jyLEAgK WATEB POLLUTION CONTROL RESEARCH SERIES 15040 EJ2 12/71
Selective Withdrawal From
A Stratified Reservoir
ENVIRONMENTAL PROTECTION AGENCY WATER QUALITY OFFICE
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Selective Withdrawal From A Stratified Reservoir
Jorg Imberger
and
Hugo B. Fischer
University of California
Berkeley, California 94720
for the
Environmental Protection Agency
Project Number 15040 E J Z
December 1970
For sale by the Superintendent of Documents, U.S. Government Printing Office
Washington, D.C., 20402 - Price $1
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EPA Review Notice
This report has been reviewed by the Environmental Protection
Agency 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 consitute endorsement or
recommendation for use.
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ABSTRACT
This study describes both theoretically and experimentally the flow
into a line sink in a linearly density stratified reservoir. The
geometry of the boundaries is simplified to a parallel walled duct with
the line sink at the center of the fluid. The primary focus is on par-
titioning the flow into distinct flow regimes, and predicting the with-
drawal layer thickness as a function of the distance from the sink.
For fluids with a Schmidt number of order unity, the withdrawal layer is
shown to be composed of distinct regions in each of which a definite
force balance prevails. The outer flow, where inertia forces are neg-
lected, changes from a parallel uniform flow upstream to a symmetric
self similar withdrawal layer near the sink. Inertia becomes a force of
equal importance to buoyancy and viscosity for distances from the sink
smaller than xc = (q/v)3'2 (vD/eg) '4 . When viewed from the outer
coordinates, this is a small distance, but in physical terms it may be
equal to, or longer than, the length of the reservoir, depending on the
sink discharge. The equations valid in this inner region, and which
have solutions capable of being matched to the outer flow, are derived.
Using the inner limit of the outer flow as the upstream boundary condi-
tion, these inner equations are approximately solved for the withdrawal
layer thickness by an integral method. The inner and outer variations
of 6 , the withdrawal layer thickness, are combined to yield a com-
posite solution, and it is seen that inclusion- of the inertia forces
yields layers thicker than a strict buoyancy-viscous force balance.
Also in terms of the inner variables, the equations, their solutions
and any experimental data are shown to be dependent only on the Schmidt
number.
Experimental laboratory experiments were carried out to verify the con-
clusions reached from the theoretical considerations. The observed
withdrawal layer thicknesses were shown to be closely predicted by the
integral solution. Furthermore, the data could be represented in terms
of the inner variables by a single curve verifying the independence of
any parameter other than the Schmidt number.
Prototype data is still sparse, but comparison with the measurements made
by the Tennessee Valley Authority indicates that in a reservoir the with-
drawal layers are approximately one and a half times as large as pre-
dicted. The difference in thickness is probably explained by the
presence of a certain amount of turbulence in the reservoir, and inter-
ference by the rear boundary of the reservoir.
This report was submitted in fulfillment of Grant No. 15040 E.J.Z. be-
tween the Environmental Protection Agency and the University of Califor-
nia, Berkeley.
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CONTENTS
Page
ABSTRACT i
FIGURES iv
TABLES v
CONCLUSIONS vl
RECOMMENDATIONS x
1. INTRODUCTION 1
2. A REVIEW OF PREVIOUS WORK 5
A Physical Discussion 5
Unsteady Flow 8
Type (b) Layers F -> 0; R = 1/G =0 9
Type (c) Layers F2/Re -* 0 ; F = 0 = 1/G 12
Type (c) Layer in a Diffusive Fluid
F = 0 ; G = 0 (1) , R -» 0 16
Numerical Studies 17
Experimental Studies 18
Summary
3. ANALYTIC CONSIDERATIONS 21
The Governing Equations 21
The Initial Value Problem 24
Structure of the Withdrawal Layer - Outer Solution ... 27
Structure of the Withdrawal Layer - Inner Equations ... 33
Structure of the Withdrawal Layer - Inner Solution ... 38
Summary of the Layer Structure - Composite Solution
for 6 42
4. EXPERIMENTAL OBJECTIVES AND APPARATUS 49
Review of Previous Experimental Data 49
Experimental Objectives 49
The Experimental Tank 49
The Salinity Probe 52
Establishment of a Linear Density Profile 55
The Discharge Meter 58
The Dye Particles 58
Photography and Timing 60
Experimental Procedure 60
Experimental Program and Data . 62
5. ANALYSIS AND DISCUSSION OF EXPERIMENTAL RESULTS 65
Analysis of Data 65
Effects of the Finite Size of the Tank 67
Comparison with the Theoretical Results and Previous
Experiments 67
11
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6. COMPARISON WITH FIELD DATA 73
Unsteady Withdrawal 73
Steady Withdrawal 73
ACKNOWLEDGEMENTS 77
REFERENCES 79
GLOSSARY OF TERMS 83
APPENDIX 85
111
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FIGURES
Page
3.1 Theoretical Model 22
3.2 Outer Solution 31
3.3 Outer Solution 32
3.4 Integral Solution Velocity Profile 43
3.5 Theoretical Withdrawal Layer Thickness 44
3.6 The Structure of the Withdrawal Layer 45
3,7 The Composite Solution: G = 2.34 46
3.8 The Composite Solution: G = 29.0 47
4.1 Experimental Data - after Koh (1966b) 50
4.2 General Layout of Experimental Tank 51
4.3 Photographs of Experimental Equipment 53
4.4 Conductance Probe and Float Circuit 54
4.5a Example of the Probe Calibration 56
4.5b Example of the Density Profile 57
4.6 Discharge Meter 59
4.7 Example of the Dye Traces 61
5.1 Experimental Withdrawal Layer Thickness: G = 29.0 ... 66
5.2 Experimental Velocity Profiles: G = 29.0 68
5.3 Variation of the Integral Solution with Kg 70
6.1 Prototype Experimental Data 75
IV
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TABLES
Page
4.1 Experimental Program 63
6.1 Field Data 74
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CONCLUSIONS
The long term objective of this research is to provide an operational
technique for managing the quality of reservoir outflow. This objective
requires a complete understanding of the details of stratified flow and
mixing in a prototype reservoir. In recent years a number of bulk man-
agement schemes have been devised and tested. These schemes are en-
couraging, but their predictions are usually restricted to the reservoir
for which they have been calibrated. This failing is predominantly due
to their primitive description of the reservoir mechanics.
The present report describes a detailed mathematical and experimental
investigation of one aspect of the management problem, the withdrawal
layer which forms at the level of the exit.
The following summarizes the conclusions drawn from a critical study of
previous work and the conclusions derived from the present study.
Prior Knowledge
Previous work on the steady flow problem may be divided into the follow-
ing two categories:
(i) The fluid is inviscid and the upstream boundary condition is given
by PT IT = constant. The layers corresponding to this class have been
designated as type (b) in Chapter 2; or
(ii) The inertia forces are completely neglected and the fluid is
assumed to be of infinite extent. The layers corresponding to this
class have been designated as type (c) in Chapter 2.
The weakness of the solutions of class (i), when applied to reservoir
flow, is that in a reservoir the upstream condition is supplied by a
solution of type (c) or by a solid vertical or sloping wall, and not
by the rather special constant head condition pT IT = constant. It
is therefore not clear where and how well this solution approximates the
flow.
The solutions of class (ii) have generally been thought to be valid at
large distances from the sink. In Section 2 it is shown, however, that
both the diffusive and non-diffusive solutions corresponding to this
class have a physically unrealistic behavior at infinity. In both cases
the work done on the fluid at infinity by the pressure anomaly is in-
finite. Furthermore, the validity of neglecting the inertia terms in
any part of the reservoir must be established.
Both classes of solution yield withdrawal layer thicknesses much smaller
than obtained from experimental or prototype observations.
VI
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Koh's (1966a) analysis may be used to estimate the time taken for steady
withdrawal layer to form. In Section 6 a comparison with some prototype
data showed that this yields good results.
Contributions of the Present Study to the Unsteady Problem
The solution by Koh (1966a) predicts, as mentioned above, times for the
development of a withdrawal layer which are in good agreement with proto-
type and experimental observations. However, the method of solution in-
volved a non-uniformity for small times and for large distances from the
sink, the effects of which were uncertain. This inconsistency in the
method of solution was removed by:
(a) considering flow confined in a duct of finite depth,
(b) by not making the Boussinesq assumption, but leaving the domain
infinite.
In both cases the time taken for the flow to reach a thickness 6S was
the same as predicted by Koh (1966a) and is given by
t = J1
/eg" Sc
Contributions of the Present Study for the Steady Withdrawal Layer
The withdrawal layer in a duct was analyzed and partitioned into differ-
ent flow regions. Each region is characterized by a particular force
balance summarized in Fig. 3.6. By matching the region where inertia
may be neglected to the thin layer zone nearer the sink, a form of non-
dimensionalization was derived, which leaves the layer dynamics dependent
only on a single parameter - the square root G of the Schmidt number.
The magnitude of the withdrawal layer thickness, determined by an inte-
gral method applied to the dimensionless equations, and the observation
that G is the only parameter remaining when the flow is viewed from the
dimensionless coordinates, was in good agreement with experimental data.
(a) Flow Regions: In Section 3 a partial answer has been found to the
intricacies of the flow structure of the withdrawal layer for Schmidt
numbers of order unity. The limit used to derive the structure shown
in Fig. 3.6 was that the Froude number and the Rayleigh number tend to
zero, but the Reynolds number q/v tends to infinity. The outer flow,
where inertia forces are neglected, changes from a parallel uniform flow
upstream to a symmetric withdrawal layer near the sink. In the limit
of zero x, the outer flow pattern asymptotes to the similarity solution
found by Koh (1966b) for a withdrawal layer in a fluid of infinite extent,
Thus, even though the problem in an infinite domain, considered by Koh
(1966b), is not well set, its solution approximates the correct solution
vii
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in a region near the sink, but sufficiently far away to avoid the in-
fluence of inertia.
Inertia is shown to become a force of equal importance to buoyancy and
viscosity for distances from the sink, smaller than
This is referred to as the inner region. When the flow reaches a dis-
tance of Order (q//Cg) from the sink, an inviscid core of thickness of
0 (q/i/eg) forms. To be able to meet the edge boundary conditions it
is probable that a further zone, consisting of a buoyancy-viscous force
balance, forms at the perimeter of the inviscid core.
(b) Integral Solution: The equations valid in the inner region, and
which have solutions capable of being matched to the outer flow, are
derived in Section 3. Using the inner limit of the outer flow as the up-
stream boundary condition, these inner equations were approximately solvec
for the withdrawal layer thickness by an integral method. The inner and
outer variations of 6 , the withdrawal Tayer thickness, were then com-
bined to yield a composite solution. This is shown graphically for dif-
ferent Froude and Schmidt numbers in Figs. 3.7 and 3.8.
(c) G as a parameter: When the outer region was matched to the inner
region it was noticed that the inner equations, in terms of the inner
stretched variables (x/x0 , z/F2) , were only dependent on the parameter G.
m
Hence any experimental data, or solution of these equations, when repre-
sented in terms of the stretched variables will depend only on G . So
far, G has been assumed to be of order unity. The flow structure, de-
veloped for Schmidt number of order unity could, however, be quite dif-
ferent when the Schmidt number is large. Extra regions in which vis-
cosity is important, but diffusion is not, would be introduced. The
new domains would, however, be governed by equations which are special
cases of the present inner equations. Hence even if G is not of order
unity, the conclusion that the flow, in terms of the inner variables,
is only dependent on the parameter G, is still valid. The integral solu-
tion and any experimental withdrawal layer thicknesses may therefore
quite generally be represented non-dimensionally with G being the only
remaining parameter.
(d) Physical properties of water: If s (z) is the variation with depth
of the concentration of a dissolved substance, for example 02 , then the
outlet concentration will be given by the weighted mean over the depth
of the hypolimnion;
JL
I P
u(z) s (z) dz
q J
viii
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If the reservoir is long enough, so that the outer region is well estab-
lished in the upper reaches of the reservoir, then this simplifies to
-^ J s (z) dz
-jfc
(e) Comparison with experimental data: Fig. 5.1 shows the experimental
variation of 6/F^j with (x/xc)^/^ . The scatter in the data is small,
indicating the independence of the results from any parameter other than
G. Comparison with the integral curve is good everywhere except near
the sink. The assumption that the velocity profile is self similar,
made in the integral method, is verified in Fig. 5.2. Koh (1966b) had
noticed this similarity for each individual discharge q, but this is now
seen to be true more generally in terms of the inner coordinates. The
data in Fig. 5.2 shows that the velocity profile remained self similar,
except for small changes in the back flow, for a variation of (x/xc) from
0.008 to 64.
(f) Comparison with prototype data: The available prototype data are
reproduced in Fig. 6.1. Included are the composite integral solution,
the linear solution, and the withdrawal layer thickness obtained from the
inviscid model. It is seen that the integral solution is an improvement
over the other two, especially as x/xc increases, but the increase in
the withdrawal layer thickness, gained by the inclusion of the inertia
terms, is still not sufficient to fully explain the observed prototype
values of 6 .
It is further noteworthy that in the prototype studies the value of xc ,
when based on molecular transport coefficients, is usually greater than
the length of the reservoir. With eddy transport coefficients which have
been adjusted so that the T.V.A. (1969 (a), (b)) data fits the integral
solution, xc becomes roughly equal to the length of a typical reser-
voir. This means that the flow in the reservoir, in either case, is
predominantly characterized by the inner solution and that insufficient
length is available for the outer solution, to which Koh's (1966b) solu-
tion is an approximation, to form. Of course, the inner solution depends
on the existence of the outer solution, so that the presence of the
reservoir end may be expected to invalidate the inner solution to some
extent, and may be responsible for the discrepancy with prototype measure-
ments. Nevertheless even in relatively short reservoirs the inner solu-
tion appears to be a reasonably correct description of the flow. The
reasons for a larger layer in the prototype, than predicted by the inte-
gral method, have therefore been restricted to the presence of small
amounts of effective eddy viscosity or the presence of a rear boundary;
probably a combination of both is needed for the full explanation.
IX
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RECOMMENDATIONS
In the process of this research the following natural extensions became
apparent:
(a) The analysis in Section 3 was carried out for the simplest geometry.
This work should be extended to include the following:
(i) end walls and a free surface
(ii) a more realistic bottom geometry
(iii) different sink positions
(iv) no slip walls.
(b) The unsteady outer flow should be reanalyzed to include viscous and
diffusive effects. This would make the patching suggested in Section 3.2
unnecessary.
(c) The structure of the withdrawal layer flow should be examined for
large values of the Schmidt number.
(d) The matching of the inviscid core IV near the sink, to the inner
boundary layer flow III, should be further studied. This will, however,
require an understanding of the behavior of the solutions of the non-
linear Euler equation.
(e) More experimental data are needed on the values of the transport
coefficients in actual reservoirs.
(f) The flow changes caused by a sudden decrease in discharge are still
unknown and should be investigated.
(g) The reverse problem, a jet moving into a stratified reservoir, appears
to have received little theoretical attention. The river inflow at the
upper end of the reservoir is such a jet, and its solution is required
for a better understanding of the overall reservoir dynamics.
