EPA-660/2-74-053
JUNE 1974
Environmental Protection Technology Series
Hypolimnetic Flow Regimes
in Lakes and Impoundments
f
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sszz
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Office of Research and Development
U.S. Environmental Protection Agency
Washington, D.C. 20460
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Research reports of the Office of Research and
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been grouped into five series. These five bread
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This report has been assigned to the ENVIRONMENTAL
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EPA-660/2-74-053
June 1974
HYPOLIMNETIC FLOW REGIMES
IN LAKES AND IMPOUNDMENTS
By
John E. Edinger
Norio Yanaglda
and
Ira M. Cohen
The University of Pennsylvania, Philadelphia
Grant No. R-800943
Project 16080 FVK
Program Element 1BB045
Project Officer
Lowell E. Leach
U.S. Environmental Protection Agency
Robert S. Kerr Environmental Research Laboratory
P. 0. Box 1198
Ada, Oklahoma 74820
Prepared for
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
WASHINGTON, D.C. 20460
For sale by the Superintendent of Documents, U.S. Government Printing Office
Washington, D.C. 20402 - Price $2.10
-------
EPA REVIEW NOTICE
This report has been reviewed by the Office of Research and Development,
EPA, and approved for publication. Approval does not signify that the
contents necessarily reflect the views and policies of the Environmental
Protection Agency, nor does mention of trade names or commercial
products constitute endorsement or recommendation for use .
ii
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ABSTRACT
The "hypolimnetic flow" is a two-layered flow with the upper layer stag-
nant. This report presents the possibility of different flow regimes
for the hypolimnetic flow which may be determined from the parameters of
slope of channel bottom, flow depth, flowrate, density difference of
water in the two layers, and channel roughness. The analysis is limited
to the steady-state case of the hypolimnetic flow and the "upper layer
analysis" in which the lower layer is stagnant. Interfacial profile
equations which predict possible existence of ten different flow re-
gimes for the hypolimnetic flow and two regimes for the upper layer
analysis were obtained form the equations of continuity and momentum
for two-layered flow.
Experimental apparatus, consisting of a large scale open-channel tilting
flume and water supply and control systems capable of circulating water
of two different temperatures was designed and constructed. The experi-
ment employed observations of a dyed flow layer and measurements of
vertical temperature distributions in the flume. It was found that
eight different flow regimes could be generated in the flume for the
hypolimnetic flow on the positive and horizontal slopes.
To use the governing differential equation of the interfacial shapes
for the prediction of the flow regimes, it is recommended to investigate
the non-established portion of the flow, the definition of the critical
depth, and the interfacial friction factor.
This report was submitted in fulfillment of Project Number 16080 FVK,
Grant Number R-800943, by the Towne School of Civil and Urban Engineer-
ing, University of Pennsylvania, under the sponsorship of the Environ-
mental Protection Agency. Work was completed as of April 1974.
111
-------
CONTENTS
Sections
I Conclusions 1
II Recommendations 3
III Introduction 5
IV Hydrodynamic Theory of Hypollmnetic Flow 7
V Interfadal ProTtle Equations 32
VI Experimental System 65
VII Experiment and Data 71
VIII References 145
IX List of Symbols 147
X Appendices
iv
-------
FIGURES
No. Page
1 Definition Sketch for Two-Layered Flow II
2 Definition Sketch for Free Surface -jg
3 Definition Sketch for Interface 20
4 Definition Sketch for Side Wall 20
5 Definition Sketch for Steady Two-Layered Flow of a Unit 32
Width
6 Values of GH 37
7 Possible Flow Regimes 39
8 Flume Structure 69
9 Water Supply System Flow Diagram 70
10 Definition Sketch for Tilted Frame 78
11 Definition Sketch for Temperature Distribution Curve 92
12 Experiment SI-3/30 97
13 Temperature Distribution Curve for SI-3/30 97
14 Experiment S2-4/30 102
15 Temperature Distribution Curve for S2-4/30 102
16 Experiment S 3-5/14 107
17 Temperature Distribution Curve for S3-5/14 107
18 Experiment Ml-4/21 112
19 Temperature Distribution Curve for Ml-4/21 112
20 Experiment M2-4/18 119
21 Temperature Distribution Curve for M2-4/18
-------
Page
22 Experiment M3-5/18
23 Temperature Distribution Curve for M3-5/18 124
24 Experiment HI -4/23 129
25 Temperature Distribution Curve for HI -4/23 12g
26 Experiment H2-5/23 13g
27 Temperature Distribution Curve for H2-5/23
vi
-------
TABLES
1 Number of Experiments Conducted 90
2 Experimental Data and Results of Method 1 and 2 for 93
SI -3/30
3 Interfacial Profile Calculation for SI -3/30 99
4 Experimental Data and Results of Method 1 and 2 for 103
S2-4/30
5 Interfacial Profile Calculation for S2-4/30 104
6 Experimental Data and Results of Method 1 and 2 for 108
S3-5/14
7 Interfacial Profile Calculation for S3-5/14 109
8 Experimental Data and Results of Method 1 and 2 for 113
Ml -4/21
9 Interfacial Profile Calculation for Ml -4/21 114
10 Experimental Data and Results of Method 1 and 2 for 120
M2-4/18
11 Interfacial Profile Calculation for M2-4/18 121
12 Experimental Data and Results of Method 1 and 2 for 125
M3-5/18
13 Interfacial Profile Calculation for M3-5/18 126
14 Experimental Data and Results of Method 1 and 2 for 130
HI -4/23
15 Interfacial Profile Calculation for HI -4/23 131
16 Experimental Data and Results of Method 1 and 2 for 137
H2-5/23
17 Interfacial Profile Calculation for H2-5/23 138
18 Values of mhc and ^ 143
-------
SECTION I
CONCLUSIONS
The interfacial profile equations were developed for steady hypolimnetic
flow, including the upper layer analysis, of a unit width. These can be
understood with a slight extension of the knowledge of open-channel
hydraulics and predict the existence of twelve different flow regimes.
A large scale experimental flume which is capable of handling the re-
quirements to produce the hypolimnetic flow by temperature difference
was constructed.
The existence of eight different flow regimes on the positive and hori-
zontal slopes was verified in the experimental flume.
-------
.SECTION II
RECOMMENDATIONS
This investigation was limited to the laboratory study of the established
hypolimnetic flow. It was not within the scope of the investigation to
develop the method of the accurate prediction of the interfacial shapes
for the different flow regimes. It is recommended that the following
items should be investigated to fulfill the above purpose.
1. The location and size of the frontal zone which exists at the point
where the flowing layer goes under the stagnant upper layer.
2. The location and depth of the vena contracta occurring after the
gate.
3. The method to obtain the value of the interfacial friction factor.
4. The definition of the stratified critical depth.
5. The effect of the Internal circulation in the layer of no net flow-
rate, although it did have little effect in this study.
6. The magnitude of the momentum coefficient in the flowing layer.
-------
SECTION III
INTRODUCTION
This report presents the verification of the existence of the different
flow regimes of the hypoHmnetlc flow by the experiment using a large
scale experimental apparatus.
The hydraulic analysis of the hypoHmnetlc flow was reported earlier.
However, the analysis is presented again tn this report, since further
analysis has been done, and to provide the references for the equations
used 1n the data analysis.
Section 4 deals with the derivation of the equations of continuity and
motion for one-dimensional, unsteady, non-uniform two-layered flow, which
are the base for the interfacial profile equations derived in section 5.
In section 5, interfacial profile equations for the hypolimnetic flow of
a unit width are derived for the steady state case. These Interfacial
profile equations suggest many possible flow regimes. The term "upper
layer analysis" 1s used for the case with stagnant lower layer.
The uniformity of velocity distribution is assumed in both layers in the
derivation. Then, the change in the interfacial profile equations is
studied with the assumption of non-uniform velocity distribution in the
flowing layer. For convenience of computation and to obtain a more uni-
versal expression of results, non-dimensionalized forms of the inter-
facial and free surface equations are included. In addition, the effect
of the following factors on the form of the interfacial profile equations
is examined: a circulation of water in the layer with no net flowrate
and the wall of the experimental flume.
The free surface profile equation for open-channel flow on a horizontal
slope 1s derived and is used as a basis for estimating the roughness of
the flume section.
-------
Section 6 is concerned with the design and construction of the experi-
mental system. The conditions required for the system are listed.
Section 7 deals with the experiment and the data. After the reason for
the change of the original plan of the experiment is explained, the
experimental procedure is described in detail. Inconveniences and de-
ficiencies of the experimental system found during the experiment and
crude discussions on the accuracy of parameters are included.
The verification of the existence of the predicted flow regimes is done
by visual observations, photographs, and temperature distribution curves.
Some discrepancies between the stratified critical depth determined by'
its definition using the data and the stratified critical depth expected
from the Interfacial profile were found. Further, reliable values for
bottom friction factor and a good method to estimate interfacial friction
factor could not be obtained. Therefore, a method to estimate the bottom
friction factor was developed. An attempt was made to correct the strati-
fied critical depth and estimate the interfacial friction factor by using
the interfacial profile equation, the interface and data obtained, and
the estimated bottom friction factor. The results of the experiments
and this attempt are shown in Sec. 7.3. Observations obtained throughout
the experiment are described in Sec. 7.1. Observations for each profile
designation are In Sec. 7.3
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SECTION IV
HYDRODYNAMIC THEORY OF HYPOLIMNETIC FLOW
Schijf and Schonfeld1 presented the equations of continuity and motion
for one-dimensional unsteady non-uniform two-layered flow of a unit
width which are the base for this investigation. The equations may be
derived from the most general expressions of conservation of mass and
linear momentum and described in the following sections.
The concept of continuum field theories is premised for the derivation.
4.1 EQUATIONS OF CONTINUITY
If there are no sources or sinks in the material volume of fluid under
consideration, and mass transfer due to diffusion 1s ignored, then con-
servation of mass can be written as
where D/Dt = material derivative
V (t) = material volume
P * p(x_,t) • density of the homogeneous fluid at x., spatial
coordinates and t, time, the underbar Indicates
vector
V = volume
-------
2
The Reynolds transport theorem can be stated as
where $ = arbitrary seal or or vector
= partial derivative
= surface bounding Vm(t)
y_ = v{x.»t) = velocity of the fluid at x^ and t
n_ = unit normal vector
A = area
If p is substituted for $ in Eq. 2 and Eq. 1 is used,
If use is made of the divergence theorem,
where w = arbitrary vector or tensor
V = gradient vector operator,
Eq. 3 can be written, by letting f = pv^ as
(5)
Since V (t) is arbitrary,
8
-------
ap _ , } Q . .
at* ^PY ( '
Eq. 6 can also be written as
D* « /,\
—+pV-y=0 (7)
Dt
Assume an incompressible flow, which is usually done for water; then,
Dp/Dt=0. This states that the density of a fluid particle does not
change along its particle path. Since the density of water is mainly
determined by its temperature and substances dissolved (or perhaps
suspended) in it, this assumption is equivalent to the neglect of the
molecular transport of energy and of the dissolved substances across
the boundaries of fluid particles. Then
In order to take into consideration the turbulence usually existing in
open-channel flow, instantaneous values of flow properties are written
in terms of mean values and fluctuating components. Substituting
y_=£+y_' (in which "overbar" indicates mean value and "prime" denotes
fluctuating components) into Eq. 8 and taking the average yields
as the mass conservation equation for an incompressible flow. [For
reference, the condition of incompressibility expressed in the average
can be obtained by taking the average of Eq. 6 and substituting Eq. 9:
9p _
—+y-Vp+V./o«y'=0 (10)
-------
The integration of Eq. 9 over Va(t), which is a control volume moving
through the space 1n an arbitrary manner, is carried out in order that
the equation can be used for a specific problem:
(ID
From Eq. 11 on, the overbar indicating mean value on y_will be understood.
If the divergence theorem with v in place of ? in Eq. 4 and the general
2 ~
transport theorem,
02)
where d/dt = total derivative
Aa(t) = surface bounding V_(t)
d o
w - mean velocity of points on the surface of Va(t)
with substituting 1 for *, are used, Eq. 11 may be written as
(13)
Assume, in a channel, the existence of the two distinct flow layers of
different densities separated by an interface across which little mix-
ing occurs. It is also assumed that the free surface and the interface
are horizontal in the direction of the width of the channel. A control
volume can be described as in Fig. 1.
10
-------
-Free surface-'
^
\ -b(x,z)
^\ -\ yv
Interface— \ >6
3^.....^,
Fig. 1. Definition sketch for two-layered flow.
In Fig. 1, x = longitudinal distance (in the direction of the
flow
y = transversal distance (in the direction of the
width of the channel)
z = vertical distance (in the direction normal to
the x direction)
h = h(x,t) = depth of a layer
b = b(x,z) = width of a channel
subscripts 1 and 2 indicate upper and lower layer, respective-
ly.
A (t) can be divided into the following three areas:
a
0) A (t) = area of entrances and exits through which the flow enters or
leaves the control volume (cross section);
(2) A (t) = area of moving boundary (free surface and interface);
(3) A = area of fixed boundary (bottom and other wetted perimeter).
11
-------
If there is no appreciable precipitation on or evaporation from the free
surface, and no mixing occurs at the interface, the velocity w=y_ can be
assumed on Ag(t). If there is no inflow or outflow, including ground-
water flow at the wetted perimeter, w=y_=0 on Ag may be used. Let Ae(t)
be fixed (w=0), and Eq. 13 reduces to
04)
For the upper layer, it is
— / / N*
dtJVal(t) Ael
Leibnitz's rule is used to calculate the total derivative of integrals
The first of Eq. 15:
d . d .
— /« ,v\dv=(in th® limit )dx — JVX bdz
dt Val(t) dt Jh2
_ 3(h.,+h?) 3h
=dx[b(h1+h2 )— -± — 2--b(h2 ) —
e t 9 1
The second term of Eq. 15:
d ,hl+h2
•*
,
. .v-ndA=(in the limit)dx— /h bdz
^ * Q^v ^
^
"2
+dxb(h2)
12
-------
where = = average of the velocity in the direction of
}\ f\
the flow integrated over a horizontal lamina,
Assume 3b(x,z)/3x = 3b(x,z)/3z=0, or b(x,z)=constant and cancel bdx.
Then,
3h, h
-0 (16)
If it is assumed that
««U1(x.t) (17)
h,+h23 9U, (18)
and /hX -fdz!=hlir
"2 3x <>x
where U, = average velocity in the upper layer, Eq. 16 can be written in
the following form.
It can also be written as
3h. a
h,U,)=0 (20)
Similarly, for the lower layer (integrating from 0 to h,,), if the assump-
tions,
=U2(x,t) (21)
r 2
/n — dz=ho — -
0 3x x- 2g
and
13
-------
are made,
3h 3U2
3x
and also *t 3x
U2)»0 (24)
2 2
4.2 EQUATIONS OF MOTION
The conservation of linear momentum can be stated as
(25)
where F= body force per mass and J_= stress tenor.
If the gravitational force 1s considered as the only body force, £=£,
which is the gravitational acceleration vector. Using the Reynolds
transport theorem (*-py^ in Eq. 2) and the divergence theorem U=pyy_ and
• J in Eq. 4) and also the relation
where P - pressure
1 = unit tensor
g = shear stress tenor
yields
(26)
14
-------
Since Vm(t) is arbitrary,
(pvv+PI-g)-pg=0 (27)
ot
Before accounting for the turbulence, assume the incompressible flow of
a constant viscosity, Newtonian fluid. Then,
V-2=fi72y (28)
where jj = viscosity
Substituting v=v+v/ , P-P+P1, P= p+p' , and 2=0+2' [and therefore v-
v+y/)] into Eq. 27 and taking the average yeilds
_____ _ _ _
+V- (pvv+pv'v'+p1 vy'+pfy*v+/o'vlv'
-g)-pg=0 (29)
If afpVj/at is ignored and let i=g-pv.' v.1 -p 'vv_' -p~IV1 £-p ' v_' v_' so that
may be called the stress considering the turbulence, Eq. 29 becomes
0 (30)
where overbars for the mean are understood.
By integrating Eq. 30 over V" (t) and using the divergence theorem
a
(V/=pvy_+PI-i in Eq. 4) and the general transport theorem ($=pv_ in Eq. 12),
15
-------
d
T«ndA
- * V I ¥ \* i«» TV j. I x. \p v v v — •» /" ijun-rj . / , \ rJL« I1OA—J ,. /*.\T«n
dt Va^' AaVT:' Aalt' " ~ Aa(t;« -
";Va(t)^dV=0 (3D
Again, assume the existence of the two-layered flow stated in Sec. 4.1
and a control volume shown in Fig. 1, and take the same division of A_(t)
a
and boundary conditions; then,
d
(t)PndA-/ (fc)I. ndA
'a
(t)P2dv=o (32)
The change in variables in the transverse section of the channel is not
considered hence forth.
In order to consider the z component of Eq. 32, dot j<, which is a unit
vector in the z direction, with Eq. 32 to obtain
(33)
'a
where subscript z indicates the z component of a vector.
Assume that
(1) the channel bottom is practically straight in the direction of the
flow;
(2) the mean particle paths of the flow are nearly parallel to each other
and to the channel bottom; and
(3) the slope of the channel bottom is less than 1/10.
Since we can define
16
-------
where £ = unit vector in the perpendicular direction
g = acceleration of gravity,
where 9 = angle between the x axis and the horizontal
Eq. 33 is applied to a differential volume dxdydz.
Cancelling dxdydx yields
(34)
where the first subscript for T indicates the plane on which the stress
acts and the second one denotes the direction in which the stress acts.
It is assumed that
0 and
(Of course, 3x /ay=0, since the change in the y direction is not con-
sidered.)
Further, for the upper layer, the density of the water is assumed to be
P-, and constant in the z direction. In addition, the surface and inter-
facial tension can be ignored. The, what is left is
n (35)
Eq. 35 can be solved with a boundary condition, P=Pg= atmospheric pres-
sure at z=h,-Hi2, to yield
P=P0-»-p1g(h1+h2-z) (36)
For the lower layer, it is assumed that P2 is constant in the z direction,
and the interfacial tension can be ignored. In a similar way,
17
-------
Take the x component of Eq. 32.
d
(38)
where j= unit vector in the x direction, the subscript x indicates the x
component of a vector.
Each term of Eq. 38 for the upper layer is examined. They are expressed
in the limit.
d d h,+h2
. Pbdz
al
— „ ,4.N-.v v 1._
dt val(t) x dt h2 L x
-r-(p1b)dz
*
-dxp1(h2)b(h2)-^
A h, +ho _
2
18
-------
-dxPl(h2)b(h2)-^
[. 36 is
*alx"' ~alvu'~ - * * - - used.)
At the cross sections [Ael(t)]r
*l+h2 3
12 3x
+dxP0b(h1-i-h2) i—2—dx(P0+p1gh1)b(h2)
At the free surface [As(t)]:
See Fig. 2. Since 6 is small according to the assumptions that the
mean particle paths of the flow are nearly parallel to each other and
to the channel bottom,
4.B=cos(90°+6)=-sin6*-tan6=—
I—6
' — Free surface
Fig. 2. Definition sketch for the free surface
Therefore,
3hl+2 dA=_dxp ,
jAc(t) 0 3x 0 1 ^ 3
5
19
-------
At the interface
-0 *
interface
Fig. 3. Definition sketch for the interface.
i-B=cos(90 -e)=sine*stan€=3h2/3x
AA (t)
At the side walls
It is assumed that the change in width, both in the flow and z direct-
ions, is gradual.
j¥
i= unit vector in
the y direction
n«-j
Fig. 4, Definition sketch for the side wall
?b(x,z)
9b
20
-------
At A(t)i
el
d hi
/. ,..r dA=dx— /, x
Ael(t) »c dx h2
h.+h., a
-dx/,1 Z-i-(T b)dz
-dxTvx(h2)b(h2)
xx ^ z
At A
5
3(h,-».h7>
,--
*dxTzx(h1+h2)b(h1+h2)-dxTxx(h1-t-h2)b(h1+h2) - —
At Ai(t)i
At Awl(t)i
3b(x,z) 3b(x,z)
Awl(t)1' 3x >xx ^ 9z
3b 3b
(We define the sign of t in this manner.)
yx
21
-------
,dV
9x=g-1=-gp-i=-gsinO«-gtan9=gSb> in which Sb=-tan0=slope of the channel
bottom.
h1
It is assumed that
(1) 9pj/3t=3p^/3x=0(3p.|/3z is already assumed. For reference, v-fTv^
in Eq. Al.2-10, since3p,/3y=0.);
(2) ab(x,z)/ax=3b(x,z)/3z=0, or b(x,z)=constant;
(3) 3PQ/3x=0; and
(4) 3rxx/3x=0.
Use the following definitions:
TDK(h1+h2).r8. rzx(h2)=T., and Jh*
where TS= surface shear stress
TJ = interfacial shear stress
TW = wall shear stress
W
W-j = wetted perimeter corresponding to the upper layer
Then, Eq. 38 takes the following form for the upper layer, by cancelling
bdx.
3(h,+h2)
) > at -p
1+ 2^hi+h2-r>dz-ra+Ti+Tw^-'9l9sl^l-0 (39)
22
-------
If it is assumed that
2
t» at x
? a -> i
6—clzs=h,——-
i2 ax x x a^
then, Eq, 39 can be written as
It can also be written as
3h-
(42)
^xCh^hj )>=<- '-—T (43)
3U,2 23hl 3hl 3h
--+ piui
W ^ t M f \
-± -^,ghTSb=0 (45)
B j. rfb *-l^ 1 £>
Subtracting the equation of continuity (Eq. 19) multiplied by p-,1^ from
Eq. 44 yields
3h0
Wi
i_. «v,.sb=0 (46)
23
-------
If Eq. 46 is divided by p,gh,, applied to a unit width of the
Ti-Ts
flow (T =0), and the definition
W
is used,
the equation of motion for the upper layer is obtained as
13U, U, 3U, 3h, 3h7
-- -+— - ~+ — ~+ - -+S~-S>=0
g at g 3x 3x ax fl °
For the lower layer, similar operation can be used. Some of terms are
listed below.
. J ,
is used.)
At the cross sections [Ae2(t)j:
d h
At the interface [A
i'n=cos(90u+«)=-sine«-tane=-9h2/3x
2
3x
At the bottom [Au]: n_=-k_, so that l-n=0.
At the side walls [Aw2(t)]: The same as for the upper layer, except
for integrating from 0 to h.
AtAfi2(t):
24
-------
ho a ?
-
3h
At AjCt):
At Ab:
3x
At A^(t): ^e same as
Assumptiopns and definitions in addition to the ones for the upper layer
are
3p2/at=3p2/3x=0, Tzx(0)=rb= bottom shear stress,
and
where vT2= wetted perimeter corresponding to the lower layer except for
the bottom.
Eq. 38 becomes, for the lower layer
If it is assumed that, in addition to Eq. 41,
h a au2
(50)
(51)
25
-------
and =u2 (52)
Eq. 49 can be written as
V \^ r2y 2 b~w (53)
o
aCUoh-,) 2
-------
p P2_(P2_P )
— =— - - - — — =1 -- , in which
If it is assumed that
p /p =I-(AP/P) can be used. [For example, the density difference is
approximately 0.001 gm/cm3 between water of temperation 20°C (68°F) and
25°c (77°F) and 0.026 gm/cm3 between fresh water and sea water at 150C
(59°F)].
Now the equation of motion for the lower layer is
s,,-.,
g 3t g 3x p x c
As a consequence, p in Eq. 58 can be used in Sfl(Eq. 47) and Sf2 (Eq. 56)-
4.3 SUMMARY OF THE ASSUMPTIONS USED IN SEC. 4.1 AND 4.2
1. Continuum field theories are used.
2. The flow of water is the incompressible flow of a constant viscosity,
Newtonian fluid.
3. Only gravity is considered as body force.
4. The surface and interfacial tensions are ignored.
5. Only the mean values of fluid properties are used to consider the
turbulence in the flow. (Related to this, 3(pT7r)/3t and vVy1
are ignored. )
6. There exist, in a channel, two distinct flow layers of different
densities separated by an interface across which little mixing occurs.
7. The free surface and the interface are horizontal in the direction of
width of the channel.
8. There is no appreciable precipitation on or evaporation from the free
surface, and no inflow or outflow, including groundwater at the wetted
perimeter.
9. The variation in the transverse direction of the channel is not con-
sidered.
27
-------
10. The channel bottom is practically straight in the direction of the
flow.
11. The mean particle paths of the flow are nearly parallel to each other
and to the channel bottom.
12. The slope of the channel bottom is less than 1/10.
13. Only a unit width of the flow 1s considered.
14. The density of water is constant in a layer.
15. The density difference between the layers is small.
16. 3vz/1t=0, and 3(vzvx)/3x=0.
17.
^O, 3Txz/3x=0, and 3rzz/az=o.
-=U1(x,t) for the upper layer.
-U2(x,t) for the lower layer.
20. h
21 h
- and
22' 2
2
23. h,+h73 - 3U,2 h2a
r i 2_a_ 2>dz=h i_ and /0^~
^2 3x 3x u *x
The assumptions 6 and 15 can be considered to differ from the assump-
tions generally used for the gradually varied open-channel flow.
4.4 NON-UNIFORM VELOCITY DISTRIBUTION CONSIDERED IN A LAYER
In the derivation of the equation of motion shown in Sec 4.2, it is
assumed that the velocity is distributed uniformly in both layers.
If non-uniform velocity distribution is assumed in the flowing layer,
-------
2 2
the assumption = made in Sec. 4.2 becomes approximate and it is
A /N
necessary to have a correction factor, g, in the assumption:
(60)
o
3 may be called the momentum coefficient. The range of e is 1<3<1.2,
2 ^
although the value of the upper limit is not definite. ' The effect of
the introduction of 3 on the form of the equation of motion is now exam-
ined. This examination is done only for the steady-state case, since the
derivation is not easy for the unsteady-state case.
