CHARACTERISTICS OF
FLOW IN CHANNEL
ABRUPT EXPANSIONS
VINCENT P. LOKROU
1979
"
COLORADO
STATE
UNIVERSITY
,
.....
CED79-SÛVPL7
DISSERTATION
CHARACTERI5TICS OF FLOW IN CHANNEL ABRUPT EXPANSIONS
Submitted by
Vincent P. Lokrou
Civil Engineering Oepartment
In partial fulfil1ment of the requirements
for the Oegree of Doctor of Philosophy
Colorado State University
Fort Collins, Colorado
FaU, 1979
'.' ' --r -'
CED79-S0VPL7
1
COLORADO STATE UNIVERSITY
f
•1,
Fa 11
WE HEREBY RECOMlIEND 1lIAT THE TIIESIS PREPARED UNDER OUR SUPER\\'ISIQ)i
DY
Vincent P. Lokrou
1
1
CHARACTERISTICS OF FLOW IN CHANNEL ABRUPT EXP~'SIO~S
E~TITLED __--"'=====~~=:::._======_'_==::..:..:=
__
1
1
i
BE ACCEPTED AS FULFILLING IN PART REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PlilLOSOPHY
ii
ABSTRACT OF DISSERTATION
CHARACTERISTICS OF FLOW IN CWl~NEL ABRUPT EXPANSIONS
Theoretical and experimental investigations of the characteristics
of the subcritical turbulent mixing fLaw in channel abrupt expansions
"ere conducted.
In the theoretical derivations, a specialized farm of the ~avier-
Stokes equations believed more suitable for expansion mixing fLaw was
defined.
A similarity approach was used ta approximate empirico-
analytical and numerical solutions of the governing equations for mean
velocity profiles in mast fLaw regions, except near the reattachment
zone.
Both the analytical and numerical solutions were in good agree-
ment \\~ith experimental profiles in respective cases of two-dimensional
single-step or uncoupled (or weakly-interfering) double-step flow. as
weIl as axisymmetric expansion flow.
The primary aim of the laboratory investigation was ta provide
pertinent information about the mixing process in the expanded flow
region.
The experiments encompassed measurements of mean and turbulent
velocity profiles using three symmetrical (AR = 1.342, 2.038 and 4.235
respectively) and two nonsymmetrical (AR ~ 1.618 and 2.752 respectively),
step configurations.
The range of Reynolds nw~bers (based on inlet
5
width and average velocity) investigated "as from 4.36 x 10
to
5
7.95 x 10.
For symmetrical step configurations, the main observations
revealed no fundamental difference with the results of previous experi-
ments conducted in much smaller scale models.' For the range of
unsymmetrical step configurations considered (minimum area ratio = 1.618),
the flow was always deflected toward the smallest step region.
The
short staIL (corresponding to the smallest step) and the long staIL
iii
(corresponding to the largest step) were stable in the sense that for
a given steady state mean flow through the test section, the flow
pattern was fixed, and no external interference could modify it.
Vincent P. Lokrou
Civil Engineering Department '
Colorado State University
Fort Collins, Colorado
80523
Pail, 1979
iv
AC K~OWL EDGMENTS
The writer wi s hes ta express his profound indebtedness ta his major
professor, Dr. H. Iv. Shen whos e inspiration, guidance, encouragement
and invaluable criticism made every stage of this investigation a
rewarding experience.
The author also wishes ta express his appreciation ta the members
of hi s committee:
Dr. L. V. Baldwin, Dr. Y. H. Chen, Dr. R. B. Kelman
and Professor V. A. Sandborn for their helpful comments and review of
the dissertation.
Special appreciation is also extended ta Dr. P. C. Duchateau for
his mast helpful comments and suggestions on the derivations of mast of
the mathematical models.
Heartfelt thanks are due Dr. and Mrs. Morel-Seytoux and children
(Amy, Claire, Sylvie and Marie) for their invaluable efforts ta make
bath the authar and his wife's long sojourn in Fart Collins a very
pleasa~t and memorable stay.
The author recalls the dedicated efforts of the members of his
family and particularly of his mother for their nûmeraus sacrifices
over the years, and of his wife, Abraha-Marie for her unfailing patience
and understanding.
Financial support for this study from the ~ational Science
Foundation (contract:
Ë~G 76-05773) and the USAID (contract AFR-538/
Task arder no. 4) are gratefully acknowl edged .
Pi.na.Ll y •. the wri t.er wi shes to acknowledge wi t h deepest gratitude
the sponsorship of the African American Institute (AFGRAG Pe l tovsh i.p) ,
and the financial support of Colorado State University (Colorado
FeIlowship), during the entire graduate program.
v
This work is dedicated to the author's father, Georges Tapé Lokrou,
who provided unlimited inspiration, and endured incommensurable sacri-
fices to assure his children's education, but never lived to see the
attainment of their academic goals.
vi
TABLE OF CONTENTS
Chapter
1
Introduction . . . .
1
II
Review of Literature
4
III
Theoretical Analysis
II
3.1
Governing Equations
II
3.2
Shear Stress Estimation
21
3.2.1
Eddy Viscosity Approximation
21
3.2.2
Explicit Functional Approximation
of the Reynolds Stress .
26
IV
Development of Comput at i.ona I Net hods
31
4.1
Analytical Considerations . . .
31
4.1.1
Self-preserving Development
38
4.1.2
Approximate Solutions
51
4.2
Numerical Mode Ls
. . • • • • •
58
4.2.1
Variable Eddy Viscosity
59
4.2.2
Numerical Solutions
64
4.2.3
Numerical Evaluation of the
Cross-stream Velocity
71
V
Experimental Facility and Procedure
76
5.1
Description of Apparatus
76
5.1.1
Water Supply Line
76
5.1.2
Stilling Tank
76
5.1.3
Adjustable Converging Section
7B
5.1.4
Backward-facing Steps
7B
5.1.5
Channel Section
.
7B
5.2
Data Collection Techniques
. . . .
79
S.2.1
Flow Visualization Technique
BI
5.2.2
Hot-film Anemometry
. . . .
B3
5.2.2.1
Instrumentation . .
83
5.2.2.2
Calibration of the Hot-film
86
VI
Experimental Results and Discussion
93
6.1
Inlet Conditions
. . . . .
93
6.2
Mean-velocity Measurements
93
6.2.1
Visual Measurements
93
6.2.2
Quantitative Measurements
93
6.3
Turbulence Neasurement s . . . . .
119
6.4
Overall Characteristics of the Eddies
121
Vll
Canelus ion .
. . . .
127
7.1
Summary and Discussion of the Theoretical
Analysis
.
127
7.2
Summary and Discussion of the Experimental
Investigation .
. . . . . . .
129
7.3
Recommendations for Further Investigations
130
vii
Chapter
Page
REFERENCES
132
APPENDIX A
133
Al
Nondimensionalization of the
Equation of Mo t i on
133
A2
Nondimensionalization of the Energy Equation
140
APPENDIX B . • • • • • • • • • • • •
143
ln
The Shooting Ne thod . . . . . .
143
B2
Fourth Order Runge-Kutta Method
143
B3
Me t hod of Solution
145
APPENDIX C . . . . . . .
148
Cl
CALLING Pro gram for Runge-Kutta
Integration Routine
148
C2
Function lRUNGE . . . . . . . .
150
C3
Pro gram for Computing the Potential
Flow Parame ter
A and the Shape Parameter
H
153
viii
-
LIST OF FIGURES
Figure
3.1
Nomenclature. • .
3.2
Schematic of fLaw past abrupt expansion in channel.
15
3.3
Eddy viscosity before reattachment (computed from
data of Hsu) . . . . .
24
3.4
Eddy viscosity downstream of reattachment (adapted
from Kim et al.).
. • .
.
25
3.5
Turbulent shear stress (adapted from Hsu)
27
3.6
TUrbulent shear stress (adapted from Kim et al.).
28
3.7
Experimental and computed shear stress distributions
(from wygnans k i and Fiedler).
.
30
4.1
Flow width as a function of longitudinal distance
(data of Hsu)
32
4.2
Flow width as a function of longitudinal distance
(data of Chaturvedi).
. . .
33
4.3
Flow width as a functioo of longitudinal distance
(present experiment).
. .
.
34
4.4'
Nondimensional mean velocity distribution (data
of Hsu)
. . .
. .
.
35
4.5
~ondimensional mean velocity distribution (data
of Chaturved L) . . .
36
4.6
Nondimensional mean velocity distribution (present
exper Lment )
.
37
4.7
Comparison of the mean velocity profile of fully
stalled flow and the equivalent laminar profile
(data of Hsu)
. . . . . . . . . .
44
4.8
Comparison of the mean velocity profi le of fully
stalled flow and the equivalent laminar profile
(data of Chaturvedi)
.
45
4.9
Comparison of the mean velocity profile of fully
stalled flow and the equivalent laminar profile
(present experiment) . . .
46
4.10
Longitudinal variation of the potential core
ve Ioc Lt.y (data of Hsu) . .
48
ix
LIST OF FIGURES (continued)
Figure
4.11
Longitudinal variation of the potential core
velocity (data of Chaturvedi)
.
49
4. 12
Longitudinal variation of the potential core velocity
(present experiment) .
50
4.13
Comparison of experimental mean velocity profile and
empirical Gaussian and power Law distributions (data
of Hsu). . . . . . . . . . . . . . . . . . . • .
55
4. 14
Comparison of experimental mean velocity profile and
empirical Gaussian and power law distributions (data
of Chaturvedi) . . . . . . . . . . . . . . . . .
56
4. 15
Comparison of experimental mean velocity profile and
~mpirical Gaussian and power law distributions
(present experiment) . . . .
.
.
57
4.16
Characteristic eddy function in plane jet (from:
B. G. Van der Hegge Zijnen).
61
4.17
Characteristic eddy function in backward facing
step flow (from: Kim et al.)
62
4.18
Comparison of computed mean velocity distribution
and experimental data.
Constant eddy-viscosity
approach--R
= 22 (data of Hsu)
.
67
5
4.19
Comparison of computed mean velocity distribution
and experimental data.
Constant eddy-viscosity
approach-o-R
= 36 (data of Chaturvedi) • . . . . .
68
5
4.20
Comparison of computed mean velocity distribution
and experimental data.
Constant-~ddy viscosity
approach--R
= 22 (present exper-iment )
.
69
5
4.21
Comparison of computed mean velocity distribution
and experimental data.
~lixing length approach--
C = 0.021 (data of Hsu) . . . . . . . . . . . • . .
72
4.22
Cornparison of computed mean velocity distribution
and experimental data.
Nf x ing length approach--
C =:= 0.0197 (data of Chaturved i )
.
73
4.23
Comparison of computed mean velocity distribution
and experimental data.
jtt xing length approach--
C = 0.021 (present experiment)
74
4.24
Cross-stream velocity profiles
76
x
LIST OF FIGURES (continued)
Figure
5.1
Overall view of experimental facility
ï7
5.2
Experimental flume
80
5.3
Dye visualization .
82
5.4a
Electronic equipment
84
5.4b
Conical hot-film
85
5.5
Calibration curve of hot-film probe
87
5.6
Schematic of the velocity calibration system
90
5.7
Velocity profiles across the .99 centimeter
calibration orifice. . . .
. . . . .
92
6.1
Flow visualization-velocity front near the
step interface (AR = 1.618)
95
6.2
Flow visualization-velocity fronts at successive
locations in the test section (AR = 1.618)
96
6.3
Flow visualization-eddy coupling effects
(AR = 4.235)
.
. .
97
6.4
Asymmetrical velocity front in a symmetrical step
flow
. . . .
98
6.5
Mean velocity profiles (AR = 1.342, DT = 71.45 cm).
100
6.6
Mean velocity profiles (AR = 1.342, DT = 46.69 cm).
101
6.7
Mean velocity profiles (AR = 1.618, DT = 71.45 cm).
102
6.8
Mean velocity profiles (AR = 1.618, DT = 46.69 cm).
103
6.9
Mean velocity profiles (AR = 2.038, DT = 46.69 cm).
104
6.10
Mean velocity profiles (AR = 2.752, DT = 71.45 cm).
105
6.1I
Vertical velocity profile near step interface
(AR = 1.342, y/H = 1)
.
106
6.12
Vertical velocity profile near step interface
(AR = 1.342, y/H = 1.5)
107
6.13
Vertical velocity profile near step interface
(AR = 1.342, y/H = 2)
.
108
xi
LIST OF FIGURES (continued)
Figure
6.l4a
Vertical velocity profile near step interface
(AR = 1.342, y/H = 2.5) .
109
6.14b
Vertical velocity profile near step interface
(AR = 1. 342, y/H = centerline). . .
110
6.15
Potential core shadot...gr aph in high step
configuration (AR = 2.038)
111
6.l6a
Nean ve l oc Lty profiles, AR = 2.0, DT = 50.80 cm
(experiment in 60.96 cm flume)
113
6.16b
Mean velocity profiles. AR = 2.0 J DT = 50.80 cm
(experiment in 60.96 cm flume)
· · · . · · .
114
6. 16c
Mean veloci ty profiles, AR = 2 .0, DT = 50.80 cm
(experiment in 60.96 cm flume)
· · ·
· ·
115
6.17a
Mean velocity profiles, AR = 2.0, DT = 33.02 cm
(experiment in 60.96 cm flume)
· · · . · · .
116
6.17b
Mean velocity profiles, AR = 2.0, DT = 33.02 cm
(experiment in 60.96 cm flume)
117
6.17c
Mean velocity profiles, AR = 2.0, Dr = 33.02 cm
(exper iment in 60.96 cm flume)
118
6.18-
Hot-film sensitivity curve
120
6.19
Evaluation of velocity fluctuations from a hot-film
anemometer output . . . . . . . . .
120
6.20
Cross-sectional turbulence intensity variations
examp le profile
122
6.21
Turbulent velocity profiles along the channel
6.22
Shadowgraph of vortices in the developing flow
region
125
B. 1
Flow diagram of function Runge
144
B.2
Ccmput at ional sequences in the shooting method of
integration .
147
xii
LIST Of SYMBOLS
Symbol
Definition
Dimension
A
coefficient in nond i.mens t.one l momcn tum equation
area ratio
A'
constant in King's law
a
constant exponent in expression of potential
.'
core veloci ty
constant characteristic of jet size in shear
stress explicit formula
B
coefficient in nondimensional momentum equation
channel width at the entrance nonexpanded section
L
B'
·constant in King's law
c
coefficient in mixing length function
I l = C(x-xo)
rate of spread of mixing zone
coefficient in expression Rf potential core
velocity
lll(x) ~ C2(x-xc)
coefficient in eddy funetion
~T(x,n) = C
) a+l s Cn)
3(x-xo
c'
coefficient in nondimensional momentum equation
C'
constant in shear stress explicit formula
l
cn
integration constant
l
cn
integration constant
2
cn
integration constant
3
C"
integration constant
4
il
jet diameter
L
il
di~meter of entrance section in axisymmetric
a
abrupt expansion
L
0,".
lIT
Jl
xiii
Symbol
Definition
Dimension
total flow depth
width of potential core
L
E
mean anemometer bridge voltage
volts
E'
coefficient in nondimensional momentum equation
e'
root-mean-square (rms) of the anemomet er
bridge voltage
volts
F
integral velocity-deficit function
F(n) = J f(n)dn
o
F.
instantaneous body force in the i t h direction
l
r
time-averaged body force in the
t h
i
direction
1
F'.F" ...
f i.r s t , second, etc. derivatives of
F
f
nondimensional velocity-deficit function
t h
turbulent body force in the i
direction
fl.f" ...
first, second. etc. derivatives of
f
g
nondimensional shear stress function
substitute function used in Runge-Kutta
integration procedure
substitute function used in Runge-Kutta
integration procedure
substitute function used in Runge-Kutta
integration procedure
nondimensional turbulent kinetic energy function
nondimensional normal stress function along
ox
nondimensional normal stress function along
oy
g'
first derivative of nondimensional shear-stress
function
g'
first derivative of nondimensional normal stress
r
along
ox
xiv
Symbol
Definition
Dimension
lower limit of initial value
upper limit of initial value
initial condition in "shooting methodlt of
integration
H
step height
L
h
step height
L
step height in asymmetric configuration
L
step height in asymmetric configuration
L
K
overheat ratio
K =
L
length scale for variation of mean quantities in
longitudinal direction
L
length scale for variation of mean quantities in
transverse direction
l
t. (x)
L
o
L
•
exponent in King's Law
n
posltlve exponent in general eddy
n
polynomial function
eCnJ-[(n-nw)(np-n)]
p
2
instantaneous pressure force
,.\\( LT
2
time-averaged pressure force
M/LT
2
p
turbulent pressure force
WLT
turbulent velocity scale
turbulent kinetic energy
electrical resistance of probe at fluid
temperature
ohms
electrical resistance of probe at operating
temperature
ohms
flow constant
xv
Symbol
Definition
Dimension
lower limit of initial value
upper limit of initial value
initial condition in "shooting method" of
integration
H
step height
L
h
step height
L
step height in asymmetric configuration
L
step height in asymmetric configuration
L
K
overheat ratio
L
length scale for variation of mean quantities ln
longitudinal direction
L
length scale for variation of mean quantities in
transverse direction
!
t. (x)
L
o
L
rn
exponent in Kingls law
n
positive exponent in general eddy
TI
polynomial function
e (ol - (Co-o) CO -0) 1
p
2
p
instantaneous pressure force
>l/LT
2
p
time-averaged pressure force
M/LT
2
p
turbulent pressure force
M/LT
turbulent velocity scale
L/T
turbulent kinetic energy
L2/T2
electrical resistance of probe at fluid
temperature
ohms
electrical resistance of probe at operating
temperature
ohms
flow constant
xv
Symbot
Definition
Dimension
t
temperature of fluid
e
t
sensor operating temperature
Oc
5
V.
.
1
.
.
h
. th d i
.
Instantaneous ve aClty ln t e l
lrectlon
LIT
l
maximum veLocity at a transverse section
of constant
x
LIT
v .
minimum velocity at a transverse section
mln.
of constant
x
LIT
streamwise mean velocity
LIT
-V
estimate of
Ü
(order of magnitude)
LIT
total variation of mean velocity over a
transverse section of constant
x,
V
= V
-V.
