally, we have used the
velocity profile found in a straight
data on the shift
tube at high a for the
entrance condition for a Squire and Winter type calculation.
In such a profile,
the
maximum
axial velocity occurs in a
circumferential ring near the
wall. The resultant secondary
flow exhibits 4 vortices
bears a strong resemblance to
and
the pattern derived by Lyne.)
The
following
table
compares
the
secondary
flow
estimates for steady Dean type flow (we estimate W ~ wox and
assume Dean's
solution
applies
Squire and Winter's inviscid
flow result :
in
a quasi-steady sense),
theory, and Lyne's oscillating
57
vsec/W
Lyne
Dean
Ax/R
(Ax/R)a2
(high alpha)
(quasi-steady)
Note that all three
estimates
the
stroke
nondimensional
geometry,
this
is
also
Squire & Winter
(2a/U)(aU/az)(Ax/R)
(inviscid entry)
are directly proportional to
length,
Ax/R.
directly
proportional
(For
a
given
to
the
nondimensional stroke length used in this study, Ax/d.) Thus
the
stroke
length
is
an
indicator
of
secondary
flow
magnitude. The adverse consequence of this result is that it
is therefore difficult to
determine which factors limit the
magnitude of the secondary flow in our experiments by simply
looking at stroke length dependence.
The viscous entry
inlet profile) has
Berger (1975),
flow
been
Singh
problem
studied
(1974)
(assuming
now a flat
theoretically
by Yao and
,and
others.
Yao (1974) has
developed a physical argument,
similar to the discussion of
steady secondary
for
flow
above,
curved tube entry length in
high
the
evaluation
of the
Dn flow. The entry length
is shorter than that of a straight tube (-a 2/U) because the
secondary flow which forms
section.
increases mixing into the cross-
Yao's derivation is as follows:
As the blunt profile enters the curve, a boundary layer
forms immediately which grows
as
6e
=vxW
where W is the
58
inlet axial velocity and
x
is
the distance into the bend.
From continuity,
W/x
inside
the
~
vsec/a,
entrance
secondary flow
boundary
velocity.
The
layer,
axial
where
velocity
vsec
is the
in the core
creates an inward pressure gradient equal to -pW 2 /R which is
balanced in the boundary layer by
)
-W2 /R ~ v ( vsec /6e 2
From the momentum equation in the direction of the radius of
curvature, the inertial term
is
balanced
by the terms for
centrifugal acceleration and pressure
Vsec W
Wc
R
x
-
lP
p dr
where Wc is the axial
velocity
in
the core which has been
accelerated due to the displacement of the boundary layer:
2
Wc
a
W
(a-be)
2
Solving for the entry length x Yao obtains
59
Stewartson et al
(1980)
They found that, for high
the boundary layer entry problem.
Dn, the secondary boundary
layers "collide" along the inner
of the entrance to the curved
bend at a distance downstream
tube 0.943 vai.
point at
This
which
collision
boundary
becomes so rapid that
wall
become
a numerical solution to
performed
layer
may be interpreted as the
growth
at
the inner bend
inertial effects perpendicular to the
important,
violating
the
boundary
layer
assumption. An inward facing jet forms, similar in nature to
the outward jet formed at
the equator of a rotating sphere.
The axial shear goes to zero at this point.
Soh and Berger's (1984) full three-dimensional solution
shows this effect in
more
detail.
Dn=680 calculation shows axial
a similar development to
calculation,
but
not
shear vs. distance following
Stewartson
quite
at
this
secondary velocity reaches a
et al's boundary layer
reaching
minimum slightly downstream, at
It is approximately
Their result for a high
zero
about 2*VNaR.
distance
and
having a
(see Fig 3.6.)
downstream that the
maximum, whereafter it weakens
gradually to the fully developed value.
Olson and Snyder's (1983,
show a similar peak
in
same axial distance
downstream,
1985) entry flow experiments
secondary
vorticity at roughly the
2/aR.
Olson and Snyder's
experiments covered a range of Dn=106-510.
Soh and Berger's secondary
secondary vortex at
this
close to the inner bend.
axial
It
flow
results show that the
location
is centered very
appears that the outward flow,
60
fed
fluid
from
by
the
sufficient momentum to
secondary
move
directly
flow
layer, lacks
across the center of
spread out. Very close to the
the tube and seems instead to
inner bend, the
boundary
separates
and
forms a smaller votex
rotating slowly in the opposite direction. Proceeding around
the bend, the
flow
pattern
moves
still centered primarily toward
profile associated with this
outward somewhat but is
the
flow
inside bend. The axial
has
a region of reduced
axial velocity near the inner bend. (see Fig 3.7)
Using flow visualization methods,
have observed
secondary
flow
patterns
described, referring to
the
inner wall vortices."
Agrawal
patterns
et
laser anemometry two components
observed similar
Scarton et al (1977)
secondary
of
flow
hot-wire technique, claimed to
of
the
type just
they saw as "trapped
al
(1978) measured by
entry flow velocity and
patterns.
observe
4
Olson, using a
to 6 vortices in
entry flow.
Ito (1960) has shown
entrance to a curved
tube
diameters upstream. (Soh
the shape
of
the
by
pressure measurement that the
changes
and
upstream
the
flow up to several
Berger remark that specifying
velocity
profile is, strictly
speaking, mathematically impossible.) No data on the changes
to the velocity field upstream are available.
6. Pulsating flow.
A number of authors
have
curved tube, for which there
the flow
in
addition
to
studied pulsating flows in a
is
the
a steady mean component to
oscillatory
component. The
61
motivation is often
to
model
arch.
is
a
Since
there
like the those
done a large
described
appear
a
the
flow is
show characteristics very much
above.
Blennerhassett (1975) has
curved
uniformly
similar
unique and remain to
component,
numerical calculations for various
of
number
flow in the aortic
does not necessarily reverse.
studies
pulsating flows in
results
steady
it
generally different, e.g.
However a number of
arterial
to
be
Yamane's
verified
tube.
results,
Some of his
others are
by experiment. Chang and
Tarbell's (1985) numerical results show vortices and a local
reduction in axial velocity
near
the inner bend. They find
secondary flow patterns containing up to 14 vortices.
Talbot
and
Gong
(1983)
measured
2
components
of
velocity of a pulsating entry flow, using laser velocimetry.
They looked at
2
flow
conditions,
the first being quasi-
steady, the second being clearly unsteady. The unsteady flow
condition
above
exhibited
during
phases
the
the
inner
deceleration
secondary
flow
bend
only.
structures described
During
moved
across
the acceleration
the
whole tube
diameter and the axial profile was generally more blunt.
The
following
chapters
experiments and examine them
theory.
present
in
results
of
our
the framework of the above
62
IV. Qualitative Results.
our qualitative observations of
This chapter describes
oscillatory flow in
a
bifurcation
based
on videotape and
still photo data, of a large range of flow conditions. These
observations allow us to
begin
to build a general, overall
picture of the important
fluid
dynamic phenomena.
detail is provided
the
with
addition
Greater
of the quantitative
results in the next chapter.
As described
in
experiments covered
0.75 - 20. The
the
a
previous
range of
chapter, the videotape
a
=
2.3 - 21.3 and Ax/d =
visual characteristics of the flows suggest
flow
patterns are shown
4
schematically
in Figure 4.1 and are described below:
1.
Quasi-Reversible
particles
with
categories.
The
a grouping of
each
are
seen
cycle of
by playing back the
the
particle
paths
flow
to
lowest
oscillation.
this flow regime,
same
path
(This effect is best seen
videotape at high speed.) Consequently,
traced
during
of
Ax/d=0.75
value, a=2.3, no
10 cycles. At =21.3,
In
retrace virtually the
apparently just the reverse
regime is seen for
.
net
the
motion is negligible but a
expiratory
flow
are
inspiratory flow. This flow
for
all
values of a. At the
motion can be seen even after
highest value, the cycle-to-cycle
slight net change of in position
roughly one-tenth of the stroke length can be detected after
several cycles for some of the particles.
63
There is no apparent secondary motion; the particles in
all these cases seem to
follow the wall curvature. There is
no lateral overlap of particle paths.
At a=21.3
it
is
possible
Stokes layer along the
wall
particles in the core. As
increasingly
difficult
all particles come
This behavior
out
particles in the
of phase with the
a decreases, this effect becomes
detect,
and
is
no
longer
end inspiration and end expiration,
to
seems
see
moving
to
noticeable at a=2.3. At
to
rest
simultaneously
entirely
consistent
flow in a straight tube,
at this low a.
with oscillatory
suggesting little effect of airway
geometry for these flow conditions.
2.
Quasi-steady
regime are shown in
flow.
Fig.
Streak
4.2.
In
photographs
of this
this regime a secondary
flow pattern of twin helices forms in the curved sections of
each daughter branch.
side, the particle
When
paths
which overlap, forming
viewed
look
from
like
crosshatch
above or from the
many smooth "zigzags"
patterns. These patterns
resemble those formed in curved tubes and those reported for
steady bifurcation
flow
helices trail into
the
tubes during
curved section merge
Schroter
straight
inspiration
cross-section. During
(e.g.,
and
appear
expiration,
in
the
portions
two
parent
to
and Sudlow). The
of the daughter
fill
vortices
tube
the entire
from each
to form a four-
vortex pattern which continues into the straight tube, again
filling the entire
cross-section.
expiration, the secondary
motions
In either inspiration or
are
convected about one
64
stroke length past the curved region.
The fluid moving from
a straight section into the curved section is always free of
secondary flow.
The
quasi-steady
nature
of
the
flow is
indicated by: (i) the overall secondary flow patterns during
acceleration resemble those
seen during deceleration (i.e.,
Fig. 4.2b resembles Fig.
4.2d
4.2h);
entire
flow
velocity
at
and
(ii)
the
secondary, goes to zero
cycle (Figs. 4.2a
and
and Fig. 4.2f resembles Fig.
4.2e).
lowest value of a, 2.3. At
This
field,
primary
and
the same times in the
regime
occurs for the
low Ax/d it overlaps with regime
(1).
3. Transitional
flow.
An
increase
in
either a or
stroke length from either of the above regimes yields a flow
designated as transitional. During most of the cycle in this
regime, the
flow
field
resembles
(Figs. 4.3b,c,d,f,g,h).
As
in
due to secondary convective
axial flow.
However,
regime (2), lateral mixing
motion is strongest during peak
near flow turnaround (end inspiration
or end expiration) there
outside wall of the
the quasi-steady regime
is
an
curved
obvious
section,
region, near the
where the fluid turns
around earlier in the cycle than the rest of the flow. These
"turnaround" regions can be seen
in Figs. 4.3a and 4.3e. As
flow turnaround
one
is
approached,
moving helically along
bend. As the flow
this region of the
the
begins
inside
wall
to
can observe particles
toward
the inside of the
turn around, the particles in
bend
do
not proceed across the
cross-section; instead they stay near the inside of the bend
65
as they decelerate to
zero
and
begin to accelerate in the
opposite direction.
4. Confined vortex flow.
the highest values of both
an
acceleration
phase
different from those
a
seen
develops
patterns
are
flow regime, seen for
and Ax/d, is distinguished by
having
the flow as it
seen
This
flow
patterns dramatically
during deceleration. We describe
through
the
a half cycle; the general
same
in
both
inspiratory
and
expiratory directions.
During
the
beginning
structure to the secondary
of
acceleration
no
apparent
motion is seen, nevertheless the
magnitude of the motion is significant (Figs. 4.4b & f). the
characteristic eddy size during this phase is small compared
to the
tube
motion
to
radius,
disipate
causing
this
rapidly
as
Organized helical patterns
do
they are apparent by the time
reached and persist during
a different appearance than
form
apparently unorganized
the
flow
accelerates.
as the cycle proceeds;
peak flow (Figs. 4.4c & g) is
deceleration. However, they have
those
seen in the quasi-steady
case in that they are confined
to a region near the outside
wall. This region of secondary
motion is clearly visible in
Figs. 4.4d & h. The region begins near the downstream end of
the curved section, grows to fill a region at most much less
than half the tube
at constant size as
cross-section, then continues to persist
it
proceeds into the straight section.
At flow turnaround, when the bulk
axial flow goes to zero,
the secondary flow is no longer confined and exhibits strong
66
lateral
mixing
the
over
Figs. 4.4a & e.
entire
Although
cross-section,
shown in
the secondary flow at this point
in the cycle has arisen from
becomes much more
complex.
to determine any
organized
a well organized flow, it soon
Eventually it becomes difficult
pattern.
This mixing is strong
enough to persist somewhat into the acceleration phase.
Because the accelerating flow patterns are so different
from the
ones
seen
during
deceleration,
this
regime is
clearly unsteady.
Transitions between above flow
display in Fig. 4.5 the values
flow regimes occur.
It
can
regimes are gradual. We
of
be
a and Ax/d for which the
seen that Reversible flow
occurs for low values of stroke length (Ax/d < 1) and Quasisteady flow occurs for low a
Ax/d both increase,
we
(a
pass
through the Transitional flow
characterized by the turnaround
Confined Vortex regime.
which
secondary
flow
Also
was
< 2) values. As both a and
zone, and finally reach the
shown
seen
turnaround. This occurred for
a
are the conditions for
to
>
persist
4
and
through flow
Ax/d >5. In the
range Ax/d 5, the secondary flows are weak during the entire
cycle.
As described
above,
even though the secondary flow
patterns in inspiration and expiration are similar, we never
observed a steady,
through flow
motions
dies
organized
turnaround.
out
secondary flow that persisted
Instead,
completely
apparently unorganized pattern.
or
either
breaks
the secondary
down
into
an
67
This division of regimes
derived from numerical
has resemblance to categories
results
presented
by
Yamane et al
for the case of oscillating flow in a uniformly curved tube.