Then, Eq. 42 for the upper layer becomes
hj+h2 d 2 dUx2
h2 dx V* l dx (61)
Therefore, Eq. 44 can be written as
du,2 o
-------
If it is applied only to a unit width of the flow with the definition of
Sf-i (Eq. 47), the equation of motion for the upper layer is obtained to
be
pU,dU, dh, dh2 o _
£-1—1+—i+—^+Sfl-Sb=0 (65)
g dx dx dx
Similarly, the equation of motion for the lower layer becomes
dh, dh-
g dx p dx dx
4.5 EQUATIONS OF CONTINUITY AND MOTION FOR ONE-DIMENSIONAL UNSTEADY NON-
UNIFORM OPEN-CHANNEL FLOW
The method of derivation used for two-layered flow can naturally be used
for open-channel flow. Using the assumptions mentioned for the two-
layered flow, for an open-channel flow with a constant width,
equation of continuity:
0 (67)
3t 3x
and equation of motion:
13U U3U 3h T,-T_+T,_ (W/b)
g3x 3x pgh
sb=0 (68)
D
For a unit width of the flow, Eq. 68 can be written as
lau uau 3h
+sf-Sfr=0 (69)
g3t f b k '
with a definition sf=b~s (70)
f pgh
If the discussion on the momentum coefficient is applied, $, mentioned
in Sec. 4.4, Eq. 68 becomes, for the steady-state case,
30
-------
g dx dx pgh
For a unit width of the flow, it reduces to
31
-Sb=0 (72)
gdx dx
-------
SECTION V
INTERFACIAL PROFILE EQUATIONS
5.1 INTERFACIAL PROFILE EQUATIONS FOR HYPOLIMNETIC FLOW OF A UNIT WIDTH
One-dimensional equations of continuity and motion for the steady two-
layered flow can be written from the equations for the unsteady-state
case (eqs. 20, 24, 48, and 59). These equations are listed below with
the definition sketch.
Free surface
Interface
Bottom
Fig. 5. Definition sketch for steady two-layered flow of a unit width.
Equation of continuity for the upper layer:
Equation of continuity for the lower layer:
Equation of motion for the upper layer:
dh, dh-.
g dx dx dx
;fl-sb=o
(73)
(74)
(75)
32
-------
Equation of motion for the lower layer:
_2—2 + (i-—)—1+—2+Sf 2-sb=0 (76)
g dx p dx dx
where p = (p,+p2)/2
Ap = p-i ~Po
q = flowrate per unit width
Sf is the friction slope, which is defined by the relations
and Sf2
T. and T. may be given empirically as
(77)
(78)
8
(79)
8
where f = friction factor at the channel bottom, which is called
b
"bottom friction factor"
f. = friction factor at the interface, which is called "interfacial
friction factor
can be ignored for the case in which the wind effect on the free sur-
s
face is small.
For the hypolimnetic flow, the upper layer is assumed to have no motion,
i.e., U, = 0. (See Sec. 5.4 for a discussion on the case in which the
internal circulation of water is allowed in a layer having no net flow
rate.) Then, Eqs. 75 and 76 can be written by substituting U2 = q2/h2
from Eq. 74, as follows.
33
-------
dh,
Sb=° (81)
p dx gh dx
where subscript L Indicates the hypollmnetic flow.
T and T may be written, from Eqs. 79 and 80, as
T .u 2 (83)
034
and Ts~"
Eqs. 77 and 78 can now be written as
and
(85)
(86)
Substituting dh-j/dx from Eq. 81, and Eqs. 85 and 86 into Eq. 82 and re-
arranging It yields
(Ap/p)gh23 dx
2
" b 8(Ap/p)gh23 H-h2
where H = h,+h2 = total depth.
Since Ap/p 1s expected to be at most 0.03,* the term (AP/P)
* This is for a case of fresh and sea water. For water with temperature
difference of 16.7°C (30°F), AP/P is at most 0.005.
34
-------
may be ignored in Eq. 87, provided that f^ is not very large compared
to fb< Then Eq. 87 becomes
Up/p)gh23 dx
Positive Slope Flow
When Sfa is positive, Eq. 88 can be rewritten as
2
(4p/p)gh3 dx
-S li
b 8(Ap/p)gSbh23 H-h2 (* '
The stratified critical depth for the lower layer, h 2 is defined by the
relation
(sc)
The stratified normal depth for the lower layer, h «, may be defined by
the relation
A factor related to the stratified normal depth for the lower layer, G,
is defined by the relation
(H.fi/fb)H»h2 K-
"(H-fi/fb)H-hn2H-h2
35
-------
l-(h2/H)
If the above definitions are used, an equation of interfacial profiles
for the hypolimnetic flow can be obtained as
dh2 o 1- (Gh^/J^ )3
- SsrSy. - -3^ - ^
dx bl-(hc2/h2)3
It is almost the same in the form as the equation presented by Chow
for the free surface profiles of gradually varied open-channel flow.
The term Gh « is now examined. From Eqs. 91 and 92,
(92)
If q2, S., f. and ffa are assumed to be constant, change in Ghn2 depends
only on {H(fi/fb)/l>(h2/H)}'I/3, which is later called GH. Examples
of the values of 6n are given in Fig. 6. G^ is fairly constant when
h2/H < 0.8 especially for smaller f-j/fu.. The term Ghn2 may be called
"virtual stratified normal depth."
General characteristics of interfacial profiles for the positive slope
flow can be found from Eq. 93.
As Ghn2/h2 approaches 1, change in the lower layer depth, dh2/dx, goes
to zero. This means that the interface becomes parallel to the channel
bottom if the lower layer depth equals the virtual stratified normal
depth. When h c2/h2 approaches 1, dh2/dx goes to infinity. This indi-
cates that the interface will intersect the stratified critical depth
sharply and disturbances can be expected at the interface.
(Where the interface is too steep, the method used in this study is not
applicable, since streamlines are assumed to be nearly parallel to each
other and to the channel bottom.) If h2 goes to zero, it is easily de-
duced that dh2/dx = (fb+f^)/8, which is a small angle to the channel
bottom. If h2 becomes much larger than Ghn2 and hc2, dh2/dx approaches
36
-------
3.0-
2.5-
GH 2.0-
1.5
1.0
fj/fb= 0.00
0.5
0.0 0.2 0.4 0.6
h2/H
0.8
1.0
Fig. 6
Values of
37
-------
S. . This means that the interface becomes close to the horizontal in
D
cases in which the lower layer can extend toward the downstream with its
depth increasing. In addition, as can be observed from the values of.G^,
if h2 approaches H, dhp/dx may go to infinity. This indicates that the
interface may intersect the free surface sharply. Therefore disturbance
can be expected at the frontal zone of the upper layer where the two lay-
ers start to establish themselves.
The position of the interface relative to the stratified critical depth
and the virtual stratified normal depth will dictate whether dh2/dx
assumes a positive or negative value, i.e., whether the lower layer depth
increases or decreases in the downstream direction. Equation 94 shows
that it depends on the magnitude of GH[fb/(8$b)]1/3 whether Ghn2 is
greater or less than hc2> that is, whether the flow is on a mild or steep
slope.
From these discussions, it can be expected that there would exist the fol-
lowing interfacial profiles, which are illustrated in Fig. 7.
1. Mild slope flow (denoted by "M"), when Ghn2>hc2:
Ml profile, when n2>Ghn?>flc2'
This profile could represent a density flow in an impoundment created by
a dam downstream on a mild slope. Note that the downstream end of the
interface approaches the horizontal. Mixing between the two layers would
not occur quickly.
M2 profile, when Gnn2>tVhc2;
This could be a case of the discharge of lighter water into running denser
water on a mild slope. The lighter water forms a pool. Mixing between
the two layers can be expected downstream.
M3 profile when Gh 2>h 2>h2;
This could be the discharge of denser water into the lower part of the
lighter water pool from an opening lower than the stratified critical
depth on a mild slope. An internal jump may occur downstream and mixing
between the layers may result.
38
-------
Name of flow Desig- Relation of h2 Sketch of profile
nation to hc2 and Ghn2
Mild slope Ml h2>Ghn2>hc2
M2
M3
Steep slope SI h2>hc2>Ghn2
52 hc2>h2>Ghn2
S3 hc2>Ghn2>h2
Fig. 7 Possible flow regimes.
39
-------
Name of flow Desig- Relation of
Sketch of profile
Horizontal
slope
nation
HI
to
h2>hc2
H2
Adverse
slope
Upper layer
analysis
Al and A2 are essentially the same as
H flow above.
hci>hi
U2
Notet Depth in U flow is measured from the free surface.
Notation: h= depth of a layer; hc= stratified critical
depth; Ghn2= virtual stratified normal depth;
subscripts: 1= upper layer; 2= lower layer.
Legendi —=^- =
free surface;
bottom;
virtual stratified
normal depth line;
= interface;
= flow;
= stratified critical
depth line.
Fig. 7
Continued.
40
-------
2. Steep slope flow (indicated by "S"), when nc2>Gnn2:
SI profile, when h2>hc2>(^n2'
This case might represent the flow of denser water into an impoundment
created by a dam in a steep slope when the thickness of lower layer is
larger than the stratified critical depth.
S2 profile, when nc2>n2>^'1n2'
This might be the case of the density flow often observed in lakes and
impoundments, if denser water happens to start out at a height between
the stratified critical and virtual stratified normal depths. It may
also represent the discharge of lighter water into running denser water
on a steep slope. In either case, the two layers would not mix with each
other quickly.
S3 profile, when nc2>Gnn2>n2'
This might be the discharge of denser water into the lower portion of the
lighter water pool from an opening lower than the virtual stratified
normal depth on a steep slope. The two layers would not mix with ease.
Horizontal Slope Flow (denoted by "H")
When the flow is on a horizontal channel, the interfacial profile equa-
tion, Eq. 93, can be rewritten as follows.
r ___
(ap/p)gh23 dx 8(AP/p)gh23 H-h2
If a definition
_
H " H-h2 l-(h2/H)
is introduced*, then Eq. 93 becomes, for the H profile,
3
* GH is the G for the H profile. G^ can be obtained by letting hp2
in Eq. 92.
41
-------
dh2_ _bHc2/2 (97)
dx~~8 l-(hc2/h2)3
with the definition of hc2> Eq. 90.
It is easily observed that only the stratified critical depth is import-
ant to the horizontal slope flow. Applying arguments similar to the ones
for the positive slope flow, the following two possible profiles were
found.
HI profile, when hp>h pi
This could represent the discharge of lighter water into the running
3
denser water. It is a profile similar to the M2 profile. Chow would
prefer the term "H2" profile; however, the term "HI" profile was chosen
to show that there exist only two different profiles for the H profile.
A
Oil slicks show a similar profile.
H2 profile, when nC2>n2;
This profile is similar to the M3 profile.
Adverse Slope Flow (denoted by "A")
Adverse slope means the flow on a slope which has a negative value. The
same Eq. 93 can be used for this flow; however, the positive value of the
slope is used for convenience.
Sb=-|sb| (98)
Substitute this into Eqs. 93 and 94 and
dh
j I Ml i t \. J\- \ -> V-'-'/
where - - — • *'"» - (1°0)
42
-------
Subsequently, only the stratified critical depth is an important factor
for the A profile.
Al profile, when h2>h 2;
This is the same as the HI profile, except that it is on the adverse
slope.
A2 profile, when h o>hp>
This profile is similar to the H2 profile.
Free Surface Profile Equation for the Hypolimnetic Flow
The equation for the total depth profile can be found by using Eqs. 81
and 85
001,
dx 8 p h2(H-h2)
This indicates that the free surface is approximately horizontal and shows
a very slight increase toward the downstream direction, except when h«
approaches either zero or the total depth.
Upper Layer Analysis (Designated by "U")
For this analysis, it is assumed that Ik = 0. Then, Eqs. 75 and 76 can
be written in the following form by substituting U^ = q-i/hn from Eq. 73,
CN dh^ dh-.
Ap dh, dh?
1->
where subscript U indicates the upper layer analysis.
If TS is neglected again, Eqs. 77 and 78 can be written as
43
-------
and
Eliminating dh2/dx from Eqs. 102 and 103, substituting Eqs. 104 and 105,
and rearranging them yields
r. _V_.fhl- fiq!H __ , %
L Jdx 8Ah3h (106)
If the stratified critical depth for the upper layer, hcl , is defined by
the relation
(107)
cl (Ap/p)g
an equation of the interfacial profiles for the upper layer analysis can
be obtained, in terms of h-j, as
As in the cases of the horizontal and adverse slope flow profiles, only
the stratified critical depth is important for this upper layer analysis.
Thus, two profiles appear possible. The interface becomes vertical at
the points where h^ approaches either hcl or H. Eq. 108 can be applied
on any sloped bottom, since S^ is not included in it.
Ul profile, when hcl>h1;
This could represent the release of less dense water on a pool of denser
water. Mixing of the two layers may be expected downstream.
U2 profile, when h^h^;
This case could represent the familiar shape of the arrested salt wedge
found in estuaries. As pointed out above, the shape of the wedge could
be a sinuous curve even on a sloped bottom although it is well known that
the wedge takes the sinuous form on the horizontal slope.
44
-------
Free Surface Profile Equation for the Upper Layer Analysis
The equation of total depth change can be obtained by using Eqs. 102,
104 and 108.
dH_ fiAP(hcl/h1)l-hhcl/h(H-h1) (log)
dx~ b~ 1-(
This indicates that the free surface would remain approximately horizont-
al and show a very slight increase towards the downstream direction for
the Ul profile and a very slight decrease for the U2 profile, except when
h, = h , , h, = H, and h-| = 0.
5.2 NON-UNIFORM VELOCITY DISTRIBUTION CONSIDERED IN THE FLOWING LAYER
When non-uniform velocity distribution is assumed in a layer, the momen-
tum coefficient, e, may be included as discussed in Sec. 4.4. Then,
the equations of motion can be written as follows.
The equation of motion for the upper layer:
3U,dU<, dh, dh?
I_l — 1+ — 1+ — Z+s s -o
g dx dx dx fl to (HO)
The equation of motion for the lower layer:
dh
g dx p dx dx
Then, it is easy to derive interfacial profile equations by following
the method of Sec. 5.1.
45
-------
Positive Slope Flow
If the same definitions are used for the stratified normal depth, criti-
cal depth, and G, the interfacial profile equation for the positive slope
flow can be written as
dx" bl-3(hc2/h2)3
If a definition
is used, Eq. 112 becomes
dh2
where Ghn2MshC2M I --[l* f 1 1 1/3 (115)
*n2M c2M 8pSb^ i-(h2/H)
Note that Ghn2M = Ghn2>
Horizontal Slope Flow
Similarly, the interfacial profile equation for the horizontal slope flow
can be
^2^ fb(GHhc2/h2) (n 5.)
dx ~8 l-3(
With the definition hc2M = 31/3hc2,
dx
Adverse Slope Flow
46
-------
018)
dx l-3(hc2/h2)
(119)
, *b r, fi/fb nil/3
where Ghn2Masft^2M I _ _ i _ i L1+:—- — J) (120)
Free Surface Profile Equation for the Hypolimnetic Flow
The free surface profile equation for the hypolimnetic flow is not in-
fluenced by 3» but will change its appearance slightly.
,
b 8 p 0h22(H-h2)
Upper Layer Analysis
A similar derivation results in the following interfacial profile equa
tion for the upper layer analysis.
If a definition
hclM=p hcl (123)
is used, Eq. 122 becomes
dx 83
Free Surface Profile Equation for the Upper Layer Analysis
The free surface profile equation for the upper layer analysis can be
written as
47
-------
dH.o fi&p *• /hi>3 i+hdMVCsh!2 (H-hj) ] (125)
dx b 8 p 3 i-toclJ/V
This indicates that the free surface would remain approximately horizontal.
5.3 NON-DIMENSIONAL INTERFACIAL AND FREE SURFACE PROFILE EQUATIONS
For convenience of computation and to obtain a more universal expression
of results, the interfacial and free surface profile equations obtained
in the previous sections can be non-dimensionalized.
Non-dimensional depth of a running layer, total depth, virtual stratified
normal depth, and distance along a channel are defined as functions of the
stratified critical depth as follows.
(126)
(127)
Ghn2N*Ghn2/hc for P°sitive sl°Pe flow (128)
Sb x/h for positive and
D u adverse slope flow v«Z9)
and Xjjsx/h- for horizontal slope now and
upper layer analysis
Also, if the momentum coefficient, 3, is included in the equations,
{132)
HNMsH/hcM (133)
Ghn2MNisGhn2M/hc2M for Positive slope flow (134)
* SK x/h-,., for positive and
CM r . , __ /iic\
adverse slope flow (135)
48
-------
and x^TM^x^M for horizontal slope flow and
and ~MM CM upper layer analysis
For Positive Slope Flow
Interfacial profile equation:
"Ghn2N (137)
dxN
where
If the momentum coefficient, e, is included,
3-GK -,^,3
dh2NH h2NM*Ghn2MN
~
Free surface profile equation:
dH
f b fi/fb i
MN^-Ci] (140)
—T~~. T ^— M41)
(H —h )
With e,
~sl+~Tf27^ * \~ (H2)
"2NM ^MNM n2NM'
For Horizontal Slope Flow
Interfacial profile equation:
•Q
**-• Cf HN /T y» o \
-i^r- («3)
49
-------
3s+ fi/gb (144)
G sl+
HN 1-
With e,
dh2NM Cfl/(8g^GHNM3 (145)
" ~~
f i/f
where
Free surface profile equation:
dHjj (fi/8)(Ap/P)
^
With p,
For Adverse Slope flow
Interfacial profile equation:
With e,
Free surface profile equation:
**N~
With 3,
3 3
dh2NM h2NM "^"i^MN
50
-------
dHNM (V8HWp)/(sbg) (152)
™ """' ' "" " «»^* JL ^r *v
h2NM (
For Upper Layer Analysis
Interfacial profile equation:
(153)
°*N x "IN
With e,
dhlMM [f4/(8P)l/[1-(hlNM/HN'M^
.LiMn__ x^ xiii i tit *
*l-«r 1 —>1 3
o^NM nlNM
Free surface profile equation:
11N (HN"hlN
With
3
1NM
5.4 INTERFACIAL PROFILE EQUATIONS FOR HYPOLIMNETIC FLOW OF A UNIT WIDTH,
WHEN INTERNAL CIRCULATION IS CONSIDERED WITHIN A LAYER OF NO NET
FLOWRATE
In the previous section, one of the layers is assumed to have no motion
at all. It is, however, learned [Ippen , Harleman , and Polk et al ]
that circulation of water within the layer having no flow rate can be
generated by the moving water of the other layer and its average flow
rate in the layer is very close to zero. Therefore, the layer can be
said to have no net flow rate.
51
-------
However, in this case, the assumptions related to the homogenuity in the
layer of no net flow rate (assumptions 19, 20 and 23 listed in Sec. 4,3)
cannot be made. The changes in the forms of the interfacial profile equa
tions derived in the previous section are studied below.
If the assumptions mentioned above are not used for the upper layer and
homogenuity of velocity is still assumed for the lower layer, equations
of continuity and motion for this analysis can be written as follows
(see Eqs. 16 and 39). Equation of continuity for the upper layer:
d(h,+h7) dh2
^ 2 --^=0 (157)
Equation of continuity for the lower layer:
h2U2=q2=constant
Equation of motion for the upper layer:
V^) 2dh2
dx gh^ dx
dh
Equation of motion for the lower layer:
A dh,
__ -- — _ rL-,
g dx p dx dx f2rL b
where = average velocity in the x direction integrated over lamina
/v
at depth h,
and Sf2rL=— -3— (162)
52
-------
Subscript r in f^ ans S.e . indicates that f. is influenced by circulation
within a layer: it is assumed that the definitions of t. and T., Eqs. 79
and 80, can still be used for this case and it may be considered that,
after stable circulation is established in a layer, the internal friction
would be different from that for the case of no circulation due to the
motion of the lower part of the upper layer.
An assumption of no net flow rate can be expressed as
;hl+h2dz=o
h2
Naturally, this assumption satisifies Eq. 157, since it says
>dz=0
--
dx h2
Subtracting [Eq. 157 x (1/gh, ) ] from Eq. 159 yields
I h.+h2 3-1 ,hl+h2 9
-t-/-1 — 2dz -- /hi — dz
h ^ x 2 h
1 7 _d(h1+h2)
*— [2-] - J—2-
»*
*sflrL-sbg°
If following definitions are used,
1
(165)
53
-------
1 h, +h~
and - " x i
3x
h,+h-
(166)
where $r is used for the friction slope caused by the circulation, Eq. 164
is now
d(h,+h7)
(1+B)— l_2-,SflrL-Sb+Sr
From Eqs. 158 and 160,
A dh, q.2
-------
Up/p)(l+B) fb h 2
where D~= — - -- — (— — )
(Ap/p)+B 8Sb
Xll
fir/fb
(h /H) 1+B .
l-Up/p)Srl (172)
and p~B+Up/P) Sb
It was not attempted to define the stratified normal depth for this
equation.
For the horizontal slope flow, the derivation of the interfacial profile
equation is done as follows. Sb = 0 is used in Eqs. 167 and 168 and the
same procedure as used for the sloped flow is followed.
dx8 l-(hc2/h2)
(173)
l-(h2/H)
WhCre EH^V8)<1+B> °74)
For the adverse slope flow, Eq. 170 for the positive slope flow can be
used.
The free surface profile equation can be obtained from Eqs. 161 and 167,
]
^
dx 1*B 8
If the free surface is found to remain horizontal in experiments, this
equation can indicate that the effect of B and S^ may be ignored. Both
B and S are expected to assume positive value and, if so, dH/dx
should be smaller than that for the case without considering the internal
circulation.
55
-------
Upper Layer Analysis
For this analysis, the equations of continuity and motion are as follows.
Equation of continuity for the upper layer:
lllulss
gh2 ° 3x x gh2 dx
A« dh.
where SflrU=8gh 3 (18°)
(181)
and
8gh12h2
Again, it is assumed that T^ can be expressed in terms of f. , q,, h-,,
and h~. The subscript r in f. indicates the effect of the circulation
as before. Subtracting Eqs. 177 multiplied by /(ghp) from Eq.
Aw £
179 gives
56
-------
3 7 h2
dz) (183)
Eq. 182 is
AD dh-, . .
(1— £) + -s+S+S=0 (184)
Also, from Eqs. 176 and 178,
q, ^ dh. dh, dh
(185)
The interfacial profile equation can be found, from Eqs. 180, 181, 184
and 185 and the definition of the stratified critical depth for the upper
layer, as
3
dhl (fir/8)(hcl/h1)3[l-(h1/H)(fbr/fir)]
•5T -- [l-^/h^lCl-^/H)]
Sr2
_
where B°"J (187)
For the free surface profile equation, Eqs. 176 and 186 are used to find
dH firap hcl 3l-(h1/H)+(hcl/h1)3(h1/H)(l-fbr/fir)
" b"~" 3
dx
3
-------
As can be seen, the forms of the interfacial and free surface profile
equations are not as simple as the previous ones because of the terms
arising from the internal circulation. However, the general forms of the
equations remain unchanged. Furthermore, it is learned where the effect
of the circulation appears in the equations.
5.5 INTERFACIAL PROFILE EQUATIONS FOR HYPOLIMNETIC FLOW IN A RECTANGULAR
EXPERIMENTAL FLUME
In this section, the change in the form of the interfacial profile equa-
tion is studied when the effect of the side walls of a rectangular flume
is considered. The width of the flume is designated by "b" and assumed
to be constant. The change does not occur in the equations of continuity
and motion, Eqs. 73, 74, 75 and 76, but in the definitions of the friction
slope, which are designated by S~.
T.-T+T (w./b)
-
sflv
tlv
(w2/b)
and 2
where T = wall shear stress
w
W, = wetted perimeter corresponding to the upper layer
Wp = wetted perimeter corresponding to the lower layer, except for
the bottom.
If it is assumed that
T =^U 2 (191)
the friction slopes, Sf, can be written as
58
-------
(192)
("3)
since Wy=2.h~ and T =0 can be assumed for the upper layer. By a similar
derivation shown in Sec. 5.1,
o
H
(Ap/p)gh23 <*x * 8(AP/p)gSbh23
] 094)
H-h2
The stratified normal depth for the lower layer including the wall fric-
tion, h_9.., is defined by
Note that the stratified critical depth has not changed. Also
(196)
l-(h2/H) ffab 1-(hn2v/H^ fbb
Then,
8Sy. 1— (h-5/H) fl»b
O f. O
Now the equation of the interfacial profile for the hypolimnetic flow can
be written in the form
59
-------
dx D l-(
If this equation is compared with Eq. 93, it is easily found that the
difference between the two is the term 2h2fw/(ft)b).
For the horizontal slope flow, GH, now becomes GHW defined by
,f.
l-(h2/H) fbb
And
dh2 f (
(199)
On the adverse slope,
dh-> , , l+(G,JHM-},tJ/h-j)
(201)
dx l-(hc2/h2)
r fh
where "v^C'^*^ (202)
The free surface profile equation remains the same regardless of the in
clusion of the wall friction.
Upper Layer Analysis
For this analysis, the friction slope may be stated as
v
flwu
and sf2ir-8 h 2h (2°4)
Then,
60
-------
dx 8(Ap/P)gh13h2
With the stratified critical depth (Eq. 107)
dh^ (f./8)(hcl/h )' 1 ^h.f (2Q6)
dx [l-d-^j/h-L)3] Ll-(h1/H) ftb
The free surface profile equation may be written as
dH_s f±*p (h^/h,)3 hcl3 2f^
dx b 8 P l-(hcl/h1):$L hj^^CH-h-L) qb
5.6 FREE SURFACE PROFILE EQUATION FOR OPEN-CHANNEL FLOW ON A HORIZONTAL
SLOPE IN A RECTANGULAR EXPERIMENTAL FLUME
Free Surface Profile Equation
For a channel with constant width, equations of continuity and motion for
steady state open-channel flow are as follows. (See Sec. 4.5).
Equation of continuity:
hU»q=constant (208)
Equation of motion:
UdU dh
-—+—+sf-sb=o (209)
g dx dx v '
Tb-T+(r W/b)
where Sf« p s - (210)
pgh
If it is assumed that
P ->
VgV (211)
-r =-f
v 8
61
-------
and Tg=o,
Eq. 210
q2[fbH-(2hfv/b)] (213)
where b = width of the flume and h is used for W.
If an equivalent friction factor for the bottom and the wall, f , can be
defined by
2h v
— )sf+_S: (214)
becomes
8gh3
The free surface profile equation can be derived easily from Eqs. 208
and 209 by eliminating U.
dh l-(Sf/Sb)
(216)
For the horizontal slope flow,
dh
The critical depth is defined by
(218)
Then, the free surface flow profile for the horizontal slope flow can be
expressed as
dh Sf
62
-------
where - —"— (220)
If the momentum coefficient in this discussion is included, Eq. 219 can
be written as
_ Sf (221)
"
with the same definition of Sf.
Relation Between Hydraulic Radius and Friction Factor
Recalling the assumption that, in open-channel flow, the uniform flow
forumula can be used for the gradually varied flow, some relations be
tween hydraulic radius and friction factor are derived for later use.
For uniform flow, in Eq. 209
»f = Sb
From the assumption in Eq. 215,
q2 8g bh
UZ=-== - Sy. -
h2 f q bb-«-2h
Since hydraulic radius, R, is
bh
R = - ,
b+2h
bR (224)
q
If it is assumed that this uniform formula can be applied to^the hydraulic
radius corresponding to the bottom, Rfe, and to the wall, RW, relations
rf-rVb
fb
63
-------
and
can be obtained.