S
max
mln
Ut (X, t)
or
free stream velocity
LIT
Ut (x)
u.-
.th
time-averaged velo city in the l
direction
LIT
l
ent rance velocity
LIT
.th
u.
turbulent velocity in the l
direction
LIT
l
u
mean velocity scale
LIT
o
u'
streamwise turbulent velocity
LIT
2
2
normal Re~~olds stress along
ex
L IT
2IT2
shear stress
L
2/T 2
shear stress
L
nondimensional velocity deficit
v
mean transverse veLocity
LIT
v'
c~oss-stream turbulent velocity
LIT
normal stress along
oy
shea r stress
xvi
Symbol
Definition
Dimension
Il
cntrance section half-width
L
o
w
mean vertical velocity
LIT
w'
vertical turbulent velocity
normal stress along
oz
x
longitudinal distance along channel from
step interface
L
virtual origin of abscissa
L
y
lateral distance from the centerline of the
channel or fram one of the sicle walls
L
y.
lateral posltLon in the mixing zone
(excluding potential core)
y* ~ y - d(x)
L
z
vertical distance from water surface
L
Greek Symbols
constant
B = 2a/(a+l)
6n1
st ep-œ i ze in Runge-Kut ta a Igor-i thm
boundary layer thickness
L
initial boundary layer thickness
L
2/T
eddy viscosity
L
nondimensional lateral coordinate
s = I-n
n
nondimensional lateral coordinate
~ = y*/io
nondimensional lateral coordinate
YI = I-Ca+I)VIRSC1"
8
eddy function
8',El"
derivatives of
e
dynamic viscosity of fluid
kinematic viscosity of fluid
p
density of fluid
xvii
Symbol
Defini tian
Dimension
o
shape parameter
shear stress
,= p u'v'
~
substitute veloçity-defect funcLion
41' .<fI"...
de r i va t Ives of
<fi
41
Gau5sian error integral
1
xviii
CHAPTER 1
Introduction
Hydraulicians have always showed a great interest for flows in
channel transitions in general, and in expansions (graduaI or sudden)
in particular.
In mast cases, los ses generated by flow separation
.
-
and excessive turbulence in sudden expansions are undesirable for they
impair the performance of the channel.
In sorne instances, howeve r ,
they are beneficially u~ed ta "kill" exces s energy de t r iment a l ta chan-
nel stability (stilling basins, flow controls) or in the case of closed,
conduit flow, ta deter potential cavitation damages (sudden expansion
downstream of valves) or in hydraulic machineries (diffusers) ta
enhance pump or turbine performances.
For mast practical purposes, the flow characteristics in expansions
-
are usually determined based on energy principle and/or momentum equa-
tian.
Such a method yields average flow velocity, pressure, shear
stress, etc., ~n those cross-sections of the expanding reach where the
Flow has been fully es t ab li shed (Hcndcr son , 1966).
Un fo r t una te Ly , i t
does not provide information about the flow pattern ln the zone of
separation.
In practice, abrupt expansions as weIl as other transi-
tians of similar hydraulic properties (miter bendsJ are avoided when-
ever possible in the designs of artificial channels because of the
trernendous hydraulic lasses they induce.
The lasses are generated in
the standing eddies where return flaw with high level of turbulence
takes place.
These eddy-zones have been referrred ta as zones of "de ad
2
water" erroneously one might argue, considering the tremendous
turbulence activities there. but in a mass transfer viewpoint they seem
ta behave as such. relative ta the whole reach.
The zone of main flow
displays changing characteristics in the dOlffistream direction and does
not lend itself ta a one-dimensional treatment as used in the zone of
es t ab Li shed flows.
Ltt appe ar s , ther efor e , that the flow characteristics
as computed by the one-dimensional energy and momentum equations, are
inadequate ta accurately describe whatever takes place in the separa-
tion zone in terms of me an flow characteristics.
Due ta the complexity of the flONS in abrupt expans i ons and more
particularly in the case of free surface low velocity flows. only
limited contributions in terms of laboratory modeling and ~athematical
simulations of this class of flows appear in the literature.
The purposes herein are ta investigate experimentally and
analytically the flow behavior in double-step channel expansions in
the case of relatively low velocities and small aspect ratios (small
depth ta width ratios).
The primary con cern will be the flow in the
mainstream, rather than 10 the standing eddies prope r .
Hcweve r , con-
t rary to sorne be i Iefs , the separation st r eamLi ne does not co inc Lde wi t h
the zero velocity loci for the me an longitudinal velocity profiles, 50
that a complete description of the main stream involves encro~ching
on that part of the standing eddy adjacent ta the t rougb flow where the
flow neves downs t ream (Fig. 3.1).
The aim of this study is twofold;
1) to introduce a simple
mathematical analysis of the subcritical turbulent mixing flow in chan-
nel abrupt expansions (single or double-step with no strong eddy inter-
ference) designed to predict the velocity variations across and along
o
most of the mixing region.
This mathematical treatment will be based on
a semi-empirical ~pproach leading to a simple form of the Navier-Stokes
equations to be solved numerically with the aid of the digital cOQPuter.
and 2) to present a series of experimental investigations of the expan-
sion flow fOT different step configurations and dynamical conditions
with emphasis placed on the mixing process in the thTough-flow-region.
CHAPTER II
Review of Literature
Engineering consideration of the sudden expansion flow gcometry
dates back ta 1766 and Bordat s (1) original ana Lys i s .
The Borda-Carnot
equation based on the principle of conservation of mass and momentum of
the flow as it enters the expansion interface and after it has com-
pleted the mixing process has been widely used by experimenters (2, 3,
4. 5) throughout the years ta der ive useful information about head
los ses in sudden expansion flow.
As s.t a ted in the introductory chap te r ,
this type of treatment does not il1uminate any of the details of the
mixing proces5 going on between the two control sections.
The earliest systematic study of the interna! mechanics of the
energy changes in the expansion f10w was·conducted by Kalinske (6) and
pub.Lished in 1944.
In the cont ext of a genera1 experimental s tudy of
the problern of hydraulic conversion of kinetic energy to potential
energy. which was initiated by the ASCE Comm~ttee on Hydraulic Research
in 1935. the author collected series of data pertaining to ~ean pressure
change. cross-sectiona1 mean velocity and turbulence distributions in
expanding pipe f1ows, using flow visualization techniques.
A ti~ed
tracing of immersed particles as shown by successive photographie
frames was used to determine point velocities across and along the flow,
then turbulence levels were estimated through local time averaging
(corresponding to a mere super-i mpos Lt i on of success-ive frames).
Bootha (7) conducted a series of experiments on high velocity
(supercritica1) flows in ,open channel abrupt expansion.
He based his
5
analytical description of the problem on the elementary theory of wave
propagation to isolate the pertinent parameters for supercritical flow
in sudden expansions. name Ly the Proude numbcr and the br cadt h-dcpt h
ratio of the approaching f Iow , and a Lso the ratio of the dep th at a
point of ioterest to the original depth.
His method was later used by
wat t s (8) who combined it wi th Albertson et al.
(9) jet-flow type of
analysis to determine flow patterns at culvert outlets.
Howeve r , the
behavior of supercritical flow being drastically different from its
subcri tical coun te rpart 1S, the re sut ts and ana Iys Ls of bath aut hor s cao
only be of limited he Ip in dealing w.i th subcr i t i cnt sudden expansion
flow.
Hsu (10) first made extensive maasur ement s of mean flow and turbu-
lence characteristics with hot-wire anemometry in a two-dimensiona1
(width ratio:
2 to 3) abrupt expansion of an air-tunnel for subcritical
flow at a relatively high Reynolds number
(S,x 105).
Although some of
his experimental data (shear stress data, for example) show noticeable
discrepancies (too high) with many existing ones. his theoretic~l deri~
vat ions and experimental estimations of the dynamic equations governing
the mixing process in the f10w of a sudden expansion could reasonably
be extended to the similar flow of water in open channel of interest in
the present study.
It was but twe Ive years after Hsu's efforts t hut
Chaturvedi (11) conducted experiments on the similar case of close
conduit sudden expansion.
He used an app'roach similar to Hsu t s to
derive an impulse-momentum relationship describing the force-field in
the mixing region. and a work-energy equation describing the dynamic
equilibrium of the motion.
On the account of the complexity of the
phenomenon and the lack of simple mathematica1 expression of the
6
dynamics of the motion. he presented the significant factors involved
in the farm of their spatial distrib~tion.
Lipstein (12) used 10 dif-
ferent wiùth ratios ta investigate the important aspect of statie pres-
sure recovery in sudden expansion duc! flow.
His data display a strik-
iog similarity in pressure recovery for a certain range of diameter
ratios, and show good agreement with well~known existing theories (such
as the Borda-Carnot equation) especf.a l Ly at small di.amete r ratios.
His
mean velocity data illustrate quite weIl the separation pattern and the
fLaw process in the mixing region. especially in the through-flow
region.
But due ta difficulties inherent in the measurement of the low
ve Ioc i t i.es , the author's r-eturn flow data are r'epor ted l y only of
qualitative interest.
It is noteworthy that low-velocity axisymmetric flow displays
remarkable similarities with subcritical channel flow for expansion
geometry, as reported by Rouse et al. (13).
However. the presence of
a free surface and ~he connoted asymmetry in the fLaw pattern at the
abrupt expansion wou1d require a somewhat different and more invo1ved
treatment than reported in (11) and (12J.
The most significant contribution in the study of expansion ftow ln
open channel until recently was by Abbott (14J and ~bbott and Kline
(15).
They have investigated subcritical flow models in both single
and double step configurations. for a much wider range of area ratios
(width ratios) than previously reported.
They first reported extensive
studies of the size and structure of the stalls as influenced by the
geometry of the expansion (configurations, width ratios). and experi-
mental data supporting the quasi-two-dimensionality of the throug:h-f1ow
Abbott's (14) theoretica1 ana1ysis of the prob1em uses an iteration
7
method ta solve the Navier-Stokes equations used in their simple farm
of the two-dimensional boundary layer type of approxination.
The pro-
cedure requ i res that sui table Iong i tud i n a I pressure gradient and shee.r
stress distribution as well as an adequate first approximation velocity
profile be assumed.
The author's theoretical rasults show fair agree-
ment with his experimental findings.-
However , an obvi ous weaknes s in
his analysis that might be accountable for sorne of the discrepancies
pertains to his assumed boundary conditions.
In effeet, the dividing
streamline has been assumed ta coincide with the zero-veloeity loci due
ta the difficulty inherent in velocity measurements within the staIl
where the actual zero-velocity loci are located.
This misconception is
certainly one of the rcasons behind the autnor's refutal of similarity
approach in abrupt expansion flow.
Hung (16) and Giaquinta (17) have developed interesting computer
models of the axisymmetric (and two-dimensional (16)) abrupt expansions
for laminar flow at very low Reynolds numbers (120 to ~OO) based on
specific forms of the ~avier-Stokes equations (i.e., the vorticity
transport equation and the stream function equation).
Unfortunately,
the Navier·Stokes equations for turbulent flows are impracticable as
to their solution by finite numerical methods.
1n effect, sueh an
approach must represent the dissipation process in detail, since other-
wise there will be no way of getting rid of the energy whi.ch is con-
tinually being injccted into the turbulence.
Strictly speaking, this
implies that the mesh interval for the applied fini te difference must
be somewhat smaller than the average eddy size in order ta achieve
numerical stability (l~).
This poses a problem of storage capacity
and treatment power with present-day digital computers.
Fortunately
•
enough. high Reynolds number flows that are of practical interest in
mast engineering problems are wholly determined by eddies mueh larger
than the dissipation eddies except near ta solid walls.
Computer models
of 5uch flows can be successfully developed provided the dissipation
mechanism is properly devised.
Eddy viscosity models are used for that
matter.
Deardorff (19) has studied ,a three-dimensional internal fLaw
using methods previously developed for meteorological calculations.
His approach distinguishes between grid scales, or eddies which can be
represented by his f i n i t e difference me sh , and sub-grid s ca Ie s (or
smaller eddies) for which eddy viscosity assumptions are introduced.
Deardorff's numerical results are not in good agreement with his exper-
imental data, probably due to two factors:
(1) the Ei ni t e difference
schemes (explicit) do not allow a great accuracy for this type of prob-
lem; (2) the eddy viscosity treatment of the subgrid scales is not
adequate enough in most flow regions (near the wall and away from the
wall).
The author's approach, however. appears as an attractive case
of the numerical solution of the ~avier-Stokes equations in turbulent
f Iows , with al! the difficulties involved.
NeLlor (20). Pierce and
Klinksiek (21), Klinksiek and Pierce (22) and East and Klinksiek (23),
among many others, have introduced more or less improved eddy viscosity
models and numerical schemes in the solution of the Navier-Stokes equ~
tians for three-dimensional (and two-dimensional). incompressible,
turbulent boundar-y Layers .
WeIl before these autho r-s , Abbot t
(14) had
used the eddy viscosity h}~othesis and an iterative method to solve the
two-dimensional boundary layer equations adapted through appropriate
boundar-y and initial conditions
to desc r.lbe the stalled flow in a chan-
nel abrupt expansion.
The discrepancies between his numerical results
and his experimental data seem to stem from his particular choice of
the approximating function for the shear distribution.
It is note-
worthy. however, that Abbott's carly endeavor led the ~ay to numerical
approach in the treatment of turbulent stalled flows.
Unfortunately,
the numerous difficulties involved in the theoretical as well as experi-
mental investigations of abrupt exp~nsion flows in open channels did
not incite further attempts to improve Abbott's ~odel since.
Only very
rccently have Kim. Kline, and Johnston (24) conducted experiments on
one-step flow separation in a rectangular air-duct ,that revisited the
efforts by Hsu, Abbott and sorne others.
They launched a fairly com-
prehensive investigation of flow characteristics in the s epar-at.ed shear
layer. the reattachment zone, and the redeveloping boundary layer after
reattachment.
Their experimental endeavor included collections of me an
data, cross-stream pressure distribution and streamwise pressure distri-
bution as well, and measurements of turbulent quantities (normal
stresses. Reynolds stresses) and a l so a study of the behavior of the
turbulence phenomenon (intermi ttency) in the abrupt expansion flow
region. with the aid of fairly sophisticated instrumentations.
They
developed a flow computation mode I that uses the so-called "zonal
method" involving a division of the flow mto four modules each of
which is handled by a different technique--namely, potential flow.
attached boundary layer. free shear layer. and reattaching boundary
layer.
Unfort unat e Iy , it appears that the interaction be tween different
flow modu Ie s r equ i res rather Invo Ived c Iosu re models that ma)' affect
the practicality of their model.
AnaIyt i.ca I investigations of channel expans i.on flaw have been
attempted eventually but only for very large width ratios for which the
10
wall interference is nearly nonexistent, sa that the flow behaves like
a s e l f-pr-ese rv ing free jet flow (9)
(o. g .• river flO\\~in~ .int o large
reservoir or delta).
Abi-amovi ch
(25) us ed a mixed ana Iyt Lca I approach
ta solve the problem of channel flow behind bluff body (or around bridge
pile).
He distinguishes between a zone of jet-like flow extending up
ta a d i s t ance about f i ve step heights Er-oui the "stern" of the 'Jody. and
a non-similar region t.hat comprises the reat t achmen t
rone .
He applied
jet flow analysis in the former region and a solution method involving
the hydrodynamics of an ideal fluid (conformaI mapping technique) in
the latter region.
His results are in good agreement with his exp~ri
mental clata.
However, since no solid wall interference exists in this
type of fION but instead the interaction of the other eddy of the pre-
vailing horseshoe vortex, his results should be regarded with caution
here.
In summary, open channel abrupt expansion flow has received very
limited attention unlike its axisymmetric (laminar or turbulent) counter-
part in the literature.
In particular, due to its practical application
in f l ui ci cs , laminar axisymmetric or tvc-d imens i ona I expansion fIow in
ducts received broad attention; ~nd more or less sophisticated computer
models have been develo?ed throughout the years (16,17) that predict
rather weIl the behavior of this type of flow.
On the contrary. turbu-
lent abrupt expansion flows in open channels still need ta be further
investigated in order ta improve the very few existing theoretical and
experirncntàl models in arder to advance our understanding of this very
important class of flows.
CHAPTER III
Theoretical Analysis
3.1
Governing Equations
Flow in open channel transitions (expansions or contractions) are
naturally described br the very general Navier-Stokes equations as
presented by Chow (1950) and Henderson (1966).
Abbott (1961) and
Klinsiek and Pierce (1973) have reported the more restrictive but aiso
more frequently encountered cases of steady two-dimensional incompres-
sible equations of motion as applied in boundary layer flow and consid-
ered ta approximate very weIL the through-flow in a separated flow
region.
In this chapter, the equations governing the through-flow in
channel sudden expansion are derived based on an arder of magnitude
investigation of the general Navier-Stokes equations for turbulent flow
as reEorted by Hinte (1975).
These equations are derived from the
general expression of the Navier-Stokes equation for steady incompres-
sible flow through a time smoothing procedure due to Osborne Reynolds.
In Figure 3.1, part of the nomenclature to be used herein is summari~ed.
x
= the longitudinal distance along the channel
y
= the lateral distance either from the centerline of the
channel or from one of the vertical side walls
Z : the vertical distance from -the water surface
BI = the channel width at the entrance non-expanded section
U
P, Fi
= instantaneous velocity, pressure and body forces,
i,
respectively
12
0
"
"~~
-...
-00
,(
0
-
5
"
"
-
-
'"
"
0
",
"
'"'
'~
~
U , ï', F.
= time-averaged velocity, pressure, and body forces
i
l
= turbulent ve1ocity, pressure and body force
U i ' P. f.l
fluctuations
d(x)
~ width of the potentia1 core
t
(x) = width of mixing region
o
p
= density of the water
u
= dynamic viscosity of the water
~
= kinematic viscosity of the water
The equations of motion for a steady turbulent flow of an
incompressible f1uid in compact tensor notations according to Hinte
(1975) are as fol1ows:
aï'
a
=
- - +
u. u _)
ax.
ax.
+ F.
(3. 1)
l
1
l
1
1
i = i , 2, 3; j = l, 2, 3
(3. Z)
.
where by definition:
U.l
p = p
P
i = l . 2 . 3 ; j = i , 2. 3
14
The equivalent cartesian notations that will be used hereafter
are:
x = Xl' Y = x ' Z = x
for the longitudinal laterai and vertical
2
3
coordinates, and U = U
V = U
W = U for the streamwise, cr05S-
l,
2'
3
stream and vertical velocities respectively.
The turbulent velocity
fluctuations
u
u
u
convert ta u', yI. Wl respectively.
1'
2'
3
Ta apply the Dutlined equations of motion ta our specifie situa-
tion would require basic as sumpt i ons as inspired by our speci fic chan-
nel geornetry and boundary conditions.
These assumptions can he outlined
as follows:
(1)
The flow takes place in a rectangular open channel with
negligible longitudinal slope.
(2)
The test section of the channel 1S made up of an inlet sec-
tion, a backward-facing step and a downstream section of
larger width (see Fig. 3-2).
