 Title Page 
 Acknowledgement 
 Table of Contents 
 List of Tables 
 List of Figures 
 List of symbols and abbreviati... 
 Abstract 
 Introduction 
 Considerations for application... 
 Dispersive mass transport in an... 
 Dispersive mass transport in rectangular... 
 Comparison of results with field... 
 Summary and conclusions 
 Appendices 
 References 
 Biographical sketch 

Full Citation 
Material Information 

Title: 
Dispersive mass transport in oscillatory and unidirectional flows 

Physical Description: 
xvi, 143 leaves. : illus. ; 28 cm. 

Language: 
English 

Creator: 
Taylor, Robert Bruce, 1942 

Publication Date: 
1974 

Copyright Date: 
1974 
Subjects 

Subject: 
Hydraulics ( lcsh ) Fluid dynamics ( lcsh ) Onedimensional flow ( lcsh ) Civil and Coastal Engineering thesis Ph. D ( lcsh ) Dissertations, Academic  Civil and Coastal Engineering  UF ( lcsh ) 

Genre: 
bibliography ( marcgt ) nonfiction ( marcgt ) 
Notes 

Thesis: 
Thesis  University of FLorida. 

Bibliography: 
Bibliography: leaves 140142. 

General Note: 
Typescript. 
Record Information 

Bibliographic ID: 
UF00098181 

Volume ID: 
VID00001 

Source Institution: 
University of Florida 

Holding Location: 
University of Florida 

Rights Management: 
All rights reserved by the source institution and holding location. 

Resource Identifier: 
alephbibnum  000580815 oclc  14087798 notis  ADA8920 

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Table of Contents 
Title Page
Page i
Page ii
Acknowledgement
Page iii
Table of Contents
Page iv
Page v
List of Tables
Page vi
List of Figures
Page vii
Page viii
List of symbols and abbreviations
Page ix
Page x
Page xi
Page xii
Page xiii
Page xiv
Abstract
Page xv
Page xvi
Introduction
Page 1
Page 2
Page 3
Page 4
Page 5
Page 6
Page 7
Page 8
Page 9
Page 10
Page 11
Page 12
Page 13
Page 14
Page 15
Page 16
Page 17
Page 18
Page 19
Page 20
Page 21
Considerations for application of Taylor’s dispersion analysis to free surface flows
Page 22
Page 23
Page 24
Page 25
Page 26
Page 27
Page 28
Page 29
Page 30
Page 31
Page 32
Page 33
Dispersive mass transport in an infinitely wide rectangular channel
Page 34
Page 35
Page 36
Page 37
Page 38
Page 39
Page 40
Page 41
Page 42
Page 43
Page 44
Page 45
Page 46
Page 47
Page 48
Page 49
Page 50
Page 51
Page 52
Page 53
Page 54
Page 55
Page 56
Page 57
Page 58
Page 59
Page 60
Page 61
Page 62
Page 63
Page 64
Page 65
Dispersive mass transport in rectangular channels of finite width
Page 66
Page 67
Page 68
Page 69
Page 70
Page 71
Page 72
Page 73
Page 74
Page 75
Page 76
Page 77
Page 78
Page 79
Page 80
Page 81
Page 82
Page 83
Page 84
Page 85
Page 86
Page 87
Page 88
Page 89
Page 90
Page 91
Page 92
Page 93
Page 94
Page 95
Page 96
Page 97
Page 98
Page 99
Comparison of results with field and laboratory data
Page 100
Page 101
Page 102
Page 103
Page 104
Page 105
Page 106
Page 107
Page 108
Page 109
Page 110
Page 111
Page 112
Page 113
Page 114
Page 115
Summary and conclusions
Page 116
Page 117
Page 118
Page 119
Appendices
Page 120
Page 121
Page 122
Page 123
Page 124
Page 125
Page 126
Page 127
Page 128
Page 129
Page 130
Page 131
Page 132
Page 133
Page 134
Page 135
Page 136
Page 137
Page 138
Page 139
References
Page 140
Page 141
Page 142
Biographical sketch
Page 143
Page 144
Page 145