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1. INTRODUCTION
For centuries surface water reservoirs have been constructed as a
means of storing water for flood control and water supply. As such
their function is well understood and their design is no longer a
field of new research. Only in the past few decades, however, have
engineers become aware of a major, sometimes helpful and sometimes
deleterious side effect of reservoirs, namely their effect on water
quality. A surface reservoir represents not only a given number of
cubic feet of stored water, but also an inventory of dissolved oxygen,
heat, nutrients, minerals, turbidity, plants, and animals, all dis-
tributed in an uneven manner throughout the reservoir in a delicate
balance whose maintenance is vital to the health of the reservoir
environment and the river downstream. Engineers know a great deal
about managing the amount of water in a reservoir, but very little
about managing its quality. Properly managed, reservoirs can provide
a useful means of regulating the quality of water in a stream, in-
cluding its temperature, dissolved oxygen, and other constituents, in
whatever way will be most beneficial to the downstream users. Im-
properly managed a reservoir can contribute to the destruction of a
river environment by permanently changing its temperature, oxygen
content, and turbidity, with disastrous effects on the downstream
ecology. Much remains to be learned about how the flow in a reservoir
can be controlled to produce the desired quality of outflow. This
report presents a contribution to our understanding of one aspect of
the problem, namely prediction of the thickness of the withdrawal
layer when the reservoir is stratified.
Most large surface reservoirs are density stratified throughout most
or all of the year. The surface, or epilimnion water, consisting of
the upper 15 feet or .so, is heated by the sun and becomes lighter than
the water beneath. Below this there is a sharp drop in temperature,
called the thermocline. The deep water, called the hypolimnion, has a
slight, nearly linear stable temperature gradient, meaning that the
density of the water increases continuously as one goes down in the
reservoir. The effects of this slight stratification are profound,
because the flow patterns found in stratified water are totally dif-
ferent from those found in water of constant density. In stratified
systems flow in the vertical direction requires the expenditure of
energy, whereas flow in the horizontal direction does not; hence flows
in stratified waters tend to extend over great horizontal distances,
while being confined to narrow vertical layers. When water is taken
out of a stratified reservoir the discharge tends to come only from
the level of the exit, flowing in a layer which extends over the en-
tire length of the reservoir but only over a fraction of the depth.
The temperature of the outflow, as well as the concentration of
dissolved constituents such as oxygen and nutrients, is that of the
water in the reservoir at the exit level. Management of the tempera-
ture and quality of water downstream of the reservoir can be achieved
by varying the level of the exit to select the desired quality.
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The long-term objective of this research is to provide an opera-
tional technique for managing the quality of reservoir outflow.
This objective, a substantial undertaking, will require a complete
understanding of the details of stratified flow and mixing in an
arbitrarily shaped reservoir. The complexities are enormous, and
may never be understood in complete detail. Nevertheless, in recent
years several bulk management schemes have been put into use, for
instance those described by Orlob and Selna (1970) and Beard and
Wiley (1970). In these schemes a computer is told as much as is
presently known about reservoir mechanics, that the incoming river
water sinks till it reaches its level of buoyancy, that the outflow
comes from the level of the exit, that solar energy is supplied at
the water surface, and that some type of vertical mixing takes place.
The computer is given an historical sequence of in and outflows and
is programmed to keep track of temperature and water quality as a
function of depth in the reservoir. These schemes have considerable
promise, their only failing being in the primitiveness of their
description of the reservoir mechanics. Thus Beard and Wiley (1970)
find that the coefficients required for their program vary from
reservoir to reservoir by a substantial factor, and that the varia-
tion cannot be predicted within the present state of knowledge.
Improvement in the accuracy of schemes such as this one will come
only by improving our understanding of the basic mechanics of the
processes involved.
The present report describes a mathematical and experimental investi-
gation of one aspect of the management problem, the withdrawal layer
which forms at the level of the exit. Even in this detail the
mathematical difficulties are so formidable that the reservoir must be
idealized to obtain a solution. Mathematically the reservoir is con-
sidered to be infinitely long and of constant width and depth, and the
density profile is taken to be linear. The exit is assumed to be
located halfway from the surface to the bottom, and all walls are
assumed to be stress-free. Within these limitations a mathematical
solution can be obtained; the details of the derivation are described
in Section 3. Sections 4 and 5 describe a laboratory experiment and
compare the results of the experiment to the analysis of Section 3;
the comparison is extremely good. In Section 6 the theory is compared
with the few available measurements in prototype reservoirs; the
thickness of the predicted layer is less than is observed, but not by
a large amount, and the result is much closer than those of previous
theories. The most important reason for the error is believed to be
the mathematical assertion of an infinitely long reservoir; indeed,
one significant result of the mathematics is that prototype reservoirs
are in general short compared to the length required for complete
development of the withdrawal layer. A major recommendation for fur-
ther research is study of the influence of the sloping bottom of the
reservoir on the formation of the layers.
The major analytical technique used in this report is the method of
matched asymptotic expansions, a method developed by applied
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mathematicians only within the past approximately 20 years for the
solution of problems in boundary layer theory. The method has proved
a powerful tool in many branches of fluid mechanics, and appears to
be of significant usefulness in many civil engineering problems, even
though it obviously will not be familiar to many practicing engineers.
The authors hope that a detailed understanding of the mathematics will
not be essential to understanding the results of the study; those
less interested in the mathematics may skip directly to the descrip-
tion of the discussion of the experiments, and their results on the
comparison of the theoretical predictions with field observations.
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2. A REVIEW OF PREVIOUS WORK
The following is a review of previous work on selective withdrawal
by a line sink. The physical limitations of the various solutions
are investigated and their applicability to experimental work is
discussed. The comments here complement the recent survey article
by Brooks and Koh (1969).
A Physical Discussion
A sink placed in a homogeneous fluid of infinite extent induces a
radial motion. When the same sink is placed in a stably stratified
fluid a thin layer of fluid is observed to move, from both the left
and the right, towards the sink. It is this layer which is of
interest here. Before proceeding, however, to discuss specific known
solutions of the above phenomenon it is perhaps helpful to look at
the general implications of a variable density.
Let
where
PT = PQ + Pe(z) + p(x,z,t),
p = total density
p = mean over the depth of the density
p (z) = density above p when motion is absent
p(x,z,t) = density changes induced by the motion
then a measure of the scale of the density gradients is given by
p dz '
o
The principal flow modification a variable density induces, concerns
the creation of a non-trivial body force. The main force, gravity
acting on p , leads only to a passive hydrostatic pressure which may
be removed from the equations of motion. However, if say a particle
of volume 6v moves a vertical distance of order L, then its density
anomaly with the surroundings will be of order ELp . This induces a
body force, in the direction of the particle's original level, of
order £L p g&v. This force, which now actively enters any vertical
momentum balance, tends to inhibit vertical motion. If the stratifi-
cation is strong this force will dominate the other prevailing forces,
and the flow will be layered or jet like.
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Moment of momentum considerations may be used to show that the above
force not only inhibits vertical motion, but also induces a vorticity
in the fluid motion. Consider again a sphere 6v of a stratified
fluid. The pressure force acts, to first order, through the center of
volume, but the body force vector passes through the center of gravity.
In general, as the sphere has more of its mass concentrated in the
lower half, the two centers will not coincide and the sphere will tend
to rotate. The production of vorticity by this applied moment yields
baroclinic motion and the usual notions about inviscide fluids exe-
cuting irrotational motion must be discarded.
If this buoyancy force is important in an unsteady fluid motion, the
time scale will be fixed by the period of the internal waves present.
The natural period of the fundamental oscillation may be found by
analyzing the motion of the model sphere after it is displaced a ver-
tical distance z from its original position, in a linear density
field. The net force experienced by this particle is of the order 6v
e Po gz towards its original position. Disregarding other forces, the
equation of motion governing the particle is,
. . d z
g&vep z = p 6v -
° ° dt;
or
d2z f
5- = egz i
dt2
and the solution is given by
-i/eg t o+i/eg *
z = Ae + Be
Hence the time scale T of the internal oscillations is l//eg . The
frequency /Eg is generally known as the Brunt-Vaisalla frequency. If
the fluid is forced to respond at a much faster rate, then the buoyancy
forces may be overwhelmed and the fluid will respond as a homogeneous
mass. The equations of motion are then elliptic.
The other modification a variable density causes is that the inertial
resistance of the fluid is now a function of the vertical height. Two
particles separated by a vertical distance L will have a density
difference of the order £L. If both of these particles are placed
under a similar pressure gradient, their inertial resistance will be
correspondingly different; the lighter particle will accelerate faster
than its heavier counterpart. The importance of this effect is related
directly to the magnitude of eL. In a reservoir L is bounded by 2i,
the depth of hypolimnion, making 2t,e of 0(1CT3). The smallness of EL
implies that the percentage difference in the^inertial resistance is
small, which in turn means that the term PT in the equations of
motion may be approximated by p . This is what is generally known
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as the Boussinesq approximation and it will be used throughout this
report.
The above discussion shows that the pressure gradient induced by
switching on a sink, meets only a small viscous resistance in the
horizontal direction, but it is counteracted by viscosity and
buoyancy in the vertical direction. As the horizontal velocity at
the level of the sink increases, the velocity everywhere else will
decrease, since the discharge q remains constant. This results in
a thin horizontal velocity jet, with an associated vorticity con-
centration at the level of the sink. As the layer narrows, the
velocity and velocity gradients increase and inertia and viscous
forces become important.
In contrast to boundary layer theory in a homogeneous fluid, the thin
vorticity concentrations in a stratified fluid may be sustained by
three distinct mechanisms:
(a) Cross stream diffusion of vorticity is balanced by longitudi-
nal convection (ordinary boundary layer).
(b) Production of vorticity by the baroclinicity is balanced by
downstream convection (inviscide buoyancy layer).
(c) Cross stream diffusion of vorticity is balanced by the creation
of vorticity of opposite sign by the baroclinicity (creeping
flow layer).
A layer of type (a) is usually relevant only when a wall is present
and the velocity of the fluid is high enough to overcome any
buoyancy effects. Redekopp (1970) has shown how this type of layer
may, however, form an obstruction with its displacement thickness,
for flow in which buoyancy is important. The other two layers both
involve an active participation of the buoyancy forces; they are the
ones which constitute the withdrawal layer considered here.
The existence of these layers requires, as already mentioned, that
the buoyancy forces be strong enough to make vertical displacements
within the layer small compared to the reservoir depth. Dimen-
sionally this means that when assessed with respect to the reservoir
depth, the buoyancy forces should be larger than either the inertial
or viscous forces, but that also there be a scale present, smaller
than the depth, based on which the forces balance. The quotient of
the relevant forces to the buoyancy forces, when all are assessed
with respect to the reservoir depth, then form the small parameters
usually associated with a thin layer theory. For layers of type (b),
this yields the quotient
->
Du
o v n 9 9
inertia forces o Dt _ _U q _ 2
buoyancy forces ~ 6V e I (>n g ~ firl2 ~ fi
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where F is called the densimetric Froude number. Similarly for
layers of type (c) the parameter becomes
2
viscous forces _ (J,U6v 1 ___ VU _ F
buoyancy forces ~ .2 SveAp g .3 R '
Xj o G g* e
£u
where R = = Reynolds number.
e v
When diffusion of the stratifying species is non-zero, then the speed
of propagation of the species is of order D/A . If U is of the same
order or smaller than this, the adjustment in density by diffusion is
important and the relevant number becomes the Rayleight number R,
where
2 ' I
In general there are thus three independent parameters F, R and R
which are important in the study of withdrawal layers. The internal
Froude number F is the relevant parameter for layers of type (b) ,
while F /Rg and R are the parameters governing the non-diffusive and
diffusive layers of type (c) . Sometimes it is more convenient to
deal with the alternative triplet of parameters F, R, and G, where
G = (v/D)2. Known solutions may be categorized into one of the
following classes:
F -» 0 ; R = 1/G = 0
F = 0 ; F2/Re -» 0 ; 1/G = 0
F = 0 ; R -» 0 ; G = 0(1) ,
where the arrow -» 0 indicates that the parameter is small, but non-
zero, and the solution is the limiting solution as the parameter
approaches zero.
Unsteady Flow
The unsteady flow development appears to have received very little
attention. The analysis by Koh (1966a) seems to be the only one on
the subject. Koh (1966a) deals with the unsteady flow due to a
variable sink strength q(t) in a fluid of infinite extent and linear
stratification. The analysis neglects all forces except the unsteady
~<~^
inertia force, p -5 , pressure and buoyancy, and from above it
may be anticipated that the time taken for the flow to reach steady
state is k//Fg , where k is a constant to be determined.
For the case where the sink is suddenly brought to full strength q,
the solution for small time is near radial flow, while for large time
-------
^ ______ c
, . cos 6 ;n Q2 sin Bt
u(x,z,t) ~ q ^-Vl-P - 5 ~ <*
where 6 = artan - , f3 - sin 8 and t* = t/e"g . The solution, for
large time, thus collapses to a thin line moving with infinite speed
towards the sink. This may be expected as all the forces which could
counteract buoyancy have been neglected. Koh (1966a) evaluated the
constant k by defining the layer thickness to be given by the 1st
velocity reversal and patching the above solution onto a steady layer
thickness 6 . The velocity u is zero when
Pt* = TT
hence let
6 t*
s
- = Tt
X
or
ou
In assuming that -^- is multiplied by PQ, rather than pT ,
(1966a) assumed a Boussinesq approximation. However the solution is
radial for small times and thus the scale of the motion L becomes
infinite a large distance from the sink. This means that £L is not
small and the Boussinesq approximation should not have been made. To
see the effect of this non-uniformity, the full problem was analyzed
by Imberger (1970) . Although the flow pattern obtained by him was
different, the time estimate for the withdrawal layer formation was
insensitive to the non-uniformity and is the same as that found by
Koh (1966a) . Another way to overcome the non-uniformity is to put
the sink between parallel walls of finite spacing - this is done in
Section 3.
Type (b) Layers: F -» 0; R = 1/G = 0
The fluid is inviscid and non diffusive and the depth of the
fluid is large enough so that F is small, causing the flowing layer
to separate from the walls. With R = 1/G = 0, the equations of mo-
tion reduce to the Euler equations for a stratified fluid and
= 0.0; the density is conserved along streamlines. The equations
may be rearranged to form a single equation for the streamfunction
(Yih (1965) , p. 76) .
, (2.D
0 d\f d
/ PT
where ty = / \|r and H is the total head. The flow is inviscid and
o
-------
the only generation of vorticity is by the baroclinicity effect.
The equation really states:
V ill' - Pseudo = Pseudo vorticity + Pseudo vorticity in-
vorticity generated in the herited by the flow
particles flow from any upstream
history shear.
dp
For the case -TT = 0, Fraenkel (1961) and Childress (1966) have
deduced some general results regarding the formation of vorticities
in corners, but otherwise little is known about a general solution
of this equation.
T" H fT
Yin (1965), pp. 75-140, discusses the case when T- and r- are
linear functions of l|f. For a sink between parallel walls, the follow-
ing two upstream conditions meet this requirement.
2
(a) Upstream density has a linear variation and pU = constant.
oo
(b) Upstream flow velocity and density are parabolic.
In case (a), the solution depends on the Froude number
F = u//₯g t . For F larger than 1/TT the solution indicates a motion
towards the sink over the complete depth of the channel, the only
exception to this being a large eddy in the corner above the sink.
This eddy grows as F decreases and when F = 1/rr it extends to infinity
thus violating the assumed upstream boundary condition. List (1970)
has investigated the corresponding problem in three dimensions and
found that no solution satisfying the above upstream boundary condi-
tion exists for any value of F = q//eg~ J&2 . This may be explained
simply by noting that for a finite q, u -» 0 as r -» °°, hence
F = u//£g J&2 -» 0, and by the two dimensional result, no solution
exists. Case (b) always falls into the category F > I/TT and there-
fore is only relevant for large discharges where no separate layer
forms and the fluid dynamics is only slightly dependent on the
buoyancy forces.
In an effort to extend the above work to layered flow, that is where
F ^ , Trustrum (1964) linearized the convective acceleration term
about the upstream uniform velocity. This may be compared with the
Oseen approximation in slow viscous flow. She then investigated the
time dependent problem for suddenly switching on the sink. For F >
the solution converges in the limit t -» °°, to flows which are very
similar to those found by Yih (1965). However, as F -» I/TT or F < I/TT,
the eddies in the upstream corner extend to x = °° as t -» » and the
original assumption involved in the linearization is violated every-
where. Thus, the analysis suggests uniqueness for F > I/TT, but
nothing is learned for the case F ^ I/TT.