Then, a relation
bh Rb
—
fa f_(b+2h) fK
*1 M O
may be obtained.
Furthermore, a relation
may be found if the definition of the hydraulic radius is considered
64
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SECTION VI
EXPERIMENTAL SYSTEM
Prior to the design of the experimental system, all requirements and
restructions were considered in order to construct it in an organized
manner.
6.1 CONSIDERATION OF PARAMETERS INVOLVED
As shown in Sec. 5.1, the positions of the interface relative to the
stratified critical and virtual stratified normal depths produce the
various interfacial profiles. The interfacial profile equations show
the parameters to be included in the design and construct an experimental
system for the verification of the proposed interfacial profiles.
The definitions of the stratified critical and virtual stratified normal
depths in the lower layer analysis and the stratified critical depth for
the upper layer analysis are listed below.
q2 -i1/3 (229)
-r
-L
(230)
1/3 (231)
The stratified critical depths (Eqs. 229 and 231) can be determined when
the flow rate of the flowing layer and density of the two layers are
known.
The parameters needed for the virtual stratified normal depth (Eq. 230)
are: the slope of the channel bottom, the total depth of the flow, the
65
-------
depth of one of the layers, the friction factors at the bottom and at the
interface.
The experimental system should be able to provide various flow rates,
densities of the water, and slopes of the channel bottom. Further, arti-
ficial roughness is necessary to provide bottom friction and to offer
somewhat easier determination of its magnitude.
6.2 REQUIREMENTS AND RESTRICTIONS FOR THE EXPERIMENTAL SYSTEM
As indicated in the previous section, the experimental system is designed
to handle the following parameters: slope of channel bottom, density of
water, depth of flow, flow rate, roughness of a channel, and end condi-
tions.
As pointed out in the introduction, most of the studies which have been
published on the two-layered flow were limited to the case of uniform
total depth in the flow, i.e., to the horizontal slope flow. In this
study, the flume can be tilted to give the desired slope to the flume
bottom.
In addition, the previous studies have used chemicals, such as salt, to
obtain density differences in the flow. This experiment uses the
difference in density produced by temperature differences.
The depth of one of the layers, or the position of the interface is deter-
mined by measuring the vertical temperature distribution in the flow.
The space available for all structures, piping and equipment was ap-
proximately 19.2 meters (63 feet) long, 3.05 meters (10 feet) wide and
3.36 meters (11 feet) high, including unusuable space which is round stair-
wells for two spiral staircases of approximately 1.83 meter (6 foot) dia-
meter. The system includes a flume with a tilting device, a system to
supply water of two different temperatures (or of different chemical
concentrations) at different flowrates, and equipment to measure the
temperature of water (or chemical concentration).
66
-------
Preliminary studies proved that the flume needed "inlet-outlet boxes" at
both ends where water enters and leaves the flume and a "supporting
frame" to support a load -- that is, water, the flume, and the boxes.
Therefore, the flume and its accompanying structure is called the "flume
structure." The system to supply water is called the "water supply
system," The equipment to measure the temperature is called the "tem-
perature measuring equipment."
Detailed requirements and restrictions for the experimental system are
listed in the following sections.
Flume Structure
1. The maximum gross space available for the flume structure is approxi-
mately 12.2 meters (40 feet) long, 1.22 meters (4 feet) wide, and 3.36
meters (11 feet) high. This height includes the clearance required
when one end of the flume is tilted up.
2. The walls of the flume are transparent so that the state of flow can
be observed from the sides of the flume.
3. The inside of the flume is as smooth as possible.
4, The flume is made in sections to provide control weirs, side inlet
and outlet devices, and changes of geometry, when necessary.
5. Two inlet-outlet boxes with the depth deeper and the width wider than
the flume are placed at both ends of the flume to give the water
supply system a chance to stabilize before flowing into the flume,
and to provide a drain.
6. The deflection of the supporting frame is as small as possible.
The deflections of the frame at the center and both ends are approxi-
mately equal.
7. The supporting frame is made in sections for ease of construction and
transport.
8, The supporting frame is able to support the flume with a width of
approximately 0.914 meters (3 feet).
9. A "tilting device" is able to give a slope of approximately 1/10 to
the flume bottom.
67
-------
10. Bottom roughness provides friction much larger than that found in
nature since the flow rate obtainable in the flume is limited. The
bottom roughness is removable from the flume.
11. A gate can be installed at several points along the flume, and its
opening is adjustable,
Temperature Measuring jquipment
1. The temperature of the water in the system supplying the flume and
the air temperature is recorded in order to determine if the flow in
the flume has reached a nearly-steady state.
2. The temperature of the water in the flume is measurable vertically
and longitudinally to find the temperature distribution profile of
the flow.
6.3 DESIGN AND CONSTRUCTION OF THE EXPERIMENTAL SYSTEM
The experimental system is designed and constructed to fulfill the re-
quirements and restrictions are stated in Sec. 6.2.
A detailed sketch of the flume is shown in Figure 8. A schematic dia-
gram of the water supply system is shown in Figure 9. Further details of
o
the system are given in Yanagida.
68
-------
-T-1.22m-r-1.22m-u—
cr>
approx. I2.26m<40.15')-
U
o
FFIHFEJ
inr
View A-A
Viev/ B-B
View C-C
Fig. 8
Flurne structure.
-------
10
Cool water
sub-system
HT"
!l-''*T~
Jl*
ii_Water ;:
chiller
^U*
. is/n';otT
Warm water 4
sub-system
1
Temperature subsystem
City
service
«i water
p r ? T
i
— /
Cool water /
tank
"hi
^
W
1/4" i-i»|#
Water j
heater
Circulation subsystem
Warm water
tank
Legend
Material:
-**- PVC
CPVC
•*>*- copper or
other metal
— — reinforced
rubber hose
Pipes are 3", unless
otherwise stated.
- V'O.D. clear
plastic glass
tube
Valves have screwed
joints, except for
flanged
solder-jointed
Fittingsj
s*a ball valve
A diaphragm valve
M butterfly valve
step control
* butterfly valve
infinite control
* automatic
control
valve
ME- check valve
/ angle valve
» strainer
» reducer
g iVrotometer
•- pump
\ mixer
I to drain
Fig.
Water supply system flow diagram.
70
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SECTION VII
EXPERIMENT AND DATA
7.1 EXPERIMENT
Experimental Plan
The principal object of this experiment is to generate as many of the pre-
dicted interfacial profiles as possible in the experimental flume.
Prior to the experiment, an experimental plan was formulated out as fol-
lows, assuming that each profile could be generated if the proper values
were given to the parameters included in the interfacial profile equat-
ions.
For each profile, a set of parameters was found to satisfy the requirement
that it could be observed within the length of the flume. The parameters
are the slope of the flume bottom, the flowrate of moving layers, the
temperature of the two layers to determine their density, the total depth
of the flow, the depth of one of the layers, and the interfacial and
bottom friction factors.
The selection of the parameters was done through trial and error by num-
erically integrating the proposed interfacial profile equations with
the free surface profile equations.* The temperature of the two layers
was chosen so that the difference in the average temperature of the two
layers was at least 3.3°C (6°F).6 The bottom friction factor was esti-
mated by the formula suggested by Ghosh and Roy. For the interfacial
friction factor, the value calculated from the friction factor for the
smooth channel of open-channel flow was substituted after Polk et al.
* The computation used the 4-th order Milne predictor-corrector method.
Since the same program is used in Sec. 7.2, see Appendix C.
71
-------
Care was taken to give sufficient flowrate to the moving layer so that
its flow would be in the turbulent region if it was a non-stratified
flow. The effect of side walls was ignored at that time.
However, the following observations were obtained from the preliminary
experiment for the lower layer analysis.
1. An attempt to determine the depth of the lower layer with an under-
flow gate resulted in having the effect of changing the SI or S2
profiles into the S3 profile. The gate seemed to be helpful to pro-
duce the S3, M3, and H2 profiles.
2. The SI, S2, Ml, M2, and HI profiles could be generated without the
gate mainly by changing the flowrate and the slope.
Thereafter, the experiment was conducted by trial and error, using the
obtained parameters as a starting point. The experiment was conducted
only for the important cases of the lower layer analysis.
Experimental Procedure
After testing several different methods of introducing the water into the
flume, most of the experiments were conducted by the following procedure.
The term "cool water" is used for the lower layer water; "warm water" is
used for the upper layer.
Prior to the experiment, the connection between the flume and the water
supply system was established by means of reinforced hoses. For the S,
M, and H profiles, the discharging line of the cool water system was
connected to the upstream end of the flume. Its return line started
from the downstream end of the flume. The ends of both the discharging
and the return lines of the warm water supply are placed in the upper
part of the downstream inlet-outlet box. They are equipped with baffle
boards to prevent the force of the flow from mixing the two layers.
Preparation -
The flume is tilted to the desired slope. The flume must be raised to
an elevation slightly higher than the elevation desired in the experiment,
since the weight of water lowers the flume slightly.
72
-------
The ends of the pipes of the warm water circulation subsystem are placed
at the desired position in the inlet-outlet box,
The desired total depth is marked OM the side wall of the flume..
The main switches of the electricity for the pumps are on; the thermister
thermometers are warming up. Each valve should be examined if it is open
or closed correctly before each specific operation.
The main valve of the service water is opened.
Making the Cool Water -
The cool water tank is filled with the service water. To measure the
temperature of-this water, it is circulated in the temperature sub-
system through the short cut line by the pump of the water chiller.
To do this, the temperature control of the chiller is set so high that it
does not chill the water, while the setting of the cool water control
valve is adjusted low to open the valve fully. Desired temperature is a
few degrees [1.1 to 1.7°C (2 to 3°F)] lower than that to be used in the
experiment. If its temperature is close to the desired one, the circula-
tion is stopped and the water in the tank is fed into the flume through
the by-pass line in the rotometer manifold. Initially, the discharge
pump has to be run with the valve in the by-pass line being half-closed
for a very short period to expel the air in the line from the air bleeder.
After that, the water can be fed to the flume by gravity, if the water
level in the tank is maintained higher than the level in the flume by
supplying the service water into the tank. Of course, the drain valves
have to be closed. However, both valves were kept open until some
water was discarded from the drains to clean the flume.
If the temperature of the service water is higher than that desired, the
water chillder has to fae used. If it does not require too much cooling,
the water can be fed continuously to the flume while cooling it by the
circulation through the short cut line. However, when considerable cool-
ing is necessary, a batch method has to be used; that is, one tankful of
water is cooled down through circulation and then fed to the flume. This
procedure is repeated until the water reaches the desired level in the
-------
flume. The desired level is a little higher than the total depth to be
used in the experiment, since it is easier to adjust the water level by
reducing the water level than by increasing it.
If the service water temperature is lower than desired, the warm water
temperature subsystem must warm the water. In a similar manner, the ser-
vice water put into the warm water tank is warmed by steam while circula-
ting through the short cut line. If the desired temperature is reached,
this warmed water has to be sent to the cool water tank before being
introduced to the flume. This inconvenience arises because the end of
the warm water discharge line is in the air with the baffle board which
splashes water until it is submerged in the water. Both batch and con-
tinuous feeding to the flume can be used depending on the degree of
warming.
Making the Warm Water -
The water for the upper layer should have a temperature 1.7 to 2.2°C
(3 to 4 degrees F) higher than that we desire to have in the flume during
the experiment. As described previously, the service water is warmed by
steam injected at the in-line type heater, while circulated by the dis-
charge pump through the short cut line. After the desired temperature is
obtained, the value of one of the rotometers is opened to introduce the
water from the end of the discharge pipe submerged in the cool water.
This introduction of the warm water into the flume is done very slowly.
At the same time, the drain valve at the upstream end of the flume is
slightly opened to discard the cool water already in the flume. The cool
water is removed at about the same flow rate as the introduction of the
warm water to maintain a static water level in the flume. This process
has to be repeated if the flume is not filled to the desired level by one
tankful of water.
The preparation of water for the two layers takes 1-1/2 to 2 hours.
74
-------
Before Running the Water -
The warm water, with the temperature a Tittle higher [0.6 to 1.1 °C
(1 to 2°F)] than we desire to have in the flume, is made in the warm
water tank. All the valves in the circulation subsystems are open and
all air bleeder cocks are open to expel the air. Then the water level
in the flume is brought down to the marked total depth by discarding the
cool water from the bottom drain of the flume.
Thermister thermometers have to be calibrated, if necessary. The camera
should be ready. The slope of the flume is recorded.
Running the Water -
The mixer in the warm water is started. The main steam valve is opened.
The two pumps in the warm water subsystem are started. First, both flow
rates are brought to 0.000631 to 0.000758 m3/sec (10 to 12 GPM) for a
short while; then, they are reduced to around 0.000316 m3/sec (5 GPM).
After these flow rates become relatively constant, the two pumps 1n the
•cool water subsystem come into action. The mixer in the cool water tank
is on and the water chiller must start to work with the appropriate set-
ting of the cool water control valve. The flow rate of the cool water
should be brought gradually to the desired rate. The temperature probe
bar is submerged in the upper layer to measure the temperature at a cer-
tain point of the layer. This probe and two probes in the warm and cool
water supply are connected to the auto scan type thermometer.
The flow rate of the 4 pumps usually fluctuates and changes considerably
in the beginning of their operation. These have to be adjusted by watch-
ing the level of the water in the flume and in both tanks. Adjustment
of the flow rate has to be made until these water levels can be con-
sidered to be nearly constant.
Also, the temperature of the discharging water has to be adjusted. If the
warm water tends to lose heat, steam has to be injected. If the warm
water temperature rises, the cool water has to be brought from the cool
water tank. In this case, approximately the same amount of the warm
water has to be returned to the cool water tank to maintain the water
level. The cool water temperature was fairly well maintained by the cool
75
-------
water control valve.
If the temperature at the three points becomes almost constant, then
dye (methyl green) is put into the cool water tank. After dye spread
through the flume and reaches the return line (dye can be seen through
the glass of the rotometer), photographs are taken,
The air temperature probe is connected to the scan type thermometer in
place of the probe which has been measuring the temperature at a point
of the upper layer. Now, the probes in the probe bar are connected to
the thermometer with a rotary switch. It took about 1 to 1-1/2 hours
to achieve a nearly steady state.
Measuring the Temperature of the Flow in the Flume -
The vertical distribution of the temperature in the flume was measured at
the middle of each section of the flume during most of the experiment.
The probe bar is fixed to the holder so that the top probe in the bar is
slightly below the free surface. After several minutes, the probes reach
the temperature of their surrounding, and the recorder is set to record-
ing position and the rotary switch is turned at specific intervals to
profuce the record of the vertical temperature distribution. Since only
11 probes are in the bar, the probe bar has to be lowered to measure the
lower part of the flow by using a guide mark put on the bar. If the
probe bar is lowered, the temperature is recorded after the transient
period. This procedure is repeated, if necessary. Meanwhile, the visual
observations of the state of the flow are made.
The flow rate and water level have to be checked and adjusted, if nec-
essary, before going to the next measuring point. These measurements
take about 1 hour.
Stopping the Run -
The main steam valve is closed. The mixer in the warm water tank is stop-
ped. The two pumps in the warm water subsystem are stopped. The water
chiller is shut. The mixer in the cool water tank is stopped. The main
valve for the service water is closed. The flow rate of the two pumps in
the cool water subsystem is lowered gradually to a small rate and the
76
-------
pumps are shut. The water in the flume is drained slowly from the two
drains. The water in both tanks is drained. The electric source is shut.
Deficiencies and Inconveniences of the System
Although the experimental system works well in general, a few deficiencies
and inconveniences were noticed during the experiment. They are listed
below.
1. The supporting frame is slightly twisted toward the upstream end to
produce a slope of approximately 3.18 to 6.35 mm (1/8" to 1/4") per
0.305 meter (1 foot) in the direction of the width of the flume.
This was found by observing that only one of the two jacks started
to rise about 6.4 mm (1/4") and then both jacks went up with the
same speed when the frame was lifted from the horizontal position.
This situation may be due to inaccuracy in the construction of the
yoke-shaped upstream end's support or unexpected force exerted by
aluminum welding. A few attempts were made to correct the problem,
but they were of little help. The fact that the bottom of the flume
at the upstream end is not horizontal in the direction of the width
is an inconvenience when reading the depth and determining the slope
of the flume. However, the slope is not so large that it does not
influence the results of the experiment.
2. The automatic temperature control valve for the warm water (steam re-
gulating valve) is not reliable. This seems to be due to overdesign
for small flow rate of warm water circulation occurring at the experi-
ment corresponding to the lower layer analysis. One correctional
measure was taken, in which the flow rate through the heater is in-
creased by returning a portion of discharging water through the heat-
er into the warm water tank, and it did help a little. However,
manual control of the steam was used for most of the experiment.
3. The flowmeters, which must be checked quite often during an experi-
ment, are located far apart between the warm and cool water subsys-
tem because the limitations of space and the requirements of the
piping.
77
-------
4. The roughness plates made of plastic glass sheet warped in the
water. They had to be attached to the flume bottom with adhesive
to prevent them from disturbing the flow.
5. When the water in one subsystem was senttothe other subsystem
through the 1-1/2" pipe to maintain the temperature of the water
in the tank, the adjustment of its flow rate was quite difficult
and took time to obtain constancy. This is because these lines are
not provided with flow rate measuring devices.
6. The gate can be installed only at limited points of the flume be-
cause the gate hits lighting fixtures or a concrete girder on the
ceiling when the flume is tilted.
Accuracy of Parameters
This is a rather crude discussion of the accuracy of the parameters mea-
sured in the experiment.
1. Slope
The slope is measured by finding the elevation from the horizontal of a
point on the flume located at a distance of 7.63 mm (25 feet) from the
center of the pivot toward the upstream end.
The point on the flume is 1.22 m (4 feet) above the level of the center
of the pivot. This is because of the height of a leveling instrument.
These relations are shown schematically in Fig. 10.
A point on
the flume
The point after tilting
The center of
the pivot
Fig. 10. Definition sketch for the tilted frame.
78
-------
The slope is obtained by E/L [L = 7.63 mm (25 feet)]. If trigonometry
is used, we easily find E/L = sin9-(l-cos9)tany. TanG is at most 1/20
and tany = 4/25 in this experiment. Since the slope is equal to tanQ
by its definition, the slope obtained by E/L generates an error of
-0.0003 at the maximum.
Other errors come from the measurement of L and the reading of E. Also,
as mentioned in the deficiencies of the system, the bottom of the flume,
where the E is measured, is not horizontal in the direction of the width.
Although the averaged bottom elevation is adjusted to the horizontal, it
certainly introduced more error. These errors could amount to ±0.5%.
2. Flow Rate
The flow rate measured by a rotometer was examined by leading the water
to a weighing tank after going through the rotometer while measuring
the time. Errors were found to be at most ±1% of the flow rate for flow
rates greater than 0.00378 m3/sec (60 6PM). For 0.00252 m3/sec, it is
about ±2.5% of the flow rate. For 0.00158 m3/sec (25 GPM), it could be
approximately ±4.5% of the flow rate.
3. Temperature
The manufacturer of the thermister probes states that ±1% of 50°C is the
error for this temperature reading. The recorder used for the tempera-
ture in the flume has .iG.5% error. Therefore, the accuracy of the tem-
perature measured by the instrument is ±1.5% of 50°C. Furthermore, water
temperature in the flume could not be maintained constant during an
experiment. Although data to be used in later analysis were chosen by
the standard of less variation in the temperature in the flume, the
temperature averaged in the lower layer may have an error of ±0.2°C
around 18.0°C (64.4°F). Therefore, the error in obtaining temperature
could amount to ±1°C at the most. This could cause a maximum error of
±0.027% in density and of ±55% in the value of Ap/p.
If the errors of the flow rate and density difference are used, the error
in the calculated stratified critical depth can be estimated. These
errors are included in the tables shown in Sec. 7.3.
79
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4. Depth
The fluctuation of the total depth during an experiment was maintained
within ±0.79mm (1/32 inch). However, the total depth was measured at
two points close to the ends of the flume after the free surface was
found to be practically horizontal; the total depth at the other points
and the lower layer depth were obtained from the temperature distribution
curve on which the total depth obtained at the two points were plotted.
This may introduce ±2.54 mm (0.1 inch) of error. Therefore, it is
reasonable to consider that the depth used in data analysis has approxi-
mately ±3.18 mm (1/8") of error.
5. Width
The width of the flume could be 304.8 ± 1.59 millimeters (12 ± 1/16 in-
ches). It is difficult to examine its width close to the bottom since
its depth is 0.914 meters (3 feet). If the above estimate is used, the
error may be ±0.52%.
General Observations
The observations made through the entire experiment are listed in the
following. Specific observations on each interfacial profile are
described in Sec. 7.3.
1. The two layers of different temperatures c~;. ..
-------
appears rather quickly.
5. The change in the flow rate of the warm water does not change the
position of the interface if the flow rate remains within approxi-
mately 1/5 of the flow rate of the lower layer. The flow rate of
about 0.000316 m3/sec (5 GPM) was used for the upper layer circula-
tion in most of the experiments.
6. The position of the ends of the discharging and return lines of the
warir, water supply does not influence the position of the interface
unless they are located close to the interface or in the lower
layer.
7. Changes in the total depth of the flow and in the temperature of the
water influence the position of the interface.
8. The interface is almost straight or slightly dipped at the center in
the direction of the width of the flow in the experimental flume.
9. When the gate is used, it introduces more disturbance just behind
the gate. However, the temperature of the layers are maintained
better than in the cases without the gate.
10. When the head of the warm water layer is seen in the flume, it is
very difficult to determine where the stable two layers start.
7.2 DATA ANALYSIS
Data used in the analysis were selected mainly by examining how well the
temperature of the water in both tanks was maintained during the experi-
ment. Then, the location of the interface was determined by connecting
inflection points on vertical temperature distribution curves. The
temperature of the two layers was calculated from the area between the
temperature distribution curve and the vertical line through the in-
flection point. Water density in the two layers was found from the
table of density of water; then,
-------
i. Although an open-channel flow experiment was conducted to determine
the bottom friction factor for this experimental flume, only the
value for very shallow depth could be obtained, since only a fairly
crude method was available. Therefore, a method to estimate the
values of bottom friction factor for deeper depth must be found.
2. A good method to estimate the value of the interfacial friction fac-
tor is not available. Thus, it must be estimated.
3. It seemed that most of the stratified critical depth calculated from
its definition (Eq. 30) was lower than the data indicated.
Some methods to cope with these problems were implemented and are des-
cribed later; however, the outline of methods is given in the following,
since these problems are related to each other.
Eqs. 198 and 200 are used as the interfacial profile equations and re-
written for the experimental flume (b=l) as follows.
For the positive slope flow:
dho 3-"^GvJln2w//^12^ (232)
—5, •
dx b l-(
, V(fb*2h2fv) -nl/3
L
8Sb l-(h2/H)
(233)
82
-------
where f = equivalent friction factor.
For the horizontal slope flow:
where fbGHv ^b* *2h2fv
cbc 8 l-(hc2/h2)3
fi
^ /H)
(234)
l-(h2/H)
(as,
If the relation between the equivalent friction factor, bottom friction
"factor and wall friction factor obtained for open-channel flow can be
used for the two-layer flow, and the term ffa is used for f (l+2h2), then
Eqs. 232 and 234 become the same forms as Eqs. 93 and 97.
Therefore, if the value of f from the open-channel experiment can be
obtained, the estimated value of the stratified critical depth and the
interfacial friction factor can be found by the following trial and
error method.
1. Integrate numerically the interfacial profile equations (eqs. 93
and 97) with the free surface profile equations (Eq. 101), using
a set of the values of the stratified critical depth and the inter-
facial friction factor,
2. Compare the interface thus obtained with the interface obtained by
the experiment.
3. If the two interfaces are quite different, choose another set of the
values of the stratified critical depth and the interfacial friction
factor and integrate.
4. This is repeated until the two interfaces reasonable match.
83
-------
Bottom Friction Factor
An attempt was made to find the value of the bottom friction factor for
the experimental flume using Eq. 219 with Eq. 220, that is,
dh f.JH-Zh) (lv/h)3
(236)
When open-channel flow was run on the horizontal slope in the experi-
mental flume, it was observed that the downstream depth was slightly
lower than the upstream depth. Therefore, this can be considered as an
H2 profile according to Chow.3 (This is similar to an HI profile in
this study).
Unfortunately, the flow rate designed for the two-layered flow is very
small for open-channel flow in the experimental flume and better equip-
ment to measure the depth is not available. Fairly reliable data was
obtained only from the measurement at approximately 102 millimeters (4
inches) of depth for three different flow rates. After compensating for
the irregularity of the channel bottom, the free surface profile was
obtained for this flow. Numerically integrating Eq. 236 with different
equivalent friction factors gave the calculated free surface profile.
(The Milne predictor-corrector method is used to calculate the profile.
See Appendix A.) Comparison between this and the profile obtained from
the experiment allows to estimate the equivalent friction factor for a
102 millimeter (4 inch) depth of flow as approximately 0.135. Further-
more, the effect of the momentum coefficient on this result was exam-
ined. (See Eq. 221). Whe 3=1.2 was assumed, it did not change the
above value. B=1.2 could be the largest value occurring in the straight
experimental flume.
However, equivalent friction factors are necessary for greater depths
of the flow. Therefore, the following method was developed to estimate
the equivalent friction factors.
According to Ghosh and Roy, the friction factor for roughwalled
channel can be expressed as
84
-------
(23.7)
rff
where f = friction factor for roughwalled channel
R = hydraulic radius
r = height of roughness element
s = width of roughness element
1 = longitudinal spacing of roughness element
m = transverse spacing of roughness element
C = a constant to be determined
Also,a is defined for a rectangular channel as
)-£, if h<| (238)
where h = depth of flow
b = width of the channel
For the experimental flume, Eq. 2 can be written as
<240>
where f^ = bottom friction factor
Ru = bottom hydraulic radius
K = C-2.031og [r (rs/lm)].
Since b=l ,
a=ln(l+2h)-h, if h<0.5 (241)
=ln[l+(0.5/h)]-0.5, if h>0.5 (242)
Now, if the relations derived in Sec. 5.6 are used (Eqs. 214, 227, and
228),
85
-------
Rb*h(l-2Rw) (243)
fb.fq(l+2h)-2hfw
and therefore fb-fq
-------
This equation can be solved by an approximate method* to obtain f (l+2h)
as a function of q, v, and h for each experiment. At the same time, Re..
W
can be calculated and examined if it is between 750 and 25,000. q and v
are considered to be constant for an experiment. The average hg is
used for h. One value of f (l+2h) was chosen for each experiment byex-
amining the average depth for that'experiment. The values of f (l+2h)
(=f^+2hfw) used in the data analysis together with the values of the
interfacial friction factor used in the analysis, are included in the
tables comparing data with the methods estimating the stratified critical
depth shown in Sec. 7.3.