(3)
The flow separates at the sharP edge of the step and reattaches
ta the vertical side wall sorne distance downstream.
The flow
in the zone of separation behaves like a free shear flow made
up of two mixing layers generating from the sharp edges of the
steps. evolving as such until mutual interference around the
centerline on one side. ~nd wall interference on the other.
sorne distance downstream (this is with respect ta the x-y
coo rddnate s) .
(4)
\\Hth respect to the z coordinate (vertical) the flow can be
·viewed as a fIat plate boundary layer type of flow.
(S)
In principle the free water surface is susceptible of rising
linearly from the inlet section downstrcam until the flow is
15
1
"
1
•
____
1
~
,
----
0
c
/
,t- - --- -,- '-"i'
"-
U
"
l
,
 - -~ - -)'-=__---i'>--..I
, 1
l
'\\
-
N
,~
1
\\
1,"
_~I-_ .:-- ---I--..J,.'
-ë
....
1
/ '
",
,
, ,
------'1-- ......
1
/
1
•
,
;<
u
"
--------~;
u
----....
,,"
1
- ---'\\,,"
,
S-
,
..~
r
,
/
0
, '
u
/
1
,;'
\\
,
.,
- ,
>.------,-
•c,
1
•0~
1
...
...
\\
0
\\
~ u,u,e0
1
l
'
-C
1
/
'J
en
l
,'
l
'
1
-,
1
/
.
1 /
~
OL..
-I ~.V
c
1
..0,.,,
0.
1
1
1
lb
fully reestablished.
Un der rnost subcritical channel flow
conditions, however. this effect 1s actually of Little or no
significance.
Equations (3.1) and (3.2) will he simplified into boundary layer
type of equations on the basis of the aforementioned assumptions.
Actually the flow in channel abrupt"expansion can be regarded as a frce
turbulent flow bounded by non-turbulent (potcntial) flow with velocity
gradients small compared with those in the turbulent flow.
A mean flow
direction is selected as the coordinatc direction Ox.
As a general
consideration, gradients of rnean values on the Ox direction are con-
siderably leS5 than in the yaz plane, i.e., the length scale for vari-
ation of mean quantities in the Ox direction, say L, is an order of
magnitude greater than the length scale of variation in transverse
directions, say i.
Defining
U
as the free stream velocity and
l
u = CU
- u . ) as the total variation of mean velocity over a trans-
s
max
mm
verse section of constant x, an order of magnitude analysis can be per-
formed on the Navier-Stokes equations toward restricting them to the
case of boundary layer type of flow, describing as weIl free turbulent
shear flow of channel abrupt expansion (10, 11, 14).
The general th ree-
dimensiona1 continuity equation is:
au
aW
-
+
+
ax
ez = o
or, in terms of the velocity deficit (U - UI):
aV
aw
+
+
+
ay
ïZ = 0
(3.3)
'(Û-U )
U
I
If
= OtT),
3x
'V
U
dU
U
I
dU
dUI] 1:
= OtT + dx) ... V =:. O((T + _I)e! = O[U +L-
oy
dx
s
dx
L
'W-
U
dU
U
dU
dU
I
1:
= 0(2. + dx) + w= O(e ~ + _1)1:] = O[U +L-lJ
(3.4)
Oz
L
dx
s
dx
L
Equation 3.4 indicates that unless the free stream velocity. Ur' changes
by a large factor in the distance
L, transverse velocities are smaller
1:
than longitudinal velocities by. at least, a factor of arder
[".
If
u
is the scale velocity for the fluctuations, the y-momentum equation
is eva!uated as follows:
- av
- 'V
- av aû'V'
av"
U - + V - + w -
+
+ - +
<Ix
ay
'z
dX
3y
2
2
al'
a2y
a y
a y
•
-. " [-2· ----, + -J
(3.5)
ay
2
ax
ay
az
2U
2
U
d
U
av
>
f
2
d
1:
ax
L
.(_I)J
U
U
[ -
U
• L
- "
[Us
(_1)]
2
L
2
2
dx
dx
L
U
dU
U
dU
-
av
s
I
I
dU! 2 1:
v -
1:] [(2- •
I:]/t
(U
• L
ay
!CL· dx)
dx )
=
dx )
L
s
;:z-
-
aV
dU! 2 1:
w-
(U
• L - )
-
az
s
dx
L2
aP
?
'z
au
2
'v·
u
f
.
, j (
r: 'r
;}Vi7
·2
u
aY
r
av
2
'w,
u
az
t:
18
2;;;
u 2
a
Us
dUI
f
dUI
;:2
v
s
v
v [ - + - ]
U
2
L
dx
L2 = (U ;:) [ - +
dx]
L
s
2
ax
s
L
a2;;;
U
dUI
U 2
dU
s
1
v
s
I
v - -
v [L
+ dx ] t : (U ;:l (-L- + Us dx ]
a/
5
U 2
a2w
U
dUI
1
dUI
V
[
5
+
5
V -:-r
V
dx J r : (U ;:) [T+ U
dx J
(3.6)
L
5
az
s
The vi.s cous terms are obviously ~ma11, for in turbulent El ows the
U ;:
t
Reynolds number (2-) is necessarily large.
In the limitas - + 0, the
V
L
second and third terms of (3.6) are negligi~le, and unless
2
2 d U
, L2
I
U (Us + L
- - ) Ls of the same arder as u - -
(wht ch for a fact does
2
;:,
dx
oceur only in distorted wakesJ. the first term is aiso negligible as
2
t
u
"[-+-0.
There must be at least one term of the same arder (r-) as bath
transversal turbulent terms in arder ta balance the equation.
Inspec-
tian of (3.6) shows that only the pressure term is available for that
purpose.
Therefore:
r
av,2
aV""'W'
l
'P
- - +
ay
az
=- p ay
Then,
2
~=
ay
o (r)
and
(3.7)
where
Pl
is the free-stream pressure at the particular section.
Note
that a sirnilar treatment of the z-mamentum equatian ~ill lead ta the
same result (3.7).
For mast fla~s of interest. the ambient flaw is
irratatianal with constant total head. and then.
(3.8)
The streamwise equatian:
19
- aü
- aû
- aü
u - + v - . w - . alIT
-. a~' aU\\;'
-.-
ax
av
'Oz
x
ê
ay
z
ê
l
ap
-
~ + \\1
P ax
(3.9)
With the knowledge that
U = U1(x)
and with (3.7) in mind we cao
1
rewrite (3.8) as follows:
-
;h12
.-. auv'
ay .
+ \\\\'
-
ax
1 a
p ax
•
v
(3.9)
And the orders of magnitude for the different terms:
_ U
acû-u )
1
U
U~
ax
L
a (Û-U )
[Us • L (dU
1
1
1/dx)
V
U
Jy
L
s
_ a CÛ-U )
[U
• L (dU
]
1
s
1/dx)
W-
U
Oz
L
s
a~
2
u
!
- - '
ax
t: L
2
3u'v'
u
ay
r
au'w'
az
dU1
U
-
s dx
1 a
p ax
20
v
U 2
s
(ut) T
s
U 2
v
s
(U i) T
(3.10)
s
l
As -+ a,the fourth and eighth and ninth terms are negligible, and the
L
5treamwîse equation reduces ta:
- aü
.2-
u -
- au
- au
au'w'
+ v -
w -
'--!!.)
ax
+
+
(3.11)
ay
az
+
az
,,2
For the sake of simplicity we will assume that the rate of change of U
with respect ta the z coordinate (depth) will be somewhat slower than
the rates of change in the othe r two directions.
In fact , experimental
data ta be presented herein~below will tend ta support this assumption.
It will aIso he assumed that
i S stnall.
And Equation (3.11)
will be reduced ta the following two-dimensional faTm:
-
-
- au
- au
-
au'V'
u - + v - · u
ax
ay
+ v
(3.12)
1
ay
For channel flows at high Reynolds number the vîscous term is usually
relatively small. maioIy away from the wall.
Consequently, in the flow
region that is of interest in this study the equations of motion
reduce to the further simplified form as follows:
U-momentum:
o aü +
'Ü
V c
ax
ay =
(3.13a)
Continuity:
au
av
- +
ax
ay = 0
(3.13b)
-
. _ - - - - - - - - - , - - -
21
Next, we shall attempt ta solve the equations of motion (3.13a and b)
for the boundary conditions, characteristic of flow in a two-dimensional
channel abrupt expansion.
In general. the problem of finding exact
solutions of the Navier-Stokes equation presents insurmountable mathe-
matical difficulties.
This i5, primarily, a consequence of their being
nonlinear, sa that the application of the principle of superposition,
which serves 50 weIl in the case of frictionless potential motions, or
creeping motions with negligible inertia, is excluded.
In addition,
and perhaps most important of aIl in turbulent flows, a c losur e p rob Iem
exists, as a consequence of the pressure gradient and shear stress
terms present in the momentum equation.
To make the problem determin-
ate requires that either the spatial distributions of
U
and
u'v'
l
be known from experimental data. or explicit relationships between
Ul
and
x
on one hand. and
u1v'
and the streamwise velocity U on the
ether, exist.
The follewing paragraphs will address this problem.
For
most flow problems only numerical solutions (or step-by-step integra-
tien) of the Navier-Stokes equations prove obtainable. provided a suit-
able discretization scheme is available that approximates the solution
at predefined nodal points with an adequate degree of accuracy.
The
present chapter will deal with presenting approximate numerical and
empirico-analytical evaluation of equations (3.13a and b) for the
aforementiened f10w conditions.
3.2
Shear Stress Estimation
3.2.1
Eddy Viscosity Approximation
J. Boussinesq (1877) was the first ta introduce the concept of
eddy viscosity in his search for a suitable relation between the
apparent shearing stress
-pu'v'
and the me an velocity.
In analogy
22
with the coefficient of viscosity in Newtonls law of fluid friction (or
Stokes' law for laminar flow) he defined a turbulent exchange coeffi-
cient (often referred ta as "apparent, l'or "v i r tua l ;" or "eddy vis-
cosity") as follows:
au
-p u'v '
= P €:r
(3.14)
ay
The underlying idea is that in turbulent flow certain lumps of fluid of
macroscopic size are moving as a whole and thus work in similar fashion
as the free molecules of a gas.
The transversal distance
il traveled
by a lump of fluid before 105ing its individuality is known as the
Prandtl's mixing length. in honor of L. Prandtl who first developed this
hypothesis in 1925.
The transverse motion of a lump of fluid causes a
turbulent velocity fluctuation generated by the difference in its own
x-momentum and that of the surrounding fluid.
Schlichting (1968) has
estimated the fluctuation velocity
u'
ta be equal to:
u' =
Sa that:
au
-p u'v' =
P cr ay
Hence:
(3.15)
Hsu {16) has calculated the distribution of
z' (re ferred ta by him as
!m) near and shortly downst ream of the r-eat.t achment zone .
It di sp Iay s
a maximum near the plane of maximum shearing and goes to :.:ero as TI or
,u
ay goes to zero.
The eddy viscosity values can be dcduced from the
values of i' and
VI -
/
v' 2 as indicated by equation (3.15). This is
shown in Table 3.1 for Hsu's data.
Two plots of
€T
as a function of
the lateral coordinate in Figures 3.3 and 3.4 show data by Hsu and Kim
23
Table 3.1.
Eddy viscosity data (adapted fram Hsu).
Location
x/H
r/ H
1
v'
p
_ v
I<TI = 1 v'
ID
ID
7
fps
f" Is
0.375
0.0050
1.410
0.00705
0.525
0.020
2.100
0.0420
0.675
O. 022
3.510
0.0772
6.0
0.825
0.040
4.050
0.1620
0.975
0.030
3.750
0.1125
1.1625
0.005
2.700
0.0135
0.075
0.0050
1.050
0.00525
0.225
0.0150
1. 950
0.02930
0.375
0.060
2.550
0.1530
0.525
0.090
2.850
0.2570
7.50
0.675
0.085
3.150
0.2680
0.825
0.063
'3.750
0.2363
1.115
0.040
2.250
0.0900
1.275
0.0150
1.500
0.0225
In general, the eJdy viscosity
ET
is a function of both y and x
as indicated in Figures 3.3 and 3.4. ln some shear flows, however, the
assumption of constant eddy viscosity independent of either y or bath
y and x has been used successfully ta approximate self-preserving fLaw
characteristics.
24
<>
<li
'"
'"
" Il
<>
'"~ '">;-
'",
0
<l
"
o
.2
ù
"~~
~ "
" ~
0:=
"~ ..c
"o •
.. ~~"
"'''
o
>-.
~
0
"n "
,,", I.l.-.
oÙ"
~
o
-ri
...
> 0
>,g.
o
-e 0
-e Ù
'..:J '--'
o
25
.20
J6
~.'2
.~
1- ~I
'"
1-
.s.:
.08
•
X/H
0
1.61
• 8.'S
0
.04
/!;
/QjJ
• IJ
• ".61
o
·2
·6
.8
1.0
Figure 3.4.
Eddy viscosity downs t ream of r eat tuchmen t
(adapted from Kim et al.).
26
3.2.2
E~licit Functional Approximation of the Reynolds Stress
A simple procedure for estimatîng the shear stress term in the
equations of motion wou Id be, of course, by way of measuring
u'v'
in
the mixing flaw region as was done by Hsu and Kim et al.
Their d~ta
presented in Figures 3.5 and 3.6 show the strong dependence of
u'v'
upon
y
and
x.
Approximate functional forms of
u'v'
can be inferred
from data of this k i nd in order- ta c i r-cumvent d i Ef i cu l t i e s inherent in
mas! methods of point data entry in computational procedures.
Abbo!t
(14) assumed a fairly simple distribution, linear in
x
and parabolic
in
Y. ta approximate data by Hsu (1950) and Tani (1957).
His parabolic
fit, unfortunately, significantly overestimates the shear stress ln
most flow regions except near- the plane of maximum shear i ng .
A higher
degree polynomial seems to be in order for a better fit.
However, the
final choice of the functional form should be contingent upon a proper
trade-off between the gain in accuracy of the computational procedure
and the degree of difficulty thereby introduced.
On the basis of
similarity argument as will be seen in later sections of this study.
Van der Hegge Zijnen (1958) has derived a shear-stress distribu~ion for
plane jet by introducing the adequate ve~ocity distribution into the
equation of motion.
With ~he assumption of a Gaussian error distribu-
tion for the me an velocity, the corresponding shear-stress distribution
is of the form:
1
Iii
=
[3. l6)
218.' T
where
a'::: constant, charac t er i s t i c of the jet s i ze , and
41
i.s the
1
Gaussian error integral.
Also, Hinze (1975) has derived a power-law
BéFORE
A F TER
REA T TA Cf/MEN T
1.50
r..
X!H= 1.50
"l X/rl= 6.00
0
X!H= 3.00
•
X/H= 1.50
0
X!H= 4.50
•
YII/
~
~
,"l
0·6
0.4
0.2
o
0.2
0.4
0·6
0.8
U'v'
102
---;-;rX
Uo
Figure 3.5.
Turbulent shca r stress [udup t ed from IJsu).
X!H= 6.18
X!H=7.61
'·5
'.5
1·0
10
Y/H
~/H
~
œ
05
os
, - - - -
,
o
oorJj
0.010
0.015
o
ooa>
0·0/0
0.0/5
-u' v'
-u---:V-
Ul
-=
u,;
BEFORE REATTACHMENT
AF"TER REATTACHNENT
Figure 3.ü.
Turbulent shcar- stress (adupt.cd from Kim et a L, }.
29
distribution for the shear-stress in a round-jet in accordance with the
velocity distribution, as fOllows:
1
-n
(3.17)
2
where
n • .-L
and
C
x+x
t
~ constant.
1
o
Figure 3.7 shows the strikingly goad agreement between measured and
computed shear stress distributions as reported br Wygnanski and
Fiedler (1969).
An approximation based on eddy viscosity approach has aiso been
derived br Van der Hegge Zijnen (1958) in the following manner:
aU/umax
(3.18a)
an
And with the assumption of Gaussian veloçity profile,
<T
, 2
-an
= - 2 ::-;;;'- a' n
x
e
(3.18b)
Umax
<T
where
varies very slowly on mast of the jet width.
x Umax
Equations 3.16 to 3.18 p rov i.de compact fo rtn expressions for the
shear stress distribution in turbulent shear flows.
They. however , sug-
gest that the flow must display a certain degree of self-preserving
behavior.
As will he seen later, two-dimensional and axis~nooetric
abrupt e~ansion flows herein considered are self-preserving in the
velocity profile on rnost of the mixing region.
Self-preservation in
the shear stress. however. is nearly impossible to achieve, especially
in the mixing region of a sudden expansion flow.
Equations 3.16 and
3·18b can therefore he used only as a first approximation in this study.
30
D
X (D = 50
o
X (0 =.0
y
XID =-'5
COMPUTED
ao
,.0
D...
O.
0.05
O.fO
0.15
0.2
,,~
y/x
y
"_/.0
Figure 3.7.
Experimental and computed ~hear stress
distributions (f'rom lvygnan s k i and Fiedler).
CfjAPTER IV
Development of Computational Net hod s
4.1
Analytical Considerations
Any attempt to solve equations 3.13a and b analytical1y via simi-
l ar i t y approach as in boundary layer. flow or f ree shear fl6w is usually
hindered by the fact th~t the boundaries of the mixing zone are diffi~
cult to delineate.
This is especially 50 because the loci of zero
ve loc i ty is located in the standing eddy whe re the excessive turbulence
level makes mean flow measurements rather tedious.
Hence, for the sake
of simplicity the loci of zero velocity has been sometimes assumed to
co i nc i de with the concave dividing
s t reaml i.ne
(Abbot t , 1961).
As a
consequence. the spread of the mixing layer is very unlikely to be a
linear function of the downstream distance (~). one of the requirements
for self-preserving behavior of the mixing flow as will be seen in sub-
seq~ent sections.
Ho~ever, when measurements in the standing eddies
are available that determine the location of the zero velocity loci. as
reported by Hsu (10) and Chaturvedi (11), for example. it is possible
to show that the mi~ing region spreads relatively linearly in most of
the flow region (Fïgures 4.1, 4.2, and 4.3).
Se Iec t i ng the po ten t i.al
core velocity as the scaling velocity and the mi~ing flow width as a
length scale, this author has reduced the velocity data of Hsu and
Chaturvedi as weIl as his own to fall along nondimensional average pro-
files (with reasonable accuracies) (Figures 4.4, 4.5, and 4.6).
This
fact lets us believe that similarity approximation can be extended with
a fairly good chance of success to the two-dimensional abrupt expansion
flow.
The approacb will be outlined in the ne xt few paragraphs.
3Z
33
~
.'-e•>-,~•
<>
-"
..;
u
~
~
•~
,-a~
•uC
•
~
'1".
