/ /
CHARACTERISTICS OF TRANSVERSE MIXING IN OPEN-CHANNEL FLOWS
by
Jo sephat K. Okoye
W. M. Keck Laboratory of Hydraulics and Water Resources
Division of Engineering and Applied Science
CALIFORNIA INSTITUTE OF TECHNOLOGY
Pasadena, California
ENVIRONMENTAL ENGINEERING LIBRARY
REFERENCE ROOM COpy RETURN TO ROO\.1 136
W. M. Keel<. £:1g:,n~erir:!? l : 1~h"T, ' torjes
California Institute of T echnoJogy
Report N o. KH-R-23 Nove mbe r 1970
S OF RANSVERSE MIXING
LF
y
" Jose K Okoye
P ect rvisor:
Norman H. Brooks Professor of Environmental Science
Fede
and Civil e
Funded y
Water Q ant No.
Administration 7 DGY
Pasadena, California
3 19
r expresses e
Dr H. Brooks sugge s
a source unwave advice assistance, and
The writer also wishes to thank A Vanoni
Dr Fred
E,
Raichlen fo their kind advice and assistance and
List r his comments the the thes
For his invaluable assistance and instruction in the des
and
to Mr.
The as sistance of
is so
the laboratory , the writer is espec
F. , superviso r the s and labo
rt L. Gre
app rec iated.
in the construction the
The writer also wishes to thank A. Green ., for
the the s Carl T Eastvedt
Mrs rae the manusc
Rankin and Mrs.
sec ra-
s
sis the
eInents for
Science.
Ins
ee of
9 ,as a
of Technology
eto of s
e
source a rm
trace was
a ro source at
Hon was measured at several
us s
Tracer concentration was
e -ave
s transverse
transverse co
st near the wate
In contrast s
lent transve e AU,-,,"'-U,",
shear
ratio A. ==
For
e
was
ed in two
, and
the
was
a
Tracer concentra-
ream source
es
, its
coef
that the
level and was
was atest"
the
a constant
where W ==
reased 24
whe neve
s
the
ection the intens
related
the
at
-average
the res
their
s
es
dens of the concentration
were calculate ribution the
dete
es were inter-
to the transverse
TABLE s
r
s pz
3 69
3.C s en-70
3. C@ 5 rim ental
4 LA es I and 73
4 A" for the 73
4.A.L Flumes-- ss sties
a" The 85- ern flume 73
The 1l0-em 77
e. Ro the 110-em bottom
4.A.2 entration Detection 82
a. The 82
b The 84
e" B eire 86
d. 88
e data
4. e
4. The race
4"
"
s
4
4.C
c
run
4. B. 3" Studies
B.40 Veloc Measurements and
4,.B 5.
Data
4" C. 1" Recorded
4"C.2"
c
Data
suIts
1
11
5
3
2
5
5
5 C with Source 4
Transverse
the 1
c. r
1
5. 2 the Transverse .LVJ.L-"-~H~
Source Due to Shear and
5
the Tracer the Axial
Vertical at
race sectio
Caused
entration
ss
Tracer Axis
155
63
169
70
6
6 A.
6
A 2.
6.A 3
6
-xi-
he Transverse of the
1
a. Overall characte s
b. Prediction of the extreme limit of the 20
c the mean pos e
d. Growth of the variance re
s
the
occurrence
at s
z 204
20
209
2 3
2
Transverse of of Variation 2
2 232
C 3 of Variations 236
sion e 23
7 .l.V ... LV .. ..,. R Y A 242
A. Results Related to e I 242
Bo< Results to Phase II 245
LIST SYMBOLS 250
LIST OF 2
3
4 5
on
the 9
2 Sketch 95
3 in the 01
4 s 06
5 and s 111
4. 6 Cross-
ec 2
$ 7
5
4
4
2
4
2
1
33
5 2
34
5
at
42
5
52
54
rse
15
• 2
1
:::
2
22
5.
24
25
13 on the vertical tracer
cone
5 •
conc entration levels of
51 •
o-concentration contours s-s sses show points where tracer
values in
51
Iso-concentration contours cross- sectional
detected. lines
Crosses show s where tracer was on the contour
ss-sectional
1
8
6
202
re
2
2
2 7
3
224
4
225
15
229
1
3
3 2
2
2
5 4
5 5
ure:ments the
expe
the
data
the variances the :mo:ment :method and 2
related :::
expe
55
98
102
130
1
9
5
-xxi-
LIST OF TA
Variance (12 the transverse concen-tration distributions at various distances ~
levels '1; experiments included
A 2 rence to riments and 26
characteristics of a
mine, approp
the parameters
tracer
e
STUDY
inve
shear
measurements and
s
c s
ve se
seeks to deter
pertinent
the rate transverse
at ambient
e
into an open-channel
e
ss-wise
a
es
re r
r the
cross wise cone
and 1 grow as the z-powe
is the the
the more veloc
results" view the dead-
s a
ave
theoretic
the
:=
re
labo
and d = the uri
s expe
values of e and
s seeks determine mo re accurate
unive al r for p
e fo r various Detailed meas urements of time-
concentration are made in order to establish both sectional
and ral di
measurements,
tracer
reported
within the Such
r some wind- expe riments ,
for Davar are non-existent in open-channel
The attenuation of the axial entration
is also to check the power-law exponent
ion$ Ve rtical variation
characte process is a1
examine
e
seco rtant
rk
ifiusion time
o e 50 pre-
is to
s
e.
rse
and
re
wakes behind rs
1 ,or the
• and r and Head
< s dete rmined. The
are an
asurements ared
conc recorded at
s
rans
The
model
ea. Phase
from
two separate rs are set
and at the same time avoid r
rs derived
iders
than divide the
the
a
es
summarizes the ast studies.
tions, and the experimental
is structured identic
relate to concentration
The expe
cause the
are gene
data r
to
relevant to e r 3
exc that considerations
cover es and II
nts and the experirrlental procedure
ses The distinction in methods of
del
r 5 contains experimental esults and a discussion of
these res se te 6 presents results
ct
at
time-ave
bient vela
is a s
The source is es s
in
theoretical and experimental inve
pe
2"A PAST
2
mass
ective of this e of the s
ervation
is
cted continuous
and the
channel.
the
at am-
are discussed" The ex
s also presented ..
The statement
the
dis ransported and
environment .. of s
an remental mas s balanc e the
mass conse
+ S., 2
=
r
Fo sediment rec can be
a s if se sses
s ass to
states that there is an exact
e between mas s the control
the control volume ..
e and the rate
mass
exact and
CLL,U.!.;"/,; to
For an
List
the bas is
r indeed
ssible
is the
all per-
processes ..
where ~p is small see
and thus becomes app
::: 2
ress mass conservat
+ 2
s
:::
Fa case
a
and
whe re the time ave are
an time base
ove s
2
res
+
and
a
= +
+
dt
+
dt
is the period
The pe
remain invariant
can each be de-
ave
rave
ement s
2.
, and I
U. 1
must be so
s taken
sense s'" 2 7 the
:::::
cts
s
te a form s to the terms of
a
s the transfer coef-
ficient of mass that of momentum The res is
that the turbulent mass is related
the time- ave
rt co
concentration
EO ,such that
I I ::::
where s either a constant or a se
ij::::123 Since EO is
entire
retained as a tens
cuss r
tu
espec
leal
a later section
and 2 .. 9
the local gradient
means of the turbulent-
2.
rder tensor with
to remain constant in the
ations of
r
is
• 2 9 will be
<J) are
molec
however,
was
that E »
The
that each
es
streaITl accelerated ITlolecular
to at
s assuITled so
is ITluch r than the ITlolecular diITlen-
sion ITluch sITlaller than the sITlalle s of e
Molecular ITl is thus restricted to within each the
will be effected redistribution of the
eddies $ The
are the s aITle dens
If
ope rations pe
the result:
r turbulent
s also occur at very low conc
, p, as the
and
concentration is denoted by c, all
on 4> can, s , be ied to c with
c + ::: D 2.
where represents the ITlass transfer coefficient characteristic of
the lon the
and C is the tITle-ave
2
as
becoITles
"* :::
naliz
s
a
=
in the tu environITlent,
concentration defined in the sense
axes are r
tensor for
2
the
= 0
co
Be + =
s rmed convective-
sian coordinates oriented as shown in the
the
ae a +u
ion iun takes the rm
+ v ae + w ae =
2 2
ion
carte -
sketch , 2~
2 3
where u v. and ware, re • the time-ave veloc
in the x
are the mass transfer
and z directions, and
ients of the tracer
re directions x y and z
For the a a
shear as is s 1. the
=w=O
"slim or the
=
s e,
es
• and
in the
point source in a
as can be
r
2
=
(!)
~ cd
I I I I I '7j
:2 "0 I ~
til Vi I >.
I !-;
.,j..>
(\)
I S I
0 (!)
b.O c 0 a::
4-! 0
Q) "0 "iii
N
-13-
oC + 2 .. u
The in 2. 6 arises mo because the
characteristics and variation , and u are either unknown
or too to permit a To
ilitate the , a power law, relates u,
and to powers of y t has been several inve
for atmo diffusion 23), Davies , Yih
25), and Smith )). Smith ), , in an to solve
2 .. 16 for a point source in the atmo re
assumed that u 0: + = == Dz(y + 1-1J. re h:::: height
the source and IJ. and are constants <> He the
c
C :::: • 1
where 0 is the Dirac delta ion and , the source
below,
ac 0 at y::::O :::: 2 1
c -+ 0 as y, I z I - co 2
lve , material conservation s that the
source
co dz :::: :::: constant. 2
-co
that, r :::: t an exact solution could be de
c e s were
s the ze and sec
t C were c ated
, and
s of Smith's is is
that a mathematical the sens
the variations the rs h, and iJ. which
turn, indicate the effects of source
and shear ..
, the d c ients,
Another used r solution the convective- ion
is a rm of the moment- introduced
2 and extended 2 for the
an instantaneous po source Basic
the relation
y z dz 2 21 -00
is ove the and
the various moments
the rs of an instan-
s
=
and
ic s
- 5-
a
r the various
attained if the
t »
a
dz ::> 2
an unded re,
asserted that the
2~ 23)
where y = d is the he below which the material is and D
is a
expres sions
is
e
ion He us
r the centroid and variance the concen-
the moment Aris and Saffman was used
a the direction
transve rse distr
rated later~
s
the method
of the continuous
bas
s
s
terms mean square and the correla-
:::
W ::: C the at time t, and
+ ::: value w the same He
deduced is and a
dstic late can be
1 2 :::2
r time T such that
dt ::: constant.
In 2,,25 = mean square the lateral of the
ct use of 2.25 the calc has been
the cannot be measured but esti-
r 2
be
Neve
p
r s one is to
the case of e,
c stence of a diffusion
Inean cone a voluIne
Inarked
p = -x 2.2
where P is the that the the position
vector x lies within the Inarked at tiIne
to whether x
The
is or zero ac
the volUIne V
where
dens
is a
the ralized
where
tiIne
c
and
the
within or outside
::::
- ~ I , t) is the
x x at tiIne t ..
+ 2
s
a solution of
::: 2: 2 3
where
(2.3
Thus. under the conditions a ral ion co ient
can be
This res
while
ficient
s
c
such that
=2 2" 32)
is consistent with T r s one-dimensional • but
24 res T * to be constant, existence the coef-
::: j = 3 requires that have a Gaussian
rmore has been shown that if the coordinates a
are ass to vary
random
ients can be
time accord
r v::: w:::
to a
is a Fokker
s such the
2 25 all
deduc-
can be represented
a co ela
- 9-
the dens r the particle dis
tends a the ransvers
as demonstrated ation of the Central
rem and
without recourse a
ient s a Fickian can be
rm 2.32 even
Table 2 .. 1 summarizes
ficient made several
The re obtained
and (3
Elder ass that the
small t.
measurements the transverse
inve
veloc
u=
rs in various flow
and used Fischer
the s for the present
distribution was
+- in 1'], K
a
in the
coef-
and
so that
whe re in is the natural the veloc
s
n
= > and = mean flow For the linear shear distribution.
p = - v'u 2. - =
re s r vertic
2 3
(A)
(B)
hid where h"injection height
Depth~integrated valut',
asurernents the
on the water surface at a point on the
channel bottom and d=£1ow depth.
= D lu.d for floating particles. p '"
the ambient water; thus producing a density flow.
ansverse
Test Reach
W
(7) (8)
v e d T rae e r s
2.2 0.36
490 18,30
18.3 2.42
9200 305
1.5 0.69
2.38
1.2
67.4
14.1
306.0
}S.
14.7
ion co
Mean Velocity
u cm/sec
(lO)
21.6
64.6
66,3
o
94. :;
3-80.8
5
'1.5 \ 0.76 \ 7.3 - 10. 2 \ 22.9-1 S. 3
10,000 ! 226 175,0
30.0 34.
30.8
33.0
35,0
<'; N/A" not available.
the dissolved tracer ano 19 particle experiments
listed arc the
120 experiments reported by
o
the ve rtical diffus becomes
u
-2 -
= -;- G
o 2 3
ssum that the -avera transverse ient,
could be expressed in a rm similar to
that, for isot
since the von
turbulence,
K = 6"
;:::;; 0.067
constant, K;:::;; 0,40.
o 2.36, Elder predicted
2,3
To evaluate exper , Elder measured the
inte d concentration distr r both and a fixed,
continuous source in an open-channel 1. 0 cm
He calculated the lateral half-width
points where C e s one
the source. From the growth
us the modified Einstein ion:
1 -:::::"2 u
ral distance between
at various distances
with x, he calculated
2.3
where is the ave variance the transverse di
the tracer concentration his data the dim ens ss
:::::
has been be in error
ized
over the in a
s was about 2 meters
vario downs reampos ionso a
, he d ave ero ss -wise
concentration so His calc values of 0 from
to , 33 fo normal , 9 cm to 7.32 cm 0
iseher measured the transverse of Rhodamine WT
trace ected continuous into a anal 8.3 meters wide in a
6 cm He calculated variances dire from meas-
ured concentration distr
For both centerline and bank
the normalized transverse
across the channel at various stations.
cHons the tracer he that
ient 0=0.24.
In a similar experiment, Yotsukura et aL measured the
cted continuou on the su
the Misso iver The source was located at the river center line
downstream of the Blair Neb. and measurements were
made ove a km stretch of the river. The transverse coefficient
Einstein
riments des
was :;::;: O.
diffus ion in
s on
at various
3
meas 3 s
e ion s co be
s
::: 36 E 2.3
where E is the mean rate of energy diss per unit mass of
and
Orlob, is
a charaete stie scale which acco
• however
Batchelor has
s
expe
-power the It is
o 2 0 39 IS expe riments
shown that
res that 0- Z «
of such a
s
where st
's expe
ern and the tes reac was
, among othe r
e
s the late
a
ats
dete
24-
releas a the ed of water s e veloc
gr on a horizontal g id.
asurements re made 4cm 3cm
He calc ate
ficient was similar
ass
the
rian correlation coef
coefficient and reported that for
his data ::: Q 20 ~
From these and other experiments listed in Table 2.1, it is
a bs e rved that, r dis solved tracers,
ficient e ranges from 083 measured
to 0 73 measured Glover
es is
dimens
Pien
ss coef-
1) in a laboratory
River. For
r the listed
experiments all of which were pe rmed in ry s.
Further discussion
in r 5
2 B NA
fic
the measured values of e will be presented
INVESTIGA
res spec
the turbulent
the
= +
ient which is bas ic to
The
of mate al
the scalar dist
coef-
2
25-
co or the ransfe r coeffic Lent
o r tensor
Prandtl theory as a ical
's theory the one-dimen-
s case can be summarized as
-u = 2.4 )
where u and v' are veloc the x and y directions
and = turbulent II ion II Lent in the y-
direction.
i i) cc £. (2.42)
where £. and v,'_ are r a characteristic Ie and veloc--,'
the eddie sand
cc .£ 2.43)
Prandtl s sal become evident when
stulates 2 4, .42 and 2 43 to real 2 4
fo assumes s excur-
small to the characte stic
mater be mixed. However Batchelor
measurements ional wakes and jets,
the 0 ns.
meas
as s
entative
ate
there re reg
up a gradient- a
res that t
has shown that
exhibit
The
zero s , the
26
that e s we e apprec
e the mean veloc
! s measurements also showed that
wake would red
rect contradictio of randt IS theo ry which re-
ients be sitive. Furthermore Starr
to o 2.4 , a wide range of fluid s ems
viscosities.
theory also icts that in the re of
coeffic ient is also zero-- a contradiction
to the fact that transverse of material occurs even if the
transverse velo ient is zero. The neither the s
no Prandt1!s theory ies in the central re n a rm
open c re 'T ~
The prec ections thus reclude use of the
coeffic c to establish existence the
transverse coefficient Rather Batchelor's
3 mathematical res scussed in Section 2 A 3 will be d.
In a e s us
channel bottom Thus if the cross-wise distri
ncentration s Gaussian, then there
exists a ransve cient
is
an
-2
channel ess from the sidewall rs it
be s and are of z Thus
2 6 becomes
= + (
where C = C ,y
the moment of Ads 26), an ion for
the second moment can be derived each term of
2@44 and over z. For the left-hand side,
co dz
co
:::: dz
= 2 4
nco re the se moment :::: dz. The first term the
-co s comes
dz ::: dz
::: 2
:::
-28-
and twice one obtains
III :::: dz - c 2.4 -00
s therefore that
the tails of C-d are such that the material
:::: c :::: 0 as z - ± 00,
00
il) the dz conve s so that C - 0
as z -+ ± 00
and
2 47 es to
III :::: 2 (2.
where the zero moment :::: C dz is the total material at any
Y of a s
Thus all terms
+ 2 2
and the transverse can be calcu-
lated the
:::: 2 50
e
e
2 5
expres ed
2.
:::: stres s $ :::: the
2
2.
of
efficient
convection and shear to the transvers e
The first term 2 53a expresses the con
veloc at any
must be the veloc at that level
The combined the ve and the
al dist is expressed the last term 2 53a.
It shows that as as 11 =/: 0, this combined ct can be a
the determination of e r for each of
the three cases: 11 = 0, 11 = 1, and =0, e can be calc
without the value of
2.53 a r e as a
11 at any distance the source. It re s that the vari-
and M be known
trate that even at
measurements. The equa-
tions also distances where a 11 = 0,
is constant with
manner that compensates
= a constant at all
the veloc e at that
2. 53a is the
the shear
e at any s
is for the dete
gene each te
nume
varies with 11 in a
If = 0, and
assumes the
of e For the
can be calc
and
be
s ex-
s
gr can
2.53a the operation
- K 2
If is once with respect to , the last term
2 54 is zero As s U..iJLU.JLLF; r that rial is uniform
over the so that , and that the rate growth the
se moment with distance is also uniform with so that
'* g then
2
= 2.5
re mean variance the c ss-
e e
2 5 a
e
-32-
reference to expe measurements in Se 5
of
Distributions characte of a
ent shear is that near the source the vertical dis
that
The
within the
and u are
is skewed o This is due
HUlHl.J..Vrm with
is illustrated in
in a
that the is
2" 2. It is assumed
<> 2.33) and that
in 2" 51). The source is
located at = lie the tracer is concentrated at the source
and represented .u ... u- ....... ~ as a Dirac delta
r release, the source mate
over some vertical distance vertical Since
as a consequence of shear and
s stretched
is para-
< Q 50,
vertical ransport more pronounced than below the
tracer release The result is that a amount mater
be
m
out over a
race
the level
downstream
distance than below s a
maximum material concentration a
source is smalle
s £ increases the ess
ase
e
3 -
(!) ()
1-1 ;j 0 00
1-1 cd (!) $::i
.. (!) ()
1-1 "H ;j 0 0
00
1-1 S 0.. 0 1-1
"H
(!) ()
(!) $::i :> cd ...., - 00
;,:::;,
-34-
2 2 As ~ increases r, the rc and the
location the source 2 2 would
at the flow bed material distribution is
rm as shown in s of 2.,2 In re
may recover m zero to a value which could even be r
than before
be due to se
is attained
current or other irre
on the
effect
The distributions shown in
of tracer
rise to the water s
over the
be measured.
2 .. 2 are
near r
before mate becomes
at The ing"
The material was numeric calcu-
lated at several g us the forc 2.2
Various values were used The theoretical results were com-
with measurements in the as cussed Section 5. E
2. C
this
time ave
was
ector
were des
values
e
s
TIVE
exper to the first e
to calculate rs derived from
trace conc ral a tracer
at
5-
chart or r data ret and
ment conc
reco was e calc
the experiments was to determine the
ient both as a value and as a space vari-
able Fo calc e a • 2 55 was used. The variance
the cross wise concentration distribution was calculated the
z dz zC dz
:::
C dz C dz
values x Then the average was deter-
mined the summat
::: 2
i=
fo a fixed x re e the c s8-wise
were establis
ve x
::: 2
2.
-36-
Values of 8 were then calculated for vari.ous unifonYl flow
laborato s an to
ve
values
measurements and to add more rmation to the
rature For one eries of experiments,
the was intensified a r of stone s on the
of the flume For the rest of the experiments the flume
bottom was essent raulic
Dete rmination of e s; as a function of £ and 11 consisted
in numer
The varianc e
given station,
each term
was calculated at
was
Thus for that station
e = +-1.
- K
these calculations the ral
ion 8 with increas
mean tracer cone
£ is ne
a modified form • 2.53b.
values of x and y. At a
y and a curve fitted to the
e, e was calculated
2.
d.
Paramete
The decay the
g was measured at fixed levels
the s s
were e shed
s
ous levels es s
are ass
ITleasureITlents fo ur values ~the
£. curves
one Thus
£ at vari
the transverse concen-
identical at all levels, the pro-
cess scribed as one-diITlensional with a charac-
ristic transverse ITl
2 58 The
where is
z =: The source st
=:
=:
=:
re
area
=:
expressed ac
=:
solution is
o
e
C assuITled
is defined
,y dz
are re e
e
of
=------
as = s
cated
veloc
to
6
- 8
indee occurs, the tracer concentration
attenuate the relat was ve
measurements pres
62 also offers a method of c the
ave coefficient a at a known
am the source and ass the var
ance Such calc we re made and c to
the moment method Section 2 C 2
values were measured
at s ient points to permit of the on both
cross- sectional and c ro ss- S Q The iso-concentration con-
tours
process
vertical
the areas
the
calculate
were used to ualize the s the
entration maps were also constructed on the
the axis" a dist
levels the
rs" From the experiments, the variation
and was determined.
meas ements were made to ve the assumed
the ve establis sovels
zones the wall , and
various ume c s s sections
se
-39-
c ss- concentration es were and 8 calculated
as a local and as a
the
A
constant" Near
were measured and
behavior was also
Iso-concentration contours were on cross-
se • transverse s.. Material distribution
the was e hed and vela measurements made.
3
ST ST ST IONS
A CT
The second e of this deal the ral fluctu-
ations of tracer concentration at various points within the
some theoretical models have been proposed r describ
this process, rimental ve rification is scarce. This
r reviews the pertinent wo rk thus far reported in the literature,
presents some leal ramifications, and outlines the experimental
ective e II.
3.A, PAST STUDIES
a method introduced Cs 1 ,a
conservation is derived the mean square concentration
at a fixed po Ess the time mean
conservation is subtracted from the instantaneous
the s veloc concentra-
tion. , s 2 8 • 2 5 one obtains:
+ + = 3. )
where <p! = mean
ss =
::: 3.
If • 30 s mult ! and ied, res
can be averaged the sense o 2. the
e for msf:
+ ::: 3.
re the rate of diss ion msf, is shown Batchelo r et aL
) to be related to the molecular
::: (3.
Hinze that if veloc and concentration fluctuations
are at a grid of a w tunnel can be
expres sed as
re s a concentration microscale. For this case, as the
decay kinetic energy a d, Hence
::
re ::
s e a
-42-
= - 2 + a 3
where u == and = Thus assum that i) the
ion relations is val for the concentration and msf,
the marked-particle sis can be Leo ¢ is ed
• and the is "slim, 11 the result:
= _ D BC , 3.
-r-;z = -D s
and
as 2D + ] + ( 3. 0 u =
where D and are ivities for concentration and msf respec-
s == is the mean square the rna concentra-
C is time-ave cone axes x y
and z are oriented as shown in 2. 1
Cs po stulated that for the case of the point
source may be determined the relation
s 3 =
re a dec scale e s
self-similar ies,
x
re c ated curves
a
a
transverse s-p
value and that the dec rate m r the
rate turbulence
3 The
model, sented s a r pre-
the properties int concentration s@
3 $ 1 for the one- ess
as s ume s that the emitted a continuous source in a
rm is a superposition of an infinite number el
3. element s allowed to grow in the z sks as
rection as ion time increases but the elemental thickness, dx
is constant and u dt thus ass ne ion
The centro the material
the disk wanders back and rth in the z a random
manner
repre
as
=
the
::: :::
z u
)
z
u u e
45-
tration has been rated ove Over many realiza-
mean of the concentration dis is
c 00
= • 3 00
where is the function desc the va of
ove the trials.
trials ~
where
the
3 13 can be
c
, and
, f
rm of f is the ens of
as a convolution r which
= + • 1
are, res the variances the
g. Ass that the transverse distribution
of mean concentration is sian and that the of the
instantaneous concentration distribution is random so that
also
this
the
a
sian,
sian.
3. 3 becomes a Weierstrass rm. For
the inversion theorem that f is also
the model rd predicted
distribution of the concentration
He e that
-average represented
= +
a
a
3 A 3 Pr
Table 3. izes lite rature on
ITleasured paraITleters related to ral fluctuations of a scalar
prope at a point ithin a turbulent flow field, the s les
Gibson and Schwarz 50) and Lee and Brodkey (5 we re conducted
in a wate r ITlediuITl No ITleasureITlent in an open-channel shea flow
has be en repo rted.
The scare of experiITlental in fact thea retic info -
is due not to the inlITlate these
but to the difHc of and ITlea8Ur
paraITlete r8 ,
paraITleters
us r practical purposes and then fa pertinent ions.
For • it is ext diffic ult to a peak fa r a randoITl
variable without SOITle iHc ation e spec with re to the
specifies a the observation tiITle. ractice one
whic is assuITled to the ITlean value ove r an
11 S per twice the standard of the
fluctuation both which ITlust then be dete rITlined Even when
rs are well- • pe rtinent equ are ated
anisotropy and nonhoITlogene ust as in the tiITle-ave relation-
the added terITlS which account interactions between
veloc and scalar and for iss as shown
3
re is
sponse cu detec
e s
SonTce
et ( 521
Smith (54)
Gosline 1
,and Gartrell
Lee
NOTES:
Environrnent
1 atrnosphere
pipe
Reported by Gifford (16)
" Not available
24.4
76.2
60.0
s
NO,
N/A 2
~lo~rescent 1 pigment
oil-fog
NaC!
Gentian violet
ements of concentration
Sampling Frequency
amples/sec
O. 1
continuous saITlpling
O. 1
N/A
continuous record
continuous record
continuous record on wave analyzer
Test Parameter Reach Investigated Result
1620 ~=~~~to~a:~rage P a decays as a
ground level
~SOlineSOf
244
2 to 12
I ground-level concentration
frequency, f of various concentration levels, c
P a at ground
level
radial distribution of msf 5
decay of msf with
(i) Decay of ms£ with x (ii) radial distribution of rms value of c!
P decays ini-a -I
as x and then remains constant after x""' 3. 2 km
P a""" 2.0 for x
values betvteen 50 50 m and 200 m from the source
~, Other measurements of peak-to-mean concentration ratios are reported by Gifford (53)
4 Reported by Csanady ( 15)
is mean square concentration fluctuation.
Source located at x 0.0 ro.
ilarge
I 5an1(' x
small Hence for ve R s r flows 1 spec
Gibson and Schwarz ) must
be used fluctuations This is espec
necess because the peaks are us
associate the r
Most measurements made in the atmo
s of the s
re relate to the
-average concentration ratio 1 r elevated sources under
various atmo ric stabil conditionso As expected, there is con
but some gross trends can be siderable scatter in the data
detected. Measurements et ale 5
rd 1 ), indicated that near the source
and Wanta and Gartrell
at ground level
dec as x Data Wanta and Gartrell also showed that
an value 2 0 x=3.2krn
the rce I S measurements rd that
for x between 50 m 200 at source he ess
constant and to The measurements also nstrated
that for the same s of x, was cons ide r at the
at the source level
riments rmed Hinz e s
as r t e where
were generated at the same w
s a a
e
er e
x and that at a fixed section the radial ist
ne s als
s were self-s lOUS values of
s of and othe expe are given in
Table 3
work has been done in concentration
problems c Fa theoretic es 1 two models
for continuous s have been advanced: a
lnodel dese the conservation of the mean square
for concentration expressed in 3.10, and Gifford I s
model 3. L
is meagre and to
measurements -average conc in
trace s re These ric observations
-1 0:: X and far away attains an e
about 2 One meas urement in a wate r tunnel
ms£ decays as x for grid-generated e
ms£
als
s
-50-
3. B. LYTICA GATIONS
A which varies with Y and z and
sacco to s model is cons ide red Let ,z, t)
be the instantaneous concentration distribution at the fixed station Xl
at the instant The centroid of f is located at z ;y, t) and its c
variance is so that f is represented ic as
f:::: f[Xl;Y' ) ; 3. 6)
The centroid f is c by
00 00
z , t) :::: c
zf f dz • 1 -00 00
and the variance
00 00
, t) :::: - Z f dz. 1 c
-00
r a number of re izations, the ensemble average can be re-
s inte Thus the mean concentration at fixed
Xl becomes
The cent
z o
C :::: im T-oo
and variance (f 2
00
:::: z
are resented
where
e can
z
00
::: C dzo
centroidal oscillation becomes
T-oo c
a 3.2 , one obtains
00
I+ Z c
T £ dz
dz dt
dt
:::: £ dz and z ::: Z - Z
00
be c
:::
e
z c
dz ::::
.23
C
a
2
3
.23
2 z t) - Z ] dt
c :: lhn +
00 , t) dt
3
If ,t) is invariant with time
:: + (3.25)
where is the ensemble average the variance the
taneous conc entration f at the level y taken over a
r of trials"
3.25 demonstrates that the between variances
for the which varies in the transverse and
ve al directions is similar to the
a s in the transverse n. In the
more case two-dimens var , the variance the
distribution does not have to be constant with time as
s model assume The rement is that the total
amount mate the clo ud at any level. y, remain
constant with time.
latte is re ized if f is as
value the instantaneous concentration
Therefore F be ed defined
:: dz 2
s c
as
ave
shows for
instantaneous concen-
rated over the This
is acco a
level above the on the
the This te pr esents
a monstrous des and data- red uction First, each
must be ed thro calibration to deter-
mine the concentration dist at each instant exposure in
order calc the variance the and to locate its
centro I-fundreds s must be s ed to obtain
a representative ensemble sens
gene s s serio s
reliable measurements can be made.
s s was
x::::::
se
, OSC s back and fo the transverse dire
x and y, there exists a se
There
the
z-dire concentration is exceeded
a ime of observation. This section
termed an inte see fo F 3
This idea was Townsend 0, 45 who, while
be
to ve
wake a
s theo ry of loc al
observed that away
in the
wake center the
was inte He the inte
r, as the fraction of which the
and calculated its
vectors. The conc
and
which re
to the
derivatives
has since been extended to
rrsin and r ) who built a circuit
and measured intermitte rs
et who determined "
direct meas urements and hot-wire s s;
and to an
used a
as relates co
compress wake Demetriades 1 who
of B s circuit to di measure "
measurements made a
Table 3.2 Some expe ntal
has me
we c
z
So r c
0)
and Kistler (II)
(
(56)
Head (14)
(12)
Flow Investigated
plane wake
plane wake
turbulent boundary layer
easurelTIents of
Experimental Arrangement
a cylinder 0.953 em in an air strearrl of velocity 1280 eml sec
a cylinder O. 159 em in an air streanl of velocity 1280 cmlsec
in a
Reach of Measurement
m
1. 52
1. 51
measurenl.cnts at only one section
3.21 m from channel
entrance
(a) turbulent boundary I a wind tunnel with a 2. 59 layer
(b) round jet (c) plane wake
plane jet
wall jet
turbulent boundary layers
axisymmetric cOlnpressible wake
working section 6 em x 61 em
divergent wind tunnel with smooth
38-cm wide air
blown £ron1 a m slot
parallel to wall of a wind tunnel
wind tunnel with a flexible upper wall
0.396 em diameter rod in air stream at Mach number 3
3.56
0.66
0.92
not given
0.40
ParalTleter Measured
spatial derivatives of velocity
6 - si'~nal 2
velocity fluct.uation
for
velocity fluctuation
(i) velocity fluctuation (ii) L - signal
(; -(i) differenti circuit, and (ii) photoprobe
6 - signal
'I
In investigations listed here, 'Y is the ratio of the duration of turbulent motion to the total san1pling period at a fixed point.
signal is unity when motion is turbulent and zero when it is non-turbulent.