Full Text 
DISPERSIVE MASS TRANSPORT IN OSCILLATORY
AND UNIDIRECTIONAL FLOWS
by
ROBERT BRUCE TAYLOR III
A DISSERTATION PRESENTED TO THE GRADUATE
COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
ACKNOWLEDGMENTS
The author would like to thank Dr. Robert G. Dean, Professor of the
Department of Civil and Coastal Engineering, who supervised the research.
Dr. Dean's many hours of encouragement and patient teaching will not be
forgotten. Appreciation is also extended to Dr. Wayne C. Huber, Associate
Professor of the Department of Environmental Engineering Sciences, whose
interest and suggestions were of a great help to the author.
Ms. Pat Hulit and Mrs. Wanda Smith worked many arduous hours on
short notice to type a highly professional final draft of the manuscript.
Special thanks are given to Mrs. Evelyn Hill, who assisted in many ways
including the typing of the rough draft of the manuscript, and to Ms.
Denise Frank, who did the drafting.
This research was sponsored by the Florida Power and Light Company
through a grant to the Coastal and Oceanographic Engineering Laboratory,
University of Florida. Both the funds made available by the Sponsor and
the use of the COE Laboratory facilities are greatly appreciated. All
computer work associated with this project was done on the Northeast
Regional Data Center's IBM System/370165.
TABLE OF CONTENTS
Acknowledgments . . . . . .
List of Tables . . . . . . .
List of Figures . . . . . .
Key to Symbols and Abbreviations . .
Abstract . . . . . . . .
Page
. iii
. vi
, vii
Six
Sxv
CHAPTER
I INTRODUCTION . . . . . . . . . . . .
IA General Background . .
IB Discussion of Previous Work
Mass Transport . . .
on D spearss .ve
on Dispersive
. . . . . . .
IB.1 Unidirectional Flow . . . . . .
IB.2 Oscillatory Flow . . . . . . . .
IC Description and Scope of the Present Work . . .
II CONSIDERATIONS FOR APPLICATION OF TAYLOR'S
DISPERSION ANALYSIS TO FREE SURFACE FLOWS. . . . .
IIA General . . . .
IIB Unidirectional Flow . .
IIC Oscillatory Flow . . .
IID Applications of Dispersion
by Taylor's Method . .
22
. . . . . 22
. . . . . . 27
. . . . . . 30
Coefficients Predicted
. . . . . . 33
III DISPERSIVE MASS TRANSPORT IN AN INFINITELY
WIDE RECTANGULAR CHANNEL . . . . . . . . .
IIIA Steady Unidirectional Flow . . . . . .
IIIA.1 Velocity Distribution. .
IIIA.2 Concentration Distribution
IIIA.3 Dispersive Mass Transport.
IIIB Oscillatory Flow. . . . . . . . . .
IIIB.1 Velocity Distribution. .
IIIB.2 Concentration Distribution
IIIB.3 Dispersive Mass Transport.
IIIC Discussion of Two Dimensional Shear Flow Results.
IIIC.I Dispersion in Unidirectional Flow as
a Limit of the Oscillatory Flow Case . .
IIIC.2 Characteristics of Dispersive Mass
Transport in Oscillatory Flow . . .
'
I I f I I f I I
I ) I I I I I
CHAPTER Page
IV DISPERSIVE MASS TRANSPORT IN RECTANGULAR
CHANNELS OF FINITE WIDTH. . . . . . . . .. 66
IVA Oscillatory Flow. . . . . . . . ... 66
IVA.1 Velocity Distribution . . . . .. 67
IVA.2 Concentration Distribution. . . . .. 71
IVA.3 Dispersive Mass Transport . . . ... 74
IVB Steady Unidirectional Flow . . . . ... 90
IVB.1 Velocity Distribution . . . . .. 90
IVB.2 Concentration Distribution. . . . ... 93
IVB.3 Dispersive Mass Transport . . . ... 95
V COMPARISON OF RESULTS WITH FIELD AND 100
LABORATORY DATA . . . . . . . . . . .
VA Unidirectional Flow. . . . . . . . ... 100
VB Oscillatory Flow . . . . . . . .... 106
VI SUMMARY AND CONCLUSIONS . . . . . . .... 116
APPENDIX
A DERIVATION OF NORMALIZING EXPRESSIONS FOR TWO DIMENSIONAL
OSCILLATORY SHEAR FLOW SOLUTION . . . . .... 120
A1 Surface Amplitude of Velocity, umax . . . .. 120
A2 Surface Excursion Length, L . . . . . . 124
B MATHEMATICAL DETAILS FOR THREE DIMENSIONAL
OSCILLATORY SHEAR FLOW DISPERSION . . . . .... 125
B1 Velocity Solution . . . . . . . .. 125
B2 Concentration Solution . . . . . . .. 128
B3 Dispersive Mass Transport. . . . . . ... 134
C DERIVATION OF umax NORMALIZING EXPRESSION FOR