Yamane's categories (I)-(VII)
Reversible region
are
corresponds
negligible curvature
effect,
"viscosity
dominated
regimes.
coincides
the
with
two regimes of
one
unsteady
(III)
viscous, and unsteady
case falls into Yamane's
and
The
"viscosity
Transitional
regime
forces
and the
viscous and unsteady (II).
this
Yamane
Fig. 4.6. The
Yamane's
(I)"
dominated (IV)"
in
to
other being transitional between
The Quasi-steady regimes for
shown
are
in
all
convection
regime roughly
which centrifugal,
important (VI). The
Confined vortex regime corresponds to Yamane's regime (VII),
dominated by both unsteady and centrifugal forces.
The 18 flow conditions
on Yamane's in Fig. 4.7.
of
this study are superimposed
The only discrepancy between their
categories and the ones seen
in
this study is the location
of the division between categories (VI) and (VII), analogous
to the change from
the
Transitional regime to the Confined
vortex regime. Overall, however, the correspondence is quite
good.
Unfortunately,
Yamane
et
al
present
velocity field
results for only 6 flow conditions. Of these we note at this
point only that for
discussed in Chapter
a
confined vortex/category (VII) flow,
III,
vortex structure that is
the bend during
most
of
their
shifted
the
results
show a secondary
over toward the inside of
flow
cycle, indicating some
68
resemblance to the confined vortex regime seen here. They do
not, however, present results showing a strong mixing across
the entire cross-section during flow turnaround.
Secondary flow estimate
By judging the angle of
their flow direction, it
the magnitude of the
is
This estimate is made
data,
with
particle path
following
angles
possible to estimate, roughly,
secondary
flow.
the
the particle paths relative to
in
flow
relative to the axial
from the top view of the video
results
the
parent
(0P
and
and
Od represent
daughter tubes,
respectively, and the estimated error is about 50):
69
a
Ax/d
Op
Od
.75
00
0.0
0.0
5
5
00
5
0.1
0.1
10
5
5-10
0.1
0.10-0.18
20
15
10-15
0.27
0.18-0.27
4
.75
0
0
4
5
10
5-10
0.0
0.18
4
10
15-20 20
0.27-0.36
4
20
25
20
0.47
0 .0
0 .10-0.18
0 .36
0 .36
6
.75
0
0
6
5
10
10-15
6
10
20
6
20
20
25
20
0 .0
0 .18
0 .36
0 .47
12
0.75
0
0.0
12
5
15
0.10
12
10
15
0.47
12
20
0
5
25
30
25-30
0.58
0.0
0.27
0.27
0.47-0.58
21.3
0.75
0.0
5
0
5
0.0
21.3
0
5
0.1
0.1
21.3
10
30
25
0.58
0.47
2. 3
2. 3
2. 3
2. 3
Vseo/Vaxial,p Vseo/Vaxial,d
0.0
0.18-0.27
0.36
0.36
70
V. Quantitative Results.
In this chapter are
dimensional
streak
presented
measurement
regime example, overviews
are presented first,
slices
of
the
of
technique.
For
each flow
the entire bifurcation volume
followed
data.
the results from the 3-
The
by
an
examination of axial
parameter
values
for
the
quantitative data sets are as follows:
Designation
Flow regime
1
quasi-reversible
Ax/d
a value
2.3
0.75
21.3
0.75
quasi-steady
2
quasi-reversible
unsteady
3
quasi-steady
2.3
19.5
21.3
9.7
w/curvature effects
4
unsteady
w/confined vortex
Case I.
Raw Data.
We first
3-dimensional streaks.
indicating
their
present data as projections of the
The
streaks
are
The
length
direction.
directly proportional
to
the
velocity
there is secondary vortical motion,
shown
of
it
as arrows,
a
streak is
represents. If
it is exhibited just as
71
it is in the qualitative results of the previous chapter: In
a view from
the
top
or
location go in slightly
side,
streaks
in the same axial
different directions, which results
in a cross-hatching appearance.
The
data
presented
in
this
qualitative still photographs, as
results are quantitative, they
manner
resembles
the
expected. But since these
will
add more detail to the
fluid mechanical description of the flow.
Fig. 5.1 shows the
top
views
case 1. Peak inspiratory flow
near the wall
are
shorter
of the streak data from
is seen in Fig. 5.la. Streaks
than
the
indicating lower velocity near the
ones
in the center,
wall. The streaks in the
parent tube are essentially
parallel
particles approach the flow
divider and curved section, the
streaks
acquire
a
orthogonal to the
lateral
local
component
radius
emerge from the curved section
of
of
entire
streaks,
the tube axis. As
which
keeps
them
curvature. The streaks
parallel
daughter tube. Throughout the
no cross-hatching
to
to the axis of the
flow volume, there is
indicating
no
evidence of
secondary motion.
The view of peak
very similar to
expiratory
inspiratory
flow in Fig. 5.1b appears
flow
view, with the direction
reversed.
Secondary flow magnitude.
development of
secondary
displayed as a function
conditions and phases.
flow
of
It
It
is useful to follow the
by
examining its magnitude
axial
should
distance for the various
be
pointed out that the
72
concept of
vicinity
secondary
of
the
flow
is
somewhat
bifurcation
since,
ambiguous
due
to
in the
the complex
geometry, the primary flow direction is not well-defined. We
use for a secondary flow
parallels
the
outer
spatial location
reference a coordinate system that
wall
is
of
still
Fig
carina (x=4-5)
exhibit
motion due to
the
2.4.
change
in
sufficiently unambiguous to
particles near the
some
artifactual secondary
shape
of the cross-section.
motion displayed this way is
be
the data is, as described in
although
using the cartesian
Thus,
Aside from this, the secondary
to 0.5 mm in identifying
bifurcation,
identified
coordinate system of
will
the
useful.
The noise level in
Chapter 2, due to errors of up
streak endpoints.
(The nonuniform
distribution of data can be misleading. It should be kept in
mind that the characteristic
of
interest
is the amount of
secondary flow reached at any given x neighborhood.)
The
Case
evidence of
1
flow
secondary
reasons discussed in
shorter for this
condition
motion
in
Chapter
case
and
exhibited
the
3,
we
the
secondary
Looking at
the
Fig. 5.2(a) and
flow
can
therefore
readily
inspiratory
(b)
secondary motion is
we
below
and
see
For the
expect the noise
That there is little
be
seen
expiratory
that
this
data.
little
streaks are somewhat
level to be somewhat higher, near 25%.
or no
raw
very
the
level
level
in Fig. 5.1.
flow cases in
of apparent
throughout the data
region, as expected.
(For the
remaining
flow
conditions
longer and the expected error is about 10%.)
the
streaks are
73
Axial and secondary
velocity.
sectional slices through the
plots of axial velocity and
Data in a given
slice
We
have taken cross-
data region to produce contour
vector plots of secondary flow.
includes all streaks whose midpoints
fall within the slice. In the parent tube, the slice extends
from x=O to x=3, for which
the parent tube is essentially a
straight tube. The daughter tube
slice extends 3cm past the
end of the curved section. The graphs are produced using the
techniques described in Chapter III.
Fig. 5.3 shows contours of axial velocity in the parent
tube for the quasi-steady,
The axial profile
shape
this case, essentially
quasi-reversible flow of Case 1.
is, within the estimated error for
parabolic
expiratory directions. Secondary
for
both inspiratory and
velocity (not shown) falls
within the noise level, as stated above.
Case 2..
Raw Data.
large
In Figs. 5.4a and 5.4b it can be seen that a
number
indicating a
of
streaks
flat
axial
are
velocity
character of the flow seems
is no
evidence
of
nearly
quite
secondary
pattern closely resembles a
the
profile.
same
length,
Otherwise the
similar to Case 1. There
motion,
and the inspiratory
reversed expiratory pattern. At
a = 21.3 and Ax/d=0.75 a thin, unsteady boundary layer could
be seen from the
video
data.
boundary layer is not apparent
The
unsteady nature of this
in the streak data, probably
because the velocity of an unsteady layer is proportionately
74
not very different from the
core velocity when the phase is
near peak flow, as it is in these measurements.
Secondary flow magnitude.
This is another case having
negligible secondary flow. The graphs
out, exhibiting secondary motion
the divider region,
where
in Fig. 5.5 bear this
less
geometry
than 10% except near
changes
cause a false
secondary level of up to 20%.
Axial and
from Case 2,
secondary
unsteady
Fig. 5.6. The
axial
velocity.
Cross-sectional data
quasi-reversible
profiles
in
flow,
is shown in
each direction are quite
flat and secondary velocity
is negligible. Because the data
consists of somewhat longer
streaks
some evidence
of
the
curved
in this case, there is
region,
seen
as horizontal
secondary motion at the outer walls.
Case 3,.
Raw data.
which
place
This
it
on
flow
the
condition
border
has parameter values
between
quasi-steady and
unsteady in Yamane's categories. Fig. 5.7a shows the flow at
peak inspiratory velocity (00
in the parent tube
is
phase angle). The flow coming
fairly straight, showing no evidence
of secondary flow. Near x=4 one can see the streaks begin to
move laterally, following the
By x=5 the
cross-hatching
curvature of the bifurcation.
appearance
of secondary flow is
evident. The cross-hatched character is spread fairly evenly
across the
cross-section
extends
into the straight tube. There
the smallest being close
is seen.
to
past
the curved section
is a range of velocities with
the
walls. No flow separation
75
Decelerating flow at
450
past
peak
is shown in Fig.
5.7b. The description of the flow is exactly the same as for
peak flow, the only
difference
(and thus all velocities)
being
that all the streaks
somewhat reduced. This is as
are
expected for a quasi-steady flow.
the
In Fig. 5.7c
flow
peak expiratory flow.
Now
daughter tube. There is
accelerating, 450 prior to
is
flow
the
no
is
secondary
entering from the
flow coming into the
straight tube, and the flow remains free of secondary motion
until partway
symmetric
around
bend,
cross-hatched
downstream into the
effect
the
is
much
parent
reduced
description
tube.
from
applies
at
x=4. There a
and
proceeds
By x=1 the cross-hatched
suggesting
that
the
Again no separation is seen.
equally
expected, the velocity patterns
magnitude
begins
x-3,
decelerating expiratory flow, seen
increased
roughly
pattern
secondary motion is dying out.
This
at
peak
well
in
are
to
peak
and
Figs. 5.7d and e. As
the same, showing only
flow,
decreasing
again
in
decelerating flow.
Accelerating inspiratory
shown
in
Figure
5.7f.
The
flow
at
flow
does
450
before peak is
largely
resemble
inspiratory flow at the other two phases. However the crosshatched effect in the daughter tube seems somewhat reduced.
Secondary
flow
magnitude.
inspiratory acceleration
flow in the
parent
section, near x=3,
(a),
tube.
the
At
level
we
the
of
(Fig.
see
5.8)
During
neglgible secondary
beginning of the curved
lateral motion begins to
76
rise.
It
reaches
about
gradually thereafter.
20%
peak
450 later
secondary motion begins
rises to more than
of
at
30%,
x-4.5 and drops
(b), during peak flow, the
about
and
by
the same axial location,
drops off slightly going into
the daughter tube. The secondary motion is still significant
as the flow exits the straight
During deceleration (c),
flow is found in
the
magnitude compared
another
same
to
tube out of the data region.
450
later, the secondary
region, only slightly lesser in
peak
flow,
and
persists
into the
daughter tube.
During expiration, the level of secondary motion is low
at all locations during acceleration (d), with only a slight
rise to 20% near
x=3.5.
As
peak
flow (f) is reached, the
daughter tube secondary motion remains low, but a rapid rise
appears at x-5.5. It reaches perhaps 35% before dropping off
as the flow enters the straight parent tube. By the time the
flow exits
the
measurement
level is greatly
(g) exhibits a
reduced.
very
region,
The
gradual
close to 25% somewhere in the
the
secondary motion
pattern during deceleration
rise
which
seems to peak at
range x=3-4, and falls off in
the parent tube.
During both
during
inspiration
acceleration
is
deceleration, suggesting
However,
the
magnitude
slightly higher than
this flow condition.
similar
some
expiration,
to
degree
during
during
bulk flow, perhaps due to
and
the
one
the pattern
seen during
of quasi-steadiness.
deceleration
is
always
acceleration at the equivalent
the borderline unsteady nature of
77
Axial and secondary
flow.
Fig.
velocity contours and secondary
flow
tube during the inspiratory half
phases,
the
axial
velocity
velocity in the center
of
is
the
with a
skewing
slight
axisymmetric
with
peak
tube. Secondary velocity is
a straight tube flow.
The corresponding data for
The
vectors in the parent
of the cycle. At all three
minimal. The flow resembles low
in Fig. 5.10.
5.9 shows the axial
the
daughter tube is shown
axial profiles are nearly axisymmetric,
toward
the
inside
wall, which is
toward the outside of the bend.
These axial velocity results
appear different from the
results of other researchers, described in Chapter III.
All
the studies observe an axial peak on the outside of the bend
and an "M-shaped"
shape of the
vertical
peak.
All
profile
those
due
to the horseshoe-
studies, however, have much
higher Dean and/or Reynolds number values. Nevertheless, the
results are rather curious and
will be discussed further in
the next chapter.
The secondary motion during the 3 phases of inspiration
in the daughter tubes is the 2-cell vortical pattern seen in
curved tube flow. The
acceleration,
peak
pattern
flow,
and
is
essentially the same for
deceleration,
however
the
intensity is greatest at peak flow and next strongest during
deceleration.
Velocity data for Case 3
Figs. 5.11 and 5.12.