'I
concentration is exceeded at that point The inte
r may thus be considered to ate the fraction that the
is on one side 0 other the position Z e Thus is
a distr function express ed as
= < 2
where Z is the instantaneous z - po s ition of the
c
u ~ o U
>
the
re
If as shown
=
Time, t
record at a
3 2 is the period dur which
when c at a z. then follow-
.2
and Pz
and
an
that
:::
:::
are re
and
::: ~ 0
if c
c
s
t) >
,t) -<
the e facto becomes
of the
the threshold conc
:::
are shown
concentration
measurements
pe
for the
r s
densities
The
r a s
re
z
,,31
.3
s
T I I
T T \r I I I I
58
. M
any s rs
factor ac the
remains at a over the
central c re ze at the extreme
Thus a section is charac-
terized a central core record
the thre ~ 1 a inte and an
zone where the thre concentration is never attained.. It
must be noted that the central core s not exist at s
x. It should also be out that is not a z
as s 3.3 and 3 Rather , y, z) , and
- ,y,z;
atment has x and y
YI and
Z s x and
Since is a as it
exists a cor dens
s
=
s
60
6
;z) dz
Yl
;z) dz
and the corre variance, the inte re is
expressed s
Yl;Z dz
3 3
It is noted that the
00
'Yl;Z dz = (3.3 o
the central core continuous record exists at the station under
consideration.
If
the characte riz of the
self- similar as one suspe , then a desc
of reg will e the pararn-
eters are spec
as
(\J -... Q)
+
of the inner core at z = ~, the extreme the
front pos The p ~ to
and back 0 oscillate as a al function time, or
the inte
r each model, the corre
factor distribution
derived in
s that
l' the sinusoidal model.
and r the no
=2
1 =2:
as shown in
the
achieved as becomes
• For the tr
+
random
+ erf 2
where L the characteristic half-width of inte
e Q! = e 1'0
=
e are fo
=
e
of Q!
dx,
I:::;
Iz I ::> vV
cases
is
oscillation,
.3
.3
.3
Thus if s elf- s ) holds true the of the inte r-
at eve section on the :mode the
frontal scillation. If this space-ti:me :motion repre
the sentable, then so characteristic transverse dis
inte r.
The :mean nt pos ition. ess locates the
inte
its :mean pos , then
where
terized
= 0.50. The width
L as shown in
If the is ical
co incide s with the percentile
the inte zone can be charac-
However the distribution
approaches its extre:mes • the variance,
characteristic. In such a case.
would have to be deter:mined experi-
beco:mes a :more appropriate
the li:mits z::::!::' and z:::
or assu:med be ated at so:me factor the standard
deviation
OB
3 The second of this rs fro:m
the first the :method data is In general tracer
:measured situ co
ed at a selected rate and
e expe
facto c u-
a
e
dete
Front
r
where the
space-thne
cone
was
e
re
eve ral stations
that ins tantaneo us
the
between the observable
s
the
s the thre
earlier the
the z-dire
were made use
rwise stated
a
There-
and the re are N s s the period.
at othe values z
established for the
ic characteristics
model we re de the s
mean pos ition of the nt was calculated
::: z )
::: z i •
i::: 1
whe re M is the number where was meas
usual if the inner core of continuous record exists
the
3 43. the variance
i :::
the inte
::: --:-----
• 3. 3 5 and th e
was
was
The
3.
As
an e , I z
re
the source
were asce rtained.
rates the
re s a repre-
sentative curve
result of s s
at all of x
a
z
z to zero.
W the
characte ris tic
the sweeps
ze occurrence
the inter
varies
/ 4
s
:::: the characte
scale s
::::
at the fixed level Fo as in an ess
wide channel, is constant so that both
grow at the same rate"
method r calc
3~28
If the period over
ceeded is denoted as shown in F 3,,2
occurrence II excess) and is the
occurrence. "then Pl and pz are the
occurrence and non-occurrence
The rest here not to use
to dete rmine the s of the
a the
3" 28 to
densities
ristics
periods occurrence
non-occurrence were
6" time
rs an alternate
is ex-
the
non-
ities of
but
the
the pe
see r
and
were selected The occurrences of
+ ~ and
the pe
The s
= pz ==
s of c
occurrence and non-occurrence was calculated = = and the
= Thus
where is the
at various
s
ns + =
T ~T = L
nu:mber periods
densities
and
rate
, pz and were deter-
values of z where ::::; 0 50.
:model was
fro:m an elevated
24 fra:mes pe second.
the instantaneous was located
s
the centroid was
the :mean concentration was
ss- e
s
2 :::::
f
and, from 3 22,
::::: c
where n is the number s e From 3.25,
total
thus c
e
at
was
distances
s of the variances were
the source.
n
mine the usual
the variation
es used
points were also
s and to relate calc
msf as outlined in Section 3 A
ed info so that
mean concentration C was determined
C ::::: di)
i::::: 1
the deviation the mean the ral
c ::::: C
S :::: c
c
ed to deter
s to
statistical
set as c i)
51)
c
.5
1
::: '2
ral
::: 5
re the total r po s the conc entration
s recorded over the total s pe and N s
The -average was c ulated the relation
::: .5
where is the st of c within the pe d.
some expe iments, the fr dens r the
densities
was as
was a total of 20
From the value c where
the mean g
S :::-~--..,.;;.::. 3 5
e
reve the nature the
power
The expe
one the
and
in the
a
a
a random
appropriate
measurements
the
osc
were
tracer conc
po
s each model is
theoretic
to
as due to each
the
to
ns are examined
r-
enees
A EXPE s
were e flumes des as -ern and
s are
scribe
Overall view of the 8 stream
power acks was alculated ement
ve mo
the channel The was c rated to
the nearest
The wate pump a P.
speed delivers water from the outlet tank the
an 8 inch 20" 3 with an 8 X 5 inch
meter
Water is dis
an inlet
zontal baffle
ire into the
box, and a set
water dis
into the
not attached A
ve sc reens at the
downstream the box were used to smoothen the dis
into the channel, and to
r
wave
rm
number experiments a
across the
horizontal
downstream the screens to reduce inlet
The flume sides and are HLau,,", steel
as the west • 6 m downstream
The re
a
espec
• 4~ 2. The instrument ed in
used r
As shown in 4 2 and the s are
mounted on the instrument car s verniers
for meas ald ements to Transverse
positions were measured the nearest
s s were used r meas tance
supported s c s
r s e the
nearest 5 cme The steel
the cloth
the
4,,3, is40m
wide and 6 cm deep"
et al ..
and maintaine d
s
was used for e
characte
O-cm shown sc
cross s
desc has been
s
the
the
in
Ocm
The rests on a central and screw
t two and the r two am
The electric motors and thus
Flume was calculated
a vernier
The vernier scale was read the nearest
us a water s
the main
continuous
the ve
r c
the
0025
it was was related to the
7 @ 7
re the s
WEIR
RESERVOIR RETURN
STAINLESS STEEL SIDES ""
1 0-
ve
the
ove an
To maximiz e the
the rise the tracer concent
ment res
s
the
tank and a set three
am
the the water
the outlet
water used and
B
minimize
an experi-
4
tanks the outlet
tank water
conne reser-
s
5. on the flume bottom
e
re
the
value the
bottom at that po
the mean stone thicknes s r the ent
c For zero was c
we
record
a stone
to the
measurements
to be
be cm.
-
s
The
electrodes
a
where it was
cas was
was
set
us s
cher
des The
consisted
kovar s
as the
red
Rez
three
which
end
cas
connector
5 thick
copper
soldered
the
",u>.u.'""U,i:> metallic
were
a r shielded cable
the connecto
rec
elements
re
to the
842
connector
circuit
from
s shown the
the s
are
the exc
c
the via
and the
The
be re
robe sensi
e at
8
s
g
s was
recorded as the true
were imme sed in the wate
Se< The s
the combined
rre-
a
instrument
e
the recorder was
e
the
as the
when two or more
the
the
was
UT A-
a car-
a
s
a at a selected
to be c
permitted the
desired.
where the
fed
pass
A
be
carrier p
ave s
ave over a -s ec pe
In experiments
was
Data
4. 00 ata ac
~ SYNCHRONOUS IJ.J
(f) x MEMORY TAPE TRANSPORT I- IJ.J
...J...J (A) ::::> a.. 1J.Ja..
~ z-z z~ AID DIGITAL <.!) «::::> CONVERTER MULTIPLEXER 0 ::I:~ ...J u<.!) SAMPLE AND MEMORY « '0 z <D...J HOLD (B) « «
z «
I , ~ ....... I
DIGITAL HEADER CLOCK CONSTANT
MANUAL INPUTS
Fig. 4. 11. Flow diagram of the digital data system.
-92-
the analog voltage was digitized, was selected by the operator by use
of the BASE FREQ and DIVIDE BY controls. The maximum sampling
rate was 1600 samples per second (s/sec) for one channel. Thus if
eight channels were being used, the maximum rate was 200 sl sec per
channel.
Sarnpled voltage was fed to the AID converter where it was
converted to a binary signal and then to binary- coded-decimal (BCD).
Meanwhile the header data received by the control logic were coded
appropriately. The header data consisted of (i) identification infor
mation from the analog multiplexer, (ii) digital clock data, (iii) a
four-digit number termed the header constant, and (iv) manual identi
fication inputs. The BCD of the converter and the header data. from
the control logic were received by a digital multiplexer and trans
ferred to one of two memory units for storage. Each unit has a
capacity of 1024 tape characters.
Since the system utilized a. synchronous tape transport, as
opposed to an incremental recorder, data were first collected in one
memory unit at the sampling rate, and then transferred to the tape
at the maximum transport rate of the recorder. Meanwhile the other
memory unit accepted data from the digital multiplexer while data in
the full unit were being recorded on tape. Thus no information was
lost while digitized data were being recorded on tape.
Digitized information was packed on the magnetic tape in a
language and format compatible with the IBM 360/75 high speed com
puter. Data from a set of measurements. such as concentration
monitored at a fixed point over a given length of time, were stored on
-93-
tape as a file. The files were separated from each other by END OF
FILE marks. Each file was composed of records separated from
one another by INTER RECORD GAPS. A record length was 1024
tape characters, and compris ed the storage of each memory unit.
In each record, the first sixteen tape characters identified the header
data from the control logic, the remaining 1008 characters were
digitized data. Since voltages were recorded as 3-digit values, each
sample consisted of three tape characters; thus 336 samples were
stored in every record.
By use of appropriate subroutines. the recorded information
was conveniently retrieved by the IBM 360/75 computer; thus digitized
concentration data were available for reduction and analysis by the
main computer program.
4. A. 3. Velocity Measuring Device. Water velocities were
measured with a 1/8 inch (0.32 cm) diameter Prandtl pitot static
tube with a dynamic head opening of 0.107 cm. The pressure differ
ence between the static and dynamic heads was measured by a pres
sure transducer built by the Pace Manufacturing Co., Los Angeles t
California .. The pressure difference deflected a 0.0102 cm diaphragm
which in turn induced a voltage that was measured by a Sanborn analog
recorder.
The transducer was calibrated by inducing pressure differ
ences across the diaphragm, and measuring the pressure differential
in manometer pots equipped with micrometer scales. Pressure heads
were recorded to within 0.00025 cm. Calibrations made before and
-94-
at the end of an experiment agreed to within two per cent.
The veloc ity was obtained by po sitioning the pitot tube at the
desired point within the flow field, and recording the output on the
strip chart of the Sanborn recorder for about 30 seconds. Since the
output was averaged over a one-second time constant by the recorder,
variations about the rilean value were generally less than 15% of the
total stylus deflection from zero.. The mean deflection was obtained
by placing a straight line on a transparent scale over the record, and
estimating an average value by eye ..
4.A.4. The Tracer Injection System. Figure 4 .. 12 is a sketch
of the tracer injection system.. The tracer was a sodium chloride
solution colored blue with dye primarily for visual effects. The solu
tion density was restored to approximately unity by addition of
methanol. The tracer was stored in a five-gallon constant head tank
placed on a support base about 2.5 m above the flume bottom at the
injection station.
With a plug valve completely open, tracer flow rate was con
trolled by use of a metering valve, and measured by a precision flow
rator which used a floating ball as a flow rate indicator. The tracer
flowed through a Poly-flo connector to a copper tube which was sol
dered to a stain1es s steel injecto r. The internal diameter of the in
jector was 0.267 cm, and the tracer was injected at ambient velocity
parallel to the water flow. The vertical position of the injector was
determined to within 0.001 ft (0.030 cm) by a vertical displacement
scale mounted to the instrument carriage. The injector was always
5-GALLON FLASK
TRACER
GLASS TUBES (077 - em ad)
(DYED NaC I~, SOLUTION)
FLASK SUPPORT
CONSTANT HEAD STORAGE
4em
~2.5m
TYGON TUBING
POLY-FLO TUBING
METERING~
VALVE "'"
Flume BOTTOM
~PLUME
Fig. 4. 12. Sketch of the tracer injection system.
/~ PRECISION FLOWMETER
(40em3/see max)
-- VERTICAL DISPLACEMENT SCALE
COPPER TUBE (062-em a d)
ST AI NLESS STEEL INJECTOR (0267- em i d)
(NOT TO SCALE)
I --0 \J1 I
-96-
located at the flume center line.
The flowmeter was calibrated by keeping the plug valve wide
open and, using the metering valve for control, recording tracer
discharge for various flowrator readings. With the injector cross
sectional area known, the mean tracer flow velocity was calculated~
and a curve of tracer velocity versus flowrator reading was developed.
It was found that within the range of the change in tracer temperature
experienced from one experiment to another and within the sensitivity
of the flowmeter, a single calibration curve was adequate for a set of
experimental runs 9 A new curve was developed only when any section
of the injector system was alteredo
During an experiment, the injection velocity, which closely
equaled the local water flow velocity, was first determined. The cor
responding flowrator reading was then obtained from the calibration
curve. With the injecto r set at the desired depth, and the plug valve
wide open, the flowrator was set at the required reading using the
metering valve. Tracer flow was subsequently controlled only by
completely opening or closing the plug valve.
4 .. A.5 0 Photo Analysis Equipment. The motion picture of
the fluctuating plume was taken by a Bolex camera motorized to
operate at exactly 24 frames per second. The developed negative
was stored in reels each with 30.5 m of film.
The film was analyzed by projecting the picture on the screen
of a film scoring viewer. The film could be projected automatically
at a selected speed, or manually one frame at a time. With the
-97-
picture on the screen, a cross-hairline was ITlanually used to deter
ITline the crosswise positions of the edges of the plUITle at that parti
cular instant of exposure. A transverse scale located in the picture
field was used to calibrate the scale on the screen.. Transverse dis
placeITlents in the fluITle were deterITlined to within 1 ITlITl.
Since the ITlotion picture studies were conducted in the 85-.cITl
fluITle, the longitudinal tape located within the channel was used to
ITleasure values of x" Elapsed tiITle was ITleasured either by a ten
second sweep clock in the caITlera view or by counting the filITl
fraITles.
4.B .. EXPERIMENTAL PROCEDURE
4. B. 1. Identific ation Code for FluITles and ExperiITlents ..
Since experiITlental nUITlbers will be used in this section as exaITlples
to illustrate typical ITleasureITlents or operations, it is necessary that
the code for identification of the flUITles and the experiITlents be ex
plained.. A sUITlITlary of the classification is shown in Table 4.1.
The flUITle identification code consists of a letter-figure COITl
bination with the letter referring to the roughness of the fluITle bottoITl,
and the figure to a particular flUITle. The letter S indicates that the
flUITle bottoITl was hydraulically SITlooth during the experiITlents, and
R ITleans that the fluITle bottoITl was roughened with rocks. The 85-cITl
fluITle is identified by the figure 1, and the 110-cITl flUITle by 2. Thus,
the code S2 refers to an experiITlent perforITled in the 110-cITl fluITle
with the flUITle bottoITl hydraulically SITlooth.
All experiITlents were grouped in "series I. each cons isting of
Table 4" 1 ~ Classification of flumes and experiments
Flume Flume Data SERIES Flume Bottom Identification Digitized?
Roughness Code
SOO 8S-cm Smooth S1 no
600 8S-cm Smooth S1 no
700 110-cm Smooth S2 no
800 ttO-cm Smooth S2 yes
900 1tO-cm Smooth S2 yes
400 ttO-cm Rough R2 yes
300 8S-cm Smooth S1 no
-"'----
Mode of Analysis
Time-averaged calculations only
Time-averaged and intermittency analysis
Same as in Series 600
Time-averaged, intermittency, and statistical analys es from digitized data
Cros s-correlation analyses
Same as in Series 800
Fluctuating plume studies using motion pictures
-_.-
I -.0 00 I
-99-
one or ITlore experiITlents referred to as "runs. I. The first digit of a
run nUITlber refers to the series, the second and third digits to the
experiITlent nUITlber within the series .. For exaITlple, RUN 512 de
notes experiITlent 12 in the 500 series. Each run, except in Series
300, consisted of an entire experiITlent beginning with uniforITl flow
establishITlent and concluding with concentration rneasureITlents.
If velocity ITleaSureITlents were also ITlade and velocity contour
ITlaps subsequently developed, the letter V was affixed to the run
nUITlber. For exaITlple, run nUITlbers 506V, 708V, and 404V indicate
that velocity ITlaps were developed for the corresponding runs 506,
708, and 404.
Velocity contours were developed in Series 500 for the 85-cITl
flUITle with the SITlooth bottoITl, in Series 700 for the 110-cITl fluITle
also with SITlooth boundaries, and in Series 400 for the 110-cITI flUITIe
with the bottoITI roughened with rocks.
4. B. 2. Typical in situ MeasureITIent
a. EstablishITIent of uniforITI flow and calculation of hydraulic paraITI
eters. By adjusting discharge and fluITle slope, the water depth was set at
a desired value. UniforITI flow conditions were assUITled to prevail
when the flow depths at various stations agreed to within ± 0.02 CITI.
In the 85-cITI fluITIe the energy slope Sf was deterITIined by
fitting a straight line to the plot of the difference between the still
water level and the flowing water surface elevation versus x. The
slope of this line was used as the energy slope. Since the deviations
of the flow depth at various stations froITl the ITIean depth we re
-100-
generally SITlall and randoITl, corrections for differences in the veloc-
ity head did not significantly alter the value of Sf thus calculated.
A typical energy slope deterITlination is shown in Figure 4.13. In
general Sf was different frOITl the fluITle bottoITl slope by between
1 % and 10%"
The i1UITle rails of the 110-cITl flUITle were so precisely posi-
tioned that the slope of the rails was exactly the saITle as the fluITle
slope S deterITlined by Eq. 4" 1. The value of S at the condition o 0
of uniforITl flow was used as the energy slope. Velocity head cor-
rections were ITlade for the sITlallest flow depths.
The norITlal depth, d, used for subsequent calculations is the
ITlean value of the flow depths ITleasured at the various stations for the
uniforITl flow condition. For experiITlents where the f!tune bottoITl
was rough, the ITlean stone thicknes s used was 1.66 CITl, i. e. 92.3%
of the thickness calculated froITl point gage ITleasureITlents. This cor-
rection was obtained froITl velocity ITleasureITlents. The depth y =
1.66 CITl above the flUITle bottoITl represented the average height at which
flow velocity was effectively zero.
The ITlean velocity, u was evaluated by the relation
u = a/A, (4.2)
where a is the discharge, and A the flow cros s-sectional areaj
the ITlean shear velocity. u* by
u* = .,; grSf ' (4.3)
where r is the hydraulic radius; the friction factor. f * by
+ 1.2
+O.B
E +0.4 u ..
en w 0 u Z <!
-0.4 r-en -0
--1 <! -4.BO u r-ffi -5.20 >
-5.60
-6.00 L
no &0 BD ~Flume Entrance
RUN 50B Flume S I
d= 5.26 cm,u=41.7cm/sec 1R* = 1,130
10.0 12.0
STATION, m
STILL WATER LEVEL
ENERGY SLOPE, Sf = 0.001000
Flume BOTTOM SLOPE, So= 0.001100 (plotted)
=0.001050 (from slope gage)
14D 16.0 IB.Ot Channel End
Fig. 4.13. DeterITlination of the energy slope Sf in the 85-cITl fluITle; RUN 508.
...... o ...... ,
-102-
(4.4)
the Froude nwnber, IF by
IF = u Igd
(4.5)
the Reynolds number, IR by
(4.6)
where v is the kinematic viscosity at the measured water tempera-
ture for the particular run; and the friction Reynolds number. IR* by
(4.7)
When velocity measurements were made. the von Karman constant,
K was determined from the slope of the velocity profile by the rela-
tion
log Y1.... K = 2. 30 u * 10 YI
Uz - Ul (4~ 8)
where Uz and Ul are the mean velocities at Yz and Yl respectively.
For experiments in the flume with the rough bottom, the bed shear
velocity, u':<b was calculated by the side-wall correction method
of Vanoni and Brooks (59).
b. Tracer preparation. A tracer batch was made by dissolving
approximately 613 gm salt (NaCl), 2.58 kg methanol (Tech grade) •
-103-
and 20.30 gm 7-K blue dye in 11.13 kg laboratory water. The re
sulting solution consisted of approximately 4.34% NaGl t 18.32%
methanol, and 0.14% dye by weight. The sodium chloride was used
as the primary ionizing agent, the methanol was added to restore
the specific gravity of the tracer solution to approximately unity. and
the dye was introduced essentially for visual observation.. The tracer
batch was usually immersed in a reservoir of flume water for at
least 24 hours so that, during an experiment, tracer and flume water
temperatures were approximately equaL It was found that generally
the specific gravity of the flume water was slightly greater than that
of the solution with a discrepancy of about 0 0 02% in most cases.
Since tracer conductivity was the distinguishing property to be
detected during the experiments, the conductivities of the constituents
of the tracer solution were measured with one of the probes. Table
4.2 summaries the results. It is evident that the Tech grade methanol
was essentially non-ionized. and that both the NaGl and the blue dye had
the same order of specific conductivities {at 1 % solution about twenty
times the value for the flume water}. Since the amount of NaGl
used in the tracer solution was 31 times that of the dye, and their
conductivitie s were in the ratio of 3.52 NaGl to 1 of dye, it meant
that the conductivity, above the flume water background, measured
during the experiments was due essentially to both NaGl and the dye
in the ratio of 109 to 1. This ratio assumes that the turbulent mixing
characteristics of NaGl and the dye were the same. and that there
was a linear relationship between conductivity and concentration for
each constituent.
-104-
Table 4.2
Relative Conductivities of the Constituents of the Tracer Solution
Equivalent Concentration Conductivity
Solute Solvent by weight, Relative to*
I % Zero Load
I Arbitrary Units
None Distille~ 0 1.0 Water
Natural City Water unknown ~236
Salts Supply
NaCl Distilled 1 8.88X10 3 I
Water
7-K Distilled 1 2.50X10 3
blue dye Water
Methanol Distilled 1 1.7 (Tech Grade) Water
Notes: *Zero Load occurred when the probe was left in air with the electrodes dry.
tThe distilled water was essentially de-ionized.
Just prior to the beginning of experimental measurements, the
specific gravities and temperatures of the tracer solution and the
flume water were measured. If these values agreed to within accept-
able limits, the tracer storage flask was set up as shown in Figure
4. 12. The solution was supplied to the injector at a constant rate
from the constant head reservoir.
-105-
Co Calibration of probes. The purpose of the probe calibration was
(i) to deterll1ine if, within the lill1its of the concentration values to be
ll1easured during the experill1ents, there was a consistent (and per
haps linear) relationship between NaCI concentration and recorder
deflection (or conductivity), (U) to evaluate the proportionality
constant for each probe, and (iii) to check for the existence of
ground loops in the bridge circuits. The probes were separately
ill1ll1ersed in standard solutions having known concentrations of NaCI.
Corresponding deflections of the analog recorder were ll1easured.
Plots of recorder deflection versus NaCI concentration were developed
as shown in Figure 4.14 for RUN 7090 The calibration curves for the
various probes were linear and converged to a single point.
The probes were also ill1ll1ersed separately into a glass beaker
of a sall1ple of flull1e water, and then together into the flull1e water in
the flull1e, to check for ground loops and interactions between probes.
There was no ll1easurable difference between the recorder deflections
for the flull1e water in the beaker and the sall1e in the flull1e. This
indicated cOll1plete elill1ination of ground loops by the isolation trans
forll1er o
Calibration curves developed for various experill1ents were
virtually identical for the sall1e probes and preall1plifiers 0 During
any given experill1ent, the variation of flull1e water tell1perature was
less than 10 C; thus a set of calibration plots recorded at the beginning
of a run was used for ll1easurell1ents during the entire run.
en t-Z ::)
m a:: <t .. z o tU W -.J 1.1.. W Q
a:: w Q a:: o u w a:: (!)
9 <t Z <t
80~1----------T---------~----------T---------~----------T-~
60
40
20
o~
RUN 709 Flume S2
d =21.97 em, ti= 30.5 em/sec 3 u. = 1.40 em/sec t [R. = 3.082 X 10
200 400
PROBEN7
600 800 ~OO
No CI CONCENTRATION, ppm Fig. 4.14. Calibration curves for the conductivity probes.
....... o 0'
-107-
d. A typical run. A typical run in which data were recorded on the
strip chart and also digitized by the AID converter, will be outlined.
First the stations and the flow levels at which measurements would
be made were determined. In most experiments six stations and four
levels located at yld = 0.850,0.632,0.368, and 0.095 were used.
The tracer injector was placed usually at y Id = 1 Ie (= 0.368)
and about 15 meters from the flume entrance for the 110-cm flume.
In the 85-cm flume, the injector was stationed approximately 9 m
from the channel entrance. The tracer injection velocity was set at
the mean flow velocity, u.
The probes were calibrated and placed in the flume. Usually
three probes were used. If the flume water conductivity was exces
sive. the output voltage of the recorder was reduced to a convenient
level with a zero suppression control. A sine wave was always gen
erated by a function generator, observed on an oscilloscope during
the entire run, and fed to the first channel of the AID converter.
This offered a check on the integrity of the data recorded on the other
channels.
The probes were placed at the first level of measurement
(usually the level of tracer injection) at the station nearest the tracer
source. They were then set at the desired values of z (transverse
position) and the background concentration recorded for about 10
seconds. Then the tracer was injected continuously at the constant
velocity, u, and tracer concentration recorded for about 30 seconds
after allowing an initial period of process establishment. Tracer
supply was then shut off, and the probes moved to the next set of z
-108-
values. The procedure was then repeated. Adjacent probes were
overlapped at one point as a check on the conversion constants used
for inter-relating various probe measurements.
After the measurements were completed at the first station,
the probes were moved downstream to the next station until all
stations were covered. The probes were then set at another level,
and measurements made; this time moving upstream from one station
to the next. The proces s was then repeated moving downstream for
the third level and upstream for the fourth.
If the data were not digitized, an averaging switch was used
to automatically average the signal recorded on the strip chart over
a one- second period. This es sentially smoothed the record, and
facilitated the measurement of the time-mean concentration which
was obtained by simply laying a straight edge on the record, and
averaging by eye.
When data we re digitized, all calculations were made from the
magnetic tape, and the analog record, which was not averaged, pro
vided only a check on the digitized information.. At the end of the
experiment, all the header data for the entire experiment, including
the number of records in every file, were first retrieved. The result
was compared to a log which was drawn up during the experiment.
Then the mean value of the background concentration for each meas
urement was calculated. With these values and the conversion
factors for the probes now available, the entire experimental
measurements were analyzed.
For any fixed point, the instantaneous tracer concentration,
-109-
c(i) due to the point source of tracer was computed by the relation
(4.9)
where K, the conversion factor, is the ratio of slope of the calibra-
tion line for probe number 5 to that for the particular probe detecting
the concentration, at is the attenuation value for the analog channel
connected to the particular probe, Cb
is the average background
concentration, and cd(i) is the ith concentration value recorded on
tape for the particular channel within the sampling period. With
Eq. 4.9, therefore, concentration values for all probes were calcu-
lated relative to probe 5 set at attenuation 20 t thus giving a truncation
level of approxi mately 0.3 ppm equivalent conductivity of NaCI (1. e.
7 X 10-4 % of initial NaCI concentration).
4.B.3. Photo Studies. The photo studies were conducted in
the 85 - cm fl ume fo r only one unifo rm flow depth, d = 17 .. 0 cm.
To photograph the fluctuating plume, the motion picture camera was
mounted on the instrument carriage with the camera lens 2.13 m
above the water surface on the flume center line.. The camera made
24 exposures per second.
The plume was photographed at eight stations located 0.55,
0.70, 0.85, 0.95, 1. 10, 1. 28, 1.46, and 1.62 m downstream of the
source. At each station, 33 seconds of data were collected.
For each frame, the z values of the plume boundaries were
measured, and the difference between them used as the plume width
at that instant of exposure.. Readings were made at an interval of
4 frames (i. e. 1/6-sec). This was considered adequate because
-110-
the expected characteristic frequency of the pluITle ITleander was of
the order of one Hz.
4. B. 4. Velocity MeasureITlents and Results. Water flow
velocities were ITleasured priITlarily to (i) obtain velocity profiles ~
(ii) cOITlpute the von Karman constant, K, and (iii) develop velocity
contours as a check on the two -diITlensionality of the flow.. As a
secondary objective, velocity ITleasureITlents were ITlade to cOITlpute
flow discharge by integration of the velocity distribution in the cross
section.
The velocity profiles were logarithITlic, and of the forITl ex
pressed by Eq. 2.33. ExaITlples of profiles ITleasured during RUN
706V for which d = 2.75 CITl are shown in Figure 4 .. 15. The transverse
positions z where the ITleasureITlents were ITlade and the values of
the von KarITlan constant K are also included for the two stations
x = 400 CITl and x = 1900 CITl~ It is evident irOITl Figure 4 .. 15 and the
SUITlITlary of hydraulic data of Table 5.1 that 0.30:S K <: 0.43.
Velocity contours were plotted during RUN 506 V in flUITle
S1 for d = 2095 cm, and during RUNS 706V and 708V in flume S2
for the hydraulically smooth bottom with d = 2.75 cm and 17,,31 CITl
respectively. With fluITle 2 roughened on the bottom with rocks,
velocity contours were mapped during RUN 404V where d = 10.36 cm.
Contours obtained for RUNS 404 V and 708 V are shown in F; gure 4" 16
with one station of each run selected as an exaITlple ..
Both Figures 4.15 and 4.16 indicate that within the central
region of the flUITle where concentration measureITlents were confined,
1.00
0.80
0.60
0.40
~ 0.20
0.10
0.08
0.06
0.04
1 1.00
0.80
0.60
0.40
~ 0.20
0.10
0.08
0.06
0.04
20 20 20 20 20 20 20 20
20 20
VELOCITY, u,cm/sec
20 20 20 30
RUN 706V Flume 52
d=2.75 em,u =29.4 em/sec u.= 1.65 em/see,lR. =464
x=400cm
x =1900em
20 20 30
Fig. 4.15. Velocity profiles at two stations and several sections in flume S2; RUN 706V.
.......
.......
.......
E u
>-
I fCl. W o
E u
>. I fCl. W o
12
20
RUN 404V Flume R2 STATION 15.00ml12
'- 56-- • -56:::::::="" ~ •• ,. ~
'- 52 48 52 - _____ 8
4844 44- ~ =¥,\4 t'§= 38 30 t • i 30 38 • =-I 20 I I * . I 20~ i I
"" -20 //// -10 0 10 20 /,/~ 30 40 //// -50 -40 -30 50
TRANSVERSE DISTANCE, Z, em
20 RUN 708V Flume S2 STATION 15.00m
L 7 /:7 7 7./ /. 7 + " 42 ,:7 + \ \ \ \ \t \ \ J 16
~41 12 40 40~ 38---.-----"---__ 8
= : ~~~=- 30 32::=C::=------:------. : r~ ~- 34 -~ ---------. ~ f' -55 -50 // -40 //// -30 -20 ,if/? -10 0 //// 10 20 /1// 30 L 40 5 55 01) =24- :-=r28 I 24- 5 28 ! I t "/ • + 0
TRANSVERSE DISTANCE, Z, em
Fig. 4.16. Velocity contours at a station 15 m. from. the £lum.e entrance for RUNS 404V
and 708V. Cross-m.arks indicate points where velocities were m.easured.
Velocities on isovels in cm./sec.
t-'
t-'
N
-113-
the flow was essentially uniform across the flume except perhaps for
RUN 708V where the aspect ratio, d/W was largest. In particular,
it was found that within the central 800/0 of the flume, the maximum
deviation of u , the depth integrated velocity at a given z, from its a
cross-wise mean value was 3.1,0.7,6.4, am 3.5% for RUNS 506V,
706V, 708V, and 404V respectively.
It was found that the discharge calculated by integration of
velocities over fixed cross sections was always less than the metered
discharge by less than 30/0. This discrepancy was acc rued in the
integration of the velocity distribution near the flume walls. The
metered discharge was therefore considered a better measure of the
mean flow and was used for calculation of the mean velocity u as
expressed in Eq. 4 .. 2 ..
4. Eo 5. Probe Response to High Frequency Loading. To
measure concentration fluctuations, it was necessary to determine
probe response to a rapidly changing load.. Manufacturer specifica-
tions for the response of the recorder system could not be used
because the response of the conductivity probe depends on the trans-
ducer used and the cell volume surrounding the electrodes (see, for
example, Gibson and Schwarz (50), or Lamb, Manning, and Wilhelm
(60)).. For a given transducer, the larger the electrodes, the slower
is the response.
In this study, the probe response was determined by subjecting
the probes to a nearly instantaneous load, and recording the resulting
transient response of each probe on the magnetic tape. The probes
-114-
were placed in a tank of salt water such that initially the lower tips
of the electrodes were about 1 mm above the water surface with the
probe axes inclined at about 73 0 to the horizontal. Using a labora
tory "tsunami generator, " the water level was raised rapidly so that
the electrodes were completely immersed in 6.4 milliseconds. Once
immersed, the electrodes were left in the water at least until the
final deflection of the recorder was attained. Simultaneously the
output of each probe was digitized at 800 samples/sec, and recorded
on the magnetic tape.