.'-a
~
•.
~l::l.Q
'
-a
..
,
~
~
" eo
~
'"0c0
"~
"3
'~
•
<>
'n
•
-
-~-o'.
•
•
0
~
c,
"~
u
-,
c
'"
'.~
'"
<>
<i
,,0
< ;
0
.,
..
3·0
l O/H Z·O
o
,.0
'"...
o
o
o
/.0
2.0
J.O
x/H
40
5.0
60
'.0
Figure 4.3.
Flow
as func t i
of
i
inaI distance (present
w i . d t h
o u
I o n g
t u d
c x p c
r
I m c n t
l
.
3S
36
~
.,
-e
0
,.
",~
q
-ü
".; "....
~
0
" ,1 "
"~
"~
" " "
>< - -
"
-a
" "
~
o
• <1
<1
.c
.,
•
"
"~
"
-a
>-
".
•
<l
"
.,uo~o
.<l
,.
.,0 e
."
.....
.,.;,-~.........
"
C)
0<l
-"
.<1
.
e
<l
~
~
<l
o•<1
<l
o
•
.
,
C>
C>
C>
C>
..
'"
-;
"
"
~ "
>:
">: ">:
- ~ --..
><
><
><
><
• !> ~ •
C>
C>
C>
C>
..;
..
.. '"..
'"
" " " " 11
>:
>:
>:
>:
>:
- ~
><
><
- -;;-
><
><
0
<l
0
• •
C>
C>
38
4.1.1
Self-preserving Development
The similarity hypothesis in turbulent flows is introduced by
Townsend in these terms:
"In a developi..ng flow, the transverse distributions of mean
velocity and other mean quantities change with distance dO~TI
stream. but it is often assum~d that the distributions retain
the same functional forms. merely changing their transverse
length scale and the scales of the mean-value quantities ... ,lt
The similarity or self-preservation principle is without an! doubt
a convenient mathematical hypothesis which serves the purpose of Teduc-
i ng the number of independent var Lab Ies in a somewhat t nt rac t ab Ie par-
tial differential equation ta eventually transform it into an ordinary
differential equation usually less difficult to handle.
On top of being
a convenient mathematical hypothesis
though, the assumption embodies
J
the princip le of moving equilibrium that supposes that the flow at any
section is not determined by conditions at initigtion or sorne distance
upstream. and 50 the flow lS geometrically similar at aIl sections,
depending only on one or two simple parameters.
lf a mean velocity scale is chosen as
U J
a turbulent velocity
o
scale as
and a length sc a Le as ta. the as sumpt Lon of self-
preservation is expressed as fo l Iovs :
u = U + u
f(y';l)
(4.1a)
1
o
0
u'v' =
(4.1b)
where y* =
d(x) + y and U is the free-stream velccity.
Letting
1
y*/t
= n,
and substituting in equations 3.13a and b, and after
o
reorgani~ation (see Appendix A for details):
39
du
du
2
(_0 f
' u _ o _ f
dx
o dx
u
dru t )
o
0
0
r
(4.20)
dx
o
Or by dividing through by q 2/ t
:
o
0
urt
du
u
dt
_ _
0
(~f
o
0
nf ' )
2
d x
:Cclx
qo
o
u t.
du
2
+~_o_f
= 0
(4.2b)
2 dx
qo
where the primes signify differentiation with respect ta Il.
For similarity condition ta exist. it is necessary that the non-
dimensional coefficients of the various terms in the equation be either
zero or independent of x (constant).
Therefore:
ut dUr
u t
du
0
0
-2-d'X :: A
0
0
0
D'
- 2 - dx
-
qo
qo
2
Utta du
u
dio
0
B
0
- 2 - dX::
E'
- 2 dx
=
qo
qo
U1ll
di
o
0
c'
- Z clx
=
qo
Fur tuermo re , the turbulent kinetic energy equation that r-eads :
il ~ (l <7) • - a
1
::ï7 au
,........- aV
,
1
aü
ax
v - (- <7) • u
-
+
v'" -
• uv
-
2
ay
2
ax
3y
3y
3
1 -
1
+
(- pv ' + -~)+E=O
(4.3)
ay
p
2
can he transformed in a fashion similar to the equation of motion (sec
Appendix A for details) to yield:
.u
(ng'-2g+2g)
r
.5
t
2
u
dq
o
0
+ - - - fg
f'g
2
dx
r
2
q
dl.
du
u
dl.
1
o
2
0
f '
- . " . - u - n s : + q
(-"- f
o
0
f')
2
r
c,
cdx
"""T
o
dx
dx
n
o
o
(4,4a)
Dividing through by
(ng'-2g+2g)
r
5
t
d ( u l. )
I
o 0
2
dx
l.
du
(-". _ 0 f
ua dx
qo
+ -
k'
(4,4b)
uo
Here again, the self-preservation arglli~ent requires that the
coefficients be either zero or îndependent of x.
Obviously the coeffi~
cient of the energy dissipation term g/u
is no t
zero.
It fo Ltows t he.t
o
q
is proportional to ù
and will he replaced by it in the analysis.
o
0
The coefflcients in equation 4.2b become:
41
t: du
c:
ri
0
0
= u<lX
o
E' =
(4.5)
One way of satisfying çonditions (4.5) 1S by letting:
a
(4.6.)
Cl
(x -
x. )
o
and
l
0
(x - x )
o
a
(4.6b)
where Xc 1S sorne reference abscissa (virtual origin) and a is any con-
s t ant pos Lt i.ve or negative, integer or not.
Hence. Equation 4,2b becomes after rearrangement:
2
VI paf - (a ... 1) n,f'] + ai
- (a + 1) f" J fdn = .&..'-
(4.7)
o
Cl
with
n = 0 +
f(O) = l, and n = 1 +
f(l) = 0.0
dl
U
0
1
whete
=
( _1
Cl
and
V
V
0) .
Actually in the
<lX
= -
c
c
present
1
1
ua
case V = -l, for the maximum velDeity deficit u
= U
U1 ' is at t a ined
1
0
at V = 0, in the
l
flD\\~ region wh i
the
of
s e
E - p r - e s e
r v t
n g
c h
e x c
I u d e
s
z
o n e
r~turn flow.
The self-preserving through-flow behaves as a zero wall
streSs boundary layer.
The momentum integral will provide an additional relation necessary
to make the exponent
a. det.ermi nat;e .
Integrating Equation 4.7 over
the self-preserving layer lassuming that the contribution of the retulTI
flow is negligible):
42
œ
œ
œ
0
2aV
fdn - Vl(a .1)
of' d n - (a • 1)
[f'
I
f d"l] dn
f
f
f
f
0
0
0
0
œ
œ
• a
f2 dn
K-'- dn = 0
f
• f CI
0
0
Using integration by parts for the second and third terms:
œ
œ
œ
o œ
nf' dn = ;1 fi
-
fdn =
fdn
f
0
f
f
0
0
0
and
œ
n
œ
œ
"
0
[f'
fdn J dn =
f2
[f YfdnlJ:
dn = -
f2 d"
f
f
l
f
f
0
0
0
0
0
The equation simplifies as fo Ll ows :
œ
œ
œ
.2a\\'lf fdn +V1(a+l)
fdn • (a • 1)
f2 d"
f
f
0
0
0
œ
œ
0 '
• a
f2 d" •
~ dn = 0
f
f CI
0
0
Assuming a symmetrical (or nearly symmetrical) distribution for
the shear stress that goes ta zero at
n = 0
and
n ~ ~. the last term
is vanishingly small.
Hence , the momentum integral reduces ta:
(3aV1· VI) Il • (2a • 1) 1
= 0
(4.8)
2
where
œ
1
= J [f(n)]n dn
n
0
43
Solving for
a:
v I
i-
l 1
a =
(4.9)
21 2
1
and the exponent
a
must lie be tween
a =
for
V
very small J
2
1
i.e., jet in still fluid, and
(II - 12)
a =
(31
- 21
1
2)
for
VI = -l, the practical limit of velocity defect (N.B.,
1
since f(n) ~ 1, Il > I
for plane flows.
Z)'
Approximate values of a computed from three sets of data by Hsu ,
Chaturvedi and this author (hereafter referred ta as data of SET!. SET2
and 5ET3. respectively), range from -0.175 ta -0.387.
These results are
in accordance with results by Townsend (1976) for flows with small ta
2ero wall shear.
Sandborn (1968) suggested that the velocity profile should resemble
the equivalent laminar steady separation profile (laminar-turbulent
analogy) and more likely sa inasmuch as the eddy viscosity approach
is applicable over the whole mixing region.
The available velo city data are compared with laminar separation
ve10city profile (Figs. 4.7. 4.8, and 4.9).
The agreement is rather
remarkable and seems to follow Sandborn's laminar-turbulent analogy
in fully developed turbulent separation (staIl mode1).
Furthermore.
1a = -2(1 - 1
- 1
for axisymmetric flow with
1
2)/(31 1
2)
In = J~[f(n)]n ndn
o
44
.6
.4
.2
...
HW (1950)
-
L AN/IVAR ,o&?OFIl.E(H-Jftl
o.
o.
0.2
0.4
0.6
0.6
1.0
Yi! o
Figure 4.7.
Comparison of the me an velocity profile of
fully-stalled flow and the equ iva l errt Iami.nar-
profile (data of Hsu).
45
•
·8
·6
4
e
.2
'NA TURVEDI (/962)
_LAMINAR PROFILE(H=S.J4)
Q.
Q.
Q2
0.4
0.6
0.8
1.0
0/ t o
figure 4.8.
Co~parison of the me an velocity profile of fully
stalled flow and the equivalcnt laminar profile
(data of Chaturvedi).
o
.8
c
-6
-.9_
U1
-4
.2
o PRESENT EXPElilMENT
-
LAM/NAR PROFlLE(Hd.6~
o
o.
0.2
0.4
0.6
0.8
1.0
YI i o
Figure 4.9.
Compar i son of the mean vc l cc t t y profile o f fully
s t a l Icd f l ow and the equi.va l cnt l am i na r profile
(present experimcnt).
47
the computed potential core velocity profiles
U (x)
are compared with
1
measured values in Figs. 4.10, 4.11, and 4.12.
In mast cases greater
discrepancies betNeen experimental data and theoretical values accur
downs t reum ta ward the r-eat t achmen t
zone. wher e i nc i.den t a Ll y , the pro-
posed theory breaks clown due ta a pronounced curvature of the flaw and
a high level of flow instability.
Using an eddy viscosity mode! with constant ~T ta estimate gr.
reduces equ3tion 4.7 ta the following second arder ordinary differen-
tial equation (ODE).
2
1
V
-
Ca + 1)"f'j + af
-
Ca + l)f' J fd" =
f"
(4.10)
1[2af
RsC l
The boundary conditions are:
f = 1.0
and
f' = ~
at
n = 0, f + 0.0
at
n + l
where
R = (U1lo)!(E )
is a turbulent Reynolds number,
s
r
also called flow constant (Townsend 1976) and characteristic of the
kind of flow.
Equation 4.10 is one form of the so-called Falkner-Skan equation.
honoring V. M. Falkner and S. W. Skan. who first deduced it.
The solutions of the original Pa l kne r-Bkan equation pertaining to
the laminar boundary layer in fluid motion were investigated in detail
by D. R. Hartree {1937).
They de scribe similar profiles generated in
both favorable and adverse pressure gradients.
In the context of
investigating possible limiting forms of pressure gradient and regions
of validity of similarity solutions for the boundary-layer equations.
Brown and ~tewartson (1965) have discussed solutions of the original
Falk~er-Skan equation.
Aiso Schlichting {1968) has presented a ~ide
range of practical cases of occurrence of the Falkner-Skan problem in
laminar flow.
48
'"'"
....~
-,
~
'-
"
0:
0:
'" <:>
'"
~
'"
0
;:
c
'"
~
>
•
<:>
-e
~
~
>-
u
,"'J
0
~
,>
0
"
0
c
-
'"
"
..
"u=Uug,
~
u
,
'"u
~
0
§
<:>
"u
..;
~
"""
>
-""
"u"
'"
ë
~
~
~
~
"'c","
-
<:>
<:>
'"
<:>
-:
<l
:s,~
• EXPERINENT
_ _
THE ORY
,,0
•
•
•
..'!L
~
OD
Uc
"j
,
o
'.0
XIH
2.0
s.o
Figure 4.11.
i
i nu
variation of the
t
i
core
i t
i ) .
L o n g
t u d
I
p o
e n r
a
I
v e
I o c
y
( d a t
a
o f
C h n t
u r v
e d
--
1
• E XPERIItJENT
THEORy
1·0
~
•
•
'1
•
•
~
o
Ua
•
0·5
o
o
1·0
Z.O
J.O
'.0
5.0
6.0
7.0
8.0
X/H
Figure 4.12.
Lcn g i r ud iua l vu r ia t l on of the por eut i a I core ve l oc i r y (present cxper-tment j .
~
51
In turbulent flow. the Falkner-Skan cquation is primarily used ta
describe a family of turbulent boundary layers -in steady, plane flow
known as "equ i Hb r Ium l ayers" and first experimentally evi.denced by
Clauser (1954).
Thesc self-preserving turbulent flows have bcen theor-
etically treated by TOWTIsend (1956, 1976), Clauser (1956) and Coles
(1957).
Rotta (1962) presented a survey of the conditions for sorne
of the mast characteristic equilibrium solutions.
4.1.2
Appro~imate Solutions
Equation 4.10 is a nonlinear second arder ordinary differential
equation (ODE) whose analytical evaluation is rather tedious.
In
general, the faTm of the solution depends on bath a and Vi'
If the
velocity defect is smalt (VI large and negative). the nonlinear terms
are negligible and equation 4.10 reduces ta:
(2af - (a
(4.11)
If we put
fil
,-(a
:=
+ 1) V R
1
Cl
s
"
af
1
df
- - =
"'<fil
'°1
/-(a i- l) V R
1
Cl
s
,2 f
-1
,2 f
- =
, 2
(a + 1) V1 R
~
s Cl
"1
n
and equation 4.t1 becomes:
(4.12)
~here
S": 2a/(a + 1)
Equation 4.12 is one form of Hermite's equation.
Its solution behaves
as
52
-(~+1)
( 1
Il
exp - -
1
2
for large positive v~lues of
III (asymptotic solution).
Practically, mast interest is attached ta boundary layers in zero
or adverse pressure gradients.
For the aero pressure gradient f l ows ,
a = 0; hence
B = 0, and equation 4.12 simplifies as follows:
f"(ll) + 1)
f'(n)
= 0
(4.13)
1
1
1
The method of separation of variables leads ta:
f"(n )
1
~'(01)
~(n1)
= -0 1
Integration of this expression yields:
1
2
"2 111
tHll ) = Cil e
1
1
Using the first boundary condition as expressed in the foregoing ta
determine the constant of in r egr at Lon Cî. we have:
1
2
2" nI
A second integration leads ta:
+ Cil
2
where Cï i s a second constant of integration ta be eva Luat ed ".i th the
aid of the second boundary condition.
~nence.
n
1
2
=(0.-2
1
2
t
2 t
dt -
dt
o
53
or
dt
(4.14)
Note that
a
is negative constant since the function
f
is monotonously
decreasing.
Therefore. e~ression 4.14 cao he rewritten as:
= 101 f e
dt
nI
Defining a new variable as:
the distribution function becomes:
(4.15)
For a self-preserving boundary layer in zero pressure gradient.
101.= 2
(Townsend 1976).
,
The ot he r mast interesting case of boundary layer f Iow in adverse
pressure gradient Ca < 0) is the zero-wall stress flo~ (separating
flow) for whi.ch a = -1/3 and B = -1.
ln this case equation 4.12
becomes:
f"(O) + 0
f'(n) + f(n
= 0
(4.16 )
1
1
1
1)
which is rewritten:
f " + ( ll f ) ' = O
1
Integrating once:
f" + Il
f = e"
(4.17)
1
3
S4
where
C"
is a constant of integration.
If it is assumed that the
3
slope of the velocity at
ni = 1
tends ta zero, C3= O. Then using
the method of separation of variables as before, and integrating a
second time:
(4.18)
With the first boundary condition
f(O) = 1.0, C~ = l, and the velocity
deficit distribution is a simple Gaussian distribution.
An approxima-
tion to the solution for intermediate cases is an interpolation farm
given by:
~
2)
2)
f(n) = cr exp(- .!- n
+
(1 - crJl~) 1/2
exp( - .!- t
dt
(4.19)
2
,
J
2
n
where
cr
is a shape parameter which lies in the range o - i , and shoutd
be chosen to make the assumed distribution resemble as closely as
possible a solution of the Falkner-Skan equation (4.10).
'If the velocity defect is not small, a solution to equation 4.10
can still be approximated by one of the small-defect solutions.
Usually, however, the value of the exponent
a
differs.
Figures 4.13, 4.14, and 4.15 show empirical fits of Gaussian
distribution functions through data of SET1, SET2, and SET3, respect-
ively.
The respective Gaussian distributions for the three data sets
are:
1
2
ï (n/0. 391)
f(n) = e
(4.20a)
i (n/0. 272)2
(4.20b)
55
/.0
1
2
-Z("/O.391)
_ _ ft") = e
,
5 1 ~ 7
-c '87
----ft") = (1 - Tl ,..'!)-
'c
'c
\\ .
D. \\~
,
~'J
f ("l
Q'
",~'c,
o o
J
Figure 4.13.
Comparison of experimental mean velocity profile and
empirical Gaussian and power law distributions (data of
Hsu) ,
.
56
,
,
I."",",~---------'-------;----:;----'
-Î(O!O.272)-
E(ol = e
U
=
[1 _(1_0)7/2]2
UI
" Il
,, Il,,
Efol O,,
011
o
O.,
1.0
n
figure 4.14.
Comparison oE experimental mean velocity profile and
emplrical G~ussian and power-law distributions (data of
Chaturvedi) .
57
'·0 _.-------------~----_._---~-_,
-,Co/O. 358)"
_____ fCo) , e -
----fCo) ,
3/ 2)2
0_0
fCo) 0.5
o
0.5
10
o
Figure 4.15.
Comparison of experi~ental mean velocity profile and
empirical Gaussian and power law distributions
(present experiment).
58
and
1 (n/ 0 . 358 ) 2
2
f(n) = e
(4.20e)
The distributions show a fairly good agreement wi t h the respective
experimental distributions.
Greatcr discrepancies accur close to the
potential core boundaries (zone of vanishing velocity deficit) espe-
cially for SETI and SET3.
There the actual value of the apparent
coefficient of eddy diffusion
is much smaller than the assurned
constant value.
Power-law distributions usually approximate better the tail values
in a shear flow than Gaussian distributions.