10
-------
The fact that the corner eddy extends in both Yih' s (1965) and
Trustrum's (1964) solutions to infinity, has led Kao (1965 and 1970)
to postulate a slip line within the flow field, separating the
flowing fluid from a stagnant upper layer. The results of Kao' s
work, especially the later paper, suggests that the Froude number,
based on the flowing depth, remains constant and equal to 1/n,
independent of the discharge q.
Two serious questions remain to be answered. Firstly, what happens
to the corner eddies as F decreases below I/TT? To say that these
are statically unstable is rather vague at best. They are the re-
sult of a dynamic force balance in which the centrifugal force
plays an important part. To suddenly neglect this in the eddy
cannot be justified. What is required is an analysis of their
stability to small (or large) disturbances. Segur (1970) has re-
cently carried out such an analysis and he found the flow to be
stable if
< n
gz sT - a? < ° '
where r is the local radius of curvature-. By this criterion the
slow moving eddies are in fact stable and it is not clear how they
degenerate into a stagnant fluid as the Froude number decreases
below I/TF. It could be that viscous forces predominate within the
eddy, but this does not appear to have received any attention.
The second question concerns the non-uniformity of the flow as
x -» oo when viscosity is included. The hope in solving any inviscid
problem is that the solution so obtained is in fact the solution of
the full Navier-Stokes equations in the limit as v -» 0. However,
in the slipline solution, viscosity would be important around the
vortex sheets and the two viscous layers (top and bottom of the in-
viscid layer) would coalesce at some x. For x greater than this,
viscosity would be important throughout the whole layer. This in
turn could lead to a different upstream boundary condition for the
inviscid core, which if Fraenkel's (1961) work is a guide could
have first order effects on the slipline solution. What is there-
fore needed is that these three layers be matched in the sense of
matched asymptotic expansions. The difficulty here is that this
requires not only a general solution of the equation governing the
inviscid core, but also a solution to the coalesced flow. So far,
neither is known. In Section 3, an approximate procedure is
suggested.
The experiments of Debler (1959) are tailored to the slipline solu-
tion, in that the upstream boundary condition was not allowed to
develop naturally but was forced in the manner suggested by Yih
(1965), to be of the form p U2 = constant. Hence, although the ex-
periments were in excellent check of Kao1 s (1970) solution, they do
not throw any light on whether the slip line solution with p_,U2 =
11
-------
constant upstream, is the correct limiting solution close to the sink
for a long withdrawal layer. Long here means one in which viscous
forces are important upstream.
F2
Type (c) Layer : - -» 0 ; F = 0 = l/G.
R
The fluid is non diffusive and the discharge is small enough so that
the inertia terms may be neglected completely. This results in a
p2
linear equation, which for the case -* 0, possesses a thin layer
like solution. e
Gelhar and Mascolo (1966) investigated such a layer imbedded in a
fluid of infinite extent, by making a boundary layer assumption.
DpT
Further, as the fluid is non diffusive, - 1 = 0 and p is not deter-
Dt T
mined until it is specified somewhere in the flow field. The authors
assumed
(2-2a)
where f is a constant. This leads to a single equation for l|f, viz.
o~ ill f S\li
i; 3L . i_ UT _ n ^o 9H^
rr + -v u . \£..6\»
~ 4 v ox
Sz
2
The thickness of the layer, in terms of F /R , may be deduced from
the following argument. When z -» «, the vertical velocity w -> 0,
which means that for conservation of mass
I
udz = l|r(oo) - ty(-oo) = - q (2.3)
Hence let
lit* = - ; z* = r and
,
where L is taken, in the absence of a duct width, as the distance of
the point of consideration from the sink. Then
3 *4 r * = ° (2>4)
fL Bz Sx
Q Q ^Ap AP
Also ^24^ ^ which implies -^- s p < -^- and f is constant.
12
-------
«
where Ap is the density jump across the layer. Hence Eq . (2.4)
becomes
R ,. *4 -*
e cz ox
where F = p q /g&pL and R = . Now rescaling z* , such that
o e v
z** - **
z - 6
and 6 = \ / leads to a single parameter free equation
/
e
(2.5)
Gelhar and Mascolo (1966) found the solution to this equation by
postulating a similarity solution
x*n
(2,6)
The condition §*(<*>) - ^(*(-<») = - 1 implies m = 0 and to satisfy
Eq. (2.5) n must be equal to 1/4 and the function f must satisfy
the equation
4f(lV) - Tjf' = 0 , (2.7)
where
**
The solution was evaluated numerically, with the help of basic power
series and the reader is referred to the author' s paper for further
details. The solution showed a large center line jet moving towards
the sink, with a small amount of backflow at the edges.
In a recent paper, Gill and Smith (1970) have given analytic solu-
tions for the more general equation
4f(tv) - Tjf' + mf = 0 . (2.8)
They found that
13
-------
f =[ p-*-1 e^ dp , (2.9)
m.k «)
where C are four different paths in the complex plane, yielding four
independent solutions. For the sink flow m = 0 and the appropriate
real solution reduce to
f = A J (Tl) (2.10)
where J is defined in the paper, and A is a constant adjusted so
that ^(oo) - $(~°°) = 1. For the asymptotic properties and a plot of
JQ, the reader is referred to the original paper.
It is important to note that as well as dT^ . (2.12)
oo
CD
Hence for J to be constant 1 p(T))dT) must be equal to zero. That
_oo
this is indeed so, is not difficult to show. From the momentum
equation, with p ~ x* p(T|)
14
-------
p - Tip' = 4f'" (2.13)
Integrating this between !)=-<» and * yields
ao
2 ! pdTl - Tip = 4(f <-«) - f (-»)) - 0
J l
i.e.
DO
pdTl = 0 .
CD
[Conversely, in the problem of a wake in front of a plate, Long
(1959) and others assume J^O, i.e. m = - 5. This means that
q = Ot which again is evident from an examination of their solutions,]
Although pdz = 0 implies that ! pgdz = potential energy = 0
O tj
-00 -03
i.e. the potential energy of the flow is zero, the energy balance of
the flow reads; the work done by the pressure is equal to the loss
in unit potential energy pwdzdx plus the work done against viscous
v
dissipation. The pressure work is porportional to
r r
up dz - ! up I dz
J 'x=x <; H'x=x
L. £
OO
CD
= (x* - x|) f f'(Tl) P(T,) dT\
1 2. u
(2.14)
If x^ =0 and x^ ~* ^ then an infinite amount of work must be done on
the flow at infinity rather than at the sink. In other words, the
flow is being pushed from infinity rather than being sucked towards
the sink.
Another difficulty with the solution is the matter of how this sort
of flow gets started. Gelhar and Mascolo (1966) assume that
= get (2.15)
where t is the width of the opening. This has no physical basis,
Further, assuming that the flow started from an abrupt density
variation, with the sink at the density discontinuity is not valid
either, as with no diffusion the density profile could not adjust
15
-------
to a smooth variation.
These didfficulties are crystalized somewhat by noting that as the
layer grows, since q is constant, the velocity u will decrease. At
some point u will be small enough to be of 0(D/L) and then diffusion
will modify the solution. Hence, the above non diffusive solution
may be regarded as a solution for intermediate x. But again
matching is absent.
Type (c) Layer in a Diffusive Fluid: F - 0; G = 0(1) , R -» 0
The flow is slow enough so that inertia may be neglected, and the
diffusion of the species is important. From the previous discussion
this may be seen to be a limit of the full equations very far from
the sink. In Section 3 a rigorous matching is carried out.
Koh (1966b) has investigated the sink flow in this limit, in an
infinite domain with linear stratification. (He also considers the
axi-symmetric case.) In a similar way to the above the
equations of motion may be reduced to a single equation for
the stream function ty =
q
(2.16)
-v **t> -x *2
§z** dx
* i /o
where z** = -=-75- . Hence 6 = 0(R ). Koh (1966b) investigated the
R1/3
solution of the above equation by postulating a similarity solution
**
t* = x*m f(Tl) ; Tl = -^-TT- (2.17)
Jc Jl / O
x*
00
and again m = 0 to satisfy I u*dz** = - 1. The resulting ordinary
J
-co
differential equation is
f(vi) + I T2f" + - r\f' = 0 (2.18)
f f'
or if g = f
g(V) +|T\2B/ +|% =0 (2.19)
Koh (1966b) obtained the solution by a numerical integration from
T| = T10 » 1 to T) = 0. He obtained the value of g, at the point T|0,
by noting that for large T\ the differential Eq. (2.19) reduces to
TjV + 4% = 0 . (2.20)
16
-------
The solution to this equation is g = . However, this variation
^ 1
for the velocity implies that the density p ~ -» oo This means
z
that an infinite amount of salt would have to be transferred from
below the level of the sink to above it. List (1968) in a similar
problem for sink flow in a porous media obtained:
p ~ - sgn z as z -» ± »
- an even worse singularity. He reasoned that the flow could there-
fore not develop unless this singularity is already present in the
fluid before the motion is started.
In an effort to resolve these difficulties, the five solutions of
Eq. (2.19) were determined analytically. A detailed discussion of
the properties of these solutions is given by Imberger (1970). It
is found that of the five solutions two are exponentially large,
two exponentially small, and one decays as 1/T] , as T| -» « . By
symmetry, the stream function must be an odd function of z, and it
is shown that this is only possible if g contains the algebraically
decaying solution - as well as the two exponentially small functions.
The physical interpretation of the 1/T] decaying solution is that
the pressure associated with it grows logarithmically as x -» » But
as J = 0, this is not acceptable. This unpleasant behavior is not
due to assuming the Boussinesq approximation, which is quite valid as
it makes little sense to keep the variation of inertia, when inertia
itself is completely neglected, but rather it is due to posing the
infinite problem with a linear density profile.
In addition, when the solution is substituted into the neglected in-
ertia terms, it is seen that near the sink they dominate the re-
tained viscous terms. Hence the solution has non uniformities both
far and near the sink. These shortcomings are investigated in
Section 3.
Numerical Studies
In an effort to obtain a solution which includes both the effects of
inertia and viscosity, Koh (1964) applied an integral technique to
the boundary layer form of the Navier-Stokes equations. The fluid
was assumed to be non-diffusive and of infinite extent. Assuming
the fluid to be non-diffusive means that the density distribution
must be specified somewhere. Koh (1964) postulated a relation simi-
lar to Eq. (2.2a):
PT = po(l - £11) , (2.21)
where T) = z/6 (x) . This leads, however, to the same difficulty in
the interpretation of e, encountered by Gelhar and Mascolo (1966).
17
-------
In recent years a number of numerical algorithms have been written
for solving the full Navier-Stokes equation when diffusion is absent.
Slotta et al. (1969) have used what is generally known as the Marker-
Cell technique to study withdrawal from a stratified fluid. They
show only results for a two layer stratification - a case for which
analytic solutions are available for inviscid fluids. They claim
that the height of the withdrawing layer d yields a Froude number
u
F = ,. °° = 0.293 . (2.22)
This is then compared with Kao's (1965) result F - . However, the
latter is derived for the blocked flow in a linearly stratified fluid
and is really not relevant to a two layered flow. It is surprising
that dg is not an independent parameter as in the analytic solutions.
Until these discrepancies are explained, comparison with other work
is difficult.
Experimental Studies
Koh (1966b) appears to be the only one who has studied selective
withdrawal experimentally in a continuously stratified reservoir,
A discussion of his results is left to Section 4, as his data was
replotted to conform with the theory developed in Section 3. The
Tennessee Valley Authority have over the last few years begun inves-
tigating withdrawal in actual reservoirs. The data is still sparse,
but a full discussion and comparison is given in Section 6.
Summary
A withdrawal layer forms when a sink is placed into a density strati-
fied fluid, because the buoyancy force inhibits vertical fluid
motion. The solution, obtained by Koh (1966a), for the transient
switch on problem, clearly shows how a density gradient causes a
withdrawal layer to form. Since there is no resistance to horizontal
motion, the horizontal velocity at the level of the sink increases
at the expense of the motion elsewhere, and a layer forms the thick-
ness of which decreases with time. The non-uniformity associated
with Koh1s (1966a) solution at large distances from the sink, due to
the use of Boussinesq approximation, was shown not to be serious, but
for completeness it is remedied in Section 3, by a slight modifica-
tion of the boundary conditions.
If viscous and convective inertia forces are included, then layers
of finite thickness form for which the available theories fall into
two classes. Firstly, there is the work of Kao (1965, 1970), in
which the fluid is assumed to be inviscid and non-diffusive and
secondly, similarity solutions have been found by Gelhar and Mascolo
18
-------
(1965) and Koh (1966b) when the fluids inertia is neglected. For
selective withdrawal in a long reservoir the flow changes, as the
fluid moves towards the sink, from one in which inertia may be
neglected to one in which viscosity and diffusion can be assumed
zero. Hence, the known solutions appear to cover the two extremes
of the withdrawal layer; the flow very close and the flow very far
from the sink. For this to be so the solutions must, however,
correspond in the language of matched asymptotic expansions to the
inner and outer expansions of the problem with respect to the
variable x. In the above review it is shown that this is not so,
and further that the inertia free solutions have physically un-
realistic pressure distributions at large distances from the sink.
What is therefore required is an analysis of the overall structure
of the layer with proper matching of the different regions. This
subject is treated in detail in Section 3.
19
-------
3. ANALYTIC CONSIDERATIONS
In Section 2 it was discussed how a density gradient inhibits vertical
motion and causes the fluid to flow in a thin horizontal layer towards
the sink. This layer is, within the Boussinesq approximation, symmetric
about the x and the z axis and is present not because the Reynolds
number is high, but because the overall Froude number is small.
In actual reservoirs the stratification is usually due to a temperature
variation with depth. Hence G is of order one. Also the parameters F
and R, when the length is based on the half depth i, (Fig. 3.1), are
always very small. Typical values are of the order 10~2 and 10~5, re-
spectively. However, the value of the Reynolds number is usually very
high. Hence in what follows, the symmetric problem shown in Fig. 3.1 is
analyzed in the limit F -> 0, R -> 0, Rg -» °° .
First a simplified unsteady problem, similar to the one considered by
Koh (1966a) is analyzed for the present geometry. The results here are
a substantiation of Koh's (1966a) solution and they have been included
mainly for completeness.
Following this the structure of a steady withdrawal layer is analyzed.
For simplicity the horizontal walls are assumed to be stress free and the
density along them is assumed to be constant (p =0). This approximates
a well mixed free surface. More complex boundary conditions could be
treated in a similar way, but boundary layers at a no-slip wall would
further complicate the analysis. Flow in a duct was chosen primarily to
overcome the logarithmic rise in pressure at infinity in Koh's (1966b)
solution.
The analysis assumes that at large distances from the sink, the flow is
uniform and the density varies linearly with depth. The method of matched
asymptotic expansions shows that the structure of the flow changes pro-
gressively from a slow uniform flow at infinity, to a layer of type (c)
closer in, which in turn develops into a jet flow in which inertia is
important. There is some evidence that very close to the sink a type (b)
layer forms.
The outside flow, where the influence of inertia is neglected, is solved
in closed-form and in the limit as x and z tend to zero the solution
approaches a similarity solution. The inner equations are solved approx-
imately by Galerkin's method and composite solution for the withdrawal
layer thickness is derived.
The Governing Equations
Changes of density in the reservoir water are either caused by tempera-
ture differences or by a variable concentration of a denser substance
such as salt. In the first case, the basic equations governing the mo-
tion will be the Navier-Stokes equation with the energy equation de-
scribing the influence of the temperature gradients on the motion. When
21
-------
r
INFLOW = q.
to
to
SINK
WALL
ZERO STRESS
/>=0
WITHDRAWAL LAYER
INFLOW=q.