Fluid
expiratory flow is shown in
entering from the daughter tube
has very little secondary flow. The axial profiles here have
78
a
slightly
more
pronounced
flow,
indicating
inspiratory
outward
an
skew
upstream
compared
effect
to
of the
curved section.
In the three
parent
tube,
phases
the
pattern commonly
Sudlow).
As
secondary
seen
in
in
the
intensity is greatest
during
shown
of
expiratory flow in the
flow
exhibits
steady
flow
inspiratory
during
deceleration.
The
4-vortex
(e.g. Schroter and
case,
peak
the
secondary
flow
flow and next strongest
axial
profile
axisymmetric during acceleration, but
is
almost
as the cycle proceeds
a vertical ridge of higher velocity flow forms in the center
of the tube. It
motion at
is
these
daughter enter
carrying the
medial
phases:
the
flows
As
the
of
higher
After
the
collide
tube,
and
quasi-steady
the
axial
in
falls
coming
from each
secondary
flows are
velocity
the
the
the
vertical direction. This axial
in steady and
flows
passing
vertically up and down. Thus
center of
with the stronger secondary
bifurcation,
fluid
direction.
secondary
associated
flow
center
toward the
divider,
and
the
continue
peak axial flow is at the
off
less
velocity
studies
rapidly in the
ridge is also seen
by Schroter and Sudlow
and Isabey and Chang.
Case 4.
Raw Data.
Figure 5.13a shows accelerating inspiration.
During this phase
exist in the entire
a small degree of secondary flow seems to
flow
field.
There
is also a somewhat
79
smaller distribution of veloticies than
This smaller velocity range can
Fig. 5.13b. By this
tube is free of
time
the
secondary
There is a slight
also
was seen in Case 3.
be seen at peak flow,
flow
coming into the parent
flow and appears quite straight.
amount
of
goes around the curve and
secondary
into
motion as the flow
the daughter tube (the high
density of data in some locations makes this harder to see).
The
secondary
flow
is
much
stronger
decelerating inspiration. It seems
downstream of the flow
divider
to
and
in
Fig.
5.13c,
be quite strong just
weaker by the time the
straight section is reached. There is also a small amount of
lateral motion in
flow
begins
the
as
parent
during
tube. During expiration, the
inspiration,
with
secondary flow
present in the entire flow field during acceleration (Figure
5.13d). At peak flow, seen
from the
daughter
in Fig. 5.13e, the flow entering
tube
into
the
straight, with secondary
flow
seeming
the flow
is
divider.
phase. At flow
from the
This
analogous
deceleration,
daughter
tube
curved
Fig.
is
still
section
is now
to emerge from near
to
the inspiratory
5.13f,
the flow coming
straight, and secondary
motion is again seen only in the parent tube.
Secondary flow
magnitude.
inspiratory acceleration
secondary motion
is
(a),
we
roughly
having a low but significant
the
value
(Fig
5.14)
see
that
same
at
Beginning at
the
level of
all locations,
of 15-20%. At peak flow
(b) the level in the parent tube is slightly diminished, but
grows
starting
at
-x=4-5
to
about
35%,
decreasing but
80
remaining significant as
the
flow
exits
During deceleration (c) the pattern
the data region.
is the same as for peak
flow, with the magnitude being surprisingly slightly higher,
up to
45%.
Note
primary motion
that
is
the
even
ratio
higher
instantaneous axial velocity
motion is
still
at
is
significant
of
secondary motion to
this
phase
since the
lower. Again the secondary
at
the
most downstream data
region.
Acceleration in
the
expiratory
half-cycle
(d), like
inspiratory acceleration, has a significant level throughout
all locations.
However,
the
level
is
somewhat higher,
ranging 20-25% in most axial locations. At peak flow (e) the
secondary motion is barely above noise level in the daughter
tube, but
grows
rapidly
to
upstream into the parent
daughter level is still
45%
tube.
low,
near
x=4-5, falling off
During deceleration (f) the
but
x=5 reaches even higher levels
the secondary level past
than in peak flow, over 45%,
falling off slightly in the parent tube.
Axial and
Secondary
velocity.
inspiratory acceleration,
the parent tube is
fairly
straight tube flow. The
enhanced
momentum
contribute
to
the
flat,
in
flatness
of
As
the
flow
deceleration, regions
expected for a high a
motion, which due to the
the
the
dore
axial
region,
profile,
may
is
does not exhibit an organized
develops
of
5.15). During
profile entering from
as
secondary
significant in magnitude, but
pattern.
axial
transport
the
(Fig.
lower
through
peak
flow
and
velocity gradually develop
81
near
the
upper
and
lower
walls
as
the
secondary flow
decreases.
Inspiratory flow in the daughter tube (Fig. 5.16) shows
considerably
velocity is
more
During
variation.
nearly
outside wall. There
flat,
is
but
it
but
the
higher toward the
slightly
secondary
wall toward the inside,
acceleration
motion
from the outside
does not extend across the
entire cross-section. By the
time
peak, the axial maximum
been displaced inward, and the
secondary flow shows
has
somewhat
the flow has reached its
more
of
a curved tube type
vortical structure, but again not filling the cross-section.
At deceleration the axial
shape, having
a
profile
region
of
outside wall. The secondary
has acquired a horseshoe
reduced
motion
axial
flow
near the
is strong, exhibiting a
roughly vortical pattern in each half of the tube, but again
does not extend all the way across the cross-section.
The description of expiratory flow (Figs 5.17, 5.18) is
little different from inspiratory
daughters reversed. The axial
during flow acceleration is
flow is
significant,
During peak and
but
somewhat higher at peak
profile
fairly
with
decelerating
skewed somewhat toward the
with the parent and
in the daughter tube
flat, and the secondary
no
phases,
apparent organization.
the axial profile is
outside. The secondary motion is
flow but is diminished considerably
by deceleration. There appears to
to the motion.
flow
be a slight outward trend
82
In the parent
also
fairly
flat
tube
the
accelerating axial profile is
with
significant
motion. At peak flow, most of
unorganized secondary
the profile is still flat but
small low velocity regions have formed at the outside walls.
The secondary motion has acquired a 4 vortex pattern similar
to the expiratory flow pattern
is a stagnant region in
the
in Case 3, except that there
center
inspiratory phases, the secondary
through the
entire
motion does not penetrate
cross-section.
low velocity regions are larger
of the tube. As in the
During deceleration the
and the secondary motion is
stronger. The stagnant region at the center remains.
B. Stereo views.
The raw data can best
stereo
views,
shown
description are the
in
be appreciated by examination of
Figs.
same
as
5.19-5.22.
for
the
stereo pair, the left and
right
the left and right
respectively.
eyes,
extend toward the observer,
out
data is often too dense
view
to
top
The
data
and
views. In each
graphs should be viewed by
of
The daughter tubes
the page.
Because the
easily, half the data has
been removed in Case II inspiration and in Cases III and IV,
except for Case IV, decelerating expiration.
83
VI. Discussion.
In this chapter we examine the data in the light of the
related
theoretical
Chapter
III
description.
and
numerical
and
develop
further
First, we
review
the
studies
the
discussed
fluid
mechanical
major results from the
experimental data:
1. Four flow regimes were identified.
length > 1
3.
the
There is never
flows
are
significant for stroke
.
2. Secondary flows
a steady mean 2nd component,
either
dissipate
(regimes
1-3) or
break down in structure (confined vortex regime,
high alpha and stroke).
4. Secondary flows can
be
as
large as 45% of
the peak bulk flow.
5.
Secondary flow
patterns
are
of two types:
either quasi-steady or confined vortex type.
6. An unsteady region
wall in
the
appears near the outside
Transitional
and
Confined vortex
regimes.
7. The Confined
mixing,
is
new
bifurcation flows.
vortex
for
in
regime, with end cycle
unsteady
bronchial
84
It can be seen that
motions; these
will
discussion
the
of
many results involve the secondary
be
considered
fluid
first,
mechanics
of
the
followed
by a
various flow
regimes.
A. Secondary flow considerations.
For each flow condition,
peak value of
the
the
secondary
ratio of the approximate
flow
to
velocity has already been reported.
magnitude of
the
motion
in
a
tube.
We now characterize the
more
calculating the average vorticity
the peak bulk axial
objective
manner by
in a cross-section of the
The calculation requires that
we be able to find the
inlet axial velocity gradient in
order to predict the axial
vorticity change, which we
calculate directly. We can
also
estimate the velocity gradient coming from the daughter tube
in
expiratory
flow,
however
entrance geometry is much
The average
vorticity
more
in
the
integrating for the circulation
one quadrant of the
tube.
for
inspiratory
flow
the
complex and is not treated.
parent
along
The
tube
was
found by
a path that includes
vorticity found in this
region represents the amount generated in half of the curved
section.
The streak data
is
order to increase data density.
reflected
as in Chapter V in
The circulation integral is
then
F
=
f u . ds;
where s is the variable
along the integration path, and the
average vorticity is then found from
85
S= F/A;
where A is the area
Olson, we adjust
the
maximum vorticity.
Flow
within the integration path. Similar to
integration
path slightly to produce
The results are:
Vsec/Vpeak
1
2
0.2
0.2
3,accel
3,peak
3,decel
0.15
0.35
0.22
2.9
10.9
5.4
4,accel
4,peak
4,decel
0.25
---
0.45
0.48
6.3
3.8
0.0
0.4
Note that the accelerating phase of flow condition 4 did not
exhibit
an
organized
secondary
flow
vortex
structure,
therefore the vorticity calculation is not appropriate.
We can
now
consider
the
appropriateness
of certain
curved tube approximations discussed in Chapter III.
Comparison to uniformly curved tube.
The results of Yamane et
al, described in Chapter III,
included a graph of the maximum of the secondary flow stream
function, 4ma, which
plotted was
= 1-50
was
and
shown
Dn
approximation to the secondary
result for the 2 flow
=
in
Figure 3.5. The range
5-500. We used ' ma/a as an
flow magnitude to find their
conditions which fall into this range
86
of a and Dn.
The
results
are listed below, along with the
maximum secondary flow observed:
flow condition
Dnpeak
Yse
observed
1
2
2.3
---
0.2
2
160
21.3
0.1
0.2
3
51
2.3
0.8
0.35
Here vsec has been
Yamane
secYamane
et
al
normalized
found
very
conditions of Case 2 (his
observe.
by
low
the peak axial velocity.
secondary
regime
motion
for the
III), and this is what we
Case 1 is off their chart, but by extrapolation it
also falls into regime
motion, as
III
expected.
and also exhibits low secondary
The
Yamane
prediction
conditions, however, is much
higher
than is observed here.
The reason is most likely
that
is of course finite, whereas
endless
curved
tube.
the bend in the bifurcation
Yamane
The
for Case 3
et al calculate for an
finite
bend
can
affect the
development of secondary flow in two ways: (i) it must be of
sufficient length for secondary
it is short
compared
to
the
motion
to develop; (ii) if
stroke length, the secondary
motion will decay while it travels through straight sections
of tube.
We first consider (i):
suggests that the
The
secondary
regardless of whether the
disscussion in Chapter III
flow
is proportional to Ax/R,
motion arises from either steady,
87
entrance, or oscillatory flow
this scaling directly,
(derived later in
tube studies.
we
this
In
conditions.
will use a nondimensionalization
chapter)
this
Rather than use
that
rescaling,
is common in curved
the
stroke length is
nondimensionalized by 2N, where the factor of 2 is include
for convenient conparison
to
the
secondary flow development in
arguments and entry flow
a
entry length for maximum
curved tube.
Dimensional
for significant secondary flow development is that Ax/2VaR
1.
In the
bifurcation
of this study, 2x/a
or about 40 0
an arc angle of 2/1/8 radians,
angle of the curved section
length of the curved
secondary flow
is
section
350,
But
experience that flow development,
following 2 extremes:
corresponds to
Since the arc
.
close to 2Vai, so the
is sufficient for significant
development.
at least as long as 2VaR.
-
studies suggest that a requirement
in
order
for fluid to
the stroke length must be
This brings us to (ii), and the
First,
assume
less than the curved section length.
the stroke length is
Then the fluid in the
curved section never leaves that region, and the behavior is
like Yamane's regions II or III.
This is true for Cases 1
and 2, which have Ax/d=0.75. For the geometry of this study,
d=2.54 and VaR=2.69; thus d is
within
6% of VaR and we can
consider them equal. Then Ax/d=0.75 implies Ax/(2Vai) -0.37.
The stroke
length
is
much
shorter
than
the
length for
secondary flow development; thus the secondary flow does not
become strong, independent
This is a
physical
of
whether
explanation
for
the bend is finite.
the
low magnitude of
88
secondary motion in Yamane's
lie
regions
a
above
regions
II
and III.
a2/Dn=1.31,
having
line
These 2
which
corresponds to Ax/(2Va)-1/1.31=0.76<1.
The second extreme is a stroke length
the situation for Cases 3
and
5,
respectively.
spends most of its
and
4
Under
time
secondary motion can decay
the
conditions,
-
10
the fluid
straight sections and the
diffusively.
decays fully depends on the
2vaR, which is
which have Ax/(2vWa)
these
in
>>
Whether or not it
ratio of viscous diffusion time
to cycle time. The viscous
diffusion time tvis is estimated
by a2 /v and the cycle time tc is 11w. Then
tvis
t
a21
tc
1/w
Thus, assuming a large
the secondary
flow
stroke
has
reentering the curved
that the secondary
=a2
a2
-=
=
length, small a implies that
time
to
section,
flow
decay
and
survives
completely before
larger
a values means
through flow turnaround.