THREE DIMENSIONAL SHEAR FLOW SOLUTION . . . ... .137
REFERENCES. . . . . . . . . ... ....... 140
BIOGRAPHICAL SKETCH . . . . . . . .... .... 143
LIST OF TABLES
Table Page
1 PREDICTED RESONANT PEAK DATA FOR LONGITUDINAL
DISPERSION COEFFICIENT IN THREE DIMENSIONAL
OSCILLATORY SHEAR FLOW. . . . . . . .. 83
2 LIMITING WIDTHTODEPTH RATIOS FOR APPLICATION
OF THE TWO DIMENSIONAL OSCILLATORY SHEAR FLOW
DISPERSION COEFFICIENT. . . . . . . . 87
3 OBSERVED DISPERSION DATA AND PREDICTED T
VALUES USING (11159) . . . . . . .... 106
4 OBSERVED DISPERSION DATA AND PREDICTED
VALUES FOR THE MERSEY NARROWS . . . . . .. 110
5 OBSERVED DISPERSION DATA AS REPORTED BY
SEGALL AND CORRESPONDING PREDICTED T VALUES 113
LIST OF FIGURES
Figure Page
1 DEFINITION SKETCH. .. . . . . . . .22
2 DEFINITION SKETCH FOR TWO DIMENSIONAL
SHEAR FLOW . . . . . . . .. .. . 34
3 NONDIMENSIONAL DISPERSION COEFFICIENT
T/EL, AS A FUNCTION OF T'. . . . . . . 53
4 DISPERSION IN UNIDIRECTIONAL FLOW AS A LIMIT
OF THE OSCILLATORY FLOW CASE . . . . . .. 55
5 COMPARISON OF DISPERSION FOR DIURNAL AND SEMIDIURNAL
PERIODS OF OSCILLATION WITH SAME umax . . . ... 58
6 COMPARISON OF DISPERSION FOR DIURNAL AND SEMIDIURNAL
PERIODS OF OSCILLATION WITH SAME SURFACE EXCURSION
LENGTH . . . . . . . . . .. . 60
7 NONDIMENSIONAL LONGITUDINAL DISPERSION COEFFICIENT
E', AS A FUNCTION OF T' FOR TWO DIMENSIONAL
x z
OSCILLATORY SHEAR FLOW . . . . . . . .. 61
8 NONDIMENSIONAL VELOCITY AND CONCENTRATION VARIATION
COMPONENT PROFILES FOR T' = 0.5. . . . . . 63
9 NONDIMENSIONAL VELOCITY AND CONCENTRATION VARIATION
COMPONENT PROFILES FOR T' = 1.58 . . . . . 64
10 NONDIMENSIONAL VELOCITY AND CONCENTRATION VARIATION
COMPONENT PROFILES FOR T' = 5.0. . . . . . 65
11 DEFINITION SKETCH FOR THREE DIMENSIONAL SHEAR FLOW 67
12 NONDIMENSIONAL LONGITUDINAL DISPERSION COEFFICIENT
E',AS A FUNCTION OF T' FOR FULL RANGE OF T' VALUES ... 79
x z C
13 NONDIMENSIONAL LONGITUDINAL DISPERSION COEFFICIENT
E', AS A FUNCTION OF T' FOR T' = 1.0, 0.67, 0.33
xz c
AND 0.1. . . . . . . . . . . . 81
14 NONDIMENSIONAL LONGITUDINAL DISPERSION COEFFICIENT
E' AS A FUNCTION OF T' FOR T' = 0.1, 0.01, 0.001,
x z c
0.0001, and <1 x 106. . . . .............. 82
LIST OF FIGURES (Continued)
Figure Page
15 NONDIMENSIONAL VERTICAL MIXING TIME, T', AT
WHICH MAXIMUM E' OCCURS AS A FUNCTION OF T' . 84
16 PEAK Ex AS A FUNCTION OF T'. . . . . .... 84
17 SYMMETRIC DISPLAY OF ISOLINES OF E' x 105
x
IN THREE DIMENSIONAL OSCILLATORY SHEAR FLOW. ... 89
18 NONDIMENSIONAL LONGITUDINAL DISPERSION COEFFICIENT,
E', AS A FUNCTION OF T' FOR THREE DIMENSIONAL
UNIDIRECTIONAL SHEAR FLOW. . . . . . ... 98
19 COMPARISON OF PREDICTED AND OBSERVED LONGITUDINAL
DISPERSION IN THREE DIMENSIONAL UNIDIRECTIONAL
SHEAR FLOW . . . . . . . . ... . 101
20 COMPARISON OF PREDICTED AND OBSERVED LONGITUDINAL
DISPERSION FOR THE ENGLISH CHANNEL, LIVERPOOL BAY,
AND CUMBERLAND COAST . . . . . . . .. 109
21 COMPARISON OF PREDICTED AND OBSERVED LONGITUDINAL
DISPERSION FOR THE MERSEY ESTUARY. . . . .. 112
22 COMPARISON OF PREDICTED AND OBSERVED LONGITUDINAL
DISPERSION USING SEGALL'S DATA (19). . . . .. 114
viii
LIST OF SYMBOLS AND ABBREVIATIONS
a Pipe radius
an,a ,amn, Fourier coefficients
am' a~k
A Flow crosssectional area
Ai General function used by Bowden
B General function used in examination of conditions for
which concept of dispersion is valid in oscillatory
free surface flow
B1 General function used by Bowden
c Concentration of substance
c Mean concentration of substance over flow cross section
c" Variation of substance concentration from crosssectional
mean value
c" Separation variable representing spatially dependent
s portion of c"
cND Nondimensional form of c"
cm Centimeters
D Molecular diffusion coefficient in radial direction
Dt Molecular diffusion coefficient in direction transverse
to principal flow axis
e Vertical molecular or eddy diffusivity used by Elder