Interfacial Friction Factor
Few studies have been conducted on interfacial friction factor and no one
has offered a definite method to estimate its value.13' 14' 15> 16> 17
Therefore, an attempt was made to estimate the magnitude of the inter-
facial friction factor from the obtained data by numerically computing
the interfacial profiles with different values of the interfacial friction
factor and comparing these calculated profiles with the data. As a basis
for estimating value of the interfacial friction factor, the value cal-
culated from the Blasius equation with the Reynolds number defined by the
hydraulic radius, 0.5/(l+h), is used, namely,
fi=0.223/Rei°-25 (251)
*An iterative method to solve this equation, though somewhat primitive,
was developed by the author. The computer program is shown in the
Appendix B.
+The values of v are obtained from the values of viscosity found in
Reference 11.
Jin this hydraulic radius, 2(l+h) is used for the wetted perimeter to re-
flect the effect of the lower layer running under the upper layer
surrounded by the walls and the bottom.
87
-------
__ ___
where Reis~^ 2"l+h 2v"(i+h) (252)
Unfortunately, as stated in the beginning of Sec, 7.2, the stratified
critical depth has to be estimated as well. Therefore, an accurate es-
timation of the interfacial friction factor cannot be expected.
As shown in the next section, two methods to estimate the stratified cri-
tical depth and thus two different values of the interfacial friction
factor can be obtained.
Since the calculation of the interfacial friction factor by Eq. 251 uses
the values of q/v as the equivalent friction factor, the calculation of
the interfacial friction factor is included in the computer program cal-
culating the equivalent friction factor. (See the Appendix B.)
Stratified Critical Depth
As stated at the beginning of Sec. 7.2, the indication was that the stra-
tified critical depth calculated by its definition (Eq. 90) is smaller
than the data suggests for most of the experiments conducted. The vari-
ables governing this critical depth are the flow rate, the density dif-
ference, and the momentum coefficient, if it is included in the critical
depth (Eq. 113). Errors in the flow rate and the density difference are
mentioned in Sec. 7.1. If the stratified critical depth indicated by the
data exceeds the value including the error, then the definition of the
interface and the method of obtaining the temperature of the two layers
may have to be reconsidered.
When the word interface appears in the derivation of the interfacial
profile equation, that interface must lie between "momentum layers"*
(Carstens18). However, the interface found from this experiment may be
called the interface between "temperature layers." In fact, it was
*The momentum layers are the layers with alternative flow direction, that
is, a layer is referred to the one between 0 points in the velocity
distribution.
88
-------
assumed that these two interfaces were very close to each other. Accord-
18
ing to Carstens , they do not necessarily coincide with each other.
An attempt to see the coincidence between two interfaces was not made
with equipment measuring velocity. However, visual observations of
the movement of small bubbles in the flow suggested that these inter-
faces were practically identical, as far as the experiment in this study
is concerned.
As for the averaging of temperature in the layers, no other method seems
to be more justified.
Two methods of finding the value of the stratified critical depth have
been attempted and the results are shown in Sec. 7,3. Both methods use
the profiles obtained in the experiment to estimate both the stratified
critical depth and the interfacial friction factor. The interfacial and
free surface profile equations in non-dimensional form are used for
the computation. (See Sec. 5.3 for the equations.) The following para-
meters are substituted into these equations: (1) estimated stratified
critical depth, (2) estimated interfacial friction factir, (3) bottom
friction factor obtained by the method described in the bottom friction
factor, (4) lower layer depth at the point in the flume where the two
layers are considered to be fully established in the profile obtained by
the experiment, (5) total depth at that point, and (6) density difference.
Estimated stratified critical depth is expressed in the form of the stra-
tified critical depth calculated by the definition multiplied by some
constant. Similarly, estimated interfacial friction factor takes the form
of the value obtained by the method shown earlier multiplied by a con-
stant. Numerically integrated interfacial profiles are compared with the
profiles obtained in the experiment to find the combination of the stra-
tified critical depth and the interfacial friction factor which produce a
calculated interfacial profile best fit to the profile in the data. (See
the Appendix C for the computer program.)
In the first method (method 1), it is assumed that the density difference
could be different than the one obtained by the method described at the
89
-------
beginning of Sec. 7.2. Therefore, density difference is expressed by
where h, * multiple for the stratified critical depth. It gives
hcs»hc'ohc
(253)
(254)
where Qhc a the stratified critical depth calculated by its definition.
In the second method (method 2), we assume that the ignored momentum co-
efficient, e, is the factor responsible for the smaller stratified cri-
tical depth. For this method,
»v*1/3 (255)
The result is shown 1n Sec. 7.3.
7.3 RESULTS OF THE EXPERIMENT AND THE DATA ANALYSIS
The Number of the Experiments Conducted
Some preliminary experiments were not directed to produce specific pro-
files. They are classified into certain profiles from observation. Ex-
periments done without the service of the cool water control valve be-
cause of Its malfunction are included in the total number of experiments.
Table 1
NUMBER OF EXPERIMENTS CONDUCTED
Profile
SI
S2
S3
HI
M2
M3
HI
H2
Total
Number
4
19
9
6
4
2
2
1
47
Number used for
1
3
4
2
3
2
2
1
analysis
18
90
-------
Experimental Data and Its Analysis
Data and their analyses for different profiles are presented in the fol-
lowing order: SI, S2, S3, Ml, M2, M3, HI, and H2 profiles. In every
profile classification, each experiment used for the analysis is present-
ed by:
1. Observation of the experiment for the profile;
2. An experimental photograph and a figure of vertical temperature dis-
tribution curves at several points along the flume;
[Each figure, table, and computer print-out is marked by a profile
identification, as Sl-3/30. "SI" is profile designation described
in Fig. 7. "3/30" is the date of the experiment (in 1973).
The number seen in photos is Expt. No. It does not agree with the
number of the experiments actually conducted: this number was intro-
duced to identify the photos after a part of the preliminary experi-
ment was conducted. The photos are not clear since they include the
reflection of items located on the other side of the room and the
shadow of the railing for the probe bar.
The figure includes an interfacial line obtained by connecting in-
flection points of the temperature distribution curve, and virtual
stratified normal depth line and stratified critical depth line ob-
tained by the method 1. These lines are shown only for the portion
used for the analysis. The longitudinal distance in the figures of
the vertical temperature distribution is shorter than that in the
photos. Each flume section is 1.22 meters (4 feet) long. There-
fore, only the 7.32 meter (24 foot) length of the flume is shown in
the figures. Further, depth is very much exaggerated compared to the
longitudinal distance in the figures],
3. A table of experimental data and results of the methods 1 and 2 for
comparison between them;
[Lower layer depth and total depth are obtained from temperature dis-
tribution curves. The results of the method 1 and 2 are chosen from
the print-outs of interfacial profile computation which follow this
table.]
91
-------
4. Computer printouts of interfacial profile for the method 1 and 2.
[A printout expressed 1n inches for depth and in feet for distance
along the flow precedes a printout expressed in a non-dimensional
form. The computer program shown in Appendix C gives printouts
showing both dimensional and non-dimensional length on one sheet.
They are, however, shown In separate pages as described, obtained by
modifying the program slightly.]
Explanation of symbols and legend used in these figures and table follows.
Fig. 11 Definition Sketch for Temperature
Distribution Curve
[1] • free surface;
[2] = Interface obtained by connecting inflection points in the tempera-
ture distribution curve;
[3] » bottom;
92
-------
[4] = stratified critical depth line calculated by the method
1. The stratified critical depth line from method 2 is not shown
to ensure simplicity;
[5] = virtual stratified normal depth line calculated by the method 1;
[6] = temperature of water measured at 25.4 millimeter (one inch) in-
tervals of the depth at the center of each flume section except
when the gate is used. Since the gate was Inserted at the center
of flume section 3, the measurement was taken 76.2 mm (3") be-
hind the gate for the section. Measurement was also taken at at
least one point between sections 3 and 4, but their curves are
not shown in the figure because of the limited space;
[7] = to show depth from the bottom for reference;
[8] = point of the flume where temperature measurement was taken;
[9] = starting point of the calculation;
[10] = gate;
[11] = gate opening;
[12] = highest water temperature in a curve, in °C (in °F);
[13] = lowest water temperature in a curve, in °C (in °F); and
[14] = name of a flume section, which is at the center of the section in
the figure.
Symbols are listed alphabetically. Parentheses and brackets after ex-
planation are for the tables; the values in parentheses are in British
units; the values in brackets are non-dimensional.
Symbols expressed in capital letters (usually shown in parentheses) are
for computer printouts. If "I" is found in these symbols, they are in
expressed in inches.
3 (=BETA)= momentum coefficient obtained by method 2;
DTDHI = difference between total depth and horizontal drawn through the
free surface at the starting point, in Inches;
DX = integration interval, 1n feet;
dh2/dx(=HP) = interfacial slope (dh2/dxN);
93
-------
dh2/dxN(sHpN) - non-dimensionalized dh /dx
=dh2/(dxSb) for positive slope flow (dh2/dxN=
dh2/dx for horizontal slope flow);
AP/p(«DELTA) - 0(Ap/p)/mhc3 1n method 1;
fb(=FB) = bottom friction factor estimated by the method in Sec. 7.2;
fi (=fl) a interfacial friction factor obtained by method 1 or
2 - ^Yo^* 1" which fj is substituted by trial and error;
Ghn2(=GHNI) = virtual stratified normal depth found by method 1 or 2,
in meters (in inches) [non-dimensional];
GhB2N(=GHNN) = non-dimensionalized Ghn2 = Gnn2/nc2;
H(=TDI) = total depth, in meters (in inches) [non-dimensional;: if there
are two separated by a comma, the former is for method 1, and the lat-
ter for method 2];
HN(=TDN) = non-dimensionalized H = H/hc2;
HP * see dh2/dx;
HPN = see dh2/dxN;
h2(=HI) = lower layer dpeth, in meters (in inches) [non-dimensional;
the former is for method 1 and the latter for method 2];
h2^(=HN) = non-dimensionalized hp=h2/h 2;
hc2(=HC2I) = stratified critical depth obtained by method 1 or 2, 1n
meters (in inches) = mhc*0hC2» 1%n which ^ is substituted by trial
and error;
mft(=MFI)= see f^
mhc(=MHc) . see hc2;
0Ap/p(=ODELTA) = relative density difference, or the ratio of density
difference between two layers to mean density of two layers, calcula-
ted from I, and T2;
Qf^ = interfacial friction factor estimated in Sec. 7.2;
Ohc2 (-OHC2I) = stratified critical depth calculated by its definition
hc23=q2/[(0Ap/p)g], in meters (in inches), j$=est1mated maximum error;
q(=Q) ~ flow rate, in m3/sec (in cfs);
S(j(sSB) = slope of channel bottom;
TQ = average air temperature during the experiment, in °C (in °F);
T-|("T1) ~ averaged temperature in the upper layer, in °C (in °F);
94
-------
T?(=T2) - averaged temperature in the lower layer, in °C (in °F);
Ts = average temperature of water to be supplied into the flume, in
oc (in °F), subscript 1 for the upper layer and 2 for the lower layer;
TDI = see H;
TON = see HN;
x(=X) = distance along the flow from the starting point of the computa-
tion in meters (in feet) [non-dimensional for method 1, XN for method
2]; and
xN(=XN) = non-dimensionalized x
= S.x/h ~ for positive slope flow
= x/h 2 for horizontal slope flow.
95
-------
•SI profile
The profiles designated as SI were produced with the slope considered
to be steep and the flow rate low. The observation, the photograph,
and the temperature distribution of both SI-3/30 shown here and the
other three experiments have the following general tendencies.
The lower layer begins very markedly, but gradually becomes ambiguous
and deeper toward downstream, and disappears. Meanwhile, another inter-
face which is definitely situated above the original interface and al-
most horizontal appears, gradually becomes clear, and finally becomes the
only interface at the end of the flume.
The latter interface may have the characteristics predicted for the SI
profile. If this assumption is correct, the internal jump, which was
expected to occur from the analogy to open-channel flow, is replaced by
the gradual change in the depth of the lower layer for this two-layered
flow.
96
-------
Sl-3/30
HlllM
Fig. 12.
Experiment Sl-3/30.
•23.2
(73. i
ma
23.;
(73
23.
C (73.9
.8°F) '
=17.4°C
63.3°F)
Sb= 0.0408
457m
(18")
17.16
63.7)
•23.;
C73J
.305n
(12"]
.152n
(6-)
23.2
(73.6
Interf^e
•(Computeid
7.6' Computed
63 .'7)
)
.'23.(2
(-73.
Ghn2
17,6
63.7)
17
.6
63.7)
Fig. 13. Temperature distribution curve for Sl-3/30.
97
-------
Table 2. EXPERIMENTAL DATA AND RESULTS
OF METHODS 1 AND 2 FOR Sl-3/30.
Sl-3/30
S»
Expt. NO. 14
.0408
q*.00158(.0556)
TS2-16.4(61.5)
T2»18.4(65.1)
fb/(8Sb)*'833
Ohc2«.145(5.69)±28%
Tsls24.3(75.7)
^=22.7(72.9)
fb».272
T0*20.0(68.0)
^dp/ps.000912
X
h2
H
0.00
.437(17.2)
[2.77,2.93]
. 592(23. 3)^
[3.75,3.96]
1.22(4.00)
[.316, .334]
.488(19.2)
[3.10,3.27]
.643(25.3)
[4.08,4.31]
2.44(8.00)
[.632, .667]
.533(21.0)
[3.39,3.58]
.693(27. 3)n
[4.40,4.65]
Method 1
hc2a.!57(6.20)
Ghn2
dh2/dx
H
. 437(17. 2)[2. 77]
. 155(6. 10)[. 984]
,04090[l,002]
. 591(23. 3)[3. 75]
mhc=1.090
,,,£^0.250
,487(19.2)[3.09]
,155(6.12)[.987]
,04086[l.00l]
, 640(25.2)[4.07]
ap/pa.000704
fj/f^.0369
.536(21.1)[3.41]
.156(6.14)[.990]
,04083[l.00l]
.690(27.2)[4.38]
Method 2
hc2*. 149(5. 87)
Ghn2
dh2/dx
H
,437(17.2)[2.93]
. 146(5. 73)[. 976]
,04092[l.003]
. 591(23. 3)[3. 96]
f^aO.400
ID 1
.487(19.2)[3.26]
.146(5.76)[.981]
,04087[l,002]
.640(25.2)[4.29]
3=1.10
. 536(21. 1)[3. 60]
. 147(5. 70)[. 986]
. 04084 [l. 001 ]
. 690(27. 2)[4. 63]
98
-------
Table 3. INTERFACIAL PROFILE CALCULATION FOR SI-3/30.
Sl-3/30 ACC= 4.0 . [1ET.HQD 1
Tl= 22.7C= 72.9F TP= l8,4C= 6.5. IF Q= *
SB= 4.060E-02 FR= 2.719E-01 OOFLTA= 9
OHC2I= 5.688E 00 OFl= 4.015E-02 .125*FR/SR= fl.330E-P1
HC2I= 6.200E 00 MHC-.= 1.090 HELTAr 7.04PE-04
FI= 1.0Q4E-02 nFI= 0.250 FT/FH= /S.692E-0?
X HI GHNI JO I H.P._ P_TPHI..
O.OOOE-Ol 1.720F 01 6.097E 00 2.3?5E m «t.09pE-0? 0._PpE-Ol
2.000E 00 1.818F 01 6.108E 00 2.423E 01 4.nfl7E-OP -1.:
00 1.916F 01 6.119E 00 2.521E nl ^
6".OOOE 00 2.014F 01 6.1P9E 00 2.619E 01 t.OflfE-0? 5.72E-Of,
DOUBLED INTERVAL OV= 1.0000E 00
8.000E 00 2.112F 01 S.luOE QO 2.717E 01 ^
INTERFACE BECOMES HORIZONTAL
si-3/30 """ METHOD ?
HC2I= 5.872E 00 MHC= 1.032 RETAr l.mn
.FI= 1.606^-02 "HFI= 0.400 FT/FB= R
. __. „. ^HNI Tcl Hp DTDHI
O.OOOE-Ol 1.720E HI 5.730E 00 2.325E HI 4.092E-3? O.OOE-01
2.000E 00 1.81RF 01 5.70.5E 00 2.423E 01 4.069E-0? 1.49£-OA
tf.OOOE 00 1.916E.01 5.760E 00 2.521E 01 4.0B7E-n? 1.91f-ns
6.000E 00 2.0J,4j:_OJL 5.774E_yP 2_, 619JE __ 0_1 4^ .OfiSE-'T.?....?^.^.^-?^.
DOUBLED i"NTERVAL'' DX= ^.oTiooE 06
8.000E 00 2.112F 01 5.7ft9E 00 2.717E 01 4
INTERFACE BECOMES HORIZONTAL
99
-------
Table 3. CONTINUED.
_Sl-3/30
ACC = Jf • 0
_J_I= _2_2_.7C= 72.9F
SB= H.080E-02
OHC2I= 5.688E On
NON-
T_?= l6.t»C= 65»1F_
FR= 2.719E-01
QFI =
METHOD 1
ODELTA=
.12_5*F_B/S.B=
HC2I= 6.200E 00
. . — - - - — • — -- -.--.---- — • —
FI= 1.00t*E-02
o.
1.
«*.
fi.
OOOE-OI
579E-01
738E-01
DOUBLED
317E-01
2
2
3
HN
.774E
.932E
.091E
INTERVAL
3.407E
«HC= _1.090
MF I =0.250
00
00
00
00
DXN =
00
9
9
9
9
9
GHNN
.835E-01
.852E-01
.869E-01
.886E-01
3.1587E-01
.903E-01
TON
3.750E
3.908E
4 . 066E
4.382E
PFLTA= 7.
FI/FB= S.
00
"b'b
no
00
i .
1 .
i.
i.
ntf?F.
KPN
OQ2E
00?E
001_£_
OOlE
001E
,nu
00
on
^on~
on
INTERFACE BECOMES HORIZONTAL
Si-3/30
HC2I= 5.872E 00
FI= 1.606E-02
__ ___
HN
O.OOOE-01
1.668E-01
3.335E-01
5.003E-01
2.9?9E 00
3.097E 00
3.26UE 00
3.131E 00
MHC= 1.032
MFI= 0,1*00
9.758E-01
9.809E-01
9.83HE-01
DOUBLED INTERVAL .DXNs 3.335tE-Ol
6.671E-01 3.59flE 00 9.859E-01 4.627E 00.
INTERFACE BECOMES HORIZONTAL
METHOD ?
RETA= 1.1 on
FT/FB= "S.907E-OP
"TON HPN
1 .003E PO
l.QO_2£..pn
1.002E 00
I.OOIE en
I.OOIE on
00
*.127E 00
4.293E 00
00
JOQ
-------
.52 profile
These profiles were produced on the steep slope and with a fairly high
flow rate. In most of the experiment, the two layers start with a fair-
ly unstable zone of approximately 8 to 12 feet, which may be called the
"frontal zone." This frontal zone is followed by the lower layer with
almost constant depth. The temperature distribution curve shows that
the lower layer depth increases gradually toward the end of the flume.
It seems that this profile reaches the virtual stratified normal depth
very quickly after the unstable frontal zone without having much of the
descending curve which is characteristic of the 52 curve predicted.
The question is then how the gradual increase in the lower'layer depth
can be explained. The following was attempted, although it did little
to explain this effect.
If the lower layer depth is equal to the virtual stratified normal depth
at this portion of the flow, we may calculate the value of f^ by sub-
stituting Ghn2 in place of h2 in GH (Eq. 94) and using the values of
h2, H, Ap/p, and f.. Unfortunately, the accuracy of the value of Ap/p
is not enough to verify the fact that AP/P decreases along the flow,
which can be expected from the decrease in temperature differenc be-
tween the two layers. Therefore, the values of f.. are calculated, using
the constant stratified critical depth. However, this trial finds a
very rapid increase in f1 (doubled between the two sections) which may
be considered as unlikely to occur. Then, the accuracy of the estimate
of the value of ffa may be questioned.
Another question is what happens if the increase in the lower layer
depth continues. Since the length of the flume is limited, the experi-
ment did not find any definite answer. From the 51 experiment, it
may be stated that the 52 flow would eventually become the profile simi-
lar to the SI flow.
101
-------
S2-4/30
Fig. 14.
Experiment S2-4/30.
Tmin=fl6.6°C
61.9°F)
Sb= 0.03
Fig. 15. Temperature distribution
curve for S2-4/30.
102
-------
S2-4/30
Table 4. EXPERIMENTAL DATA AND RESULTS
OF METHODS 1 AND 2 FOR S2-4/30.
Expt. No. 26
Sb*.03
Tsls24.6(76.3)
T1s2l.8(71.2)
fb=.2ll
q=.00631(.2228)
Ts2sl6.5(61.7)
T2=17.1(62.8)
fb/(8Sb)«.881
.359(14.1)±26%
ofi!
s.0278
T0=20.4(68.7)
.000944
X
h2
H
0.00
.399(15.7)
[.983,. 995]
.574(22.6)
[1.41,1.43]
1.22(4.00)
[.902,. 914]
.401(15.8)
[.989,1.00]
.610(24.0)
[1.50,1.52]
2.44(8.00)
[1.80,1.83]
.409(16.1)
[1.01,1.02]
.645(25.4)
[l.59,1.6l]
Method 1
Ghn2
dh2/dx
H
hc2=.406(15.98)
fi=.00444
.399(15.70)[.983]
.398(15.66)[.980]
-,0042[-.140]
.574(22.6)[l.4l]
fi/fb*'0210
Method 2
Ghn2
dh2/dx
H
hc2=.401(15.78)
f^.0350
.399(15.70)[.995]
.398(15.66)[.992]
-.Ol49[-.495]
.574(22.6)[l.43]
mhc«1.116
Mf.al.260
103
-------
Table 5. INTERFACIAL PROFILE CALCULATION FOR S2-4/30.
SP-4/30
ACC=
MFTHOD i
Tl= 21.8C= 71. 2F
SB= 3.GOOE-02
GHC?I= l.*HfE 01
T?=
Q=
HC?T = 1 .5
Fj = 'I
X
01
FR= 2
OFlr_.2.777E-02
HHC= 1.1.30...
= 0.160
OOFLTA= 9
. 1 2 5 * F B../SB = _ B
ngL.TAs ft
FI/FB= g
- 0 J
Hi
O.OOOE-Ol 1.570E 01
H APPROACHES GHM
S2-4/30
HC?I= 1.576E 01
X HI
O.OOOE-01 1.570E 01
H APPROACHES
GHNI Tpl
01 2.260E 01 -4.
HP
MHC= 1.116
MFI=1.260
GHNI TDl
01 2.260E 01
METHOD ?
RFTA:s_l._3_90
FY/FR= 1.655E-01
HP^ DIQHI
704
-------
Table 5. CONTINUED.
Sp-4/30 ACC= t.O NON-DIMEhSloNALlZF.n ...METHOD. 1
Tl= 21.8C= 71.2F T? = 17.1C= 6?.QF Q= g»?2_BE^Ol
SB= 3.oooE-02
OHC2I= 1.414E 01
FR= 2.114E-01
OFI= 2.777E-02 .125*FB/S.B = . 8.808E-01
HC2I = 1.596E 01
FI= J*_._M; 43^E - p 3
XN
MHC= It 130
P_0 1
TON HPN
00 -<+
105
-------
•S3 profile
This profile was produced with the use of the gate. The interface dips
down slightly after the gate to have the portion which may be called
vena contracta (after the term used in open-channel flow), then curves
up, and finally the lower layer depth becomes almost constant.
When the gate opening was set to the lower layer depth found in the
preliminary calculation of a profile, that flow which occurred in the
flume was totally different from the expected one; that is, the lower
layer depth was much larger than calculated. This may be due to the
turbulence in the flow following the gate produced by the gate.
106
-------
S3-5/14
Fig. 16. Experiment S3-5/14.
T -25
'
>C
DF)
254rt
(10
25.8
25.8
78.4
,457m
(18")
Computed Ijic2
.305i
Interface (12V)
.25.9
(.78.6)
Cmjjuted
.25.9
(•78.6)
Sb= 0.03
Fig. 17. Temperature distribution curve for S3-5/14.
107
-------
Table 6. EXPERIMENTAL DATA AND RESULTS
OF METHODS 1 AND 2 FOR S3-5/14.
S3-5/14 Expt. No. 32
Tgi«26.4(79.5)
^825.6(78.1)
fb*.208
qs.00378(.!337)
TS2*18.0(64.4)
T2*19.4(66.9)
fb/(8Sb)«.867
Ohc2*.217(8.54)±15*
^..0301
T0*22.8(73.0)
Ap/pa.00154
X
h2
H
0.00
.274(10. 8)
[.766,. 760]
.559(22.0)
[1.56,1.55]
1.22(4.00)
[.102..101]
.320(12.6)
[.894, .887]
.594(23.4)
[1.66,1.65]
2.44(8.00)
[.204, .203]
.351(13. 8)
[.979,. 971]
.632(24.9)
[1.76,1.75]
Method 1
hc2=. 358 (14.1)
Ghn2
dh2/dx
H
. 274(10. 8)[. 766]
. 355(14. 0)[. 993]
.0288 [.960]
. 558(22. 0)[l. 56]
n^c*1'650
. 309(12. 2)[. 864]
. 356(14. 0)[. 995]
,0287[,956]
. 594(23. 4)[l. 66]
. 000343
. 344(13. 5)[. 961]
. 357(14. 1)[. 997]
,0276[.919]
. 631(24. 8)[l. 76]
Method 2
Ghn2
dh2/dx
H
hc2*.361(14.2)
fjs.406
.274(10.8)[.760]
.350(13.8)[.970]
.0254[.845]
.558(22.0)[l.55]
mhc«1.663
mfi=13-50
.306(12.0)[.847]
.354(14.0)[.982]
,0259[.863]
,594(23.4)[l.65]
8=4.60
fi/f^l.954
.338(13.3)[.935]
.358(14.1)[.993]
,0267[.89l]
.631(24.8)[l.75]
108
-------
Table 7. INTERFACIALPROFILE (^LCULATIONFOR S3-5/14.
S3-5/14
Tl= 25.6C=
ACC= 3.0
78. IF
SB= 3,QOOE-02
OHC2I= B.543E HO
HC?I= 5.U10E 01
Fl= 1.35UE-02
X
O.OOOE-
2.000E
6.000E
8.000E
l.COOE
IT IS
S3-5/14
HC2I= 1
Fl= 4.0
01 1
CO 1
00 1
00 1
00 1
01 1
r-fO
HI
.080E
.218E
.287E
LONGER
01
01
01
01
01
01
A
1.