This result correlates well with the observations of Chapter
IV, where it was
seen
that
for
Ax/d
>
5, secondary flow
persists for a > 6 and decays between cycles for a B 4. Thus
the critical value of a for secondary flow persistance seems
to be a
-
5.
Even for cases where
the
situation is different from
have seen that
the
flow
does
reversal
the
secondary
not
secondary flow persists, the
uniformly curved tube.
We
motion that survives through
retain
its
organized
vortical
89
structure.
Instead, it
which both causes
rapidly and
may
the
has
an apparently random structure
secondary
conceivably
motion
interfere
flow development of the next
cycle.
to dissipate more
with the secondary
Certainly there is no
steady secondary component
as
results.
expected that the secondary flow
So it is
to
magnitude in Case 4 is
be
less
there
is
in Yamane et al's
than what one would obtain for
either the uniformly curved
tube
or
the curved tube entry
flow calculation.
Comparison to steady curved entrance flow.
As discussed above, Case
3
is categorized as a steady
flow with significant curvature effects.
average vorticity found at the
the expected value
from
the
presented in Chapter III.
We can compare the
beginning of this chapter to
theory
of Squire and Winter,
The Squire and Winter formula is
as follows:
=
-2 4 (aU/az) cos 0,
Since we are interested
in
an approximate average over the
cross-section, for aU/az we use the very crude approximation
of simply
radius.
dividing
the
peak
axial
velocity
by the tube
The results are as follows:
Flow condition
3, accelerating
3, peak
3, decelerating
&calculated
ESquire & Winter
3.1
8.2
10.9
11.6
5.4
8.2
90
The agreement is best for
in
the
secondary
peak expiratory flow. As was seen
magnitude
secondary motion during
results
acceleration
deceleration, perhaps due to
organized
secondary
acceleration and
vorticity less
the
motion
deceleration
than
predicted;
unsteadiness of the flow,
of
Chapter
is
V,
the
less than that of
near total destruction of
at
flow
show
reversal.
slightly
this
may
be
Both
less average
due
to the
which is more strongly manifested
at these phases, or it may
be due to the inaccuracy of this
simple calculation.
Importance of the energy required to generate secondary
flow.
In Chapter III it
was
indicate that the peak
noted
pressure
that Yamane et al's data
gradient required to drive
oscillatory flow in a curved tube begins to deviate from the
gradient required for a straight
that this value also
coincides
the
magnitude.
secondary
contains
curved
phenomena
would
flow
sections,
occur.
it
We
increased pressure requirement
in energy
dissipation)
is
due
tube for a 2 /Dn - 1.31, and
with a dramatic increase in
Since
is
wish
the
likely
to
bifurcation
that
the
determine
same
if this
(associated with an increase
to
the
work
required to
generate the secondary flow.
From Uchida we find the time-mean rate of external work
to drive straight tube oscillatory flow:
We =
7
I
U2 a DD(a);
91
D(a)
where
DD(a)
=
[(1-2C/a)2 + (2D/a) 2
D(a)
-
ber a ber'a + bei a bei' a
ber 2 a + bei2 a
C(a)
-
ber a bei'a - bei a ber' a
ber 2 a + bei2 a
and primes denote derivatives with respect to a.
mean work
equals
the
internal
energy is recovered over a
the importance of the
cycle.)
secondary
the dissipation of energy due
The former we determine by
energy contained in
the
assuming that this
twice during each
dissipation
we form a ratio of
the secondary flow to We*
estimating the amount of kinetic
secondary
entire
cycle,
since kinetic
To obtain a measure of
motion
to
(The time-
amount
at
disorganized motion described
flow
of
flow
above.
at
its peak,
and
energy is dissipated
turnaround,
due to the
Thus the energy per
unit length per cycle that is used to generate the secondary
flow is estimated by
Ws
=
2(1/2)
p Vs 2 A f,
92
where Vs is an estimate of the secondary velocity magnitude,
A is
the
cross-sectional
area,
Introducing the definition of
Ws
Then
=
the
and
f
is the frequency.
this becomes
2(1/2) A Vs2 a2
ratio
of
secondary
flow
work
to
pressure or
dissipative work for a straight tube is
Ws /We
=
s/U)2 a DD1(a).
(1/r)
The quantity a DD-
(a)
is
ranging from 0 to 25 as
a
a
nearly
goes
linear function of a,
from 0 to 20.
area average secondary flow observed
the axial bulk flow, therefore for a
-0.8.
This estimate may
The highest
is less than 0.3 times
-
20, Ws /We
is at most
be somewhat low since it considers
dissipation only at
flow
that the additional
energy
turnaround. We therefore conclude
required
for the generation of
secondary flows can be significant compared to the time-mean
energy requirement.
B. Discussion of flow regimes.
We have seen that the flow patterns and regimes seen in
oscillatory
bifurcation
similar flows through
flows
curved
these regimes is begun with
show
tubes.
in
common with
The consideration of
a very basic examination of the
forces that determine the flow.
of-magnitude description of
much
the
This is done by an orderterms of the Navier-Stokes
93
equation for flow
in
the
cross-section
of a curved tube,
which can be written as follows:
I
II
(unsteady)
III
+ (convective) + (centrifugal)
IV
-
V
(pressure) + (viscous)
The order-of-magnitude of each term is shown below:
II
Wv
III
v* 2
U*2
a
R
+
I
V
*
IV
(pressure)
+
v
-
a2
The following transformation is made, following Berger et al
(1983),
which
results
commonly used form:
in
dimensionless
groupings
in
a
*
*
94
v
V
R 1/2
-
=
U
(-)
U
U
a
=
U
0
0
which yields the following:
1/2
[w(aR)]
2
+
2
v
+
U
U
0
=
(modified press.) +
V
f
R 1/2]
(
aU
v
a
0
The left and right bracketed terms are known respectively as
the
Strouhal
number
and
the
previously
identified Dean
number:
1/2
1/2
St
w(aR)
t
]
l|v
=
d
(aR)
U
=x
0
~x
unsteady forces
curved tube convective forces
aU
0
Dn
=
1v
a 1/2
(-)
R
curved tube convective forces
viscous
forces
95
have used the fact that for
where in the Strouhal number we
the tube diameter.
the bifurcation geometry, (aR) 1/2
is just the inverse of the
number
we see that the Strouhal
dimensionless stroke length.
A
third parameter, a, may be
obtained by taking the square root
Dn.
Clearly, since
flow must be
there
of the product of St and
are
three important forces, the
by
two independent parameters.
characterized
The Strouhal number by
So
itself
validity of the quasi-steady
determines in most cases the
assumption.
For an important
intermediate range of St, however, where many HFV conditions
occur, whether the character of the flow is of quasi-steady,
transitional,
or
confined
vortex
separately by either a or Dn.
and Chang
introduce
their
parameters of Chapter III)
fact St
and
a
taken
nature
is
determined
It is ironic that both Pedley
versions
as
together
of
St
replacements
consitute
(the
f and E
for a, when in
a complete flow
description.
We
now
discuss
the
flow
regimes
in
terms
of the
dominance of these forces:
1. Quasi-Reversible. examples: Case 1, Case 2.
This regime fluid mechanically ecompasses two extremes.
At low a and
low
Ax/d,
like
Case
1, the flow is viscous
dominated and therefore approaches a truly reversible Stokes
flow. The frequency is
sufficiently
has plenty of time to
diffuse
axial flow is so
that
low
low that wall momentum
into the core fluid, and the
centrifugal forces never become
important. Centrifugally-driven secondary velocity therefore
96
inviscid Squire and Winter mechanism
The
does not arise.
for generating secondary motion due to inlet axial vorticity
does not play a role because of the short stroke length.
As we move up
forces become
a,
in
large
so
enough
Lyne-type secondary motion are
axial accelerations dominate
layer is so thin, the
Case 2, the centrifugal
in
as
the relatively small
that
probably present, but strong
the
flow.
Because the Stokes
axial profile is quite flat, although
it does seem to adjust
slightly
the cross-section, indicated by
to
the change in shape of
the slight elliptical shape
of the expiratory profile. Curvature effects, which comprise
the
rest
of
the
nonlinear
negligible compared
to
the
words, flow in the curved
inertial
axial
terms,
are
also
accelerations. In other
section falls into Yamane et al's
region III. Thus the flow is nearly reversible.
2. Quasi-steady. example: Case 3.
As described in
bifurcation work has
the
introduction, nearly all previous
been
done
flow. We limit discussion to
on
steady or quasi-steady
what appear to be the dominant
mechanisms in our experiments.
During inspiratory flow, the entering profile resembles
that of straight pipe flow. The changing geometry downstream
does not seem to affect the upstream flow. The daughter tube
velocity
field,
as
described
somewhat curious. The
secondary
the straight
has
section
pattern as expected.
the
However,
in
the
last
chapter,
is
flow pattern seen entering
2-vortex
the
axial
curved
tube type
profile shape is
97
less easily understood. Since the flow divider has split the
parent profile, initially
region will have
formed off the
an
inward
flow
layer at the outer
the
axially
skew,
divider
wall.
peak
in the curved
with the boundary layer
being
However,
much
at
thinner than the
the entrance to the
straight daughter tube just past the curved section, we find
a nearly axisymmetric profile.
This result
is
not
inconsistent
with
previous work
since the values of
the governing parameters are different.
The Reynolds number
in
Case
3
based
on peak velocity is
Re=215.
Re=480 in the work of Isabey and Chang,
used by
Schroter
and
Sudlow,
and
Re=352,
although
the
lowest
value
presented in his thesis
used by Schroter
and
is
Re=468.
Sudlow
was
this study's (a/R=1/6); Olson's
on the daughter tube.
on the daughter
greater than
the
Olson
for
Re=290 was
went
as low as
which
data
is
However, the curvature
much larger (a/R=1) than
was closer (a/R=1/7), based
The corresponding Dean numbers, based
tube,
are
peak
215
value
condition. Thus one would
the case of this study,
and 143 respectively, much
of
51
at
the present flow
expect weaker secondary motion in
and
therefore a lesser tendency to
sustain the axial peak toward the outside of the bend. Turni
ng to expiratory flow, we
in
the
daughter
straight daughter
again find unusual axial profiles
branches.
tubes
shows
outside of the bifurcation,
an upstream
effect.
The
(Ito
flow
a
which
has
coming
slight
out
skew
of the
toward the
may be representative of
demonstrated
an upstream
effect of pressure due to the presence of a bend.)
98
At the low a
high
and
while
in
travelling
as
it
The fluid
sections.
straight
the
motion
regains secondary
to lose its secondary motion
fluid
sufficient time for the
in this regime, there is
Ax/d
350 bend and
the
rounds
enters the parent tube. The secondary flow pattern is the 4cell
expiratory
flow
shape.
motion is slightly greater
about
35%
of
peak
vertical ridge,
than
flow.
probably
The
it was during inspiration,
The
due
of secondary
magnitude
profile
axial
to
the
secondary convection and the lack
exhibits a
combined
effect of
of wall resistance in the
center of the tube.
3. Transitional flow.
(No quantitative example.)
The flow
of
Yamane's
parameters
categories
VI
centrifugal forces are
and
both
superficially quasi-steady
because of
the
these
greater
results
VII,
where
important.
during
put them into
and
These flows appear
most
importance
unsteady
of
of
the cycle. But
centrifugal forces
compared to the category just described, it is expected that
the secondary
motion
will
stronger secondary flows
horseshoe shaped
axial
be
stronger.
would
be
profile
described in Chapter III.
the
in
Coupled with the
development of the
the
curved section,
This horseshoe-like profile has a
higher velocity core near the
inside, with a lower velocity
region at the outside wall. Since unsteadiness is important,
the highly
inertial
core
lags
the
time-varying pressure
gradient somewhat compared to the more viscous dominated low
velocity region.
Unsteadiness
effects
are
strongest at
99
times in the
these times
cycle
that
near
flow
temporal
accelerations
convective accelerations are
turnaround that we see the
the
outside
wall
turnaround,
lowest.
since it is at
are
Thus
highest and
it is at flow
low velocity viscous region near
responding
the
the
reversing pressure
gradient somewhat before the more inertial core region.
In this
transitional
forces are not
strong
regime,
enough
to
sustain the low velocity
region for a significant
portion
mechanisms
important
become
more
convective and unsteady
of
the flow cycle. These
in
the
following flow
regime.
4. Confined vortex flow.
The regime which will probably
mass transfer modelling
trapped vortices.
is
There
the
one exhibiting confined or
are features of three-dimensional
and time dependent nature,
occurs is in direct
have the most impact on
and
contrast
the
manner in which mixing
to previous studies of quasi-
steady mixing.
We begin by
describing
(The same scenario
inspiratory and
of
the
flow during acceleration.
mechanisms
expiratory
directions.)
curved section from the straight
residual and apparently
seems
to
occur in both
Fluid
enters the
daughter tube. It has some
unstructured
secondary motion left
over from the previous cycle. The axial profile, hovever, is
the
basically
Womersley
flow
flat
in
shape
a
one
straight
expects
tube.
from
high-alpha
During
initial
acceleration, the residual secondary motion seems to obscure
100
any
organized
fluid with
from
negligible
region, and the
uniform
pattern
axial
velocity in the
developing.
secondary
flow
develops
profile
into
core
sets
curvature direction.
In
motion is small, this
motion
much
a
up
the
bend.
somewhat
Here
the
downstream,
singularity
begins to move toward
fluid,
being
from
the
curvature
across the
and
the
does
But this
not
have
it is decelerated by
and
does
it
pair
fluid here
tube.
layer,
Instead,
one
Stewartson-type
III),
of
gradient
vortices",
where axial
thickens until,
a
therefore
cross-section.