EL Longitudinal dispersion coefficient for unidirectional
flow
Ex Longitudinal dispersion coefficient for oscillatory flow,
temporally dependent
E Time averaged longitudinal dispersion coefficient for
oscillatory flow with an infinitely large period of oscil
lation, used by Holley et al.
LIST OF SYMBOLS AND ABBREVIATIONS (Continued)
E' Nondimensional time averaged longitudinal dispersion
x coefficient for oscillatory flow
T Longitudinal dispersion coefficient averaged over
period of oscillation
E' Nondimensional longitudinal dispersion coefficient
for unidirectional flow
f Summation index
fDW DarcyWeisbach friction factor
fl General function used in examination of conditions
for which concept of dispersion is valid in oscillatory
free surface flow
f, General function used in examination of conditions
for which concept of dispersion is valid in unidirectional
free surface flow
ft Feet
F Function used to simplify umax normalization of two
dimensional oscillatory shear flow solution
FI Functional distribution of turbulent pipe flow used
by Taylor
g Function used to simplify velocity solution for three
dimensional oscillatory shear flow
g, General function used in examination of conditions for
which concept of dispersion is valid in oscillatory free
surface flow
g2,G General functions used in examination of conditions for
which concept of dispersion is valid in unidirectional free
surface flow
h Water depth
i /5
11,12 Functions used to simplify derivation of umax normalization
for three dimensional oscillatory shear flow solution
j Summation index
k Summation index
LIST OF SYMBOLS AND ABBREVIATIONS (Continued)
K Pressure gradient modulus
K Amplitude of eddy diffusion coefficient used by Fukuoka
Kz Vertical coefficient of eddy diffusion
K Lateral coefficient of eddy diffusion
K Mean crosssectional value of K
z z
K Mean crosssectional value of K
k Summation index
Distance from point of maximum surface velocity to most
distant bank, used by Fischer
L Surface excursion length
m Summation index
m Mass transport
T Average mass transport over one period of oscillation
M Meters
n Summation index
osc. Oscillatory
p Summation index
P Pressure
P1 General function used by Bowden
q Summation index
q" Depth integrated u" used by Fischer
Qi General function used by Bowden
r Cartesian coordinate in radial direction
rl,r2 Functions used to simplify presentation of E' for three
dimensional oscillatory shear flow
Re Real part of complex function
LIST OF SYMBOLS AND ABBREVIATIONS (Continued)
RH Hydraulic radius
RI,R2 Functions used to simplify development of three dimensional
oscillatory shear flow solution for u", c", and T
Rmn Rpq Functions used to simplify development of three dimensional
Rp unidirectional shear flow solution for u", c", and EL.
iRk
sec Seconds
t Time
t' Variable of integration
T Period of oscillation
Tcy Lateral mixing time
Tcz Vertical mixing time
T Semidiurnal tidal period, 44,712 sec.
T' Nondimensional vertical mixing time used by Holley et al.
T' Relative mixing time
T' Relative mixing time for which two dimensional oscillatory
Cm E' solution is valid
x
T' Nondimensional lateral mixing time
y
T' Nondimensional vertical mixing time
u Velocity component along principal flow axis, x
uf Unidirectional flow component used by Fukuoka
umax Temporal and spatial maximum of u
u Separation variable representing spatially dependent
portion of u
ut Oscillatory flow component used by Fukuoka
u Mean value of u over flow cross section
u" Variation of u from crosssectional mean value
LIST OF SYMBOLS AND ABBREVIATIONS (Continued)
UND Nondimensional form of u"
u" Separation variable representing spatially dependent
s portion of u"
u, Shear velocity
Uo Surface amplitude of unidirectional flow component
used by Okubo
unidir Unidirectional
Vo Surface amplitude of oscillatory flow component
used by Okubo
w Width of channel
x Cartesian coordinate directed along principal flow
axis of the channel