The transient response curves of the probes, as recorded on
tape, are plotted in Figure 401 7(a). The input (or excitation) which
was approximately linear is also shown. The relative amplitude of
the ordinate is the ratio of the concentration at any time to the final
concentration value after an infinitely long time. Also included in
Figure 4. 17(a) is the rise time, re due to the excitation load. The
rise time was defined as the time to attain final deflection, as meas
ured by the best straight-line approximation to the initial rise of the
transient curve. Values of rt
were 14.9.13.2, and 11.0 millisecs
for probes 5, 4, and 6 respectively.
The frequency re spons e of the system is the relation between
the relative amplitude of the output and the frequency of a cyclic
loading imposed on the probe. Since this relationship could not be
phys !cally determined directly. it was calculated from the known
input signal and the corresponding transient response curves shown
in Figure 4.17(a). A method proposed by Walters and Rea (61) was
used.
w 0 ::)
r---.J 0... ~ « w > r-« --.J W 0::
3 ~
0::
w 0 ::)
r---.J 0... ~ « w > r-<I: -.J w 0::
-115-
1.20 (0)
100 INPUT SIGNAL
080 (excitation -PROBE TRANSIENTS load)
0.60
0.40 NOTE: rt = RI SE TIME,millisec
020
0.00 0 0 0 10 20 30
TIME, millisec
1.1
(b) -TO ZERO
1.0 --TO ZERO --TO ZERO
0
0
08 PROBE 4
06
0.4
02
00 2 5 10 20 50 100 20 50 100 20 50 100
EXCITATION FREQUENCY, w Hertz
Fig. 4.17. Response of the conductivity probes to high frequency
loading showing (a) the transient response curves, and
(b) the corresponding frequency response.
-116-
The input and transient response functions were represented
as Fourier series. A common fundamental period, T was chosen p
for both functions such that the transient had damped to its final
amplitude at a time less than T /20 The Fourier coefficients, I(n) , p
of the input series and J(n) of the output series were then calculated,
and the transfer function R(n)ehCP(n} determined by the relation
R( } iA>(n} _ J(n} n e - I{n) , (4.10)
where R(n) is the amplitude ratio and cp(n) , the phase shift of the
th transfer function fo r the n multiple of the fundamental frequency,
wf
' and l is n. The multiple n takes values 0, ±1, ±2, ••••
Thus R(w} and cj>(w} can be evaluated as functions of the real and
imaginary parts of J(n} and I{n} with w = nWr
Figure 4.1 7{b) shows the frequency response curves calculated
for the input signal and corresponding transients of Figure 4.1 7(a}.
For the three probes, the response was flat at R(w) = 1 0 0 for the
excitation frequency w::S 6.3 Hz. Beyond this point, the curves
decreases monotonically such that the relative amplitude, R(w) was
down about 3 decibels at approximately w = 30 Hz. Near w = 100 Hz,
the curves became oscillatory.
The limitations of the probes in measuring high frequency
signals become apparent. The probes would respond to fluctuations
up to a maximum frequency of 6.3 Hz without modulation of the ampli-
tude of the input signal. For higher frequencies, the probe output
would indicate a reading reduced according to the curves of Figure
-117-
4.1 7(b). Since the expected characteristic frequency of the concen
tration variations with time was in the order of 1 Hz (from prelimi
nary 0 bservations on the analog recorder), the probe response was
considered adequate for the intermittency and statistical analyses of
the concentration fluctuations. Data were digitized at 60 samples/sec
so that the cut-off frequency was 30 Hz.. This provided a suitable
sampling rate as will be shown in the next section.
4.C. REDUCTION OF DATA
4. C. 1. Recorded Concentration Data. Figure 4. 18(a) shows
examples of concentration measurements recorded on the strip chart
of the analog recorder.. Corresponding records digitized at 60
samples/sec by the A/D converter are also plotted in Figure 4. 18(b)
for comparison. The mean background concentration was used as
zero reference. The high frequency (30 Hz) fluctuation shown in the
plot of the digitized record is a O.01-volt background noise .. This
noise is not apparent in the analog record because it was filtered off
by the inertia of the recorder stylus.
Comparison of the analog and digitized records reveals that
all peak values indicated on the analog recorder were also recorded
on tape. This was expected because the response of the stylus of the
analog recorder was slower than the response at the output jack
leading to the A/D converter. Thus, in general, the A/D converter,
operating at 60 samples/sect furnished an excellent record of the
variation of the tracer concentration.
I_II. s 10 15 20 25
TIME, SECS
a (i) Analog record for z 5 CIT}
+ + L \
i i t 1 . , I !i I
~ 'll · , I j·+.1 ~ [' I I ~ ,'t! ~ -
, '" "" lt1m+~ ~2 '.' ".1. . •. ~"~~ '. ~ ~ I! II[ . . " ~ 0 .ld Jl !·h~~"t-~-WHu#~ ~r ii' I I
:" . ~
'" "". ,
,.
b (i) Corresponding digitized record for z 5 CIT}
..tiH L!1'~;;;' ",, ,." I, .. II. ii't 1,1, '''illll+ , .. I :~;t··:; j •• , •• It·, t I It. -II • + 1+ 1-+ .......... .
JUtl illt t ~ ..... I'" m, IJ' '1 111 1 ,. iLl ,J1I'l , ......... . pi: . . ... , ... , 't', '1 . 't· t ',~. ' .. , .... .. TI ,.. • ..... 1" , , I . t i ' ',T ........ i···· 6 '1 . ..~ .' _ ....
11 I I I'tlt ~: I • ~: :: :.; I : ! I I+;. ; j : .!: :: :::: .: .. II I : I . ... ,j " . 'J t I "I, , .. , .... ,.. j' "T i:.il , ' l' ., .• j • , 1 j .. t ~.: r. '-..:" .... __
11. ' ......... , I 111'j'l .1., '~I=:"I'I~"" [" ...... " t· 1 1'1 ....... 0 .' c'" "l'li I 'i .,,, ... .
4 ~i; ': ~::: :::: t ,~: ; ;.1': :: .. >~_~_~_.~~ 11 1 i ',... . ..• t t. ~ I. II t ." t ill !llh ... .... .. • !" j" ' •• 1 ~ 1 j j t • t • • , •• • •• .
·Ut. ,Lt, ·'..1·il , ,'.; Wi,'" ' .... ~
!II ' n Ik'I':lI il' "j' ... I',..... .. : • ; I I tf t 1 . .. .. 'i'" .. ... j Ii~i :: ' If 11.l : :. . : i : : 1: :.: ':. 2 . • .,~ . __
U llt ~;ltH ,r. l!t"Jai i:: ';j;·d:> I i ;': M\ ' Jii' l~i~wi;~ ~!. I;jH~ ;iit:
!IIi I II I I. J.~ 1tJ~~ i! I! I; I;} i i jlHI 1±±lU t ilRUW!1OOW±l1jElil
, , , IIIIII~ I ~ l'tjhJ! 1 ttiil±lli±fit ~ iUi.~~ 6 10 15 20 26
TIME, SECS
a (ii) Analo g record for z ·5 CIT}
I" d; i-·lj~
,. .+.
, 1:' ~..L~\"
•• TIM(. ues
b (ii) Corresponding digitized record for z ·5 cm
Fig. 4. 18. Typical concentration data at two points as recorded by the analog recorder
shown in a(i) and a(ii). The same data digitized by the A/D converter at
60 samples/sec are correspondingly plotted in b(i) and b(ii); RUN 806.
~
~
00 I
-119-
4. C. 2. Effects of SaITlpling Period and SaITlpling Rate or:
Calculated ParaITleters.
a. Effects on the tiITle-ITlean concentration. The saITlpling period
T is the total length of tiITle over which concentration used for sub-ITl
sequent analyses was ITleasured at a fixed point.. The tiITle-ITlean
concentration C is the average over the saITlpling period T • The ITl
value of C is thus a function of the length of T ,and theoretically ITl
would approch a liITliting value as T becOITles infinite. In practice, ITl
however, the saITlples ITlust be truncated at a finite T such that for ITl
periods longer than T , C is essentially invariant. Ogura (62) ITl
showed that the deviation of C froITl the aSYITlptotic value increases
with the size of the largest eddy effecting ITlixing. Thus the greater
the flow depth the longer is the saITlpling period required to obtain a
representative value of C ..
MeasureITlents were ITlade to deterITline an appropriate value
of T • RUN 804 for which d (::: 10.84 CITl) was larger than ITlost of ITl
the other experiITlents was chosen. Concentration data digitized at
60 saITlples/sec were recorded for one minute, and C evaluated
nUITlerically with T increasing frOITl 4 to 56 seconds. The result ITl
is shown in Figure 4.19. The relationships were developed fo r
x/d::: 7.4,16.6,35.0,71.9, and 108.8 at values of z where the
interITlittency factor If ~ 0.50 for each case. All points were located
at 11 = 0.368. the level of tracer injection. The interITlittency factor
was calculated with the threshold concentration equal to the ITlean
background concentration. FroITl Figure 4.19, it was concluded that
for experiITlents where d <: 10.84 CITl, T = approxiITlately 25 seconds ITl
en .. -U ·c
.. =t Z 0..0 - ~ ..... 0 <t 0:0 ..... z z::;) Wo Uo: z<.!) o~ Uu z<t <t(J)
Ww ~:t: I .....
Ww ~> -0 ..... (J)
<t
0.30 x d = 7.4 o
0.20 o
0.10
0.00' ..J o
0.30
0.10
20 30 40 50 Tm ,sec
2.... = 16.6 d
o 0 " 7 0 0 0
60
O.OOb 10 20 30 ~O 510 60
0.10
0.06
0.02
Tm ,sec
x cr=71.9
O.OOb 10 20 30 4'0 5~ ~o PERIOD, Tm ,sec
0.10
0.06
0.02
RUN 904
Flume 52
d=10.84cm,u = 39.2cm/sec
m* = 2,026
o x d = 35,0
o Q Q
~o 0-0 0 0 0
0.00 ,-I __ ~_---I. __ ....L-__ L.-_--'-_---'
o 10 20 30 40 50 60 T m ,sec
0.06 x d = 108.8
0.02 O
_..lI°",--..,.o ... o ° _-0-
~~o
0.000 10 20 30 40 50 60
SAMPLING PERIOD, Tm ,sec
Fig. 4. 19. The effect of sampling period on time-mean concentration; RUN 904.
..... N o I
0.22
0.20
(/) .t c: :J O.IS ~ \.. e -:.e 0.16 o
.. u .. z o
0.14
~ 0.12 0:::: r-Z W u 0.10 Z o U Z O.OS <! w ~ L1.J 0.06 ~
r-0.04
-121-
~--~----------
RUN 904 Flume S2
d=10.S4 cm,u=39.2 em/sec TR* =2,026
SAMPLING PERIOD, Tm=2S sec
x/d
7.4
16,6
P---,::r----o--------<J 35.0
J::--~----I~------_n 71.9
0.02 ~-o--o----o()--------<>IOS.S
O.O-------~----------~----------~-----------------~-----------------~-------o 10 20 30 40 50 60 70
SAMPLI NG RATE, samples/sec Fig. 4.20. The effect of sampling rate on time-mean
concentration: RUN 904.
-122-
was adequate for determining C.
Figure 4.20 illustrates the effect of sampling rate S on r
computed C for the same values of x/d and z as in Figure 4019.
and T = 28 secs. The plots show that for this sampling period, a m
consistent value of C was obtained for S greater than 15 samples / r
sec for all points investigated ..
b. Effects on the intermittency factor, If. Figure 4.21 illustrates
the effect of sampling period, T on the intermittency factor m
at various x/d. The points chosen for analysis are exactly the same
as in Figure 4019. The figure shows that a fairly stable value of If
was achieved for T > 22 secs. As expected, the deviation from m
the mean value of If at large T m was greatest at higher values of
x/d where the signal-to-noise ratio was lowest.. Indeed for x/d =
108.8, a constant value of If was not actually attained within
T m = 56 secs. This meant that a large scatter in the plots of If{z)
versus z would be realized at large x/d unless longer records
were utilized"
The effect of sampling rate on If is shown in Figure 4" 22.
The points analyzed are the same as those in Figure 4" 19" The
graphs show that for Tm = 28 secs, an essentially constant If was
achieved beyond S ~ 18 sample s / sec at all stations. r
c. Choice of the sampling period and the sampling rate. The effects
of T and S on other statistical parameters were determined m r
for RUN 804. On the basis of the analyses, a sampling period of at
least 28 sees for d > 10.84 cm was used. For lower values of d,
-
-123-
0.6 x/d = 7.4
~ :UN;04 cr a Flume S 2
0.3
0.60
0.30
0.60
Fig. 4. 21.
d= 10.84 em,u = 39.2 em/sec TR* = 2 1026
SAMPLING RATE = 60 samples/sec 16.6
~ __ 35.0 0 ~_dOD---_D __ ~O~--uo~~~
71.9
108.8
10 20 30 40 50 60
SAMPLING PERIOD, Tm sec The effect of saITlpling period on the interITlittency factor; RUN 904.
O.55
V x/d
00(). 074 v .
0.45
~ 0.501 6166 is:O . .. "-"
a::: ~ u 040
Et >u z W lI-~ a::: w IZ
RUN 904 Flume S2
d=10.84 em,u=39.2 em/sec TR. =2,026
SAMPLING PERIOD, Tm =28 sec - 0 0 0108.8
10 20 30 40 50 60 SAMPLI NG RATE, samples/sec
Fig. 4. 22. The effect of saITlpling rate on the interITlittency factor: RUN 904.
...... N ~ I
-125-
T was set at 25 sees. In all experiments, S was 60 samples/sec. m r
At this rate, all fluctuations with frequencies equal to or less than
30 Hz would be recorded on tape. The maximum eddy size whose
effects were measured would have a frequency of 1 /T (~1 /2 5 Hz). m
4 .. C. 3. Choice of the Threshold Concentration Ct for
Determination of the Intermittency Factor. If.. It is immediately
apparent from Figure 3.3 that the value of If at any fixed point
depends greatly on the threshold level Ct
used. This is especially
true in regions far from the source where the s ignal .. to-noise ratio
is smallest. To illustrate the influence of Ct
on If at a fixed point,
a 56-second record in RUN 804 was analyzed for x/d = 7 .. 4,16.6,
35.0,71.9, and 108.8, all at the level 1'] = 0.368. For each station
z was chosen such that the asymptotic value of If relative to the
background concentration was approximately 0.5. At each point If
was calculated for various ratios of the threshold concentration to
the local time-mean concentration Ct/C. The result is shown in
Figure 4.23.. For clarity, only three stations are plotted. It is
observed that If rose suddenly to 1.0 for Ct/C < 0.0, and decayed
gradually as Ct/C increased beyond zero. At Ct/C = 0, the con-
centration equalled the rnean background concentration.
The resulting modification of the transverse distribution of If
by the choice of Ct/C is shown in Figure 4.24 for RUN 804 with
x/d = 7.4. The plots again indicate that substantial changes could be
introduced without a proper choice of Ct "
--.. 1.0
cr: 0 I-
~ 0.8 LL
>-u 0.6 Z w l-I-- 0.4 ~ cr: W I- 0.2 z
90 .2 0
THE
RUN 904 Flume S2
d= 10.84cm, u=39.2cm/sec TR* = 2,026
SAMPLING PERIOD, Tm = 56 sec
x/d C/C max 0 7.4 0.13 A 16.6 0.14 0 35.0 0.10
1.0 2.0 RATIO OF THE THRESHOLD TO TI ME - MEAN CONCENTRATION, ct/c
Fig. 4. 23. The effect of the choice of threshold concentration C t on the
intermittency factor. Each curve applies to a fixed point, and
C t is normalized by the local mean concentration; RUN 904.
...... N 0'
1.0. rr-t- ~-o-~ -;r ~t -0.40 I /-0.20 0.-,/ ;0.40
- I I / / ~ 0.8 , I -/ / :- RUN 804 'I • // )
gj d=10.84cm,u=39.2cm/sec, I / / ~ x/d = 7.4, Y/d =0.368, /./ ~ 0.6 ,I / / l.C I I /}/ L; " II. / Z II I / ~ I
~ 0.4 I I ~ I ~ ~ I I II / I
~ I 1 / /M" w I P V / ~ 0.2 I I / Z ~. I / pI /
// 1/
00' I I 6=. I I I I I
. -14 -12 -10 -8 -6 -4 -2 0
TRANSVERSE DISTANCE, l,cm Fig. 4.24. Modification of the transverse distribution of the intermittency factor
by the choice of the threshold concentration; RUN 804.
-128-
In the computer program used for calculating If' C t was
chosen to be slightly above the mean background to effectively elimi
nate the background noise. Thus in all calculations of If' the thresh
hold concentration was set essentially equal to the background con
centration ..
-129-
CHAPTER 5
PRESENTATION AND DISCUSSION OF
EXPERIMENTAL RESULTS (Phase I)
This chapter sUITlInarizes the hydraulic data for all experi.,
ments and presents the results of measurements relating to time
averaged concentration. Each result (or set of results) is discussed
according to the objectives outlined in Chapter 2.
5.A. HYDRAULIC DATA
Table 5.1 is a complete summary of the hydraulic data for all
experiments related to both Phases I and II. The experiments are
grouped in an increasing order of flow depths which ranged from 1.52
to 21.97 cm.
Column 1 lists the experimental runs, and Column 2 the phase
of the study undertaken: Phase I refers to time-averaged concentra
tion measurements, and Phase II to concentration fluctuation analyses.
Column 3 identifies the flumes as explained in Table 4.1. Columns 4
through 7 are explained as indicated. The shear velocities shown in
Column 8 were calculated according to Eq. 4.3. However, the bed
shear velocity was used for experiments conducted in the flume
roughened with rocks. Friction factors entered in Column 9 were
determined using values of the shear velocity given in Column 8.
The von Karman constant K, tabulated in Column 10, was
calculated for only one run in each set of hydraulically similar experi-
Table 5.1. ---.------
Phase Flllnl(> Normal Hydraulic 1..1ean Energy RUN of , Idcntif.
2 Depth Radius Velocity 5lope. Study Code
d r U 5f
em em cn1/sec
IxJ03
) -------:- r----- --,-- 1-----I--
I 2 3 4 5 6 7
507 I 51 I. 52 I. 47 31.2 3. 110 705 I 52 I. 69 I. 64 32.8 2.467 805 II 52 I. 69 I. 64 33. 5 2.464
707 I 52 2.74 2.61 50.4 2.735 706 I 52 2. 75 2.62 30.0 I. 063 R07 I, Il 52 2. 76 2. 63 49.5 2. 735 S06 I. Il 52 2. 77 2.64 29.7 I. 064 ,06 I 51 2.95 2.76 27. I 0.910
-- f-703 I 52 3.46 3.26 32.0 0.973 504 I 51 3.47 3. 21 29.9 0.631 SO) I 51 3. 57 3.29 37.2 I. 040 503 I 51 3.93 3.60 33. I 0.779
501 I 51 4. 25 3.86 32. 9 0.811
509 I SI 5.25 4.67 42.6 I. 110 5 II I 51 5. 25 4.67 42.8 0.943 'i12 I 51 5.25 4.67 42.5 I. 030 S08 I SI S.26 4.68 4 I. 7 O. 990 5 10 I 51 5.26 4.68 42.4 O. 978 hOO II 51 S.26 4.68 42.3 I. 005 (,0 I II 51 5.26 4. 68 42.3 O. 990 802 I, II 52 5. 36 4.88 43.7 0.981 702 I 52 5.41 4. 93 43.5 0.981 70 I I 52 5. 53 5.02 42.0 0.981
400 I, II R2 6.81 6.06 35. 9 3.837 L--_ _ '-------_
Summary of hydraulic data.
"-
Shear Friction von Ka'rman Froude Kinematic Ve.locity, 3
Factor 4 Constant, Nunlbcr Viscosity.
(",,2 U u",oJgr 5f f~8 u) , Fo- V
jgd em/sec em';:: /sec
Ix 10 2 ) Ix 102
)
8 9 10 II 12
2. 12 3.69 0.364 0.808 0.992 I. 99 2. 95 - 0.805 0.960 I. 99 2.83 - 0.828 0.980
2.65 2.21 0.347 0.972 0.979 I. 65 2.43 O. 408 0.578 O. 980 2.66 2. 30 - 0.951 0.978 I. 66 2.50 - 0.570 O. 978 I. 57 2.68 0.374 O. 504 0.992
I. 76 2.43 - 0.549 0.961 I. 41 I. 78 0.341 O. 512 O. 980 I. 83 I. 94 0.363 0.629 O. 975 I. 66 2. 01 0.38 I O. 533 0.980
I. 75 2.27 O. 375 0.510 0.975
2.26 2.24 - 0.594 0.992 2.08 I. 89 - 0.596 O. 953 2. 17 2.09 - 0.592 O. %0 2. 13 2.09 O. 375 0.581 0.992 2. 12 2.00 - 0.590 I. 005 2. 15 2.06 - 0.589 0.991 2. 13 2.03 - 0.589 I. 0 II 2. 17 I. 97 - 0.602 0.990 2.18 2.01 0.373 O. 598 0.982 2.20 2. 19 0.372 O. 570 0.969
5.01 15. 61 - 0.439 I. 007
I Continued)
Reynolds Friction Reynolds
Nurn.ber :;\fun1ber. 4
R= 4u~ u"d
R",o-v v
IxI0- 4 ) Ix 10- 3 )
13 14
I. 85 O. 325 2.24 0.351 2.24 O. 343
-- ~---~--
5. 37 0.741 3.21 0.464 5. 32 O. 750 3.20 0.470 3.01 0.464
-------4.34 O. (,35 3. n 0.499 5.03 0.671 4.86 0.665
5.21 O. 764
8.03 I. 193 8. 39 I. 145 8.27 I. IRS 7.87 I. 130 7.90 I. 109 7.99 I. 140 7.83 I. 109 8.62 I. 173 8. 74 I. 200 8.71 I. 254
8.64 3.388
Level of Tl'ac('r
Injection
h
15
0.368 O. }OK O. 500
O. 36R O. 368 O. 368 O. 36H O. %8
-----,-
O. 3(,8 O. 3(,8 O. 368 n.36H
O. )22
O. 368 O. 05 I 0.8S0 O. %8 0.612 ~.368 O. 368 0.368 O. 129 O. 391
O. 368
Rlll\:
I
507 705 805
707 706 K07 H06 50(, --
703 504 505 503
50 I
50!)
511 "12 50R SIO 600 (,01 802 702 70 I
405
>-'
W o I
Table 5.1 (Continued)
Phase FluD1e I'\orn1al Hydraulic Mean Energy Shear Friction von Karman Froude Kinematic Reynolds
Friction Levf'l of RUT'\" of Identif.
Depth Radius Velocity Slope. Velocity, 3 Factor 4 Constant, l\urnber Viscosity, Nurnber Reynold s Tracer RUN
Study 1 Code 2 :\'umber, '* Injection
f.=8Cf F=~ R =:!c!i!_ u",d
d r U Sf u,:,=Jgr Sf ~ R,,=-,-, r'h u hd \!
em em em/sec cD1/sec cm 2 /sec
Ix 10 3 ) Ix102) Ix 10 2 ) rxl0- 4 ) Ix 10- 3)
1'i'-~ ---I 2 3 4 5 6 7 8 9 10 11 12 13 14 1
407 I. II R2 8.66 7.48 41. 0 3. 189 5. 12 12.50 - 0.445 1. 018 12.05 4. 3S6 O. 368 407
404 I. II R2 10.36 8.72 42.8 2. 597 5. as 11. 15 O. 324 0.425 1. 052 14.19 4.973 O. 368 404 606 II 51 10.70 8.55 42.6 0.454 1. 95 1. 68 - 0.416 0.945 15.43 2.210 O. 368 (,ali 607 I, Il 51 10.70 8. S5 41. 8 0.429 1. 90 1. 65 0.333 0.408 0.955 14.95 2. 125 O. 368 60, ,04 I S2 10.81 9.03 39.2 0.390 1. 86 I. 80 0.357 0.381 0.978 14.58 2. 05 S O. 368 704 804 II S2 10.84 9.06 39.2 0.390 1.86 I. 80 - 0.380 0.996 14.25 2.026 0.368 H04 904 Il S2 10.84 9.06 39.2 0.390 I. 86 I. 80 - O. 380 0.996 14.25 2.026 0.368 '104
300 II SI 17.00 12. 10 35.4 0.210 I. 58 1. 60 - 0.274 I. 020 16.80 2.634 ~1. 00 300
406 I. II R2 17.07 13.03 35. 3 0.853 3.64 8.49 - 0.273 1. 087 16.9Z 5. 716 0.368 406 708 I S2 17.31 13. 17 34.9 0.213 1. 66 I. 81 0.332 0.268 0.961 19. Il 2.988 0.368 708 808 I. Il S2 17.32 13. 17 34.9 0.213 1. 66 1. 81 0.332 0.268 0.978 18.80 2.938 0.31,8 808
...... W ......
602 I, II SI 17.34 12.32 36. 5 0.242 1. 71 1. 75 - 0.280 0.978 18.38 3. 038 0.368 602 h03 I. II SI 17.34 12.32 36.9 0.263 1. 75 I. 80 - 0.283 0.978 18.59 3. 110 0.368 h03 604 Il SI 17.34 12.32 36.9 0.263 1. 75 I. 80 0.388 0.283 0.953 19. 07 3. 181 0.368 604 605 I. II 51 17.34 12.32 37.0 0.263 1. 75 1. 80 - 0.284 O. 963 18. 93 3. 105 0.850 60';
709 I 52 21. 97 15.70 30. 5 O. 127 1. 40 1. 68 - 0.208 0.997 19.21 3. 082 O. 368 ,09 809 I. II S2 21. 97 15. 70 30. 7 O. 126 1. 39 1. 65 - O. 209 O. 978 19. 71 3. 140 O. 368 S09
GOTES: : Phase I is related to tin1e -averaged concentration; phase II to temporal fluctuations of tracer concentration. 2 For flume identification, Sand R denote smooth boundaries and rough bottom respectively. Flurn.e 1 is 85 em wide. Flume 2 is 110 en1 wide. 3 For the 400 Series only, the listed values are for bed shear velocity calculated by a side-wall correction method; g::gravitational acceleration . . For the 400 Series only. f applies only to rough bed.
---- ---- --- -- --_ .. _-- -- - -
-132-
ITlents. For exaITlple, a K-value of 0.375 ITleasured for RUN 508
was used for RUNS 509, 510, 511, 512,600, and 601 which were
essentially identical to RUN 508.
A s indicated in ColuITlns 11, 13, and 14, all flows were sub-
critical and turbulent. The Froude nUITlber varied froITl a high value
of 0.972 for d=2.74cITl toalowofO.208for d=21.97cITl. The
Reynolds nUITlber ill was greater than 1.8 X 104
in all experiITlents,
and the ITliniITluITl value of the friction Reynolds nUITlber ill* was 325.
ColuITln 15 lists the level llh at which tracer was injected.
For ITlost experiITlents, llh = 1le = 0.368, and the injection velocity
was set at the ITlean water flow velocity.. The injection level was
specifically varied for the runs where the norITlal depth was approxi-
ITlately 5.25 CITl in order to deterITline the effect of llh on initial
distributions of tracer just downstreaITl of the source. In this set of
runs, the following values of llh were used: 0.368, 0 .. 632, 0.850,
and 0.051. The injection velocity in each case was adjusted to
approxiITlately equal the local water flow velocity.
5.B. TRANSVERSE DISTRIBUTIONS OF TIME-AVERAGED CONCENTRATION
Typical transverse distributions of the tiITle-averaged concen-
tration within the pluITle are shown in Figures 5 .. 1, 5.2, and 5.3 for
various norITlal depths. The runs shown were selected to cover
ITleaSureITlents ITlade at various levels of the flow with the fluITle
bottoITl either SITlooth or roughened with rocks.
In Figures 5. 1(a), (b), and (c), the ITleasured concentration C
-133-1.4.----r------.---r--.---,---.--.---,.-------;,---,---,-------;--,---,---,
x o E 1.2 ~ u z o ~ a:::
1.0
f- 0.8
iE u is 0.6 u o ~ 0.4 -.J <l:
~ 0.2 o z
RUN 706 Flume S2
d=2.75 em u=30.0 em/sec, u.=1.65 em/sec
IR = 464, 7]. 0.368
-30 -25 -20 -15 -10 -5 o 5 10 15
(a)
"I =0.368
LEGEND
Cmaxl Z at
SYMBOL , Cmax ' arb. units em
0 29.1 83.5 -1.0
• 65.4 55.2 0.0 <) 138.2 36.7 0.0 ~ 247.2 27.9 0.0 ~ 392.7 235 0.0 e 610.8 17.9 0.0 -- GAUSSIAN CURVES
20 25 30 35
1.4 f----r----,r--.---,---.---r--.-----r--r---.----.--.---,---.--~
S uE 1.2 "-u Zo ~ a:::
1.0
f- 0.8 Z W u is 0.6 u o ~ 0.4 -.J <l:
~ 0.2 o Z
(b)
7] = 0.800
LEGEND
Cmax • Z at
SYMBOL , Cmax' orb. units em
0 29.1 54.0 -0.8
• 65.4 52.9 0.4 <) 138.2 38.2 0.0 ~ 247.2 28.3 0.5 ~ 392.7 22.0 0.8 e 610.8 18.1 0.4 -- GAUSSIAN CURVES
1.4 f----.----.--..---,---.----,---.---.----,---r---,---.---.---.----l
x o
u E 1.2 "-u Z o ~ a:::
1.0
f- 0.8 Z W u is 0.6 U
o ~ 0.4 :::J <l:
~ 0.2 o z
Fig. 5.1.
SYMBOL
0
• <)
~
~
e
(el
7]=0.236
LEGEND
Cmox • , arb. units
29.1 92.0 65.4 52.8
138.2 37.5 247.2 28.3 392.7 23.0 610.8 183
Z at Cmox • em -1.0 0.0 0.0 0.4 0.0
I 0.0 -- GAUSSIAN CURVES
-15 -10 -5 o 5 10 15 20 25 30 35
TRANSVERSE DISTANCE FROM FLUME CENTERLlNE,Z,CM
Transverse distributions of the time-averaged
concentration measured at various distances
S = x/d downstream of source, and at various
levels of the flow: (a) n = Ilh = 0.368,
(b) n = 0.800, (c) n = 0.236; RUN 706.
99.9
~ 990~ 98.0
aY z 90.0 0
~ eoo
l, I 0:
I-Z
50.0 .• ! w <.) z 0 <.)
W 20.0 > ~ 10.0 -l ::)
:::?! ::> <.)
0.1 L-L
-5
_ 1
!
1
i I
I I I
RUN 708
Flume S2 d=17.31 em
p
u=34.9 em/sec, u.= 1.66 em/sec 7J =0.632, 7Jh 0.368
IR.= 2.988 x Id /" GAUSSIAN LINES
0 5 -5 0 5 -5 0 5 -5 0 5 -5 0 5 -5 0 5
TRANSVERSE DISTANCE. Z. FROM FLUME CENTER LINE IN CM
Fig. 5.2. Transverse distributions of the cUITlulative ITlean concentration ITleasured
at various distances S froITl the source and at the flow level'll = O. 632.
Plots on arithITletic probability paper; RUN 708.
>--' \.N ,.(:..
~
a....°50.0 ~
Z o ~ a:: 20.0 I
~ ~ Z 10.0 ~ p
~ RUN 405 Z 8 2.0 Flume R2
1.0 d = 6.81 em W 05 u=35.9 em/see,u =5.01 em/sec > . *b
I-:: 'T] = "I = 0.368 ~ h --1 0.1 IR = 3.388X 103 ~ ~ ::E / GAUSSIAN LINES ~ 001 I I I
u -8 -4 0 -8 -4 0 -8 -4 0 -8 -4 0
TRANSVERSE DISTANCE ,Z,FROM FLUME CENTER LINE IN CM
Fig. 5.3. Transverse distributions of the cumulative mean concentration measured
at various distances S from the source and at the flow level T] = 0.368.
Plots on arithmetic probability paper; RUN 405.
...... w U1 I
-136-
norITlaliz ed by the ITlaxiITlUITl concentration C located at the ITlode ITlax
of the concentration distribution is plotted against the transverse
distance, z 0 All ITleasureITlents plotted in this set of figures were
ITlade during RUN 706 in flUITle S2. The flow depth was 2.75 CITl, the
ITlean flow velocity 30.0 cITl/sec and the friction Reynolds nUITlber
ffi* = 464. Tracer was injected at 'lh = 0.368.
In each set of graphs shown in Figure 5. 1 (for exaITlple,
Figure 5. l(a)), ITleasureITlents ITlade at a fixed level of the flow are
plotted fo r various distances ~ s. The points shown are the ITleas ured
values ~ and the fitted curves are Gaussian distributions each having
a variance equal to that calculated nUITlerically froITl the ITleasured
data, and a corresponding value of C 0 The legend explains the ITlax
plotting symbols: the distances S at which the ITleasureITlents were
ITlade, the actual values of C and the transverse positions z at ITlax t
which C was located. Distributions at the levels 'l = 0.368, ITlax
0.800, and 0.236 are shown in Figure 5.1(a), 5.1(b), and 5.1(c)
respectively.