1.
1.
1.
1.
1.
S3
.t|2lE 01
63E-01
X
O.OOOE-01 1
2.000E CO 1
H.OOOE CO 1
6.000E 00 1
8.000E
l.OOOE
1.200E
IT IS
00 1
01 1
01 1
f\iO
HI
.080E
.1*UE
.203E
.265F
.329E
.395E
LONGER
01
01
01
01
01
01
01
A
1.
1 .
1.
1.
L9.4C= 66. 9F
FR= 2.080E-03
OFI= 3.010E-02
0=
ODELTA=
,125*FB/SB=
("IHC= 1.650
P1FI = 0.1*50
r,HNI
399E 01 2
401E
402E
01 2
01 2
^OfE 01 2
H-05E 01 2
-------
Table 7. CONTINUED.
ACC= 3 . 0
NON-QIHE.MSloN-ALJ.2Fr) _______________ METHOD. 1
Tl= 25.6C= 78. IF
SB= 3.000E-02
OHC2I= 8.5H3E 00
HC2I= 1.410E 01
FI= 1.354E-02
XN HM
O.OOOE-Ol 7.6A?E»Ol
5.108E-02 8.15PF-01
1.022E-01 8.641E-01
1.532E-01 9.1P8F-01
2.043E-01 9.606E-01
2.55«*E-01 1.015E 00
IT IS NO LONGER A S3
S3-5/1**
HC2I= 1.421E 01
XN HN
O.OOOE-Ol 7.601E-01
5.067E-02 8.03PE-01
1.013E-01 8.467F-01
1.520E-01 8.907E-01
2.027E-01 9.353E-01
2.53»*E-Ol 9.820E-01
3.040E-01 1.030E 00
IT IS NO LONGER A S3
T?= 19.4C= 66.
FR= 2.0QOE-01
OF_Ir 3.010E-02
MHC= 1.650
MFI= 0.^50
GHNN
9.925E-01
9.937F-oi
9.948E-01
9.959E-01
9.969E-01
9.983E-01
PROFILE FROM
MHC= 1.663
HFI=13.500
GHNN
9.701E-01 "
9.759E-01
9.817E-01
9.875E-01
9.93^E-01
9.997E-01
1.006E 00
PROFILE FROM
9F Q-
ODFLTA=
,125*FR/SB=
DFLTA=
FI/FB=
TON
1.557E no
1.608E 00
1.659E 00
1.710E 00
1.762E 00
1.813E 00
THIS POINT
1 .337E-01
A.667E-01
\
"5.428F — 0^4
A.512E-02
HPN
9^.^88E^01
9.561E-01
9.189E-01
1.115E 00
METHOD 2
RETAr 4.600
FI/FB=
TON
1.5<*5E 00
1.596E 00
1.6
-------
•Ml profile
The Ml profile was obtained with a small slope and a small flow rate.
The interface was almost parallel to the free surface in the experiment.
Ill
-------
Ml-4/21
Fig. 18. Experiment Ml-4/21
. Tm
. 23.
;(73
T =
mm .
77777
ax*
IOC . 23.
6°F) • ( 73
•
*
•
L7.4°C
63.3°F) <
' ' / / /777
1 .'23.:
8) -(73
•
•
_
17.3
63.1) (
'/77777
,61m
9J24'
•
•
' : *
7.4
63.3
/T777
(23.:
•(73,
.305n
— — _
77777
23.5
9) (74,
Interface-
•
^Compu
63.1) | i
Computed
7777777777
.'23.6
3) -(74,
-
-
.ed 'Ghn2
Z*51.
63.5)
/777777-^T-,
3)
•
.
•
.4
63.3
^j • ••
Sb= 0.0158
Fig. 19. Temperature distribution curve for Ml-4/21.
112
-------
Ml-4/21
Table 8. EXPERIMENTAL DATA AND RESULTS
OF METHODS 1 AND 2 FOR Ml-4/21.
Expt. No. 22
Sb=.0l58 q=.00158(.0556)
Tsl*24.1(75.4) TS2*16.3(61.3) TQ*21. 9(71.4)
^=23.1(73.6) T2«18.1(64.6) O^P/P*. 00106
fb=.255 £b/(8Sb)*2.02
Qhc2*. 137(5. 39)±23%
0fiS.0394
X
h2
H
0.00
. 389(15. 3)
[2.71,2.71]
.604(23. 8)n
[4.21,4.22]
2.44(8.00)
[.268,. 269]
.424(16. 7)
[2.96,2.96]
.645(25.4)
[4.49.4.51]
4.88(16.0)
f. 536, .538]
.462(18. 2)^
[3.22,3.23]
.683(26. 9)^
[4.76,4.77]
Method 1
"2
Ghn2
dh2/dx
H
hc2=.144(5.66)
f^=.00788
.389(15.3)[2.71]
.187(7.36)[l.30]
,0148[.937]
.604(23.8)[4.2l]
mV1'050
m
ffl 1
. 425(16. 7)[2. 96]
. 187(7. 36)[l. 30]
,0150[.952]
. 643(25. 3)[. 643]
.000918
fi/f^.0309
.462(18.2)[3.22]
.187(7.37)[l.30]
,0152[.962]
.681(26.8)[4.75]
Method 2
Ghn2
dh2/dx
H
hc2=.143(5.64)
fi*.0236
.389(15.3)[2.71]
.187(7.35)[l.30]
.0148[,936]
.604(23.8)[4.22]
h si.048
Rl C
mf,*0.600
Rl J.
.425(16.7)[2.97]
.187(7.38)[l.31]
.0150[.95l]
.643(25.3)[4.49]
B=1.15
. 462(18. 2)[3. 22]
. 188(7. 41)[l.3l]
.0152[.96l]
. 681(26. 8)[4. 76]
113
-------
Table 9. INTERFACIAL PROFILE^ dkLCUlATION "FOR Ml-4/21,
ACC=
Tl= 23.1C.=. 73.6F
S8= 1.56GE-02
OHC?I= 5.366E nn
HC2I = 5.655E 00
FI= 7.860E-03
HT
O.OOOE-Ol 1.53PE 01
4.000E 00 l.AOlE 01
8.000L 00 1.673E 01
1.200E 01 i.7*f6E"Ol"
QOUBLFn INTERVAL
1.600E 01 i.aiPF 01
2.400E 01 1.965F Ol
3.2HOE 01 2.113F 01
4.QOOE Ol 2.261E Ol
OOU61 Fn INTERVAL
4.300E Ci 2.mOF 01
6.tOOE 01 2.709R- 01
8.000E Ol 5.009E 01
9.600E OJ. 3.311E Ol
1.120E 02 3.61PE 01
DOUBLED INTERVAL
1.2AOE P2 3.91HF 01
1.&OOE 02 4.519F 01
1.9POE 02 5.12«*F 01
2.240E 02 5.730£ 01
DOUBLED. INTERVAL
2.560E 02 6.336E 01
3.200E 02 7.5H8E 01
INTERFACE BECOMES
=_6Ji «6F
FB= 2.5S2E-01 OHELTA= 1.Q63E-03
OF 1= 3.940E-02_, . 125*FB/SR= P.019F
MHC= 1
- -
. JMF
RHN
7.3R3E
7.3R6E
nxr a.
7.3A9E
7.*flOE
7
7
*
•
402E
DX= 1.
7,«n3E
7.U*5E
7.479E
7.502E
DX= 3.
7.A11E
7f CiiCr
• «> TF~ W
DX= 6.
7
7
•
•
781E
T= 0
I
00
00
00
•.
_•_
2
2
2
00 2
OOOOE
00
00
2
2
00 2
00 3
6GQOE
00 3
00 3
OQ 3
00
00
|l
n
2000E
00 4-
00 5
00 6
00 &
4000E
00
00
7
8
050
200
...TILL.
.532E
,60ftE
00
.683E
.835E
.987E
.138E
01
.290E
.593E
.897E
.200E
01
.807E
.020E
.627E
01
:"TE
01
01
ni
01
01
01
01
01
ni
01
ni
01
01
01
01
01
01
01
01
i
i
i
i
i
i
i
l
i
i
i
i
i
i
i
i
l
l
i
DFLTA=
FT/FB=
JiP_
.4ROE-0?
.493E-OP
.504E-0?
.513E-0?
»521F-05>
.R33E-Q?^
C ti Q F* ^ fl Q
C C ^» C ^p, rt 'j
•5A2E-02
.570E-0?
.572E-02
.574E-0?
• B>76E-0?
.S77E-0?
.578E-0?
.S78E-0?
.579E-OP
9.
0
-2
-8
-1
-2
-6
-6
-1
-1
-3
0
0
0
-1
0
0
0
0
183F.-OU
088E-02
DTnHI
.OOF-Ol
.OOE-05
.58F-06
.86E-06
.68E-06
.68F-OA
.62E-OR
.05E-05
.81F-OA
.OOE-01
.OOF-01
.OOE-Ot
.53E-OS
.OOE-01
.OOE-01
.OOE-OI
.OOE-01
HORIZONTAL
114
-------
Table 9. CONTINUED,
Ml-H/21
HC2I= B.fc*
ACC= <+.<
*3E 00
3
HHC= 1.
FI= 2.364E-02 MF
0.
1.
8.
1.
1.
2.
3.
«*.
<*.
6.
8.
9.
1.
1.
1.
1.
2.
2.
3.
X
OOOE-Ol
flOOE 00
OOOE 00
200E 01
DOUBLED
600E Ol
tOOE 01
200E 01
OOOE 01
DOUBLED
800E Ol
fOOE 01
OOOE 01
600E Ol
120E C?
DOUBI ED
230E 02
&OOE 02
920E 02
2<*OE 02
OOUBl ED
560E 02
200E 02
HI
1.530F 01
1.601F 01
1.673f 01
1.7<+5E 01
INTERVAL
1.81AE 01
1.965F Ol
2.112E 01
2.260F 01
INTERVAL
2.H09F 01
2.708F 01
S.OOAf- 01
3.30AE 01
3.610E 01
INTERVAL
3.911F 01
<*.5l6F 01
5.121F Ol
5.72&F 01
INTERVAL
6.332F Ol
7.5«mE 01
INTERFACE BECOMES
1= 0.
018
600
METHOD. ?
RFTAr 1.150
FI/FB= 9.?
fiHNI Tnl
7.SFJ1E
7.3A5E
7.378E
7.392E
DX= 8.
7.U05E
7.4.^3E
7.4ftOE
7.^A8E
DX= 1.
7.516E
7.571E
7.AP6E
7.AAOE
7.7/S1E
DX= 3.
7.7A8E
7.892E
7.995E
8.095E
DX= 6.
8.192E
A.3A1E
00 2
00 2
00 2
00 2
OOOOE
00 2
00 2
00 2
00 3
6000E
00 3
00 3
00 3
00 4
00 <*
2000F
00 4
00 5
00 6
00 6
tfOOOE
00 7
00 8
.380E
.^56E
.532E
.608E
00
.683E
.835E
• 987E
• 13flE
01
.290E
.593E
.897E
.200E
.504E
01
.807E
«f mE
.020E
.627E
01
.2?i+E
.•+47E
01
01
01
01
01
01
01
ni
01
01
ni
01
01
01-
OJ
ni
ni
01
ni
l.
l.
i.
l.
l.
l.
l.
i »
l.
i *
l .
l.
l.
i.
l.
i.
i.
i.
l.
HP
479F-
492F-
502E-
511E-
*19F-
B31F-
^S4 0 F "
*•> tt 7 p •»
552F-
PS&OE-
Fi&^F-
fififtE-
570^"
572F-
f)7P»r-
R7Af-
q>77E-
578F-
578E-
OP
02
02
OP
0?
OP
02
OP
02
IP
OP
n?
OP
02
0?
IP
n?
n?
OP
o.
2.
2.
3.
2.
5.
6.
8.
8.
9.
1.
1.
1,
1.
1.
1.
1.
1.
1.
'63E-0?.
OTDHI
OOE-01
15F-06
OOf-OR
72E-05
86E-ieS
25E-05
lOp-05
Oip-o^
30E-05
s^E-ns
0 8 £_- 0 a
37E-04
37E~0<*
53E-04
53E.-0'4
6fi£-0£i
68£-0'4
68E-0<+
83F.-04
HORIZONTAL
TT5
-------
Table 9. CONTINUED.
Ml-H/21
ACC=
NQJM - DI ME N S LQ.N AJLIZED.
HFTHOn 1
Tl = 23.1C= 73.6F
SB= l.SflOE-02
OHC2I= 5.386E 00
HC2I= 5.655E 00
FI= 7.880E-03
XN (
=__ 18. 1C=.. 6U.6F
FR= 2.552E-01 ODFLTA= 1.063E-03
OFI= 3.940E-Q2 .125*FR/SR= 9.019E On
MHC= 1.050 OELTA= q.l83E-Qt»
= 0.20CL
GHNN
FT/FB= 3.068E-0?
HPN
0
1
2
5
8
1
1
1
2
2
3
3
f
5
6
7
8
1
.OOOE-OI
.682E-01
.023E-
DOUBL
.364E-
.073E
.3M1E
DOUBI
.609E
.682E
.218E
.755E
OOUBL
.291E
,36«+E
.437E
.510E
DOUBL
01
ED
01
00
00
ED
00
00
00
00
00
ED
00
00
00
00
ED
2.705E
2.832E
2.959E
3.087F.
INTERVAL
3.216E
3.736E
3.99RE
INTERVAL
4.261E
5.3P1F
6.3A7E
INTERVAL
7.990E
9.061E
1.0 13F
INTERVAL
,5«3E 00 1.1POE
.073E 01 1.335E
INTERFACE BECOMES
00
00
00
00
DXW=
00
00
00
00
DXNs
00
00
00
00
00
DXKis
0.0
00
00
01
01
01
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
,299E
.300E
.301E
00
00
00
.302E 00
2.6821E-01
.303E
.305E
.307E
.309E
5.3642
.311E
.315E
.319E
.323E
.326E
1.072«
.330E
.338E
00
00
00
00
>E-01
00
00
00
00
00
)£ 00
00
00
00
.353E 00
2.1457E 00
.361E
.376E
00
00
ll
n
14-
5
5
5
5
6
7
7
8
9
1
1
1
1
(1208E
.3*f3E
.477E
.611E
.745E
.013E
.281E
,818E
!891E
.H27E
.963E
.500E
.573E
.065E
.172E
.279E
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
01
01
01
01
9.365E-01
9.517E-01
9.575E-01
«9.70?E-01
9.759E-01
9.803F-01
9.836E-01
9.883E-01
9.914F-01
9.934E-01
9.9H9F-01
9.959E-01
9-973E-01
9.981E-01
9.986E-01
9.989E-01
9.993E-01
HORIZONTAL
116
-------
Table 9. CONTINUED.
HC2I= 5.6*4
Fl
0.
1.
2.
**.
5.
8.
1.
1.
1.
2.
2.
3.
3.
4.
5.
6.
7.
3.
L.
= 2.364F
XN
OOOE-Ol
Sf^E-Ol
688E-01
032E-01
DOUBLED
376E-01
064E-01
075E 00
S'ffE 00
DOUBLED
613E 00
150E 00
688E 00
226E 00
763E 00
DOUBLED
30 IE 00
376E 00
151E 00
526E 00
DOUBLED
602E 00
075E 01
ACC= 4.
•3E 00
>02
HM
2.711E
2.838E
2.965E
3.093E
INTERVAL
3.222E
3.U8?F
3.743E
4.005E
INTERVAL
H.2&9E
f .798F
5 . 3 3 0 E
5.863E"
6.397E
INTERVAL
6.93PE
8.00PE
9.07UE
1.015E
INTERVAL
1.12PF
1 .337E
INTERFACE BECOMES
0
NON-DIPIENSIONALIZFD .METHOD..?....
MHC= 1.018 RETA= 1.150
KFl= 0
.600
FI/F8= 9
GHNN
00
00
00
00
DXfM =
00
00
00
00
DXMr
00
00
00
00
00
DXN =
00
00
00
01
DXN =
01
01
HORT
1.303E
1.305E
1.307E
1.310E
2.688
1.312E
1.317E
1.3P2F
1.327E
00
00
00
00
OE-Ol
00
00
00
00
-------
-M2 profile
This profile was the most clearly observed in the flume. The lower lay-
er depth decreases along the flow after the unstable frontal zone. The
higher the flow rate, the steeper the slope of the interface became.
118
-------
M2-4/18
Fig. 20.
Experiment M2-4/18.
4 Computed Ghn2
22
/////
.4 .23.3
(72.3K73.
•
17. "5 1
!63.5)
/'/V/V/V/V
Tm =23.7
9)max (74,
• •
•
7.5 T
-------
Table 10. EXPERIMENTAL DATA AND RESULTS
OF METHODS 1 AND 2 FOR M2-4/18.
M2-4/18 Expt. No. 21
Sb«.0058
Tsl»25.2(77.4)
^•22.8(73.0)
qs.00757(.2674)
TS2«17.7(63.9)
T2*17.8(64.0)
fb/(8Sb)s4.84
Ohc2*.392(l5.4)±20%
of.».0274
T0»22.2(73.8)
.00105
X
h2
H
0.00
.523(20.6)
[1.34,1.34]
.676(26.6)
LI. 73. 1.73]
.610(2.0)
f. 0090,. 0090]
.511(20.1)
[1.31,1.31]
.678(26.7)^
[1.73,1.73]
1.22(4.0)
[.0181, .0181]
.498(19.6)
[1.27,1.27]
.683(26.9)
[1.74,1.74]
Method 1
Ghn2
dh2/dx
H
hc2=.392(15.4)
^•.0206
.523(20.6)[l.34]
,742(29.2)[l.90]
-,0185[-3.19]
.676(26.6)[l.73]
mhc=i.ooo
mf-0.750
.511(20.l)[l.3l]
,736(29.0)[l.88]
-.0209[-3.60]
.679(26.7)[l.73]
Ap/pa.00105
. 498(19. 6)[l. 27]
. 730(28. 7)[l. 86]
-,0245[-4.22]
. 683(26. 9)[l. 74]
Method 2
hc2*.392(!5.4)
mV1*000
The same entry as in Method 1.
3=1.00
120
-------
Table 11. INTERFACIAL PROFILE CAUCUIATlbN FOR M2-4/18.
ACC= 3.0
HFTHOn 1
Tl= 22.6C= 73.OF
SB= 5.600E-03
OHC?I= 1.5^?E 01
HC2I= 1.542E Ol
FI= 2.055E-02
X HI
T ?= 17 «.BC = ... 6it, 0F
FR= 2.247E-01
OFI= 2.740E-02
MHC= 1.000
«FI=0.750
TD.I
g= 3.
ODELTA= 1.
DFJ-TA=
HP
O.OOOt-Ol 2.060E r>l 2.9P2E 0.1 2 ,6 fe 0 E 01 -_1 jjg 5 3_F -_0p 0 . 0 0 E - 01
2.000E 00 2.013E °1 2.897E 01 2.67<+E 01 -2.089E-0? 9.27E-Oci
3.999E 00 1.959E. PI ?.ft_7^E _0_1_ 2 . fe' 8^F_ Q 1__^,2 . ^tjSg -_0 2 1.65F-04
5.999E 00 l.«5^E- 01 2.PF11E 01 2.702E 01 -3.04^E-OP 2.63E-01*
7.999E OT 1.808E 01 2.896E Ol 2.71.6 E 01 -.H •?_?6£.Tin?L __ 3
9.999E 00 1.&54E: 01 2.797E 01 2.730E 01 -1.173E-01 4
HALVED INTERVAl OX= l.OOOOE 00 _
GHN-CUPF BECOMES NEGATIVE
PI2-4/16
HC2I= 1
Fl= 2.055E-02
X HI
MHC= 1.000
MFI=...0,75Q_
BE.T A= _i*iL!lfl_
FI/FB= 1
HP J3LTDHI
o.onoE-Oi
2.O^OE oo
3.999E 00
5.999E 00
7.999E CO
9.999E 00
2.060E
2.013E
l.?59E
1.
1
l.i
01
ni
PI
ni
01
01
2.9P2E
P..897E
01
01
01
ni _^i
01 -2.0A9E-0?
01 -2
8S1E
B98E
2.797E
01
01
01
2.66QE
2.674E
2.688E
2.702E 01 -!
2.716E 01 -t»_.3_26E-0? 5.71E-04
2.730E 01 -1.173E-01
HALVFD INTERV/AL DX= 1. OOOOE 00
GHN-CUBE BECOMES NEGATIVE
12 V
-------
AC.Cs.
T1 =_ 22.8C =_73 . 0_F_
SB= 5.800E-03
OHC?I= 1.552E 01
HC2I= 1_.54-2E._01[
Fir 2.055E-Q2
XN
Table 11. CONTINUED.
i NON-DIMEMS_IONALI_ZFD
HN
O.OOOE-Ol 1.336E 00
9.026E-03
FR= 2
OFl- 2.7*fOE-02
MHC= 1.000 ._.._DELTA =_
C= 0.750 FI/FB =
TDN
GHNM
HPN
1.305E 00
1.270E 00
1.22AE 00
1.172E 00
1.073E 00
HALVED INTERVAL DXN=
-------
•M3 profile
This profile utilized the gate. At the first glance, it is very similar
to the S3 profile. However, one notable difference between them found
during the experiment was that the wave was seen after the lower layer
reached the largest depth for this profile, whereas no significant wave
was seen 1n the S3 profile. Since the theory predicts the discontinuity
at the point where the interface meets the stratified critical depth,
this profile may be considered to be M3 profile.
The interface calculation could not produce a profile very close to the
profile obtained in the experiment. This may be due to the short length
of the curve available for the analysis. Unfortunately, the data with a
gradual ascending curve was not obtained.
123
-------
M3-5/18
Fig. 22,
Experiment M3-5/18.
1
T «24 . 7*
(76
•
,254n
(10"
Tin
/'/v/v/v/v/y/v//,
: .
5°
,
•
\
.n
h/
.Compute
L_ 24.6
•) (76,
fComputi
\Inter
•
•
17.2°C
(63.0°F
v//////
d Ghj
,610ra
3)(24
457m
(18")
5d hc;
-ace
•
•
17.4
(63.
////;
2
24.6
•JX76,
•
•
>305m
(12-
,152m
(6")
0
l//;/ ;
•
24.6
3) (76,
.
•
•
'.
1761
63.7) (
S///ss;jsj
24.7
3) (76,
•
•
•
17.7
63.9)
—
5)
.
*
m
•
17.5
•
Sb= 0.005
Fig. 23. Temperature distribution curve for M3-5/18.
124
-------
Table 12. EXPERIMENTAL DATA AND RESULTS
OF METHODS 1 AND 2 FOR M3-5/18.
M3-5/18
Expt. No. 34
Tsl»25.1(77.2)
^=24.3(75.7)
fx=.2lO
q=.00378(.!337)
TS2=17.0(62.6)
T2=17.8(64.0)
fb/(8Sb)=5.24
hc2*.223(8.79)±14%
TQ»21.8(71.2)
. 00141
X
h2
H
0.00
.300(11.8)
[.895. .911]
.668(26.3)
[1.99.2.03]
,075(.25)
[.0011,. 0012]
.307(12.1)
[.917, .934]
.668(26.3)
[1.99,2.03]
,152(.50)
[.0023, .0023]
.317(12.5)^
[.947, .965]
.669(26.3)
[l.99,2.03]
Method 1
Ghn2
dh2/dx
H
hc2«.335(13.2)
f.^.0107
.300(11.8)[.895]
.599(23.6)[l.79]
,088l[l7.6]
.668(26.3)[l.99]
rahc«1.500
. 307(12. 1)[. 917]
. 600(23. 6)[l. 79]
,108[21.6]
. 668(26. 3)[2. 00]
.000418
fA/fb««.0509
.317(12.5)[.946]
.600(23.5)[l.79]
.159[31.8]
.669(26.3)[2.00]
Method 2
Ghn2
dh2/dx
H
hc2s.329(13.0)
fi*.232
.300(11.8)[.9ll]
.560(22.0)[l.70]
.0850[l7.0]
.668(26.3)[2.03]
rahc«1.474
^.7.600
. 307(12. 1)[. 933]
. 562(22. l)[l. 71]
. 668(26. 3)[2. 03]
3*3.20
fi/fb-1.11
.318(12.5)[.967]
.566(22.3)[l.72]
.216[43.2]
.669(26.3)[2.03]
125
-------
Table 13. INTERFACIAL PRbFILE CALCULATION FOR M3-5/18.
M3-5/18
ACC= 3.0
SB= 5.000E-03
OHC2I= 8.794E 00
HC2I= 1.319E 01
FI= 1.066E-02
T?= l7.8C=
. OF
ME.THOD_I.
Q= 1.337E-01
FR= 2.096E-01
OFI= 3.047E-Q2
ODELTA= 1.41PE-.0*
. 125*FB/SB= 5.240E 00
MHC= 1.500
MFI= 0,350
DELTA= 4.184E-04
FI/FB= ei.088E-02
HI
GHNI
D.OOOE-01
1.25QE-01
2.499E-01
T.749E-01
4.998E-01
6.24SE-01
HALVED
6.873E-01
7.49BE-01
SIGN OF
1.180F 01 P.360E 01 2.630E 01
1.194F 01 2.3AOE 01 2.631E 01
1 - 2 09 F 01 _2 .361E 01 2 . 631E Ol
T DI JHJP__ DTDHI
8 • 81.1E - 0 P Q.OOE-Ol
9.637E-OP -5.56E-06
l.n79E-01 -1.11E-05
1.226E 01 2.362E 01 2.632E 01
1.248E 01 2.363E 01 2.633E..ni
1.277E 01 2.3A4E 01 2.634E 01
INTERVAL DX= 6.2500E-02
1.308F 01 2.366E Ol 2.634E 01 9.406E-01 -7.72E-06
1.P58E-01 -1.67E-05
2.629E-01 -1.26E-OR
1.391E 01
01
01 -1.351E-01 -2.67E-I
INTERFACIAL SLOPE IS CHANGED
M3-5/18
HC2I= 1.296E 01
FI= 2.316E-01
X HI
O.OOOE-01 1.180E 01
1.248E-01
2.496E-01
3.745E-01
4.993E-01
HALVED
6.P37E-01
6.659E-01
SIGiM OF
METHOD P
MHC= 1.
HFI= 7.
P
1.193E 01 2
1.209F 01 2
1.227E 01 2
1.253E 01 2
GHNI
.P04E
.POSE
.P14E
.PPOE
.PP9E
01 2
01 2
01 2
01 2
01 2
474
600
ini
.630E
.631E
.632E
.632E
.633E
INTER\//\l. DX= 6.2500E-02
1.274F 01 2.?.^7E Ol 2.633E
1.268F .01 2
1.314F 01 2
IMTERFACIAL
.PS2E
SLOPE
01 2
01 ?