"trapped inner
layer
boundary
pressure
layer,
the wall toward the inside
center
significant axial velocity,
The axial
gradient drives a secondary
Chapter
the
the
tube.
reaches
in
entry flow of a
pressure gradient in the
boundary
it
(described
like
boundary
motion which propels fluid along
of the
later,
enters the curved
curved
a
pressure
Slightly
on
not penetrate
survives
each
as
four
side of the
bifurcation.
The description to
this
point
experimental results on entry
and by Agrawal et al, and
& Berger. The
trapped
flow
is consistent with the
_into a curve by Scarton
with the numerical results of Soh
vortical
structures persist through
deceleration until very near bulk flow turnaround, where the
axial core flow is
greatly
reduced and the axial component
of the trapped region is now
significant.
is clear that the inner region
the
entire
cross-section,
and
At this point it
can now burst forth and fill
strong
lateral
mixing is
101
observed. What is not
determinable, if highly complex,
a
in
inner region occurs
of instability drives the flow into
manner, or if some sort
500,
cases are less than
Re/a for all confined vortex
of
The values
random motion.
whether this growth of the
is
clear
somewhat below Hino et
is
which
al's value of 700 for straight pipe turbulent transition, as
However, the velocity profiles
III.
described in Chapter
are quite different from those in a straight tube; therefore
the
for
conditions
instability
addition, the curvature
of
the
centrifugal instability effect.
may
different.
be
bifurcation
In
may promote a
it does not seem
However,
possible to ascertain the exact mechanism by our techniques.
In any case the
mixing
vortex patterns
are
not obviously organized and the
is
not
enough of the secondary motion
mentioned above,
As
retained.
survives to persist into the
acceleration phase of the next cycle.
It is worth detailing
the
field through flow turnaround:
the trapped regions
becomes
development of the velocity
First, the helical motion in
circular,
axial motion there has gone
to
two of the
vortex
three
confined
a=12, Ax/d=10), the inner
section,
remaining
and
the
case
distort as they
resemble
vortices.
velocity
patterns
organization.
and
In
that
cases (a=6, Ax/d=20 and
grow
expand
(alpha=21.3,
grow
zero. After this point, for
regions
vortices
to fill the cross-
accordingly.
Ax/d=10),
the
all
indicating that the
For
the
the inner regions
velocity patterns no longer
three
follow
cases,
lose
however,
all
the
apparent
102
Because of the entry
analogy to Yamane et
flow
al's
only partially valid.
nature
of this regime, the
uniformly curved tube results is
They do predict (see Chapter III) the
unsteady region that
forms
near
however their secondary flow
inward location, and
the
the
inside
pattern
of the bend,
shows only a slightly
pattern remains organized through
flow turnaround.
Finally, we
note
two
important conclusions regarding
the confined vortex flow regime:
1. The flow is strongly unsteady.
the fact that the
flow
patterns
This is evidenced by
450
before peak flow are
dramatically different from those
seen
ie at the same
The observation of strong
bulk
velocity.
unsteadiness in a bifurcation
flow
450 past peak flow,
is important because it
is unexpected based on predictions by Chang and Pedley.
2. The
different.
strongest
transport
During quasi-steady
at
the
same
strongest, and the 2
cycle.
In
confined
during high axial
go
time
to
vortex
flow
confined to isolated
mixing.
mechanisms
are
are
flow
that
zero
likely
the lateral motion is
the
at
flow,
strong,
to be quite
axial
motion
is
the same time in the
the
but
secondary motions
they are largely
regions, thus reducing cross-sectional
Instead, strong
occur at flow turnaround.
cross-sectional
mixing is seen to
This occurrence of lateral mixing
is exactly the reverse of the quasi-steady case.
C. Flow regimes encountered in the lung.
103
Based on the Weibel
made to determine
lung dimensions, calculations were
which
of
this study are encountered
the
in
flow regimes discussed in
the human lung under various
combinations of frequency and tidal
volume.
In addition to
the Weibel specifications, we have made the same assumptions
of average radius of curvature and angle of branching as for
the experimental model.
The
results are plotted on graphs
of a vs. Dean number which also display the appropriate flow
regimes, as in
Chapter
IV.
encountered during normal
10
liters/minute).
Figure
quiet
Each
the
and proceeding into the lung
for
normal
experience
quiet
represents
trachea
to
the
breathing,
quasi-steady
shows the regimes
breathing (15 breaths/min,
symbol
generation, starting with
that
6.1
flow,
Weibel
at the upper right,
left.
all
with
a
It can be seen
lung
the
generations
first
several
generations being dominated by inertial effects, and viscous
forces become
lung.
dominant
as
one
It should be recalled
further into the
from Chapter III that lines of
constant stroke length have
can see that the local
proceeds
a
slope
of 1/2, therefore one
stroke length varies very little for
the first 15 or so generations.
We next
examine
the
trends
frequency and tidal volume.
Figure
varying frequency at a fixed
that might
be
generations are
used
in
shown;
viscous-dominated regime.
associated
We
6.2 shows the effect of
tidal volume Vt= 25ml, a value
HFV.
the
with varying
Only
rest
are
the
upper
15 or so
in the quasi-steady
see that increasing frequency
104
from 1 hz to
15
hz
does
not
show a tendency to increase
unsteady effects. Instead,
all
the
into the Turnaround
zone-Confined
frequency determines only how
upper generations fall
vortex
regimes, and the
many of the upper generations
reach these regimes.
Varying tidal
volume
has
a
unsteadiness, as shown in Fig.
fixed at 5 hz while
100 ml.
the
We see that
much
6.3.
tidal
stronger effect on
Here frequency is kept
volume is varied from 10 to
the lower tidal volumes are closer the
the unsteady regimes, and increasing Vt moves all upper lung
generations
regimes
through
and
into
the
Turnaround
zone-Confined
quasi-steadiness.
unsteadiness is more strongly
The
affected
vortex
importance
of
by a 10-fold change
in Vt than by a 15-fold change in frequency.
Finally we display in Fig. 6.4 a graph showing a shaded
region representing many combinations of frequency and tidal
volume, with frequency = 5-15 hz and Vt
that the range
of
parameter
generations
nearly
transitional
and
values
parallels
confined
generations fall into either
the
quasi-steady
unsteady regime.
zone;
Fredberg
for predicting lung
Their
airways are in the
and
impedance
axial velocity profiles.
0.5-10 Hz.
one
there
The
velocity
unsteady
of
the
vortex
10-100 ml.
=
the first several
location
regimes.
of
is
Mead
We see
of
The
the
upper
these regimes or into
no
entrance
into
the
have proposed a model
which assumes Womersley-type
model is intended to apply to
description
implies that the
regime, which means that their
105
assumption is valid only for tidal volumes much smaller than
10ml.
D. Summary
The fluid
mechanical
regimes
encountered
in a model
lung bifurcation correlate well, qualitatively, with results
for curved tube studies.
relative importance of
curved tube
The
regimes are determined by the
three
forces: viscous, unsteady, or
convective
forces.
Because
there are three
competing forces, it is not possible to describe the flow by
a single ratio such as a Reynolds number or the
used by Pedley and Chang.
parameter
The flow regime must specified by
two parameters, such as a and
a and stroke length.
E
Dn, or, for a fixed geometry,
106
Mass transport implications.
VII.
This chapter
briefly
reviews
previous
experimental mass transfer results
mechanisms.
work covering
in the lung and proposed
We then consider the character of flow found in
this study and suggest a streaming-mixing model.
A. Fundamental transport mechanisms.
diffusive transport, governed
A(dC/dx), where V
is
diffusion coefficient,
area, and (dC/dx) is
(2) convective
turn form the
the
A
the
for
mass transport are: (1)
by
Fick's
mass
flow,
is
the
equation
V
=
is the molecular
cross-sectional exchange
local concentration gradient; and
transport.
basis
of
-
The fundamental mechanisms
These
fundamental mechanisms in
the transport mechanisms proposed
for HFV, which are the following:
Augmented diffusion or dispersion.
means of convection by
Transport occurs by
a nonuniform axial velocity combined
with a lateral mixing mechanism which is either diffusive in
nature
(which
kinematic,
motions.
(1953)
we
arising
The
and
refer
from
diffusive
Watson
to
as
augmented
turbulence
case
(1983),
has
who
results are expressed in the
or
or
complex secondary
been
studied by Taylor
examined
unsteady problems, respectively, in
diffusion)
the
steady and
a straight tube.
Their
form of an effective diffusion
coefficient Deff, as follows:
Deff/n = 1 + (1/48)(Ua/K) 2
(7.1)
Deff/n = 1 + f(a,Sc)(Urmsa/x) 2
(7.2)
107
Equation (7.1) is
flow through a
mean velocity
Taylor's
pipe
U.
of
of
the
circular
Equation
result for oscillating
square
expression
flow,
for Deff for steady
cross-section, radius a,
(7.2)
is
where
Urms is the root-mean-
area-averaged
axial
Watson's analogous
velocity,
So
is the
Schmidt number Sc-v/x, and f(a,Sc) is a closed-form function
which yields 1/48 for a<1
for
most
respiratory
experimentally
when
Sc -1,
gases.
verifed
by
a good approximation
Watson's
Joshi
et
al
result
has been
(1983)
for
the
circular pipe case.
The appropriateness
of
the
augmented diffusion model
for a branching system was studied experimentally by Kamm et
al (1984b), Slosberg, Paloski
(1986), and Keramidas (1986).
All
multigeneration
measured
Deff
in
a
conditions including HFV-like ranges.
transport for
gas
combinations
of
model
under
Slosberg examined the
varying diffusivities,
Paloski studied temporally asymmetric forcing functions, and
Keramidas made measurements
network.
Kamm et al's
graph of Deff
in
Figures
The
7.2,
a
results
normalized
function of a.
in
by
are
7.4.
normalizations were used by
these
is normalized by a2
than
factor of r.
Paloski
raising normalized tidal
than 2.
The molecular
Vt 2 /A2 ) plotted as a
of the other studies are shown
and
rather
shown in Figure 7.1, a
(r/2)(f
results
7.3,
more anatomically shaped
found
volume
a
Slightly
different
groups: The tidal volume
A,
resulting in an extra
better
to
the
power
law fit by
power 1.82 rather
diffusion x was subtracted from Deff
108
before
normalization
to
molecular transport;
remove
this
the
typically
changes
than 20%.
The independent
variable is
studied 3
values
the
of
Watson's theory are
Sc;
shown.
and So = 0.84, so the
a.
Thus Figure 7.4
3
effect
of
purely
Deff by less
8=a*Sol /2. Slosberg
curves
corresponding to
In Keramidas's study, n=0.179
horizontal axis is nearly the same as
can be compared to Figures 7.1, 7.2 and
7.3 with acceptable accuracy.
The Keramidas study is
distinct
model used was anatomically shaped
curvature of
ratios.
It
similar to
study.
the
is
bends
and
therefore
that
seen
with regard to radius of
parent-to-daughter
more
for
in that the branching
tube area
likely to exhibit behavior
the
bifurcation
model of this
The other studies used models having sharp bends and
parent-to-daughter
recirculating
area
separated
ratios
zones
of
1:2.
which
were
Thus
the
visualized
by
Slosberg were probably not present in the Keramidas study.
Nevertheless it can be
these studies, Deff
augmented
dependence.
is
diffusion
In fact,
seen
that
slightly
theory,
when
for the data of all
higher
and
shows
an
much
less
a-
by fVt 2 /A2 , Deff is
normalized
constant to within less than
than predicted by
order of magnitude for all
the experiments.
It is somewhat surprising
little variation when,
that
according
to
Deff should exhibit so
this study, the fluid
mechanical regimes vary significantly for the range of a and
Ax/d-1 up to 30 used
in
these
studies.
For the Kamm and
109
Slosberg branching models this behavior may be explained, in
part,
by
geometry,
nonanatomical
the
recirculation zones not present
causes
which
anatomical models.
in
The
fluid and mass transport
mechanisms are therefore likely to
be somewhat different.
The
Keramidas
more anatomically correct and
the range,
Ax/d-1-5,
that attempts
to
exhibits the same behavior in
reported.
measure
model, however, is
(Keramidas also mentions
Deff
for
a
lower
range of Vt
yielded results which were too low to be meaningful.)
Streaming transport.
This
mechanism, identified and
studied by Haselton and
Scherer, can arise from directional
assymmetry of the axial
velocity
in an oscillatory flow the
moves
to
the
bidirectional
left
undergoes a net
the
blunt
results.
leftward
walls moves rightward.
inspiratory
and
For example, if
profile is parabolic as the flow
but
motion
profile.
to
The
drift
fluid
while
Directional
expiratory
the
the
in
a
the
net
center
fluid near the
asymmetry is found in
velocity
bronchial bifurcation, as described
right,
profiles
in Chapter I.
in
a
Haselton
and Scherer, in the studies described in Chapter I, measured
the streaming velocity in branching models.
of stroke lengths
puts all those
used
in
experiments
their
The short range
single bifurcation model
in the quasi-reversible regime,
and as stated before,
the
was small, never much
greater
than one-tenth of the stroke
length and
less.
Thus
generally
agreement with the results
streaming displacement per cycle
of
that
data
is in good
this study, despite the fact
110
that their
bifurcation
sharp bends.
geometry
Transport
not quite as small.
in
included non-anatomically
their multigeneration model was
The difference could be due not only to
the multigeneration nature of the flow, but also possibly to
the
larger
stroke
length
range
or
the
non-anatomical
diameter ratio.
Grotberg
streaming
(1984)
motion
found
due
to
analytical
oscillatory
solutions
flow
in
for
a tapered
channel, which was presented as a first approximation to the
shape change of a bifurcation.
found to be fairly small,
The streaming velocity
was
less than 0.1 times the amplitude
of the oscillatory velocity.