To illustrate the forITl of the concentration distributions at
higher values of IR*, the cUITlulative concentration, Pc' was plotted
against z on arithITletic probability scales for various S in Figure
5.2 for RUN 708 where m* = 2.988 X 10 3 , and in Figure 5.3 for RUN
405 where the flUITle bottoITl was rough and m *b = 3.388 X 10 3 • The
variable P was calculated by the relation c
P (x 1 , Yl , z .) c J
-137-
where Xl and Yl are fixed values of x and y; Zo is the point
beyond which, for increasing z, C first attains a non-zero value;
zN+1 is the largest value of z at which C first becomes zero;
.6z i = zi +1 - zi' i = 0,1, ••• , N+ 1; and j = 1,2' •• 0' N. In cases where
.6z. is small (about 1 ern) and constant, and N was large, P was 1 c
calculated s imply by
(5.1b)
Straight lines representing Gaussian distributions were fitted to the
plots. A representative standard deviation
each distribution by the relation
0" was determined for p
(5.2)
where z84.1 and z15.9 are, respectively, the values of z where
the fitted Gaussian line intersects P = 84.1 and P = 15.9. As will c c
be pointed out in the next section, 0"2 in general differed from the p
variance, 0"2, calculated directly from the measured data using Eq.
5.3.
The plots shown in Figures 5.1 through 5 .. 3 demonstrate that
the transverse distribution of the time-averaged concentration was
very closely Gaus sian at all levels of measurements and for S
extending from 4.0 "to 610.8 regardless of whether the flume bottom
was smooth or roughened with rocks. Deviations from the fitted
normal distribution occurred only at large values of I z I/w where
-138-
the tails of the distributions were very near the flume wall. This is
attributed primarily to the effects of the side-wall boundary layers
generated next to the flume walls. Reflections from the side walls
also contributed, to a lesser degree, to the deviations of measured
C at the tails from the Gaus s ian distribution.
For a given experilTIent, the closeness of fit of experiITlental
points to the normal distribution was not a function of either the
distance downstream of source g 0 r the level of concentration meas-
urement Y). The measured points appeared to fit best at the lowest
and highest values of g, as shown in Figures 5.1 through 5.3. How-
eve r, the deviatio ns of the points from the Gaus sian curve at inte r-
mediate g were still generally insignificant. It is thus concluded that
the transverse distribution of C is Gaussian both very near the
source (s = 4.0) and far away, and at all levels of the flow.
5.C. VARIANCE OF THE TRANSVERSE CONCENTRATION DISTRIBUTION
5. Co 1 ~ Calculation of the Variance. The variance (1"2 of the
cros s-wise distribution of tracer concentration for fixed x and y
defined by Eq. 2.56, was calculated numerically by
(5.3)
where SMi
and FMi are, respectively, the second and first moments
of the trapezoid formed by the concentration values C(zi)' C(zi+1) and
the points zi' zi+1' and Ai is the trapezoidal area
-139-
(5.4)
In most experiments t values of C used in Eq. 5.3 were the measured
concentration values. In some cases where the experimental data
closely fitted a smooth curve, C was obtained as points on the ex-
perimental curve.
The variance a2 calculated by the moment method of Eq. 5.3
was compared with 0-2 determined from the arithmetic probability p
plots and evaluated by Eq. 5.2. The slopes of the straight lines,
d0-2 /dx and d0-2 /dx t fitted to the plots of a2 p
and a2 versus x, were p
also compared. The results are shown in Table 5.2. Four depths
of flow were chosen: 2.75, 5.53, 10.70, and 17.31 em with cor-
responding 1R *::: 464, 1254 t 2125, and 2988. For each run a different
level of measurement YJ was selected as an example. The compari-
son was made at various S ranging from 4.0 to 611.0.
As shown in the table the discrepancy between
was generally non-zero but within 140/0, and tended to increase with
increasing lR*. The errors between d0-2 /dx and d~ /dx were even p
smaller remaining within 80/0. The moment method (1. e. 0-2) was used
for the calculation of the mixing coefficients in all experiments except
the 400 series where the probability method (i.e. 0-2 ) was used. p
Table A.1 of the Appendix lists the calculated values of the
variance ~ for experiments in which concentration was measured
at more than two levels of the flow. In RUN 705 where d::: 1.69 em,
the flow depth was too small to allow measurements at more than two
Table 5.2 Comparison of the variance cr 2 computed by the moment
method and cr2
derived from the probability method p
cr:3 :3 cr:3 cr:3 cr RUN S p Error
RUN S p :3 :3 0/0 ':' :3 . 2 em em em em
1 2 3 4 5 1 2 3 4
29. 1 2.95 2.89 -2.9 14. 5 4. 59 4.71 706 65. 5 6.11 6.25 +2. 3 701 32. 5 12.89 13. 19
138. 1 16.84 16.40 -2.6 68. 7 29.88 29. 18 d=2. 75 em 247.5 36.06 36.00 -0.2 d=5. 53 em 122.9 52. 16 51. 10
393.0 59. 19 59.29 +0.2 213.3 94.85 90.25 R,:, = 464 611. 0 90. 76 92.16 +1. 5 R,:, = 1,254 303.7 140.28 140.30
(Tl=O. 368) dcr:3 (Tl=O. 855) dcr:3 dx = O. 0566 em
+2.3 dx = 0.0854 em
dcr:3 dcr:3 dt = 0.0579 em d!- = O. 0826 em
:3 cr 2 2 2 cr Error cr ()
RUN S P RUN S P 2 2 0/0 2 2 em em em em
1 2 3 4 5 1 2 3 4
4.0 2.5 2.6 +4.0 4.6 5.6 6.4 607 8.7 9. 1 8.8 -3_ 3 708 10.4 26.0 26.0
18.0 15.2 16.4 +7.9 21. 9 64. 7 65.6 d=10. 70 em 36. 7 43.3 39.8 -5.8 d=17.31 em 39.2 126.4 124.3
50. 7 54.8 58.2 +6.2 62.4 195.0 216. 1 R,:, = 2,125 69.4 80.7 77.8 -3.6 R,:, = 2,988 99.4 313.2 349.0
(Tl = 0.095) dcr 2
(11 = 0.632) dcr:3
dx = 0.109 em -- = 0.188 em 0.0 dx
dcr2 dcr 2
-E. = 0.109 em d!- = O. 203 em dx
NOTE: ~:~ The variance ()2 is used as bas e.
Error 0/0 ':'
5
+2.6 +2.3 -2. 3 -2.0 -4.8 +0.0
-3. 3
Error 0/0
5
+14.3 - 0.0 + 1. 4 - 1. 7 +l0.8 + 11. 5
+ 8.0 , ,
I
1-'-
.r::. o
-141-
levels. In the 400 series, where cr2 was calculated, concentration p
was measured only at T) = 0.368 and 0.632. The depth-averaged
variance, denoted by A VG in the table, was dete rmined as a simple
average of the non-zero values of the variance at the various levels
of measurement.
5. C. 2.. Growth of the Variance cr2 With Distance x Down-
stream of Source. Figures 5.4 through 5.7 are plots of the variance
cr2 versus x, developed for various levels of the flow T). The bottom
curve in each figure denoted as A VG is the plot of the depth-averaged
variance ~ versus x. The runs shown were selected as typical
examples covering a wide range of flow depths and IR*"
The plots of Figures 5.4 through 5.7 show that the variance
grew linearly with x both at various levels of the flow and as a depth
average. For every level, straight lines could be fitted to each plot
for x > 70 cm provided trac~r was detected at that station. Thus the
plu:me width (estiITlated as a constant facto r of cr) grew parabolically
for x >- 70 CITl for all norITlal depths covered in this study. The para-
bolic growth is in accord with Taylor's (28) one-diITlensional diffusion
theory and is illustrated byconcentration contours plotted in Figure 5.24.
The linear growth rates dcr2/dx, however, varied with levels
of ITleasureITlent T) as shown in Figures 5.4 through 5.7. The dis-
crepancies € between dcr2 /dx at various levels T) and the growth
rate d? Idx of the depth-averaged variance are listed in Table 5.3
for d= 2.75~ 5.53,10.70, and 17.31 CITl. As indicated, the absolute
value of the error varied froITl 0.0% ITleasured in RUN 706 where
C\I E u
C\I ..
b .. W <.)
Z <{
a:: ~
_142-
100.------.---------.--------.---------~----~--------~--------~----~~----~
RUN 706 80 d=2.75 cm,u = 30.0 cm/sec
u = 1.65 cm/sec,'" = 0.368 • "h
60 CR.= 464.0
0
20
0
20 LEGEND
o ... do-2
SYM "J d"X,cm
0 0.368 0.0566 ~ 0.236 0.0514
20 e 0.800 0.0515
• AVG 0.0528
o ~--------~----~---------L--------~--------~--------~----~~----~--------~ o 2 4 6 8 10 12 14 16
DISTANCE DOWNSTREAM OF SOURCE, x ,m
Fig. 5.4. Growth of the variance 02
with distance x at
various levels of the flow Ti and as a depth
average; RUN 706.
18
N E 0
N ...
b ... w u z « a:: ~
'-143 -
150 ~--~----~----~---r----'----'-----r----.----,
125
100
75
50
25
0
25
0
25
o
25
o a
RUN 701
d= 5.53 cm,u = 42.0 cm/sec u .. ~ 2.20 cm/sec,7]h= 0.368
IR = 1.254 X 103
*
2 4 6
SYM
0 ~
e
• 8 10 12
LEGEND dcr 2
7] d'X,cm
0.368 0.0724 0.095 0.0829 0.850 0.0854
AVG 0.0786
14 16
DISTANCE DOWNSTREAM OF SOURCE, x ,m
Fig. 5. 5. Growth the the variance 02 with distance x at
various levels of the flow '1 and as a depth
average; RUN 701.
18
C\J E u ..
C\J
b .. w u Z « 0::
~
-144-
100r----.----~--~----._--_,r_--_r----._--_r~
80
60
40
20
0
20
0
20
o
20
o
20
RUN 607
d=10.70 em,u=41.8 em/sec u = 1.90 em /see,'" = 0.368 * "h
IR* = 2.125 X 103
()
'TJ = 0.368
0.095
0.632
0.850
LEGEND
0.368 ~ 0.095 () 0.632 e 0.850
- ...... o ~~~ ____ J-__ ~~ __ -L ____ L-__ ~ ____ ~ __ ~~
2 345 o DISTANCE DOWNSTREAM OF SOURCE, X ,m
Fig. 5. 6. Growth of the variance 02 with distance x at various
levels of the flow II and as a depth average; RUN 607.
350
300
250
200
150
100
'" E ()
"'b 50 A
W U Z 0 « a::: 50
~
0
50
0
50
0
100
50
0 0
-145-
RUN 708
d=17.31 em,u=34.9 em/sec u.= 1.66 em/see,'7
h =0.368
IR =2.988Xld •
e
• •
2 4 6 8
o
SYM
0
~
()
e •
10 12
'7 =0.368
0.095
0.632
0.850
AVG
LEGEND dcr2
'7 (j"X,em
0.368 0.204 0.095 0.182 0.632 0.188 0.850 0.158
AVG 0.187
14 16
DISTANCE DOWNSTREAM OF SOURCE, x,m
o
18
Fig. 5.7. Growth of the variance 02 with distance x at various
levels of the flow 11 and as a depth average; RUN 708.
-146-
Table 5.3
Comparison of dcr2 /dx at Various Levels II and dcr2 /dx, the Rate of
Growth of the Depth-Mean Variance
dcr2 dcr2 Error, Mean RUN d II dx dx
E I E I cm cm cm % %
706 2.75 0.368 0.0566 0.0528 +7.2 0.236 000514 -2.6 0.800 0.0528 0.0 3.3
701 5.53 0.368 0.0724 0.0786 -7.9 0.095 0.0829 +5.5 0.850 0.0854 +8.6 7.3
607 10 .. 70 0.368 0.118 0.117 +1. 0 0.095 0.122 +4.3 0 .. 632 0.116 -1.0 0.850 0.112 -4.3 3.5
708 17.31 0.368 0.204 0.,187 +9.1 0.095 0.182 -2.7 0.632 0.,188 +0.5 0.850 0.158 -15.5 9.3
In all cases the injection level llh = 0.368.
d = 2. 75 cm to a maximum of 15 .. 5% evaluated in RUN 708 where
d = 17 .. 31 cm" The mean value of the absolute error I E I was about
50/0.
5 .. D. THE TRANSVERSE MIXING COEFFICIENT
5. D. 1.. The Depth-Averaged Mixing Coefficient, D zo
a. Calculation of the depth-averaged mixing coefficient. It has been
shown in subsection 2. B. 4 that if the variance cr2 of the transverse
distribution of tracer concentration is invariant with depth, the depth-
-147-
averaged eddy coefficient of transverse ll1ixing D can be calculated z
by Eq. 2.55. Without norll1alization, this equation reduces to
D z
(5. 5)
where u is the average velocity in the cross section, and ? is the
depth- ave raged value of 2 cr •
In this study, Eq. 5.5 was used to calculate D • As shown z
in Figure 5.8 for six runs other than those already included in Figures
5.4 through 5.7, plots of cr2 versus x were developed. Since ?
grew linearly with x fo r x > 70 Cll1, straight lines were fitted to the
plots and their slopes deterll1ined.
Eq.5.5.
Thus D z was calculated by
RUN 509 differed froll1 RUN 511 only in the height of tracer
injection, "lh. For RUN 509, "lh = 0.368 and for RUN 511, "lh = 0.05t.
As shown in Figure 5.8, dcr2 /dx is virtually identical for both runs
indicating that dcr2 /dx and therefore D was not affected by the level z
of tracer injection, "lh.
Table 5.4 sUll1ll1arizes all calculated values of D including z
other related parall1eters. In addition to the hydraulic data shown in
ColUlllns 1 through 5, the table lists: the aspect ratio X. = d/W in
Colull1n 6; the rate of growth of the depth-averaged variance dcr2 /dx
in Colull1n 7; the depth-ll1ean coefficient D in ColUllln 8; and the z
norll1alized coefficient, e = D /u""d in ColUllln 9. For experill1ents z ,-
in the flull1e with the rough bottoll1, u* was replaced by the bed shear
velocity u>!cb in the norll1alization of D z.
~
w u z « 0: ~ o w (!) « 0: w ~ I
I le.. w o
80
60
40
20
o
80
20
o 20
o
40
-148-
o ~~~----~--~----~--~----~--~----~--~ 3 4 5 6 7 8 9
DISTANCE DOWNSTREAM OF SOURCE,x,m
Fig. 5.8. Growth of the depth-averaged variance 0-2
with distance x
downstream of source for various normal depths;
RUNS 707, 702, 704, 509, 5 1 1, 603 •
-149-
b o Variation of e = 15z !u*d with the aspect ratio X. = d!W. A
similarity approach was used to develop a representation of the depth-
averaged mixing coefficient as a universal function of the hydraulic
parameters. The pertinent variables chosen were Dz' u*, d, and W ..
The choice of the independent variables, u,,_, d, and W was justified -,.
by the following reasoning: the transvers e mixing coefficient depends
on essentially two parameters--turbulence intensity and the size
scale of the eddies. The shear velocity u* has been shown by
measurements of Laufer (63) in a two-dimensional flow in a wind
tunnel to equal approximately the rInS value of the transverse velocity
fluctuation. The turbulence scale in flows with upper and lower
boundaries and side-wall confinements is limited not only by the flow
depth but also by the separation distance between the side walls. A
characteristic scale of the mLxing process is, therefore, related to
both d, the flow depth, and W, the flume width.
The other variables of the flow are either intrinsically part of
the variables already selected or insignificant in describing the mixing
process .. For example, the mean velocity, u is important only in
the transport of the marked fluid and has been utilized in the calcu-
lation of D in Eq. 5.5. The friction factor falls within the deterz
mination of u*o Since the Reynolds number was very high in all
experiments (minimum IR ~ 2 X 104), the turbulent Schmidt number
S = Dz/v was also high--a typical value of S ~ 300. Thus the kine-c c
matic viscosity will be considered important only in establishing that
the flow was indeed turbulent but will not be included as a variable.
-150-
With four variables: D z ' u*, d, and W, and two basic units:
length and time, dimens ional anal ys is yields the functional relation-
ship
(5.6a)
which could be written as
e = <f!( A) , (5.6b)
where <.I> is an unknown function, e = Dz/u*d. and A = d/W is termed
the aspect ratio.
The results listed in Table 5.4 and plotted in Figure 5.9 sup-
port the preceding relationship. Fo r example e remained es sentially
unchanged when the mean flow velocity u was almost doubled between
RUNS 706 and 707 while A was kept constant. That e was independent
of the friction Reynolds number IR* or the friction factor £* is vividly
illustrated by several results. For example, when IR* was raised
from 2.06 X 10 3 (RUN 704) to 4.97 X 10 3 (RUN 404) while A was essen-
tially unchanged (z +4% error), e increased by only 4%. Similarly
when IR* was more than doubled between RUNS 708 and 406 while A
remained within 1 % of each other, the change in e was only about 4%.
However when JR* was virtually constant in RUNS 709 and 405 (error
of + 10% in JR*). but A was decreased by 70%, "9 was increased by
about 32%. This clearly demonstrates a well defined sensitivity of e to A and little relationship of e to IR*.
Figure 5.9 also shows that e is not dependent on the flow depth
d but on the aspect ratio A.. Each pair of experiments with essentially
-151-
Table 5.4 Sununary of measured depth-averaged mixing coefficient Dz and related parameters.
_T Depth-
RUN Flume Normal Mean Shear Aspect
d0 2 qveraged Normalized RUN Depth Velocity VeLocity R~tio § Mixing Coefficient, ,~
dx Coefficient T -D
d U u* \=d/W Dz
(Eq. 5.5) 8°o-z-u*d
em em/sec em/sec (102)
em cm2 /sec (xl02)
1 2 3 4 5 6 7 8 9 1
507 SI 1. 52 31. 2 2. 12 1. 79 4.06 0.63 O. 197 507
705 S2 1. 69 32.8 1. 99 1. 53 4.82 0.79 0.235 705
707 S2 2.74 50.4 2.65 2.49 4.74 1. 19 O. 164 707
706 S2 2. 75 30.0 1. 65 2.50 5.28 0.79 O. 174 706
506 SI 2.95 27. 1 1. 57 3.47 5.80 0.79 O. 166 506
703 S2 3.46 32.0 1. 76 3.15 5.35 0.86 O. 142 703
509 SI 5.25 42.6 2.26 6. 17 6.25 1. 33 0.112 509
511 SI 5.25 42.8 2.08 6.17 6.20 1. 33 O. 122 511
512 SI 5.25 42.5 2. 17 6. 17 6.72 1.43 0.126 512
508 SI 5.26 41.7 2. 13 6. 19 5.80 1. 21 O. 108 508
510 SI 5.26 42.4 2.12 6. 19 6.87 1. 45 O. 130 510
702 S2 5.41 43.5 2. 18 4.92 7-.46 1. 62 O. 137 702
701 S2 5.53 42.0 2.20 5.03 7.86 1.65 0.135 701
405 R2 6.81 35.9 5.01 6. 19 26.8 4.81 O. 141 405
407 R2 8.66 41. 0 5. 12 7.87 29.4 6.03 O. 136 407
404 R2 10.36 42.8 5.05 9.42 35.0 7.49 O. 143 404
607 SI 10.70 41.8 1. 90 12.58 11. 7 2.09 O. 103 607
704 S2 10.81 39.2 1. 86 9.83 14.2 2.78 0.138 704
406 R2 17.07 35.3 3.64 15.52 38.2 6.74 O. 108 406
708 S2 17.31 34.9 1. 66 15.73 18.7 3.26 O. 113 708
603 SI 17.34 36.9 1. 75 20.39 15.5 2.86 0.094 603
709 S2 21. 97 30.5 1.40 19.97 21.4 3.26 O. 107 • 709
NOTES: ):~ Bed shear velocity was used for experiments in the 400 Series.
§ W = flume width.
T For the 400 Series, measurements were made at two levels: T1 = O. 368 and O. 632; ~ is the depth-averaged variance of the transverse distribution of tracer concentration.
-
1Cl) .. ~ Z w U lL. La.. w 0 u (.!)
Z X ~ Cf) Cf)
W .....J Z 0 Cf) Z W ~ 0
0.40 r- LEGEND
0 Flume SI 0.
30 t 0 Flume S2 V05 A Flume R2
0.20 f CP(A) 407 4 ?4
A ~d 704
0.10 t 607~ 603 _____ j
I K 6
0.041 I I I I I J
0.01 0.04 0.10 0.40 1.00
ASPECT RATIO, A Fig. 5.9. Variation of the depth-averaged, dim.ensionless m.ixing coefficient e
with the aspect ratio A for experim.ents perform.ed in this study.
The average value ofe for RUNS 508, 509, 510, 511, and 512 is
plotted for A = o. 062.
I ....... Ul N I
-153-
equal flow depths but perfonned in separate flUllles are lllarked a, b,
c, d, and e. As shown in the plots, a for each pair was always
lower for the higher Ao
That 15z /u*d should decrease with d/W lllay be explained as
follows. The larger the turbulence scale in the lateral direction, the
greater is the transverse spreading of lllaterial cloud. Thus if the
flow depth d is constant, D/u*d increases as W increases since
the lateral scale can increase further yet4 Therefore, Dz/u*d in
creases with decreasing A = d/W, and decreases with increasing 7\..
As A - 1, the analysis is no longer applicable because the
flow becollles strongly three dilllensional.. As A - 0, the f1ullle width,
W, is no longer the characteristic transverse length and the dilllension
less transverse coefficient e would be expected to approach a constant
value. For all cases where llleasurelllents were lllade in this study,
e was always greater than K/6, the value of D /u*d. Y
c. Dependence of e on A for experilllents perforllled by other inves
tigators. Figure 5.10 is a plot of e = Dz/u*d versus A = d/W for
both the present study and the llleasurelllents lllade by other investiga-
tors. All experilllents perforllled in the laboratory and reported by
Prych (39) t Sullivan (6), Elder (1), and Kalinske and Pien (30) lie very
close to the curve ?(A.) except in the range 0 0 07 < A < 0.11 where
llleasured values of e plot above cI>{A). No apparent reason could be
found for this discrepancy. Fo r experilllents perforllle d in the field
let> I-"' z w U LL LL W o U
(.!) Z x ~
w Cf) 0:: W > Cf) Z
~ I-Cf) Cf) W ..J Z o Ci5 z w ~ o
1.00 _I -----,----r---,--r--r--r--r--r-r-----y------,,...--.-'--r---,,...-r-..--r-,
" " " 0.40
0.10
"-"-
"-" "-""-~~(\)
"-"-
'e,
" " --......
<P(\)
-- ---0/
8. ltl ~
Laboratory Measurements ~ Elder(1) ~ Sullivan (6) 181 Kalinske and Pi en (30) 0/ Prych (39)
[J Flume S2 Present o Flume SI }
8. Flume R2 Study
Field Measurements
181
() Yotsukura et al (8) ~ Fischer (7) e Glover (42)
K '6 ------I
0.04 ::1 :-__ --L __ -'------L_..L..-...L-...LJL.L...l...-___ --L_---i_---L_--.J_L..J~...LJ
0.01 0.04 0.10
ASPECT RATIO, A
0.40 1.00
Fig. 5. 10. Variation of the depth-averaged, diITlensionles s ITlixing coefficient e with the aspect ratio A for all experiITlents perforITled in the present
and past studies.
..... U1
*" I
-155-
and reported by Yotsukura et ale (8), Fischer (7), and Glover (42),
"8 is approxiITlately twice the value that <.p(A.) predicts. Indeed the
field experiITlents tend to lie on a higher curve cI>f(A.) which also
dec reases with A..
The difference between laboratory and field experiITlents is
due to accentuated ITlixing caused by the large secondary currents and
strong lateral gradients in velocities generated by bends and non-
uniforITl c ro ss sections in natural streaITls. Fischer (64) showed
experiITlentally that a short bend in a laboratory fluITle could increase
"8 on the average about six tiITles above the value in an identical
straight channel. Further field ITleasureITlents 1 however, are needed
for a ITlore cOITlplete understanding of irregular channels.
The present experiITlents agree with previous results but cover
a wider range of the aspect ratio A.. In addition this study confirITls
5. D. 2. Variation of the Transverse Mixing Coefficient With
Depth Caused by Shear and the Non- UniforITl Distribution of the
Vertical Diffusivity With Depth. As a result of shear and the variation
of the vertical diffusivity D with depth, the transve rse ITlixing coefy
ficient D varies with depth within the flow. Two ITlethods were used z
to deterITline this variation.
The first utilized Batchelor's (33) result outlined in Chapter 2.
Stated for the one-diITlensional case, this proposition ass erts that if
on a plane of hOITlogeneous turbulence, tracer distribution is Gaus sian,
then a ITlixing coefficient D can be defined such that
-156-
1 do2 D=2 ill (5. 7)
where (Tz is the variance of the tracer distribution and t is tiIT1e.
For the uniforITl open-channel flow every transverse plane parallel to
the fluITle bottoITl is a level of hOITlogeneous turbulence. Measure-
ITlents shown in Figures 5. 1 through 5~ 3 demonstrate that at each
level of ITleasureITlent '1, the transverse distribution of tracer con-
centration is Gaus sian fo r all x. The refo re at each level one can
define a transverse coefficient D expressed as z
where (Tz is now the variance of the transverse distribution of C at
level '1. Thus as sUITling that a space-tiITle transforITlation is valid,
Eq. 5.8 can be written as
(5.9)
where u('1) is the flow velocity at the level '1. By calculating u('1)
and d(Tz (1'1) at various '1, a diITlensionless coefficient dx .,
8 (1'1) = u( '1) d(Tz (1""1) I·, 2uddx .,'
* (5.10)
is evaluated.
Figure 5.11 shows distributions of 8 1 ('1) fo r RUNS 511, 704,
and 512. As shown in the legend, the injection levels, '1h
' were
respectively 0.051,0.368, and 0.850 for RUNS 511,704, and 512.
The ITlaxiITluITl variation of 81 ('1) froITl its depth ITlean value was 24%,
~
...J w > w ...J
1.0'r----,---------"{--------IJ-------
0.8
0.6'
RUN
7'Jh
01
511 ° 0.051
It
704 0.368
f
512 0.850
L.EGEND
SYMBOL RUN
° 511 0 704
d em
5.25
10.81 I ~ o 0.4 6. 512 5.25 ...J LL
0.2
0.0' ". 0.0 0.10 0.0 0.10 0.0 0.10 0.20 0.30
NORMALIZED TRANSVERSE MIXING COEFFICIEN T, e I Fig. 5.11. Depth variation of the normalized transverse mixing coefficient 81
due to shear only; RUNS 511, 704, 512.
0.40
, >-' U1 --.) ,
-158-
21<7'0, and more than 35% for RUNS 511, 704, 512 respectively. The
large discrepancy in RUN 512 is due to the fact that when tracer is
injected near the water surface where the flow velocity is high, both
d(T2 u(l"]) and dx (l"]) are high at large l"]. Thus 91 (l"]) is much higher
at large l"] than at the lower levels of the flow.
The second method for calculating the variation of D with z
depth is based on Eq. 2.59. This method considers the effects of
both shear and the vertical variations of (T2 and of the diffusivity D y
on D • As stated in Chapter 2, numerical calculation of the derivz
atives of (T2 at a fixed l"] by incremental approximation was inade-
quate because of the generation of large numerical errors. There-
fore, an analytical method was used.
First (T2 was determined fo r various levels of measurement
l"] and distances S from the straight lines fitted to the (T2 versus x
plots (for example, Figure 5.6). At each S, (T2 was plotted against
l"]9 Using the plotted points a representative curve of (T2 as a function
of l"] was drawn. Then utilizing the method of least squares, poly-
nomials of various degrees were fitted to points on this curve until
the chi-square error was small (about 0.02). The resulting poly-
nomial was used to represent the variation of (T2 with l"] at the
particular S. As further verification of the goodness of fit. (T2 was
re-calculated using the fitted polynomial and compared with the data
points. Agreement was within 2% (for most points the error was
zero) •
A third degree polynomial was found adequate for the runs
investigated and at all S. Thus (T2 was express ed as
-159-
(5.11)
where the coefficients a O' al, a2, a3 were dete rInined for each s. Eqo 2.59 was then written in the forIn
(5.12)
Substituting Eq. 5.11 into Eqo 50 1 2 and norInalizing by u*d, it was
found that at a given S
(5.13)
Therefore,
Hence e(T]) is cOInposed of two parts: the first, edT]) , due to shear
and the second due to the interaction of the vertical distributions of
the variance 0-2 and vertical difiusivity, D •
Y
Figures 5.12 and 5013 show the Ineasured distributions of
a-2(T]) and the corresponding e{T]) calculated by Eq. 5.14 at several
1.0
(0)
OS~ \ \ \ \ , t f::'-
....J 0.6 w > I c = IS.2 liT 7 136.S \55.S 174.9 /103.4 w ....J
3 OA 0
O.J I j / / / / / I RUN 511 ....J LL Flume S I
d = 5.25cm 7]h = 0.051
r [ 0.01 l l Il I I l I I I -
0 10 20 30 40 50
VARIANCE, 0- 2 , cm2
(b)
°l \ f::'- ,= S.2 \'77 \368 1558 \ 749 /103.4 /141.5 _ Ba= 0.129 0.126 '" 0.119 0.122 • 0.123 0.123 0.125
Ld 0.6 • > w I I B= 0.122 ....J
3 OA 0 ....J LL
0.2
0.15 0.05
NORMALIZED TRANSVERSE MIXING COEFFICIENT, e Fig. 5.12. Depth variation of the normalized transverse mixing coefficient e
due to shear and the interaction between the vertical distributions
of the variance as and of the vertical diffusivity D ; RUN 511. Y
I ,...... 0' 0 I
\ '\ (0) I 1.0
0.8
~
LLl 0.6 ~ i
> w --.J
3: 0 --.J LL
~ "
--.J w > W --.J
3: 0 --.J LL
~= 7.4 16.7 35.1
0.4
0.2
0.00 - -50
1.0
0.8
0.6
0.4
0.2
0.0 0.05
16.7
0.136
62.9
-100
100.0
1 150
VARIANCE,o-2, cm2
62.9
0.138
~
200
155.2
RUN 704 Flume 52
d = 10.81 em 7Jh= 0.368
I 250
0.145
8=0.138
0.05 0.15 0.05 0.15 0.05 0.15 0.05 0.15 0.25
NORMALIZED TRANSVERSE MIXING COEFFICIENT, ()
(b)
Fig. 5.13. Depth variation of the norm.alized transverse m.ixing coefficient A
due to shear and the interaction between the vertical distributions
of the variance a2 and of the vertical diffusivity D ; RUN 704 y .
>-' 0"">-' I
-162-
So In Figure 5.12, the results for RUN 511 are shown. The flow
depth was 5.25 ern and the injection level l1h = 0.05 (i.e. very close
to the flume bottom). Although ()'"2(11) was similar at all S, the shape
of 9(11) varied with So Figure 5.12(b) shows that as S increased,
9(11) successively grew larger at higher values of 11 and decreased
at low 11 with the position of maximum 8(11) increasing from
YJ = 0.632 at S = 8.2 to 11 = 1.0 at S = 41.5. This was due to the
fact that the curvature of (T2(11) decreased with increasing S. Hence
as the curvature diminished, the shear effect 91 dominated over the
second term of Eq. 5.14. Consequently at high 11 where 91 was
large, 9 was also large, while at small 11 where 91 was small
9 was also small.
In Figure 5.13 where the flow depth was 10.81 ern and the
injection level l1h = 0.368, the distributions of 9(11) were essentially
similar at all S except at S = 155.2. This similarity was achieved
because ()'"2(11) was approximately uniform with depth for all S
except at S = 155.2. The highest value of 9 was located around
T} = O. 72 at all S. For S = 155. 2 ~ however, maximUIn 9 occurred
at 11 = 0.632 where the curvature of (T2(11) was greatest.
The mean value of 9(11) denoted by 9 was determined from a
the calculated 9 at each S. The results are shown in Figures 5.12(b)
and 5.13(b). These were compared to the depth-averaged coefficient
e (also shown in the figures) evaluated for the particular runs by use
of Eq. 5.5.
5% for all S.
Agreement between 9 at various S and a was within a
As a summary. therefore, it is noted that the variation of the
-163-
transverse m.ixing coefficient with depth can be considered as due to
either shear only or a com.bination of shear and the interaction of
the ve rtical distributions of the variance o-Z and vertical diffusivity
D. In the first case a single distribution 91(1")) can be determ.ined y
for any given experim.ent using Eq. 5.10. In the second, 9(1")) is
evaluated for each £ us ing Eq" 5 .. 14. In general the shape of 9(1"))
varies with £ depending on the vertical distribution of o-Z(1")) at
corresponding L However. if the variation of o-z(1")) with depth is
sim.ilar at all £. then with successive increm.ent in £, 9(1")) in-
creases at high values of 1") and dim.inishes at low 1"). Indeed 9 can
attain a negative value which would m.ean that at the particular £
the interaction between D and o-Z(1")) has dom.inated the effect of y
shear, and there is a resultant transfer of m.aterial vertically to adja-
cent layers.. The depth m.ean value of 9 calculated at each £ agreed
closely with e determ.ined for the entire run using Eq. 5.5.