IS
.634E
.634E
RETA= 3.200
FI/FB= 1.105E On
01
01
01
01
01
01
01
01
8
9
1
1
2
4
3
-5
HP
.497E-OP
.521E-OP
.1 08E-01
.387E-01
.159E-01
.196E-01
.303E-01
.07PE-01
0
1
3
5
8
1
1
1
DTDHI
.OOE-Ol
IfiS^Ss
.40E-OF)
.73E-05
.09E-04
.30E-04
.67E-04
CHANGED
126
-------
Table 13. CONTINUED.
. M3-5/18 ACC= 3.
Tl= 24.3C= 75. 7F
S6= 5. OOCE-03
OHC2I= 8.794E 00
HC2I = 1.319E 01
FI= 1.066E-02
XN HN
0 _NON.-PlHENS.lO.NAkI?Fn METHOD !...__
T9 = 17.8C= 64. OF Q= 1.337E-01
FR= 2.
OFI= A
MHC =
MFI =
G
096E-01
.047E-02
1.500
0.350
ODELTA= t.41?E-0^
.125*FB/SB= 5.P40E 00
DELTAr 4.184E-04
FI/FR= 5.088E-0?
TDN HPN
o.
5.
1.
1.
2.
2.
3.
3.
OOOE-Ol
137E-03
705E-03
274E-03
842E-03
HALVED
126E-03
410E-03
SIGN OF
8.945E-01
9.050E-01
9.166E-01
9.298E-01
9.458E-01
9.683E-01
INTERVAL DXN=
1.054E 00
INTERFACIAL si
i.
i.
i.
i.
i.
i.
2
1.
789E
789E
790E
00
00
00
790E 00
791E 00
79 2E 00
.8428E-04
797E 00
OPE IS
1
1
1
1
1
1
1
1
.994E
.994E
.995E
.995E
.996E
,997E
.997E
.997E
no
00
no
no
00
00
00
00
i
1
3
R
._.„
.76?E
• 927E
• 15RE
.516E
.~258E
.ft'BlE
.703E
01
01
01
01
01
0?
01
CHANGED
M3-5/18
HC2I= 1.296E Ol
FI= 2.31feE:-Ol
XN HN
0. OOOE-Ol 9.106E-01
5.779E-04
1.156E-03
1.734E-03
2.312E-03
HALVED
2.600E-03
2.888E-03
3.176E-03
SIGN OF
9.210E-01
9.3P8E-01
9.U71F-01
9.6A6E-01
INTERVAL DXW=
9.831E-C1
9.784E-01
l.OlfE: 00
INTERFACIAL si.
WHC= 1.474
«FI= 7.600
GHNN
1.701E 00
M_ETHpn_.2
RETAr 3.?0n
FT/FB= 1.105E 00
TON
P.029E 00
1.704E 00 2.030E 00
1.708E 00 P.031E 00
1.713E 00 2 TO 31 E 00
1.720F 00 2.032E 00
2.8938E-04
1.726E 00 2.032E 00
1.724E 00 2. 032E 00
1.738E 00 2.033E 00
OPE IS CHANGED
HPN
1.699F HI
1.904E 01
P.215E 01
" P.775E 01
4.317E Pi
A.392E_ 0.1
6.606E 01
-1 .014F 0?
127
-------
•HI profile
As predicted, HI profile is very similar to the M2 profile.
128
-------
Hi-4/23
Fig. 24.
Experiment Hi-4/2 3
sz_
.23.'
(74
Tmax-23.
1) (74
•
*
•
iS-M Tmi
64.6)
OG . 23.8
7°F) • (74
1
Interface
•
'Compute*
•
•
=1B.1°C
64.6°F)
3)
3 frc2
18.2
64.8
23.5
(75
— * •
•
,457n
(18")
. 305n
(12-
•
. 24.0
0) • (75
»
•
*
•
8.3 1
64.9)
. 24.2
2) (75
•
•
•
•
•
3.3 1
64.9) (
6)
-
J.3
64.9)
Fia. 25.
Temperature distribution curve for Hi-4/23.
129
-------
Table 14. EXPERIMENTAL DATA AND RESULTS
OP METHODS 1 AND 2 FOR Hl-4/23.
Hl-4/23
Expt. No. 23
q=. 00504 (.1782)
TS1»25.1(77.2) TS2«17.8(64.0)
^.23.4(74.3) T2*18.6(65.5)
fb«.255
Ohc2».297(11.7)±20*
^•.0310
T0»24.4(75.9)
ap/p*.00107
X
h2
H
0.00
.622(24.5)
[1.75.1.75]
.719(28.3)
[2.01,2.01]
1.22(4.00)
[3.42,3.42]
.612(24.1)
[1.72,1.72]
.719(28.3)
[2.01,2.01]
2.44(8.00)
[6.84,6.84]
.599(23.6)
[1.68,1.68]
.719(28.3)
[2.01,2.01]
Method 1
dh2/dx
H
hc2».356(14.0)
f^. 00620
.622(24.5)[l.75]
-.00872
.719(28.3)[2.0l]
mhc*1.200
_f4*0.200
in x
.611(24.l)[l.72]
-.00916
.719(28.3)[2.0l]
Ap/ps.000618
.600(23.6)[l.68]
-.00970
.719(28.3)[2.0l]
Method 2
. 356 (14.0)
dh2/dx
H
. 622(24. 5)[l. 75]
-.00895
. 719(28. 3)[. 719]
.1.200
mri
.611(24.l)[l.72]
-.00905
.719(28.3)[2.0l]
.600(23.6)[l.68]
-.00924
.719(28.3)[2.0l]
130
-------
Table 15. INTERFACIAL PROFILE CALCULATION FOR Hl-4/23.
Hi-4/23 ACC= 3.0 METHQD_ 1
Tl= 23.5C= 74. 3F T?= l8.6C= 65. 5F Q= 1 .782E-01
SB= O.OOOE-01 FR= 2.551E-01 OOELTAr
OHC?I= 1.169E 01 QFI= 3.099E-02 •1.?5*FB.=.,
HC?I= 1.403E 01 MHC= 1.200 nFLTA=
FI= 6.196F--03 MFI= 0.200 FI/F8=
X
O.OOOE-OI
2.000E 00
4.000E 00
6.000E 00
DOUBLED
8.000E 00
1.200E 01
1.600E 01
DOUBLFD
2.000E Ol
2.800E Ol
3.600E 01
HALVED
l.bOOE Ol
2.000E 01
2.400E Ol
2.800E Ol
3.200E 01
3.600E 01
4.000E 01
HALVED
4. 200E Ol
4 . 400E 01
SIGN OF
HI
2.4SOE 01
2.429E 01
2.407E 01
2.385E 01
INTERVAL DX=
2.3&2E Ol
2.314E 01
2.?ft2E 01
INTERVAL DX=
2.206E 01
2.074E 01
1.A93E 01
INTERVAL DX=
2.262E 01
2.206E 01
2.144E 01
2.074E 01
1.992E 01
l.fl93E 01
1.757E Ol
IMTERV/U DXr
1.656E 01
1.394E 01
I:\jTEBFAClAL Si
TOI
2.8?5E 0]
2.825E Ol
2.825E 01
2.825E Ol
4.0000E 00
2.825E Ol
2.825E 01
2.325E 01
8.0000E 00
2.825E 01
2.825E 01
2.825E 01
4.0000E 00
2.825E 01
2.825E 01
2.825E 01
2.625E 01
2.825E 01
2.825E 01
2.825E 01
2.0000E 00
2.825E Ol
2.825E 01
HP
-8.718F-03
-8.928E-03
-9.16lF.-03
-9.419F-03
-9.704E-03
-l.OSyr-O?
-1.12nE-0?
-1.227r-02
-l«5pOE_-0.? .
-2.350F-0?
-1.120F>0?
-1.227F-0?
-1.366E-02
-1.560^-02
-1.850^-02
-2.35nE-0?
-3,51fef:-02
-5.239F-0?
1.711F 00
1 . 067E-03
3.189E-Q9
Pt 430E-OP
DTDHI
-1.53E-05.
-1.53E-OS
-l.SSE-O^
-1.53E-OS
-1.53E-05
-LSSE-OS
-1.53E-OS
1.53E-OFS
3, 05£rJ0..5..
6.10E-OS
-l.BSE-O'i
-1.53E-OS
-1.53E-OS
1 ,5ilJE_"Q5
LSSE-O^
4.58E-Oci
3.05E-05
3.05E-05
3.05E-05
OPE IS CHANGEH
131
-------
Table 15. CONTINUED.
Hl-^/23
HC2I = l«t°
FI= 3.719E
X
O.OOOE-Ol
2.000E 00
H.OOOE 00
6.000E 00
DOUBLED
8.000E 00
1.200E 01
1.600E 01
DOUBLED
2.000E Ol
2.800E Ol
3.600E 01
tf^tfOOE Ol
HALVED
H.800E Ol
5.200E 01
5.600E 01
HALVFD
H.600E Ol
<+.800E 01
5.000E 01
HALVED
4.500E Ol
t.fcOOE 01
«*.700E 01
H.800E Ol
4.900E 01
5.000E 01
5.100E 01
HALVFD
5.150E 01
SIGN OF
ACC= 3.0
3E Ol
-02
Hi
2,4tiOE Ol
2.t28E Ol
2.407E 01
2.3S5E OJ
INTERVAL DX=
2.363E 0]
2.318E 01
2.271E 01
INTERVAL DX=
2.2P3E 01
2.116E 01
1.988E Ol
1.816E Ol
INTERVAL DXr
1.697E Ol
2.441E 01
2.397E 01
INTERVAL oxr
1.758E 01
1.687E 01
1.585E 01
INTERVAL DXr
1.789E Ol
1.758E 01
1.7?5E 01
l.ft87E 03
1.6H2E-" 01
1.585E 01
1 . 4-9'4-E Ol
INTERVAL DXr
1.39AE Ol
INTERFACIAL Rl
MHC= 1^0^
MFTHnn ?
RFTAr 1.730
HFI= 1.2_00 FT/FB= 1.tl58Ej-J3l
TDI
2.825F. Ol
2.825E Ol
2.825E Ol
2.825E Ol
lUOOOOE 00
2.625E Ol
2.825E 01
2.825E 01
8.0000E 00
2.825E Ol
2.825E 01
2.625E Ol
4.0000E 00
2.825E Ol
2.825E 01
2.825E 01
2.0000E 00
2.825F. 01
2.825E 01
2.825E 01
l.OOOOE 00
2.825E Ol
2.825E 01
2.825E 01
2.825E 01
2.825E 01
2.825E Ol""
2.625E 01
5.0000E-01
2.825E Ol
OPE IS CHANGED
_HE
-8.950F-03
-8.985E-03
-9.129E-03
-9.237F-03
-9.52AE-03
-9.920E-03
-1^20JLE-0?
-1.493E-02
-2.222F-02
-3.t07E-0?
-8.960E-03
-9.077F-03
-2.6tlF-0?
-3.HO&E-0?
-2.HOAE-0?
-2.641F-02
-2.95ftF-0?
-3.U06F-0?
-5.56AE-02
-1.163F-01
2.295E 00
DTDH1
-1.53F-nfi
7.63E-OS
1.37F-OU
1.98E-04
3!A1E-0«+
6.26E-04
1.05E-03
1.30F-03
l.tOE-03
1.51E-03
1.65F-03
1.36F-03
l.HOE-03
1.33E-03
1.36E-0*
1.39F-03
l.HOE-03
1.H6E-03
1.50E-03
'1.50E-03
132
-------
Table 15. CONTINUED.
H
T
l-f/23
1= 23.
ACC= 3.
: 74. 3F
SB= O.OOOE-OI
OHC?I= 1.169E 01
HC2I= 1.403E 0].
FI= 6.196E-Q3
0
1
3
b
6
1
1
1
2
3
1
1
2
2
2
3
3
3
3
Xl\i
.OOOE-OI
.711E 00
.422E 00
.133E 00
DOUBLFD
.843E 00
.027E 01
.369E 01
DOUBLED
.711E 03
.395E 01
.080E 01
HALVED
.369E 01
.711E 01
,OR3E~ 01
.395E 01
.737E 01
.080E 01
• 4?2E 01
HALVED
.593E
STGM
01
01
CF
1,
1.
1.
1.
I INTERVAL
1.
1.
1.
INTERVAL
i .
i.
i.
INTERVAL
1 .
1 .
3. *
i .
i .
i.
i.
INTERVAL
1.
q t
INTERFACI
0
NON-DIMENSIONAL IZFH
_
1
FR= 2
OF I =
MHC
MFI
HFM
747F
73 IF
700E
DXM =
613F
57?E
1 1 "*J Q r-
* * O p
DXN =
613F
572F
528F
349F
253E
nxw=
180F
93UE-
AL RL
0
0
0
0
0
0
00
3.
00
00
00
6.
00
00
00
3.
00
00
0
0
0
0
0
0
0
0
0
0
1.
00
01
OPE
8.6C= 65. 5F
.551E-01
3.099E-02
= 1.200
= 0.200
TON
2.014E
P.OlfE
2.014E
4217E 00
2.03
-------
Table 15. CONTINUED.
H i*/24 "ri~ ^ n MnM-nTMFM
HC2I = I.1* P.
FI= 3.719E
XN
O.OOOE-Ol
1.710E 00
3.420E CO
5.131E 00
DOUBLED
6.a^iE on
1.0?6E 01
1.368E 01
DOUBLED
1.710E 01
2.394E 01
3.078E 01
3.762E 01
HALVED
4.447E Cl
4.789E 01
HALVED
3.933E 01
H . lOtE 01
4.275E 01
HALVED
3.8H8E 01
3.933E 01
H.019E PI
t.lOM-E 01
**.190E 01
4.276E Cl
4.361E 01
HALVED
S I G N OF
*E Ol |^HQ= Ijt2j)0 RET As 1. 73n
-no MFT= 1.200
— U £ 1 11 A -V_»^_"w --
MM TON
i-7a,fF 00 P.013E 00
1.731E 00 2.013E 00
1 .71SF 00 2.013E 00
1.700E 00 2.013E 00
TWTFR\/Al nXW— 3.*f20M'E 00
1.68t4F 00 2.013E 00
1.65PF 00 2.013E 00
1.619E 00 2.013E 00
INTERVAL DXN- 6§8t08E 00
1.584E 00 2.013E 00
1.507E 00 2.013E 00
1.417E 00 2.013E 00
i -?9t*£ 00 2.013E 00
INTERVAL DXW= 3.tf20^E 00
1.202F 00 2.0i3E 00
1.739E 00 2.013E 00
1.708F 00 2.013E 00
INTERVAL DXN= 1.7102E 00
1.253F 00 2.013E 00
1.202E 00 2.013E 00
1 .130F 00 2.013E 00
INTERVAL DXW= 8.5510E-01
1.275F 00 2.0i3E 00
1.253F 00 2.013E 00
1 .229F 00 2.013E 00
1.202E 00 2.013E 00
1.170F 00 2.013E 00
1.130F 00 2.013E 00
1.065E 00 2.013E 00
INTERVAL DXM= 4.2755E-01
9.965F-01 2.013E 00
INTERFACIAL SJ OPE IS CHANGED
PT/FB= l.ttRAF-01
HPM
-A.985E-03
-q.OU5F-03
-9.129E-03
-9.237E-03
-9.920E-03
-1.201F-02
-2.222F-02
-A.U07F-OP
-8.960E-0?!
-q.077F-03
-?.fi«HF-02
-3.406E-02
-5.566E-C?
-9.U06E-0?
-9.956E-0?
-3.406E-02
-5.566E-02
-1.163E-01
2.295E 00
134
-------
•H2 profile
This profile is almost the same as the M3 profile.
135
-------
H2-5/23
Fig. 26. Experiment H2-5/23.
1 ...—.. •
Tmax -2£
-
.254m
(10"
T
7°<
5°
>
.„<
: 24.7
') (76.!
*
|Comput«
7
Interfc
•
17.2°C
63.0°F
5 )
457m
(18")
5d -hc;
tee •
•
17.2
1 (63
24.6
(76,
-
. 305m
(12-
.152ra
(6")
0)
24.
3) (75
17.4
(63.3)
9) (75,
•
*
L7.5-
(63/5)
9)
*
*
•
*
17.4
(63.3)
Fig. 27. Temperature distribution curve for H2-D/23.
136
-------
Table 16. EXPERIMENTAL DATA AND RESULTS
OF METHODS 1 AND 2 FOR H2-5/23.
H2-5/23 Expt. NO. 36
q«. 00442 (.1560)
TS1=24.9(76.8) TS2*16.9(62.4)
^=24.4(75.9) T2=17.9(64.2)
fb».2lO
Ohc2a.247(9.73)±14*
^=.0296
X
h2
H
0.00
.317(12.5)
[.857,. 857]
.692(27.3)
[1.87,1.87]
.152(0.5)
f. 411, .411]
.345(13.6)
[.932. .932]
.692(27.3)
[1.87,1.87]
T0»21.8(71.2)
o
-------
Table 17. INTERFACIAL PROFILE CALCULATION FOR H2-5/23.
ACC=
H2-5/23
Ti= ..2H..H Q= 75
SB= 0. OOOf-01
OHC2I= 9.730E 00
KEItiOD
FR= 2.102E-01
QFl= 2.96QE-C2
Qr l
ODELTA= 1
HC
.O
0.
1.
3.
5.
6.
7.
8.
a.
9.
7.
8.
8.
21= l.«»59E 01
= 5.920E-02
X
OOOE-Ol
250E-01
500E-01
750E-01
OOOE-Ol
250E-01
500E-01
HALVED
125E-01
750E-01
375E-01
HALVED
812E-01
125E-01
437E-01
SIGN" OF
1.
1.
1.
250E
267E
P85E
1.305E
1.328E
1.355E
„ 1.393E
INTERVAL
1.137E
1.143E
INTERVAL
1.406E
1.423E
1.770E
01
01
01
01
01
01
01
DX=
01
01
OXr
01
01
01
INTERFACIAL SI
MHC =
MFI =
2.
2.
2.
1.500
_2_.000
TDI
725E
725E
725E
01
01
01
2.725E 01
2.725E 01
2.725E 01
2.725E 01
6. 2 50 OE- 02
2.725E Oi
2.725E 01
2.725E 01
3.1250E-02
2.725E 01
2.725E
2.725E
1 OPE
01
1
1
1
1
2
3
5
7
3
5
• 6
DELTA=.
FT/FB=
HP
.074E-01
.158E-01
.268E-01
,«H9E-01
.057E-01
.ISSE-Ol _
.724E-01
.509E-OP
,900£-01
.721E-01
.057E-02
IS CHANGED
5>.ftl6E
^•.0_«L_.....
DTUHL
-1.53E-OB
-1.53E-05
-i.5^t-05
-1.53E-OS
-1.53E-05
138
-------
Table 17. CONTINUED.
H2-5/23
ACC= <4.0
KFTHQO 2
MHC= 1.500
RETA= 3.375
FI= 4.736E-01
O.JDOOE-OI
i.250E-Oi
2.500E-01
3.750E-01
5.000E-01
6.250E-01
7.500E-01
HALVED
8»125E-01
8.750E-01
9.375E-01
HALVED
7.812E-01
8.125E-01
HI
MFI=16.000
TDI
FI/FBs P.253E 00
.HP
DTDHI
SIGN OF
1.2ROE 01
1.2C.7E 01
1.285E 01
1.306E 01
1.330E 01
1.359E 01
1.401E 01
INTERVAL DXr
1.447E 01
1.308E 01
1.3POE 01
INTERVAL oxr
1.417E 01
1.<*<+8E 01
1.512E 01
INTER FACIAL si
2.725E 01
2.725E 01
2.725E 01
2.725E 01
2.725E 01
2.725E 01
2.725E 01
6.2500E-02
2.725E Ol
2.725E 01
2.725E 01
3.1250E-02
2.725E 01
2.725E 01
2.725E 01
OPE IS CHANGED
1.081E-01
1.17PE-01
1.293E-01
1.72«4-E-01
3I773E-01
1.771E 00
1.609E-01
5.227F-01
1.853E 00
-H.209£-01
"•1 .53E"05
1.5SE-05
H.58E-05
7.63E-05
O < tip w fl ll
O < lip *• fl ti
2 T ^^* ft ^
o i it c" j» fi fi
O < |i c* |Bt fi (i
2 < ^f r* TI ft tt
TJ9
-------
H2-5/23
ACC=
Table 17. CONTINUED.
NOiM-D I KEN S I.ON ALT Zr D
METHOD
Tl= 2<+.4C=_J5j_9F_
SB= O.OOOE-Ol
OHC?I= 9.730E 00
HC?I= 1.459E 01
FR= 2.102E-01
0FI= 2.960E-02
(«iHC= 1.500
MFI= 2.000
OOE LTA
L 1 . ? 5 * F B
1 .«H9E-OS
. ? . 6 2 8 E - Q.?.
.DFLT_A=_.4..?0.tfE;.-0<+.
0
1
?
3
4
5
6
f,
7
6
6
6
XM
.OOOE-OI
.OP8E-C1
.055E-01
.083E-01
.111E-01
.139E-01
.16&E-01
HALVED
.680E-01
.708E-01
HALVED
.423E-01
.680E-01
.937E-01
SlGf\r6~F
8
8
Q
~£
9
9
9
INTERVAL
9
~7
7
INTERVAL
9
Q
1
"INT ERF AC
•
*
9
.
•
•
.
•
•
•
*
•
•
565F-
679F-
804E-
941F-
098F-
28&E-
54?E-
DXN =
751F-
793F-
83?F-
DX(\!=
631E-
751F-
21?F
IAL SI
01
01
01
01
01
01
01
5.
01
01
01
2.
01
01
00
OPE
,
1
1
1
1
I
1
1387E-02
1
1
3
5694E-0?
1
1
1
TON
,8f>7E
. 8&7E
.867E
.867E
.867E
.867E
.8f,7E
.S67E
.8&7E
• 8ft7E
.8A7E
.867E
.867E
no
00
00
00
00
00
00
00
00
00
00
00
00
1 .
1 .
t .
1 .
?.
3.
5.
7»
7.
3_.
5.
-f-. •
HPfcL.
15fiE-
?6AE-
419E-
648E-
057E-
155E-
724E-
401E-
50_9E-
900£-
721E-
D57E-
ni
01
01
m
01
01
01
0 1
0?
o?
01
01
0?
IS CHANGED
140-
-------
Table 17. CONTINUED.
H2-5/23 ACC= 4.0 _N.PN."P.I.Nf.NSJ_ONAl.I2FO METHOD ?
HC2I= 1 .159E Oi flHC= 1*500 RETAs 3.37*
Fl= <*
XN
HM
TON
FI/F8= J>.253E 00
HPN
O.OOOE-OI
1.028E-01
2.055E-01
3.0fl3E-01
H.111E-01
5.139E-01
6.166E-01
HALVED
6.680E-01
7.19*fE-01
7.708E-01
HALVED
6.«+?3E-01
6.6POE-0.1
6.937E-01
SIGN OF
8.565E-01
8.6ftOE-01
8.607E-01
8.9^8E-01
9.110E-01
9.309E-01
9.596E-01
INTERVAL DXM= 5.
9.91UF-01
8.96?F«01
9.0flF-Ol
INTERVAL DXM= 2.
9.709F-01
9.918F-01
1.036E 00
INTERFAClAL SI OPE
1.867E
1.867E
1.867E
1.867E
1.867E
1.867E
1.867E
1388E-02
1.867E
1.867E
l»8fe7E
569fE-02
1.867E
1.867E
1.867E
IS CHANGED
00
00
00
00
00
00
00
00
00
00
00
00
00
1.081E-01
1.172E-01
1.293E-01
1.H62E-01
1.72»tE-Ol
?.21«»E-01
3.773E-01
1.771E 00
l»«f6?E-01
1.602E-01
5.227E-01
1.S53E 00
-4.209E-01
14T
-------
Comment on the Result of Methods 1 and 2
Calculation either by method 1 or 2 can produce different profiles close
to the profiles obtained in the experiment. This fact may elucidate that
the interfacial profile equations derived in Sec. 5 can successfully pre-
dict the existence of the different profiles which occurred in the flume.
When looking at the summary of the values of mhc and fflfj (Table 18),
the following statements can be made:
1. The values of h from the methods 1 and 2 are close.
2. Most of the values of mhc are within the maximum values of mhc
which could happen by error (second column of the table 18) for the
profiles that did not use the gate.
3. Most of the values of h are larger than 1.
4. Most of the values of f^ for the method 1 are smaller than 1,
whereas most of the values of mf^ for the method 2 are larger than
1.
From 1, 2, and 3, it can be stated that, although variation of mhc may
be explained by the error in the calculated stratified critical depth,
there must be some factor decreasing the value of the calculated stra-
tified critical depth because most of hc are larger than 1.
Methods 1 and 2 were used to increase the calculated value of the stra-
tified critical depth; however, nothing conclusive has been obtained.
16 13
According to Keulegan and Bata and Knezevich , the estimated magni-
tude of fj for a small channel and for small velocities is smaller than
the coefficient obtained for smooth-walled channel. If this is true,
method 1 is a suitable method to correct the stratified critical depth.
However, this method tends to give similar values to the f^ for the cases
when the gate must have introduced more turbulence and thus should have
larger f^. Method 2 seems to give larger fj for the cases involving the
gate than for the cases awithout the gate. However, the values of f.,
on the whole are fairly large.
Therefore, as far as the data obtained in this experimental system are
concerned, they do not offer a solid way to predict the profiles occur-
142
-------
Table 18.
VALUES OF mhc AND
Profile
Identi-
fication
Sl-3/30
S 2 -4/0 2
S 2 -4/11
S2-4/30
S3-5/02
S3 -5/04
S3- 5/0 9
S3-5/14
Ml -3/26
Ml -4/21
M2-3/21
M2-3/23
M2-4/18
M3-5/18
M3-5/21
HI -4/2 3
HI -4/25
H2-5/23
Error
in
ohc2
±28%
±35%
±25%
±26%
±24%
±15%
±16%
±15%
±27%
±23%
±41%
±35%
±20%
±14%
±15%
±20%
±20%
±14%
mh
M 1
1.090
1.020
1.150
1.130
1.650
1.400
1.450
1.650
1.600
1.050
1.160
1.490
1.000
1.500
0.900
1.200
1.050
1.500
c
M 2 (B)
1.032(1.10)
1.007(1.02)
1.150(1.52)
1.116(1.39)
1.749(5.35)
1.401(2.75)
1.409(2.80)
1.663(4.60)
1.518(3.50)
1.048(1.15)
1.310(2.25)
1.710(5.00)
1.000(1.00)
1.474(3.20)
0.951(0.86)
1.200(1.73)
1.051(1.16)
1.500(3.38)
vf
M 1
0.250
1.450
0.400
0.160
0.005
0.150
0.500
0.450
0.850
0.200
0.700
0.800
0.750
0.350
9.000
0.200
1.000
2.000
•
M 2
0.400
1.730
2.000
1.260
3.900
6.400
8.200
13.50
7.500
0.600
1.800
5.250
0.750
7.600
10.50
1.200
1.450
16.00
143
-------
ring 1n this flume.