The following power laws
follow very closely the experimental distributions.
They are:
SETI:
f(n) = (1
(4.21.)
(4.2Ib)
For SET2 the velocity profile is closely approximated by:
(4.22)
where
< = 1 - n
Figures 4.13, 4.14, and 4.15 aiso show graphical representations
of equations 4.2la. 4.21b, and 4.22.
The agreement with experimental
data is very satisfactory.
4.2
Numerical xïodel s
In the preceding paragraphs sorne basic concepts for a close form
solution of the Navier-Stokes equatiofis for channel abrupt expansion
have been laid out.
The similarity approximation led ta equation (4.10)
assuming constant eddy viscosity.
However, as outlined in 3.2.1, the
59
eddy viscosity is actually varying in space bath in transversal and
axial directions.
ln this subsection, the more general case of vari-
able eddy viscosity model is cons i.der ed , wh i ch IdU i.nvo l ve a numerical
method ta solve the similarity equation.
4.2.1
Variable Eddy Viscosity
The variable eddy viscosity h}~othesi5 makes the shear stress term
g'
~n equation (4.10) equal ta:
ET
E'
g ' -
f"+
T
f'
('.23)
- tOUI
tOUI
where the primes denote derivations with respect ta
n.
of
n ,
This suggests that
must he of the farm:
(4.24 )
where
C
is a constant and
Sen)
is an arbitrary function of
n.
ln
3
the case where
8(n) = constant, expression (4.24) reduces ta Prandtl's
(1945) new theory.
1f expression (4.23) i s Lnt.r-oduced t rrto equation
(4.7), the generalized similarity ~p~roximation reads:
2
-2af + (a + Lj nf ' + af
- Ca + l)f' f Fdn = - K 8 fil - K '3' f '
o
(4.25)
2
where
K = C
)
and
C
defines the constant in the potential
3/(C 1C 2
2
a).
core velocity expression (i.e., U (x) = C (x - xoJ
I
2
In arder ta solve (4.25). an exp l i.c i t expression for the eddy
function 'B(n)
must be stated.
Very limited information ~bout v~riable
eddy viscosity shows in the literature.
Ta this vr i.ter-t s know l edge , no
explicit models of
B(n)
ex i s t , but only very few e xperi ment a l re su l t s.
60
In eiieet Van der Hegge Zijnen (1958) has measured the distribution of
B(n)
in two plane jets of diffeTent sizes.
The trend of
B(n) vs. n
is shown in Figure 4.16 to be nearly linear.
Kim et al (ID78) obtained
the distribution of a characteristic faTm of the eddy function ln
abrupt expansion flows (before reattachment) for two difEeTen! step
configurations.
The distributions are definitely nonlinear and exhibit
a relatively sharp Tise near the centerline of the mixing region
(Figures 4.17a and 4.17b).
Based on the data of Hsu (195r) and Kim (1978), it can be inferred
that
a is sorne polynomial function of n.
A general representation
of
Ben)
can he assumed as follows:
(4.26)
where
and
are :ero eddy viscosity loci necess~rily close to
boundaries of the mixing flow region. and
n
is an arbitrary power
(n ~ 0).
If we assumed
n = l, 9(n)
is a parabola.
Both the data of
Hsu and Kim seem ta follow such a trend, except for sorne ùegree of
skewness.
Prandtl's oid theory of free turbulence (1925) offers an
alternative way of expressing the variable eddy viscosity principle.
The the ory sugge s t s that the mixi ng l eng th
I l
varies on Ly in the
longitudinal direction, and i.s proportional t o the wid th of the free
layer. i , e . ,
='
constant,
or
(4.27)
0.004
-- -
Er
----- -------
XU
--------r-----___ ------ ---
0.002
~
~
o o
0.U2
0.04
0.06
0.08
0.10
0.12
0.14
n
Figure <l.10.
Charuct ori s t Lc eddy funer Lon in plane jet
(from:
iL G. Van der Heg gc Zd j nen ) .
62
L>.
•
•
~
o XIH =
0
t.
.........
L>.
2JJ
.04
•
4Jf
o
...
589
•
.02
o
o
/.0
Scep-Height
H = 3.81 cm
o X/H= I.f
L>.
L>.
L>.
...
H
.06
•
483
...
6./ 7
• ~ ."
L>. • 0
o·
.04
L>.
0
... .........
ET
•
'7 (SC'd)
• ...
...
•OZ
...
0
'"
o
0.5
'·0
W0
Step Height
H = 2.5~ cm
Figure 4.17.
Characteristic eddy function Ln backward
facing step flow (from:
Kim et al.),
63
where
C
is a constant.
The shear stress is expressed by
T =
-
pu' VI =: -p
2
l.
laulau
-
-
1
ay ay
(4.28)
With the expression for the shear stress, the nondimensional equation
of motion is as follows:
2
.
-2af + (a + l)nf' + af
- (a + l)f' f f dn
o
Equations 4.10, 4.~5J and 4·29 constitue three different approaches in
the attempt of describing the mixingprocess in the zone of flow separa-
tion.
The constant eddy viscosity approximation of equation 4.10 has
the advantage over the other two of being mathematically less invQ!ved
ta sorne degree at the extent of being empirically estimated as shown in
(4.1).
Hoveve r , it completely misrepresents turbulent t rnn s fe r in the
highly non-homogeneou5 flo", of abrupt eÀ~ansion.
The other !WQ
approaches are inconvenient in that they cither increase the number of
terms in the equation of motion
(4.25) or they augment the degree of
nonlinearity (4.29).
An alternative ta t hi s is ta i nt roduce func t i ona l forms of the
turbulent stress similar ta (3.16) or (3,17), i n to the momen tum oO:qua-
t t on .
This met hod has the obv i ous edge cve r a l I the o the r s ta r educe
the arder of the ODE and not Lnduce further non l i.near-i t y.
As stated
earlier, no ana Lyt i ca I integration me thods for equations 4.25 and ..J..:9
are available.
Only numerical approximations can be considcr~d in this
instance, namely numerical methods capable of handling nonlinear ordi-
nat-y differential equations of a boundary value problem.
The fo l l owi ng
section deals with one such technique.
64
4.2.2
Numerical Solutions
The numerical integration of the boundary value problems
represented by equations 4.25 and 4.29 along with boundary conditions
(4.10) is not possible by mast existing methods.
In effect, a11 of
them assume the availability of initial conditions ta start the solution
procedure.
The initial condition in. the present flow situation would
be the value of the slope of the ve Loc i t y profile at the "stagnation"
point (point of zero velocity) which is not easily attainable consider-
Log the rather pocr accuracy in the measurements of low velocities.
Equations 4.25 and 4.29 under their present forms pose a structural
d i ff i cu l ty for their handling by nuraer-i ca l methods , es pec i a l l y due ta
the nonlinear terms involving integrals.
We will get rid of this
difficulty by differentiating the equation once.
This amounts to
l
letting
f(n)dn ': F(n).
Equation 4.25 in terms of the new dependent
o
variable reads:
2
-2aF' • aF'
+ [f a-Lj n - (a.I)F • KS'] F" • K8FH!
o
(4. 3D)
\\\\'ith the boundary conditions modified as fo l lows ,
n ': 0
F=O,F'=O.O
n = l
F' = 1.0
(4.31)
Equation 4.:'50 new describes the non-d Lmens i ona l ve l oc i t y profile.
The
numerical integration method most suitab1e in the evaluation of boundary
value prob Iems is an iterative procedure called the "s hoot Ing!' me thod .
Unfor-tunat e l y , when t r i.ed on equations 4.30 and 4.31, this method
proved unsuccessful due to the high nonlinearity of the equation com-
p Li.cat ed by numerical instability gener a ted by Iow values of S.
Hence ,
65
001y the special case of constant a will be treateJ here by the
aforementioned technique ~hose full description is presented in
Appendix B.
The simplified ODE i5:
2aF'
- aF,2 _ (a+l)nF" .~ (a+1) F F" "
(4. 32)
whe re R
and Cl have beert defined in pr-evi ous sections.
1f Ive i nt ro-
s
duce the following new change of variable III ~ IR
Cl (a+1) n ioto the
s
simplified (constant S) form
of 4.25 then
d i f fe ren t i a te it, l'Je ob t a i n :
'F'C,)
s F,2{, J
F"('I) • FCo, Il F"('I) '" F'" ('1)
~
(4.33)
1
- 7:
1
- ' 1 '
where B '" 2a/(a+1) as defined in the foregoing.
And the bounda ry con-
ditions now become:
III = 0,0
F=O.O,
F'=O.O
III = 1.0
F' = 1.0
(4.34)
The method of solution of 4.33 and 4.34 based on the fourth-order
Runge-Kutta procedure requîres that the third arder ODE be transfor~ed
ioto an equivalent set of three first-order equations.
This is
accomplished by defining
Whence,
(' . 35)
66
subject ta
n
= 0.0:
&1 = 0.0, &2 = 0
1
n
= 1.0:
&2 = 1.0
(4.36)
1
Starting at
nI = O. the integration of (4.36) over successive steps
fl.r}l
is achieved br the fourth-order Runge-Kutta function RUNGE using
algorithm BI (Appendix B) and a search technique ta find the appropri-
ate initial conditions for which the upper boundary condition
&2(1) : 1.0
is satisfied.
A description of the overall computational
procedure is presented in Appendix B.
The computed velocity profiles
using the described method are compared with measured distributions of
SET!, SET2, and SET3 in Figures 4.18, 4.19, and 4.20.
The agreement is
weIl within the accuracy of the measurements.
The mixing length approach of 4.28 is used next ta determine the
mean ve l.ocLt y distributions.
Using the .same procedure of transformation
of the integral differential equation to a simple nonlinear ODE as in
the above. we can rewrite 4.29 as follows:
Z
-2aF' + aF'
+ ((a+l) n - [a e L] F] F"
F'"
(4.37)
If we introduce new variables as:
ci
1/3
ci
1/3
[-2 (a+l)]
n = n
and
[-Z (.+1)1
F = F
1
1,
ZC
2C
aF1 . aF
1
a2F
--'= -
anl
an
1/3 an Z
(a+1)]
67
-
NUMER/CAL SoLuTION
•"
u
0.5
U,
o
•s:
1.0
0.5
o
Figure 4.lS.
Comparison of computed mean velocity distribution and
experimental data.
Constant eddy-viscosity approach-~
R = 22 (data of Hsu).
s
68
- - NUMERICAL SOLUTION
•Il.
•o •Il.
•
u 0.5
U,
e•
•
o
o
/.0
0.5
o
n
Figure 4.19.
Comparison of computed mean velocity distribution and
experimental data.
Constant eddy-viscosity approach--
R
= 36 (data of Chaturvedi).
5
69
1.0
•
-
NUMERtCAL SOLUT/ON
..!!.. 0,'
U,
•
8
OL
~
_=___<~
o
1.0
0>
Figure 4.20.
Comparison of computed mean velocity distribution and
experimental data.
Constant-eddy v Lsco s i t y apprcach->
R = 22 (present experiment).
5
70
3F
1
d
e~
3
2/3 dn
[ - 2 (a+l)]
2e
Hence , Equation 4.34 becomes :
:: -FI!(n )F't1 en )
1
1
1
1
(4.38)
where a = (2a)/(a+l) as hefore, subject to boundary conditions 4.34.
The equivalent set of first-order differential equations using
the following definitions:
and
g
= fll(T') )
3
1
1
is
-
Il
+ g
(4.39)
1
1
subject ta boundary conditions 4.36.
The solution of 4.39, the mixing
length approximation, is contingent upon an assumption on C. the mixing
length constant.
Measurements in mast free s heu.r flows i.ndt cut e
the value of C ta lie bet .. een 0.015 and 0.020.
ln ef fec t , For thraann
(1934) reported a value of C = 0.0165 in plo.ne jet measurements.
Kuethe (1935) estimated a (-value of 0.0174 ta ensure good agreement
between theoretical and experimental velocity profiles in fully devel-
oped round jet. while in the developing flow region the value is
significantly hi gher .
rani et al.
(1961) found C-values lying between
0.018 and 0.022 in abrupt expansion flow.
Bradshaw and Wong (1971)
71
found a significantly higher value of C (equal ta 0.030) for the same
type of flaw.
The calculations were carTied out for values of C
ranging from 0.018 ta 0.030.
The best fits for the three data sets
used (SETI. SET2, SET3) were obtained at C-values of 0.021. 0.0197,
and 0.021 respectively (Figs. 4.21, 4.22, and 4.23).
The solution technique distinguishes between two fLaw regions:
an
inner layer near the teTo velocity loci and an outer layer extending up
ta the boundary of the potential core.
The ext en t of the former has
been arbitrarily determined by the location of the inflexion point of
the velocity profile.
It is assumed that in this tail region the non-
linear terms are negligible and the solution is sought from si~plified
Fo rms of 4.35 and 4.39 re s pec t Lve l y , whcre the te rms in g; and glg3
are dropped from the third-order ODE.
The result is a strikingly
good approximation of the lower boundary profile as can be seen in
Figs. 4.18 through 4.23.
4.2.3
Numerical Evaluation of the Cross-stream Velocity
From the continuity equation, the nondimensional lQterQl velocity
has been derived as (see Appendix A):
V
-ClaT] + Cla J fdo - CI J T]f1dT]
=
(4AO)
UI
0
0
But,
J T]f1dT] = nf - J fdn
using integration by parts.
Whence,
V = Cl [van + (a+l)
(4.·11)
U
J fdn - of]
I
0
71
found a significantly higher value of C (equal ta 0.030) for the same
type of flow.
The ca l cu l at Ions were carried out for values of C
ranging from 0.018 to 0.030.
The best fits for the three data sets
used (SETI, SET2. 5ET3) were obtained at C-values of 0.021, 0.0197.
and 0.021 respectively (Figs. 4.21, 4.22, and 4.23).
The solution technique distinguishes between two fLaw regions:
an
i nne r layer near the zero veleei ty loci and an outer layer ext end i ng up
ta the boundary of the po t en t i a l core.
The e xt en t of the former has
been arbitrarily determined by the location of the inflexion point of
the velocity profile.
It is assurned that in this tail region the non-
linear terms are negligible and the solution i5 sought fram simplified
2
forms of 4.35 and 4.39 respectively, where the terms in g2 and
are dropped frorn the third-order ODE.
The result is a strikingly
good approximation of the lower boundary profile as can be seen in
Figs. 4.18 through 4.23.
4.2.3
Numerical Evaluation of the Cross-stream Velocity
From the continuity equation, the nondimensional lateral velocity
has been derived as (see Appendix A):
V
•
-Clan + Cla
(4.40)
VI
J fdn - CI J nf'dn
0
0
But,
f nf'dn '" nf - J fdn
using integration by parts.
wbence ,
v
-V • Cl [-an + (a+l) J fdn - nf]
(4.4l)
1
o
72
- - NUMERICAL SoLUTION
•
V
0·5
VI
o
1.0
0.5
'Figure 4.21.
Comparison of computed mean velücity distribution and
experimental data.
Ni x ing length app rcacb-c-C = 0.021
(data of Hsu).
73
- - NUIJER/CAL SOLUT/ON
•f>
o·
f> e •f>
•
f>
o
U
0.5
•
U1
f>
•
o
1·0
0.5
n
Figure 4.22.
Comparison of computed mean velocity distribution and
experimental data.
Mixing length approach--C = 0.0197
(data of Chaturvedi).
74
•
-
<, ,,;
NUMERICAL SOLUTION
il Q.5
'1
OL-
':-
• ~
1.0
0.5
o
n
Fi~~re 4.23.
Comparison of computed mean velocity ùistribution and
experimental data.
Mixing length approach--C = 0.021
(present experiment).
7s
In terms of the F-variable.
= C
+ (.+l)F - nF']
(4.42)
1[-.n
The computer solution of equation 4.42 requîres that it be rewritten
as:
=
(4.43)
The computed values of
V are plotted in Figure 4.24.
Data SETl
.OJ
Data SET2
Data SET3
.01
....... -.-._.-.
rï
. ------
.........
_01
Figure 4.24.
Cross-stream velocity profiles
CHAPTER V
Experimental Facility and Procedure
5.1
Description of Apparatus
The experimental part of the investigation was carried out in the
River Hechanics flume of the Colorado State University Hydraulic Labor-
atory.
A multiple-section waoden flume was built inside the 30.48 x
7.32 x 0.91 m concrete channel for that purpose.
The overal1 setup
is a recirculating open surface conduit with different sections that
will be described below.
Figure 5.1 is a schematic representation of
the overal1 view of the experimental apparatus.
5.1.1
Water Supply Line
A 12.19 m pipeline of variable cross sections (50.80 cm and
45.72 cm diameter) connects the flume ta the main slimp (one-half acre-
fû~t capacity) supplied with sediment-free Horsetooth Reservoir or
Fort Collins city water.
The pipeline is preceded by a 50.80 cm,
3/sec
125 HP, 0.68 m
cfs maximum capacity supply pump.
The flow rate
is contro11ed by a rising-stem gate valve activated by a sma11 electric
motor and is measured by a 38.1 cm calibrated orifice in the supply
1ine.
The supply line ends in a submerged manifold discharging into a
settling tank.
The supply line is sufficiently long and the terminal
manifold physically isolated from the wooden flu~e 50 as to minimi!e
vibration~ from the supply pump in the test section.
5.1.2
Stilling Tank
The influx velocities through the manifold are relatively high.
A
4.27 x 2.44 x 0.91 m head tank is designed to reùuce the velocity.
The
77
... t)ij
1
125 HP pump
,
Z
Auxil iary pump
,
3
50.60 cm pipe
,
..... lM
t,
,",..,
,
i
4
45.7Z cm pipe
.
"
5
7.32
f'
ID x 30.48 m river
; , '
1
J' '1
hi:I."h·"
,::11:1, ,,1 .. 1 ••:. ~d, .-.', ;':I:i~ l "
mechanics flume
,
, ,
" ' , "
6
Settling tank
l\\a
i,
1
,
1
,
!
9
J
1
,
5 ,11ïl
7
Adjustable converging section
,
~I
a
,
Step sect ion
!I
~'
9
Channel section
~
:
10
Measurement bridge
6
r -
Il
Tail section
1
I
• , ~ z
"
7
li ~:
'!
_..' -
,
~
,
~
6
,
,
,
,
"
4
1
,,
,
,
1
3
œ
es.,
Figure 5.1
Overall view of experimental facility.
'"
excess turbulence generated at the manifold out lets 1s dissipated
through a pebble sereen located immediately downstream of the manifold.
The relatively tranquil water enters the converging lolet section
through a 0.16 x 0.16 cm fine mesh sereen
which dissipates sorne more
excess turbulence and aiso restrains lint and particles down ta fairly
small size from entering the test section.