y Cartesian coordinate directed in lateral direction
across width of channel
z Cartesian coordinate directed in vertical direction
over depth of channel
z' Variable of integration
a Constant
al, 02, a3 Phase angles
B Shear wave parameter in two dimensional oscillatory
shear flow solution
61,62,63,64,6s Functions used to simplify three dimensional shear
flow solutions
ez Coefficient of vertical eddy viscosity
ey Coefficient of lateral eddy viscosity
Ez Mean value of e over flow cross section
y Mean value of cy over flow cross section
e,602 Functions used to simplify development of T for
three dimensional oscillatory shear flow
LIST OF SYMBOLS AND ABBREVIATIONS (Continued)
v Locus of channel perimeter
X Function used to simplify development of T for
three dimensional oscillatory shear flow
Pkk Factor used to properly account for ao0, abk, and aj' terms
in three dimensional shear flow solutions.
Cartesian coordinate directed along principal flow axis
of the channel as seen by point traveling with velocity u
p Fluid density
a Angular frequency of oscillation,
T
S Summation symbol
T Nondimensional time used by Aris
4,,' Functions used to simplify umax normalization of three
dimensional shear flow solutions
X General function
i Function used to simplify presentation of E' for three
dimensional oscillatory shear flow solution
dimensional oscillatory shear flow solution
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
DISPERSIVE MASS TRANSPORT IN OSCILLATORY
AND UNIDIRECTIONAL FLOWS
By
Robert Bruce Taylor
August, 1974
Chairman: Robert G. Dean
Major Department: Civil and Coastal Engineering
Four sets of boundary value problems for the equations of fluid
motion and transport diffusion are solved to obtain expressions for the
spatial variations of velocity and concentration in free surface flows.
The four problems investigated treat unidirectional and oscillatory
uniform shear flows in rectangular channels of infinite and finite width.
The solutions are then used to determine longitudinal dispersive mass
transport. Analytical expressions for the longitudinal dispersion
coefficient associated with infinitely wide channels are presented as
functions of the temporal and spatial maximum of velocity, channel depth,
period of oscillation oscillatoryy flow only), and depth mean vertical
coefficient of eddy diffusivity. For channels of finite width, analytical
expressions for the longitudinal dispersion coefficient are presented as
functions of the temporal and spatial maximum of velocity, channel depth,
channel width, period of oscillation oscillatoryy flow only), and cross
sectional mean values of vertical and lateral coefficients of eddy dif
fusivity.
It is found that for infinitely wide channels the longitudinal dis
persion coefficient in oscillatory flow is described by a type of resonant
interaction between the period of oscillation and the time scale of vertical
mixing. The maximum longitudinal dispersive mass transport occurs when
the ratio of the vertical mixing time to the period of oscillation equals
1.58. This functional behavior is shown to be unique and distinctly
different from the dispersive process in unidirectional flow.
For rectangular channels of finite width it is shown that the dis
persive mass transport process is symmetric, such that a channel geometry
skewed in width produces the same dispersive mass transport as a channel
geometry skewed in depth, provided that specific requirements regarding
the maximum velocity, period of oscillation, and vertical and lateral
mixing times are satisfied.
The longitudinal dispersion coefficient for oscillatory flow in
rectangular channels of finite width is presented as a family of resonance
curves described by the resonant interaction of the period of oscillation
with the vertical and lateral mixing times.
For unidirectional flow in channels of finite width it is found
that the predicted nondimensional longitudinal dispersion coefficient