5.E. NEAR-SOURCE VERTICAL DISTRIBUTION OF TRACER DUE TO SHEAR AND THE VERTICAL DIFFUSIVITY
As outlined in Chapter 2 and illustrated in Figure 2" 2. the
vertical distribution of m.aterial within the plum.e is strongly skewed
very near the source. This is due to the non-unifo rm. vertical distri-
butions of the flow velocity u, and of the vertical diffusivity D • Y
To evaluate the resultant vertical distribution of the tracer concen-
tration along any vertical plane parallel to the plum.e axis, one m.ust
solve Eq. 2.44. A s explained earlier analytical solutions have
proved inadequate.
-164-
* A num.erical solution, however, has been perform.ed for the
case of a continuous line source placed laterally at any selected
depth, TJho Since the line source is a superposition of an infinite
num.ber of point sources along a lateral line, this solution is equiva-
lent to the solution of Eq. 2.44 integrated with respect to z. The
result is an elim.ination of the effect of D with Eq. 2.44 reduced to z
(5.15)
where MO = S 00 C dz. Eq. 5 .. 15 was solved num.erically with the -00
logarithm.ic velocity distribution of Eq. 2.33, and the parabolic
variation of D expressed in the form. of Eq .. 2" 51.. The initial y
condition was:
(5 .. 16)
and the boundary conditions,
(5" 1 7)
and
(5.18)
* Coudert, J. F., "A Num.erical Solution of the Two-Dim.ens ional Diffusion Equation in a Shear Flow with Variable Diffusion Coefficient--Case of a Steady Line Source in a Stream.," W .. M. Keck Lab. of Hyd. and Water Res., Tech. Mem.o. 70- 7, California Institute of Technology, Pasadena, California, 1970.
-165-
The numerical program has been applied to RUNS 509 and 510 where
the inj ection levels 1]h we re, respectively J 0.368 and 0.632. Using
the experimental values of 1]h J U, u*, K J and d for each run,
theoretical values of
(5.19)
were calculated for values of S where measurements were made.
From continuity J
(5.20)
The results ofnurnerical calculations and the measurements are
plotted in Figure 5 9 14. For each run the forcing functions D (1]) /u. d Y >j<
and u(1])/u*, and the theoretical distributions of MO
(s,1]) are drawn
fqr successive distances S = 1.0, 8.2,17.7,36.7,55 0 9,75.1,
102.7, and 144 .. 1. For comparison, points determined from experi-
ments are plotted for the values of S where measurements were
made ..
Agreement between the nume rical solution and the experimen-
tal measurements was very close. Both showed that the vertical distri-
bution of M O(s,1]) was skewed for 0 < s -< 17.7, with the level of
maximum MO(s, 1]) rising with increasing S for RUN 510 where
1]h = 1 - 1/ e = 0.632, and falling with increasing S for RUN 509
where 1]h = 1/e = 0.368. Beyond S = 35.7, there was very little
vertical var iation of MO (s ,1]) (mean deviation;::;; 5%), and material
~ I f(L [,I o lf) lf) W --' Z
06
o 04 i]) Z w :2: i5
02
010
\ Di')1 u.d
10.0
u(7)1
--u.-
0.8
SOURCE • LEVEL
0.4
02
~.IO 17.7 751 1027 o
08 r' ··1,'"
0..6
0.4
I 1 II I I f I 1 I __ LL_~_ L __ - 20.0 0.0 2.0 4.0. 00 2.0 0.0 ? 0 on no. no. 0..0 2.0 0.0
1441
n
2.0. 010 10 r-----.--r------·
~ I f(L
~ 0.6 (f) (f) w --' z Q 0.4 lf)
Z w :2: i5
., Dy(7)1
u.d
U(7))
u.
0.8
06
SOURCE,
LEVEL
02
~.LO
0..6
0.4
0..2
367 55_9 75.1 1027 1441
o
I I 1,1 II II ILl 11-----,---10.0 20..0 0.0 2.0 4.0 0..0 20 00 2.0 0.0 00. 00. DO 00 20.
u(7))/u,
FORCING FUNCTIONS VALUES OF Mo(~,r;) AT SUCCESSIVE ~
Fig. 5. 14. Vertical distribution of ITlaterial MO (E:, 11) = MO (E;, 11)/ (Qs !'i:rd) for
(a) RUN 510 and (b) RUN 509. The curves are theoretical solutions
of Eq. 5.15 at various S. The plotted points were ITleasured.
(0)
(b)
...... 0' 0' I
-167-
distribution was essentially uniform with depth for both runs.
Figure 5.15 shows the plots of curves describing the levels
of maximum MO(S ~ T]) as calculated by the numerical analysis for
four levels of tracer injection. Theoretically the point of maxi-
mum MO(s,T]) reached the water surface at sz9 0 0 for T]h=0.850,
and at S z 31.0 for T]h = 0.632, and fell to the flume bottom at
S = 5.0 for T]h = 0.095. and at S = 19.0 for ~ = 0.368. Thus the
point of maximum MO(S ,11) arrived at the flow bottom for small T]
faster than it rose to the water surface at high T]. This is of course
due to the fact that near the flume bottom, shearing is strong and
u(T]) decreases rapidly with decreasing T]. Thus near the bed both
shear and the variation in vertical diffusivity combine to accelerate the
fall in the center of mas s of the plume. Near the water surface t
however, u(T]) is nearly uniform with depth and the contribution due
to shear is substantially reduced.
Figure 5.15 also shows plots of experimentally determined
levels of maximum MO{s, T]) for corresponding runs. It is evident
that indeed there was a general trend in the direction predicted by
theory. However after the locus of maximum values of MO(StT])
reached the free surface or the lower solid boundary, it seemed to
"bounce H back into the flow interior. This phenomenon was meas-
ured for all runs except RUN 512 (T]h = 0.850) where the level of
maximum MO(LT]) rose at a rate slower than theoretically pre
dicted but once reaching the water surface remained there for
larger S. The apparent bounce back feature is probably a result
~
J:
Ii: w o en en w -l
~ en z w :::!! o
--------Jr--------~~~-~- 7 1.0 I 7 ~ .... -- .....
---0.8
0.6
0.4
0.2
RUN 511
- ... ----.... .... .........
:--......................... "'",
................. 0-.. ........
"'", "'",
"', " "
..
"
40
DIMENSIONLESS DISTANCE,!
SYM RUN "Ih
Ii. 512 0.850 o 510 0.632 o 5090.368 v 511 0.095
--- THEORETICAL PREDICTION ---- C~VE FITTED TO
EXPERIMENTAL POINTS
..
60 80
Fig. 5.15. Theoretical levels of maximum MOP':' 11) for four levels of tracer injection
~. The plotted points were measured; RUNS 512, 510, 509, 51l.
...... 0"-00
-169-
of secondary current which at large S is strong enough to overCOITle
the gradient ElMO/Elll which approaches zero with increasing s. The
phenoITlenon was also ITleasured by Davar (9) for a pluITle generated by
a point source of gases within the wall boundary layer of a wind tunnel.
5.F. DISTRIBUTION OF THE TRACER CONCENTRATION C(S,ll,O) ALONG THE AXIAL PLANE
F.5.1. Vertical Distribution of C(Lll,O) at Various S. Since
the concentration of tracer along the pluITle axis was generally equal
to the ITlaxiITluITl value of the tiITle-averaged concentration C, it will
be assuITled that
C(S,ll,O) = C (S.ll.S) (5.21) ITlax
where C = ITlaxiITluITl value of C at sand 11, and s::: z/d. A ITlax
diITlensionless variable l3(s .1'"]) is defined such that
= C(S ,1l~O)
Q /Wud s
(5.22)
To visualize the depth variation of tracer concentration on the
vertical axial plane, values of 13 calculated froITl ITleaSureITlents
were plotted as functions of 11 at several s. In contrast to the close
agreeITlent between the experiITlental and theoretical distributions of
M O(S,1'"]) shown in Figure 5.14, curves fitted to ITleasured 13(S,1l) at
low S, were generally different froITl those of MO(s, 11) predicted by
theory. This is, of course, a reflection of the difference in the rates
of transverse ITlixing at various levels of the flow. Since ITleasure-
ITlents have deITlonstrated that the transverse distribution of C is
-170-
Gaussian at all ~ and T] (see for example Figure 5.1), combination
of Eqs 0 5.19 and 5.22 and the Gaussian distribution of C results,
for given ~ and T], in the relationship
(5. 23)
where cr(~, T]) = standard deviation of the transverse distribution of
C at the given ~ and T]. Vertical profiles of f3(~, I)) are t therefore t
the same as those of MO(g ,I)) modified by the variation of cr(g ,I)).
It was found that regardless of the modification of MO(s, T])
due to cr(S, T]), the level of maximum f3(~, T]) also rose or fell
exactly as in the measured values of MO(~tT]) shown in Figure 5.15.
The IIbouncing" phenomenon was also measured at identical stations.
5. F. 2. Longitudinal Attenuation of the Tracer Concentration
Along the Plume Axis 0 Figures 5. i 6 through 5.19 show the attenuation
of tracer concentration, expressed as f3(S t T]), along the plume axis
and at different levels of the flow for the injection level T]h = 0.850,
0.,632, 0.368, and 0.051 respectively. The curves are extrapolated
beyond S = 8.2 for ~ - O. O. The plots show that very near the
source, the decay rates were vastly different at various levels of the
flow. At all levels except T]h' f3(~, T]) initially increased with ~ and
then diminished with further increase of s. At distances greater
than a critical value denoted as ~ , the concentration decayed at a
a constant power of ~ at all values of T] for any given experiment.
Therefore the depth-averaged value "j3(~) was expressed as
40
CQ . Z 0
ti a:: f- 30 Z w U Z 0 U
-..l
~ 20 « o w N ::::i « 2ii 10 I o I Z I
/\ i \
i \ I I I ! I
-171-
RUN 512
Flume SI d = 5.25 em, U = 42.5 em/see
3 u:2.17 em/see,IR,; 1.188xI0
LEGEND
SYMBOL "J o 0.095 " 0.368 o 0.632 '" 0.850 ="J.
°0~L-~--~20----~---4~0----L----6LO--~----8~0~--~--~10~0~--L---~12~0--~--~1~40~
DIMENSIONLESS DISTANCE,'
Fig .. 5.16. Attenuation of the normalized tracer concentration f3 at four levels of the flow 11 on the vertical axial plane; RUN 512, 11h=0 .. S50
40
CQ
z o ti a:: f- 30 Z w U Z o U
-..l « X « o w N -..l « ~ a:: o z
20
10
0 0
II II II II
20 40
RUN 510
Flume SI d = 5.26 em, U = 42.4 em/sec
u =2.12 em/see,1R = 1.I09XI03
• •
60 80
DIMENSIONLESS DISTANCE,'
100
LEGEND
SYMBOL "J 0 0.095
" 0.368 0 0.632 ="J.
'" 0.850
120 140
Fig. 5.17. Attenuation of the normalized tracer concentration f3 at four levels of the flow 11 on the vertical axial plane; RUN 510, 11h = 0.632
C!:l. 40
Z o ~ cr I-15 30 U Z o U
-' <l: X <l: 20
o W N ::i <l: :. cr o z
10
0
(
I I I / I( II
/1 " / " /
0 20 40
-172-
RUN 704
Flume S2 d=I0.81 em, ii = 39.2 emlsee
u.= I.S6 em/see,lR; 2.055X Id
60 SO 100
DIMENSIONLESS DISTANCE, ~
120
LEGEND I
ISYMrL ~ I
~ 0.095 0.36S=~ •.
I 0 0.632 I 'V 0.S50
140 160
Fig. 5 .. 18. Attenuation of the normalized tracer concentration 13 at four levels of the flow II on the vertical axial plane; RUN 704, llh=0.368
~
(
I I
40 I
C!:l.
Z 0
~ cr I-Z W U Z 0 U
-' <l: X <l: 0 W
'" N ::i <l: 0 :. 10 cr 0 z
0 0 20 40
RUN 511
Flume SI d= 5.25 em, ij =42.S emlsee u=2.oSem/see,1R =1.I45X10
3
• •
60 SO
DIMENSIONLESS DISTANCE, {
100
LEGEND
SYMBOL ~
o 0.095 " 0.368 o 0.632 " 0.850 I
~.=0.051
120 140
Fig. 50' 19. Attenuation of the normalized tracer concentration p at four levels of the flow iJ on the vertical axial plane; RUN 511, llh = OO' 0 51
-173-
(5.24)
where Q:' = constant. Figures 5.16 through 5.19 show, as expected,
that ~ was smallest when the source was located near the flow a
mid-depth.
To evaluate the asymptotic decay rate, the depth mean value
of l3(s, I)), designated as j3(S), was calculated at various distances
~. A log-log plot of (3(S) versus ~ was developed and for each
experiment, the slope of the straight line fitted to the points equalled
To estimate Stan imaginary line was drawn at a tangent to the a
fitted straight line to the first point, in the neighborhood of ~ = 0,
which deviated more than 20% from the fitted asymptote. The value
of S at the point of tangency was used as S • a
Figure 5.20 shows plots of 13 versus ~ for RUNS 512, 704,
511 and 510. In the accompanying legend the flow depths d, levels
of tracer injection I)h' and exponents of the decay curve Q:' are
shown o Points of ~ are indicated on the plots. It is evident that a
the points plotted very close to the fitted straight lines and that S a
is relatively small regardless of the flow depth or the level of tracer
injection.
Table 5.5 summarizes the values of ~ and Q:' for the a
twenty experiments where measurements were made at two or more
levels of the flow. The flow depths covered ranged from 1.69 to
21.97 cm. The table includes hydraulic data tabulated in Columns 1
through 3, the level of tracer injection in Column 6 and in Column 7,
leo.. C/)
X <[
w ~ ::J -l 0....
W I I-<.9 Z 0 -l <[
Z 0
~ 0:: I-Z W u Z 0 0
0 W <.9 <[ 0:: W
~ I
I I-0.... W 0
40
20
10
8
6
4
3 2
-174-
o
-Ilr--
o
LEGEND
SYM RUN d,cm a El 512 5.25 0.465 '\-l 704 10.81 0.526 o 511 5.25 0.554 8. 510 5.26 0.530
=0.632
6 8 10 20 40 60 80 100 200
DIMENSIONLESS DISTANCE, t
Fig. 5.20. Attenuation of the depth-averaged concentration
S on the vertical axial plane for four levels of
tracer injection: RUNS 512, 704, 511, 510.
40
20
10
6
4
2
-175-
Table 5. 5 Summary of measured parameters related to the
decay of tracer concentration along z==O.
Flume Flow Tracer § t Decay
F Identii. Injection Sa RUN Code ,:~ Depth Level
'='t Exponent RUN
d llh a t cm
1 2 3 4 5 6 7 1
705 82 1. 69 0.368 60.0 55. 1 0.452 705
706 82 2.75 0.368 14.5 25.4 0.519 706
707 82 2.74 0.368 0.0 24.8 0.472 707
506 81 2.95 0.368 0.0 40.7 0.414 506
703 82 3.46 0.368 0.0 39.0 0.506 703
509 81 5.25 0.368 0.0 18. 0 0.545 509
511 81 5.25 0.051 5. 3 14.8 0.554 511
512 81 5.25 0.850 6.3 14. 5 0.465 512
508 81 5.26 0.368 0.0 15.2 0.447 508
510 81 5.26 0.632 3.4 11. 9 0.530 510
702 82 5.41 o. 129 8. 1 18. 1 0.536 702
701 82 5.53 0.391 9. 0 9.8 0.563 701
607 81 10. 70 0.368 7.8 8. 7 0.480 607
704 82 10. 81 0.368 0.0 12. 9 0.526 704
708 82 17. 31 0.368 2.9 4. 7 0.526 708
603 81 17.34 0.368 2.2 3. 7 0.602 603
709 82 21. 97 0.368 3. 5 3. 6 0.650 709
407 R2 8.66 0.368 8. 1 11. 5 0.601 407
404 R2 10.36 0.368 5. 8 15. 5 0.610 404
406 R2 17. 07 0.368 0.0 8. 8 0.676 406
NOTE8: -,- 8== smooth; R==rough; 1==85 -cm flume; 2== 110-cm flume. -.-
§ St ==value of S beyond which? grows linearly with S.
t Sa ==value of E beyond which B(S) decays at a constant rate.
For S~Sa' S(S)~S-a.
-176-
the norITlalized distance Sf beyond which 0-2 grew linearly with S.
In all cases Sa was greater than Sf' with Sf = 0.0 in several
experiITlents. The average value of Sf for d >- 2.75 CITl was 4.1.
For d = 1. 69 c ITl , S P. = 60. 0 • The average value of S was 18.3 a
decreasing frOITl a high of 55.1 for the lowest depth to a low of 3.6
for the deepest flow where d = 21.97 CITl. For a given flow, S a
was generally sITlaller when tracer was injected near the ITlid-depth
than when 'YJh
was near the water surface or the bottoITl boundary.
The rate of attenuation of 73(s), in general, was highest when
d was greatest. Where the flow depth was about 2 CITl, Q::::; 0.46.
However when d::::; 20 CITl, Q::::; 0.60.. The decay rate was also high
when the fluITle bottoITl was roughened with rocks. Thus for RUNS
404 and 704 where the flow depths were 10.36 CITl and 10.81 CITl
respectively, Q was 0.610 for RUN 404 with the rough bottoITl and
0.526 for RUN 704 for the fluITle with hydraulically SITlooth boundaries.
The average value of Q for all experiITlents was 0.534 indicating a
decay rate of 73{s) slightly greater than the (-i) -power predicted by
Eq. 2.62 for one-diITlensional transverse ITlixing.
This ITleans that although 0-2 grew linearly with x (since
Sa> Sf)' the depth ITlean concentration along z = 0 decayed at a
rate controlled by both transverse and vertical ITlixing. For a
two-diITlensional ITlodel in which concentration distribution is Gaussian
in both lateral and vertical directions with y = z = 0 coincident with
the ITlode of the distribution, the continuity equation predicts that
-177-
where D ,D are the mixing coefficients in the y and z directions y z
respectively. A(t) cc t- 1 • Thus if both D and D are constant, I-' s S Y z
This means that mixing in the two directions causes a decay rate
exponent of S twice that for one-dimensional mixing. As the
vertical distribution of tracer varies from uniform to Gaussian, a
increases from 0.50 to 1.00. Realization of a value of a greater
than 0.50 is believed therefore to be a result of the two-dimension-
ality of the mixing process. Thus the greater the flow depth the
larger is a as indicated in Table 5" 5. Roughening of the flume
bottom intensified mixing and accelerated the decay in the concen-
tration along the plume axis. Hence a is larger when the flume
bottom was roughened with rocks than when it was smooth.
To estimate the error involved in the use of Eq. 2.62 for com-
putation of the transverse mixing coefficient, RUN 702 for which
a = 0.536 was selected. This run was chosen because its measured
value of a was very close to the mean value for all experiments
which was 0.534. Using the value of /3(S) measured at S = 200,
-0 50 and assuming that (3(S) - S • ,a depth-averaged cross-wise
mixing coefficient was calculated by use of Eq. 2.62 and found to be
1.58 cm2 /sec. As expected this was less than the value of 1062
cm2 /sec determined from the linear growth of the depth-averaged
variance expressed in Eq. 5.5. The error incurred, however, was
only -2.50/0. Thus for an increase in a from 0.500 to 0.536 (a change
of about 7%), the error involved in predicting D by the one-dimen-z
sional model was only - 2.5%.
-178-
5. G. ISO-CONCENTRATION MAPS
5. G.1. Tracer Distribution on Cross-sectional Planes.
Figures 5.21,5.22, and 5.23 show the distribution of tracer on
cross-sectional planes for the injection levels 'Ilh
= 0.051, 0.368,
and 0.850 respectively. The flow depth d was 5.25 CITl for the three
cases. The iso-concentration contours were obtained froITl
curves of C versus Z ITleasured at four levels of the flow and six
or seven stations froITl the source. The crosses shown on the plots
are points where tracer concentration was ITleasured. Concen-
tration values shown on the contour lines are in arbitrary units.
Multiplication of these values by Q' , also given for each figure, c
reduces theITl to diITlensionless values of the forITl expressed in
Eq. 5.22.
The iso-concentration ITlaps clearly deITlonstrate the effect
of 'Ilh
in the concentration distribution within the pluITle. In Figure
5.21 where 'Ilh
= a. 051, the core of ITlaxiITluITl concentration quickly
dropped to the fluITle bottoITl. As x >- 543 CITl, tracer distribution
becaITle approxiITlately uniforITl with depth and the level of ITlaxiITluITl
concentration was no longer easily discernible. When 'Ilh
= 0.850,
however, the peak concentration rose to the water surface generating
basin- shaped contours which persisted for large x. As shown in
Figure 5.23, concentration distribution was non-uniforITl even for
x = 743 CITl. As expected, injection of tracer near the ITlid-depth
enhanced the attainITlent of uniforITl distribution with depth. Thus in
Figure 5.22 where 'Ilh
= 0.368 the zone of ITlaxiITluITl concentration
initially sank to the flUITle bottoITl but then rebounded to near ITlid-
E u >;
E u >;
E u >;
E u >;
E u >;
E u >;
E u >;
I I-Cl... W 0
~t x=43cm
6 x=93cm
4
2
0
6 x = 193cm
4
2 +
+
x = 293cm
+ +
+
+
x = 393cm
+ + +
+
+
x = 543cm + +
+ +
2 + +
0 + +
6 x = 743cm
4 + + +
+ +
2 + +
0 + +
-42.5 -20 -16 -12
-179-
+ ,n6j,: ///
+ + + + +
+ + + + +
+ + + + +
+ + + + + +
+ + + +
+ +
+ +
+ +
+
+
+ +
-S -4 0 4 S
TRANSVERSE DISTANCE, Z, em
RUN 511 Flume 52
d= 5.25 cm,iJ = 42.Scm / sec Injection Level '7h=0.051
:t +
+
+
+ +
+
+
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
16 20 42.5
Fig. 5.21. Iso-concentration contours on cross -sectional
planes. Cros ses show points where tracer was
detected. Concentration values on the contour
lines are rendered dimensionless (as in Eq. 5.22)
when multiplied by 2.08; RUN511,.llh
= 0.051.
E u :>;
E u >:
E u >:
E u >:
E u :>;
E u :.; I I-CL w 0
-180-
RUN 509 Flume S I
d= 5.25 cm,u = 4 2.6cm/sec Injection Level '7h =0.368
6 x= 93cm
+ + + + + + +
+ + + + + +
x= 193cm
+ + +
+ + + + + + + +
+ + + +
x=293cm
+
+ + + + + +
+
~t x=393cm
+ + + +
t x=543cm
+ + !~ + + + + + +
6 x=743cm
J 21 ) f J; 2.0 1.0
2 + + + + + + + +
+ + +
-20 -16 -12 -8 -4 0 4 8 12 16 20 42.5
TRANSVERSE DISTANCE, Z, em
Fig. 5.22. Iso-concentration contours on cross-sectional planes.
Crosses show points where tracer was detected.
Concentration values on the contour lines are rendered
dimensionless (as in Eq. 5.22) when multiplied by 1. 56;
RUN 509, ilh = O. 368.
-181-
RUN 512 Flume S I
d = 5.25 cm,u = 42. 5cm/sec Injection Level "7h=0.850 6t x=43cm ] u>.E~42 I + ~~\: + I
2 .0 4') + "1.0 / + +
t t 0.5 t ! o '~I-----L~h~??r--L----~------~/.~Wr---L----~--'~7.?7r-~------~-~7-7n/--L-----~--~»n/~----;I 6[ x=93cm ]
~: + : +
o ~1-----L----~----/.fiY/7~!------~+--~--~--~~=-~~~~--~-----L--~/.~~r!------~--__;I 6[ x=193cm ]
§ 4 :
:>; 2 +
o ~----L------L~--~------~~~~~~~~~~~~~~ ___ + __ -L ____ ~ __ 7-7 __ ~ __ __;
6t X=393Cm+ + + + J § 4 + + +
>; 2 +
o ,~ ____ L-~ __ -L __ ~~ __ ~ __ ~~~~~~r=~~~-LL-~~ __ ~ __ ~+ __ ~+~~ ______ L-__ -; 6[ X+=543~m + + J 54 + + + +
~2 + + + + +
o ~ ____ L-+~ ___ +L-__ ~~ __ L-~L--s~~-L~-L __ L-~~~~~~~-L __ ~ __ L-+~ __ -L __ __;
5~ 6[ x= 743cm ] >:4 + + + + + I + + + +
~2 + + + + +
~ 0 ~~ __ L-+~--~+~--~~~~~~~ __ ~ __ ~-L __ ~~~ __ ~~~-2-L __ ~+ __ -L~+ ____ ~+ ____ -; -42.5 -20 -16 - 8 -4 0 4 8 12 16 20 42.5
TRANSVERSE DISTANCE, z,cm
Fig. 5.23. Iso-concentration contours on cross-sectional planes.
Cros ses show points where tracer was detected.
Concentration values on the contour lines are rendered
dim.ensionless (as in Eq. 5.22) when m.ultiplied by 2. II;
RUN 512, llh = 0.850.
-182-
depth. Meanwhile the tracer :mixed quickly ove r the flow depth
giving rise to essentially vertical contour lines for x:> 543 c:m.
The iso-concentration contours of Figures 5.21 through 5.23,
the refore, confor:m with the variation of MO (~ ,YJ) dis cus sed in
Section 5.E. They also show that, on a given cross-sectional plane,
the principal axes of the cloud are directed along the y and z axes.
This indicates that at a fixed x, the z and y axes chosen in Chapter
2 for the description of the :mixing process are indeed generally
oriented in the principal directions. Consequently the :mixing coef-
ficient tensor can be diagonalized as stated in Chapter 2.
5. G. 2. Tracer Distribution on Lateral Planes Parallel to the
Flu:me Botto:m. Iso- concentration contours were <:.lso constructed at
planes parallel to the flu:me botto:m, and located at the levels of the
flu:me where tracer concentration was :measured. Figure 5.24 shows
a typical set of :maps developed for RUN 512 at the four levels
11 = 0.,850,0.,632,0.368, and 0.095. The flow depth d = 5.25 c:m,
the flu:me width W = 85 c:m, and the source was located at the flu:me
center at the level 11h = 0.850. The conversion factor Q' for the c
concentration units shown on the contour lines is 2 .. 11.
Figure 5.24 shows that. on each plane, iso-lines progressed
fro:m an elliptical distribution in the plu:me interior to near parabolic
at the boundary as C decreased. At the level of tracer injection,
the extre:me contour line (in this case C = 0.5) was wedge-shaped for
x;!s 100 c:m, beco:ming parabolic for larger x. At other levels, how-
ever, the plu:me edge (i.e. C = 0.50) were essentially parabolic
E u N
E U
N
E u N
E u N W U Z f:! (f)
is w (f) 0:: W > (f)
Z <! 0:: f--
-20
-10 SOURCE
-183-
RUN 512 Flume S I
d" 5.25 cm,u" 4 2.5cm Isec Injection Level 7)h" 0.850
~~====~0.5 ---------~ 1.0 ---~-------~ 2.0 ---~ ______ ~
0
r~~~~~~~~~=========----- 7)"0.850 2.0~-~
10 ______ ~_ I.O--_________________ ~ 0.5 --____ ~ __________ ~
20
-20
_--~-0.5 ---....!..-----------~ -10 SOURCE
4.0 0
10
_-:::::::::::::::::::::::::::==---~-I.O --~----__
fIi~2'0-----'"
7) " 0.632
2.0 ------------~ 1.0 -----'--_______ ~ 0.5 - __ -.-_______ _
20
-20
-10 SOURCE
~ 0
10 ~0'5
====-=--:::.-=--=-~_=_-j-~=== ' . 0368
========:;:::_= 1.0---.--______ _ 0.5 --____ -. ___________ .....!.
20
-20
-10
0 7) "0.095
10
200 800
Fig. 5.24. Iso-concentration contours plotted on transverse
planes (parallel to the flum.e bottom.) at four levels
of the flow. Concentration values on the contour
lines are rendered dim.ensionless (as in Eq. 5.22)
whenrrlUltipliedby2.1l; RUN512, I1h = 0.850.
-184-
for x::S 743 cm. It should be mentioned that all iso-lines of C > 0
will eventually revert to the plume axis at large x.
Further examination of Figure 5.24 reveals that the value of
x at which tracer concentration was maximum on a given transverse
plane increased from x = 0 for the plane "=,, = 0.850 to approxih
mately x= 243 cmfor ,,= 0.095. If this distance is normalized by
d and represented as g ,then for a given ", m s varies with the
m
height of tracer injection "h. Values of S were evaluated by inter-m
polation from iso-concentration contours such as those shown in
Figure 5.24. The results for three levels of tracer injection
"h = 0.850, 0.368, and 0.095, are plotted in Figure 5.25.
5.H. SUMMARY
This chapter has presented the results related to time-
averaged concentration measurements. It has been shown that the
transverse distribution of tracer for the continuous point source in
a shear flow is Gaussian for ; extending from 4 to 611. The
variance of the distribution grows linearly with x both on fixed
transverse planes and as a depth average. A coefficient of trans-
verse mixing was calculated for various levels of the flow" and as a
depth- integrated value D • z Measurements showed that
is a decaying function of the aspect ratio of the flow, A =
Dz/u,;cd
d/W. Near
source behavior of the plume agrees with theoretical prediction.
Detailed distribution of tracer within the plume was illustrated with
iso-concentration maps on cross-sectional and transverse planes.
1.0r.---------------------------------------------------------------------,
0.8 ~ .. I ~ a.. w Q 0.6 CJ) CJ) W -1 Z o CJ)
z w ~ Q
Fig. 5.25.
SYMBOL
o o l:l.
20 40
LEGEND RUN "7h 512 0.850 703 0.368 511 0.051
o
D/ MENSIONLESS DISTANCE FROM SOURCE, tm
d, em
5.25 3.46 5.25.
60
Dimensionless distances r: from the source where concentration is maximum m
at given levels of the flow il for injection levels ilh = 0.850, 0.368, and 0.095.
I ...... co U1 I
-186-
The next step is to examine the contribution of the temporal
variations of trace r concentration and of plume boundaries to the
overall transverse mixing. Results of this phase are presented in
the next chapter.
-187-
CHAPTER 6
PRESENTATION AND DISCUSSION OF
EXPERIMENTAL RESULTS (Phase II)
This chapter summarizes the experimental investigation of the
temporal fluctuations of concentrat1on~ as outlined in Chapter 3. First
the fluctuating plume front model is assumed and the motion of the
plume front was studied by use of the intermittency factor concept.
Then the fluctuating plume model proposed by Gifford (16) was applied
to the photo studies and the variances characteriz ing the motion of the
plume centroid and the instantaneous concentration distributions were
calculated. Finally the result s of the statistical analyses of the vari-
ation of tracer concentration at fixed points within the plume are pre-
sentedo A discussion of the results is included wherever appropriate.
60 A. PARAMETERS ASSOCIATED WITH THE PLUME FRONT OSCILLATION MODEL
6"A.1. The Transverse Distribution of the Intermittency
Factor. The intermittency factor, as defined by Ego 3 .. 32, indicates
the fraction of the total sampling time that a fixed point is within a
fluctuating plume. It is obtained by integrating the intermittency
function h{z, t) which is related to the conc entration c (z ,t) as follows:
{
1' c{z,t» Ct h{z,t}=
0, c{z, t) -< C t
(3.31)
where c{z, t) is the instantaneous concentration at the point z, and
RUN 804
[ 200 d = 10.84 em, IT = 39.2 em /see
! = 7.36, TJ = 0.368, ~= - 0.552
I f =0.433
Q.
.:=. u C ~ o ~
C I Q) o 8 100
I' u a; o o ~
S .£: C
.Q "0 c: ~
lL. >. o c:
~ ·e ~
~ ..£
I
o I--_L_AJUJ ~
i I~
Threshold
~_ L ~oneentration I ,I
_-Ar jlw
1.0
II illl, ~ il ! [1 r I I
i I
! I I I
I . t
U I I "L--~~~ __ JU ~_~
-lljj r-- T2j-----l ~
o 2 4 6 8 10 12
Time, sees
Fig. 6.1. Plots of digitized concentration data c(t) and the corresponding intermittency
function, h(t). Sampling rate = 60 samples / sec. Typical periods of
Iloccurrence ll Tl j and'hon-occurrence" T2 j are shown on H(t); RUN 804.
...... 00 00 I
-1S9-
Ct
is the threshold concentration. The threshold was chosen
-4 slightly (about 7 X 10 % of initial tracer concentration) above the
mean background concentration to eliminate background noise and
reduce the contribution due to the tails of the signals--a result of the
finite response frequency of the detection system.
Figure 6~ 1 shows plots of a measured concentration signal c(t)
recorded at a point within the plume. The corresponding intermittency
function h(t) calculated for the same signal using Eq. 3.31 is shown
directly below c(t). The intermittency factor If determined from
22 seconds of the data shown was 0.433. Also shown on the plot of
h(t) are typical values of the period of occurrence T1j. and the period
of non-occurrence TZj used for the calculation of probability densities
pdTd and pz(Tz) respectively.