144
-------
SECTION VIII
REFERENCES
1. Schijf, J.B., and J.C. Schonfeld, Theoretical considerations on the
motion of salt and fresh water. In: Proc. Minnesota Intntl. Hyd.
Conv.. Sept. 1953. p. 321-333,
2. Whitaker, S. Introduction to Fluid Mechanics. Englewood Cliffs,
N.J., Prentice-Hall, Inc., 1968.457^
3, Chow, V.-T. Open-Channel Hydraulics. New York, McGraw-Hill Book Co.,
1959.
4. Wilkinson, D.L, Limitations to length of contained oil slicks.
J. Hyd. Div.. ASCE. 99:701-712, HY5 (May) 1973.
5. Ippen, A.T. Salinity intrusion in estuaries. Chap. 13 In: Estuary
and Coastline Hydrodynamics. Ippen, A.T. (ed.). New York, McGraw-
Hill Inc., 1966.
6. Harleman, D.R.F. Mechanics of condenser-water discharge from thermal -
power plants. Chap. 5 In: Engineering Aspects of Thermal Pollution,
Nashville, Tenn., Vanderbilt Univ. Press, 1969.
7. Polk, E.M., Jr., B.A. Benedict, and F,L. Parker. Cooling water den-
sity wedges in streams. J. Hyd. Div., ASCE. 97:1639-1652, HY10
(Oct.) 1971.
8 Einstein, H.A. Formulas for the transportation of bed load. Appen-
dix II. Method of calculating the hydraulic radius in a cross sec-
tion with different roughness, Trans. ASCE. 107:575-577, 1942.
9. Yanagida, N. A Study of Interfacial Profiles in a Two-Layered Flow.
Ph.D. dissertation, University of Pennsylvania, 1974.
10 Ghosh, S.N., and N. Roy. Boundary shear distribution in open channel
flow. J. Hyd. Div., ASCE. 96:967-994, HY4 (Apr.) 1970.
11 Weast, R.C. (ed.). Handbook of Chemistry and Physics. 50th ed.
The Chemical Rubber Co., 1969-1970.
12 Keulegan , G.H. Laws of turbulent flow in open channels. J. Res.
' Natl. Bur, Stand. 21:707-741, RP1151, Dec. 1938.
145
-------
13. Bata, 6.L., and B. Knezevich. Some observations on density currents
in the laboratory and in the field. In: Proc. Minnesota Intntl.
Hyd. Conv., Sept, 1953, p. 387-400,
14. Bata, G.I. Recirculation of cooling water in rivers and canals.
J. Hyd. Div,, ASCE. 83:paper 1265 (27p.) HY3 (June) 1957.
15. Harleman, D.R.F, Stratified flow. Sec. 26 In: Handbook of Fluid
Dynamics, Streeter, V.L, (ed.) New York, McGraw-Hill Inc.,.1961.
16. Keulegan, G,H, Interfacial instability and mixing in stratified
flows, J. Res. Natl. Bur. Stand. 43:487-500, RP2040, Nov. 1949.
17. Lofquist, K. Flow and stress near an interface between stratified
liquids. Phys. Fluids. 3:158-175, 2 (Mar.-Apr.) 1960.
18. Carstens, T. Turbulent diffusion and entrainment in two-layer flow.
J. Waterways & Harbors Diy.. ASCE. 96:97-104, WW1 (Feb.) 1970,
146
-------
SECTION IX
SYMBOLS
(See Sec. 7,3 and Appendices for the symbols used in computer programs)
b - width;
f = friction factor;
g = acceleration of gravity;
h = depth of a layer;
i = with an under bar, unit vector of x directjon;
j = with an under bar, unit vector of y direction;
k - with an under bar, unit vector of z direction;
1 = longitudinal spacing of roughness element;
m = transversal spacing of roughness element;
n * with an under bar, outwardly directed unit normal vector;
p « with an under bar, unit vector in the perpendicular direction;
q « flow rate per unit width;
r = height of roughness element;
s = »Hdth of roughness element;
t = time;
v - velocity;
w = mean velocity of points on the surface of a volume;
x = longitudinal distance (in the direction of the flow); and
= with an under bar, spatial coordinates;
y = transversal distance (in the direction of the width of the flow);
z = vertical distance (in the direction normal to x);
A = area, either cross sectional of surface; and
= adverse slope flow;
B = term defined in Eq. 165;
C = constant in Eq. 237;
D = term defined in Eq. 171;
147
-------
E = term defined in Eqs. 172, 174, and 187; and
= elevation;
G = factor related to the stratified normal depth for the lower layer;
H * total depth; and
= horizontal slope flow;
I = with doube under bars, unit tensor;
K * constant in Eq. 240;
L = horizontal distance;
M * mild slope flow; and
= method;
P » pressure; and
- positive slope flow (= S and M);
R - hydraulic radius;
Re= Reynolds number;
S = slope; and
* steep slope flow;
T = with double under bars, stress tensor; and
= temperature
U = average velocity in a layer; and
« upper layer analysis;
V a volume;
W = wetted perimeter which does not include the bottom, or wall;
a = term defined in Eqs. 238 and 239;
e = momentum coefficient;
Y = angle defined in Fig. 10;
6 = angle defined in Fig. 2;
e = angle defined in Fig. 3;
e = angle defined in Fig. 4;
e = angle between the x axis and the horizontal;
* coefficient of viscosity;
148
-------
v = kinematic viscosity;
P = density, but (Pl+p2)/2 in Eq, 76;
a = shear stress;
T = stress which has taken consideration into the turbulence;
$ = arbitrary seal or or tensor;
4> = arbitrary vector or tensor;
A = difference;
v = gradient vector operator;
D/Dt = material derivative;
< > = average of a variable integrated over lamina;
1 (prime) = fluctuating component of a variable when considering the
turbulence;
"(upper bar) = mean value of a variable when considering the turbulence;
_(under bar) = vector;
.(double under bars) = tensor;
Subscripts
a = arbitrary control;
b = bottom;
c = critical;
e = entrance and exit;
f = friction;
i = interface;
m = material; and
= multiple;
n = normal;
o = original;
q = equivalent;
r = circulation;
s = surface;
w = wetted perimeter which does not include the bottom, or wall;
x = x component of a vector;
y - y component of a vector;
z = z component of a vector;
149
-------
H * horizontal slope flow;
L * hypoHmnetlc flow;
M » momentum coefficient Included;
N » non-dimensionalIzed by dividing with the critical depth;
P » positive slope flow;
S • water to be supplied Into the flume;
U = upper layer analysis;
0 = atmosphere;
1 = upper layer; and
2 » lower layer.
150
-------
SECTION X
APPENDICES
A. Computer Program "H2PROFIL" and an Example of the Print-out
B. Computer Program "ESTMFBFI" and an Example of the Print-out
C. Computer Program "STRAFLOW"
151
-------
APPENDIX A
COMPUTER PROGRAM "H2PROFIL" AND AN EXAMPLE OF THE PRINT-OUT.
This program Is to calculate the H2 profile in the open-channel flow in
a rectangular experimental flume, using the 4th-order Milne predictor-
corrector method. This is written in WATFIV, Generally, comments will
precede statements, except for GO TO statements.
Symbols
Symbols used in the program are shown in parentheses. "0" in variables
expressing length usually indicates initial values.
ACC « exponent in "UP" (see below), which is the upper limit to be used
to determine an integration interval. ("DM" is the lower limit.);
DX(=HH) = interval of an integration;
FQ = equivalent friction factor for the bottom and the wall;
HC = critical depth, in feet;
Q - flow rate, in cfs;
X = longitudinal distance from a starting point, 1n feet;
XF = limit of X to terminate a calculation;
Y(=H) = depth, in feet;
YI(=H) = depth, in inches;
YID(-HID) » difference between depth and the initial depth, in inches;
and
YP(=W>) = slope of the free surface.
Data Card Specification
1st cart Includes Q and HO in FORMAT (2E10.3).
2nd care includes FQ 1n FORMAT (E10.3). FQ is positive. When going to
next dati, let FQ have negative value. To terminate computation, let
FQ be 0.
152
-------
/PROGRAM HPPROFILtNOEXT
C
C _ *** DECLARATIONS AND INITIALlZATIOMS
ACC/5./t XF/l6./»
1 A A (/
UP= ?9./lG.**ACC
DN = uP/600. ...
C
C *** | 1ST OF FORMAJTS
100 FORMAT(2E10.3)
110 FORMAT(E10.3)
?00 FORMAT(*1*,///////
I / t • , T47 , • H2 PROFILE » « T7U« t_ACC_=_• ±F I-H
210 FORMAT(•-•,T22,•Q=«,1PE1n.3»T4?» »HC=f,F10.3)
F 0 R M A T < ' « , T 2 2 , » F Q = • «1 P_E 10 . 3 / :
> ,T76,»YIDf/)
230 FORMAT V' • »T22 « 1PE9 . 3» El<* . 3 t ?E1 2 . 3 , E13. 3 )
FORMAT ( * • « T27 . • INTERVAL IS JHALVFD. _P.X=_._!_«.lPElIlL.!LL
FOPMATC ».T27»'f INTERVAL IS DOUBLED DX= »«1PE10.1)
P60 f-ORMAT( f 1' « * *) ...
C
C *** READ Q _ANp_^J^JND^RjJ\lT_
300 READ 100« (?» HO
KOI= HO*12. . .. ._
HC= rG**2/32.17)**.3333333
PRIIVT 200, ACC •
C
C *** TO CHANGE FQ
XtO READ 110« FQ
IF(FQ) 300,1000,320
*** PRINT O, HC, AMD FQ Afin GIVE INITIAL VALUES
3PO PRINT 210, Q, HC
PRINT 220, FQ
Y (1 .1 ). = 1 • . _._
Y
-------
C *** COMPUTE SLOPE OF FREE.SURFACE.
UOO YP(K.2)= FQ*(l.+2.*Y**3>
C ***_ T_0 PRINT_INI HAL _V ALLIES . _ ._. -.
..._ iFdM.EQ.O) GO TO 999
GO TO (910,920,950,960), LI
C *** GO BACK TO INTEGRATION
C
C *** TO PRINT OUTPUT
X= YOJJT Li.)..-1.
HP= YPOUT(2)
IF(TW.EQ.O) HPO=^HP
H= YOUT(2)
HI= 12.*H
HID= HOI-HI
230, X, HP* HV Hit
_ _
~IN= 1
f-
C *** TO TERMINATE A CALCULATION
IF(X.GT.XF) GO TO 310
IF(HP*HPO.LT.O. ) GO TO 31 n
GO
_ _
*** "fiO BACK TO INTEGRATION
^ _
C- *** INTEGRATION - 4*TH ORQFR RUNGF-KUTTA
C" WITH GTLL»S CONSTANT
C ____ «*» INITIALIZE_.QJ _______________
SOO K= 1
DO 9fl5 I = lt2
905 QQ(I»= 0.
Ll= 1
L?= 1
___ _ _____________ .......
C "" *** RO TO COMPUTE THE SLOPF OF FREE SURFACE
^ __
C *** TF K=*tt GO TO THE PREniCTOR-rORRECTOR PART
910 IFCK.EQ.t) GO..T.0..9SO _______________ ______ _______ ............
K=
___
Y(K,T)= Y(K-itl)
915 YP(K.I)= YP(K-1,
J= 1
L2= 1
154
-------
C *** TFJ=5» GO TO SET THE OUTPUT OF THE INTEGRATION
IF(J.E.Q.5) GO TO 999
00 995 I = it2
DD= HH*YP(K,I)
RR= AA(J)*DD-BB(J)*QQ(I)
Y(K.T)= Y(K.I)+RR
925 QQ(T)= QQ (I ) + 3. *RR-CC ( J|*nD_.
J=
Ll=
GO TO 400.
C *** GO TO COMPUTE THE SLOPE_ OF_ FR.EE__SURFACE_
C
C
C *** INTEGRATION - <*TH ORnFR MILNE PREDICTOR-
C CORRECTOR
C" *** INITIALIZE" PMCf"
9^0 PMC= 0. .
MIL= 0
C _
C *** RET MODIFIED Y WITH PREDICTOR
K= ^ _
Y(5«1)=~Y<1«1>+HH
= Y(li2)+1.333333*HH*_(?._*YP|?*2L
-YP(3t2)+2.*YPC+«2))
?)= PRED-.9fe55l72*PMC
GO TO
C *** "fib "TO" COMPUTE THE "SLOPF OF FREF SURFACE
C *** GET IMPROVED Y WITH CORRECTOR.'
950 COR= Y<
PMC= PRED-COR
Y(5.9)=
C _ __ _____ _________ —
C *** TEST IF THE INTEGRATION INTERVAL IS O.K.
TEST= ABSd.-PRED/COR) .... _... _________
IF(TFST.GT.UP) GO TO 970
IF(TFST.LT.DN) GO TO 980 . _..
C
c *** MO CHANGE IN INTERVAL ___________
Li's U
GO TO 400 _________________________ ---------------
C *** RO TO COMPUTE THE SLOPE OF FREE SURFACE
T55
-------
C *** DETERMINE Y. ANO..YP..FQR .TH£..JNK£F.ATIjQISuQF..
C NEXT POINT
DO .965. irJ,»2 ________________________ ...... _______________________
DO 965 K=l«4
Y(K,T)= Y
965 YP(K.I)= YP(K+1.I)
K= 5 ........ ...... ... ____________ ............... -
MILs 1
L£= ? ___________ _____________________________ ......... _ __________ , ________ ,
GO TO 999
c *** go TO SET THE ouTpyT OF AiJ IMTEGPATJON
C
C *** HALVING INTERVAL ______ __.... __________
970 HH= .5*HH
JPLPIRLT _?.9Pt HH __ _. ____________________ _______________
C *** IF MlL=Ot GO To THE RUfoGF-KUTTA PART
IFd^lL.EQ-O) GO TO 900
00 975 I = l»2
975 Y(1.T)= Yd, I) _______
GO TO 900
C *** GQ TO THE Ryi\!GE-KUTTA pART
C
c *** DOUBLING INTERVAL
9flO HH= ?.*HH
PRINT 250. HH
DO 9A5 I=l«2
9«5 Y(1,T)= Y(3,J[)
GO TO 900
c *** GO TO THE RUNGE-KUTTA PART
c
C *** TO SET THE OUTPUT OF AN If-JTEG.RATlpfv
99«? YOUT = Y ( K , 2J_ ...............
YPOUT(2)= YP(k«P)
GO TO 600
C *** GO TO PRINT THF OUTPUT
C
C *** TO ELIMINATE COMPUTER P'FSSAGF OUT OF
C__ ....LAST PRINTED L..PAGE ......... __________________ ....._ ........ _.
1QOO PRIMT 260
STOP
END
-------
H2 PROFILE"
ACC=
G = 1 .1J4E-01
FC = 1.35nf-0l
HC= 7.280E-02
C.
i.
2.
4.
6.
P
X
OOOE
oo o^
r>ooE
oGGc
I
0COE
000E
OOGE
I
YP
1 .8COE
-OI
oo
no
no
NTF.HVAL
00
00
00
NTERVAL
01
01
-3.062E-OH
IS DOUBLED
IS DOUBLED
3.294E-01
3. 29If 01
3.286E-01
3.285E-01
DX= 2.0000r
3.28?E-Oi
3.276E-01
3.269E-01
DX= ^
3.263E-01
3.250E-01
3.238E-01
Y^
3.953E
3.949F.
3.94BF
3.94PF
- oo
3.93RE
3.931E
3.9?3r
- oo
3.916F.
3.901E
s.aapiE
00
on
00
on
00
oo
00
00
on
00
0
3
7
1
1
?
2
3
Pi
f.
._Y.in
.OOOF-01
.f,70F-03
.349F-03
.104>-02~
,47i»F-02
.?lg,F-0?
.962F-02
.711F-02
.P22F-02
.7H8F-02
157
-------
APPENDIX B
COMPUTER PROGRAM "ESTMFBFI" AND AN EXAMPLE OF THE PRINT-OUT.
This program is to obtain, from q/v and h, the estimate of fb value, and
the value of fj which is used as a starting point for the trial and error
method. This is written in WATFIV. Generally comments will precede
statements, except for GO TO statements.
Symbols
H = depth, in feet;
HI = depth, in inches;
HIO = starting value of depth for a computation, in inches;
SC = shape correction factor (see the program for the definition); and
QN =< q/v.
For the definitions of other terms, see the program under statement
number 300, 320, and 570.
Data Card Specification
Only one card includes profile identification, q/v, and HIO in FORMAT
(A7, 3X, E7.3, 3X, F4.1), To terminate computation, it is necessary
to have a card starting with FINISH and having Os for q/v and HIO.
158
-------
/PROGRAM ESTMFBFI«NOEXT
c
C ..... ***..n_ECLARATIpNS. .AND INITIALIZAT.IP&LS ______
CHARACTER PROFIL*7
REAL FQ<2)» FQP(2), RW<2)« TRW(2)»
1 RB<2) , F&
FORMATC • + ' »T42» • t SlfJCF 0 OOFS MOT COfjVFRGEM
FORMAT (.' + •» TH2 » « » SINCE R£ , . LE_..._n ...JLL
FORMAT( f"4-»VTtf2« ' « SINCE FR ,LF. n.»)
FORMAT( »-• «T23* «HI« .TH9» «FT» tT76« »PEI»/)
FORMAT (' • t T22 . F'f . 1 « TH5 « 1PE9 . 3 , T?3 , E9 . 3 )
270 FORMAT C •!•»••)
C
C _ _ ___
C~ *"** READ PROFILE IDENTIFI CAT TOM » Q/NUi AMD HTO
?500 READ 100* PROFIL» QN» Hln
C *** TO TERMINATE COMPUTATION
IF(PROFIL. EQ. 'FINISH* ) GO TO 999
C *** PRINT PROFILE IDENTIFICATION AMD Q/MH
GF= ON** 0.2
HI= HIO
C
C *** POR NEW Hl» DETERWTNF_ VARIABLES RELATrO TO H
3?0 H= HT/12.
HF= _n.223**p._8*H. _______ _________________________ ....__ ........ _______ ...... ____ ..... _____
GMH= "QN'/H
H2= 9.*H
P= 1 .+H2
IF(H.GT.0.5) GO To 330 ________________
SC= n.8727*( ALOG(P)-H)
_^L°_TO 350 __ ____________________ ........ ________________
330 SC= n . 8727* ( A'LOQ < P/H2 ) -0 . 5 )
159
-------
C *** INITIAL APPROXIMATION OF FQ
350 FQ(1)= FQO
DFQ= DFQO
J= 1
GO TO 560 _ _
C *** fiO TO CALCULATE D(2) AND 0(1)
C
C *** FIND IF THE CURVE RISFS OR FALLS WHEN FQ
C INCREASES AND jHE|NJ DECIDE INITIAL DIRECTION
c OF INCREMENT" OF'FG(DFQ)
360 DDr D(2)-D<1)
c *** TF OD= o, THIS D-FQ "CURVE is CONSTANT
C AND 60 TO PRINT »NO SOLUTION* _
IFfnn.EQ.O.) GO TO 786
DFQ= - SIGN <1_. »Dp)*sIGN( OFQ.D(l) )
J= "?"
IF(DFQ*DFQO.GT.Ot ) GO TO 400
GO TO 560
C *** SINCE POSITIVE INCREMENT WAS NOT RIGHT*
C GO TO CALCULATE NEW D(2)
C
C *** CHECK IF THE SIGN" OF FH2) AND 0(1) DTFFFRS
UOO IF(n(2)*D(l) ) f40,
-------
C *** .SHORTEN QFQ . F_QR_FQ._.£LO.SE...IO_ THE_LIMIT0_F ...E9__.
500 DFQ= DFQ/10.
560
C *** GO BACK TO CALCULATE 0(2)
C . _.
C *** TF FQ IS BEYOND THE LIMIT OF F0» NO SOLUTION
550 IF(FQ(1_) •GT«H.•9.99999
C
_C *** nEFINE._NEW_FQ.L2J
560 FQ(?)= FQCD^-OFQ
C *** TO GET THE SOLUTION CLOSE. ,TO
C THE LIMIT OF F Q
IF_< FQ (2 ) .GT.5. ,OR .FQ (21.LT .AO^QnOOpl )
C
C *** CALCULATE _D_ANQ
~
__
570 ~00 ~AoO =l»2
C *** TO AVOID REPEATED.
IFII.EQ«1«AND«J.EQ.2) GO TO
FQP(I)= FQ(I)*P ____________________________ ... _______
RW(I)= HF/QF/FQp GO TO 7ft5
C *** T6 "CHECK WHICH FQ "is" CLOSER "~r6" THE
DAD=
__
_ iT**" TO OBTAIN A SOLUTION
C BY THE LIMIT. OF D _
IF(nAD.GE.O.O) GO TO 610
IF(ARS(D(2) ).LT..Op01) GO JO
L= 1
GO TO 650
fi40~TFTARS(D(r)).LT..0001) GO TO 7*0
L= ? __ __ _. .... —
t€1
-------
C *** TO OBTAIN A SOLUTION „
c RY THE LIMIT OF DFQ"
650 IF(ARS(DFQ).LT.l.E-"9) GO TO (745t755)« L
GO TO (360*400). J
C *** GQ_BACK TO THE iTERAriQN
C
: *** SET-THE SOLUTION __ ___
C *** J= 3 IS TO PRINT •*« AFTER A SOLUTION
C _ TO INDICATE THAT THIS SOLUTION IS OBTAINED
C ~RY THE LIMIT OF DFQ
745 J= 3
750 AFQ= FQ(2)
AFQP= FQP(2)
ARW= RW(2)
ARB= RB(2|_ __
AFB:TFB(2)
GO TO 770
755 J= 3
760 AFQ= F<3(1) __
AFQP= FQP(l)
ARWj=_ .RW(11_ .
A'RBs RB(1)
AFB= FB(1)
C *** CALCULATE REYNOLDS NUMBER AND PRINT OUTPUT
770 REB= QNH*ARB ___ _____ __ ___
REW= QNH*ARW
_ _ Pf? INT 210« HI t AFQP, AFB, REB, REW _
C
C *** TO PRINT »*• MENTIONED ABOVE _____ ___
IF(J.NE.3) GO TO 800
PRINT 220 _________ _________________________
GO TO 800
C _ *** TO PRINT THE REASON WHY THFRF IS NO SOLUTION
7*5 J=~ 14
GO TO 790 ___ __ ________
786 J= 5
GO TO 790 ____ ____
787 J= 6
GO TO 790 _____ _____________ _ _
788 J= 7
GO TO 790 ______ ______
789 J= ft
790 PRINT 230f HI ____ ____
IF(J.EQ.**) PRINT
__. .____
IF(J.E0.6) PRINT 242
IF(J.E0.7) PRINT 243
IF(J.EQ.8) PRINT 244
-------
C *** INCREASE HI
800 HI= HI + .I
C *** GO. BACK TO CALCULATE MORE OUTPUT
IF(HI.LE.HIO-H,95) GO TO 3?0
C __
C
C *** PRINT HEADING FOR FI TABLE
PRINT 250
C *** SET HI TQ HIO_
His RIO "" " ~"
C
C *** CALCULATE FI AND REYNOLDS NUMBER
C AND PRINT THEM
900 REi= .5*QN/(l.+Hl/i2.>
Fir .223/REI**.?5
PRINT 260, Hit Fl« REI
C _
C *** INCREASE HI
HI= Hl+.l
C *** TO CALCULATE MORE OUTPUT AND GO TO NFXT DATA,
c IF Fi TABLE is DONE
IF{HI-HIO-1.95) 900»300t300
C
C *** To TAKE COMPUTER MESSAGE OUT OF
C i AST PRINTED. PAJ3E.
PRINT 270
STOP
END
JLfi3_
-------
Sl-3/30
Hi
lfi.1
18.2
18.3
18.4
18.5
18.6
18.7
18.6
18.9
19.0
19.1
19.2
19.3"
19.4
19.5
19.6
19.7
19.8
19.9
20.0
Hi.
18.1
18.2
18.3
18.4
18.5
18.6
18.7
18.8
18.9
19.0
19.1
19.2
"IT. 3
19.4
19.5
19.6
19.7
19.8
19.9
20,0
Q/MI=
FB+2H*FUi
2.643E-01
2.650E-fll
2.656E-01
2.663E-ni
2.670E-01
2.677E-ni
2.664E-ni
P.691F-01
?.698E-nl
2.705E-01
2.712E-01
2.719F-CJ1
2.726E-01
2.733E-nl
2.740E-nl
2.747E-fU
?.754E-nl
2.761F-01
2.768E-01
2*775E-nl
.... .,,-,.. - . -.
4.950E 03
FB
1.373E-01
1.372E-01
1.371E-01
1.371E-nl
i.37nE-ni
1.36qE-01
1.36BE-01
1.367E-fil
1.3b7E-Dl
1.366E-01
1.365E-01
1.364E-01
1.363E-01
1.363E-01
1.36PE-01
1.36JE-01
1.361E-nl
1.36QE-nl
1.359E-01
1.350E-01
FI
3.979E-02
3.962E-02
3.985E-02
3.989E-02
3.992E-02
3.995EI-02
3.999E-02
4.002E-02
4.005E-02
4.00RE-02
't.OllE-02
4.015E-U2
4.di6E-n2
4.021E-02
4.C24E-02
4.028E-02
4.03lE-n2
4.034E-02
" 4.037E-02
4.04PE-02
RER
03
03
2.531F H 3
~
2.515F H3
?.507E 03
P.500F 03
2.492E n"3
2.47&F n3
03
2.453E 03
03
REV
~7~."A8~3~F ~ft?
7.A67F 02
7.K51F 02
7.ft34F 02
7.P1RF 02
7.A02F 02
7.786F 02"
7.770F 02
7.754F 02
7_.73flF 02
7.722F 02
7.707r 02
0?
02
7.659F 02
7.6*»3F 02
02
)2
7.
HET
02
02
9.A67F 02
9._A34F 02
9.ft02F 02
9.770F 02
9.738F 02
_ _
9.674F" 02
9.fe43F 0?
9.f,12F H2
9.ES81F 0?
9.550F 02
9.519F P?
9.489F 02
9.459F 02
9.4"29"r" "0?
9.399F 0?
9.369F 02
"9.3l"PF 02"
9.281F 02
164
-------
APPENDIX C
COMPUTER PROGRAM "STRAFLOW"
This program is to calculate profiles of the interface and the free sur-
face of the two-layered flow, using 4th-order Milne predictor-corrector
method. This is written in WATFIV. Generally, comments will precede
statements, except for GO TO statements.