FIG. 3.1 THEORETICAL MODEL
-------
the background stratification is the result of a varying concentration
of a second species the governing equations are those of a mixture (Green
and Naghdi (1969)). The water in a reservoir can be assumed to be
incompressible and if a second species such as salt is present, its
concentration will be small. Under these circumstances, Mu'ller (1968)
has shown that the equations of motion reduce to the Navier Stokes equa-
tions for a single compound species, the density of which is governed
by Pick's law. Since the vertical motion is bounded by the duct width
2H , and as &SL « 1 , the Boussinesq approximation can be used to fur-
ther simplify the equations. The derivation of the simplified equations
under this assumption is given by Redekopp (1969) and the equations are
as follows:
V u = 0 (3.1)
Du 2 nx
P 7TT + ^P ~ ~ PgR + M-V u (3.2)
o Dt
2
w = DV p (3.3)
where
Dt dz
T(x, ±i) = 0 (3.4)
p(x, ±j&) = 0 (3.5)
u(oo)Z) = - JL i H(t), (3.6)
u(o,z) =- q6(z) H(t), (3.7)
PT = P0 + Pe(z) + p(x,z,t)
D = diffusion coefficient
p = p (z) + p(x,z,t)
dT = ~ (PO + Pe(z)) g
->
T = wall stress
23
-------
H(t) = unit step function
6(z) = Dirac delta function.
Finally, within the Boussinesq assumption, the flow will be symmetric
about the line z = 0 .
The Initial Value Problem
The aim of this section is to investigate the error induced by the
Boussinesq assumption in Koh's (1966a) estimate of the time taken for a
steady withdrawal layer to establish itself. It is shown below, within
the framework of linear inviscid theory, that the finite duct problem is
really only a slight extension of Koh's (1966a) work and that to this
accuracy, the time taken to achieve steady conditions is independent of
the wall spacing and insensitive to the Boussinesq assumption.
Within the framework of Eqs. (3.1-3.7), when the sink is opened, the
pressure field will be equivalent to one in a homogeneous fluid. Only
when the fluid begins to move will it experience the buoyancy force.
Assumed here of course, is that the pressure at the sink is not so low,
discharge so high, that inertia completely dominates the buoyancy forces,
and the flow is only a slight modification of motion in a homogeneous
fluid. Once the fluid begins to move, buoyancy tends to restore any
vertical displacement and the resulting motion is an oscillatory adjust-
ment towards the final flow. In this phase of the flow, viscous forces
play only a minor role and the flow is essentially governed by a bal-
ance of pressure, buoyancy and inertia. The time scale is thus l//Sg
as discussed in the introduction.
Let
0 * u *
A ; U = =
t* - /eg t and p*
where
P = Ag i
and
A = tt PQ q//£g JL2
Equations (3.1) to (3.7) then become:
V u* = 0 (3.8)
24
-------
where
- + F u*vi* + Vp* = - p*k + R.G. u* (3.9)
ot
-|* + F u*Vp* - w* =| ^p* (3.10)
1 <, z* ^1 and 0 < x* < °°
R ~
-~
ana
c - -
G - I-
In a reservoir F « 1 and R « 1 and thus to the zeroth order .
y . u = 0 (3.11)
£ + Vp - - pk (3.12)
|C - w = 0 (3.13)
ot
x,z e (x,z|-l < z < 1 and 0 < x < »}
u(», z, t) = iH(t)
and
u(0, z, t) = 6(z) H(t) .
(The constant density boundary condition becomes redundant as DVp = 0.)
t The asterisks have been dropped for convenience.
25
-------
Equations (3.11) - (3.13) are identical to the ones solved by Koh (1966a)
for the infinite domain, and his method of solution is adopted here.
The solution in terms of the Laplace Transform of the streamfunction i|t
is shown by Imberger (1970) to be given by:
fy = artan (coth
*
tan
_nTb
2/1+p2
(3.14)
The complete solution is now given by the Laplace inverse of Eq. (3.14)
This inversion is simplified by noting that there exist a positive
number M such that
1+P
< M
for all p with Rel(p) = c > 1 .
Hence for small x and z
artan -
Tip X
(3.15)
which is uniformly valid in p . This is the solution for the infinite
domain and Koh (1966a) has given the solution for u , viz.
u(x,z,t) =
cos 6
nr
J (ID) sin P(t-u))duu| ,
o /
(3.16)
where
= sin 6 , 6 = tan z/x,
and J is the Bessel function of zero order.
Consider x and z fixed, then as t -» »
u(x,zt)
COS
TTT
2
ilf- I
p J
J <">> sin
O
cos 6 sin
6(z) .
(3.17)
26
-------
The layer thickness, defined to be the distance to where u = 0 , is
therefore given for large t by the equation
3t = TT .
In terms of the layer thickness this means
(3.18)
(2n is obtained instead of the graphical value of 6.4 in Koh's 1966
work. )
The finite duct problem has thus the same time response as the infinite
domain. The solution is, however, seen to be non-uniform in both time
and space. The space non-uniformity is dealt with in the next section
for the steady problem and is due to the large velocities near the sink
for all times. But also as t -+
-------
where
P =
A = F e I n -
Ho
In terms of these new variables Eqs. (3.1) - (3.5) become
F(u*u** + w*u**) + p** = R.G. u**^ + R G uj^* (3.20)
F(u*w* + w*w* ) + p* = - p* + R G w* * + R G w* * (3.21)
A £j & £ £j A A.
F(u*p% + w* p*,) - w* = \ R p**z* + \ R p*^ (3.22)
where
F = - ; R =. - ;
sg
The outer region is defined as that part of the flow field in which the
inertia forces may be neglected. By an order of magnitude analysis
Imberger (1970) has shown that the inertia forces may be neglected every-
where except near the origin. The equations governing this outer flow
are given by
p** = R.G. u**z* (3.23)
P** = - p* (3.24)
(3'25)
which must now be solved in the finite duct. The "inner region" is then
that part of the flow where the solution to Eqs. (3. 23) -( 3 . 25) no longer
describes the flow. This will be near the sink.
The solution is most easily found by taking a finite Fourier Transform
in the z* direction. But first define i|r* such that
u* = fz* (3.26)
and
w* = - ** * . (3.27)
A
Then Eqs. (3.23) - (3.25) yield, on the elimination of the pressure,
28
-------
* * - °
dz Sx
2
The presence of the factor R implies a layer thickness of the order
and the appropriate scaling is given by
z** =-^ z* (3.29)
R17
and
x** = n3 x* . (3.30)
Although this scaling is used in the next subsection, the solution to
the outer problem is more conveniently represented in terms of the
scaling
Z = TT Z
and
* (3.31)
x = TT x* R . (3.32)
This also accomplishes the removal of the dependence of R of Eq.(3.27),
with the additional advantage that the wall spacing remains independent
of R . In terms of these variables the boundary value problem becomes,
5 T1 + ° ^ = 0 (3.33)
**^Q i^^j
9e Sx
- TT ^ z ^ TT and 0 <, x < °°
with boundary conditions given by
ty* = =F | z' = ± TT (a)
= ± TT (b)
(3.34)
.4 *
= 0 I = ± TT (C)
sgn z (d)
29
-------
The solution of Eq. (3.33) satisfying Eq. (3.34) and which is bounded
at x" = + °° was found by Imberger (1970) to be given by,
ir.~ i r / i -n3^ . ~ (-i)n . ~\
w(x.z) = ) I - e sin nz - sin nz I
rr L \ n n I
n=lX I
1 V 1 -n°^ . z
) - e sin nz - 7
rr L. n 2rr
n=l
) - e " " sin nz - 7^ (3.35)
Convergence of the series is very rapid everywhere, except at the origin
where it diverges. This is the singularity corresponding to the sink.
The solution given by Eq. (3.35) is shown in Fig. 3.2.
Figure 3.3 is a log-log plot of two streamlines within the forward
flowing layer. It is seen that as "X -» 0 the slope tends to 1/3, which
may be compared with Koh's (1966b) solution for the infinite domain.
To study the non-uniformitv at the sink, it is necessary to know the
limiting solution of \ji(x,z) as 'Sf -» 0 and *z -» 0 . The above indicates
that this is given by Koh's (1966b) solution. A more formal proof is
obtained by noting that the same limit is achieved by fixing x and "z
and letting the duct walls move apart. This process yields
-;
1 -p x ~ ~
e sin pz dz
P
(3.36)
00
3
1 -q ~
e sin qT| dq
o
~ ~l/3
where Tj = z/x . Imberger (1970) has shown that the above is the
solution to the corresponding problem in the infinite domain.
The outer flow therefore changes from a uniform channel flow far from
the sink to a self similar motion as the sink is approached.
Before turning to the inner flow, it should be noted that although the
simplified mass flux condition (3.1) has been used, the solution does
not imply an accumulation of mass in the duct. The rate of mass leaving
the duct at the sink is, by symmetry, equal to
30
-------
w
0
0 0.5 1.0 1.5 2.0
NON-DIMENSIONAL HORIZONTAL DISTANCE FROM THE SINK 7
f
FIG. 3.2 OUTER SOLUTION
-------
w
t\5
NON-DIMENSIONAL VERTICAL DISTANCE FROM THE SINK X = j-773-R
FIG. 3.3 OUTER SOLUTION
-------
(p (o) + p ) q .
The rate of which mass enters the duct at infinity is given by
I
PT u dz
-I
SL
n~ dz
the same as that leaving at the sink. Similarly if s(z) is the con-
centration distribution of a passive substance, such as C>2 , then the
concentration SQ at the sink is given by the mass balance:
3 q = s(z) u(z) dz
o J
-£
A
= q TTJ J s(z) dz (3.37)
-Jt
and
= s(z) dz
Structure of the Withdrawal Layer - Inner Equations
The inner region is that part of the flow in which Eq. (3.35) fails to
yield the proper description of the flow.
In the above subsection it was shown that the governing equations
within the layer in the outer region are obtained by scaling the z*
coordinate with respect to E^^ . Carrying this out for the full
33
-------
Eqs. (3.20) - (3.22) yields
^xr 'n Y_'1c'r_*p'r_'p'fr
o /« V Y jk* f *** ** Y_. **_*=!" *Y* T^T_T.T-_
^/o 'Z ZX XZZ, A ^- ^ z.
<3'38>
*** ***z**> + P*** = - P**
A A fj £-*
<3-39)
2/3
R
1/3
*
where the new density p** = R p* . The outer layer flow equations
are obtained as the limit R -» 0 and F/R2/3 -» 0 .
One method (Koh (1966b)) to compute the influence of the nonlinear con-
vection near the sink would be to expand the stream function \|r* in a
regular series in the small parameter F/R2/3 . However, an inspection
of the second term of this series shows that the series diverges when
x becomes smaller than
where
* = (F/R2/3)3/2 . (3.41)
X
C
In other words, for distances smaller than x* the influence of
inertia is not merely a correction to the outer flow, but it is the
prime force. This means that near the sink 6 = 0(p2) and new
stretched variables are needed.
Let
**
x
3/2
34
-------
and
**
*** =
(3.42)
The outer solution ty* (x* , z**> when rewritten in terms of the inner
variables is
* *
Now as F/R2/3 -» 0 , this becomes ty (i) , where
hence the inner stream-function remains unsealed. Also to be able to
match p** , it must be rescaled
_ _
1/3. 1/2
R /F
as
The equations in terms of these inner variables may thus be written:
it*** V*
2 Z
= G
(3.43)
_ r llr*
2 " Vz***x**x**
e
= ~ P***
(3.44}
R
R
35
-------
,* *** , ,|.* _ -1 *** j. i *** /:
- ? ** P #** + *** 9 P.,**-.** T r ?****** v"
xz x R/xx Gzz
e
where Rg = q/v .
In the limit as Rg -> °° Eqs. (3.43) - (3.45) reduce to: t
till - ill ill + p = G ill (3.46)
TZ ZX TX ZZ X ZZZ
p = - p (3.47)
Z
A_A._^=0 (3.48)
Yz px Yx pz Yx G pzz
which constitute the required inner equations for the limit
R-»0, F-»0, Re-*" and F/R2/3 -» 0 . Equations (3.46) - (3.48)
must now be solved with the solution (3.35) as the upstream boundary
condition.
It is interesting to note that Eqs. (3.46) - (3.48) include the outer
equations, and hence for those regions in which the wall influence is
absent, the flow only depends on the parameter G. Any experimental
flow at a given value of G is therefore self similar with respect to
the coordinates (x**, z***). The experiments described in Section 5
verify this result very well (Fig. 5.1).
The reason why Rg should be large for the above equations to be valid
is perhaps best seen by examining the width-to-length ratio of the region
of nonuniformity. The physical width of the region 6 is of order
and the physical length xc is of order F3/2VR. Hence
and hence 6/x « 1 only if Rg is large. It should be noted that
with respect to the outside scale x*, z** this region is very small
as F/R2-^3 -> 0 . However, in an actual reservoir
3/2 ,_,l/4
= 0(10 ) ft.
t Asterisks have been dropped for convenience.
36
-------
and the inertial region may occupy the whole of the reservoir. When
Re is small, Eqs. (3.46) - (3.48) are nowhere valid, and the flow
near the sink is governed by the full Navier Stokes equations.
As in the case of flow at the leading edge of a flat plate, Eqs. (3.46)
- (3.48) cannot be expected to hold very near the sink. For large IU
this non-uniformity may be examined without actually solving Eqs.
(3.46) - (3.48). Consider distances closer than 0(F1//2). Let
**
(3.49)
then Eqs. (3.43) - (3.45) become
_G_
e
***
(3.50)
***** t****x*** + ty*
***
***
tr> =o\
(3.52)
Hence the flow at the origin is inviscide to 1st order in 1/Rg and the
euations are iv ^
equations are given
(3'53)
(3.54)
t Asterisks have been omitted.
37
-------
The solution of Eqs. (3.53) - (3.55) depends on the solution of Eqs.
(3.46) - (3.48) and vice versa. Neither admits a similarity solution,
nor are they tractable by linear methods. For the very special upstream
condition of uniform flow Kao (1970) has solved Eqs. (3.53) - (3.55).
Here a slip line is required at the edge of the layer^ The^elocity
along this slip line remains constant and of order q2/(eg)^ which
means that around this slip line there is a backward growing viscous-
buoyancy layer of thickness 0(1^/3). However, at the moment it is a
matter of speculation whether these two layers could be matched onto a
solution of Eqs. (3.46) - (3.48).
Below, Eqs. (3.46) - (3.48) are solved approximately by an integral
method. For x** = °° , the withdrawal layer thickness 6 is matched
to the inner limit of the outer solution, and at x** = 0 the boundary
condition chosen is 5*** = i, consistent with the above speculations.
This is further discussed in Section 5, but as long as the solution is
not very sensitive to the exact numerical value at x** = 0 , this pro-
cedure should lead to satisfactory results.
Structure of the Withdrawal Layer - Inner Solution
In the previous section it was shown that for the flow near the sink,
Eqs. (3.46) - (3.48) must be solved using Koh's (1966b) similarity
solution as the upstream boundary condition at x** = °° . For conveni-
ence the equations are reproduced below, omitting the asterisks.
u + w = 0 (3.56)
x z
uu +wu + p =Gu (3.57)
x z x zz
PZ = - p (3.58)
u p + w n - w = i p (3.59)
Kx yz G Kzz
I
udz = - 1 (3.60)
where G = (v/D)2 .
This set of equations does not appear to possess a similarity solution
satisfying the required boundary conditions, so to obtain at least an
approximation of the layer growth, the above equations are solved using
an integral method.