Figure 6 .. 2 illustrates the transverse distribution of If for
three flow depths: d = 5.36, 10.S4, and 17.07 cm corresponding to
RUNS S02, S04, and 406 respectively. For each run the measurements
were made on the lateral plane 1"] = Y)h = O. 36S, and at distances
x = SO, lS0, 3S0, 7S0, 11S0, and 15S0 cm from the source .. In general
the plots were restricted to one side of the flow (z < 0) except for
x = SO cm where If was calculated for - 00 < Z < 00 in RUNS S02 and
S04 to demonstrate the symmetry of If{z) about Z = 0 0 It is immedi
ately evident that the If(z) -distribution was similar for all normal depths,
and at all stations. The region of intermittency (0 < If{z) < 1.0) at a
given distance from the source, however, increased with increasing
depth, d.
That the fluctuation of the plume edge was accentuated by an
3_ H
ci 0 I--u Lt >-U Z w I--I--~ a: w I--~
-190-
1.00
O.SO (0)
FLUME WALL RUN S02
0.60 d=5.36em u=43.7 em/see
0.40 IR = 1.17 X 103
• 0.20
0 -50 -40 -30 -20 -10 0 10
Z,em
1.00
O.SO (b)
RUN S04
0.60 d=10.S4em u = 39.2 em/ see
0.40 IR. = 2.03X 103
0.20
LEGEND 0 SYM x,em
-50 -40 -30 -20 -10 0 10 • SO Z,em 0 ISO
1.0 .. 3S0 () 7S0 e IISO
O.S (e) () 15S0
FLUME WALL RUN 406 0.6 d=17.07em
u = 35.3 em/see
0.4 IR =5.72 x '()3 .b
0.2
0 -55 -50 -40 -30 -20 -10 0 10
TRANSVERSE DISTANCE, z, em
Fig. 6. 2. Transverse distribution of the interm.ittency factor
for three norm.al depths d. Measurem.ents were
m.ade at the level of tracer injection Ylh = 0.368 and
at the sam.e stations for all 'runs; RUNS 802, 804, 406.
d = 2.01 ern
U = 46.9 ern/sec
m* = 350
d = 5 .. 26 ern d = 10 .. 7 ern
U = 4203 ern/sec U = 42.6 ern/sec
m*= 1,109 m*=2,210
Figo 6030 Photographs of the tracer plume taken in flume S1 for four different depths d
d = 17.34 ern
u = 3609 ern/sec
m*=3,181
....... --.0 I-"
I
-192-
increase in d is also illustrated by the photographs of Figure 6.3.
The pictures were taken in fluITle S1 and the source was located at
Station 42.0 ITl (fluITle entrance is Station 33.50 ITl). Conductivity
probes are shown just downstreaITl of Station 45.0 ITl. The photographs
show that as d increased froITl d = 2.01 CITl to d = 17.34 CITl, the
pluITle varied froITl an essentially straight ribbon to one with extensively
fluctuating edges. It should be pointed out that the pluITle edge as
ITleasured by the probes extends substantially beyond the apparent
liITlits in the photographs.
The interITlittency factor distributions plotted in Figure 6.2
were deterITlined only at the level of tracer injection. To investigate
the variation of the If distribution with depth, ITleasureITlents were
ITlade at four levels of the flow: YJ = 0 .. 095,0.368,0,,632, and 0.850
for RUN 808 where JR" = 2.938 X 10 3 , and d = 17" 32 CITl. The result--,'
ing distributions are shown in Figure 6.4. Injection level YJh
= 0.368,
and the ITleasuring station was located at x = 380 CITl. The distribution
of I/z) at YJ = 0.095 and 0.368 were virtually identical. However, as
YJ increased the core of continuous record .6 decreased owing to the
decreas e in the plUITle width as cOITlpared to the fluctuation of the pluITle
edge. At YJ = 0.850, the plUITle half-width was sITlaller than the width
of the region of inte rITlittency- - thus If < 1.00 even at z = O. The dis-
tributions, however, were siITlilar and the interITlittency region, Wf
-.6,
was essentially constant for all levels YJ. Therefore the location of
If = 0.50 approached S = 0 as YJ - 1.0. FroITl Figure 6.4 it was con-
eluded that the distribution at the flow level
typified the fl uctuation of the pI UITle front.
YJ = YJ = 0.368 best h
Therefo re further
1.0
'+--
n::: 0.8 0 I-U
~ >- 0.6 u z W l-I- 0.4 ~ n::: w I-z 0.2
RUN 808 Flume 52
d = 17.32cm, TI = 34.9cm /sec 1R*=2.94 x 103, 7Jh =0.368
x=380cm
0·Q30 -20 -10 0
TRANSVERSE DISTANCE, Z ,em
Fig. 6. 4. Transverse distribution of the interITlittency factor at various levels
of the flow 1'\. All ITleasureITlents were ITlade at x = 380 CITl froITl the
source; RUN 808.
--.0 V.l
-194-
measurements of If were confined to this level.
To establish self- similarity of the transverse distribution of
the intermittency factor, If was calculated, for RUN 808, at four
stations: x = 80, 180, 380, and 780 cm. The result shown in
Figure 6.4 was replotted in Figure 6.5 as functions of the normalized
transverse distance (z -2)/0-1" The mean position of the plume front
Z and the variance of the intermittency distribution o-~ were calcu
lated numerically by Eqs. 3.43 and 3.44 respectively. Figure 6.5
shows that all points plotted closely on the universal curve:
(6. 1)
where
This indicates that the transverse position of the plume front
was a normally distributed random variable because the intermittency
factor distribution is equivalent to the cumulative probability distribu-
tion of the position of the plume front. This distribution is similar to
those obtained by Klebanoff (13) fo r the laminar-turbulent interface of
the boundary layer, by Townsend (45) for the plane wake, by Corrsin
and Kistler (i1) for the round jet, and by Demetriades (12) for the
axisymmetric compressible wake. By comparing Eqs. 6.1 and 3.39,
it is found that the characteristic half-width of the zone of intermittency
L is given by
-H 1.0 .. a:: ~ u 0.8
Lt >-~. 0.6 w lI-~ 0.4 a:: w I-z 0.2
RUN 808 Flume 52
d=17.32 em,u=34.9 em/sec 3
IR = 2.938XIO, ." = 0.368 * "h L....---If = ~ (I+erf ~I) to z-z where ~ =-I./2C'x LEGEND
5YM x,em 'T)
o 80 0.368 • 180 0.368 () 380 0.095 C> 380 0.368 e 380 0.632 Q 380 0.850 ~ 780 0.368
o I 6 '3-#M----3 -2 -I 0 I 2 3
NORMALIZED TRANSV.ERSE DISTANCE, z;;.Z I
4
Fig. 6. S. Representation of the transverse distribution of the intermittency
factor on a universal curve. Measurements were made at four
stations and four levels at one station; RUN 808.
...... '-.0 U"1 I
-196-
L = zl2 <rI • (6. Z)
Thus the width of the intermittency region is ZL = 5064 (Tr From
Eqs. 6.1 and 3.33, the probability density function, if' for the
po sition of the plume front becomes
(6.3)
a representation symmetrical about the mean position of the plume
front z = Z. It should be noted that Eqs .. 6. 1, 6 0 2, and 6.3 are
universal relationships but both Z and (TI are functions of x.
6.A.Z. Growth of the Geometric Parameters Wi, Z, .6, and
(Tl of the Intermittency Region with distance x.
a. Overall characteristics. Figure 6.6 shows for RUN 802 the growth
with distance x of the maximum limit of the intermittency zone,
Iz I = Wf , the limit of the central core, /z 1=.6, and the mean position
of the plume front, Iz I = Z. Both Wf and .6 were determined
directly from curves of the If distribution (such as those in Figure
6 .. 2) as the values of I z I where If first attained the value of zero
and unity respectively. The mean position Z of the front was calcu-
lated numerically using Eq. 3 .. 43. In all cases the origin of the z-axis
was modified slightly to coincide with the point where C = C • max
It was found that, within the reach of the measurements
(x -< 15.8 m), W f and .6 grew at different rates. This characteristic
will be discussed further in later sections. As evident from Figure
6.6, Z ~ f(.6 + W f} indicating that the if distribution was symmetrical
E u ~
~ IN
<l -(f) o I(f)
0:: W Io « 0:: « I o o 0:: IW ~ o W <..9
40
30
20
10
W( extreme limit of the plume boundary (If=O)
Z • mean position of plume front (If 0: 0.50)
6. : outer edge of the inner core (If =1.0)
Wf
RUN 802 Flume S2
d =5.36 em, ij = 43.7 em/sec 3
u =2.17 em/sec, IR = 1.173 XIO * *
o
0 1111
o 200 400 600 800 1000 1200 1400 1600
DISTANCE FROM THE SOURCE I X ,em
Fig. 6. 6. Growth of the geom.etric characteristics of the region of
interm.ittency; RUN 802.
1800
'-' '-D -..J I
-198-
~
about 2 as implied by Eqs. 6.1 and 6.3. To further verify this
symmetry, 2 was compared to zO. 5 which denoted the value of
Iz I where If = 0.50. The comparison is shown in Table 6.1 fo r ~
RUNS 804 and 808. The deviation, E, of zO.5 from 2 was within
0.6 em for all cases. The average value of I E I was 0.4 em which
corresponded to a mean value of I E I/i = 2.7%. It was therefore
-concluded that zO.5 essentially coincided with 2 and that Wf
and
D. were equidistant from 2.. Thus once W f and 2 were known,
D. could be deduced by D. = 22 - W £"
Table 6 .. 1
~t :j: Comparison Between 2 and zO.5
- G -x 2 zOoS €u 2 zO.5 E
m em em em em em em
RUN 804 (d = 10.84 em) RUN 808 (d = 17.34 em)
0.80 6.2 5.9 -0.3 4.5 4.2 -0.3
1. 80 10 .. 6 10 .. 1 -0.5 9.5 9.4 -0.1
3.80 14.3 14.1 -0.2 18.0 18.4 0.4
7.80 25.2 25.4 0.2 25.7 25.1 -0.6
11.80 35.8 36.4 0.6
15.80 43 .. 1 43.6 0.5
t '" 2 = mean position of the plume front
:j: zO.5 = value of I z I where If = 0 .. 50
§ -E = z - 2 0.5
I
-199-
To analyze the rates of growth of Wf
, Z, and ~, it was
necessary to define a virtual origin, xv' of Wf
as the value of x
where W f = 0 0 Since, as demonstrated by Figure 6.6, the region
of intermittency extended to the plume axis near x = 0, ~ attained
a zero value at a distance xI downstream of the source. Values of
xI and Xv were determined fo r a11 experiments: the former by
extrapolating curves fitted to measured limits of the central core to
the point on the x-axis where ~ = 0, the latter by fitting parabolas
to plotted values of W f versus x by the least square method. The
result is listed in Table 602.
RUN
Table 602
Values of Xv and xI Determined for Various Experiments
Flume Flow Depth
d
Shear Velocity
Friction Reynolds Number t
IR>:<
x v
ern ern/sec ern ern
802 S2 5.36 2 .. 17 1. 17 8
804 S2 10.84 1.86 2.03 27
808 S2 17.32 1.66 2.94 41
405 R2 5.01 3.39 -66
404 R2 5.05 4.97 -10
406 R2 3.64 5.72 -5
t The bed shear velocity u>:<b was used in RUNS 405, 404, and 406
:j: x = position at which Wf(x ) = 0 by extrapolation v v
§ xI = value of x where the limit of the central core intersects the
plume axis
37
25
148
15
35
125
1---------------------------.------------- - -----
-200-
It is observed from the table that for experiments performed
in the flume with smooth boundaries x > o. When the flume bottom v
was roughened with rocks x < O. The significant point, however, v
was not the sign of x but the fact that in each set of experiments v
(rough versus smooth), x increased with the flow depth d. It was v
found that, for the smooth boundary experiments,
sented empirically by
Cv)' =O.37($i-16.SS).
x could be reprev
(6.4)
where f* = friction factor. According to Table 6.2, xI depended on
the flow depth increasing from xI ~ 30 cm for d ~ 10 cm to a maximum
value of 148 cm for d = 17032 cm, regardless of whether the flume
bottom was smooth or rough.
b. Prediction of the extreme limit of the plume boundary, W fO Dimen-
sional analysis was used to develop a universal curve representing all
experimental measurements of W f" The variables selected were W f'
the extreme limit of the plume boundary; X = x - xv' the value of x
corrected for the virtual origin xv; u* (or u*b)' the (bed) shear
velocity; and li, d, and v, the mean flow velocity, the flow depth, and
the kinematic viscosity respectively. Thus
Wf
= g(X ,d,u,. .. ,li,v)
where g represents an unknown function. Similarity argument then
predicts that
-201-
(6.6)
where gl is still an unknown function. Since both ud/v and u*d/v
were large in all experim.ents and the flow was thus fully turbulent,
it was reasoned that W f/ d depended on the roughness of the flow
boundaries but not on the value of the Reynolds num.bers. The
characteristic variable needed was therefore a frictional param.eter
which could be derived from. Eq. 6.6 as the ratio of the Reynolds
num.bers. Hence
(6. 7)
For a given experim.ent (i. e. u*/ u = constant), m.easured
values of Wf/d were plotted against X /d on log-log scales. It was
found that
(6.8)
1 where the constant exponent 'VI = z. To incorporate the
frictional param.eter, (Wf/d)Z was plotted against (X /d)(u*b / u)
using u* for the sm.ooth and u*b for the rough bottom. experim.ents.
As shown in Figure 6.7, the points fell on two well-defined parallel
lines: A, for the sm.ooth bottom. experim.ents, and B for the experi-
m.ents perform.ed when the flum.e bottom. was roughened with rocks.
By m.ultiplying (X /d)(u>''<b/ ti) by a factor Rw' however, lines A and
B collapsed into one and, as shown in Figure 6.8, all points plotted
on the universal curve
40.0
10.0
LEGEND
SYM RUN d,cm Flume o 808 17.32 S2 8 804 10.84 S2 o 802 5.36 S2 ~ 406 17.07 R2 () 404 10.36 R2 o 405 6.81 R2
-202-
B (rough)
2 (:fj 4.0
1.0
0.4 .
0.1 0.1 0.4 1.0
For smooth boundaries:
u*b= u.
4.0 10.0 40.0
(~)(U~b) 100.0
Fig. 6.7. Plots of (W/d)2 versus ('x./d)(u~~b/u) for experiITlents
perforITled in fluITle 2 with the bottOITl hydraulically
SITlooth (A) or rough (B); RUNS 808, 804, 802, 406,
404. 405.
40.0
10.0
2 (:fj 4.0
1.0
0.4 .
0.1 0.1
LEGEND
I SYM RUN d,cm Flume I o 808 i7.32 S2 {,':. 804 10.84 S2 [J 802 5.36 S 2 \'l 406 17.07 R2 <t 404 10.36 R2 o 405 6.81 R2
·0 {,':.
0.4 1.0
-203-
For smooth boundaries:
u*b= u. R =1 w
4.0 10.0 40.0 100.0
Fig. 6.8. Universal representation of the growth of the extreme
limit W f of the plume boundary (where I f = O. 0) for all
experiments; RUNS 808, 804, 802, 406, 404, 405.
-204-
(6.9)
where
R = (f /f ) 1 /4 w s r '
(6.10)
f ,f are, respectively, the mean values of the bed friction factors for s r
the smooth boundary and rough bottom experiments. For the smooth
boundary, the bed shear velocity u*b becomes the shear velocity u*
and R = 1.0. Thus the R correction is necessary only for the w w
hydraulically rough boundary flows. With Eq. 6.9, the extreme limit
Wf
of the fluctuating plume front can be predicted for given normal
flow conditions.
Co Prediction of the mean position Z of the plume edge. From
dimensional and physical reasoning similar to the arguments in the
last section, a universal curve was established for the growth of Z
with distance from the source. Log-log plots of Z/d versus X/d
for various experiments yielded a representative relationship
(6. 11)
where the exponent '12 = 2/3. This value of '12 was compared with
those previously reported in the literature for other flows. The results
are shown in Table 6.3.
From the table it is evident that '12 varied from 1.0 fo r the
rou nd jet to 1/3 for the axisymmetric compressible wake. The value
of '12 obtained in the present study was essentially equal to that
-20S-
Table 6.3
Values of the Exponents '12 and '13 for Different Flows
t :t: :t: Method of Source Kind of Flow 'I '13 Dete rmination
2 of '12 and '13 1 2 3 4 5
Towsend Two-
dimensional 1/2 ~ 1/2 Experimental (4S) wake
006S 0.70 Theoretical
Growth of approximation
the turbulent boundary
layer 0.63±001 0.67±0.1 Expe rimental
Corrsin and
Kistler ! (11 ) 1.0 1.0 Theoretical
Round I ,
jet
0.88±0.OS 1.06±0.OS Experimental
Demetriades Axisymmetric
(12 ) compressible 1/3 1/3 Experimental
wake
I
Transverse
Present growth of I a plume in a 2/3 1/3 Experimental ,
Study turbulent shear flow
t All measurements listed, except in the present study, were made for air flow in a wind tunnel
:t: (~) ~ (~ )'12 ; (~~)~(~)'I3
-206-
postulated and IT1easured by Corrsin and Kistler (11) for the growth of
a turbulent boundary laye r next to the wall of a wind tunnel. A strong
siIT1ilarity between the two processes was again indicated. The first
indication of this siIT1ilarity was the fact that the transverse distribu-
tions of the intermittency factor IT1easured by Klebanoff (13), and
Corrsin and Kistler (11) for the boundary layer were virtually identical
to those calculated for the plume front fluctuation in the present study.
To develop a universal curve, Z/d was plotted against
(X /d)2/3(u*b/ u) on log-log scales o Two parallel lines were again
found to fit the measurements; one for the SIT100th boundary experiments,
the other for the rough bottom. Further calculations showed that the
two lines IT1erged into one represented by
(6.12)
where
(6.13)
When the flume was smooth, u*b == u*, and Rz = 1"
Measured values of Z for both smooth boundary and rough
bottoIT1 experiments are plotted in Figure 6.9. It is evident that all
points closely fitted the universal curve of Eq. 6.12. For large X,
however, experiIT1ental points increasingly deviated from Eq. 6.12
showing a growth rate slower than the equation would predict. There
was a strong indication that very far froIT1 the source, Z/d grew as a
parabolic function of (X /d) just as W /d did. The present IT1easure-
~ ~N
10.0 r
LEGEND
SYM RUN d,cm Flume
4.0 r 0 808 17.32 S2 III 804 10.84 S2 0 802 5.36 S2
'"' 406 17.07 R2 () 40410.36 R2 ~ 405 6.81 R2
1.0 ~ ~ Z =331 R (K)~ u.b d . Z d u
0.41- / For smooth boundaries:
u b su * * R =1 z
0.1 0.01 0.04 0.1 0.4 1.0 4.0
2 e: = R (Xy3 U!b
u Z d U Fig. 6.9. Universal representation of the growth of the mean position of the
plume front Z (where If ~ O. 50) for all experiments; RUNS 808,
804, 802, 406, 404, 405.
~ I tv
i 0 --J I
10.0
-208-
ITlents in the fluITle, however, did not extend to distances far enough
froITl the source to conclusively verify this tendency.
Eq. 6.12 can therefore be used to calculate Z given a uniforITl
flow condition. FroITl Eq. 6.9, Wf
can also be predicted. Thus the
outer liITlit of the central core t1 ITlay be derived froITl the relation
d. Growth of the variance 01 of the region of interITlittency. The
variance ~ of the transverse distribution of the position of the pluITle
front was calculated nUITlerically by Eq. 3.44. By appropriate plots,
it was established that for each experiITlent (sITlooth boundary or rough
bottoITl), 0"1 initially grew according to the relation:
(6. 14)
where the exponent '13 = 1/3. This value of '13 was again cOITlpared
to those obtained for other flows. The cOITlparison is presented in
Table 6.3. It shows that '12 = '13 for other flows previously reported
in the literature but, in the present study, '12 = 2'13"
For all experiITlents in the fluITle, it was found that expression
6.14 could be ITlore explicitly written as
(6 ; I:: \ o ~ .J I
where, for the SITlooth boundary runs, Kl = 0.23,0.16, and 0.11 for
IR = 2938, 2026, and 1173 respectively. For the rough bottoITl bound*
ary, Kl = 0.23, 0.10, and 0.04 for IR*b = 5716,4973, and 3388
-209-
respectively. The power law relationship of Eq. 6.15 was valid only
as long as (O'"l/d) < 0.4. As (X/d) increased beyond this point, O'"l/d
began to taper off and in some cases O'"l/d actually decreased with
increasing X /d. Thus if the amplitude of the plume front oscillation
(along z for fixed x and y) is characterized by 20'" l' the maximum
relative amplitude attained in all experiments:
( 2dO'"l) :::; 008. (6.16) max
The variance O'"~ was also compared to the total variance a2
of the transverse distribution of the mean tracer concentration calcu-
lated by Eq. 5.3. The ratio (0'"/cr}2 was plotted against X/d for each
experiment. As expected the ratio decreased with increasing X /d
according to the power law
2
(0'" l) ()-1/3 - = K X. 0'" d
(6. 1 7)
The constant of proportionality K in Eq. 6.17 again decreased with
decreas ing friction Reynolds number.
6. A.3. Temporal Characteristics of the Plume Front Oscillation.
a. Frequency of "zero occurrence " w~ The rate at which the
intermittency function h(z t t' changed from zero to unity at
a fixed point was denoted as Wo (z), the frequency of ze ro occurrence
at the point z. From the definitions of liz) and wo(z) , it was rea-
soned that
-210-
{
If = 0.0 = 0 for
If = 1. 0 (6. 18)
and that as If -- 0.50, wO(z) approached a m.axim.um. value. Since the
oscillation of the plum.e front has been shown by the distribution of
If(Z) illustrated in Figure 6.5 to be a norm.ally distributed random.
function, it is expected that the distribution of wo (z) should also be
norm.al. From. a theory postulated by Rice (65), it was deduced that
wo (z) could be represented by
a distribution sim.ilar to the density function ir The m.ean value wo
corresponds to the zero-frequency at the m.ean position of the plum.e
front Z where If = 0.50"
The zero-frequency wO(z) at a fixed point z was com.puted
from. the digitized record by first counting the total num.ber of tim.es
that h(z,t) on consecutive tim.e digits, changed from. zero to unity.
Then this value was divided by the total sam.pling tim.e to give wO(z)
in Hz. It is thus evident that wo (z) is extrem.ely sensitive to the
signal to nois e ratio. At stations near the source where this ratio
was high, wo (z) was readily evaluated with reasonable reliability.
For large x, however, tracer concentration was low and so were the
signal to noise ratio and the attenuation of the recorder. Thus, the
calculated wO(z) was distorted by the background noise giving rise to
unreasonably high values espec ially in the neighborhood of If::::: 0.0.
In general, only those values of wO(z) m.easured near the source or in
-211-
the neighborhood of If = 0.50 were considered reliable and usable for
further calculations.
The transvers e distribution of the zero-frequency was studied
by plotting the cumulative values of wO(z) on arithmetic probability
scales. Examples are shown in Figure 6.10 for RUN 404. The plotted
points closely fitted the Gaussian lines drawn for each value of x. The
standard deviations of the fitted Gaussian Hnes were 2.20, 2.75, and
3.75 cm for x = 80, 180, and 380 cm respectively. The corresponding
0"1 calculated from If distributions were 2.34, 3.17, and 3 .. 88 cm.
This indicated that wO(z) was closely represented by Eq. 6.19. Typical
values of wo evaluated at If ~ 0" 50 and shown in Figure 6.10 were
approximately 4.4 Hz and essentially constant for all x.
To verify if wO(x) was due to the Karman vortex street
generated by the tracer injector, the vortex shedding frequency, n. , 1
was computed for the hydraulic conditions of RUN 404 as an example.
Using the mean flow velocity u = 42.8 cm/sec, the outside diameter
of the tracer injector di = 3.1 75 mm, and the kinematic viscosity
v = 1.052 X 10-2
cm2 /sec, the Strouhal number was found to be
0.21 u
Thus the frequency of vortex shedding by the injector ni
= 28 .. 4 Hz.
However, the values of wo (x) for plume fluctuations determined for
RUN 404 were, respectively 4.33, 4.77, 4.19 and 4.47 Hz for x = 80,
180,380, and 780 cm. Similarly for other experiments, the vortex
shedding frequency greatly exceeded the plume oscillation frequency.
As further verification, the tracer was injected on the water surface
.. >u z w ::::)
o W 0: LL o 0: W N
W > t:i --1 ::::)
~ ::::) U
-212-
TRANSVERSE DISTANCE Z,cm
Fig. 6.10. Transverse distribution of the frequency of zero
occurrence w00 Cumulative values of Wo are
plotted on arithmetic probability scales. Fitted
lines are Gaussian; RUN 404.
-213-
without the injector touching or penetrating the flow. Even then plUIne
edge fluctuation was observed.
It was thought, therefore, that the weak periodicity exhibited
by the plume front oscillation was not a response to the vortices shed
by the tracer injector. This and the fact that if{z) was Gaussian
strongly suggested that the fluctuation of the plume front was a result
of a diffusion process generally characteristic of turbulent mixing.
b. Characteristic period, wave length and amplitude of the frontal
os cillation. The fluctuation of the plume front exhibited a weak peri-
odicity. Thus a characteristic period T O{x} was defined such that
(6.20)
From Eq. 3.46, a characteristic longitudinal length scale or wave
length Lf(x) was defined by
{3.46}
where u is a convective velocity in the longitudinal direction.. Since c
tracer was injected at 11h = 0.368 and the level of concentration meas-
urements, 11 = 0.368, u was es sentially equal to the mean flow c
velocity u. Therefore the normalized {or relative} wave length
L/x) /d was evaluated by the relation
(6.21)
Table 6.4 lists the calculated values of Lf{x)/d {at several
distances x} for several runs. As indicated, Lf{x)/d was essentially
-214-
Table 6.4
Relative Wave Length of Lf(x) /d of the Plume Front Oscillation
1" I I I x, cm i Flow Average IRUN Flume Depth 80 180 380 780
Lfld 1 I d, cm I
I 1 2 3 4 5 I
I I
802 82 I 5.36 1. 49 1. 55 1
1. 04 1. 39
804 82 10.84 1.46 0.79 1. 14 1. 06 1. 11
,808 82 17.32 0.58 0.91 0.61 0.70 ,
1405 1
R2 6.81 1. 13 0.86 1. 00 I
I i404 R2 10.36 0.96 0.86 0.98 0.92 0.93
406 R2 17.07 0.32 0.50 0.36 0.39
invariant with x for any given experiment. It is evident from Column 5~
that for com.parable flow depths, the average value of the relative wave
length~ L£/d~ is smaller for the rough than for the smooth boundary
experiments. This indicates that the characteristic transverse scale
decreases with intensification of turbulence. One can speculate that
accentuation of turbulence intensity effectively breaks up the larger
eddies.
The amplitude of the plume front oscillation was represented
by 2a-r As stated in subsection 6.A.2.d, 2a-I
grew with increasing x
reaching a maximum value of 0.8 d for the same experiments shown
in Table 6.4. Thus the maximum steepnes s of the oscillating plume
-215-
front expressed as (2o-r)max/ L f was approximately unity for all
experiments. For a given experiment, therefore, the characteristic
width of the region of intermittency increased with x until the fluctu
ation amplitude 20-r was nearly equal to the characteristic wave length
of the plume front. As x increased further, the width of the inter
mittency region decreased with x.
c. Probability densities pdTd, pz(Tz), and p(T) of the intermittency
function h(z, t) at a fixed point. The probability dens ities Pl(T 1) for the
duration of occurrence TI (when c(t) > Ct), and pz(Tz) for the dura
tion of non-occurrence T z (when c(t):!S C t ) of the intermittency
function h(z, t) were determined according to the method outlined in
Chapter 3. Typical values of TI and T z are shown in Figure 6. i.
The densities PI (Td, pz(Tz) calculated for RUN 904 at g = 7.4, 16.6
71.9, and 10B.9 are plotted in Figure 6.11. For S = 7.4 and 16.6,
56 seconds (3360 samples) were analyzed, and for S = 71. 9 and 10B.9,
24 .. 3 seconds (1456 samples). Histograms representing PI (TI) are
shown as solid lines. For pz(Tz) the lines are dotted. Experimental
curves--solid for PI (Td and dotted for pz(Tz)--were fitted to the
histograms. The ordinates were normalized according to Eq. 3.47.
The time scale of the abs cis sa is in the units of 1/60 sec. Maxi-
mum values of TI were 1.1B, 2.55,1.30, and 0.B4 sec for S = 7.4,
16.6,71.9, and 10B.9 respectively.
The probability density p(T) for the combined occurrence and
non-occurrence periods is plotted in Figure 6. 12 for the same points
shown in Figure 6.11. The time scale is in units of 1/60-sec and
0.08
~ N
': 0.06 I; f--
a. U)
0.04 w f-if) Z w 0 0.02 ai 0 0: a..
00
N f-N 0.06 a.
r.:-a.
U) W f-U)
Z 0.02 w
0
ai 0 0: a..
e = 7.4
,. If = 0.565 •
· •
· 1 I
· •
f-N
N a.
r.:-
0.08
0.06
0.04
e = 16.6
If = 0.527
RUN 904 Flume S2
d= 10.84cm, u= 39.2cm/sec
1R*= 2.03x 103
I a. 1 1 1 , . _J" " , b:::-r'::_l:--t---~
~--=--
10 20
PERIODS, TI , T2
e = 71.9
If = 0.521
10 20
PERIODS, TI , T2 ,1/60-sec
0.02
30
o PI(TI) = prob. density of occurrence
r-' L_J P2(T2) = prob. density of non -occurrence
1 unit on time scale = 1/60-sec
0.06
I N
f-
~ 0.04 ~----L i\
r.:-a.
0.02
30
10 20
PERIODS. TI , T2
e = 108.9
If = 0.506
..... ----r----' I :
10 20
PERIODS, TI , T2 ,1/60-sec
30
30
Fig. 6.11. Probability densities Pl (Tl ), P2 (T2 ) of the pulse lengths (or periods)
T l , T2 of the intermittency function h(t) at various distances s; RUN 904.
I N ....... 0'
~
a.
~ t::
00+ \ CJ) z ~ >-~ :::::i
0.02 iii « CD 0 0::: 0...
~
0.
~ ~ U) Z W 0
>-~ :::::i in « CD 0 0::: 0...
00
~ = 7.4 If = 0.565
PERIOD, T
~ = 71.9
If = 0.521
20
20
PERIOD, T ,1/60-sec
~
a.
30
~ = 16.6 I f = 0.527
RUN 904 Flume S2
d= 10.84cm, u= 39.2cm/sec 1R* = 2.03 x 103
01 1: I o 10 20 30
PERIOD, T
I unit on time scale = 1/60 - sec
30
~
a.
0.06
0.04~<
0.02
~= 108.9
If= 0.506
01 0---- I o 10 20 30
PERIOD, T , 1/60-sec
Fig. 6.12. Probability density p(T) for the combined periods (or pulse
lengths) of occurrence and non-occurrence of the intermittency
function h(t) at various distances s; RUN 904.
I tv ...... -..] I
-218-
p(T) was norITlalized for each £ in accordance with Eq. 3.48.
Representative curves were fitted to each histograITl as in Figure 6. 11.
Figures 6.11 and 6.12 show that the shape of each of the distri
butions of Pl(Td, pz(Tz), and p(T) was essentially invariant with g. In
general the short periods dOITlinated long pulse lengths. The result
was a positively skewed distribution with ITlO re than 50% occurrence
within a period of 0.067 sec for IS:S; 16.6. As £ increased, the
distributions broadened as larger pulses becaITle inc reasingly ITlore
iITlportant. Since p(T) was norITlalized according to Eq. 3.48, the
broadening of the density distributions caused an attenuation of the peak
of p(T) with increasing g.
The overall shapes of the Pl(T1) and pz(Tz) distributions were
siITlilar to those calculated by Corrsin and Kistler (11) for the fluctu
ation of the edge of the turbulent boundary layer. This again indicates
a siITlilarity between the growth of the wall boundary layer and the
transverse spread of a pluITle in a turbulent shear flow.
do Cross-correlation analysis of h(z,t) for two points located at
z = Z and z = - Z. A cross-correlation analysis of the interITlittency
function was ITlade at five stations. At each station, the two points
cOITlpared were located at z::::: Z and z::::: - Z where If::::: 0.5 and at
the saITle flow level.
The procedure consisted in first deterITlining the two points at a
given station where, for 1'] = 1']h' If::::: 0.50. A probe was then placed
at each of the points and the tracer concentration at both locations were
ITlonitored siITlultaneously, digitized at 60 saITlples/sec, and stored on
-219-
two separate channels of the m.agnetic tape.
The interm.ittency functions hd - Z, t) and hz(Z, t) were com.-~
puted respectively for the probes located at z = - Z and z = Z for
the particular station. The two functions hl and hz were then com.-
pared at identical tim.es t at intervals of 1/60-sec and the result
recorded according to the following designations:
(i) ON-ON, if hl = hz = 1.0
(ii) OFF-OFF, if hl = h z = O. a
(iii) ON-OFF, if hl = 1.0, and hz = 0.0
(iv) OFF-ON, if hl = 0.0, and hz = .0
The ratio of the total num.ber of ON-ON and OFF- OFF cases to the
total num.ber of sam.ples within the sam.pling period was term.ed
INSTABILITY; the rest of the sam.pling period was called MEANDER.
Measurem.ents for the cross-correlation analysis were m.ade in
RUN 904. The flow depth was 10.84 em., the m.ean flow velocity
u = 39.2 cm./sec, and the level of tracer injection 11 h = 0.368. For
the analysis, 3360 sam.ples were used for S = 7.4, 16.6, and 35.0.