Symbols
"0" in variables usually indicates initial values. "N" in variables in-
dicates non-dimensionalized variables. Variables of length is written
in feet. If "I" is included in the variables, they are in inches.
ACC = exponent in "UP" (see below), which is the upper limit to be used
to determine an integration interval ("DM" is the lower limit).;
BETA = momentum coefficient;
BF = FACTOR/BETA;
DELTA = ODELTA/MHC**3 for method 1;
DTDH = difference between the total depth and the horizontal drawn through
the free surface at the starting point;
DX = an integration interval;
FACTOR = ,125*FB/SB for the S and M profiles and = .125*FB for the
H profile;
FB = bottom friction factor;
FI = MFI*OFI, in which MFI is substituted by trial and error;
GHN = virtual stratified normal depth;
H = depth of the lower layer;
HC2 = MHC*OHC2, in which MHC is substituted by trial and error;
HP = slope of the interface;
ODELTA = AP/P calculated from Tl and T2;
OFI = originally estimated interfacial friction factor;
165
-------
OHC2 = stratified critical depth for the Ipwer layer analysis calculat-
ed according to the definition Csee under statement number 300 in the
program, 32,17 is the value of gravitational acceleration in ft/sec2.);
Q « flow rate, in cfs;
SB - slope of the channel bottom;
Tl = mean temperature in the upper layer ("C" in Celsius and "F" in
Fahrenheit);
T2 = mean temperature in the lower layer;
TD = total depth of the flow;
TDFAC > in method 1, .125*FI*DELTA/SB for the M and S profiles and =
,125*FI*DELTA for the H profile. For method 2, the above TDFAC is
divided by BETA;
X = longitudinal distance from a starting point; and
XF = limit of X to terminate a calculation;
Data Card Specification
1st card is for profile identification, as Sl-3/30, punched from the 1st
column. 1st column indicates the slope-M, S, or H. 2nd column is either
1, 2,, or 3. Any identification can be put in from column 3 to 7.
2nd card includes DXO, FB, HOI, TDOI, and OFI in FORMAT (5E10.3).
3rd card includes SB, TC(1) - Tl in Celsius, TC(2) = T2 1n Celsius, Q,
ODELTA, and OHCI in FORMAT (6E10.3). If OHCI is not known, let it be 0,
and it will be calculated.
4th card shows method, either 1 or 2. When going to next data, let method
be 0.
5th card includes either MHC or BETA, depending on the method. Initial
value, limit value, and increment are punched in FORMAT (3E10.3).
If only one value is used, it may be punched at the initial value; if a
value smaller than that value is used for the limit value in this case,
increment should be zero.
6th card includes MFI in a similar way to the 5th card.
166
-------
/PROGRAM STRAFLpWtNOExT_
C
C *** DECLARATIONS AND INITIALIZATIONS
CHARACTER SLOPE*1« IDENT*!, DATE*5
INTEGER PROFIL _
HEAL MFIt MFIOt MFTLt MFTD, MHC, MHCQ, MHCL, FIHCD,
1 XF/1000./«
2 AACM/.St .?92ft932t1.707107,.1666667/,
3 BBJ-**}- -1 i-!-•-??2A9?_?J 1.707107, .3333353/t
4 CCT(V) / T5» . 292fl~932 «"l. 707107, . 5/
Y(5»3)•_YP( ^Sj * _YOUTC3>< YPOUT(3).
1 " GHNNCTS ) , QQ ( 3 )T TC(?), TF(2)t
2 PRED ( 3 ) » COR ( 3 ) t_P«C_m.
D AT A ( YP (K . 1) » K = 1, 5 )75*1. / ~'
C . .
C*** LIST OF FORMATS
110 FORMAT(5E10.3/6E10.3)
1?0 FORMAT(Il)
130 FORMAT(3E10.3/3E10.3)
?00 F OR H AT (_ • 1 • «////„//'.__' >
P01 FORMAT (•"""• ,'/"•" "• )
P02 FORMflTC • tT!3t Al«Al«A5)
?03 FORMATi f + •
pf)«4 FORMAT (* + t »T65t tHETHOD f_»T..lJ
P1 P FORM AT ( • 0 ' , Tl 3 ,"• fl= f , F sVl , » C= * , F* . 1, »F • . T*5, • TP=
1 __ « F 5 .1 « • C = • • t F 5 . Ll-LfJLlIfi 1 'IQ^1-!.l.Pf-?JU5
2 """ "/"• f tf I3t
3 tT56t fODEl.TA=' tElO.3
• ,Tl3t"»OHC2I = f tElO.sVTSRt »OFI=« ,E10.3)
PPO FOPMATt*+*»T52t•
F 0 R I" ft T < • 0 • « T^3j_«_HC_2l^_*_lPE10«3tTS7t »MHC=*
??ii FORM AT ("• + '» f 83 » • HC ?= » » 1PE1 0 . 3 )
?32 FORjv- aT ( * + * t T57 « » DELTA = *jJ.JPEl 0^. 3 )
? 3 3 F OR f! A T ( ' + ' « T62 « ' BE T A= • » Ffi . 3 )
935 FORMAT( f^f »T102, ' HC2/SB=» .
?^f. FORHA'TC • + • »Ti2i« »BF=' tiPE'i
237 FORMAT! ' + * «T106._*JBFJ=* «1PE10.3)
"
_ _______ _ _
F OP K A~f( ' 0 * Vfl 3 * ' F I = •". 1PE1 0 . 3 * T 37 » ' MF I = » t OPF6 . 3 )
FORMAT ( f^» «T57, »FI/FBs« t_lPElO._3)
1 » T58 « • HP • « T67 « • DTDHT •^.Tfl7«_» XN »«T98« *HN* _
2 » T107« -GHNN' .tllf, •TD^f . Tl?8, • HPMf / )
FOPMAT(»Q««Tl7«»X«fT30«*HT* « T49 y* TDI * « T56« »HP*
»XN» ,T101, '"HN» , T113, 'TON'/)
T67
-------
2AO FORMATC »»T15.'HALVED INTERVAL HX= ».1pEin.4
1 »T85«'HALVED INTERVAL DXN = «VE1 h . «*)
FORMATt1 » tTl5» 'DOUBLED INTERVAL HX= »,l.PEin.4
1
370
1
271
280
?82
1
283
284
235
28f
2«7
380
2«9
1
290
2*91
292
299
FORMAT
FORMAT
FORMAT
FORMAT
FORMAT
FORMAT
FORMAT
FORMAT
FORMAT
FORMAT
FORMAT
FORMAT
FORMAT
FORMAT
FORMAT
FORMAT
«
(
«
(
(
(
{
t
(
(
{
(
(
(
(
*
{
(
(
(
T85
t »
t »
'
»
IS
t
t
*
*
t
«
t
*
t
t
t
•]
t
»
i
t
•
t
'
i
»
»
•DOUBLED INTERVAL HXN= •tElQ.U)
T9« 1P4E13. 3 » El 2 . 2 • Tfl3 . E9 . 3 , IX . 2E1 _3 . 3 )
Tl5« 'H AHPROACHFS GHN« )
T15« 'INTERFACE BECOMES HORIZONTAL')
T15
» »
SIGN OF TfvTERFACTAL SLOPF '
CHANGED' )
t
t
»
t
«
»
t
T15
T15
"T15
T15
T15
T15
Tl5
» •
t •
« «
t »
« »
« •
t »
PROFILE
t
t
t
L1
«
»
t
t
T15
T15
T15
' *
» '
» •
» •
)
XF=' tlPEin.3. ' IS EXCEEDED')
H BECOMES NEGATIVE' )
H EXCEEDS TOTAI nEPTH')
INTERFACF PECOMEs TOO STEFP')
FREE SURFACE BECOMES TOO STEFP')
GHf\j-CURE RECO^lES NEGATIVE')
IT IS NO LONGFtt A »«Ai,Al
FROM THIS POINT' )
CONTINUE TO THE f-iEXT PAGF»)
CONTINUED FROM THE PREVTOtlS PAGE'/)
COMPUTATION IS TERMINATED')
C
C
C
300
*** READ PROFILE
READ JL°0'__s_k9PEL'
*** TO TERMTNATF
IF (SI OPE.EQ. »F» )
*** READ DATA
READ 110 « DXO» FBt HO It
SB» TC(l)t fC
IDENTIFICATION
ACC
—
COMPUTATION
GO TO 2000
* OFT«
Q« OnELTA,
OHCI
*** TO GIVE
INDE_X_
•M») PROFIL=
•S») PROFIL=
•H') PROFIL=
IFCSLOPE.CQ
IF(SIOPE.EQ
IF (SI. OPE. EO
IF(SLOPF.EO.»M».AND.IDENT
IF(St OPE.EQ.»M»
IF(St OPE.EQ.•«»
IFCSI.OPF'.EQ.>S'
IF(SI OPE.EQ.»S»
IF(SI OPF.EQ.'S'
, AND.iDENT,
•AND.IDENT,
."AN bV I DENT,
, AND.IDENT,
.AND.iDENT,
.AND.IDENT,
IF(SI OPE.EO.'H'.AND.I DENT.EQ,
*** TO CALCULATE OHCI
"~TF'("OHC"I ."E75. 0i.T~OHCI= T2. * ( ( Q**2/nDELTA/32
1 **.3333333)
JT.EQ.
JT.EQ.
JT.EQ.
JTTFQ.
n.EQ.
JT.EQ.
n.EG.
JT.EQ.
1
o
•3
1
^
J,
4
?
1
NOMPROr
NOMPROs
MOMPRO=
NOMPRO=
MOMPRO=
NOMPROr
NOMPRO=
> NOMPRO=
1
2
3
H'"'
5
6
7
8
17)
168
-------
C *** TO DETERMINE UPPER AND LOWFR LIMIT FOR
C TEST IN tFE">REbfcfORl'cORRECTOR""HEfHOO
_ _ L[P= P9 . / 1_0 . * * ACC ___ __________ _
~DN= "uP/60d".
C *** JO CONVERT TEMPERATURE FROM CELSIUS
c TO "FAHRENHEIT"
DO 310 II=1«2
310 TF(IT)= 1.8*TC< Tlj + 32.
_C _____ * * * TQ BE U_SED _J 0_ S K I P PR IN T_T_riG_nAT A _ _
IP= n
C *** TO BE USED TO PRINT OUTPUT FRpPI BOT
C TN ONE PAGE WHEN OUTPUT ~"f"s" SHORT
IPWP= 0 _
c '" ...... "~" ........
C __ *** READ METHOD ___ _ ____ ___
"~"~S50 ~REAh 120 » METHOD
C *** RO TO NEXT DATA
IF(MFTHOD.EQ.O) GO TO 300
C *** PRINT PROFILE_ IUE_NTI_FiCATIpN ANP__METHOD
IF{ IPWP.EQ.1 ) GO TO 351
PR_TJMX_2QLO_ _ _ _ _ __ _ _
PRINT 202, SLOPE* TDENf, DATE
PRINT 203, ACC
PRINT 204, METHOD
GO TO 352 ______________ _________ ........ __________ .......
351. PRINT 201
__ __ PJ? I NT 20 2 1 _» SL^OJPF « _ JJl^fiT «_ D A T F
PRINT"~204Y METHOD " .......
352 IF(TP.NE.O) GO TO 370
C *** PRINT DATA
PRINT 210, TC(1), TF(1), TCA?), 1^2), Q,
1 SB, FB, ODElTA, ........
C
C
C
IP=
*** nEFlNE VARIABLES WHICH IS NOT CHANGED
WHILE CHANGING MHC(BETA) AKD MFI
OHC= OHCI/12.
FACTOR •l?5*fj_ __ ___ - __ ...... .....
TO 360
355
360
_ __
TF(PROFILYEQ.I) GO
PRINT 221« FACTOR
DO 355 K=l,5
GHNNC(_K)= 0.
GO TO" 37"0
FACTQR= FACTpR/SB _
FRINT 220, FACTOR
169
-------
370 IF(F!FTHOP.FQ.2) GO TO 38n
*** READ HHC AND MFI FOR METHOD 1
HEAD 130» MHCO» MHcLt IHcDi
MFIOi MFIL* MFID
MHC= I*!HCO
GO TO 390
*** READ BETA AND f|FI FOR METHOD
READ 130« BFTAO, BETAL, BET AD t"
_____ HFlQt KFTLf
~= BETAO
390 FTO= HFIO*nFl
C
C *** FOR NEW HHC OR SETA, CHANGF ___
C RELATED TO'HC" AND"PRl'Frf" fHFH'~AND"
C __ rj 0 N-D_I WE N SI 0 N A L IS E VARIABLES RELATED TO I ENGTH
"FI= F'lO
MFI= MFIO _ ___ ___
IF(MFTHOD.FQ.I) GO To'ViHT
MHC= BETACR= BETA**. 3333333 _ _____
FACToR= BF= FACTOR/BETA^
HC= ?||HC*OHC _________ __
HCT=""127*HC
HNO= HOI/HCI __ _______
TDNO= TDOI/HCI
PRINT 230 1 HCI. HHC ____ , __
PRll\iT 231 1 HC
IF(MFTHOD.EQ.^_GO TO
DELTA=
PRINT 232, DELTA
GO TO 4-30
PRUviT 233, BETA
DELTA= ODELTA
i+ SO IF ( PR 0 FI L •fQ_*2)_ W_TO 440
XFAC~="HC"7S~B
UXM= DXO/XFAC
PRINT 235, XFAC
IF(MFTHOD.EQ.l) GO TO 45n
PRINT 236, BF
GO TO 450
linn " ITXNr DXO/HC
IF(MFTHOD.EQ.l) GO TO 450
PRINT 237, BF
C
C *** FOR NEW MFI, CHANGE
C FI AND PRINT THEM
H 5 0" F R I r S^F
170
-------
TDFAC= •125*FI*DELTA
IF __
FiOO IF(PROFIL.EQ.2) GO TO 510
_C ____ FOR S ANP M^PRpFILE _________ ___ _____
GHNNP. ( K ) = FACTOR* ("f." *FR ICS> U . -Y ( K , ? ) /vYk', 3 }) )
IF ( SHNNC ( K ) .L.T . 0 . ) GO TO 036
YP(K,2)= ( Y(Kt2 ) *"*3-(SHNNC (K ) ) / (VTK V? ) **3-1 . )
Y P ( K , 3 ) = 1 . +TDF A C / ( Y ( K ,2 ) * *2 * ( Y ( K_« 3 ) - Y ( K , ? ) ) ) _____________
GO TO 520
C FOR H PROFILE ___ _____
~
D
510 YP(KV2T^TACT<3R*(i.*FRTCS/f 1~.-Y ( K«? ) /Y (K
YP(K.3>=
*** TO PRINT INITIAL VALUES
520 IF(Tl C.EQ.O) GO TO 999
GO TO (910,9aO«950,960). LI
*** R"0 BACK TO IIMTEGRATION
-------
*** nlMENSIONALJZE VARIABLES !
AOO XNr YOUT(1)-1.
HN= YOUT(2)
HI= HN*HCI
TDN= YOUTC3)
TDI= TDN*HCI
HPN= VPOUT<2)
IF(IlC.EQ.n) HPNO=
TDPN- YPOUT(3>
IF«PROFIL".E0.2) GO TO 6lQ
X= XM*XFAC
6HNN= GHNNCR**. 3333333"
GHHIs GHNN*HCI
HP= HPN*SB
D T_DH T_=JDI-TDOJ-X N *HC I_
"""" 0 f DToFl = 0 .
PRINT 270 » Xi Hit GHNI « TDI * HP» DlDHJCt
1 XNt HIM i GHNM, TON, HPW
GO TO 620
x= VM*HC
DTDHT= TDI-TDOI
PRINT 271» X» Hit TDI« HP?J, DTDHT*
1 XfNit HNi TON
6?0 LC= 1
ILC= ILC+1
"*'** TLCT IS TOTAL COUNT OF LINES PRINTED
ILCT=
*** TO GO TO PRINT IN THf; NEXT PAGE
IF(T| CT.EQ.36) GO TO 800
C *** TO TERMINATE A PROFILE CALrULATION WHFN
C TT IS PRINTED T!\t 2 PAGES
IFdl CT.EQ.7B) GO TO 846
C
C *** TESTS TO" TERMINATE A CALCUI ATlON A^'D
C RIVE THE REASON
700 IFCPROFIL.E0.2) GO TO 71n
IF(ARS(HI/GHNl-l.OnO).LT..005) GO TO 83P
IF(ARS(HPN-1.000).LT..OOij GO TO 831
710 lF(hPN*HPNO.LT,C.) GO TO fi32_ __
IF(X.GT.XF) 60 TO 'ft33 ~ " ~ ' "
IF(HI.LT.O.) GO TO 634
IF(HI.GT.TDI) GC To 835
1F(ARS
-------
GO TO (750»752»754,756«758L760»762»764-) t N.QMPRQ
750 iFtHT.GE.GHNl.AND.GHNI.GE.HCl) GO TO 790
GO TO 839 ___ ___ ____ _______
752 IFJGHNl.GE.HiTAND^Hl.GE.HCf) GO TO 790
GO TO 839
754 IFtGHNl.GE.HCl»AND.HCI.GE.HI ) GO TO 790
GO TO 839
75fe lF(HI.GE.HCI.Af\!n.HcI.GE.GHNl ) GO TO 790
GO TO 839 _____ _ ___ ____ .. ........... _____________
758 TF(H7:I~.GIE:.HI.A(MD.HT.G£:.GHMI ) GO TO 790
GO TO 839
7^0 IF{HcI.GE.GHf\iI.AND.GHMI.GE.Hl) GO TO 790
GO TO 839 __ ......... _
76? IF(HI.GE.HCI) GO TO 790
GO TO ^839 ___ ______________________________
76 FI . ._._ _____________________________
IF(PROFIL.FQ.2) GO TO flih
PRINT 250 ___
GO TO 820
A10 PRINT 251 _ . . __ ....... _
A?0 GO TO (700,998) , I_C
C *** GO BACK TO CALCULATIOM AFTER HAVING
C TNTEKVAL
__ 830 PR INT _2flJO _______________________________________________ _
G~0 TO ""850"
PRINT 281 ....
GO TO 850
PRINT 282
GO TO 850
XF ...... ___________
__
GO TO 850
PRIWT 284
GO TO 850
835 PRTMT 285
GO TO 850
R36 PR7LWT_286_
GCf"fO 850
P>7>7 PRINT 287
GO TO 850
173
-------
ASP PRINT 288
60 TO 850
*3_?_PJ?INJ_?*L9* SLOPE* IDENT
GO TO 850" ~~
8fO PRINT 292 _
C *** TO PRINT OUTPUT FROM ROTH"-MEf-HODS
C TM ONE PAGE WHEN OUTPUT TS_SHORT
flpjp IFfTLCT.LT.13) GO TO 870
C
C***" TO CHANGE MFl
MFI= NFI+MFID ___
FI= MFl*OFI "
IF(MFI.LT.MFIL) GO TO <*5n
c ~~ "
C *** TO CHANGE KHC pR BETA
IF(HFTHOT57EQ.2) GO TO 860
MHC= KHC+MHCD
IF(MHC-MHCL)
flftO bETAr BETA+RETAD
IF (RF.TA-BETAL.) tfOO, UOO. 350
fl70 IPWP= 1
GO TO 350 _
c " "
C
C *** INTEGRATION -'
C WITH GILL'S CONSTANT
C *** iNItiALIZT QQ
900 K= 1 _
DO q'ns 1=113 ~
90!i CO(I)= 0.
Llr i
L2= 1
GO TO 500
C *** RO TO COMPUTE THE SLOPE OF INTERFACE
C AND FREE SURTACE ~
C
C *** TFK=4* GO TO THE
910 IF(K.EjQ,Llt) GO TO 930
~K= ~K+i
DO 915 1=1*3
Y (K . f T=~ Y (K-l,Tj
915 YP(K,I)= YP(K-1,I)
J="l""~" -""
L2= 1
-------
C *** TF J=5« GO TO SET THE OUTPUT OF THE INTERRATTQN
9?0 IF(J.EQ.5) GO TO 999
D0_9?5L I = 1_V3.__ _ __
DD = HH*YP(K,I)
RR= AA(J)*DD-BB= QQ(I)+3.*RR-CC(J)*nD
J= J + l
Ll = P
GO" f"6" 500
C _*** fiO TO COMPUTE THE SLOPE OF INTERFACE
C ""AND FREE SURFACE
C
C *** TNTE'GRAf ION - VTH ORnER >1l"l NE">REDICTOR-
C CORRECTOR METHOD
"C *** INITIALIZE PMC
930 DO 9?55 I = 2«3
935 PMC(T)= 0.
MTLs 0
C
C *** fiET MODIFIED Y WITH PREDICTOR
9H.O K= F5
Y ( 5 O ) = Y ( 4 1 1 ) +HH
DO 945 I =2* 3
PREDf I )= Y(l« I)+1.333333*HH*(f?.*YPJ2«I ) _
! -- -YP(3Vl)-»-?'.*YP(4Vl) )
_ _? ^ _T_{ 5 • I If _ P ? ED -.L9-^5-5-1-7-? * P^C( I ) _____
LT= s
GO TO 500 _ ___ _
C *** GO TO ' COMPLTfE THE SLOPE OF INTERFACE
C AND FREE SURFACE ________________________
c ' " " ..... ~" .
C *** RET iPIPROVEn Y WITH CORRECTOR ____ _
= 2 » 3
COR< T)= Y(3«
PMC(I)= PREDJI)-COR
9f>5 Y (5 ,"f ) = COR ( I")"+"."03lT4lB2fl*PHC ( I)
C
--C ***~~fE~ST IP THE INTEGRATION"INTERVAL is O.K.
AMAX= 0.
DO 9R7 I=2t3
TEST= ABS(1.-PRED(I)/COR(T))
IF ( AMAXV6E". TEST )" GO TO
A_MA_Xs_TESJ_
CONTTNUE
IF(AMAX.GT.UP) GO TO 970
IF ( AMAX .i-T. DNT GO TO 960
175
-------
*** NO CHANGE IN
Ll =
K= R "
MIL= 1 _____ ______
L2= ?
____ GO TO 999 __
C *** GO TO SET THE OUTPUT OF AN INTEGRATION
C ________
C *** HALVING INTERVAL
970 HH= .5*HH ___ _
INTr 1
GO TO 990
C~~ *** fib TO CALCULATE DX
C *** IF MIL=0. GQ TO THE RUMGE-KUTTA PART
972 IF(P>IL*EQ«0) GO TO 900
DO 975 I=lt3 _
975 Y(1,I)= "Y(^ti")
GO TO 90C
C *** GO TO THE RUNGE-KUTTA PART
C
C *** nOUDLING INTERVAL
980 HH= 3.*HH
IWTr 2
GO TO 990
C *** GO TO CALCULATE DX
9P2 DO 9fl5 I=l»3
985 Y(l.D=Y(3iI)
GO TO 900
C *** RO TO THE RltNGE-KUTTA PART
C
C *** CALCULATE Dy~
990 IF(PROFIL.EQ.2) GO TO 99? _
DX= HH*XFAC
GO TO 99*f
99? DX= HH*HC
176
-------
*** PRINT CHANGED INTERVAL
991 IF(INT.EQ.2) GO TO 996 ~
PRIN T 260 t _D_X_« _HH
GcTfo"997"
99£ PRINT 261 t CX» HH
*** rOUNT PRINTED L
997 LC= ?
ILCM = ILCM+1
ILCT= ILC+ILCM
*** T0 GO TO NEXT PRINTING PAGF
IF(T|CT.EQ.38) GO TO 800
c *** TO TERMINATE COMPUTATION OF A PROFILE
998 IF (II CT.EQ.82) GO TO 840 _ ___ ___
GO TO (972*982), INT
C _ *** J^ 1° SEI A STARTING VALUF FOR RUNGE-KUTTA PART
C '"
C *** jo SET THE OUTPUT OF _A_N_
999 YOUT(D= Y(K,1)
GHNMrB= &HNNC
-------
SELECTED WATER
RESOURCES ABSTRACTS
INPUT TRANSACTION FORM
1. Report No,
w
4. Title '
HYPOLIMNETIC FLOW REGIMES IN LAKES AND IMPOUNDMENTS,
t.
g. i ••form , Orge
7, Authorfs) . . . - u
Edlnger, J.E., Yanaglda, N., and Cohen, I.M.
5. Organization
Pennsylvania University, Philadelphia,
Dept. of Civil and Urban Engineering
SO.
11. C On t rr. >;•.:I •jJK.-'r
R- 800943
\i- £yp> ^-' j
Period O'/srerf
12, Sr-nsorin Organ tt/ca
15. Supplementary Notes
Environmental Protection Agency report number, EPA-660/2-7^-053» June 1971*
is. Abstract Tne "hypollmnetic flow" 1s a two-layered flow with the upper layer stagnant.
This report presents the possibility of different flow regimes for the hypollmnetic
flow which may be determined from the parameters of slope of channel bottom, flow
depth, flowrate, density difference of water In the two layers, and channel roughness.
The analysis Is limited to the steady-state case of the hypollmnetic flow and the
"upper layer analysis" 1n which the lower layer 1s stagnant. Interfadal profile
equations which predict possible existence of ten different flow regimes for the hypo-
limnetic flow and two regimes for the upper layer analysis were obtained from the
equations of continuity and momentum for two-layered flow. Experimental apparatus,
consisting of a large scale open-channel tilting flume and water supply and control
systems capable of circulating water of two different temperatures was designed and
constructed. The experiment employed observations of a dyed flow layer and measure-
ments of vertical temperature distributions In the flume. It was found that eight
different flow regimes could be generated 1n the flume for the hypoHmnetlc flow on
the positive and horizontal slopes.
na. Descriptors *strat1f led flow, *Flow profiles, Mnterfaces, *Research facilities,
Continuity equation, Density currents, Fluid friction, Free surfaces, Hydraulics,
Impounded waters, Momentum equation, Non-uniform flow, Saline water - fresh water
Interfaces, Steady flow, Thermal stratification.
identities *Hypol1mnet1c flow" regimes, *Exper1mental open-channel flume,
*Interfac1al profile equations.'
17c. COWRR Field & Group
A variability
Se 'rityCt ;s.
21,- No, of
Pages
Send To:
WATER RESOURCES SCIENTIFIC INFORMATION CENTER
U.S. DEPARTMENT OF THE INTERIOR
WASHINGTON. O C. 2O24O
Abstract™ Norlo Yanaglda
institution University of Pennsylvania, Philadelphia
WRSIC 102 (R6V. JUNE 1971)
U.S. GOVERNMENT PRINTING OFFICE: 1974— M4—31S>:437
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