increases without bound as the skewness of the channel geometry is corres
pondingly increased. Thus, for unidirectional flow the three dimensional
solution does not collapse in the limit to the two dimensional case as is
found for oscillatory flow.
Predicted values for the longitudinal dispersion coefficient are
compared with values based upon field and laboratory data. The compari
son shows that when reliable measurements are available for all of the
functional parameters included in the predictive solutions the agreement
between observed and predicted values is good.
CHAPTER I
INTRODUCTION
IA General Background
A growing awareness in recent years of the need for a rational approach
to the planned use and development of coastal regions has significantly
contributed to a concern for maintaining an acceptable level of water
quality within our river, estuarine, and bay system waters. To achieve
this a fundamentally sound understanding of the physical processes which
govern the mass transport characteristics of these waters is essential.
The mass transport of a substance as it applies here may be defined
as "the time averaged net transfer of substance along the principal flow
axis of the watercourse or channel." With the aid of this definition one
can intuitively reason that the mass transport mechanism for a conservative
substance is governed by the kinematic description of the flow field and
the distribution of the substance within the flow field. Unfortunately,
the complexities associated with turbulent shear flows in natural water
courses have thus far precluded a complete analytical description of either
the flow kinematics or the distribution of the substance. However, the
analytical treatment of longitudinal mass transport in steady uniform
axisymmetric shear flows by Taylor (1,2) provided for the first time a
reasonable method for predicting these phenomena.
In his analysis Taylor made the distinction between three mechanisms
of longitudinal mass transport:
(1) Convective transport of the crosssectional mean concentration
of substance by the mean crosssectional velocity,
(2) Convective transport due to the correlation over the cross
section of the spatial variations from the mean of velocity and
concentration,
(3) Diffusion of the substance along the principal flow axis (mole
cular diffusion in laminar flow and eddy diffusion in turbulent flow).
The first of these mechanisms has long been recognized as a mass transport
phenomenon and is the most amenable of the three to an analytical treat
ment. In the absence of diffusive transport due either to molecular or
turbulent motion this mechanism may be visualized as the transport along
the principal flow axis of a slug of substance uniformly mixed over the
flow cross section and undistorted by boundary shear. The third mechanism,
that due to longitudinal diffusive transport, was shown by Taylor to be
less than 1 per cent of the second which he defined as dispersive mass
transport. It is this second mechanism, longitudinal dispersive mass
transport, which is the subject of the present work.
Following Taylor's initial formulation of the dispersion mechanism
investigators began to extend his analysis to unidirectional two dimen
sional open channel flows for infinitely wide channels and subsequently
to the more realistic three dimensional case of unidirectional flow in a
channel of finite width. More recently, the emphasis of the research
effort has shifted from steady unidirectional flow to oscillatory flow
in an effort to apply the knowledge obtained from the former to develop
a predictive capability for dispersive mass transport for the latter. A
brief summary of previous efforts follows.
IB Discussion of Previous Work on
Dispersive Mass Transport
IB.1 Unidirectional Flow
As mentioned previously, Taylor (1) was the first investigator to