38
-------
Examination of the experimental data obtained by Koh (1966b) shows that
for a particular discharge, the velocity profiles are self similar even
in the inner region. In Section 5 this is confirmed in the more
general coordinates (x**, z***). This suggests postulating a velocity
variation given by
as is done in the Polhausen technique, and obtaining f from the
upstream flow. However, in that solution p ~ 1/z*** for large z***,
which is non-integrable. To be able to apply an integral method the
equations must therefore be premultiplied by a weighting function.
This is more easily done if f is an analytic function expressed in
terms of elementary functions. It is therefore convenient to let
2 _2
, -a(x) (1-k z & )
u(x,z) = ~ - 5 5-5- (3.61)
TT ti , " n~ \^
(1+z 6 >
~ 1
-b(x) z6 0.
p(x,z) = - - 5 5- (3.62)
and by continuity
e f -i . * .
w - "6 za 1"kz 6
w
: on o >
no ,n , 2 ,-23
{1 +z 6 )
The value of k is then chosen to yield the best fit to Koh's (1966b)
solution. Hie unknown functions are a (x) , b(x) and 6(x) .
Equations (3.61) - (3.63) are now substituted into the following
integral form of Eqs. (3.5?) - (3.60):
j - L. __ {(UU +wu) ~n - G « }dz=0 (3.64)
J .- 2 -2. x z z vx zzz
(1 +z o )
o
(3.65)
-------
udz = - 1 (3.66)
J
p O
The factor 1/1+z 6 is included in Eq. (3.64) to make px
integrable. Carrying out the integration yields two ordinary differen-
tial equations and one algebraic equation for a , b and 6 , viz.
-aa' /£ _k k_ \ !_ & / -Ga
TT b +21- 105 J+ 2 -^2
a5 = -- (3.69)
o K
The prime interest is to determine the growth of the layer, that is,
to find 6 as a function of x . Elimination of a and b from
Eq. (3.67) - (3.69) yields one second order equation for 6 viz.
(a 65 - o^ 6) 6" + (o^ 64 + 3a2) (6'')2 + o/3 6' + a = 0 (3.70)
where
a
1 (3-k)
128 1 (2-k)
2 3 n2 (3-k)2
/i + JL _ JE?\
I 7 21 105 /
64 1 /6 k k2\ 1 16 (2-k)
_ 8 Ilk + 32
Q4 ~ 5 3-k
The boundary condition is that 6=1 when x = 0 , and that 6
asymptotes to the solution of Eq. (3.70) with the inertia terms set to
40
-------
zero. The method of solution was to reduce this boundary value problem
to an initial value problem with the initial values of 6 , specified
at some large x , being evaluated from an asymptotic solutions of
Eq. (3.70). The equation was then integrated with a standard fifth
order Runga Kutta algorithm available in the computer center library
under the call name RK5.
To shorten the numerical integration the asymptotic series for large
x of 6(x) was computed by Imberger (1970) to the fifth term. The
series he found is given by
1/3 -1/3 -2/3 -1 -4/3
6 = K^ x + K2 x + K3 x + K4 x + K& x + --- (3.71)
where
K2 =
_ = arbitrary
2
K, =
4 5
14K" K0
JL ^
The similarity of the series (3.71) with that proposed by Koh (1966b)
should be noted. However, the above contains extra terms which are
all dependent on the arbitrary constant K3 . This free parameter is
necessary as nothing has been said about the behavior of the solution
at x = 0 . Similarly Koh's (1966b) proposed series should have an
equivalent arbitrary coefficient. An analogous problem is encountered
in rotating flow problems (Smith (1969)).
That this arbitrary coefficient must be present can also be inferred
by noting that the independent variable x is absent from Eq. (3.70).
Hence if 6 = g(x) is a solution then so is 6 = g(x + c). A formal
41
-------
substitution of x* = x + c into the series (3.71) yields the same
series with K3 replaced by K3 - cK-^/3 , but K^ and K2 remain
unchanged.
The arbitrary constant K3 was used to convert the boundary value
problem into an initial value problem. In other words, Ko was
fixed, 6 and 6' were computed from the series (3.71) for a suffi-
ciently large value of x , and the Eq. (3.70) was integrated numeri-
cally to zero. The constant K3 was adjusted until 6=1 at x = 0.
Inspection of Eq. (3.70) shows that the equation has a singular point
when
6=6 = - .
c cy
The slope of 6 becomes infinite at this value of 6 and the assump-
tion of a similar profile can no longer adequately describe the flow
here.
The value of k was chosen to make 6 =1 and K-, =3.2 the value
obtained by Koh (1966b) for the present definition of 6 . With
k = 1/3 the values K, and 6_ are 3.15 and 1.17.
i «-
Figure 3.4 shows a comparison of the velocity profile given by the
integral curve (3.61) and that obtained by Koh (1966b) . The variation
of 6 with x for G = 29.0 and G = 2.34 is shown in Fig. 3.5.
This corresponds to a salt and heat stratification, respectively.
Summary of the Layer Structure - Composite Solution for 6
The overall layer structure is summarized in Fig. 3.6. The solutions
to region I, II and VI have been found explicitly. In region III an
integral method was used to obtain the layer thickness. The solutions
to regions IV and V are not known and further work is required.
The composite expansion 6 was obtained by adding the inner and outer
solution and subtracting K^ x1/3 , the part which is common to both.
This is shown in Fig. 3.7 and Fig. 3.8 for heat and salt stratification.
Unfortunately the wall spacing in the outer problem does not scale with
F1/2 , so that the wall "moves," as F changes in an inner (x** , z***)
coordinate system. The advantage of this coordinate system, however,
is that near the sink the solution only depends on G .
42
-------
00
VELOCITY PROFILE
(after Koh 1966 b)
ASSUMED INTEGRAL PROFILE
0 0.5 1.0 1.5 2.0
NON-DIMENSIONAL VERTICAL DISTANCE FROM THE CENTERLINE
Z Umax
FIG. 3.4 INTEGRAL SOLUTION VELOCITY PROFILE
-------
INTEGRAL SOLUTION
LINEAR THEORY
O
0
01234
CUBE ROOTOF THE NON-DIMENSIONAL DISTANCE FROM THE SINK (
FIG. 3.5 THEORETICAL WITHDRAWAL LAYER THICKNESS
-------
'A/ / ////////////
SLIP WALL
SLIP LAYER>
m \
BOUNDARY t
LAYER FLOW \
VISCOUS-BUOYANCY BALANCE
(Eq. 3.41-3)
UNIFORM
PARALLEL
FLOW
SINIT
^V " X=xc «-
NNVISCID
CORE
(Eq. 3.70-2)
< x
FIG. 3.6 STRUCTURE OF THE WITHDRAWAL LAYER
-------
05
60
-------
5 uj
ox30
^£20
Q -J
i
O
/////// / / / / / / / / / / / / / / /s? / / / / / / / / / / / / / / /
INNER SOLUTION
I27T
//////////// / / / / /.
67T
0
10
15
20
CUBE ROOT OF THE NON-DIMENSIONAL DISTANCE FROM THE SINK (x/xc V*
FIG 3.8 THE COMPOSITE SOLUTION G - 2 9.0
-------
4. EXPERIMENTAL OBJECTIVES AND APPARATUS
The analytic considerations of the previous chapter showed that the
dynamics of the flow due to a withdrawal from a stratified fluid may
be represented in a scaled coordinate system. In this coordinate system
the only parameters remaining are the square root of the Schmidt number
and the depth of the tank. Hence, when viewed from this reference axes
all flows in a fluid of constant Schmidt number are dynamically similar.
Review of Previous Experimental Data
The experimental results obtained by Koh (1966b) are shown re-plotted
in Fig. 4.1. Each line represents the line of best fit to the experi-
mental points of a single run. The withdrawal layer thickness has been
reduced by 0.9, from that used by Koh (1966b) , to correspond to the
definition used in this report. It is seen that the two cases, salt
and heat stratification, fall roughly into two separate lines on the
graph. For the heat stratified mixture, the results show little scatter
and good agreement with the theoretical predictions. However, when salt
is used, although the agreement is reasonable for small values of
x/x , the data diverges somewhat for larger values. The explanation
for this is probably twofold.
(a) The data for large (x/Xc) are from the rear of the tank,
where end effects are probably affecting the data. This suspicion is
reinforced by the fact that qf , the forward discharge, varied con-
siderably from one station to the next for the same run.
(b) The definition of 6 as that point where the velocity is zero is
very difficult to measure (Section 5.1). This is especially so for
larger values of (x/x) *'* where the velocities are very slow. This
could account for some of the scatter,
Experimental Objectives
In view of the above, an experimental program was carried out in a
facility much larger than that used by Koh (1966b) to repeat and extend
his data. In this way the transient effects due to a finite sized tank
were kept to a minimum. All the experiments were conducted in a salt
stratified solution, rather than one with a temperature gradient, to
avoid the side walls induced convection boundary layers. A perfectly
still reservoir is essential, since the velocities in the withdrawal
layers are extremely small. All the experiments were carried out in
the same 42' long glass tank. For each stratification about six to nine
runs at different discharges could be made before the dye contamination
affected visibility. Four different stratifications were used.
The Experimental Tank
The experiments were performed in a 42 foot long by 4.0 foot deep by
lj foot wide tank of the University of California Richmond Field
49
-------
INTEGRAL CURVES
FOR G = 29.0
AND G = 2.34
G = 2.34
LINEAR THEORY
0
0 I 2 3 4
CUBE ROOT OF THE NON-DIMENSIONAL DISTANCE FROM THE SINK (Vxc)'3
FIG 4.1 EXPERIMENTAL DATA AFTER KOH I966b.
-------
PROBE STAND
DEPTH SCALE
CRANK
(to adjust depth
of probe )
-TROLLEY
,-PLATES TO
\STRATIFY TANK
CALIBRATION
MIXTURE
TROLLEY
SALINITY
PROBE
DISCHARGE SLOTS
FIG. 4.2 GENERAL LAYOUT OF EXPERIMENTAL TANK
-------
Station. The walls and bottom were constructed of tempered plate
glass. The tank is shown schematically in Fig. 4.2 and a photograph
of the general layout is given in Fig. 4.3.
Two trolleys were placed on rails on top of the tank. One served as
a mounting platform for the stratifier while the other held the
salinity probe and the associated vacuum pump. The discharge facil-
ity was located at one end. Mounting the salinity probe on the
trolley allowed the same probe to be used to measure the density
gradients throughout the channel, and the salinity of the discharging
water.
The Salinity Probe
To measure the concentration of salinity a conductivity probe was
built and mounted on one of the trolleys. The design of the probe
was similar to those used successfully at the Delf Hydraulics Labora-
tory, as observed by the second author during a visit in
1968. The probe was used to measure the background near linear
salinity profile. The salt used was an ordinary kiln dried commercial
product. No attempt was made to measure the very small deviations of
salinity due to the motion.
The probe and its associated circuit are shown in Fig. 4.4. A photo-
graph of its mounting on the trolley is given in Fig. 4.3(a). The
probe was constructed from a thin plastic tube, which was closed to
a fine hole of a diameter 0.040" at one end, after the electrodes had
been inserted. The two platinum electrodes, one on the inside, one
around the outside of the tube, created an electric field which had all
its field lines passing through the fine hole at the tip of the probe
(Fig. 4.3(b)). The other end of the 4' tube was connected to a small
vacuum pump which sucked fluid through the small opening at the tip.
The actual volume sucked in amounted to a few drops per second, but
the velocity induced within the contraction was quite high. In this
way the fluid is replaced almost instantaneously in the contracted
opening as the density fluctuates in front of the probe. The fact
that the electric field is concentrated in the opening, as well means
that the probe measures the salinity of the water in the opening
only, and the rapid change of fluid there makes the probe respond very
quickly to any density change. The profiles could therefore be ob-
tained quite rapidly.
To calibrate the probe, a large bottle was filled with water and salt
was added to make a mixture of a concentration corresponding to the
maximum salinity anticipated. This served as the stock solution. The
specific gravity of the mixture was then measured with a hydrometer.
This method proved to be much more accurate than weighing out a cer-
tain amount of salt, adding it to water and then neglecting the
volume of the salt. After the tank had been stratified a series of
5 dilutions of the stock solution were prepared so that their specific
gravity covered the range of the tank stratification. These solutions
52
-------
(a)
(b)
(c)
Fig. 4.3 Photographs of Experimental Equipment
53
-------
BRIDGE
CARRIER
AMPLIFIER
BRUSH INST.
R.V.4622-00
CHART
RECORDER
BRUSH INST
M.K. 220
icon
IOOA
GRID
CONDUCTANCE
PROBE
POSITION
TRANSDUCER
LOG HEED
ELECT.
POWER SUPPLY
SRC 3566
UT
k-
Ch. 1
Ch2
i
4
frl
1 W \
\ \
\ U
6v D. C.
0.02 °/0
REG
FLOAT
FIG 4-4 CONDUCTANCE PROBE
AND FLOAT CIRCUIT
-------
were then used to calibrate the probe after they had reached the
temperature of the water in the tank.
The difference of the temperature of tank water and the temperature
at which the specific gravity was measured was always small, and
since only the density gradient of the tank water is required, no
compensation was made for this temperature difference. The cali-
bration curves were nearly linear, with a slight decrease in
sensitivity for the larger concentrations. Figure 4.5(a) shows a
typical calibration curve. Over a day no drift was noticed. The
probe was, however, recalibrated directly before a density profile
was taken.
Establishment of a Linear Density Profile
The aim was to stratify the tank water so that the density of the
salt water mixture varied linearly with depth. The technique used
is essentially that reported by Clark et al. , (1967).
The whole tank was filled with fresh water, which was allowed to
come to room temperature. A little detergent was added to reduce
the formation of bubbles on the side of the glass which would other-
wise hamper the visibility. A gated partition was then introduced
vertically into the center of the tank. The partition consisted of
a piece of plywood with 9" hinged flaps at the bottom and top (Fig.
4.2). The partition was sealed to the glass walls with a soft rubber
spacer. This was sufficient to stop any transfer of water from one
side to the other.
The appropriate amount of salt was then stirred into one half of the
partition tank. To facilitate mixing the water was pumped from the
bottom to the top. Once the mixture was uniform it was allowed to
come to rest. The flaps of the partition were then opened very
slightly to allow the salty water to flow beneath the fresh water and
the fresh water to flow over the salty water. About 1/2 hour elapsed
before the whole tank had a two layer stratification in it. The
partition was then carefully removed.
To obtain a linear profile, a series of three plates, mounted on the
trolley, were then briskly moved through the interface with a crank
and pulley arrangement at the end of the tank. The largest plate
moved directly through the interface while the two smaller ones
stirred up the top and bottom layers. The wakes behind the plates
created sufficient turbulence to allow mixing. The mixing smoothed
out the interface, and after some adjustment of the position of the
plates a very linear profile was obtained throughout the whole depth.
Figure 4.2 shows the general layout of the pulley, trolley and the
mounted plates. A typical density profile is given in Fig. 4.5(b),
showing how the linearity holds right to the bottom and the free
surface.
55
-------
en
O)
PROBE CALIBRATION
1.02 1.03
SPECIFIC GRAVITY
FIG. 4.5a EXAMPLE OF THE PROBE CALIBRATION
-------
Ol
WATER LEV
CO
LU
Q_
LU
Q
0
10
20
30
40
1.00
DENSITY PROFILE
1.01 1.02
SPECIFIC GRAVITY
1.03
1.04
FIG.4.5b EXAMPLE OF THE DENSITY PROFILE
-------
The linear profile was very stable and would last unaffected for up
to two weeks. Only at the surface and bottom was there a slight
degeneration of the profile. At these extremes a layer of uniform
density formed, the thickness of which increased with time. Most
probably this was a temperature induced thermocline formation, as
the room air conditioning did not accurately regulate the air
temperature.