Streaming
displacement
is
reduced
by
any mechanism
which tends to randomize the cross-sectional location. It is
therefore optimized in
mixing
mechanism.
directionally
contrast,
of
absence
However,
symmetric,
augmented
regardless
the
if
of
the
streaming
dispersion
directional
or
diffusion or other
velocity
cannot
occur.
diffusion
symmetry,
but
field
is
In
can occur
requires
the
existence of a lateral mixing mechanism.
We can estimate
effects
follows:
such
as
the
relative importance of convective
streaming
The time scale
to
augmented
dispersion
as
for purely axial convective motion
is
L
tsu
111
where L is an
For
measure of axial velocity.
the curved section of
same as the parent
1
diameter,
tube
and u is a
we choose the length of
which is about the
bifurcation,
the
scale
length
axial
appropriate
The time scale for
d.
radial mixing is
a2
trd
~
1OLrc
If
the
mixing
radial
time
scale
concentration gradient changes
it is
convected
axially.
is
long,
the
radial
little during the time
very
diffusion exists when
Augmented
trd is comparable to or smaller than the convection time. So
the
relative
importance
of
motion to
purely
convective
1
=
convection dominates
=>
augmented diffusion
augmented diffusion is
t
2
Ax
rd
~
t
--
(
a w
)
<<
d
c
>>1
The quantity in parentheses is
parameter proposed by Schroter and
to be confused with the
in
Sudlow, discussed
gases, ic~v ,so
Chapter
a2 .
P
that the stroke length
often referred to as P2 (not
or
a
I).
For
most respiratory
The above relationship shows
or
both
must be large for
112
convective motion to be important.
We may therefore expect
a convective motion such as streaming to be important in the
confined vortex flow regime described earlier.
The
above
estimate
does
not
directly
address
the
streaming mechanism, which, averaged over time, can be small
even when the
peak
therefore make a
convection
comparison
motion
is
large.
We will
based on experimental results:
The streaming transport may be written as
~
V
u
str
where AC
is
A AC
s
the
concentration
length, and us is the
over
a mixing
streaming velocity, which we estimate
from Haselton and Scherer's
the axial velocity.
difference
results
The
to
be about 0.1 times
transport by augmented diffusion
can be written as
ad
-
dC
~ DeffA
V
dx
2
with
Deff > 0.02 u
T, where T is the cycle period,
rms
for a straight tube.
We estimate AC by
is the mixing length,
1M ~
us
rd.
trd is either (1/2)T, if mixing
the confined vortex case, or
or approximately
(1/60)0#2 T,
1m
(dC/dx), where
m
The radial mixing time
occurs on each cycle, as in
else is estimated by a2 /(lox),
when
lateral
transport is by
113
molecular diffusion.
Then
the ratio of effective transport
coefficients is
2
u
D
t
effst
s
~
rd
(
50
)
D
~.
u
rms
eff,ad
50 (0.01)
T
{1/2
or
2
(1/60)#
This estimate also
shows
that
negligible except for larger
than about
10.
So
values
again,
streaming mechanism to be
streaming
we
important
likely to be
/ ( or a), greater
of
may
is
}
expect
some sort of
in the confined vortex
regime.
B. Bronchial transport models.
Bronchial transport models
which
attempt to model HFV
conditions and which incorporate
features of both streaming
and mixing
by
have
been
proposed
Gavriely and Butler (1986),
model
consists
among
the
of
compartments
The transit time
Slosberg.
longitudinal
distribution of transit
experiments.
and
and
for a rough
profile.
to
from
a
Transport
non-uniform
are determined from washout
Butler's
cylindrical segments divided into
2 layers are allowed
Permutt et al's
along different streamlines.
distributions
Gavriely
et al (1985),
compartments.
arises
times
Permutt
model
considers
2 concentric layers.
The
move at different speeds, allowing
approximation
to
a non-uniform axial velocity
Molecular diffusion occurs within the layers while
transport between them is described by a mixing coefficient.
114
The
mixing
coefficient
is
chosen
to
satisfy
Taylor
diffusion, thus there is no provision for convective mixing.
Their results were expressed
where the tracer was
the trachea.
as
injected
Slosberg's
a tracer elimination rate,
from a specified location in
model
is based on the observation
of separation zones in his
branching system which were seen
only during inspiration.
Slosberg
region
axial
profile
during
stagnant separation zone
flow.
inspiration,
near
The expiratory flow
the
Assuming
gradient, Slosberg found
representing a
wall surrounding a core
profile was assumed to be blunt,
and lateral mixing was allowed
each half cycle.
therefore assumed a 2-
to
occur only at the end of
a locally linear concentration
the
following
expression for the
effective diffusion coefficient:
= (1/2) fVt2(Aw/Ac)(1/A 2
Deff
where A is the cross-sectional
cross-sectional
respectively.
areas
When
of
the
this
Deff
area
and
wall
is
Aw and Ac are the
and
core
normalized
as
regions
in the
preceding studies, the result is
D
A
eff
2
2
f Vt / A
w
2A
c
a constant which
velocity profile.
depends
only
on
the
shape of the axial
115
C. A suggested model.
In this section we
develop
of Slosberg's analysis (For
a more generalized version
further details, see Slosberg).
As we have focussed on the fluid mechanics in this study, we
attempt to model only
the
convective aspects of transport.
Effective dispersion coefficients
new model under some
of
the
are
predicted using this
flow conditions found in this
study.
We
begin
by
assuming
that
concentration gradient exists
C(V,t)=C(V,O), where C
volume
and
t
inspiratory
is
is
at
the
time.
end
radially
uniform
expiration, given by
species concentration, V is
After
half-cycle,
a
the
the
completion
radial
of
the
concentration
distribution is given by
C(r,8,V,T/2)
=
(7-1)
C(V-Fi(r,)Vt,O),
where T is the period, Vt
is tidal volume, and Fi(r,O) is a
shape function satisfying
1/A
f Fi(r,O)dA = 1.
This shape function can be
thought of as a normalized local
stroke length function.
axial
velocity
profile
It
related
shape.
represented by Fi.) We have
during the half-cycle.
is
assumed
to the time mean
(Hereafter
Fi(r,8)
is
nonvariation with time
Now let complete radial mixing occur
116
at
the
start
of
expiration.
The
new
concentration
distribution is given by:
C(V,T/2+)= (1/A)
where A=ra 2 .
The
f C(V-FiVt,O)dA,
redistribution
(7-2)
following expiration is
derived similarly, using an expiratory shape function Fe:
C(V,T~) = C(V+FeVt,T/2)
(7-3)
Substituting equation (7-2), this becomes
C(V,T+) = (1/A)
We wish to find
the
f C(V-[Fi- Fe]Vt,O)dA.
net
(7-4)
species volume transported past a
location Vo. The total flow to the right is
Vr = (1/A) f FiVt[ C(Vo,O) + C(Vo-FiVt,O)]f dA;
(7-5)
2
where f is the frequency.
is given by:
Similarly, the flow to the left
117
V
=
(1/A)f FeVt[ C(Vo,T/2) +C(Vo+FeVt,T/2)]fdA.
(7-6)
2
An expression for
Deff
can
following expression for
the
be
found
by substituting the
concentration gradient at end
expiration, assumed to be linear:
C(V,O) -
1 -
V(dC/dV).
(7-7)
Making use of the fact that the area average of F(r) is
unity, we find
Deff
-
where FF = (1/A)f
2
(1/2) (fVt 2
/A
F2 dA,
area average of the square of
the shape function.
the
These
)
[FFi + FFe -
equations
2];
(7-8)
revert to the forms
found in Slosberg when
Fi =
0
(a-)<r<a; in a wall region of thickness 6
A/Ac
0<r<(a-); in the core region of area Ac.
{
and
Fe = 1,
a constant.
118
through 7-8) to find Deff for
We next employ equations (7-1
some conditions of interest.
1. Parabolic profile.
This simple, non-blunt profile
is examined for comparison
purposes.
The shape function is
assumed to be:
Fi - Fe = F(r) = 2 [ 1 - (r/a) 2
which has an area average of 1, as it should.
average
F2
of
Deff,1/(fVt 2 /A 2 )
is
=
FFi
FFe
=
(7-9)
Then the area
1.33,
=
yielding
(1.33+1.33-2)/2 = 0.33.
2. Blunt profile
with
linear
Stokes boundary layers.
The boundary layers are of thickness (v/w) 1/2,
with a linear
progression from the wall to the edge of the boundary layer.
It is intended that
this
be
a simple approximation to the
Uchida solution, which resembles
this
shape at high a, and
which yields the Watson result for Deff.
The shape function
is
C
(a-r)/(a-6)<r<a;
F(r){
(7-10)
C
where b -
o<r<(a-b);
(v/w)1/2 and C
=
[1 - a-1
4
1-
-
+
3a
FFi
=
+
(-2/3)]
1
2
2a
FFe = FF =
1
1
+
[1 a
2
2]
3a
1.
Then
119
and Deff is
found
from
comparison between the
(7-8).
Eq.
Deff
from
this
Watson's solution, assuming So - v /n
that for a
5,
'
Figure
~
7.5 shows a
simple analysis to
1.
It may be seen
the Deff from the relatively simple result
of Eqs (7-11) and (7-8) closely resembles Watson's solution.
3. Simplified Confined vortex calculation.
is described in
Chapter
IV
and
The confined vortex regime occurs
Ax/d.
It
exhibits
acceleration,
a
while
fairly
during
diminished velocity is seen
motion is large within
thus there is
little
this
inspiratory
and
at
for
high values of a and
blunt
axial profile during
region
areas.
the
end
expiratory
a
region
of
the outer wall. Secondary
cross-sectional
region to higher velocity
mixing occurs only
is reviewed briefly here.
deceleration
near
This regime
but is low outside it,
convection from this
Strong lateral convective
of
flow
each
half-cycle.
patterns
appear
The
very
similar.
With these characteristics in mind, the shape functions
are chosen as follows:
accelerating flow:
Fa(r)
=
0
decelerating flow:Fd(r)
=
1.
blunt profile
(a-6)<r<a
low velocity
{
A/Ac
0<r<(a-6) core region
120
Note that these conditions
one difference
being
are quite similar to Slosberg's,
that
the
profiles are the same; thus
motion without the mixing
half-cycle.
Employing
inspiratory and expiratory
there would be no net streaming
that
these
occurs
at
the end of each
conditions in equations (7-1)
through (7-7), we find
Deff,3
=
(1/4) (fVt 2 /A2 ) (Aw/Ac);
(7-12)
where Aw is now the cross-sectional area of the low velocity
region.
4. Integration of velocity data.
shape functions and
data of Case 4.
has been shown
Deff
This
to
values
We
now calculate the
based on the experimental
data, in the confined vortex regime,
exhibit
highest
lateral mixing at flow
turnaround and reduced lateral mixing during the rest of the
cycle.
Our model therefore bears a close resembalnce to the
actual
characteristics
spatial resolution
coarse and based on
only modest
of
of
these
the
The
shape
w
=
u
- f
2
dt.
Up
temporal and
will
be rather
data, we therefore expect
functions
integrating the velocity field, as follows:
F
The
integrations
interpolated
accuracy.
flow.
are
found by
121
The velocity is
known
450
direction and
at
from
six
each
phases:
peak.
peak
We
flow in each
also assume u=O
uniformly across the cross-section at flow turnaround.
This
is not strictly true for Case IV, but it is assumed that the
strong mixing at
flow
axial velocity.
The
turnaround
integral
will
above
tend
to blunt the
is calculated by the
trapezoidal rule:
M-1
F - (At/2) { u0 + um +
}
Z [u(t )]
(7-13)
i=0
then the
area
averages
of
trapezoidal integration
F
of
and
velocity
smoothed and interpolated onto a
step
the
calculation
is
The calculated
This error
is
due
check
which
grid.
by
is
has been
At each time
determining
the
compared to the true
is
generally lower by 5-15%.
part
to
finite
in
The data
data
also found by
value
(around 5%) and in part to
Chapter II.
are
polar
instantaneous bulk velocity, which
value.
FF
is
integration error
the types of errors discussed in
rescaled
to correct for the bulk
velocity before the shape functions are found.
The results, for a slice
in
the parent tube with Deff* =
Deff/[f(Vt/A)2 ], are as follows:
expt
E)e
E
4
.947
ZL
EV-Fa Deff*
.885
.948
.877
0.12
122
From
Keramidas'
data
typical
expected to be about
0.1,
so the agreement with experiment
is reasonably good.
approximately
measurements of
Deff*
are
calculation is quite sensitive:
the
factor squared and unity.
to one, so is its
for
This is somewhat fortuitous, since from
shape factor arguments, the
Deff* is
values
difference
Since
square.
10%
between
the shape
the shape factor is close
Thus, an error in the velocity
would
mean
a
possible corresponding
error of 0.10 in the shape factor, and therefore an error of
0.10 in Deff*, or a 100% error (this is apart from numerical
errors of multiple integration,
For reasonable
accuracy,
the
which can be considerable).
precision
in
velocity data
required is v5%, a degree of accuracy barely achieved in any
bifurcation
velocity
measurements
to
calculation achieves better
accuracy
speculated that
the
errors
in
date.
than
velocity
As
our
expected, it is
measurements are
random and tend to cancel.
There is no functional
dependence
on
a in this model.
The data in Figures 7.1-7.3 exhibits a degree of
which appears to
be
intermediate
and a constant Deff which is
likely,
the
between
Watson's
transport
between the Watson model
predicted by this model.
mechanisms
augmented
dependence
are
diffusion
also
and
Most
intermediate
the streaming-
mixing model of this study.