define the mechanism of dispersive mass transport. In his paper Taylor
considered the case of a passive, conservative substance transported by
a laminar steady flow in a circular tube. A passive substance is one
which is present in small enough concentrations so as not to alter either
the velocity flow field or the magnitude of the diffusion coefficient; a
conservative substance is one which has a fixed total mass present for all
time in the system being considered. All substances referred to in the
present work should be considered to be both passive and conservative.
To begin his analysis, Taylor transformed the Eulerian form of the trans
port diffusion equation to one with the origin moving with the cross
sectional mean velocity by defining
= x ut (I1)
u = u + u" (12)
where u is the crosssectional mean velocity, u" is the variation from
the mean of the velocity in the radial direction due to the shear profile,
and x is the coordinate along the axis of the tube positive in the direction
of flow. Neglecting molecular diffusion along the axis of the tube he was
then able to write
D
( U ) + C (r ) (13)
at'S, u 5 r Sr Pr
where,
() = time rate of change of concentration as seen from moving
coordinate system
ac ac
t x
Dr = molecular diffusion coefficient in radial direction.
If (13) is integrated over the crosssectional area of the tube
and the concentration, c, decomposed into its crosssectional mean and
variation component, by the expression
c = E + c" (14)
then (13) reduces to
 T (I5)
For standardization throughout the text overbars shall be considered
to represent crosssectional averaging whereas the symbol < >T shall be
considered to represent temporal averaging over the periodic interval T.
The form of (15) led Taylor to define the convective transport shown as
a Fickian flux which he called dispersion, or
E 1 u"c"dA (16)
EL
where EL is the longitudinal dispersion coefficient for steady unidirec
tional flow and A is the flow crosssectional area. As indicated by (16)
the heart of Taylor's analysis lies in the determination of the spatial
variations of velocity and concentration, u" and c" respectively. The
velocity variation was readily determined from the parabolic velocity
profile. To determine c" Taylor returned to (13), decomposed c by (14)
and made the following assumptions:
(1) Steady state conditions exist with respect to the moving coor
dinate system, i.e. ( ) = 0.
(2) c" is not a function of 5.
(3) is constant.
DE,
These assumptions reduced (13) to
u r 3c ) (17)
which Taylor then solved for c", performed the integration of u"c" in
(16) and produced the following expression for the longitudinal disper
sion coefficient
a2u2
E max (18)
EL 192 D
where a is the radius of the tube, and umax is the centerline velocity.
It is important to note that the dispersion coefficient varies directly
with the ratio a2/Dr which represents the time scale of mixing of the
substance over the flow cross section due to transverse diffusivity.
Using the definition given in (16) for EL, (15) may then be written as
D ELC 2 c (19)
t ) = L S2
Equation (19), with EL given by (I8), shows that the crosssectional mean
concentration of a substance is effectively diffused longitudinally about
a point traveling with the mean velocity by the combined effect of the
velocity shear profile and the transverse diffusivity. Taylor correctly
cautioned, however, that in applying (19) the time scale of convective
transport associated with u" must be significantly greater than the time
scale of crosssectional mixing due to transverse diffusivity. A further
discussion of the assumptions made by Taylor and the application of dis
persion coefficients obtained in this manner will be presented in Chapter
II.
Having looked at laminar flows Taylor (2) extended his analysis to
axisymmetric turbulent shear flow in a pipe. For u" Taylor used an exper
imentally verified velocity distribution of the form
umax u(r)
ax = Fi(r) (110)