For S = 71.9 and 108.9, 1456 sam.ple s were analyzed. The results
of the calculations are sum.m.arized in Table 6.5. ~
Colum.n 1 identifies the positions of the probes: Z and - Z.
Colum.n 2 lists the values of If at each po into Although thes e values
were slightly greater or less than 0.50, they were considered suffi-
dently close to 0.50 for the purposes of the analysis. The slope of the
transverse distribution of If was very steep near If = 0.50 (especially for
sm.aJ.l s). Thus a slight deviation of If from. O. 50 caus ed negligible
erro rs in the estim.ation of the m.ean front pas ition. If, fa r a particular
~osition InterlTI. of the Factor
Probe If
1 2
~
-Z 0.531 ~
Z 0.478
~
-z 0.494 ~
Z 0.452
~
-z 0.557 ~
Z 0.621
~
-z 0.521 ,.., Z 0.611
~
-z 0.506 ~
z 0.490
Table 6. 5. Cro ss-Correlation Analys is of h(z, t) at the Two Points z :::: - Z and z:::::l (RUN 904)
Zero Per Cent Per Cent Freq. ON-ON OFF-OFF ON-OFF OFF-ON INSTA- MEANDER wO,Hz BILITY
3 4 5 6 7 8 9
£ = 7.4
I I I I I I I 3.86 872 841 913 734 50.98 49.02
3.00
£ = 16.6
I I I I I I I 2.54 642 825 1018 875 43 .. 66 56.34
3.41
£ = 35.0
I I I I I I I 4.82
1120 524 750 966 48.93 51.07
4.07
£ = 71. 9
I I I I I I I 4.77 438 247 320 451 47.05 52.95
4.98
£=108.9
I I I I I I I 4.98 370 376 367 343 51.24 48.76
6.29
I N N o I
-221-
probe, If > 0.50, the number of ON events would be increased and the
OFF events decreased for that probe. When If < 0.50, the reverse
would be true.
The frequency of zero occurrence Wo calculated from the inter
mittency function is tabulated in Column 3. The total number of ON-ON 1
OFF-OFF, ON-OFF ~ and OFF- ON events is listed respectively in
Columns 4, 5, 6, and 70 The fraction of INST ABILITY and that of
MEANDER both expressed in per cent are entered in Columns 8 and 9
respectively.
Cross-correlation analysis was undertaken to verify which of
two hypotheses governed the fluctuation of the plume front. The first
is the instability (or equilibrium) hypothesis, described by Townsend
(66). It postulates that the indentations of the plume edge result from
a growth-decay cycle of the large eddies which effect plume spreading
and mac roscale mixing. Thus the region of intermittency would be a
consequence of a pulsating motion of the plume boundaries transverse
to the flow direction. If mixing is due, entirely, to this hypothesis,
then the parameter INSTABILITY of Table 6.5 would be nearly 100%,
and MEANDER would have a value near zero.
The second hypothesis, advanced by Gifford (16), assumes that,
at any given S (expecially for large S), the plume width is essentially
constant. Fluctuation of the plume edges is therefore due to the
meandering of the entire plume with the plume centroid wandering
randomly along a transverse line normal to the flow. This model has
been dis cus sed in Chapter 3 and illustrated in Figure 3.1. If the plume
front variation is completely due to this reasoning, MEANDER of
-222-
Table 605 would be 1000/0 and INSTABILITY zero ..
A careful scrutiny of Table 6.5, however, reveals that at all ~,
INSTABILITY:::::: MEANDER:::::: 0.50, (6.22)
indicating that pulsating and meandering motions contributed to the
wrinkle of the plume edge.. Thus for ~:> 7.4 neither hypothesis
seemed to dominate the other" Further examination of the table shows
that, for each S, there was a fairly uniform distribution of events
in the ON-ON, OFF-OFF, ON-OFF, and OFF-ON slots. Thecloser
the values of If fo r hI and hz were near 0.50 (for example at
S = 108 .. 9). the better was the uniformity .. This meant, therefore,
that the opposite edges of the plume also fluctuated independently of
each other and that this test could not distinguish plume edge fluctu
ation from a purely random proces s.
FroITl the results of Table 6.5 and the transverse distribution
of the interITlittency factor shown in Figure 6.5, it is concluded that
the fluctuation of plume edge for S:> 7.4 seems to be a randoITl motion
with a Gaussian distribution about the ITlean front position. The fluctu
ation is due to both the meandering of the plume centroid and the
growth- decay cycle of the overall plume width. However, if the chan
nel alignment is not straight or the flow cross section is not uniform,
strong lateral gradients of velocity would intensify plume meandering.
This would thus dominate the growth-decay cycle as the principal
mixing mechanisITl and indeed increase the overall width of the plume
and eventually the transverse mixing coefficient.
-223-
6. B. ANALYSIS OF PLUME VARIATION USING THE FLUCTUATING PLUME MODEL
Figure 6. 13 shows the temporal variation of the instantaneous
plume boundaries at fixed stations. The data were obtained during
RUN 300 from the motion picture study described in Chapter 3. In
Figure 6 0 13, the points are plotted at intervals of 1/6-sec and at each
station, five seconds of data is shown as illustration. Calculations
were made at S =3.2,4.1,5.0,5 0 7,6.5.7.5,8.6, and 9.5 with the
flow depth d = 17.00 cm. As explained earlier, plume boundaries
became indistinct at greater distances from the source.
A casual observation of Figure 6.13 seemed to indicate that
plume meandering completely dominated the variation of the plume
width. This was certainly true for s"'::: 6.5. where the mean size of
the plume width was still small compared to the flow depth. However,
as S increased further, plume width variation became increasingly
important. This is illustrated in Figure 6.14 which shows the temporal
variation of the instantaneous plume width at fixed stations for the same
plume boundaries shown in Figure 6.13. Figure 6.14 clearly demon-
strates that the variation of the plume width was certainly not small
compared to the oscillation of the plume centroid for 705::;; s ::::s: 9.5,
and that, for all s. the variation was weakly periodic suggesting a
cyclic phenomenon characteristic of the instability hypothesis.
From data similar to those plotted in Figure 6.13, various
parameters associated with the fluctuating plume and described in
Chapter 3 were calculated. Table 6.6 lists the results. The mean
width W of the instantaneous plume tabulated in Column 1, was a
E 0
N
w U Z
~ CJ)
0
w CJ)
0:: W > CJ)
z « 0:: .-
-4 0 4 8
-4 0 4 8
-12 -8 -4
0 4 8
12
0
-224-
! = 3.2
! = 4.1
2 3
TI ME t secs
RUN 300 Flume S1
d = 17.00em, IT = 35.4 em/sec 1R* = 2.63 x 10 3
4 5
Fig. 6. 13. Temporal variation of the plume boundaries at
fixed stations downstream of the source. Five
seconds of data plotted at intervals of 1/6-sec
is shown at each station; RUN 300.
8
E 4 0
:r: 0 I- 8 0 - 4 ~
w 0 ~ => -I 8 Q..
en 4 => 0 ~ 12 Z
~ 8
Z 4 « I- 0 en 16 Z
12
8
4
0 0
Fig. 6. 14.
2 TI ME ,sees
3
RUN 300 Flume SI
d = 17.00em, U = 35.4 em/sec 1R*= 2.63 x 103
4
TeITlporal variation of the instantaneous pluITle width at fixed stations
for the pluITle boundaries shown in Figure 6.13; RUN 300.
5
I N N Ul I
-226-
Table 6.6
Parameters As sociated With the Fluctuation Plume (RUN 300)
t W t § tt :j::j: § § tT2
tT 2
S W a tT 2 ? tT 2 tT 2 ~
W
a d w f g M ;rr M g
cm cm2 cm2 cm2 cm2
1 2 3 4 5 6 7 8
3.2 3.28 o. 19 0.48 0.70 1. 65 2.35 () 7() v. 'v 0.29
4. 1 3.62 0.21 0.96 0.88 2.14 3.02 0.71 0.45
5.0 4.96 0.29 1. 51 1. 63 2.71 4.34 0.62 0.56
5.7 5.44 0.32 1. 83 1. 97 3.51 5.48 0.64 0.52
6.5 5.60 0.33 2.48 2. 12 3.71 5.83 0.63 0.67
7.5 5.70 0.34 3. 18 2.23 3.20 5.43 0.59 0.99
8.6 7.98 0.47 8. 12 4.49 8.32 12.81 0.65 0.98
9.5 7.92 0.47 6.68 4.33 10.17 14.50 0.70 0.66
t VI! = mean plume width (Eq. 6. 23) a
:j: Flow depth d= 17.00 cm
§ tT
2 = variance of the plume width (.sq. 6.24) w
tt~ = mean variance of the instantaneous concentration distribution tT f (Eq. 3.49)
:j::j: tT
2 g
= variance of the fluctuation of the plume centroid (Eq. 3.50)
§ § 2 . f h t' d' "b' (E tT M = varlance 0 t e mean concentra lon lstrl utlon q. 3.25)
calculated by
W a
-227-
N
=1 )' NL;
i= 1
W(i) (6. 23)
h W( ')' ttl idth h . th. . tId were 1 = lns an aneous p UIT1e w at tel tlIT1e ln erva , an
N = total nUIT1ber of picture fraIT1es analyzed. In general, N was near
100, and the tiIT1e interval was 1/6- sec.. ColuIT1n 2 shows the pluIT1e
width Wa/d relative to the flow depth.
width was evaluated by
The variance 0"2 of the plUIT1e w
N
0"2 = 1. )' W2 (1) _ W2 .. W NL..../ a
i= 1
Values of 0"2 are listed in ColuIT1n 3 .. w
(6.24)
The variances ~ of the instantaneous concentration distribu-
tion, ~g of the pluIT1e centroid, and 0" ~ of the IT1ean concentration
distribution calculated, respectively, by Eqs. 3 .. 49, 3 .. 50, and 3. 14
are tabulated in ColuIT1ns 4, 5, and 6. The ratios of the variances
0"2/0"2 and 0"2 /0"2 are entered in ColuIT1ns 7 and 8 .. It should be IT1en-g M w g
tioned that O"~ = 7, the depth-averaged variance of the transverse
distxibution of the IT1ean tracer concentration calculated in
Phase I.
Table 6.6 shows that the IT1ean width of the instantaneous pluIT1e
and all variances tabulated generally increased with ~o The variance
a-2 due to the fluctuation of the pluIT1e centroid was consistently greater g
than the IT1ean variance ~ of the instantaneous cross-wise distribution
of the tracer concentration at all s. Hence the ratio of 0"2 to the total g
variance O"~ was always greater than 0.50 (an average value of 0.66).
-228-
Therefore for S <: 9.5 (i. e. x <: 161.5 CITl), approxiITlately 660/0 of
the cross-wise ITlixing coefficient was due solely to the transverse
oscillation of the pluITle centroid.
That the pluITle width variation becaITle increasingly significant
with increasing S is illustrated by the values of the ratio of the vari-
ances 0-2 and 0-
2 listed in ColUITln 8. As S increased froITl 3.2 to w g
8 .. 6, 0-2
/0-2 steadily grew frOITl 0.29 to 0.98. w g
This indicated that,
except very near the source, teITlporal variation of the pluITle width at
a fixed station was significantly effective in deterITlining the overall
size of the pluITle.
The fluctuating pluITle analysis therefore shows that near the
source, transverse os cillation of the pluITle centroid is the dOITlinating
ITlotion of the plume. As x increases however, pluITle width variation
becoITles increasingly iITlportant such that near x/d = 8, the variances
of the pluITle width and the centroidal ITlotion are approximately equal.
6. C. CONCENTRATION VARIATIONS AT FIXED POINTS WITHIN THE PLUME
6 .. C .. 1. Transverse Distribution of the Intensity of Concentration
Variation.. Figure 6.15 shows the transverse distribution of the rITlS
0- of the concentration variation at various distances froITl the source. s
RUN 804 is used as an exaITlple, and all ITleaSureITlents were ITlade at
the level of tracer injection YJh
= 0.368. All distributions were assuITled
syITlITletric about the pluITle axis S = o.
It is evident that at a fixed distance i; frOITl the source, each
distribution was flat-topped with 0-S
reITlaining at a fairly constant value
~ Z ::J
al 0::: «
~
bCll ~
Z o t( ::J Io ::J ...J LL
W 0::: « ::J a Cf)
I
Z « w ~
I
Io o 0:::
4.00.-
o
1.00
0.40
0.10
e
0.04
•
0.01
o
Fig. 6. 15.
e
e
L_____ I'
2 3
Flume S2 E RUN 804
d=lo .. 84em,U:39.2 em/sec IR* = 2.03 X Id~
--------4
LEGEND
SYM e-o 7.4 D. 16.6 G 35.0 e 71.9 • 145.8
NORMALIZED TRANSVERSE DISTANCE, -~
Transverse distribution of the rms (J of the concentration fluctuation s
at various distances S from the source. Measurernents made at the
level of tracer injection 'Ilh
= 0.368; RUN 804.
5
I N N -..D I
-230-
near the plume axis and then dropping off at larger distances from the
axis. The value of lsi = IZ/dl whe re 0- began to decrease inc reas ed s
with distance from the source. At ~ = 145.8, 0- increased slightly s
with increasing I s I before decreasing at large distances from the plume
axis. The 0- distributions of Figure 6.15 are similar to those measured s
by Lee and Brodkey (51) for a continuous point source in a pipe flow.
Along the plume axis z = 0, 0- S decayed as a power of S such
that 0- (O) 0:: S -1.5. Thus the mean square fluctuation (msf) would s
decay as
(6.25)
-3 0 This compares with s{O) 0:: S .. measured by Becker et al. and re-
ported by Csanady (15) for a continuous point source in an air stream
of a pipe flow. Measurements by Lee and Brodkey (51) for a continu-
ous point source in a turbulent water flow in a pipe indicated an attenu
ation rate s (O) 0:: s -1. 9 0 For grid-generated turbulence, s ex: S -3/2.
Transverse distributions of the coefficient of variation C = 0- /C v s
were also developed. Typical plots are shown in Figure 6.16. Again
the measurements were made in RUN 804 at the level of tracer injecti::m
1'Jh = 0.368. The plots show that at a given S, C was minimum but nonv
zero (about 1 0 0 or less) at or very near the plume axis, and increased
with the transverse distance from the plume axis. The minimum value
of C near or along the plume axis was denoted as Vlo A maximum v
value Vz of C was attained near the edge of the plume at a point W v c
from the point of minimum non-zero C • As Is I increased further, C . v v
decreased--very sharply near the source and less distinctly far away.
40.0
> U
~ 10.0 Z 0
~ f------. We
0:: 4.0 ~
LL 0
I-Z W U l.o~ LL LL W 0 u
0.4
0.1 a
Fig. 6. 16.
·1
00 /"
4 '0
W' ~ ~u
~ • 2 ...JJ: Wb 0::-~
0
/ e ----- '<iI
40
Flume S2 E RUN804
d= 10.84em,u = 39.2 em/sec
IR = 2.03 X Id *
~
80 120 160
NORMALIZED DISTANCE, ,
LECEND
SYM e-O 7.4 {;. 16.6 o 35.0 e 71.9 '<iI 108.9 • 145.8
-'--______ ~ __ .. J
234
NORMALIZED TRANSVERSE DISTANCE, -~
5
200
Transverse distributions of the coefficient of variation C = CJ / C at several S. v s
MeasurelTIents lTIade at the level 1l = llh = O. 368; RUN 804.
I N W ...... I
-232-
The relative effective width W /d of the C distribution c v
grew with S. As shown in the inset of Figure 6.16, this growth
was parabolic - -an indication that V z probably occurred at a fixed
position relative to the transverse distribution of the intermittency
Further investigation revealed that for all S, C = Vz v
where If = 0.05. From Figure 6e 5, this corresponds to the point
where
z - Z
0"1 (6.26)
which is a pos ition near the extreme outer edge of the plume.
6. C. 2. Distribution of the Peak-to-Average Ratio t .!:a.
Figure 6.17 shows the transverse distribution of the peak-to-average
ratio P for S -< 0 at S = 7.4, 16.6,35.0, 71.9, 108.9, and 145.8 a
as calculated in RUN 804. The flow depth was 10.84 cm and all
measurements were made at the level of tracer injection llh = 0.368.
The distributions were very closely similar to those of C shown v
in Figure 6.16.. At a given S, P increased from a low value P 1 a
(about 5 .. 0 at S = 7.4 for RUN 804) at or very near the plume axis to
a maximum P z (about 126 for S = 7.4 for RUN 804) near the plume
edge.
The effective width W of the P distribution measured as the p a
transverse distance between the points of P 1 and P z increased with
distance from the source. As shown in the inset of Figure 6.17, the
growth rate was parabolic--such that at a given S , W :::: W • As a p c
result of the close similarity between the two sets of distributions,
400
(L0 100 ~
0
tr 0::
W <.9
40
« 0:: W
~ I
0 l-
I 10 y: « w (L
4
o
f--- Wp -----1
e
.. _- !
2
"0
w'a. ~~
4
I-- ~ 2 <I::r: ..-11-wo 0::-
~
3
E RUN 804 Flume S2
d=IO .. 84em,u=39.2 em/sec IFV2.03Xld
~
40 80 120 160 200
NORMALIZED DISTANCE, ~
'V ___
•
4
LEGEND
SYM ~ o 7.4 6 16.6 o 35.0 e 71.9 'V 108.9 • 145.8
5
NORMALIZED TRANSVERSE DISTANCE, -~
Fig. 6.17. Transverse distributions of the peak-to-average ratios P a at several S.
Measurem.ents m.ade at the level'll = 'llh = 0.368; RUN 804.
I N v.> v.> I
-234-
points of maxinlUm and minimum P coincided with those of C • a v
This means therefo re that within the plume, the region of high (or low)
peak values relative to the mean concentration C is coincident with
the region of high {or low} msf relative to C.
It was found that the peak-to-average ratios P l (along S = O)
and P z (along S = W /d), attenuated with distance from the source. p
As shown in Figure 6.18, both Vi and Vz decayed approxim.ately as
the (-0. 80) -power of s. The peak-to- average ratios PI and P z also
decayed as S -0.80 but PI approached an asymptotic value fairly
rapidly. In the present study P l -- 1.10 usually for ~:> 100. Theo-
retically the ultimate asymptote of P is 1.0 as ~ -- m. a
Comparison of the preceding results with the summary shown
in Table 3.1 reveals that P decayed more rapidly in atmospheric a
turbulence than in the present flume measurements. In the atmosphere,
P 0:£-1.0. Along s=O, C(=C )0:£-1.0. Hencetheinstan-a max
taneous peak concentration along the plume axis varies as £ - 2.0.
From the present study, PI 0: £-0.80 and C 0: £-0.53" max
Therefore
the instantaneous peak concentration along ~ = 0 varies approximately
r -1. 33 as s The accelerated decay rate in the atmo sphere is due to
both three-dimensionality and the enormous range of eddy sizes found
in the atmo sphere.
For mixing in the atmosphere, the plume size is such that the
eddies effecting the mixing are within the inertial subrange. Thus
" -1 us ing Eq. 3.15 proposed by Gifford (16, 53), P 0: S since
a
-' 2 Ij~ + Ij 0: £z f g and ~ 0: £3 (see for example Okubo (67)). At large
£, the plume siz e is so large that both Ij} + ~g and ~ are propo r-
o...N
0...-..
CI)
o
100.. IQ II II rI.o
40
P2 (along z=Wp)
RUN 804 Flume S2
d =10.84em, u= 39.2 em/sec lR* = 2.03 )( 103 0.4
VI
~ a::: 10 0.1
w (!) <! a::: w ~
1
~ I ~ <! w 0...
4 V2
(alongz=Wc)
0.5L I I I I III I I I I III
I 4 10 40 100 400
DIMENSIONLESS DISTANCE FROM THE SOURCE, e Fig. 6. 18. Attenuation of the peak-to-average ratios P 1 (along z = 0), P 2
4.0
V2
1.0
0.5 1000
(along z = W ), and of the coefficients of variation V1 (along z = 0),
V 2 (along z =p W ) with t:; RUN 804. c
I N VJ U1
-236-
tional to ~. Hence as ~ -+- 00, P - 1. o. a In the flume, the plume
width is of the same order as the flow dimensions and hence of the
scale of trubulence. Again as ~ - 00, P - 1.0. a But the range of
~ where eddies are within the inertial s ubrange is very small (very
close to the source). Hence the range of ~ over which measurements
were made in the fl ume is intermediate between P z 0: ~ -1. a and
.n P z 0: S ~ , as the present results indicate.
6. C. 3. Probability Density Functions of Concentration Vari
ations. The probability dens ity function g( c ') for concentration fluctu-
ations at fixed po ints was determined as outlined in Chapte r 3. It was
found that a more adequate analysis required a higher sampling rate
and greater probe sensitivity than were used during the experiments.
Nevertheless, some broad conclusions could be drawn from the
present study. It was discovered, for example, that various forms of
g(c ') prevailed at different parts of the plume. Very near the source,
g(c ') was strongly skewed and Rayleigh type indicating a dominance of
low concentration levels at points where u- was high. Farther away s
from the source and within the plume interior, g(c ') was generally
symmetrical about c' = 0, and was either bimodal or weakly Gaussian.
Along the plume edges both near the source and far away, g(c ') was
again strongly skewed.
In view of the complicated nature of the fluctuations, it was felt
that a more sensitive detection system be used for further detailed
study of the probability dens ity of the fluctuation of concentration.
-237-
6. D. SUMMARY DISCUSSION (Phase II)
The teITlpo ral variation of tracer concentration was analyzed
using three ITlodels. The first was a fluctuating pluITle front ITlodel
which utilized the concept of interITlittency to distinguish three regions
of the pluITle cross section:
a. a central core of continuous record above the threshold
concentration,
b. an interITlediate region where tracer concentration was only
interITlittently above the threshold t
c. an outer region where the threshold was never exceeded.
The second ITlodel considered the entire pluITle as fluctuating back and
forth transverse to the direction of flow. The variances of the
instantaneous t transverse concentration distribution ;r. of the fluctu
ation of the pluITle centroid 0-2
• and of the instantaneous pluITle width g
variation 0-2 were deterITlined and related to each other. w
The third ITlodel deterITlined the intensity and probability density
of concentration variation at fixed points within the pluITle. The rITlS-
value and the teITlporal coefficient of variation were deterITlined. Peak-
to-average values were calculated.
At any given level of the flow, a pluITle cross section could,
therefore, be characterized by four paraITleters: (i) the ITlean tracer
concentration C above the threshold, Ce (ii) the intermittency factor If
which denotes the fraction of the total tiITle that tracer concentration
exceeds C • (iii) the rITls-value 0- which indicates the intensity of t s
concentration fluctuations about the ITlean C. and (iv) the frequency of
zero occurrence Wo which is the frequency at which the front C = C t
-238-
sweeps by a fixed point in one direction only. Typical transverse
distributions of thes e variables normaliz ed by their maximum values
at the fixed station and level of flow are plotted in Figure 6.19. All
values were measured in RUN 804 at the level of tracer injection and
at a distance x/d = 16 .. 6 from the source.
Figure 6.19 shows the relative positions of the various dis-
tributions from each other. The curve of the rms distribution IT (s) s
lay between those of the mean concentration C( s) and the intermit-
tency factor If(s) ~ The 50% intermittency factor occurred at a point
where the rms-value was 32% and the mean tracer concentration only
12% of their respective maximum values along the plume axis. The
point of maximum Wo was us ually close to its theoretical point of
occurrence: If = 0.50.
The intermittency factor concept was found to be a very us eful
tool for analyzing concentration fluctuation. It not only furnished
values of the duration and frequency of occurrence of concentration
above the threshold but also provided a link between the fluctuation
studies of Phase II and the analysis of the mean concentration of
Phase I. Since the extreme limit, Wf
, of the plume boundary where
If = 0 and the standard deviation IT of the transverse distribution of
the mean tracer concentration grew parabolically, the two parameters
were related as sirllple ratios. It was found that lor the smooth-
boundary experiments,
(6.27)
and fo r the rough- bottom flume,
I If (,1 ----, I.Oar-: -0- --0 - - / _ 0___ v-~, ' ~
RUN904 FI~rne S2,~= 1~.84em t -16,6, 7J -7Jh - 0.368
CJ)
W -1 co <l: 0.8 a: ~ CJ) CJ) 0.6 w -1 Z o CJ) Z 0.4 w ~ Q
0.2
6 o~. '\ I 'e ~ 0 I \ ~'.6 I \
CTS(~) \ 'k \ '\ \
~' , ~ I \ ' 6. r \ "
\.[ ,
--0- Mean Cone., C (~) • ---I:r- • rms Cone., CTs (~ )
- 0- Interm. Factor, If(~)
--e-- Zero Freq., wo(~)
r~· \ ,~ I ". \ '~WO(~) I ~ \ \
,'- . ""-~ \ I' .~ '~
/ .~-~, / ~
0.01 _e- ........ ' ~t]'> D o 0.5 1.0 1.5
NORMAL IZED TRANSVERSE DISTANCE, -- t Fig. 6. 19. Typical transverse distributions of the mean concentration C, the rms of concentration
fluctuation as, the intermittency factor If' and the frequency of zero occurrence Wo at a fixed distance x/d = 16.6 from the source; RUN 804.
, N vv --0 ,
-240-
(6.2S)
The intennittency factor analysis) however, could not dis-
tinguish the transverse fluctuation of the plume front from a purely
random phenomenon. It showed nevertheless that for x/d > 7.4,
the motion of the opposite edges of the plume was not predominantly
due to either of two hypotheses: the meandering of the entire plume
or the pulsation of the plume widtho
The fluctuating plume model showed that near the source
(x/d!5: 9.5). the fluctuation of the plume centroid contributed sub-
stantially to the overall width of the plume at any fixed station with
a2 / ~::::: O. 660 The temporal variation of the plume width became g
increasingly important with distance from the source such that as x/d
increased from 3.2 to S06, o-Z /;Z increased from 0 .. 29 to 0.9S. w
Statistical analysis revealed that at a given station, the fluctu-
ation intensity of tracer concentration was greater near the plume
axis and dec reased at large transverse distances z from the axis
(io e. near the plume edges). The result was a flat top distribution
illustrated in Figures 6. 15 and 6.19. Thus the coefficient of variation
C was low near the plume axis, increased to a maximum Vz near v
the plume edges and then dropped off for further increase in z.. Simi-
larly the peak-to-average ratio P at a given station was srnall a
(= Pl) near or at z = 0, increased to a maximum P z near the plume
edges and decreased as z increased further. It was found that both
Vz and P z occurred where If = 0.05.
It should also be noted that near the source (x = SO cm for
-241-
example) P l ::::; 700 and P z could be as high as 200. Both P l and
P z attenuated as the (-0 .. 8) -power of x, and at large x PI ap
proached an asymptotic value of approximately L 10 0
-242-
CHAPTER 7
SUMMARY AND CONCL USIONS
A set of experiments was performed in an open-channel to
establish the characteristics of transverse mixing in a turbulent shear
flow. A neutrally-buoyant tracer was continuously injected at ambient
velocity at a point within the flow. Injection was parallel to the flow.
Tracer concentrations were measured at various locations down-
stream of the source and analyzed in two phases. In Phase I, distri-
butions of the time-averaged concentration were studied. In Phase II,
concentration variations were analyzed.. A summary of the principal
results is given below ..
7.A. RESULTS RELATED TO PHASE I
1. The transverse distribution of the time-averaged tracer
concentration C was Gaussian at all levels of the flow. This self-
similarity was preserved at all distances x downstream of the source
(extending from 4 to 611 times the depth) regardless of whether the
flow boundaries were hydraulically smooth or rough. (Figures 5.1,
5.2, and 5.3).
2. The variance 0-2 of the transverse distribution of C
(evaluated numerically by Eq. 5.3) grew linearly with x at all levels
of the flow. The depth-mean value -;z of the variance also grew as a
linear function of x. (Figures 5.4 through 5.8).
3. A depth-averaged coefficient of transverse mixing D was z
-243-
defined and calculated by the relation
(5. 5)
where u = mean flow velocity at a flume cross section. A generalized
form of the transverse mixing coefficient was derived from the con-
vective-diffusion equation and evaluated by
1 [oa-Z a ( oa-Z)~ D (x, y) = - u - - - D - , z 2 ax oy y 8y
(7. 1)
where u is the flow velocity at the flow level Y t and D is the vertiy
cal diffusivity. Since the transverse distribution of C was Gaussian
at a given flow level y, a mixing coefficient could be defined for that
level (of homogeneous turbulence) by
(5.9)
4. The normalized depth-averaged transverse mixing coefficient
e = Dz/u*d was found to depend on the aspect ratio, A = d/W, where
d = flow depth, W = flume width, and U,,- = (bed) shear velocity. -"
The
dimensionless coefficient "8 decreased with increasing A ranging
from 0.24 at A = 0.015 to 0.093 at A = 0.200. Comparison of the
present results with measurements by past investigators showed that at
a given A, "8 for field experiments was about twice the value measured
in laboratory flumes. (Figures 5.9 and 5.10).
5. The transverse mixing coefficient D evaluated by either z
Eq. 7.1 or Eq. 5.9 varied over the depth tending to be greatest near
-244-
the water surface where the flow velocity was highest. (Figures 5. 11
through 5. 13) •
6" Near source behavior of the pluITle was studied by calculation
of the depth variation of ITlaterial within the pluITle. NUITlerical solu-
tion of the convective-diffusion equation assuITling that the vertical
distribution of D was parabolic agreed very closely with experiITlental y
ITleaSureITlents for various levels of tracer injection T]h" This in-
directly confirITls that the vertical profile of D is indeed parabolic. y
(Figures 5.14 and 5.15)"
7. The vertical distribution of the norITlalized concentration
{3 along the pluITle axis was skewed because of the vertical variations
of u, D, Y
and D • z
The po lnt of ITlaxiITluITl {3 initially ro s e to the
water surface when tracer was injected at levels T]h:> 0.632, and
dropped to the fluITle bottoITl fo r T]h -< 0.50. On attaining the liITliting
level (T] = 1 .. 0 or 0.0), the point of ITlaxiITluITl f3 "rebounded" into the
flow interior.
8. The ITlaxiITluITl ITlean concentration C(x,y,O) along the pluITle
axis, attenuated, for a given experiITlent, as a power of x at all levels
of the flow.. The attenuation for x/d > 18.3, could be represented by
-Q' C(x,y,O) a: x • The exponent Q' approached 0.50 for low aspect
ratios A. and increased with increasing A.. The attenuation rate was
accentuated by fluITle boundary roughness with Q':::::: 0.61 when the fluITle
bottoITl was roughened with rocks. (Figures 5.16 through 5.20;
Table 5.5).
9. Iso-concentration ITlaps developed on cross-sectional and
lateral planes (parallel to the fluITle bottoITl) were used to establish
-245-
detailed distribution of ITlaterial within the pluITle, and to locate zones
of high tracer concentration. (Figures 5.21 through 5.25).
7.B. RESULTS RELATED TO PHASE II
The teITlporal variation of concentration was analyzed by three
ITlethods. The first is the interITlittency factor ITlodel which defined an
concentration exceeded the threshold Ct
(or background) concentration
at a given point. The second characterized the entire pluITle as a cloud
fluctuating across the flow direction. Motion pictures of the pluITle
were taken froITl above the water surface and the pluITle boundaries
used to cOITlpute different variances. In the third ITlethod, fluctuations
of the tracer concentration at fixed points were analyzed by the usual
statistical ITlethod. The principal results of all analyses are SUITl-
ITlariz ed below.
1. By utilizing the interITlittency factor technique, the pluITle
cross section was characterized by three regions: a central core where
tracer concentration was always greater than the threshold Ct
(If = 1.0) ,
an interITlediate region of interITlittency where concentration was only
interITlittently above Ct
(0 < If < 1.0), and an outer zone where the
threshold was never exceeded. (Figure 3.4).
2e At given distance x downstreaITl of the source and level of
flow y, the transverse distribution of the interITlittency factor was
self-siITlilar and could be represented universally by the error function
relationship:
where
-246-
z - Z z;, =
I .rz (J"I
(6. 1)
Z = rrlean position of the plum.e front (or plurrle edge where the concen-
tration equalled Ct) calculated by Eq .. 3 .. 43 , and (J"I = standard devi
ation of the If-distribution calculated by Eq .. 3.44. Eq. 6.1 indicates
therefore that the position of the plurrle front is normally distributed.
(Figure 6.5).
3. The georrletric characteristics of the plum.e front fluctuation
were studied. The extrerrle lirrlit of the interrrlittency region where
If = 0.0 was denoted by Wf ; the outer edge of the inner core of con
tinuous record by ~j the rrlean position of the plurrle front where
If ~ 0.50 by Zj and the variance of the If-distribution by (J"i. Since
the fluctuations of the plurrle fronts were as sUrrled sYrrlrrletric about
the plum.e axis z = 0, ITleaSurerrlents were rrlade only on one side and
Wi' ~, and Z were rrleasured frorrl Z = O. Frorrl dirrlensional analysis
it was found that rrleasured values of Wf closely fitted the curve:
3.58 (6.9)
1/4 where R ;::: (f If ) ,and f , f are, respectively, the rrlean values w s r s r
of the bed friction factors for the srrlooth boundary and rough bottorrl
experirrlents, X = value of x corrected slightly for the virtual origin
of W f' and u*b;::: bed shear velocity. For the srrlooth boundary, u'!<b
-247-
become s u* and Rw = 1. O.