Summary.
In this
chapter we have reviewed experimental
and theoretical examinations
systems.
of
mass transfer in lung-like
A streaming-mixing model is proposed, based on the
123
characteristics of flow of this study.
It is found that the
degree of transport depends on the extent to which the axial
profile is "peaked"; eg for
0 in (7-12) and FFi
=
no transport would
occur.
FFe
even if the inspiratory
the same.
Finally,
performed
by
Agreement
with
=
a
=
1 in (7-8), hence Deff = 0 and
Finite
and
integrating
the
a completely blunt profile Aw
transport is predicted
expiratory shape functions are
numerical
velocity
experimental
calculation
data
data
of
from
of Deff is
Case
4.
Keramidas
is
reasonably good despite the sensitivity of the calculation.
124
VIII. Conclusions.
Oscillatory flow in
a
studied over a wide range
bronchial
of
flow regimes encountered in
according to
convective
the
curved
high
a and Dean number.
The
the bifurcation are categorized
dominance
tube
bifurcation has been
of
unsteady,
forces,
and
viscous, and/or
these
flow
regimes
closely resemble those found for studies of uniformly curved
tubes.
It is noted that,
as in the oscillatory flow curved
tube problem, the presence of three forces requires that the
flow regime be specified
by two independent parameters, not
just a single parameter,
as
suggested by previous studies.
One regime, characterized by a confined vortex structure and
by strong convective mixing at
previously unreported.
end cycle, is believed to be
The range
encountered by a typical human
volumes and frequencies is
the flow is found
to
be
of regimes expected to be
lung
under a range of tidal
determined.
much
The unsteadiness of
more dependent on the tidal
volume than the frequency.
A streaming-mixing
mass
based on the description
study.
Reasonable
of
transfer
model is suggested,
convective flows found in this
agreement
to
experimental
found, despite the sensitivity
of
the calculation to small
studies is
errors.
Previous work on
oscillatory
curved tubes is found to
be
the fluid
the
mechanics
in
flow
and
entry flow in
relevant to the description of
bifurcation.
This relevance
125
suggests that somewhat
flow through
simpler
tube
with
finite
bifurcation, would
shed
further
problem.
experiments and analyses on
bends,
light
resembling
half a
on the bifurcation
126
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133
Appendix
Videotape examples of the 4 flow regixes.
The attached videotape contains samples of videotaped
data of each of the four flow regimes discussed in this
study. They are as follows:
start
A/d
1. Quasi-reversible.
a
4
0.75
0:00
0: 50 sec
2. Quasi-steady.
2.3
19.6
0:50
1:58
3. Transitional.
6
5.0
1:58
2:15
4. Confined vortex.
12
21.3
19.7
2:15
2:37
9.7
2:42
2:55
e
s
The image is as seen in Figure 4.3. A top view of the
bifurcation is seen in the upper part of the screen; the
mirror image side view is in the lower area.
The particles
appear as bright orange dots on a faintly bluish background.
The blue color is due to a slight fluorescence of the
acrylic plastic bifurcation.
The reversibility of regime 1 is easily seen by playing
the tape back at twice normal speed, while details in the
other regimes can best be observed by playing the tape at
reduced speed.
134
high velocity
"horseshoe"
va
horizontal axis
parent tube
daughter branches
Figure 1.1
Qualitative picture of inspiratory flow in
a bifurcation with a Poiseuille entry profile. The 2-vortex
secondary flow pattern in the daughter tube is shown in the
lower branch; the upper branch shows the "horseshoe" shape
of the axial profile.
135
-4-
90
NT
"I
Figure 1.2
Qualitative picture of expiratory flow.
A
4-vortex pattern can be seen in the parent tube.
The shaded
area is a region of higher axial velocity that extends
vertically.
136
L
L
0,2
A
p
venlral
A
ventral
',
p
1.4
I
M
i i
i
OL
i iI
M
Figure 1.3
Steady inspiratory
flow velocity data
measured in a
daughter
branch, having an additional
bifurcation upstream.
Re-460. Left:
secondary velocity
field. Right: axial velocity contours.
All velocities
referenced to the local mean velocity.
The outside of the
bend, or inside of the bifurcation, is at the bottom. (from
Isabey and Chang, 1982).
L
L
02.
0.52
0.7-
16
A
P
dorsai
R
/1.8,
dorsal
A
R
Figure 1.4
Steady expiratory flow in the trachea.
Re=1060. The figure is oriented with the daughter tubes in
the plane of the L-R axis. Left: secondary velocity field.
Right: axial velocity contours.
All velocities referenced
to the local mean velocity. (Isabey and Chang, 1982).
137
a6700
0.5a
R = 6a
Idealized
Figure 2.1
bifurcation, after .Pedley.
dimensions
of
a
lung
138
Experimental parameter range
102
5
101
go
15
100
2
00
0
a
20
10-1
1C -1
101
100
102
The range of oc and Ax/d
Solid diamonds:
Figure 2.2
encountered in the Weibel lung model under HFV conditions,
frequency - 5
hz
and
tidal
volume
=
30
ml.
The zeroth
generation begins at the right, the highest generations are
at the lower left.
Open squares: The range of parameter
values covered in this study.
139
APPARATUS
camera
tank for index-
matching
Yoke
0
Piston
U
Reservoirs
side-view mirror
Bifurcation
-
Scotch
fluid
Figure 2.3
Diagram of experimental apparatus.
The
mirror provides an additional side view image to the camera
above.
140
0
-
calibration point
Figure 2.4
Coordinate system used in the experiments.
141
1.2
1.0
_
.6
-
.8
.2
a
0
.2.
7:
-. 6
-. 8
-1.0
-1.2
F
1
1
1.0
-. 8
-. 6
-. 4
-. 2
i 7777~
i
0
.2
?RAOIAL 0IST
.4
.6
.8
1.0
Experimental data for oscillatory flow in
Figure 2.5
a straight tube compared to the theoretical solution of
Uchida. Dots are experimental data at 6 past peak flow,
squares are data at 51 past peak. Uchida solutions: 60--A-Bcurve; 510--D- curve.
142
z
I
R
a
Figure 3.1
Toroidal coordinate
curved tube geometries.
system
for
use with
VSW
W
J-
R
Figure 3.2
curved tube.
Illustration of
pressure
balance
in a
143
20
10
VI
o
I
C6)
IV
10
50
100
200
1000
Dn
viscous
-
convective-centrifugal
Figure 3.3 Flow regimes showing the dominance of various forces:
I-viscous, II-viscous/unsteady, III-unsteady, IV-viscous/convective,
V-convective, VI-viscous/unsteady/convective, VII-unsteady/convective.
The convective forces include centrifugal forces. (After Yamane, et al.)
144
a2.8
5
nO
)
UWJL
IN
OUT IN
CUT IN
OUT
-ax= 227
S70
13 5
a =30.2
ctx7.9
-5.4
325
385
3.8
440
900
7.0
-5.5
690
1.3
-2.5
6
(a
305
Secondary vorticeo (De = 150)
2
2
2
a =2.B
a =3 0.2
az 7.91
45
1
1
0
e
.90
138
135!
45
wt=0'
wt=0
0
0
0
Wt0
180
180"
180
3158
225'
225
315
-1
3150
-1
-1
270
2700
225
270'
-2
I
IN
0
r
1
-2
OUT
-2
1
IN
0
r
1r
OUT
0
IN
Axial velocity profile s (De
r
OUT
= 150)
Figure 3.4
Plots of
(above) and axial velocity
secondary flow streamfunction
profiles in the plane of the
curve (below)
a and Dn values.
1985).
for 3 sets
of
(Yamane et al,
145
so
0j
20
10
00
500
.00
I
*
50
10
20
1
S
20 I0
Streng~th of
Figure 3.5
streamfunction, W
1985).
Maximum
max,
so
10 ti 00 I 50
100
Da
200
Soo
secondary vortices
value
versus
s
200
100
of
a
and
the
Dn.
secondary
flow
(Yamane et al,
146
U
R
Vsec
Figure 3.6
Diagram showing secondary
from vorticity at the inlet to a bend.
flow
arising
147
-0
0.7
-
(w)&Q
0.8-
0.6
0.5
0.4-
0.3-.--
'
0.2
0.1
0y
Figure 3.7
1
2
3
5
4
6
7
8
Variation of axial shear at the inner bend
with axial distance, rn-(1-r)(0.5
1/2
Dn)/.
Soh and Berger's
result is the solid line;
Stewartson et al's boundary layer
calculation is the dashed
line.
normalized by the bulk
a/R-1/7.
axial
(Soh and Berger,
The
velocity.
1984).
axial velocity w is
Dn=680.3, Re=900,
148
0.40
C5 C;
'6
C;
= 41.5*
Figure 3.8
(a)
Figure 3.8
(b)
Figure 3.8
(a)secondary velocity field,
41.50 past
entry.
(b)axial velocity contours, 2670 past entry. Flat
inlet profile, Dn=680.3, Re=900,
a/R=1/7. Velocities are
normalized to the bulk axial velocity. The outside of the
bend is to the right.
(Soh and Berger, 1984).
149
end expiration
end inspiration
Figure 4.1 (c)
expiration
in spiration
6
K)
Figure 4.1 (d)
150
Figure 4.1 (a)
expiration
in1_
spiration
(t
Figure 4.1 (b)
Figure 4.1
Diagrams of the
4 flow regimes. (a)
Reversible flow.
(b) Quasi-steady flow. Secondary flow
patterns in the cross-section are shown. (c) Transitional
flow exhibiting turnaround zone. (d) Confined vortex flow.
151
Figure 4.2 (a)
Figure 4.2
Particle streak
photographs of quasisteady flow. a-2.3, Ax/d-19.5 (Rectangular
lines at the flow
divider are due to a seam in the
construction. Some
calibration marks can be seen.) (a) End expiration.
(b) 450
past end expiration. (o) Peak inspiratory
flow. Cd) 450 past
peak inspiration. (e) End inspiration.
(f) 450 past end
inspiration. (g) Peak expiratory flow.
(h) 450 past peak
expiration.
152
Figure 4.2 (b)
153
Figure 4.2 (c)
154
Figure 4.2 (d)
155
Figure 4.2 (e)
156
Figure 4.2 (f)
157
Figure 4.2 (g)
158
Figure 4.2 (h)
159
Figure 4.3 (a)
Figure 4.3
Transitional flow. Photographs taken
from
videotape data. Half the bifurcation
can be seen in a top
view above, a side view image is
below. a=6, Ax/d=9.6 (a)
End expiration.
(b) 450 past end expiration.
(c) Peak
inspiratory flow. (d) 450 past
peak inspiration. (e) End
inspiration.
(f) 450
past
end inspiration.
(g)
Peak
expiratory flow. (h) 450 past peak
expiration.
160
Figure 4.3 (b)
Figure 4.3 (c)
161
Figure 4. 3 (d)
Figure 4.3 (e)
162
Figure 4.3 (f)
Figure 4.3 (g)
163
Figure 4. 3 (h)
164
M
NZ
Figure 4.4 (a)
Figure 4.4
Confined vortex flow. Streak photographs
of top view. a-21.3, Ax/d-9.7 (a) End expiration. (b) 450
past end expiration. (c) Peak inspiratory flow. (d) 450 past
peak inspiration. (e) End inspiration. (f) 450 past end
inspiration. (g) Peak expiratory flow. (h) 450 past peak
expiration.
54
to
PI
166
Figure 4.4 (c)
167
Figure 4.4 (d)
0D
(D
Did
I-..
0%
169
Figure 4.4 (f)
-~
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-~.
~.-
$4
qoo
1k4
r--
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sinalnum~~~~~~~~~se,==nhwa.~~~~~~
------- - ---- -"
IN
A
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0n
-- -- -----
---
172
102
Confined
101
rtex
a
Quasi-
0
steady
an
a
E
Transitional
100
Quasi-reversible
I
...
100
.I2,1
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LL
101
102
Fig. 4.5 Flow regimes as a function of c and stroke length A x / d
10 2
%v
I
IV
V
\
101
Quasisteady
.
10-1 1
10- 1
Confined
vi
0
10
E
a
a
Transitional
II
100
Iu
10-1
0 0 ]
a E
Quasi-reversib e
100
101
III
102
Fig. 4.6 Yamane et al flow regimes superimposed on the flow
regimes found in this study.
173
20
10
cc
IIVII
5
a
2
I
V
IV
1
10
50
100
200
1000
Dn
Fig. 4.7 Results of this study plotted on Yamane's flow regime chart.
M quasi-reversible, (a
CV confined vortex.
quasi-steady, V
transitional,
174
Fge.(
Figure 5.1 (a)
Figure 5. 1 (b)
Figure 5.1
Top view of streak data.
Regime 1, quasisteady and quasi-reversible. a=2.3, Ax/d=0.75
. (a)peak
inspiration. (b) peak expiration.
175
.5
I
.40
.35
-M
.30
.20
. .25
*.15
.10
1
-78
2
3
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S
.
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:
i 52
.
-
0. 0
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.05
()
X
xI
9
Figure 5.2 (a)
.50w
.- as
.40
.3ss
0
0
0.
.30
0
0
.20
'S
.15
.25
. 10
.05
15
Figure 5.2 (b)
Figure 5.2
Secondary flow magnitude vs distance. The
dashed line represents the expected noise level.
Regime 1.
inspiratory flow.
a=2.3,
Ax/d=0.75
(a) peak
(b) peak
expiratory flow.
176
peak expiration
peak inspiration
CM...MM
0..
.00a
To
,.,M
CNTUM INTERVUt W 0.3DO
Figure 5.3
P13.31-
Axial
1.9683
CGN".,U
velocity
'a,.