5.1.3
Adjustable Converging Section
The coriverging section was constructed of one-haif inch thick
plywood overlaid on an elliptical-shaped truss of rigid beams.
The
contraction was graduaI enough (radius of curvature = 7.01 m) ta pre-
vent separation.
The accelerating flaw in the converging section will
minimize the boundary layer thickness.
The converging inlet section is
4.27 m long and is prolonged by a 30.48 cm Plexiglas straight section
part of the backward-facing step.
5.1.4
Backward-facing Steps
Each backward-facing side step Nas built of a 1.905 cm thick
Pléxig1as and was 91.44 cm long.
The s t ep was pLeced in an L ta the
straight inlet section.
The ~hole inlet-step setup is laterally mov-
able sa as ta a l Iow adjus tment s of the s tep height.
Only double-step
configurations (S~TImetrical and nonsymmetrical) were used in this study.
5.1.5
Channel Section
Two 22.86
III
long x 1.905 cm thick vertical wal l s made up of
pl~iood extended downstream of the step face ta a tail-end chute that
conveys the water back to the one-half acre-foot sump.
The walls were
retained parallel to each ether by oblique trusses bolted on cqually
spaced 2 x 4 traverses.
The f100r of the whole structure was laid on
the 2 x 4 traverses and extended from the upstream end of the converging
section down to the tail-e~d section of the 2.195 m wide channel.
A
79
taîlgate supported by a driving cable around a crankshaft 50 as to
allow different positioning for water level regulation is located
91.44 cm upstream of the tail end.
A 1.219 m Cipolletti weir was in-
stalled at the hindmost end of the flume.
It was capable of measuring
3/sec.
flows up to 0.793 m
The water head on the weir wa~ recorded via a weir_gage on the
inside face of the right wall of the channel (about 91.44 cm upstream
of the Cipalletti). and a stilling weIl (10.16 cm diameter) with a
hook gage on the outside face.
The role of this tail-end flow measuring
system was ta monitor the leakage level in the flume.
Incidentally.
the leakage was maintained minimum through a meticulous stripping of
aIl joints with silicone sealant. and an overall lining with epoxy paint.
Figure S.l exhibits the essential features of the closed circuit
system including the supply line. and Figure S.2 is a detailed repre-
sentation of the experimental flume proper.
s.z Data Collection Techniques
The aim of the experimental investigation was primarily to provide
qualitative and quantitative information about the mixing process that
takes place in a channel abrupt expansion.
The investigation was
implemented through two main techniques:
1.
Flow visualization based on the turbulent mixing of colored
influx (dye) and powdered influx (aluminum) for the overall
pattern of the mixing process.
2.
: Hot-film anemometry for quantitative measurement of mean and
turbulent velocities.
A description (not necessarily exhaustive) of both techniques is
presented herein.
80
Figure 5.2.
Expe r iment a I flume.
81
S.2.1
Flow Visualization Techniques
a.
Dye Injection
n~o poraus strips were built near the sharp cdges of bath Plexiglas
steps consisting of 0.21 cm hales (4 + 4 peT strip) drilled at angles
of incidence 90° ta the oncoming flow and 45° ta the separated flow,
respectively.
Quarter-inch (0.635 ~m) plastic injection tubes were
connected ta the hales.
The whole setup constituted a multi-st~ge
dye injection apparatus a110wing alternate or simultaneous injections
at diffeTent vertical locations in the flow.
The primary purpose of
su ch a s e tup ',,;as ta compare surface and subsur face flows in the mixing
region.
No noticeable differences were recorded.
The technique, how-
ever, had obvious shortcomings when dealing with deep flow of a some-
what opaque water. or relatively high velocities.
The use of skimmed
milk in lieu of dye as injection liquid has Led to better results in
some instances (Frey and Vasuki, 1966).
Direct point injections fro~
pressurized injection tanks at the origins of the two mixing layers
helped vi sua l i ze the boundar i e s of the poten t i a l core (see Fig. 5.3).
The same point injection technique vas us ed to locate the 'rea t t achment
zones of the eddde s , where the flow is high1y unsteady.
The r ca r t ach-
ment point is therefore a statistical average roint where the fION maves
in the downstream direction most of the time.
The dye injection method was inadequate to properly visualize the
eddying motion (formation and decay of vortices) in the separation Zone
due to the fact that it would mix thoroughly with the turbulent stream,
very pr ematur e Ly.
Sorne other markers had to be cons Ide red for that
matter.
82
Figure 5.3.
Dye visualizatioo.
83
b.
Aluminum Powder
The po~èered influx technique was effectively used by Frey and
Vasuki (1966) ta study El.ow development in abrupt enlargements, bath
for unsteady and steady state situations.
When tried in this experiment,
the technique offered sorne degree of success in showing the mechanism
of redistribution of f I ow pas! the .-backward facing s t eps .
A light but
sustained sprinkle of the powder near the step edge was uscd ta visual-
i ze the formation, growth , and decuy of the flow vo r t i ce s in the mixing
region.
A line influx applied immediately upstream of the step in the
approaching section was used ta follo~ the evolvement of the mean
velocity front before and after the steps.
Valuable qualitative infor-
mation (ta be discussed later) pertaining to the overall pattern of the
flow in the zone of staIl was obtained via this technique.
5.2.2
Hot-film Anemometry
5.2.2.1
Instrumentation
Mean velocity and turbulence measurements were performed using
hot-film anemometry.
The instrumentation comprised a Thermo-Sys t ems,
Inc . (TSI) Model 1050 cons t arrt vt cmperut ur e anemometer , a T51 Node I 1076
digital true Rl\\1S/DC/~lean Square Voltmeter (see Fig. 5.4a).
The pr imary
use of the voltmeter per t a ined ta the DC output as a back-up unit for
the analog readout of the anemometer unit.
The turbulence level was
recorded by a Brüel and Kjoer type 2409 Electronic True ~\\IS-meter.
In
f l ows of high turbulence Iev eLs , as vas the case he re , this type of
RJ.\\lS-mete"(· coupled with a Hew l et t-Packur d electrosensitive s t r i.p chart
recorder allowed an casier estimation of the root-mean-square of the
fluctuating and D.C. voltages.
The sensing unit consisted of a
ruggedized TSI Madel 1231W conical probe bent 90° (see Fig. 5.4b).
rhe
84
Figure 5. 4a.
Electronic equipment.
Stainless
Steel
.06 Dia.
(1. 5)
\\
\\ ~-
~
I~t-rilm Sensor near
Tip of Cane Cee ted
~
~
~-=-ç.~
wi th In te rs t i tially
Bouded Quartz
œ
#
~
~
~
.r-:
Quartz Rad
Figure 5.4b.
Conical hot-film.
86
probe was mountecl on a support system including a 91.44 cm TS! Model 1150
probe suppor t pro t ec t ed by a 91.44 cm TS! \\Iouel 1158 Locking Sj eeve .
The
probe-support ensemble "as held by a Plexiglas mounting black tightly
screwed on a metai plate carriage ta recluce vibrations.
Point velocity
and turbulence measurements were carried out transversally simply by
moving the carriage ac ros s the flow section.
Longitudinal measur ement s
were macle possible by an electrically driven carriage (measurement
bridge) moving up and clown the whole length of the test section.
5.2.2.2
Calibration of the HotFfilm
a.
Background
Extensive discussions of the heat transfer characteristi,s of
hot-film sen so r s have been presented in Baldwin et al.
(1960).
McQuivey (1967), and Sandborn (1972).
They expressed the relation5hip
between the mean voltage output of a constant temperature hot-film
anemometer in a flow with constant properties. and the local rnean flow
velocity as:
(5.1 )
whe re :
E = mean voltage
U = local mean velocity
AI, BI. m = coefficients
Thi5 non-linear relationship represented in Figure S.J
is the
ca l i.br-at-Ion equat Ion of the hot-film.
The calibration curvc is a func-
tian of the overheat ratio K, which i5 an expre5sion of the temperat.ur~
rcsponse of the sensor material.
Thus.
1 • .c (t
-
t
)
(5.2)
5
e
87
E,Oit,
~::.4.66 n
VELOCfTY, cmr
Figure S.S.
Calibration curve of hot-film probe.
88
where
RH" operating (hot) resistance of sensor. Re = resistance of
senSOT at the temperature of the fluid (te)' ~C = temperature coeffi-
cient of resistance for the sensor material referenced ta tempçrature
t
,
and t
= sensor operating temperature.
Operation of the probe :lt
e
s
high ov er-he at ratios normally will resul t in an Lncrea se in the sensi-
tivity of the anemometer, provided ideal ~ater quality prevails, and
boiling is avoided.
b.
Sorne Causes of Inaccur acy in Data and Corrective Heasur-e s .
The main difficulties encountered in measurements in water flow
are:
(1)
Contaminants collecting on the sensor and causing a calibra-
tion drift;
(2)
Dissolved air coming out of solution ta form bubhles on the
heated sensor surface, thus capable of causing a decrease of
the anemometer output voltage- until swept away by the stream;
(3)
Temperature fluctuationS.
When the temperature of the water
changes, so does the response of the hot-film and the cali-
bration curve can be significantly affected.
(4)
Lastly, electrolysis of the water caused by e l ect r i ca l volt-
age on a poorly insulated probe surface.
The effect of the
electrolysis is to increase the anemometer voltage.
The occurrence of electrolysis of the water was not experienceJ
during the course of this investigation and will not be Jiscussed
further .,'
Contaminants are usually avoided by filtering the water to remove
aIl particles above l micron in diameter.
This could not be done in
the present experiment, but the problem was somewhat circumvented by the
89
choice of a conical probe less susceptible of accumulating "dirt" than
larger diameter sensors.
Besicles, a calibration check station was set
up in the fully established flow region of the downstream section,
where the anemometer OC output was regularly checked against the output
of an OTT propeller flowmeter in arder ta deteet eventual probe cali-
bration drifts.
The sump water was kept; as c Iean as possible by
renewing the st orage water, as needed.
Dissolved air coming out of solution or trapped air were dealt
with by lowering the sensor operating temperature.
Law sensor tempera-
ture is otherwise undesirable. for it decreases the sensitivity of the
probe response ta velocity while the temperature sensitivity becomes
more important.
The water tempe rature effects can be corrected by the use of a
temperature compensation circuitry that automatically adjusts the probe
response 50 that a constant output is obtained for ~ fixed flow rate
under. fluctuating temperature conditions.
This approach is usually
prohibitive in cost. and was not considered here.
A second approach of
temperature e ffec t r emovaI was p roposed by Tan-At Icha t et al.
(1973)
and consists of maintaining the overheat resistance difference, ~ - Re'
constant.
This involves continuously monitoring the changes in the
cold resistance Re and adjusting the operating resistance ~ conse-
quently.
Unfortunately, the measurement accuracy achieved by this tech-
n~que was not very satisfactory for the range of flow conditions con-
s i dored in this s tudy .
The me t ho d cdop t ed in the present investigation
consisted of simply calibrating the probe for a wide range of tempera-
ture levels encompassing thase most likely to prevail in the actual
experimentation conditions.
Careful extrapolations were performed for
smal~,temperature deviations.
This approach proved quite satisfactory
90
1 - - -
1
Pump
7
Hot film
2
Constant head tank
8
Free overfall to tailbox
3-
U-manometer
9
Probe supporting unit
4
Gate valve
ID
Tailbox water level
5
Inlet section
l i
Tailbox
6
0.99 cm orifice
Figure 5.6.
Schematic of the velocity calibration system.
91
as long as the temperature changes were not drastic.
Since a nearly
constant room temperature was generally maintained in the laboratory,
this was not a serious matter of concern.
c.
Velocity Calibration Facility
The calibration facility compr i s ed a cylindrical chamber ui t h a
0.99 mm diameter internally rounded orifice supplied by a steady dis-
charge from a Constant head tank.
The orifice fla ... spouted into a
second cylindrical tank with a free overflow, and thus was maintained
at a constant head.
The discharge through the orifice was regulated by
a gate valve and measured via a calibrated differential manometer with
a co Io red gage liquid immiscible in wat e r .
Figure 5.6 is a sche~atic of the velocity calibrating system.
The
centerline velocity of the jet was computed from the continuity equation
that assumes the profile across the orifice uniform.
The assumption of uniform cross-sectional velocity implies that
the probe is located in the immediate vicinity of the orifice.
However,
if it is presumed that the pressure distribution is essentially hydro-
static throughout the lone of motion, the potential core velocity re-
mains constant a I l along (Alber t son et al. > 1948) 1 and the s afe region
of measurements extends. in princip le, downstream the whole length of
the potential core, provided the probe is located in the potential core.
The length of the potential core as reported by Albertson et al., is
about
5 do. where
do
is the orifice diameter.
In any event. the
optimum calibration 1S relatively close to the orifice.
Thus. the c~li
bration velocity profiles were taken at a distance about
1.5 do
from
the orifice.
Figure 5.7 shows the velocity profiles at the location
for the range of discharges that was considered in the present
ca Lib rat ion.
1·6
1·2
oB
04
- 0.4
1.2
/6
o
20
40
60
80
/00
VêLOClry(,m/,)
Figure S.7.
Velocity profiles across the .99 centimeter
calibration orifice.
CHAPTER VI
Experimental Results and Discussion
6.1
Iolet Conditions
The experiment was performed in the 2.195 m flurne ùescribed in the
foregoing for 5 different step configurations symmetrical and asymmetri-
1
cal.
The range of area-ratios
employed W3S from 1.342 up ta 4.235. this
corresponded ta a minimum step height of 27.94 cm ta a maximum of 83.82
2
cm.
The range of Reynolds numbers
thus investigated was from 4.36 x
105 ta 7.95 x 105.
These Reynolds numbers were large enough ta warrant
the development of fully turbulent separations at the step, i.e., the
boundary layer is turbulent priar to separation, as opposed ta laminar-
turbulent separation (transitional stail) as encountered elsewhere.
Furthermore, aIL the flaw conditions herein investigated fall into the
category of "overwhe Iming" perturbation, following a definition by
Bradshaw and Wong (1972) who classified the flow over a backward-facing
step into three categories according to the strength of the perturbation:
(i) weak perturbation. h/ô «1;
(ii)
strong perturbation. h/ô ~l;
o
0
(iii) overwhelming perturbation. h/ô »1, where h is the step height and
o
Ô
the boundary layer thickness at the s tep .
In the case of "overwhe lm-
o
Lng" perturbation. the flow can be t reat ed as a fully dev e Iop ed mixing
layer flow.
The converging inlet geometry as described in 5.1.3 kept the
IThe are a ratio for double-step configuration is defined as AR =
(2Wo+hl+h2)/(2Wo) where Wo is one-half the in let channel width and hl
and h2 the height of the individual step.
~nen hl ~ h2 (symmetrical
configuration), the definition is analogous to Abbott's (14).
2Reynolds number based on the inlet channel width and the entrance
,average velocity.
94
growth of the sicle wall boundary layers ta a mInimum in the entrance
section.
As a consequence a potential core prevailed in mast of the
entrance section. as verified by bath visual and quantitative measure-
ments ta be presented in later sections.
The existence of the potential
•
core is manifested in a uniform (or nearly uniform) inlet velocity pro-
file (figures 6.1 and 6.:5a. for e xamp Ie},
6.2
Mean-velocity Measurements
6.2.1
Visual Measurements
The aluminum powder technique was used ta visualize the overall
Mean velocity fTont at different locations in tho mixing region.
Figure
6.1 shows the shape of the velocity profile at the expansion exit for
the asyrnmetrical double step configuration at a width ratio of 1.618.
Figures 6.2a, 6.2b
and 6.2c represent the velocity fronts at three
J
different locations in the expansion.
ln Figures 6.3a. 6.3b, and 6.3c
strong interference of the largest eddy on the other results in an
obviously dissymmetric profile.
Note that the dissymmetry in the profile
1S primarily due to the interference between eddies and is not directly
related to the dissymmetry in the step configuration.
ln other vcrds ,
two perfectly symmetrical steps will generate an aS~TImetric velocity pro-
file provided the area ratio is large enough ta induce mutual interaction
between eddies (see Figure 6.4).
The area ratio of 1.5 is reportedly
the threshold value beyond which eddy coupling occurs.
6.2.2 .Quantitative Measurements
Local mean velocity measurements were performed with the hot-film
probe in the transversal, longitudinal. and vertical directions.
Trans-
verse measurements were taken every 13.97 cm.
The longitudinal stations
were chosen according to the step heights. and in general an incremental
95
Figure 6.1.
Flow visualization-velocity front
near the step interface (AR = 1.618).
96
Figure 6.2.
visualization-velocity fronts at successive locations
AR ~ 1.618 •
a
b
c
91
Figure 6.3.
Flow visuaiization-eddy coupling effects {AR = 4.235).
a
c
98
Figure 6.4.
AS~lliTIetric vetocity front ~n a
symmetrical step flow.
99
Iength of 1.5 h was us ed .
In case of a mixed configuration. the
sma Lles t step height was used as the i ncr emen t a I unit until r-ea t t achment
of the corresponding sma l Ies t eddy, chen the largest step was used as
p
i
i
t
t
t h e
r e v a
l
n g
u n i t ,
h e r e a f t e
r
•
I n
t h e
v e r t i c a l ,
m e a s u r - c m e n
s
w e r
e
t aken ever y 7.62 Cm s t ar-t i ng 2.54 cm be l ow the surface clown ta 2.54 cm
above the hed, in mast cases.
The v€rtical
measurements ~ere intended
t.o show the ex tent of the deviations from a twc-d imens iona I f low behav i or
since the analyses of Chap t er IV l'lere en t Lre l y based on the C',yO-
dtmens i ona I flow as sumpt i on .
le was Found chat the dev i at ions from
two-dimensionality in mast flcw situations l'lere not critical (except
in the reattachment region, exc1uded from the ~na1ysis for thi3 reason,
among others).
The ~ransverse ve10city measurcments failed to encompass
the zones of low ve10cities and return f1ow, because of high instabilÎ-
t i es
tuere , and sene unavo i dab le i nadequacy in the experimental s e t-up ,
that made it difficu1t ta access these regions.
Under these circum-
stances, only forward-flow data could he secured with reasonab1e accur-
acy.
Figures 5.S t hr ougb 6.l0 show the s e ve Ioc i t y measu remcut s for a l I
the s tep configurations (excep t AR :: -1.235, wher e hot f i ue measur ement s
c
wer e hindered
by high l eve I of sur-Face ins tab i Li t i es j .
Figures 5.11
1
t hrough 6.14 are t yp i ca I vcr t i.ca I profiles for half the f l ov \\dc.lth.