The Discharge Meter
To ensure constant monitoring of the discharge flowing out through
the sink, a float type discharge meter was installed. This provided
a means of adjusting the flow rate at a desired value and also to
check its constancy as the experiment proceeded. The actual dis-
charge used in the reduction of the data was, however, obtained by
integrating the velocity profiles. The procedure used is explained
in Section 5.
The flow from the slotted discharge tube was drained off from both
sides of the tank into two plastic tubes. The two tubes were joined
with a T section and the flow entered the discharge meter through a
gate valve behind the T. The general layout is shown in Fig. 4.6
and the circuit used is depicted in Fig. 4.4. On entering the meter
the water spilled over a cup down a pipe into the float chamber. The
float motion activated a pully arrangement which in turn was moni-
tored by a potentiometer.
The Dye Particles
The velocities induced by the fluid withdrawal are very small. The
magnitudes are of the order of a few centimeters per minute. For
such low velocities the only suitable method of measurement appears
to be that of dyeing parts of the fluid and taking photographs of the
deformation of the dye lines. The method used was to drop crystals
of dye into the center of the tank at incremented distances from the
sink. The purple streaks left in the wake of the falling crystals
were then photographed repeatedly.
Of the dyes available, it was found that potassium permanganate was
the most suitable for the present experiment. In contrast to organic
dyes, this has the great advantage that it forms very fine intense
streaks, the lifetime of which are only about 1/4 hour. Their rapid
decay, which is due to oxidation, allows experiments to be repeated
quite quickly.
The crystals used were about 1/4 - 1/2 mm. in diameter and care was
taken to choose the most spherical ones only. Elongated or plate-like
crystals form spiral paths which makes the velocity measurement more
tedious. A razor blade was used to split larger crystals down to
the required size and shape. The cutting of larger crystals also
exposed fresh permanganate and much more intense streaks could be
58
-------
T
'POTENTI METER
WIRE
ROD
PLASTIC
FLOAT
WATER
CONSTANT HEAD
OVERFLOW
^
'? ,
\
TANK
OUTLET
FIG 4.6 DISCHARGE METER
59
-------
obtained. The crystals were placed, after cutting, into a small re-
cess in a wooden dowling at a spacing of about 10 cm. This wooden
bar was then placed on top of the tank. To introduce the crystals
into the water, the bar was turned through 180°.
Photography and Timing
The motion of the dye streaks was recorded photographically. The
time between photographs was fixed by tripping the camera shutter
and the event marker on the recorder, simultaneously.
A Rolleflex S.L. 66 with an 80 mm lens was used throughout the ex-
periments. The camera was fixed at one of two predetermined posi-
tions on a sturdy bench. The positions were so chosen that two
photographs covered dye lines ranging in distance from 0 to 160 cm
from the sink. The film used was Plus x professional and this was
developed in D 76 to a characteristic value of .65. The negatives
were then enlarged to 1/2 full scale on ll" x 14" Resisto-Rapid
photographic paper. The plastic base of this paper makes it dimen-
sionally stable. Successive photographs were aligned over each
other, by aligning the images of a series of marks on the glass walls
of the tank. The profiles were then pricked through with a sharp
needle. A typical dye profile is shown in Fig. 4.7.
To fix the scale of the photographs a 200 cm measuring scale was
held into the water at the center of the tank, where the dye lines
normally are. This was then photographed. As the position of the
bench and the tank remained fixed, this calibration had to be done
only once. Also in this way parallax errors do not occur. The
scale negative was used to set the enlarger to 1/2 full scale. The
alignment marks on the side of the tank offered spot checks on the
scale of each photograph.
The camera shutter was released with an electric relay, which also
activated the event marker on the Brush recorder. The time between
successive photographs was thus displayed by the distance between
marks on the chart. Times from 10 to 200 sec. were typical. The
accuracy, estimated by comparison with measurements with a stop watch,
was near 0.3 sec.
Experimental Procedure
After the tank had been filled with tap water and allowed to stand
for a day or so, the procedure for each experimental run was identi-
cal and may be summarized as follows:
(a) A small amount of Photoflo detergent was added and the
tank water was stirred.
(b) The temperature of the room and the water was measured.
The difference was kept less than 2°C.
60
-------
Fig. 4.7 Example of Dye Traces
'
-------
(c) The wooden partition was inserted at the center of the tank.
(d) The required amount of salt was dumped into one side.
(e) The salt was mixed by first agitating the water with a broom
and then circulating the mixture with a small pump, mounted
on the trolley. After the mixture was uniform it was allowed
to come to rest.
(f) The flaps on the partition were opened about l". After about
1/2 hour the interface was continuous past the partition and
the latter was removed.
(g) The stratifying plates were pulled through the interface.
(h) Calibration solutions (dilutions of the stock solution)
were prepared and their temperature was checked against that
of the tank water.
(i) The probe was calibrated and then used to obtain a density
profile of the tank water.
(j) The gate valve at the discharge end was set to yield the
required flow.
(k) Crystals of permanganate were cut to the right size and placed
on the wooden trough on the top of the tank.
(1) The camera was loaded and mounted in the rear position.
The lights were adjusted.
(m) The rear crystals were dropped and the dye lines were photo-
graphed. The time between (j) and (m) was at least 1/2 an hour.
(n) The camera and lights were moved to the front window and the
procedure listed in (m) was repeated.
(o) The salinity of the discharge water was measured.
(p) The gate valve was reset to a new flow rate and the procedure
was begun at (j) again.
Experimental Program and Data
The experimental program is summarized in Table 4.1. All the data,
giving the withdrawal layer thicknesses at different distances from
the sink, are reproduced in Appendix I. The aim was to vary G and q
to yield a good variation of xc. The effect of different viscosities
or diffusivities was not investigated. The specific gravity of the
discharge water was always equal to that corresponding to the specific
gravity of the tank water at the level of the sink and it is therefore
not included in the table.
62
-------
TABLE 4.1. EXPERIMENTAL PROGRAM (Key to table on following page)
Exp.
No.
1-1
1-2
1-3
2-1
2-2
2-3
3-1
3-2
3-3
3-4
3-5
3-6
4-1
4-2
4-3
4-5
4-6
4-7
4-8
4-9
T °r
*W
14°C
14°
14°
20°
20°
20°
14°
15°
15°
18°
18°
18°
14°
14°
16°
16°
16°
14°
14°
14°
V
in-5 2 -1
10 cm sec
1120
1120
1120
1000
1000
1000
1120
1100
1100
1040
1040
1040
1120
1120
1080
1080
1080
1120
1120
1120
D
nn-5 2 -1
10 cm sec
1.25
1.25
1. 25
1.25
1.25
1. 25
1.25
1. 25
1.25
1.25
1.25
1,25
1.25
1.25
1.75
1.25
1.25
1.25
1.75
1. 25
I0"5cm"1
2.98
2.98
2.98
25.7
25.7
25.7
29.9
29.9
29.9
29.9
29.9
29.9
35.3
35.3
35.3
35.3
35.3
35.3
35.3
35.3
q
2 -1
cm sec
.210
.320
.454
.006
9
?
.005
.013
.031
.064
.118
.185
.006
.005
.007
.009
.012
.017
.022
.037
%
2 -1
cm sec
.225
.366
.496
.012
.062
.272
.007
.014
.033
.073
.143
.233
.007
.007
.010
.011
.013
.020
.024
.044
G
approx.
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
F
m
2
cm
1.31
2.14
2.90
.024
.124
.543
.013
.026
,062
.136
.264
.429
.012
.012
.017
.019
.022
.034
.041
.075
a3
o
,~3 -2
10 cm
.458
.458
.458
1.46
1.46
1.46
1.45
1.46
1.46
1.50
1.50
1.50
1.57
1.57
1.60
1.60
1.60
1.57
1.57
1.57
X
c
cm
688
1439
2260
5.4
64.3
584.0
2.18
6.43
22.8
25.0
204.0
420.0
2.14
2.22
3.81
4.38
5.85
9.90
13.40
32.2
w
-------
Column 1: The first number corresponds to the density stratification,
while the second designates the run number for that
stratification.
Column 2: T is the temperature of the water in degrees centigrade.
Column 3: v is the kinematic viscosity. The values are those of
pure water at T °C.
w
Column 4: D is the molecular diffusivity of salt in water.
i dP
1 e
Column 5: e = -- computed by the formula.
p dz
o
Sp. Gravity at z - Sp. Gravity at z
Column 6: q is the discharge measured with the discharge meter
divided by the width of the channel.
Column 7: q is the forward discharge obtained by integration of
the velocity profiles (Section 4.2).
1/2
Column 8: G is the mean value of (v/D) for all the runs.
Column 9: F is the Froude number q //eg .
m m
3
Column 10: a is the inverse of the Rayleigh number. This is used
to facilitate comparison with Koh' s 1966b data (Sg/vD)5
Column 11: x is the critical distance below which inertial effects
become important. (p^/2 Q>3)
m o
64
-------
5. ANALYSIS AND DISCUSSION OF EXPERIMENTAL RESULTS
Analysis of Data
The half scale photographs of the dye lines at consecutive times
formed the unprocessed velocity data. The time between photo-
graphs was obtained by measuring the length of chart between
notches at the event marker trace. The reduction of this data may
be divided into three parts,
(a) Discharge calculations: The no slip condition on the side walls
forced the layer to have a parabolic velocity variation across the
width of the tank, rather than a uniform distribution as assumed
in the theory. For this reason the local discharge q , obtained by
integrating the velocity profile, was used in the reduction of the
data. To obtain qm, successive photographs were aligned and the
top profile was pricked through onto the other photograph. The
area between the two curves of 6 roughly equally spaced profiles was
then measured with a planitneter. The total net area was used rather
than just the forward flowing part. To within 10% the areas were
the same and the mean of the 6 areas was used to calculate the
discharge by the formula
q = mean area T time between photographs.
(b) Withdrawal layer thickness 6: In the integral solution 6 was
defined by the velocity profile
2 2
a(x) 1-kz 6
when z = 6
TT 2s;-2,
(1+z 6 )
1
u 12
max
hence a convenient definition for 6 in the experimental data is that
width at which the velocity has dropped to 1/12 the center line
value. This value of 5 was also much better defined on the photo-
graphs than the point where the velocity is zero. The profile had
a large non-zero slope here, and it formed a definite intersection
with the constant line of u = u . The values of 6, measured as
J_ ^ TrlclX
described, and the above value of qm were then used to set up a table
as shown in Appendix I. The actual quantities plotted on Fig. 5.1 are:
f^l/3 3/2
versus I) , where x = F
\x / ' cm
c
65
-------
03
CT)
CO
CO
or
LJ
INTEGRAL SOLUTION
10
CE
Q
X
o
CO
o
z
o
0
i 1
0.2
t
0.67
1.0
1.62
2.0
3.0
4.0 f
4.13
CUBE ROOT OF THE NON-DIMENSIONAL DISTANCE FROM THE SINK (*/x )/3
v
FIG.5.1 EXPERIMENTAL WITHDRAWAL LAYER THICKNESS! G =29.0
-------
1/3
(c) Velocity Profiles: At four values of (x/x ) the actual
velocity profiles were reduced. To obtain the velocity profile, the
image of the dye lines was transferred onto a transparent graph
paper. From this the ordinates were read off and divided by the
time between photographs to yield velocities. The above velocity
profiles were all plotted on a single graph by normalizing the
maximum velocity and the discharge to 1. This is shown in Fig.
5.2. The solid line again corresponds to the integral velocity
distribution. Rewritten to have u/u as the ordinant and zu
., , ., . , max max
as the abscissa this reads:
1 - .367(zu )2
u max
max (l+l.l(zu ))
max
Effects of the Finite Size of the Tank
Figures 5.1 and 5. 2 show little scatter, indicating that random
errors due to parallax, enlarging, and aligning of photographs are
quite small. Systematic errors are, however, possible as the tank
has a finite width and a finite length. The possibility of a
systematic error of this sort is now assessed.
The finite width introduces a parabolic profile in the y direction.
Koh (1966b) has made a detailed analysis of the effects of the para-
bolic profile on the withdrawal layer thickness. As long as the
scale in the vertical direction, 26, is smaller than the width, and
provided the local centerline discharge qm is used, the error intro-
duced will be small. In the present experiments the ratio of thick-
ness to width of the layer was kept to less than 0.1 by taking data
only in the first 6 feet of the tank.
The tank was long enough so that the measured discharges in the test
section were independent of the distance from the sink. This means
that the layer spread sufficiently to prevent any influence of the
back wall. To ensure that the layer far upstream was inertia free,
the discharge q was limited to make x less than the length of the
tank. In the experiments xc was kept, with one exception, to below
1/6 of the tank length.
With the above precautions, it is reasonable to assume that the data
is a true representation of flow in a long reservoir.
Comparison With The Theoretical Results And Previous Experiments
The present experimental results have been summarized on Figs. 5.1
and 5.2. The data from Koh (1966b) was replotted to conform with
theoretical scaling as shown in Fig. 4.1. Comparison shows excellent
agreement up to (x/xc) ' of approximately 2. For larger values of
this parameter Koh (I966b) found values of 5 slightly larger than
67
-------
en
CO
X
O
E
-1.0
o
O
>
_l
<
o
M
cc
O
O
c/)
Z
LU 0
O
_____ !
INTEGRAL PROFILE
I-0.367 (-
0
>'] J
KEY SYMBOL
I
2 o
3 A
4 n
4.13
1.62
0.67
0.20
* ff
0.5
.0
1.5
2.0
NON-DIMENSIONAL VERTICAL DISTANCE FROM THE CENTERLINE
FIG 5.2 EXPERIMENTAL VELOCITY PROFILES! G = 29.0
Z U
2.5
max
-------
those measured by the author. However, as already pointed out, this
difference is not significant as it is most probably due to end wall
effects in the former's experiments.
Theory showed that the flow should be independent of any parameter
except G, when the x and z coordinates are scaled correctly. This
is verified in Figs. 5.1 and 4.1 where the data is seen to fall all
onto one curve for a particular G. The above observation is equally
true for .the salt stratification where G is quite large, even though
the theory was developed for G of order one. This may be explained
by noting that the inner Eqs. (3.46) - (3.48) are merely a boundary
layer type simplification of the full equations and therefore still
contain a representation of all the forces. Thus even if there are
for large G, regions in which diffusion is absent, the governing
equations there would be special cases of the more general Eqs.
(3.46) - (3.48).
For the same reason a large value of G does not invalidate the inte-
gral solution results. A comparison between the theory and experi-
ment is shown in Fig. 5.1 and it can be considered good, everywhere
except near the sink. The difficulty here results from the still
unknown behavior of the flow very near the sink. There is a tempta-
tion to adjust K3, the floating upstream constant, to improve the fit
near the origin. Figure 5.3 shows curves for different values of K3
and it is seen that the integral curve for K3 = -82.2 fits the data
better than the originally adopted curve, which has K3 = -81.8. This
is an encouraging flexibility of the method, but the original choice
of K3 remains the only theoretically justifiable one until a more
definite solution for the flow very close to the sink is obtained, to
which the integral solution may be matched.