.
.OO0
0 TO
2.1000
contours
C.TOW
INTERy
in
OF
0.30=
PT13.33= 2.1255
the
parent
tube. Velocities are normalized by peak bulk axial velocity.
Regime 1, a-2.3, Ax/d=0.75.
177
Figure 5.4 (a)
Figure 5.4 (b)
Figure 5.4
Top
view
of
unsteady and quasi-reversible.
inspiration.
streak
a=21.3,
(b) peak expiration.
data.
6x/d=0.75
Regime
.
2,
(a)peak
178
. 45
-40
. 35
.30
0
.25
.20
.15
.10
-
-
-
o
~
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34
2
67a
X
Figure 5.5 (a)
.50
.40-
35.30
0-
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Sn
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0
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-
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1
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x
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Figure 5.5 (b)
Figure 5.5
Secondary
Regime 2, a-21.3, Az/d=0.75
peak expiratory flow.
flow
.
magnitude
vs
distance.
(a) peak inspiratory flow. (b)
179
peak
W&TA
MWU4
O.OMM
inspiration
Tg
1.3000
CgeTOm
INTEMRVO. I
0.10M
peak expiration
PT(B.3I:
0.
1.2s55
.
W
W
s
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9
0.20000
r
t-32j2
t3.31,
-
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Figure 5.6 (a)
I(tTM
Figure 5.6 (b)
Figure 5.6
Regime 2, a=21. 3, Ax/d=0.75
.(a) Axial
velocity contours in the parent tube.
(b)Secondary flow
field. The distance between nodes corresponds to 0.25 times
the peak bulk axial velocity.
180
Figure 5.7 (a)
Figure 5.7 (b)
Figure 5.7
Top view of streak data.
steady,
curvature
effects.
a=2.3,
Regime 3, quasiZ6x/d=19.5
(a)accelerating flow, 450 before peak inspiration. (b) peak
inspiration. (c) 450 past peak inspiration. (d) accelerating
flow, 450 before peak expiration.
45 past peak expiration.
(e) peak expiration. (f)
181
y
Figure 5.7 (co)
y
Figure 5.7 (d)
182
I
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Figure 5. 7 (e)
y
)
Figure 5.7 (f
i
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183
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Figure 5.8 (a)
00
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x
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7
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9
Figure 5.8 (b)
Figure 5.8
Secondary flow
magnitude vs distance.
Regime 3, a-2.3, Ax/d-19.5
.
(a)accelerating flow, 450
before peak inspiration. (b) peak inspiration. (c) 450 pa
peak inspiration.
(d) accelerating flow,
450 before peak
exyiration.
(e) peak
expiration.
(f)
450
past peak
expiration.
55.-
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Figure 5.9 (a)
Figure 5.9
Axial
velocity
contours
(above) and
secondary flow field (below). Parent tube, Regime 3, a=2.3,
Ax/d-19.5
(a)
accelerating
flow,
450
before
peak
inspiration.
(b) peak inspiration.
(c) 450 past peak
inspiration.
187
8000
.400
COMMAf~o
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Figure 5. 9 (b)
188
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-
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189
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IN
OUT
KUIMN
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Figure 5.10 (a)
(above) and
contours
Figure 5.10 Axial
velocity
Daughter tube, Regime 3,
secondary flow field (below).
(a) accelerating flow, 450 before peak
a-2.3, Ax/d-19.5.
(c) 450 past peak
(b) peak inspiration.
inspiration.
inspiration.
190
CONTRA
FRM 0.000E00
Is
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ITERV
o
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OUT
Figure
5. 10 (b)
IN
191
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low CNax INTERVALt
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Figure 5.10()
192
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Figure 5.11 (a)
Figure 5.11 Axial
velocity
contours
(above) and
secondary flow field (below). Parent tube, Regime 3, a=2.3,
Ax/d-19.5 (a) accelerating flow, 450 before peak expiration.
(b) peak expiration. (c) 450 past peak expiration.
193
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FR9 TO
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C-TU
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Figure 5.11 (b)
--
194
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Figure 5.12 (a)
Axial
Figure 5.12
secondary flow
a-2.3, dx/d-19.5
expiration.
expiration.
(b)
field
.
(below).
(a)
peak
(above)
contours
velocity
Daughter
and
tube, Regime 3,
accelerating flow, 450 before peak
expiration.
(c)
450
past
peak
196
C--
-m c.am
- ..=a
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Figure 5.12 (b)
N
197
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Figure 5.12 (c)
198
I
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r
I
x II
Figure 5.13 (a)
A
Figure 5.13 (b)
Figure 5.13 Top
unsteady,
curvature
view
of
effects.
streak
data.
a-21.3,
Regime
4,
,6x/d-9. 7.
peak inspiration. (b) peak
inspiration. (c) 450 past peak inspiration. (d) accelerating
flow, 450 before peak expiration. (e) peak expiration. (f)
450 past peak expiration.
(a)accelerating flow, 450 before
199
y
Figure 5.13 (c)
Figure 5.13 (d)
200
yI
Figure 5.13 (e)
y
I
Figure 5.13()
201
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2
3
5
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--
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6789
x
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Figure 5.14 (b)
Figure 5.14
Secondary
flow
magnitude
vs
distance.
. (a)accelerating flow, 450
Regime 4,
a-21.3,
6x/d-9.7
before peak inspiration. (b)
peak inspiration. (c) 450 past
peak inspiration.
expiration.
expiration.
(e)
(d)
accelerating
flow,
peak
expiration.
(f)
45 0 before peak
450
past
peak
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Figure 5.15 (a)
Figure 5.15 Axial
velocity
(above) and
contours
secondary flow field (below).
Parent tube, Regime 4,
a-21.3, Ax/d-9.7 (a) accelerating flow, 450 before peak
inspiration.
(b) peak inspiration.
(c) 450 past peak
inspiration.
205
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Figure
5.15 (b)
206
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Figure 5.15 (c)
207
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OUT
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Figure 5.16 (a)
Figure 5.16 Axial
velocity
contours
(above) and
secondary flow field (below).
Daughter tube, Regime 4,
a-21.3, Ax/d-9.7 .
(a) accelerating flow, 450 before peak
inspiration.
(b) peak inspiration.
(c) 450 past peak
inspiration.
208
CBdTu" "uM
o.m-W TiI.~
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Figure 5.16 (b)
209
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Figure 5.16 (c)
210
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A
CNT&A FM
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Figure 5.17 (a)
(above) and
contours
Figure 5.17 Axial
velocity
Parent tube, Regime 4,
secondary flow field (below).
a-21.3, Ax/d-9.7 (a) accelerating flow, 450 before peak
(b) peak
expiration.
expiration.
450 past peak
(c)
expiration.
211
Co4IB
/
0.I0
CRITI
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W
0. ID0
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Figure 5.17 (b)
212
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-
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Figure 5.17 (c)
N-
213
OUT
T
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. .6O.
I
Figure 5.18
secondary flow
a-21.3,
6x/d-9.7
expiration.
expiration.
(b)
field
.
(below).
(a)
peak
(a)
(above)
contours
velocity
Axial
Figure 5.18
IUN
E-
O
Daughter
tube,
and
Regime 4,
accelerating flow, 45 0 before peak
expiration.
(c)
450
past
peak
214
CONTSA FRBM
O.
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to
orn00
Cfam
T EMyq
OUT
UNOW
0.
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I N
Fi
ur
5.18
215
Coo"m PIM
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0.00E0
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0.7000
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1000
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S
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I
I
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=
I
N
N
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Figure 5.18 (c)
216
A
I
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ig
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a 41 ~
JLU
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Figure 5.19 (a)
I
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~-~~1
Figure 5.19 (b)
Regime 1, quasi-steady and
Figure 5.19 Stereo pairs.
. (a) peak inspiration.
quasi-reversible. a=2.3, Ax/d=0.75
(b) peak expiration.
217
I
___________________________________________________________________
I
Figure 5.20 (a)
I.
I
___________________________________________________
/1
.1
t
i\l/
I
/
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I
I
Figure 5.20 (b)
unsteady and
Regime 2,
Figure 5.20 Stereo pairs.
quasi-reversible. a-21.3, Ax/d=0.75
. (a)peak inspiration.
(b) peak expiration.
218
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Figure 5.21 (a)
I
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41
4-
-.1
1
7
I
Figure 5.21 (b)
Figure 5.21 Stereo pa irs.
Regime 3, quasi-steady,
curvature effects. a-2.3, A x/d-19.5 . (a)accelerating flow,
450 before peak inspiration. (b) peak inspiration. (c) 450
past peak inspiration.
(d) accelerating flow, 450 before
peak expiration.
(e) peak expiration.
(f)
450 past peak
expiration.
219
I
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4-
I
Figure 5.21 (c)
t
t%\
Figure 5.21
(
4-
*'L\
~k
'
~
Y
d)
i~'
220
I
(
FI.
--
I
Figure 5.21 (e)
I-
'N'.
~z~i
ell
v'0-
i
Figur'e 5.21 (f)
5.21 (f
221
I
I
/
11
~.
r,
\
4-
~1
___________
I
~
Ad4~i
ThAN ~
v-li
ii~
/,AI
Figure 5.22 (a)
~
I'
Figure 5.22
Figure 5.22 (b)
unsteady,
Regime
4,
pairs.
(a)accelerating flow,
curvature effects. a=21.3, Ax/d=9.7
450 before peak inspiration. (b) peak inspiration. (c) 450
past peak inspiration.
(d) accelerating
flow, 450 before
peak expiration.
(e) peak expiration.
(f) 450 past peak
expiration.
Stereo
222
I
I
I
i
ll
- IV
I
Figure 5.22 (c)
Al
'
'-/T
I
Figure 5.22 (d)
'
/t"
223
I
'4
IV.
Figure 5.22 (e)
____________
I-
I
Figure 5. 22 (f)
224
Quiet breathing
102
C-V
101
Q-R
1 00
T
10-1
--S
10- 2 - ... a
10-3 10-
2
. .
..
...... -I
10-1
100
10
102
- - - 1
104
103
Dn
Flow regimes encountered in the Weibel
Figure 6.1
lung in normal quiet breathing.
(15 breaths/minute, 10
liters/minute). Each symbol represents a Weibel generation,
from the zeroth generation at the right and proceeding into
the lung to the left.
225
Increasing frequency, Vt=25
102
C-V
101
Q-R
T
Dn,f15,Vt25
Dn, flO,Vt25
Dn,f5,Vt25
-e-
0C
+
-+-
100.
-0- Dn, fl,Vt25
O-S
10-1 1
10 0
101
104
102
Dn
Figure 6.2
of
Effect
varying
frequency
on
flow
regimes encountered in the Weibel lung.
Increasing Vt, f=15
102
C-V
101
Q-R
T-
-0- Dn,f15,VtlOO
Dn, f15Vt50
Dn,f15,Vt25
-0- Dn,f15,VtlO
+-
-+-
100
Q-S
10-1 1
10 0
.
101
.
.f...
102
--.
103
104
Dn
Figure 6.3
Effect of varying tidal
regimes encountered in the Weibel lung.
volume
on flow
226
HFV range
102
-
C-V
Q-R
101
a
-
.8S
100
10-
10C 0
101
102
104
103
Dn
Figure 6.4
Flow
lung for a wide range
tidal volume-1O-lOOml.
regimes
encountered
of HFV conditions.
in
the Weibel
Frequency=5-15 Hz,
227
0.1
STRAIGHT TUBE,
THEORY
Deff
7
2
f
Vt
2
0.0{
A2
a1
10
5
0.0011r
0.1
0.5
1
5
10
15
50
Trad/T
A comparison of our results for both Series A and Series B to
the theoretical prediction for a straight tube with sinusoidal flow (solid
line). 82 percent of the data fall within the shaded zone.
Figure 7.1
Experimentally
determined
normalized
transport coefficient for a branching network, compared to
Watson's theory for a straight tube,
as a function of a.
(Kamm et al, 1984b).
228
10-1
)
-
A -STRAIGHT TUBE PREDICTION (S
H4)
6
B -STRAIGHT TUBE PREDICTION (air- CH4)
C -STRAIGHT TUBE PREDICTION (He-C l1
14)
0 - it r 0-C4
A
0
o
0
-
He -CH4
-SI'
Regression
10
0
0N &Reresio
60
4
line for He -C14
Regression line for air4
line for SI'6 -0CH 4
09
10-3
10 I
, ,i
102
BETA
2
Graph of dimensionless transport coefficient
showing branching tube results a constant
factor higher than the straight tube prediction
for #>4.
Figure 7.2
Normalized transport coefficient for a
branching network as a function of # 2
compared to Watson's
theory.
(Slosberg, 1983).
229
De f-f
f
2 (Vt
to-i
VTMI
/a3) .82
3i
I
o-2
1 -1
Figure 7.3
branching network,
10
Normalized
to
transport
as a function of
#2.
10z
coefficient for
(Paloski, 1986).
a
230
10~1
I)
-
S
.
.:
-
.* l
.
10-2
3
.
.
.
10*
*.
:3U;
-
10
I
10
ic
I
I
1
1
1
1
I
102
I
1
1
W
?:
101
Figure 7.4
Normalized transport coefficient
anatomical branching network,
function
as
a
(Keramidas, 1986).
for an
2
231
Streaming-mixing compared to Watson
101
100
I
CH
-- Def f*,
- Def f*,
1 0-1
St reami x
Watson
4
1
10-3 1
100
Figure 7.5
101
Comparison
coefficient predicted by
mixing model with a
102
of
Watson
the
to
normalized
transport
that of the streaming-
blunt velocity profile with approximate
Stokes boundary layers.