The profiles are shawn ta be s ynmet r i ca I wh en eddy coup Ling ef fec t
is
absent, and strongly dissymmetric l'hen the step heights are increased.
whi l e the potentia1 core rcmc ins ncarLy straight. and per-s Lst s cve r a
considerable ùistance in the case of uncoupled eddies, it is relativ~ly
short and def l ec t ed in high step configuration flONS (Figure 6.15).
IThe other half was omitteù because similar in character.
100
Q<
c
J.O
•
XIH= 0.0
,
0
l'
X /Ho
.6.
X/H=JO
0
X/H:4.'
'" $.
~
• X/Ho 60
"l
xlHo 7.'
••
J·O
,
2·0
Figure 6.5.
Hean velocity pr o f i l es (AR'" 1.342, DT=?l . ..!..') cu) .
101
,
-~
--- --.
(0
--- ~ c--,
-
•
--.,.,~o \\J
e.o
Id
9
J.O
• XI H: 00
..
0
oç
a
A/il= /,5
,
\\
,
• X/H=.J.O
1
t:
a
»[» = 4.;
• KIH = 60
,
'il
xlH = 1.5
).0
a
lA
20
\\t
"
·0
.
.
~ 1
~
'=
v
0
~
o
,
0.2
0.4
06
0.8
r.c
U
Uo
Figure 6.6.
i t y
f i
(AR ~ 1.,).12, DT=,~6.W
N e a n
v e
L c c
n r o
l
e
s
c
m } .
102
1·0
2.0
•
XJ.-I ~ 0·0
0
X/H = 1.$
J·O
...
X~ ~ JO
0
XIH ::::4.,
,
• X/H=6.0
>:
~,
xl~ ~ r.s
'"~ >:fH == 9·0
o
).'/ H =: /0.5
J·O
• X/H :::: 12.0
o
X/H ='J.5
~
00
'"
~0'
?--:::4' o
0
la
.;~
o
0.2
0.4
U
0.6
08
/.0
Uo
Figure 6.7.
~lean velocity profiles (AR'" I.GIS. Dr = 71. ~5 cm).
103
, "
,
, , -......-....
.......
-
,'"
..-............ ;:::. '''',
.-
J·O
"
Z·O
• X!H=O.O
o
X·'H:::f.5
,
J·O
4
xiH .:: 3·0
X/H::4.5
0
,
• X)H=6.0
~,
"7
X!H=l5
• X/ri =9.0
o
X IH::: laS
• x /H =/2.0
J·O
X/H=13.5
1')
z·
•
0 4 "
'·0
0
.."
; /
•
o
04
0.6
0.8
1.0
U
Ua
Figure 6.8.
Mean ve Loc i t y profiles (AR'" 1.618, D
cm).
1=46.69
lU4
1·0
z.
JO
•
X/H =ao
a
xl H =I~
•
XfH =3.0
~(
0
X/H =4.5
• X/H =6.0
'"
X/H = 7.$
~
X/H ::. 9.iJ
J.O
0
X/H=I0.5
2.0
1.0
o
o
0.4
0.8
1.0
U
Ua
Figure 6.9.
Hean velocity profiles (AR = 2.038, D = 46.69 cm).
T
"0
2()
J.()
J-()
•
XfH = 0.0
0
XIH = 1·5
.. X/H =J.O
2.0
0
X{H =4.5
• X/H ='.0
'Cl
X/H =T-5
• X/H =9.0
1·0
o
X/H =/0.5
• X/H=12.0
o
X{H ~IJ.5
0
0.2
0.4
il
O.s
0.8
1.0
Uo
Figure 6.10.
Mean ve Ioc i t y profiles (AR = 2.752. D
cm).
T=71.69
106
w.s
o
aB
o
z
·0,4
Q2
e
e
o
0.2
0.4
0.6
0.8
1.0
u
U,
Figure 6.11.
Vertical vclocity profile near step interface
(AR = 1.342, y/H = 1).
107
w.s
o
08
o.s
0.2
o
Q2
0.4
o.•
0.8
,.0
il
U,
Figur~ 6.12.
Vertical velocit, profile near step interface
CAR = 1.342, y/H = 1.5).
lU8
IlIS
o
o
a2
il
V,
Figure 6.13.
Vertical velocicy profile near step interface
(AR = 1.342, y/f l = 2.0).
109
'"s
as
o
a6
02
p
J
o
0.2
o.•
0.6
0.8
1.0
U
U,
Figure 6.14a., Vertical ve Ioc Lry profile ne a r s ceo i n t e r fuc c
(AR" 1.342, y/H' l.5).
uu
11(5
(0
o
.0,4
a2
b
Figure 6.14b.
Vertical velocity profile near step interface
(AR = 1.342, y/H = centerline).
III
Figure 6.15.
Potential core shadowgraph in hlgh
step configuration (AR" 2.038).
IlZ
The strength of the mutual interference of the eddies was observed ta
be not ooly a function of the area ratio but of the flow rate as well~
i.e., the mixing rate.
The effect of the flow depths on the mixîng
proc es s was i nves t i.gat ed ,
This vas clone systematically in a l l the
exper iment s by selecting two different f Iow depths for a constant dis-
charge, ta yield Fraude number ratios of 2: 1.
In a11 cases. higher
mixing process manifested in higher degree of inst3bility (mutual tnter-
ference) of the eddies vas ob served at higher Fraude nuraber s .
In order ta secure information on the Tetum flow zone that was not
obtainable in the 2.19 ID flurne. parallel experiments were performed in a
smaller 60.96 cm flurne where a plexiglas scaled model of the converging
section was built.
Only one step configuration (symmetrical AR := 2.0)
was used.
The results are shown in Figures 6.16 through 6.17c.
The magnitude of the reversed flow ve10cities as reported by others
for sing1e-step flow (Hsu, 1961; Kim et al., 1978) is of the order 10 ta
25 percent of U
•
ln the present case of double-step flow, this was
_
max
true for the reversed flow of the largest eddy, but the order of magni-
tude was significantly higher (up to 32%) in the smaller eddy.
However,
it should be pointed out that the uncertainty in the measurement is high
in this region of flow.
Figures 6.16 through 6.17c clearly illustrate the overall pattern
of the separation and reattachment process.
The length of the short est
eddy is equal to 4.5 times the height of the corresponding step, and
about twice that length in the larger cddy (8.75 h at Froude number :=
0.16 and 10.0 h at Froude number:= 0.32).
After reattac~~ent. the velocity profiles readjust gradually until
recavery of their normal boundary layer shapes.
The distance required
113
-------------
'.0
20
J.O
• X/H =00
0
XIH = '.0
... X/H =2.0
~'
0
XIH =J,O
• X/H =4.0
"7
X/H .4.5
J.O
20
',0
0.2
OA
0.6
0.8
/.0
JL
(Jo
Figure 6.16a.
Mean velocity profiles; AR,. 2.0, Dr = 50.80 cm
(experiment in 60.96 cm flume).
114
1.0
20
Ji)
• X/H = $.0
0
X/H =5.0
...
~{
X/H
=- 7.0
0
X/H
=8.0
• X/H ="'''
Ji)
2.0,
la
o.,
0·6
0.8
1.0
v
Va
Figure 6.16b.
Mean ve Ioc i tv pr-of i Ies , A.R ::: 2.0. 0T.:::50.80 cm
(experiment in 60.96 cm flume).
Il S
Z.Q
~o
o
X/H =: .9.0
?-[
• XIH =: KJ.o
>;
'" X/H :::/J-O
~o
20
/0
_.2
a
0.2
0.4
0.6
0.8
/.0
Lu.
figure 6.16c.
Nean ve l oc i t y profiles, AR:: 2.0. DT "'50.80 cm
(exneriment in 60.96 cm flume).
!lb
1.0
2.0
.<:17 ...
/0
o
0.2
0.4
0.8
1.0
U
Uo
Figure 6.17a.
Nean ve l oc i t y profiles, AR = 2.0, 0T=33.02 cm
(experiment in 60.96 cm flume).
117
e
1.0
20
JO
.. XjH =5·0
€I
XIH =60
~r
• XjH =7.0
"-
o
XIH =/3.0
• XIH =9·0
JO
20
/.0
o
O,,
0.8
1.0
il
Uo
Figure 6,17b.
Hean ve loc Ltv profiles, AR ::. 2.0, DT = 33.02 cm
(experiment in 60.96 cm flurne).
!lB
/·0
2.
J.O
<1>
X{H. /0.0
XIH
= /l0
•
J.O
2.0
/.0
o
0.2
o.,
0.6
0.8
.1.0
.s:
U.
Figure 6.17c.
Mean velocity profiles, AR = 1.0, DT = 3:).02 cm
(experiment in 60.96 cm flume)
119
for total shape recovery 15 a function of the entrance conditions and
the strength of recirculation in the separation flow region. which deter-
nines how far downstream the decaying, large wake-like eddies persist.
3ince attention was primarily focused on the fully separated region of
the step flaw, in this study it is impossible to pravide mOre detailed
information regarding the recovery zoné.
6.3
Turbulence Neasu rement s
The technique of evaluating velocity turbulence fluctuations from
the voltage fluctuations (anemometer output) has been fully described
in McQuivey (1967) and in Sandborn (1972), and only a brief reminder
will be given here.
For a film sensor oriented perpendicular ta the mean flow, the
relation between the voltage and the velocity is given by:
(6. 1)
where
e.::;:Je. 2,
root-mean-square (RHS) of the voltage fluctuations
u'=Jurz , RM5 of the ve10city fluctuations
dE
dU ~ sensitivity.
Equation 6.1 assumes a linear sensitivity of the calibration curve over
the range of the voltage and velocity fluctuations.
The technique
involves determining the sensitivity dE/dU graphical1y, as the tangent
ta the calibration curve (Figure S.Sa).
From the plot of dE/dU vs. a
(Figure 6.18), the
. . .
dE
b
sens i t i vi t y dU can
e obtained, then u 1 calcu1ated
from equation 6.1.
A short version of the procedure as used herein was
given by Sandborn (1972) and simply amounts ta p1acing the measured
120
magnitude of e' symmetrica1 ab ove and be10w the corresponding OC
output E.
The value of u' is derived from the projections to the U-axis
of the e' bandwidth on the calibration curve (Figure 6.19).
IT
Figure 6.18.
Hot-film sensitivity curve.
-----~---------- 1
____ ye,2
_
E
1
1
_____ L_
1
1
1
1
1
1
:
1
1
j"
1
1----
',1 , -
-----.1
1
1
1
1
1
1
1
1
1
1
1
1
1
U~
1
1
1
l-tean ve lod t y
Figure 6.19.
Evaluation of velocity fluctuations
from a hot-film anemometer output.
121
A typical profile of the turbulence intensity u'/U abtained in the
lresent experiment is as in Figure 6.20.
The turbulence intensity is
rery low in the potential core regian, and increases sharply (ta reach
:nfinite values, in principle) as the line of zero velocity is approached.
~ctually the turbulence levei u' (or u'/U ) is a better representation
o
)f the turbulent activity in the mixing regian, because expressing an
lbsolute magnitude and therefore making longitudinal variations more
)bvious.
Figure 6.21 shows the turbulence level u' in the 55.88 cm
symmetrical configuration flow.
The maximum u' lies aIcng the line of
Daximum shear that coincides initially with the dividing streamline,
then gradually diverges outwards and shifts back inwards past the zone
of reattachment (Figure 6.21) to adjust to the developing wall boundary
layer.
6.4
Dverall Characteristics of the Eddies
Some of the structural characteristics of the standing eddies
were made more or less obvious by the visual investigations.
They
relate to the dimensions (reattachment length) and the typical phenomena
that make up the eddying motion.
The dye influx is applied from downstream in successive trials
until the point of reattachment is discovered that is characterizeJ by
an oscillating flow alternately backward and forward.
The length of
the eddy determined in this fashion was equal to 7.5 h when the eddies
were uncoupled (symmetrical ~ = 1.342), and 4.5 h and 8.73 h to 10 h
for the smal1est and largest eddy respectively in the caSe of interfering
symmetrical configuration (AR = 2.038).
The eddy is made up of three distinct zones evidenced by different
activities traced by the aluminurn powder influx.
122
'.0-
J.O
2.0
1
1.0,
o
o.t
OZ
o.s
0.4
05
Figure 6.20.
Cross-sectional turbulence intensity variations -
example profile.
l23
,
/.,
•
•
•
•
•
•
•
•
•
•
•
o
..
.~
•
o
-
••
's
••
•
•
•
•
•
•
•
•
•
•
•
•
•
•
, s
o
~
•
•-
••
•
••
, Figure 6.21.
Turbulent velocity profiles along the channel.
124
(1)
A corner vortex contiguous to the step face where the
nacroscopic motion is a mere circulatory one.
A sustained transfer of
)owder particles from the corner vortex ta the through-flow is noticeable.
~vide~cing the entrainment process of the flow.
The length of this
:orner eddy zone is of the arder of the step height.
(2)
The maln body of the eddy is· an aval cell of fon~ard-backward
loving fluid.
The forward flow sicle is characterized br strong vortices
:enerated at the step edge growing in si~e as they proceed downstream
Figure 6.22).
I~hat happens to the large eddies in the appr-oach i ng
ayer at reattachment is still an abject of controversy.
Bradshaw and
ong suggested that they were tom into two parts (shear layer oifurca-
ion); others believe in the hypothesis of the large eddies moving
Iternately downstream and upstream without splitting~ with the possibil-
ty that bifurcation might be in arder sorne of the time, in a random
ishion.~
No attempt to confirm or infirm either hypothesis was made
1 this study since interest was mainly focused on the mixing process
~fore reattachment.
Only a few visual observations (by no means con-
.usive) were made. that seemed to indicate a definite change in eharac-
:r of the eddies both in size (smaller) and in trajectory.
The size
·duetion seems to give sorne indication of bifurcation of the eddies to
me extent.
(3)
The reattachment zone is marked by high instabilities.
The
rbulence level increases as this region 1s appraached. then decreases
ereafter, ln .consistency with pr-evi ous i nve s t i gat Lons (Hsu, 1960;
adshaw and Wong. 1972; Kim et al., 1978).
3radshID~ and Wang (1972)
im et al. (1978) quating Chandrsuda (1972).
lZS
Figure 6.22.
Shadowgraph of vortices in the
developing flow region.
126
explained that a change in the turbulence length scale was induced as
a consequence of the aforementioned bifurcation phenomenon, causing
the change in the turbulence leve1.
OIAPTER VII
Conclusion
A theoretical and experimental study of the mixing process in a
channel abrupt expansion flow has been pre sent ed .
:oiost efforts we r e
primarily directed toward understanding the mean and turbulent f l ow
characteristics in the transitional section until reattachment of the
bcunda ry layer to the channel side wal Is .
As a consequence, very Little
attention was devoted to the relaxation (redevelopment) zone leading to
the fully reestablished boundary layer flow, although sorne "memory" of
the separated flow persists throughout this region.
A summary of the
main results of the study is given herein.
7.1
Summary and Discussion of the Theoretical Analysis
The boundary layer type equations governing the flow of fluid in
a tw~dimensional abrupt expansion has been derived from the general
Navier-Stokes equations through an order of magnitude analysis.
The
flow has been treated on the basis that high Reynolds numbers prevailed
and that the commanding shear forces were the Reynolds stresses.
The
effect of pressure variations along .1 transverse direction has been
assurned negligible.
As a consequence, the longitudinal pressure vari-
ation was derived from the potential flow approximation.
The existence
of a potential core where the velocity is believed to vary with longi-
tudinal distance onLy accounts for sorne of the simplification that made
the ana lys i s tractable.
The ana LysLs itself cons Lst ed in the applica-
tion of the self-preservation princip le to the developing flow.
This
approach had been inspired by the fact that in three different instances,
128
mean velocity data collected 3t different cross-sections in the
developing fLaw ~egion had been made ta fall along a single average pro-
file, in each case.
Thus, the partial differential equations describing
the flow were transformed into ordinary differential equations less
difficult to handle in mast instances.
A representation of the turbulent
shear stress ~as needed in arder ta resolve the closure problem.
Phenomenological concepts were invoked whereby the stress was related
ta the rate of strain {au/ay).
Bath the eddy vi scc s i t y appr-oach and
the mixing length approach were considered in the study.
The constant
eddy viscosity assumption along with the "small de fec t '
as sumpt Icn led
ta exact solution of the Gaussian error t}~e, in fairly good agreement
with the experimental results.
Numerieal approximations based on the
"shoot i ng method" of solution of ODE, ve re deve Ioped in the cas e of
constant eddy viscosity and mixing length approximations.
The agree~
ment with aetual data was satisfactory, for aIl practical purposes.
The-general variable eddy viscosity approximation was also reviewed.
but an attempt to extend the nurnerical model ta this situation failed
due ta numerical instabilities e~perienced at very low eddy viscosity
values.
The advantages of the model developed ln this study are twofold:
(1)
It 1S r e Lat t ve l y simple.
The c Efec t of the flow g eomet r y
that makes most finite difference methods ill-fitted or difficult ta
adapt is not present here.
(2)
It Ls fast and reasonab Ly accurat.e .
Once a "good" initial
guess is available. the solution converges relatively fast to the true
value.
The main inconvenienee is one primarily characteristic of most
iterative methods:
the neces s i t y of a "g ood" initial gues s .
The method was not tested for a ~road range of step flows.
A
strong eddy coupling that would induce a crucial deflection of the
potential core ta the extent ta cause the mixing layer spread rate
dl /dx ta be not cOnstant, would cause the model to fail.
o
7.2
SUMmary and Discussion of the Experimental Tnvestigation
The large seale and dt svncnetr l c S'tep configuration ef fec t s wer-e
investigated for flow in channel abrupt expansion.
The main observa-
tians revealed no fundamental difference with previous results.
Namely,
(1)
The turbulent flow with a strong perturbation was roughly
independent of Keynolds number. the boundary layer thickness and step
height, for flaws of equal depths.
(2)
The eoupling phenomenon manifested in disymmetric eddies even
for symmetrical step configuration was not observed for the smallest
area ratio investigatcd of 1,342, conforming to previous results br
Abbott (14).
A test of verification of the threshold value of 1.S
b6yoncl which eddy coupling takes place as observed by Abbott in the
symmetrical step configuration flow was not attempted.
However, two
srmmetrica1 configurations of are a ratios 2.04 and 4.235 were investi-
gated, and in bath cases eddy interaction *as observed.
For a non-
symmetrical configuration of area ratio 1.62, the eddy coupling was
apparent and the length of the smallest step eduy shortened consider-
ably (about 4.5 h instead of 7.S h fOr the undisturbed eddy) , although
the largest eddy did not seem ta have elongated considerably.
lt seems
3.S
though.-the l arges t eddy was the i nt eract i ng one, and that the small-
est eddy was being interacted upon.