The depth of the experimental tank was 4 feet, which was large com-
pared with the values of F1/2 used, so that the top and bottom
boundaries did not influence the flow for the experimental values of
(x/xc). This meant that the linear or outer solution, as derived
in Section 3.3 and included in Fig. 5.1, was identical with Koh1s
(1966b) solution. Inspection of Fig. 5.1 shows that the integral
solution, which reflects the effects of the inertial terms in the
equations of motion, predicts layer thicknesses more in agreement
with the experimental data than either the solution of Koh (1966b) or
the solution 6/F1/2 =1.35 found by Kao (1970).
m
By examining Koh1s (1966b) data Brooks and Koh (1969) anticipated to
some extent the theoretical scaling. In Fig. 9 of that reference
Koh's (1966b) data is replotted in terms of
a q
versus
2/3
X x
o
69
-------
01234
CUBE ROOT OF THE NON-DIMENSIONAL DISTANCE FROM THE SINK (Mcc)'/3
FIG.5.3 VARIATION OF THE INTEGRAL SOLUTION WITH K3
oo
-------
A little rearranging shows that this is in the present notation
3.57 f
1/3
X i
c
versus
x .2/3
PN<
\x 1
The only real difference between the above and that established in
Section 3 is the factor (x/xc)1//3 in the ordinant.
It was hoped that the velocity profile measurements would indicate
some of the structure changes very close to the sink, and clarified
the nature of the inviscid core. Figure 5.2 shows, however, that
even for x of the order of F1'2 the velocity profiles are self simi-
lar to those for large (x/xc). The only change is that the back
flow velocity decreases, in relation to the center line velocity, as
x is decreased or, equivalently, as q is increased. Further work is
needed both experimentally and theoretically to describe the flow
close to the sink.
71
-------
6. COMPARISON WITH FIELD DATA
The tests conducted by the Tennessee Valley Authority on some of their
reservoirs appears to be the only source of actual reservoir data.
The work summarized in the T.V.A. reports (1969(a),(b)) is concerned
with the measurement of the velocity, temperature, and water oxygen
content distribution in a reservoir from which water is being drained.
The investigations involved the Fontana and Cherokee Reservoirs with
further work in progress in the Douglas and Barron-Nolin Reservoirs.
The difficulty of measuring velocities is compounded by the unsteady
nature of the operating discharge and the irregular geometry of most
reservoirs. Over the first 20 miles the Cherokee Reservoir is in the
form of a 1 mile wide trough making it the most suitable example for
a comparison with the simplified geometry of the theory. Altogether,
there are data from 5 steady runs, 4 from the Cherokee and one from
the Fontana reservoir. The complete data is summarized in Table 5.1.
The investigations dealing with the transient response are still very
limited; only one complete run has been published.
Unsteady Withdrawal
In Section 3 it was shown that the time scale of the reservoir res-
ponse , to an increase in turbine discharge, was of the order of
l//£g. The actual time it will take to form a withdrawal layer of
thickness 6 at a distance x from the sink was found to be given by
S
1 TTX
t = /^F 6
The value of 6g is obtained from the composite solution given in Fig.
3.7. The turbine response for the two partial tests given was very
close to a step function, and the time taken to reach steady state,
at the station corresponding to the data 8-16-67, was of the order of
12 hours. By comparison the theoretical value yields
1 8980 1 , ,
t = == TT = 13.6 hrs.
/? QvTn~4 lo.bXl.bU dbUU
i*J O . O A 1 U
The agreement is rather encouraging.
Steady Withdrawal
Figure 6.1 shows the available prototype data, plotted non-dimension-
ally with respect to molecular transport properties, as described in
Section 3. The half layer thickness chosen was that which corres-
ponded to a velocity 1/12 the maximum velocity. The theoretical
curve is shown for comparison. The prototype values are seen to be
50-100% larger than those predicted. This is especially noticeable
since the experimental results in Fig. 5.1 are always a little smaller
73
-------
TABLE 6.1. PROTOTYPE DATA FROM T.V.A. (1969(a) & (b)
Date
T°C
V
io"5
ft*
-1
sec
D
TO'5
ft*
-1
sec
G
TO'5
ft'1
eg
10"1
-i
sec
3
ao
TO3
ft'2
g
ft2
-1
sec
F
m
ft2
rJ
m
ft.
X
c
IO6
ft
6
ft
V
F^
m
X
ft
(x/xc)
2*
ft
2V
p2
m
Cherokee
8-17-57
9-20-67
8-16-67
8-24-67
9-14-67
25
20
20
Ex
20
0.900
0.980
0.980
traneou
0.980
0.144
0.144
0.144
s curre
0.144
12
0.36
1.2
nts.
1.65
3.8
1.1
3.8
5.3
5.2
2.8
5.2
6.1
5.6
5.2
6.75
3.8
288
495
346
165
17.0
22.2
18.6
12.8
25.5
30.8
33.3
12.8
40
40
40
30
2.4
1.8
2.2
2.3
2,100
2,100
8,980
52,800
.049
.041
.065
.16
150
140
140
120
8.9
6.3
7.5
9.4
Font ana
9-9-66
20
.980
.144
1.11
3.6
5.1
7.6
400
20
30.6
45
2.3
3,430
.022
330
16.5
-------
Ol
U. \ o
0 ao £
ten
2E
tn i-
LU a: |
-5 LU '
Q
i
0
x =i
(0.065,1.6)
DATE RESERVOIR SOURCE
LAYER THICKNESS
IN AN INVISCID FLUID
(BROOKS 8 KOH 1969)
V
0
o
D
9-9-66
9-10-67
8-17-67
8-16-67
FONTANA ^jT.V.A. WATER
CHEROKEE RESOURCES
CHEROKEE\RESEARCH
CHEROKEE! REPORT N0.4
A 9-14-67 CHEROKEEJ AND 13
0 0.05 O.I 0.15 0.2
CUBE ROOT OF THE NON-DIMENSIONAL DISTANCE FROM THE SINK (Vx,
FIG. 6.1 PROTOTYPE EXPERIMENTAL DATA
-------
than the integral curve for such small values of
Column 11 of Table 6.1 shows that for molecular transport coeffi-
cients, the corresponding values of xc for both the Cherokee and
Fontana data are about 10-100 times larger than the length of the
reservoirs. This means that the inertial region III of Fig. 3.6
extends over the whole reservoir. Consequently the outer solution
is valid nowhere in the reservoir. Brooks and Koh (1969), in an
effort to explain the difference between the observed withdrawal
layer thickness and that predicted by Koh (I966b) , postulated eddy
transport coefficients much larger than their molecular counter-
parts. The use of Koh's (1966b) solution is now seen to be
incorrect, but a similar procedure can be used for the present
theory. An increase in v and D, decreases xc and this moves the
data points to the right in Fig. 6.1. With the assumption that
v = D, a typical value of the eddy transport values, sufficient to
move the data onto the theoretical curve is 10~3 ft2 sec" . This
is an increase over the molecular values of 102, considerably less
than the 103 postulated by Brooks and Koh (1969) on the basis of Koh's
(1966b) theory. The value of xc, corresponding to such transport
coefficients and the mean of the data for the first three runs listed
in Table 6.1, is approximately 23 miles, which is roughly the length
of the reservoir. This means therefore that the use of the higher
transport coefficients still requires the inclusion of inertial
effects.
The weakness of the above procedure is that as yet, there is no ex-
planation of a mechanism which could sustain these larger eddy
transport coefficients. Measurements (T.V.A. 1969b) of the local
Richardson number with depth, for a typical run indicate that this
varies between 10 and 10,000; values much higher than required for
local shear instability.
The large values of x encountered in reservoirs with peak dis-
charges means that the end walls cannot be neglected and these could
have 1st order effects on the thickness of the layer. An end wall
stops the withdrawal layer and much larger reverse flows are induced.
A large discharge could cause a vigorous layer to extend to the rear
wall and this would induce a reverse layer above and below the center
layer. Schiff (1966) has found, in a related problem, that this in-
duced snaking of the flow with possible mixing at the boundaries leads
to increased diffusion of the species.
In summary it can be said that the theory, although yielding good
agreement with laboratory experiments, is still not able to completely
explain prototype behavior. However, it does yield a rational basis
from which to proceed to include the effects of larger eddy transport
coefficients and end wall effects.
76
-------
ACKNOWLEDGEMENTS
The authors would Like to thank Dr. G. Corcos and Dr. A, Lewy for
reading the manuscript and offering their valuable suggestions.
Appreciation is also due to Mr. K. Bremel who designed and superx^ised
the construction of the experimental tank; Mr. E. Parscale, who
built most of the experimental equipment; Mr. W. Kot, who prepared
the drawings; and Mrs. D. Aoki and Mrs. D. Haasarud, who typed the
manuscript.
The first writer wishes to thank the Robert and Maude Gledden Fund of
the University of Western Australia for making it possible for him to
come to Berkeley by providing a fellowship for two years. Furthermore,
he would like to thank the University of California for providing him
with a Science Fellowship in the third year.
The initial portion of the research was supported by funds from the
Water Resources Center of the University of California under project
W-204. The support of the project by the Environmental Protection
Agency and the help provided by Mr. Phillip C. Woods, Grant Project
Officer, is acknowledged -with sincere thanks.
77
-------
REFERENCES
Beard, L., and Wiley, 1970. An approach to reservoir temperature
analysis, Water Resources Research (in press).
Bellman, R., 1953. Stability theory of differential equations.
McGraw-Hill, New York.
Brooks, N. H., and Koh, R. C. Y. , 1969. Selective withdrawal from
density-stratified reservoirs. Journal of the Hydraulics Div.,
Proc. ASCE, Vol. 95, HY4, pp. 1369-1400.
Childress, S., 1966. Solutions of Euler's equations containing
finite eddies. The Physics of Fluids, Vol. 9, No. 5,
pp. 860-872.
Clark, C. B., Stockhausen, P. J., and Kennedy, J. F., 1967. A method
for generating linear density profiles in laboratory tanks.
Journal of Geophysical Research, Vol. 72, No. 4, pp. 1393-1395.
Debler, W. R. , 1959. Stratified flow into a line sink. Journal Eng.
Mech. Div., Proc. ASCE, Vol. 85, pp: 673-695.
Fraenkel, L. E., 1961. On corner eddies in plane inviscid shear
flow. Journal of Fluid Mechanics, Vol. 11, Pt. 3, pp. 400-406.
Gelhar, L. W. , and Mascolo, D. M., 1966. Non-diffusive characteris-
tics of slow viscous stratified flow towards a line sink.
M.I.T. Hydrodynamics Laboratory Report No. 88, pp. 1-39.
Gill, A. E., and Smith, R. K., 1970. On similarity solutions of
the differential equation ^zzzz + tyx = 0. Proc. Camb. Phil.
Soc., Vol. 67, pp. 163-171.
Green, A. E., and Naghdi, P. M., 1969. On basic equations for mix-
tures. Univ. of Calif. College of Eng. Report No. AM-69-4,
pp. 1-19.
Imberger, J., 1970. Selective withdrawal from a stratified reser-
voir. Ph.D. Thesis, Univ. of Calif., Berkeley, Dec. 1970.
Kao, T. W., 1965. A free-streamline solution for stratified flow
into a line sink. Journal of Fluid Mechanics, Vol. 21, Pt. 3,
pp. 535-543.
Kao, T. W., 1970. A free-streamline theory for inviscid stratified
flow. Physics of Fluids, Vol. 13, No. 3, pp. 558-564.
Koh, R. C. Y., 1964. Karman integral solution to the problem of
selective withdrawal. California Institute of Tech., W. M.
Keck Laboratory Tech. Memo. 64-14, pp. 1-16.
79
-------
Koh, R. C. Y. , 1966a. Unsteady stratified flow into a line sink.
Journal of Hydraulic Research, Vol. 4, No. 2, pp. 21-34.
Koh, R. C. Y. , 1966b. Viscous stratified flow towards a sink.
Journal of Fluid Mechanics, Vol. 24, Ft. 3, pp. 555-575.
Krenkel, P. A., Thackston, E. L. , and Parker, F. L. , 1969.
Impoundment and temperature effect on waste assimilation,
Journal of the Sanitary Eng. Div., Proc. A.S.C.E., Vol. 95,
No. SA1, pp. 37-64.
List, E. J. , 1968. A two-dimensional sink in a density-stratified
porous medium. Journal of Fluid Mechanics, Vol. 33, Pt. 3,
pp. 529-543.
List, E. J., 1970. Private communication.
Long, R. R. , 1959. The motion of fluids with density stratification.
Journal Geophys. Res., Vol. 64, pp. 2151-2163.
Lorence, K. Z., 1952. King Solomon's Ring. Thomas Y. Crowell
Company, New York.
Miles, J. W. , 1961. On the stability of heterogeneous shear flows.
Journal of Fluid Mechanics, Vol. 10, Pt. 4, pp. 496-508.
Mliller, I. , 1968. A thermodynamic theory of mixtures of fluids
Arch. Rational Mech. Anal., Vol. 28, pp. 1-39.
Orlob, G. T. , and Selna, L. G., 1970. Temperature variations in
deep reservoirs. Journal of the Hydraulics Div., Proc. ASCE,
Vol. 96, HY2, pp. 391-410.
Redekopp, L. G. , 1969. Theoretical analysis of boundary layer
regions in stratified flow. Univ. of Calif. Los Angeles,
School of Eng. and App. Sc. Report No. 69-23, pp. 1-156.
Redekopp, L. G., 1970. The development of horizontal boundary layers
in stratified flow. Journal Fluid Mechanics, Vol. 42, Pt. 3,
pp. 513-527.
Schiff, L. I., 1966. Lateral boundary mixing in a simple model of
ocean convection. Deep Sea Research, Vol. 13, pp. 621-626.
Segur, H., 1970. Private communication.
Stotta, L. S. , Elwin, E. H. , Mercier, H. T. , and Terry, M.D. , 1969.
Stratified Reservoir Currents. Oregon State Univ. Eng. Exp.
Station, Bull. No. 44.
80
-------
Smith, R. K., 1969. On the effects of vorticity entrainment in
zonal jet flows. Journal of Atmospheric Sciences, V:>1. 26,
No. 6, pp. 1233-1237.
Spiegel, E. A. and Veronis, G., 1960. On the Boussinesq approxi-
mation for a compressible fluid. Astrophysics! Journal,
Vol. 131, pp. 442-447.
Tennessee Valley Authority, 1969. The prediction of withdrawal
layer thickness in density stratified reservoirs. Water
Resources Research Lab. Report, No. 4, pp. 1-8.
Tennessee Valley Authority, 1969. Cherokee Reservoir selective
withdrawal. Water Resources Research Lab, Report No. 13,
pp. 1-30.
Trustrum, K., 1964. Rotating and stratified flow. Journal of
Fluid Mechanics, Vol. 19, Pt. 2, pp. 415-432.
Yin, C. S., 1965. Dynamics of Nonhomogeneous Fluids. The MacMillan
Co., New York.
81
-------
GLOSSARY OF TERMS
D - Molecular diffusion coefficient of salt or heat
q
F,F = Densimetric Froude number - - -
H
G = Square root of the Schmidt number I (v/D)2 1
g = Acceleration due to gravity
J = Momentum flux
K. = Coefficients in the asymptotic expansion of the withdrawal
layer thickness
L = A typical length scale of the motion
9j = Half depth of the reservoir
P = Scale of the pressure perturbation
p = Total pressure
p = Hydrostatic pressure
p = Pressure due to the motion
q,q = Discharge into the sink from the right; measured q
\2 T /,,r>\2
R = Reynolds number ( )
e \v/
D n v. K
R = Rayleigh number I 1 or
2 2 2N^
Radius vector ( (x + z ) j
T = Brunt Vaisala period ( l/(£g)2 1
t = Time
u
= Velocity vector 1 (u , v, w) j
x = Horizontal distance from the sink
(\
;
I I
5k 3k (X
x = Scaled horizontal distance from the sink I T-
\x*
c
83
-------
I ^jjc \
*** = Scaled horizontal distance from the sink y- x j
F3/2\ f 3/2
x
x* x = Length of the non-uniformity I
c c SK
z = Vertical distance from the sink
z* = Scaled vertical distance from the sink
( z* >
z** = Scaled vertical distance from the sink I- , j
*D*-/ ** *
z*** = Scaled vertical distance from the sink f
a = (g/vD)1/6
o
6 = Withdrawal layer half thickness
fl
£ = Normalized background density gradient (-
o
A = Scale of the density perturbation
p, = Molecular viscosity
v = Kinematic molecular viscosity Ij
o
p = Total value of the density of the water
T
p = Mean of the density of the water over the depth
o
p = Background density
e
p = Density perturbation due to the motion
/ \
p* = Scaled density (-T- i
p** = Scaled density (R p*)
/Fl/2 \
p*** = Scaled density - p**
T = Surface stress
l|f - Stream function
l|f = Scaled stream function I
84
-------
APPENDIX I
EXPERIMENTAL WITHDRAWAL LAYER THICKNESS DATA
85
-------
CO
e a F x 6
o m c
Exp . No .