Dimensional analysis also showed that the mean position of the
plume front could be represented by
(6.12)
where R = (£ /(J 1 /3" Since Z was half-way between the outer edge Z s- L·
~
of the central core 6 and Wf , 6 could be calculated by 6 = 2Z - Wf
•
The variance o-~ of the If-distribution initially grew as the
2/3-power of X /d and attained a maximum value 20-I/d::::; 0.8.
(Figures 6.6,6.7,6.8, and 6.9)~
4. The frequency at which the plume front sweeps by a point
was denoted as the frequency of "zero occurrence" wO. It was found
that Wo (z) was normally distributed about z = Z attaining a maximum
value Wo at z = Z and becoming zero at If = 0 and If = 1. O.
Calculations showed that this frequency was unrelated to the vortex
shedding of the tracer injector but that plume front oscillation was a
result of a diffusive type process characteristic of turbulent mixing.
The characteristic wave length of the front oscillation was approxi-
mately equal to the flow depth. (Figure 6.1 OJ Table 6.4).
5. The intermittency factor measurements were linked to
Phase I of this study by the fact that W f could be related to the
standard deviation 0- of the transverse distribution of the mean con-
centration. For the smooth-boundary experiments,
Wf - = 1.75 20-
(6.27)
-248-
and for the rough-bottom flume
(6.28)
6. A cross-correlation analysis was made of the opposite
edges of the plume by comparing the ON (when c > Ct) and OFF (when
c <: Ct ) events of the two points where z = Z and z = -Z. This was
done to test if plume front oscillation was a result of a pulsating motion
of the plume width or a meandering of the entire plume. The test could
not distinguish the plume front fluctuation from a purely random be-
havior, and indicated that neither motion dominated the other for
x/d> 7.4. (Table 6.5).
7.. From analysis of the motion pictures of the plume, the
second method of analysis showed however that very near the source
(x/d < 6 .. 5) ,oscillation of the plume centroid was a dominating contri-
butor to the total variance of the cross-wise mean concentration
distribution. But as x increased, plume width variation became
increasingly important so that at about x/d = 10.0. the variance (JZ w
due to plume width variation was approximately equal to the variance
~ of the oscillation of the plume centroid. g
approximately 0.66. (Figures 6.13, 6.14;
For x/d <: 10, (J~/(JZ
T a bl e s 6 .. 5, 6 .. 6) •
was
8. Statistical analysis revealed that the transverse distribution
of the rms -value (J of the concentration fluctuation was roughly sel£- simis
lar at all x. The rms-value was highest near the plume axis and decayed
near the edges--giving rise to a "flat top" distribution. The msf along
the plume axis decayed as the (-3) -power of x as compared to the
-249-
(- 3/2) -power m.easured for grid- generated turbulence. (Figure 6. 15).
The transverse distribution of the coefficient of variation
e = 0- Ie was very sim.ilar to that of the peak-to-average ratio P v s a
with each variable increasing from. a low value near the plum.e axis
to a m.axim.um. near the plum.e edges. Maxlm.um. P (denoted as P z) a
and m.axim.um. e v (denoted as Vz) occurred where If = o. 05. The
low values of P and C near or along the plurne axis were designated a v
PI and Vz respectively. It was found that P l , P z , VI' and Vz
attenuated approximately as the (-0. 8)-power of x with PI approach-
ing an asym.ptotic value of 1. 10.. Near the source PI could be as
large as 10 and P z as 200.. (Figures 6.16, 6.17, and 6.18).
9. Statistical analysis also showed that various form.s of the
probability density function g(c ') could be m.easured at different parts
of the plum.e. In m.ost sections of the plum.e, g(c ') was strongly
skewed and it was only very near the plum.e axis thatg(c ') becam.e
moderately sym.m.etric about c' = 0.. Better instrum.entation and
further study are. however, recom.m.ended.
10. Typical transverse distributions and the relative positions
of the m.ean concentration e; the rm.s-value 0- ; the interm.ittency s
factor If' and the zero frequency wo are sum.m.arized by the plots
shown in Figure 6 .. 19.
-250-
LIST OF SYMBOLS
at attenuation values on analog recorder
AO cross-sectional area of tracer injector
c instantaneous tracer concentration
C tiITle ITlean value of the tracer concentration
C I c-C: concentration fluctuation
C ITlax
Co
Ct
background concentration
peak value of C at given x and y
initial tracer concentration at injection
threshold concentration
C <r IC: coefficient of variation v s
d
d .. (t) lJ
D
D p
D s
D .. lJ
D ,D ,D x Y z
D ,D Y z
e
E
E .. lJ
f(xl;y, z, t)
norITlal depth
generalized dispersion; Eq. 2 .. 28
characteristic lateral ITllxing coefficient
transverse ITlixing coefficient for floating particles
diffusion coefficient for ITlsf
ITlass transfer coeffic ient tensor
turbulent mixing coefficient in the x, y; and z directions respectively
depth-averaged values of D , and D respectively y z
2.718 •••
mean rate of energy dis sipation per unit mas s of fluid
eddy viscosity tensor
instantaneous concentration distribution at fixed station Xl
mean friction factor for smooth flume boundaries
f r
IF
F(y, t)
g
g(c I)
h(z • t)
I(n) • J(n)
K
K z
L
Mzn
n. 1
p(T)
-251-
bed friction factor
mean value of the friction factor for smooth-boundary experiments in Phase II
mean value of the friction factor for rough-bottom experiments in Phase II
Fro ude number
SOO f(Xl;Y.Z .t) dz -00
acceleration due to gravity
frequency dens tty of the concentration fluctuation
intermittency function at a fixed point z defined by Eq. 3.31 oIf(z)
oz : probability density function
intermittency factor defined as the function of the total time that a threshold concentration is exceeded at a point z
Fourier coefficients
a constant
displacement of the plume centroid in the z-direction
characteristic length scales
characteristic half-width of the intermittency region
longitudinal length scale of the oscillation of the plume front
mean value of Lf
th q moment of the transverse distribution of C (p = 0, 1 , 2 • • • ~ ); E q • 2. 2 1
Mz/dz
MO(~,TJ)/(Qs/ud); Eq. 5 0 19
frequency of vortex shedding
probability density of both "occurrence" and "nonoccurrence If of the intermittency function
P(~, t)
p a
p c
Q
-252-
probability that a point defined by position vector x lies within a marked fluid at time t
peak-to-average ratio
cumulative value of the mean concentration; Eq. 5.1
peak-to-average ratio along the plume axis
maximum value of P at given x and y a
probability density of the "occurrence" pulse lengths
probability density of the "non-occurrence II pulse lengths
flow discharge
Qs
source strength for tracer; Eq. 2.61
r
R
IR
R(t I)
R(w)
hydraulic radius
rise time of the probes
source strengh/unit volume
Reynolds number
Lagrangian correlation coefficient; Eqo 2.24
amplitude ratio of excitation frequency w
IR* friction Reynolds number defined by Eq .. 4.7
R (f /f )1/4 w s r
R (f /f ) 1/3 z s r
s C 12 , mean square fluctuation
skewnes s facto r
s sink strength/unit volume
ene rgy slope
flume slope
time
arbitrary time bas e
T m
Tz
u,v,w
u
u a
-253-
sampling period or period of averaging; Eq. 2.7
pulse length (of "occurrence'l) when the threshold is exceeded
puls e length (of "non-occurrence I') when the threshold is not exc eeded
period of the plume front oscillation at point z
mean value of T O(z)
time-averaged velocity components in the x, y, and z directions respectively
mean velocity through flume cross section
.JT o/p: shear velocity
depth-integrated flow velocity at a lateral position z
bed shear velocity
instantaneous fluid velocity in the xi direction, i = 1.2,3 ,
u t fluctuating component of ui
' i = 1 t 2,3
u n
u max
W
W a
W{i)
injection velocity of tracer
maximum velocity at the water surface
time-mean velocity in the Xi direction; i = 1,2,3
a characteristic velocity
coefficient of variation along the plume axis
maximum value of C at given x and y v
flume width
time-averaged value of the instantaneous plume width
extreme limit of the plume boundary where If = 0.0; Fig. 3 .. 5
effective width of the transverse distribution of concentration at the instant i
x
X. 1
x v
-254-
Cartes ian coordinate in the direction of flow
coordinate in the ith direction for i = 1,2,3
po int where the inner core of continuous record (If = 1.0) inters ects the plume axis
value of x at the virtual 0 rigin of W f
X.(t) 1
ith component of the fluid particle displacement in time t
y
z
z
Cartes ian coordinate in the vertical direction
Cartesian coordinate transverse to flow
centroid of the time-mean concentration distribution at station Xl
value of z where If = 0.50
centroid of the instantaneous concentration distribution at station Xl
z-value of the mean position of the plume front at a given station and flow level; Fig. 3.5
z-position of the plume edge (front) at the instant t
exponent of S for attenuation of the mean concentration along plume axis; Eq. 5.24
C{S ,1],0) I(Q 1 lid), normalized concentration along the s
plume axis; Eq. 5.22
'j3( s) depth- averaged value of ~(S, 1])
" intermittency factor as it relates to turbulent or nonturbulent motion
exponent:
exponent:
Z ex: X "z 0- ex: x"3
I
6{) Diract delta function
D. outer edge of the inner core of continuous record (where I
f= 1.0); Fig. 3.5
E error
-255-
to.. turbulent transport coefficient for a scalar 1J
E molec ular diffusivity m
s z/d: dimensionless transverse distance from plume axis
'lH
y /d: dimensionless vertical distance from flume bottom
dirnensionless height of tracer injection from flume bottom
'lm value of 'l where, fo r a given station, MO is maximwn
e D /u*d: dimensionless transverse mixing coefficient z .
K
v
depth-averaged value of e for entire reach of measurement
D /u .... d P .,.
depth-averaged value of e at a given station x
dimensionless transverse mixing coefficient due to shear only; Eqo 5 .. 10
von Karman constant
d/W: the aspect ratio
concentration microscale
kinematic viscosity
x/d: dimensionless or normalized distance from the source
value of ~ beyond which 13(~) decays at a constant power of S
S 1 value of S beyond which 0-2 grows linearly with ~
7T 3.14159 •••
p fluid density
~ variance of the transverse distribution of the mean tracer concentration; Eq. 2.56
~ depth-averaged value of 0-2
0-2
g
T
<I> r< >..)
cj>(W)
<I> (>..)
x
w
-256-
variance of the instantaneous transvers e concentration distribution
variance of the transverse fluctuation of the plu:me centroid
variance of the distribution of the inter:mittency factor; Eq. 3.44
1 + a-Zg
0-2/d2: nor:malized variance
variance of the transverse distribution deter:mined by use of the probability :method; Eqo 5.2
root-:mean- square value of the concentration fluctuation
variance of the plu:me width variation
local shear stress
:mean botto:m shear stress
scalar concentration
ti:me :mean value of cj>
curve fitted to plot of "8 versus >.. for field experi:ments
phase shift of excitation frequency w
curve fitted to plot of "8 versus >.. for laboratory experi:ments
x-x : value of x corrected for the virtual origin of the Vw r<x) distribution
rate of dissipation of :mean square concentration fluctuation
excitation frequency
frequency of zero occurrence at point z
Wo value of Wo where the inter:mittency factor = 0.50
-257-
LIST OF REFERENCES
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2. Fischer, H. B., "Longitudinal Dispersion in Laboratory and Natural Streams!!, Report No. KH-R-12, W. M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology, Pasadena, California, 1966.
3. Smith, F. B., l!The Diffusion of Smoke from a Continuous Elevated Point-Source Into a Turbulent Atmosphere ' !, Journal Fluid Mechanics, Vol. 2, pp. 49-76,1957.
4. Vanoni, V. A., llTransportation of Suspended Sediment by Water l!, Transactions ASCE, Vol. Ill, pp. 67-133, 1946.
5. AI-Saffar, A. M., !'Eddy Diffusion and Mass Transfer in Open Channel Flow ll
, Ph. D. Thesis, University of California, Berkeley, California, 1964.
6. Sullivan, P. J., "Dispersion in a Turbulent Shear Flow", Ph. D. Thesis, University of Cambridge, 1968.
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8. Yotsukura, N., Fischer, H. B. and Sayre, W. W., "Measurement of Mixing Characteristics of the Missouri River Between Sioux City, Iowa, and Plattmouth, Nebraska " , U. S. Geol. Survey Water -Supply Paper 1899 - G, 1970.
9. Davar, K. S., "Diffusion From a Point Source Within a Turbulent Boundary Layer '!, Ph. D. Thesis, Colorado State University, Fort Collins, Colorado, 1968.
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-258-
LIST OF REFERENCES (Continued)
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23. Sutton, O. G., liThe Problem. of Diffusion in the Lower Atm.osphere!l, Quart. Jour. Royal Meteorol. Society, Vol. 73, pp. 257-281, 1947.
-259-
LIST OF REFERENCES (Continued)
24. Davies, D. R., "Three-Dimensional Turbulence and Evaporation in the Lower Atmosphere, I and II", Quart. Jour. of Mech. and Applied Math., Vol. 3, pp. 51-73, 1950.
25. Yih, C. S., "Similarity Solution of a Specialized Diffusion Equation", Trans. American Geophysical Union, Vol. 33, pp. 356-360, 1952.
26. Aris, R., lIOn the Dispersion of a Solute in a, Fluid Flowing Through a Tube", Proc. Royal Society, Ser. A, Vol. 235, pp. 67-77, 1956.
27. Saffman, P. G., "The Effect of Wind Shear on Horizontal Spread From an Instantaneous Ground Source", Quart. Jour. Royal Meteorol. Society, Vol. 88, pp. 382 - 393, 1962.
28. Taylor, G. 1., "Diffusion by Continuous Movements", Proc. London Math. Society, Ser. 2, Vol. 20, pp. 196-212, 1921.
29. Van Driest, E. R., "An Experimental Investigation of Turbulence Mixing as a Factor in the Transportation of Sediment in Open Channel Flow", Ph. D. Thesis, California Institute of Technology, Pasadena, California, 1940.
30. Kalinske, A. A. and Pien, C. L., "Eddy Diffusion", Industrial and Engineering Chemistry, Vol. 36, pp. 220 -223, 1944.
31. Orlob, G. T., "Eddy Diffusion in Open Channel Flow", University of California, Berkeley, Sanitary Engineering Research Lab., Contribution 19, 1958.
32. Orlob, G. T., "Eddy Diffusion in Homogeneous Turbulence!', Jour. Hydraulics Div., ASCE, Vol. 85, No. HY9, pp. 75-101, 1959.
33. Batchelor, G. K., !tDi£fusion in a Field of Homogeneous Turbulence!t, Australian Jour. of Scientific Research, Ser. A, Vol. 2, pp. 437-450, 1949.
34. Monin, A. S., t'Survey of Atmosphereic Diffusion", Advances in Geophysics, Vol. 6, pp. 29-40, 1959.
35. Sayre, W. W., and Chang, F. M., "A Laboratory Investigation of Open-Channel Dispersion Processes for Dissolved,Suspended, and Floating Dispersants", U. S. Geol. Survey Prof. Paper 433-E, 1968.
-260-
LIST OF REFERENCES (Continued)
36. Kolrnogoroff, A. N., "The Local Structure of Turbulence in Incompressible Viscous Fluid For Very Large Reynolds Numbers ll
, Acad. Sci. URSS (Doklady), Vol. 30, No.4, pp. 301-305, 1941.
37. Batchelor, G. K., "The Application of the Similarity Theory of Turbulence to Atrnosphereic Diffusion", Quart. Jour. Royal Meteorol. Society, Vol. 76, pp. 133-146, 1950.
38. Sayre, W. W., and Chamberlain, A. R., l'Exploratory Laboratory Study of Lateral Turbulent Diffusion at the Surface of an Alluvial Channell', U. S. Geol. Survey Cir cular 484, 1964.
39. Prych, E. A., "Effects of Density Differences on Lateral Mixing in Open-Channel Flows", Report No. KH-R-21, W. M. Keck Laboratory of Hydraulics and Water Resour ces, California Institute of Technology, Pasadena, California, 1970.
40. Engelund, F., "Dispersion of Floating Particles in Uniform Channel Flow", Jour. Hydraulics Div., ASCE, Vol. 95, No. 4, pp. 1149-1162, 1969.
41. Pien, C. L., "Investigation of Turbulence and Suspended Material Transportation in Open Channels ", Ph. D. Dissertation, Graduate College of the State University of Iowa, Iowa City, Iowa, 1941.
42. Glover, R. E., "Dispersion of Dissolved or Suspended Materials in Flowing Str earns", U. S. Geol. Survey Prof. Paper 433 - B, 1964.
43. Patterson, C. C. and Gloyna, E. F., "Dispersion Measurement in Open Channels 1', Jour. Sanitary Engineering Div., ASCE, Vol. 91, No. SA3, pp. 17-29, 1965.
44. Batchelor, G. K., "Note on Free Turbulent Flows With Special Reference to the Two-Dimensional Wake l', Jour. Aero. Sciences, Vol. 17, pp. 441-445, 1950.
45. Townsend, A. A., liThe Fully Turbulent Wake of a Circular Cylinder ", Australian Jour. of Scientific Research, Ser. A, Vol. 2, pp. 451-468, 1949.
46. Starr, V. P., JlThe Physics of Negative Viscosity Phenomenal', McGraw-Hill, New York, N. Y., 1968.
-261-
LIST OF REFERENCES (Continued)
47. Batchelor, G. K., Townsend, A. A., and Howells, 1. D., "Small Scale Variation of Convected Quantities Like Temperature in Turbulent Fluid ll
, Journal Fluid Mechanics, Vol. 5, pp. 113-134, 1959.
48. Hinze, J. 0., IITurbulence ll , Mc Graw-Hill, New York, N. Y., 1959.
49. Becker, H. A., Rosensweig, R. E., and Gwozdz, J. R., IITur_ bulent Dispersion in a Pipe Flow ll , Report AFCRL-63-727, Fuel Research Laboratories, M.1. T., Cambridge, Mass., 1963.
50. Gibson, C. H., and Schwarz, W. H., IIDetection of Conductivity Fluctuations in a Turbulent Field II , Journal Fluid Mechanics, Vol. 16, pp. 357-364, 1963.
51. Lee, J. and Brodkey, R. S., IITurbulent Motion and Mixing in a Pipe II, American Institute of Chern. Engineering Journal, Vol. 10, No.2, pp. 187-193, 1964.
52. Lowry, P., Mazzarella, D. A., and Smith, M. E., IIGround Level Measurem_ents of Oil-Fog Emitted From a HundredMeter Chimneyll, Meteorlogical Monographs, Vol. 1, No.4, pp. 30-35, 1951.
53. Gifford, F., "Peak to Average Concentration Ratios According to a Fluctuating Plume Dispersion Model ll , International Jour. of Air Pollution, Vol. 3, No.4, pp. 253 -260. 1960.
54. Smith, M. E., liThe Variation of Effluent Concentrations From an Elevated Point Source ll , Archives of Industrial Health, Vol. 14, pp. 56-58, 1956.
55. Bradbury, L. J. S., lIThe Structure of a Self-Preserving Turbulent Plane Jetll, Journal Fluid Mechanics, Vol. 23, pp. 31-64, 1965.
56. Gartshore, I. S., IIAn Experimental Examination of the LargeEddy Equilibrium Hypothesis 11
, Journal Fluid Mechanics, Vol. 24, pp. 89-98, 1966.
57. Sandborn, V. A., "Measurement of Intermittency of Turbulent Motion in a Boundary Layer", Journal Fluid Mechanics, Vol. 6, pp. 221-240, 1959.
-262-
LIST OF REFERENCES (Continued)
58. Vanoni, V. A., Brooks, N. H., and Raichlen, F., "A 40-Meter Precision Tilting FluITle 1', Tech. MeITlo. No. 67 - 3, W. M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology, Pasadena, California, 1967.
59. Vanoni, V. A., and Brooks, N. H., "Laboratory Studies of the Roughness and Suspended Load of Alluvial StreaITls II, Report M. R. D. SediITlent Series No. 11, SediITlentation Laboratory; California Institute of Technology, Pasadena, California, pp. 100-106, 1957.
60. LaITlb, D. E., Manning, F. S., and WilhelITl, R. H., "Measurement of Concentration Fluctuations With an Electrical Conductivity Probe II, AITlerican Institute of CheITl. Engineering Journal, Vol. 6, No.4, pp. 682 - 685, 1960.
61. Walters, E. R., and Rea, J. B., I'Determination of Frequency Characteristics FroITl Response to Arbitrary Input", Jour. Aero. Sciences, Vol. 17, pp. 446-452, 1950.
62. Ogura, Y., "The Influence of Finite Observation Intervals on the MeasureITlent of Turbulent Diffusion Parameters II, Jour. of Meteorology, Vol. 14, pp. 176-181, 1957.
63. Laufer, John, "Investigation of Turbulent Flow in a Two-Dimensional Channel", National Advisory COITlITlittee for Aeronautic (NACA) Report No. 1053, 1951.
64. Fischer, H. B., i'The Effect of Bends on Dispersion in Streams", Water Resources Research, Vol. 5, No.2, pp. 496 - 506, 1969.
65. Rice, S. 0., 1'Mathematical Analysis of RandoITl Noise", The Bell SysteITl Technical Journal, Vol. 24, No.1, pp. 46 -1 08, 1945.
66. Townsend, A. A., lIThe MechanisITl of EntrainITlent in Free Turbulent Flows ",Journal Fluid Mechanics, Vol. 26, pp. 689 - 715, 1966.
67. Okubo, Akira, "A Review of Theoretical Models for Turbulent Diffusion in the Sea", Journ. of the Oceanographical Society of Japan, 20th Anniversary Vol., pp. 286-320, 1962.
-263-
APPENDIX
-264-
Table A. 1 Variances 02
(CITl2) of the transverse concentration
distributions at various distances: and flow levels "I;
all experiITlents included.
RUN
1'1':' S 61. 1 94. 1 159. 9 225.8 324.1 455. 1
(d, CITl)
507 0.368 4. 98 6.45 14. 70 14. 03 17. 23 38. 92 (1. 52)
c: 47.4 106. 5 224.9 402.4 639. 1 994. 1 '=' Tl
705 0.368 2.0 6.6 14.3 32.2 51. 4 67.8 (1. 69) O. 750 3.2 7.8 15. 1 31. 3 50.0 70.2
AVGt 2.6 7.2 14.7 31. 8 50. 7 69.0
S 29. 1 65.5 138.2 247.3 392. 7 610.9 Tl
706 0.368 2. 95 6. 11 16.84 36.06 59. 19 90. 76 (2. 75) 0.236 3. 52 8.02 18. 72 35.66 49.86 85.85
0.800 3.68 9.77 20.05 34.47 54.09 87.42 AVG 3.38 7.97 18.54 35.40 54.38 88.01
707 0.368 3.82 8. 52 15. 92 30. 59 52.41 79.94 (2. 75) 0.250 3. 10 7.45 16.33 30.58 47.71 74.82
0.750 3.28 7.93 16. 36 36.29 51. 85 82.37 AVG 3.40 7.97 16.20 32.49 50.66 79.04
~ 31. 9 65.4 99.3 184.0 252.0 Tl
506 0.30 5.59 10.43 15. 50 40.18 41. 49 (2. 95) 0.89 6. 55 12. 59 16. 19 25.42 44.40
AVG 6.06 11. 51 15.85 32.80 42.95
~ 23. 1 52.0 109.8 196. 5 312. 1 485. 5 " 'll
703 0.368 3.40 9.34 20. 52 35. 16 53.38 90.77 (3.46) 0.095 3.66 8. 73 20.81 36. 71 58.37 90.06
0.632 3.84 8.69 19.90 35. 90 54. 79 87.89 0.860 3.65 10.44 20.85 33.88 56. 19 95. 10 AVG 3.64 9.30 20. 52 35.41 55.68 90. 96
-265-
Table A. 1 (Continued)
RUN I r= 8. 2 17. 7 36.8 55.8 74.9 103.4 141. 5 (d, ern) ':0
I 1
509 O. 368 5.43 12. 51 18. 07 23.46 36.47 38.42 (5.25) i O. 095 7.78 13. 67 17. 97 27.60 35.25 38.56
, O. 290 8.33 12.49 18.49 25.24 30.67 39.73 0.480 8.50 16. 31 17. 60 26.44 34.07 47.50 0.760 7.40 11. 40 19.32 26.27 39. 10 47.58 AVG 7.49 13. 28 18.29 25.80 35. 11 42.36
511 O. 095 1. 49 5.23 12.69 16.68 20.27 36. 89 37.68 1
(5.25 ) O. 368 1. 59 6.40 13.28 19.74 23.27 34. 62 44.39 0.632 2.40 5.21 12.34 19. 06 22.71 31. 33 42. 12 0.850 0 3.68 12.08 14.49 24.90 33.99 48.73 AVG 1. 37 5. 13 12. 60 17. 49 22. 79 34.20 43.23
512 0.850 1. 30 4. 72 14.37 29.70 47.86 58.34 (5. 25) 0.095 0 6.38 13.20 25.25 29.03 48.88
O. 368 2.29 4. 72 10.37 22.45 35. 13 37.40 0.632 1. 56 4.28 11. 82 24.09 41. 82 39.27 AVG 1. 29 5.03 12.44 25.37 38.46 45.97'
I 508 0.368 5. 21 13.43 18.43 31. 62 31. 79
(5.26) 0.190 5.70 11. 99 18.47 30.62 44.90 0.855 4. 91 15. 11 17.40 29.48 41. 70 AVG 5.27 13. 51 18. 10 30.57 39.46
1
510 0.632 1. 82 5.02 13. 91 20.86 22.62 31. 10 43. 16 I (5. 26) O. 095 1. 63 5. 53 13.74 16.04 20.69 31. 45 49. 55
0.368 1. 61 6.52 11. 29 28.90 23.41 36. 75 61. 00 0.850 1. 42 5.22 13. 14 22.19 22. 15 42. 96 54. 76 AVG 1. 62 5.57 13.02 22.00 22.22 35.57 52. 12
702 I" 14.8 33.3 70.2 125. 7 199.6 310. 5 ':0
(5.41 ) 1
O. 129 4.10 13.09 27.85 57. 14 82.72 128.95 I 0.391 4.82 14.45 28.29 47.62 76. 10 126. 74
0.663 5.26 13.48 27.92 51. 94 84.67 123.49 I 0.855 4.94 11. 67 26. 52 48.43 78.04 131. 91 I
AVG 4.7813.17 27.65 51. 28 80.38 127.27 ~
Table A. 1 (Continued)
IRUN
I (d, em.)
701 (5. 53)
O. 391 O. 128 0.855 AVG
14. 5
5. 10 4. 79 4.59 4.83
-266-
32.5
13. 52 o
12.83 13. 21
68. 7
25.84 28.37 29.88 28.03
9.2 20.8 43.8
407 (8.66)
404
Il 0.368 0.632 AVG
~ Il
0.368
AVG ~36$32 I
I s Il
607 0.368 (l0.70) 0.095
0.632 0.850 AVG
c; Il
704 0.368 (l0.81) 0.095
0.632 0.850 AVG
21. 2 21. 2 21. 2
7.7
23.2 20.3 21. 8
4.0
2.78 2.47 2.29
0 2. 51
7.4
8.65 7.70 7.32 5.54 7.30
57. 8 49.0 53.4
17. 4
70.5 60.8 65.7
8. 7
6.25 9.06 6. 03 2. 51 5.96
16. 7
23.47 24. 11 24. 53 25.56 24.42
100. 0 84.6 92. 3
36. 7
105.2 121. 0 113. 1
18. 0
19. 10 15.24 18. II 13.44 16.47
35. 1
46.20 57.84 48.84 54.20 51. 77
48.67 o
52. 16 50.42
90.0
207.4 210.3 208.9
75.2
262. 1 228.0 245. 1
36. 7
44.36 43.25 48.05 33.83 42.37
62.9
79.39 100.73 98.30 99.90 94. 58
213.3
84.81 92.01 94.85 90. 56
136. 1
354.0 376. 1 365.0
113. 9
421.0 396.0 409.0
50.7
61. 53 54. 76 59. 13 55. 13 57.64
100.0
133.89 147.11 157.30 155.24 148.39
303.7 '
121.201 137.24 140.38 132.91
1
I 69.4 1
83.86 80.65 78. 31 82.38 81. 30
155.2 I
I
241. 071 224. 09 1
252.45/ 239.83 2 39. 36 1
-267-
Table A. 1 (Continued)
~ RUN I
(d, ern) " 4. 7 10. 6 22.3 45. 7 63. 3 I
11 I I
406 0.368 26.0 64.0 149.0 303.0 420.0 I (17.07) 0.632 33. 6 64. 0 154. 0 289. 0 396.0
I I
AVG 29.8 64.0 151. 5 296. 0 408.0
I I
;: / A J '±.O
, f"\ A lV.,± 'l' ()
L..L. 7 /..') 11 00 L1 i
39.2 , VLJ. ---r / /. -~
11 708 0.368 8.08 33. 21 79.64 102.28 231. 94 336.46
(17.31) 0.095 6.30 26. 15 68.50 120.30 222.51 306.63 0.632 5.47 26.03 64.68 126.39 194.95 313.21 O. 850 0 0 62.62 100.63 161.81 257.80 AVG 6.61 28.46 68.86 112. 40 202.80 303. 53
S 2. 5 5. 4 11. 1 22.7 31. 3 42.8 11
603 0.368 3. 15 9.34 25.65 43.78 64. 58 95. 75 (17.34) 0.095 0 7.57 27.03 54.90 89.74 124.40
0.632 0 8.63 29.80 58.66 87.52 124.86 0.850 0 0 25.65 49.19 74. 58 105.49 AVG 3. 15 6.39 27.03 51. 63 79. 11 112. 63
I
E 3. 6 8.2 17. 3 31. 0 49.2 I 11 I
I
709 0.368 7.91 27. 16 75.60 149. 09 189.29 I (21. 97) 0.095 0 26.71 69.60 131. 89 216.42 I
0.850 0 0 57. 18 141. 00 198.70
I AVG 7. 91 26.94 67.46 140.66 201. 47
I
NOTES: I I -', The first value of 11 listed is the level of inj eetion 11h . For I -.-j
RUN 511, 11h = O. 051. j
!
i I
2 I
t A VG = averages of the non-zero values of 0- l I I
-268-
Table A. 2. Reference guide to Experiments and Figures
RUN
507 705
707 706 706V 806
I 506
703
509
511
512
508 510
802
702 701
405
407
404
404V 607 704
804
Flume Ident t Code
Sl S2
S2 S2 S2 S2 Sl
S2
Sl
Sl
Sl
S1 Sl
S2
S2 S2
R2
R2
S2
R2 Sl S2
S2
Normal Depth
d cm
1. 52 1..69
2.75 2.75 2.77 2.95
5.25
5.25
5.26 5.26
5.36
5.41 5.53
6.81
8.66
10.36
10.36 10 .. 70 10.81
10.84
Figures Where Experiment is
Cited
5.9 5.9
r= 0 C f"\ Jo 0, :>0 /
5.1,5.4,509 4015 4.18 5 .. 9
5.8,5.9,5.14, 5 .. 15,5.22 50 8, 5.9,50 11 , 5 0 12,5,. 15,5 .. 19, 5 0 20,5,. 21,5025 5,,9,5.11,5.15, 5. 16, 50 20, 5. 23 , 5.24,50 25 4.13,5.9 5.9,5 0 14,5 .. 15, 5.17,5.20 6,,2,6.6,6.7, 6 .. 8,6.9 5.8,5.9 5.5, 5.8,5.9
5.3,6,,7,6.8, 6.9
5.9
6.9,6.10 4.16 5.6,5.9 5.8,5.11, 5,,13,5.18,5.20 4.24,6 01,6.2, 6.7,6.8,6.9, 6.15,6.16,6.17, 6.18,6.19
Corresponding Pages
152 152
148~ 152 133, 142, 152 111 118 152
152, 185
148, 168, 148, 160, 174, 152, 171 , 183, 101, 152, 171 , 190,
152, 166, 180 152, 168, 149, 157, 174, 185 152 166, 174 197, 207 152
157, I 172, 185 168, 181,
168,
202 I 203,
148, 143, 148, 152
135, 202,203, 207
i 52
152,202,203, 207, 212 112 144, 152 148, 1 57 , 161,172,174 127, 188, 190, 202,203,207, 229,231,233, 235, 239
-269-
Table A, 20 (Continued)
RUN Flume Normal Figures Where Corresponding Ident f Depth Expe riment is Pc..ges Code d Cited
ern
904 S2 10.84 4.19,4.20,4 .. 21, 120, 121, 123, 4 .. 22,4 .. 23,6011, 124, 126, 216, 6.12 217
300 S1 17000 6,,13,6 .. 14 224" 225
406 R2 17.07 5.9,6~2,607, 152, 190, 202 6 .. 8, 6 .. 9 203 t 207
708 S2 17 .. 31 5.2,5 .. 7,5.9 134, 145, 152 708V S2 17.31 4" 16 112 808 S2 17 .. 32 6.4, 6. 5,6. 7, 193, 195, 202
6.8,609 203, 207 603 S1 17034 5 .. 8,5.9 148, 152
709 S2 21.97 4 0 14,509 106, 152
t For flume identification, S denotes smooth boundaries and R rough bottom. Flume 1 is 85-em side, Flume 2 110-em.