~~ile in the case of symmetrica1
configuration, the long and short stalls could be interchanged by a
mere deflection of the oncoming flaw at the step face toward the long
130
sta11 (Abbett, 1961), thi~ could not be achieved in the case of
asymmetrical step for the range of flow explored in this experiment.
For the symmetrical step configuration, the highest values of velocities
were experienced in the shortest staIl sicle in conformity with previous
investigations (Figures 6-16a through 6-17c).
In the case of disymmetric
geometry. the ~easurements performed here seemed to indicate that this
was still the case.
However. the range of configurations considered
was tao limited. and more asymmetric combinations should be explored
for more conclusive remarks.
The effec! of flow depth On the ~haracteristics of the staiis was
investigated through a range of depth-velocity combinat ions for Froude
number values varying from 0.10 ta 0.43.
The observations seemed to
indicate no noticeable change in the length of the small eddy. but the
larger staIl appeared to elongate somewhat in aIL cases explored.
Measurements in the return flow regions tended ta indicate slightly
higher magnitude of velocity for double-step configuration as compared
to single-step flows of previous investigations.
7.3
Recommendations for Further Investigations
The dynani.c s of the mix mg region in a channel abrupt expansion
1s of a very complex nature, and despite a few laudable past endcavors.
it is still not weIl understood.
In the present study, a simple compu-
tational method has been devised that predicts with adequate accuracy
the mean velocity profiles in the two-dimensional mixing region before
the reattachment zone.
However. both the simplifying assumptions intro-
duced in the formulation of the different mode15 and the numerical pro-
cedures, have inherent shortcomings that account for the discrepancies
between theory and experiment.
The pheno~enological theories as used
1
!
131
,
frein to formulate the shear stress may not be the best approach in
.e present case of fully stalled flaw.
A simplified approach was sug-
~sted by Sandborn (private communication) that would neglect the
,
~lative contribution of the turbulent shear stress. Appropriate modi-
i
tcations of the present numerical method need be introduced to handle
he lower arder nonlinear ordinary differential equations thereby gener-
,
j~ed. Further efforts should be directed toward extending the mathe-
1
~tical treatment of the mixing fLaw to the reattachment zone and the
,
)elaxation region where the developed self-preservation argument does
,
lot hold.
!
Regarding the experimental study. and provided that a more suîtable
~d reliable instrumentation is available. detailed measurements should
~e made in the recirculating region for a better understanding of the
ntrainment process.
More asymmetrical step configurations should be
tudied to further understanding of the flow pattern and especially
o determine the threshold are a ratio that induces eddy coupling.
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Chaturvedf , H. C.
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j
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Pl et.che r , R. H.
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Prandtl, L.
"Ueber ein neues Formelsystem fiir die aus geb i.Lde t en
Turbu l.enz ;"
Nachr . Akad. \\Jiss. Gottingen, Math. Phys . , KI.,
pp. 6-19, 1945.
24.
Richardson, E. V., and McQuivey, R. S.
"Heasur ement of Turbulence
'in vate r ."
ASCE Journal of the Hydraulics Division, Vol. 94, No.
HY2, pp. 411-430, 1968.
25.
Rotta, J. C.
l'Turbulent Boundary Laye r s in Incompressible Flow."
Progress in Aero. Science, Vol. 2, pp. 1-219) Per.gamon Press, 1962.
26.
Sandbo rn , V. A., and Lin, C. Y.
"On Turbulent Bounda ry L.lyer
Separation."
Journal fluid Mechanics, Vol. 32, Part 2, pp. 293-304,
1968.
27.
Sandborn, V. A.
"Resistance Temperative 'I'rensduce r s . Il
Het ro Iogy
Press, Fort Collins, Colorado, 1972.
28.
Schlichting, H.
"Bounda ry Layer Theory."
Six Edition, HcGraw-Hi l I
Book Company, 1968.
29.
Tan-at.Lcha t., J., Nagib, H. M., and Pbus te r , J. W.
"On the Inter-
pretation of the Output of Hot-Film Anemometers and a Scheme of
Dynamic Compensation for Water Temperature Variation."
Proceedings
of the Third Symposium on Turbulence in Liquids, University of
Missouri. Ralla, September, 1973.
136
30.
Tani, 1.
"Experimental Investigation of Flow Separation Ove r a
Step."
Boundary Layer Rese a r ch , Freiburg Symposium, Sp r i nge r ,
Germany, 1957.
-
31.
Tovnsend , A. A.
"The Structure of Turbulent Shea r FloW'."
First
Edition, 1956, Second Edition) 1976, Cambridge Honog r aphs on
Hechanics and Applied Mathematics, Cambridge University Press,
Cambridge.
32.
Van Der Hegge Zijnen. B. G.
"Hea su rement.s of the VeLoci t y
Distribution in a Plane Turbulent Jet of Air."
Applied Scientific
Research. Section A, Vol. 7, pp. 256-276, 1958.
33.
Wygnanski, I, and Fdedl.e r , H.
"Sorne Hea surement;a in Se l f-Pr ese rv-
iog Jet. rt
Journal Fluid Mechanics, Vol. 38, Part 3, pp. 577~612.
1969.
137
APPENDIX A
Al.
Nondimensionalization of the Equation of Motion
The momentum equation (Equation 3.13a):
The continuity equation (Equation 3.13b):
oU
àV
- + - " D
ax
êy
Let Ü .. U + u
f(n)
l
o
z
u'v' .. qo g(n)
where u
= u (x) and Q
= Q (x) are mean velocity scale and turbulent
o
o
0
0
velocity scale respectively. and n = yi! ; i
= i
(x) 1S a length
o
0
0
sca Ie .
The momentum equation is rewritten as:
a
- a
[UI + U
f(nJ]
v - [UI
f(n) J
o
+ U
ax
ay
o
dUI
a
2
= U - -
[qo g(n)]
1 dx
ay
After expansion:
du o
H(,)
dx
f(n)
ax
du0
2
af(n)
af (n)
+ U
f2 Cn) + U
f(n)
+ V u
0
dx
0
ax
0
ay
dUI
2 ag (nl
= U1 dx - qo ay
But by applying the chain rule of differentiation:
138
a(ylt )
dl.
o
n
0
ax
:ç dx
and
acr
also
==
ay
an
-.
ay
Introducing those values into the equation:
du
Uju
dt
o
o
0
dX
fCn)
----r- dx l'l f' (n)
o
2
u
dl.
u
o
0
r - n f f ' + vrf'CTl)
o dx
o
2
. - qot: g' (n)
o
where the primes signify differentiation with respect to 11.
From the continuity equation:
avaIT
- =
ay
ax
or
aV
a
dU
du
j
a
- =
+ U
=
- u ax:
ay
[U t
f (n) ]
0
f(n)
dx
- dx
f(n)
ax
0
0
dU
du
u
j
o dt
o
o
=
dx
- dx
f(n) + - - n f'
t
dx
0
Integration with respect to y:
y dU
du
Y u
dl
V :: - f
0
o
d/ dYl
Jdx fdYl + J-ç dx Tl fi dYl
0
0
dUt
du
n
o dl.0 r
[ fdn +
l'lf' d11
V
_ 0 t:
u - -
. - I)l.° dx -
odx
dx
0
139
Substituting in the momentum equation:
dU
du
l
u "i dl
0
o
0
u d
f(n) + U
dx
f(nl
l
-r- dx n f' (n)
o x
0
2
du
u
dl
uodU 1
+ U
--"-
dx
é (n)
0
0
f f' -
n f ' (n)
0
r dx n
dx
0
u 2 dl
n
duo
In
I
u -
f'
fdn + --.!è _0_ f'
nf' dn
odx
t
dx
o
o
o
2
. - qor g' (n)
o
Rearranging.
du
u
dl
(_0 f
o
0
f')
- - - n
dx
t
dx
o
2
du
u
dl
o
--.!è __
I
+ U
0
f'
(-n f
nf'dnl
c <lX
t
dx
+
c
2
qo
_ 0 f"
l
du
- u
fd", ::
dx
r g' (nl
o
o
The expression within parenthe se s in the faurth term,
(vnf + J nf'dnl
can he reduced by using an integration by parts.
Let
o
u = f
-+-
du = f'dn
dv = dn -+- V = Tl
But
JUdV = uv - J vdu
Therefore.
J fdTl = Tlf - J nf'dTl
The equat lan becomes :
140
dU
du
u
dt
1
u
(f - nf') + U
(
0
f
0
0
-
- r - nf' )
0
dx
1
dx
dx
0
2
du
2
u
dt
du0
r
+ U
--"- f
o
0
fi J fdn
U
f'
fdn
0
dx
r dx
0
dT
J
0
2
qo
r g' Cn)
o
or
du
u
dt
Cf - nf') + u
(_0 f
~ __
o nf')
1 . dx
t
dx
o
du
2
du
_ _
0
f
o
+ U o dx
dx ) f' J f dn
2
qo
=
-
g'
t o
The coefficient of the fourth term can be rewritten as follows:
2
u
dt
du
u
d Cu i
)
o
0
o
. . - - - + u
---.2. ""
0
0
.(.
dx
0
dx
.i
dx
o
e
Finally the non-dimensional equation of motion reads:
u
di
...2. __
0
f')
.t
dx
Il
o
du
2
u
dru t )
J
_ _
0
f
+ U
.,(2-
0
0
f"
f cl ... '"
o dx
c
dx
"
o
A2.
Nondimensionalization of the Energy Equation
The turbulent kinetic energy equation for the two-dimensional
flow i s :
-
- a
-:ï
-:i"
C.!. q2)
+ li L
au
av
C.!. q2)
-r-. au
U -
+ U
- + V
- + uv -
x 2
ay 2
ax
ay
ay
a
1 -
1 2 : )
+ -(-pv'+ - q v
+ E • 0
ay p
2
141
Let us define additional similarity distributions as follows:
2
tIV'= qo g(fl)
-
2
2 = q
g (n)
q
0
r
-:z
2
u
=
( n)
qo gs
;r
2
v
=
g Ln)
qo
t
1 -
1 -2-,
- pv' + - q v ""
p
2
1
3
< • t
qo h(n)
o
Substituting ioto the turbulent kinetic energy equation.
,
[U
+ U
f(n)]
1
o
'x
g(nl]+V
t:
aV
- +
ay
3
,
3
qo
+ 'Y (qo ken)] + r
h(n) = 0
o
or,
,
2
2
U
dQo
U
- dt
u
dq
1
1 qo
0
+ - - - g
-
+ -2. __
0
f gr
Z dx
r
"'2" r dx
fl s;
2
dx
o
2 dt
2
u
dU
o qo
0
qo
l
duo J
Z r
f g'
fd r,
dx
n
+
t:
sr: g~ [-nt dx - t
0
0
dx
0
0
2
udt
dU
du
dt
o
qo
0
2
l
+
f
2
0
nf' dn] + q
f gs-
o
f'
~gs + qo
- u
dx
dx
0
dx
t
1'1
g s
0
0
u
dU
du
u
dt
0
2
2
l
fig
0
+
g t (-
f + -2. __
0
f "]
+r qo
qo
d"X - dx
t
dx
n
0
0
3
3
qo
qo
+ T' kT + rh = 0
0
0
1
1
,
142
earranging.
u
dq2
o
0
+ -
-
fg r -
2
dx
u
dt
o
0
t: dX nf')
o
The second t.e rm within the bracket i.s eC\\.ual ta:
dt0
f
dto
= u
u
- - 1lf ... nf') dn
0
dx
n
o dx
The equation 15 rewritten as:
d 2
2 dt
1 dU
U
I
2
'0 + .'g )
1
qo
qo
0
r _.
-
t
ng')
2dX
(l1g~ -
qo
" 5
2" (dX gr
dx
r
0
2
2
u o 2
u
dq
1 q
d(u t. )
o
0
0
f' g
o
0
f
g' 1f dn
+
I r
+ r
qo
2d:X gr
dx
r
0
0
u
dt
o
0
2
nfg' + q
(g
-
r -
nf')
dx
r
0
s
o
APPENDIX •
NUMERICAL ALGORITHMS
81.
The Shooting ~lethod
The shooting method reduces the solution of a boundary value
problem ta the iterative solution of an initial value problem.
The
approach involves a trial-and-erro~ procedure.
That boundary point
having the nost known conditions 15 selected as the initial point.
Any other missing conditions are assumed, namely the s!ope of the curve.
The equation will then he solved as an initial value problem u5ing the
fourth-order Runge-Kutta method ta he described in the next section.
Unless the computed solution agrees with the known boundary conditions
(mast unlikely on the first try). the initial conditions are adjusted
and the problem is solved again.
The process is repeated until the
assumed initial conditions yield. within specified tolerances, a solu-
tion that agrees with the known boundary conditions.
A half-interval
~ethod of root-finding is uscd to estimate the optimum value for the
initial condition.
The shooting method is best described by these lines from C. F.
Gerald (1973):
" . . . it resembles an artillery p rob I em ,
One sets the eleva-
tion of the gun and fires a preliminary round at a t~rget.
After
successive shots have straddled the target. one ze roes in on. it
by using intermediate values of the gun's elevation."
B2.
Fourth Order Runge-Kutta Method
The Runge-Kutta methods are one-step procedures which involve only
first-orde~ derivativcs evaluations.
They produce results equivalent
in accuracy ta the higher-order Taylor formulas.
•
éNTRy
m-m+1
..
m
' - -
-
'llo TO
r>.
PASS
1
RETURN
\\.~.1
-,
1 .1
ï
- - - - - - - - - - - - -
- ï
-
~
..1.
} . ( . _
} - .
Y..,.-YJ
r
>.
r>
( 2
J-I,2.
'/>J-f.
H 22 )- x- x't ---..{RETURN
' - . /
... n
YJ -Yaay.+.!lFJ
./
\\.
1 .1
• 2
' - -
r ---------------,
~
..i.
••
3
r< 1 1 >--
k
/
-,
~
J: 12
,/>, -q.,'2fJ
33
RETUR~
\\ . . 1
... n
y. - y
• nf·
"
UV~
2 J
\\.1.1
- .
----
' - -
r----------------·
J.
..J-
~~ -flJ + 2F,j
~
X J:I,2.)-.
---..{
44
) - .
x-x,.!!.. HRETUR~
'"
4
.)
... , n
Yi -Y"vJ + hFj
2
1
-
' - -
r - - - - - - - - - - - - - - - ,
~
i
-
~ r, 2}-. Y'-Y"'J
r >;
S
J-l. •
f-+( ss l--
m-O
\\ .
... ,0
.n (q. . • F. J
\\ . . 1
H RETUR~
_
6
J
J
_
0
• m is Preser 10 Zero
Figure B.l.
Flow diagram of function Runge ,
j
1
•ij
14>
j
The fourth-order Runge-Kutta with accuracy equivalent to Tayloyls
4
expansion of the dependent variable y(x) retaining term5 in h
requires
the estimation of the first derivative f(x,y) at four values of the
independent variable x on the interval x.
c x c X. i '
The fourth-order
l
-
-
1""
Runge-Kutta algorithms to be used here are given by
k, • f()Ci'Yi),
,
k
= f('i + 2" h ,
2
Yi + '[ h k,l
,
i
1
= f('i + 2" h ,
3
Yi + '[ h 1 2)
i
1
• f(x
+ 2"
h 1
4
i
h,
(B2.')
Yi +
3)
The local truncation error of the fourth~order Runge-Kutta et is
of the fo ra:
(B2.2)
where K depends (in a complicated way usually) upon f(x,y) and it5
5
higher-order partial derivatives.
For h sufficiently small (et -Kh )
it is possible to find hounds for K.
The chaice of a reasonable step
si:e requîres that an estimate of the errOr being committed in inte-
grating across one step be known.
The 5tep si~e should be small enough
ta achi eve required accuracy but at the s aae time it should be as large
as possible ta control round-off and minimize computing time.
AlI Runge-Kutta methods are convergent, i.e .• Hm (y. - y(x.J) = o.
h-e
j,
i.
Their st~bility depends on the type of problem.
BJ.
Hethod of Solution
Since the initial conditions Cat ~ = 0) are available (~.3t) for
gl and g2 only, we search for a value of &3 at ~ ~ 0 that will generate
a solution that yîelds g2 = l at ~ = l~
The half-înterval method îs
used for the searching process.
The technique begins by assuming two
limits. g3L and g3R' between which the missing initial condition. g30'
is thought ta be.
The solution of 4.27 is performed iteratively n
times. improving at each iteration the value chosen for g30'
At each
iteration &30 i5 set equal to (&3L·+ g3R)/2.
One of the limits g3L or
g3R has been adjusted according to the half-interval method.
The
criterion for the adjustment is whether or not the computed g2(1)
exceeds 1.
If it does. the upper lîmit g3R i5 equated to the current
midpoint g30' and g3L is not changed.
However. if the computed &2(1)
îs less than the actual boundary value, the next g3L is equated to the
current g30 and g3R is left unchanged.
Figure B~l shows a diagram that
i11ustrates the essential basic steps in the computatîonal procedure.
Figure B-2 is a flow diagram schematizing the essential steps of the
Runge-Kutta algorithm and the calling program as used in this study.
r-----------------------------------------~'
1
r--------------------------i :
1
"1
- 0
Q.L+9.1t
1
1
Compute Flow
O,I.! Ou' ar]"
ITER= 1,2... ) - . j 9 -0
9.. -
2
1
1
Porcmeters:
n
•• ,. n
,
1
1
Q,C"R.,C.
Q. -
0
Q. -Qao
1
1
1
1
1
1
1
1
8
1
1
1
1
1
1
T
1
1
CALL on
RUNGE to
1
1
ADVANCE "1, 9(1 O. 1 O.
"'<"'IIIU
1
1
~
ta 1he End of 0 STEP
"-
1
1
....
F
1
1
1
1
1
1
O < J
2
1
1
1
ï
-
- - - - - - _ _ - - - - - - -
.J
,
,
1
F
1
r - - - - - - - - -
...J
Ou - 9'0
1
1
1
,
1
T
9. -0,0
L
P i gur e B.2.
Compur ar icna I sequences in the shoo r Ing method of integration.
148
Table' Cl.
CALLING Pro gram for Runge-Kutta!ntegration Routine
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T.bl, q. (continu.dl
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150
Table Cl'.
function: IRUNGE
l~lEb[~ rV~CIIO. I~UNGEI~.,.~.~.~1
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Tablo C2 (con~inued)
Constant Eddy-Viscosity Approximation
ITEIt
• 22
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Il E1-1
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Table C2 (continued}
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153
T.bI~ C3.
Program for Computing the Potential Flow Parameter
A and
the Shape Parame ter
H
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154
Table C3 (continued)
1
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