10"5 cm"1 103 cm"2 cm cm cm
1-1 2.98 0.458 1.145 688 1.0
1.3
1.5
1.7
2.0
2.1
2.3
2.7
2.8
2.9
3.3
3.5
1-2 2.98 0.458 1.46 1439 1.6
1.8
2.3
2.4
6/F 2
m
0.87
1.14
1.31
1.49
1.75
1.83
2.00
2.36
2.44
2.54
2.88
3.06
1.10
1.23
1.57
1.64
X
cm
1.0
3.4
6.0
8.3
13.6
15.2
20.8
29.0
37.4
43.4
52.0
60.8
2.6
7.0
10.8
12.6
<*/*/ 3
0.113
0.170
0.206
0.229
0.270
0.280
0.315
0.348
0.378
0.398
0.422
0.445
0.106
0.169
0.196
0.206
-------
3 *
e a F x 6
Exp. No. o m c
-5 -1 3 -2
10 cm 10 cm cm cm cm
2.4
2.6
2.9
3.0
3.3
3.3
00
3.5
1-3 2.98 0.458 1.70 2260 1.6
1.9
2.3
2.8
3.4
3.5
3.5
3.7
3.8
6/F *
m
1.64
1.78
1.99
2.06
2.26
2.26
2.40
0.94
1.12
1.35
1.65
2.00
2.06
2.06
2.18
2.24
x
cm
13.6
17.4
24.4
30.8
38.0
47.0
53.6
2.6
4.0
8.0
16.4
22.6
30.2
39.4
47.0
57.1
(x/xc)1 3
0.215
0.230
0.255
0.278
0.298
0.319
0.333
0.105
0.121
0.152
0.194
0.215
0.237
0.259
0.275
0.294
-------
e a F 2 x 6
o m c
Exp. No. ' * , ^ 2
10~° cm 10 cm cm cm cm
2-1 25.7 1.46 0.16 5.4 1.10
1.2
1.25
1.47
1.50
1.45
oo
00 2-2 25.7 1.46 0.35 64.3 1.02
1.20
1.28
1.34
1.58
1.80
2-3 25.7 1.46 0.74 584.0 1.53
1.64
1.77
1.93
6/F 2
m
7.10
7.75
8.07
9.49
9.68
9.37
2.90
3.41
3.64
3.80
4.48
5.11
2.08
2.22
2.40
2.69
X
cm
16.8
22.0
26.1
32.9
37.6
43.5
4.6
16.8
21.4
24.0
44.6
47.3
4.6
9.4
13.0
17.6
/ / x1/3
(x/xc)
1.46
1.59
1.69
1.82
1.91
2.00
0.424
0.644
0.693
0.720
0.886
0.902
0.200
0.252
0.280
0.310
-------
oo
3 \
e a F x 6
,-, o m c
Exp . No .
51 *? 2
10 cm 10 cm cm cm cm
2.07
2.17
2.20
2.21
2.46
2.46
2.70
2.80
3-1 29.9 1.45 0.11 2.18 0.83
0.80
0.90
1.05
1.20
1.35
1.40
1.50
6/F 2
m
2.82
2.94
2.98
3.00
3.34
3.34
3.66
3.79
7.30
7.01
7.92
9.22
10.56
11.80
12.21
13.11
X
cm
20.4
25.2
27.0
30.2
37.0
39.6
44.4
46.0
6.8
14.2
19.4
28.4
33.8
44.4
52.2
111.4
ii \l/3
(x/xc)
0.327
0.350
0.353
0.373
0.398
0.408
0.424
0.429
1.46
1.87
2.07
2.36
2.49
2.73
2.88
3.74
-------
e a F 2 x 6
o m c
Exp . No .
^ 1 ^ 2
10 cm" 10 cm cm cm cm
1.50
1.60
1.70
1.75
1.79
1.82
>o
o
1.80
3-2 29.9 1.46 0.16 6.43 0.65
0.77
0.92
1.12
1.20
1.30
1.38
1.40
1.45
6/F 2
m
13.11
14.00
14.92
15.31
15.64
15.89
15.80
3.99
4.72
5.64
6.87
7.36
7.98
8.46
8.60
8.87
x
cm
118.4
125.4
132.4
141.0
148.2
156.0
163.0
2.8
8.6
14.6
21.4
28.4
33.0
43.4
47.0
53.2
(x/xc)1/3
3.78
3.86
3.93
4.00
4.09
4.15
4.21
0.76
1.10
1.31
1.49
1.64
1.72
1.89
1.94
2.02
-------
3 \
e a F 2 x 6
Exp. No. o m c
5 1 3 2
10 cm 10 cm cm cm cm
1.50
1.75
1.86
1.82
1.82
1.85
1.85
1.85
1.80
2.00
2.00
3-3 29.9 1.46 0.25 22.8 1.00
1.15
1.20
1.35
1.40
6/F 2
m
9.20
10.71
11.41
11.12
11.12
11.31
11.31
11.31
11.00
12.22
12.22
4.00
4.62
4.80
5.44
5.62
x
cm
60.6
108.2
115.1
119.4
125.3
130.0
136.2
142.4
148.0
156.0
160.0
7.0
14.6
19.8
25.6
29.0
(x/x//3
2.11
2.56
2.62
2.64
2.69
2.72
2.76
2.79
2.84
2.89
2.92
0.67
0.86
0.95
1.04
1.08
-------
3 2 c
e a F x 6
o m c
Exp. No. ,00
«^~O "* - ,-,«5 <3
10 cm 10 cm cm cm cm
1.50
1.50
1.61
3-4 29.9 1.50 0.37 75.0 0.98
1.20
1.46
co
NJ
1.62
1.70
1.92
1.90
2.12
2.10
2.30
2.35
2.38
2.40
6/F 2
m
6.00
6.00
6.42
2.66
3.25
3.95
4.39
4.61
5.20
5.15
5.75
5.69
6.24
6.36
6.45
6.50
X
cm
38.0
41.8
50.8
2.4
8.8
15.8
21.6
29.1
38.4
42.4
52.4
59.6
106.9
116.0
119.0
124.6
(x/xc)1/3
1.19
1.23
1.31
0.32
0.49
0.60
0.66
0.73
0.80
0.83
0.89
0.93
1.12
1.16
1.17
1.18
-------
D 1.
O o c
e a F x o
TH o m c
Exp. No. _5 _1 _2
10 cm 10 cm cm cm cm
2.45
2.40
2.40
2.55
2.65
2.65
co
W 3-5 29.9 1.50 0.51 204.0 0.65
1.00
1.20
1.35
1.40
1.70
1.80
2.00
2.20
2.25
6/F 2
m
6.64
6.50
6.50
6.90
7.16
7.16
1.27
1.95
2.34
2.64
2.73
3.32
3.51
3.90
4.29
4.39
X
cm
130.4
132.0
141.6
149.0
155.0
165.0
0.6
3.8
7.0
9.0
13.2
21.8
31.8
36.4
45.8
52.0
t / \1/3
(x/xc)
1.20
1.20
1.24
1.26
1.27
1.30
0.14
0.27
0.32
0.35
0.40
0.47
0.54
0.56
0.61
0.63
-------
ID
e a F 2 x 6
o m c
Exp. No. 3-2
10 cm" 10 cm cm cm cm
2.25
2.45
2.45
2.55
2.70
2.75
2.80
2.90
3.20
3.10
3-6 29.9 1.50 0.65 420.0 1.00
1.00
1.20
1.30
1.35
1.40
6/F 2
m
4.39
4.78
4.78
4.97
5.26
5.36
5.47
5.65
6.24
6.04
1.53
1.53
1.84
1.99
2.06
2.14
X
cm
61.0
111.6
117.4
124.0
130.4
133.6
140.4
147.8
156.2
160.5
3.2
4.8
10.0
12.6
16.4
20.6
(*/*/ 3
0.67
0.82
0.83
0.85
0.86
0.87
0.82
0.90
0.91
0.92
0.20
0.23
0.29
0.31
0.34
0.37
-------
tn
3 I
e a F x 6
,-, .. o m c
Exp. No. _<1 3 _2
10 cm 10 cm cm cm cm
1.65
1.65
1.75
1.80
2.00
2.70
2.70
2.80
2.90
3.00
3.00
3.20
4-1 35.3 1.57 0.11 2.14 0.55
0.68
0.75
0.90
5/F 2
m
2.52
2.52
2.68
2.75
3.06
4.13
4.13
4.29
4.44
4.59
4.59
4.89
4.96
6.15
6.78
8.13
x
cm
28.6
33.6
40.4
48.4
60.0
109.4
115.0
123.2
132.8
141.4
148.3
158.0
2.3
4.9
9.0
17.0
<*/v1/3
0.41
0.43
0.46
0.49
0.52
0.64
0.65
0.66
0.68
0.70
0.71
0.72
1.04
1.32
1.61
1.99
-------
e a F x 6
Exp . No . °
10~5 cm"1 103 cm"2 cm cm cm
1.00
1.15
1.22
1.30
1.52
1.61
CO
1.68
1.65
1.70
1.72
4-2 35.3 1.57 0.11 2.22 0.40
0.65
0.78
0.84
0.93
1.05
6/F *
m
9.03
10.41
11.00
11.72
13.72
14.53
15.18
14.90
15.35
15.50
3.57
5.80
6.96
7.50
8.30
9.40
X
cm
24.8
32.8
44.2
52.8
116.0
124.8
135.1
143.6
153.6
163.4
0.4
4.4
6.1
9.0
14.4
23.6
(x/xc)
2.26
2.48
2.74
2.91
3.79
3.90
3.98
4.06
4.15
4.24
0.56
1.25
1.40
1.59
1.86
2.20
-------
(0
3 -
e a F 2 x 6
Exp. No. o m c
10~5 cm"1 103 cm"2 cm cm cm
1.14
1.19
1.25
1.55
1.60
1.64
1.60
1.65
1.66
1.68
1.67
4-3 35.3 1.60 0.13 3.81 0.75
0.85
0.85
0.90
1.00
6/F 2
m
10.20
10.61
11.22
13.83
14.31
14.61
14.30
14.70
14.82
15.00
14.93
5.63
6.36
6.36
6.75
7.50
x
cm
31.4
37.0
48.2
112.4
119.0
124.8
131.3
137.6
143.7
154.0
165.0
4.3
5.3
6.9
8.2
17.1
, i \ J. / »J
(x/x )
c
2.42
2.55
2.79
3.70
3.77
3.83
3.90
3.96
4.01
4.11
4.20
1.04
1.12
1.22
1.29
1.64
-------
<£>
oo
e a F * x 6
Exp. No. ° _2
10 cm 10 cm cm cm cm
1.10
1.15
1.20
1.30
1.55
1.55
1.60
1.65
1.65
1.62
1.70
1.65
1.80
1.70
1.85
4-4 35.3 1.60 0.14 4.38 0.60
6/F *
m
8.25
8.60
9.00
9.75
11.60
11.60
12.00
12.41
12.41
12.20
12.75
12.41
13.55
12.75
13.81
4.20
X
cm
23.8
31.1
39.2
48.0
110.7
115.4
119.5
126.2
130.5
137.3
143.7
151.2
154.8
161.2
166.9
2.6
(X/XC}1 3
1.84
2.01
2.17
2.32
3.07
3.12
3.15
3.21
3.24
3.30
3.35
3.41
3.44
3.48
3.52
0.84
-------
-------
e c£ Fm* xc 6
Exp. No. _ _ 3 ° _2
10 cm 10 cm cm cm cm
4-5 35.3 1.60 0.15 5.85 0.68
0.70
0.80
0.85
0.95
1.05
r-1
o
0 1.10
1.15
1.20
1.25
1.35
1.60
1.60
1.65
1.75
1.75
6/Fn,2
4.60
4.70
5.30
5.70
6.35
7.00
7.38
7.70
8.04
8.36
9.00
10.71
10.71
11.00
11.72
11.72
X
cm
1.8
2.8
3.4
7.0
10.8
17.2
21.6
32.3
39.4
47.1
51.4
108.1
114.0
128.9
133.0
145.0
(x/xc)
0.68
0.78
0.83
1.06
1.22
1.43
1.54
1.77
1.89
2.00
2.05
2.64
2.69
2.80
2.84
2. 92
-------
3 *
e a F 2 x 6
EXP' N°- -5 -1 3 ° -2
10 cm 10 cm cm cm cm
1.65
1.70
1.80
4-6 35.3 1.57 0.18 9.90 0.60
0.78
0.92
1.00
1.10
1.22
1.35
1.40
1.42
1.88
1.88
2.00
1.95
6/F 2
m
11.00
11.40
12.00
3.30
4.20
5.00
5.40
6.00
6.60
7.30
7.60
7.70
10.21
10.21
10.81
10.62
x
cm
148.9
155.1
159.1
1.6
4.8
8.4
14.1
20.8
29.8
41.6
49.6
57.4
111.1
116.2
121.4
127.2
(x/x )
c
2.94
2.98
3.00
0.55
0.78
0.95
1.13
1.28
1.44
1.61
1.71
1.80
2.24
2.27
2.31
2.34
-------
e ot F x 6
EXP' N°- -5 -1 3 ° -2
10 cm 10 cm cm cm cm
1.98
2.00
2.02
2.05
2.05
M 4-7 35.3 1.57 0.20 13.40 0.60
o
to
0.78
0.87
0.90
1.25
1.20
1.30
1.38
1.42
1.40
1.50
6/F *
m
10.71
10.83
10.90
11.10
11.10
2.94
3.83
4.27
4.41
6.14
5.89
6.38
6.77
6.97
6.90
7.35
X
cm
136.0
142.3
148.6
161.8
164.8
1.8
4.4
7.0
9.4
15.6
20.3
26.8
33.4
40.9
46.0
51.6
(x/x )1/3
c
2.39
2.43
2.46
2.54
2.56
0.51
0.69
0.81
0.89
1.05
1.15
1.26
1.35
1.45
1.51
1.57
-------
e a F xc 6
Exp. No. 3 ° -2
10 cm 10 cm cm cm cm
1.50
4-8 35.3 1.57 0.27 32.20 0.70
1.10
1.15
1.20
1.30
o
1.38
1.40
1.45
1.70
1.85
1.85
1.90
1.95
1.95
2.00
6/F *
m
7.35
2.56
4.02
4.20
4.38
4.75
5.04
5.10
5.28
6.20
6.75
6.75
6.84
7.12
7.12
7.30
X
cm
57.0
2.3
8.8
14.3
16.6
25.0
31.0
36.3
45.4
58.3
106.8
112.7
119.5
127.4
130.9
136.7
(x/xc)1/3
1.62
0.41
0.65
0.76
0.80
0.92
0.99
1.04
1.12
1.22
1.50
1.52
1.55
1.58
1.59
1.62
-------
e a F x 6
_ o m c
Exp. No. _2
10 cm" 10 cm~ cm cm cm
2.05
2.05
2.05
2.35
6/F *
m
7.50
7.50
7.50
8.57
X
cm
144.8
150.7
157.3
166.8
<*/v1/3
1.65
1.67
1.70
1.73
------- |