where ux is the centerline velocity, u, the shear or friction velocity,
and Fi(r) an experimentally determined function. Then by assuming Reynolds
analogy which states that the transfer of mass, heat, and momentum by
turbulence are exactly analogous, and the same conditions required to
obtain (17) he obtained a solution for c" and numerically performed the
integration of u"c" shown in (16). The result was the following expression
for the longitudinal dispersion coefficient in turbulent pipe flow
EL = 10.06 a u, (111)
where a is the radius of the pipe. By assuming isotropy of the turbulent
fluctuations he was then able to estimate the additional longitudinal
transport due to eddy diffusion along the flow axis. Adding this to
(111), the corrected dispersion coefficient becomes
EL = 10.1 a u, (112)
Shortly after Taylor's discovery and formulation of dispersive mass
transport, Aris (3) presented a very elegant analysis which supported
Taylor's theory for unidirectional steady flows and rigorously defined
the conditions under which Taylor's assumptions were valid. Rather than
obtaining solutions for u" and c" to compute the dispersive mass trans
port, Aris chose to apply a moment analysis of the longitudinal distribution
of the crosssectional mean concentration of substance about a point
moving with the mean velocity. The significance of Aris' analysis lies
in its generality, and the insight that it yields for looking at the
precise conditions under which an ordered correlation of u"c" over the
crosssectional flow area may be represented by the implied stationary
random process of a Fickian flux. Aris' development begins with the
basic Eulerian form of the transport diffusion equation for an arbitrary
flow geometry and an arbitrary initial distribution of substance. He
then proceeds to develop expressions for the moments up through the third
of the crosssectional mean concentration about with no assumptions
regarding steady state, or the functional form of the concentration c.
This generality allows him to obtain time dependent solutions which show
that after an initial transient period, steady state conditions will be
asymptotically approached in which the crosssectional mean concentration,
c, disperses about a point traveling at the mean speed of the flow in a
Gaussian manner and that the rate of growth of the variance is constant
with time. In other words the critical assumption in Taylor's analysis
is the one concerning steady state conditions with respect to the moving
coordinate system, i.e. () = 0, which by Aris' analysis must be satis
fied if the definition of the convective mass transport of a substance as
a Fickian flux about the mean speed of flow is to hold. The dimensionless
time parameter governing the dispersive process is defined by Aris as
Dtt
i a (113)
where t represents time; Dt is the molecular diffusion coefficient in the
transverse direction; and a is a characteristic length of the flow cross
section. The ratio a2/Dt, as noted previously, represents the time scale
required for the substance to mix over the flow cross section due to trans
verse diffusion. Thus, steady state conditions are approached for large
t, or when the convective flow time is large compared to the time required
for crosssectional mixing. This supports Taylor's comments regarding the
application of dispersion coefficients to predict concentration distributions.
In Chapter II it will be shown how the assumption of steady state condi
tions supports Taylor's other assumption regarding the functional dependence
of c for both steady unidirectional and oscillatory flows. Aris' method
of analysis has been used by other investigators to obtain expressions
for the dispersion coefficient and for brevity later in the text will be
referred to as the "method of moments." Although it remains a very useful
method for determining expressions for longitudinal dispersion coefficients
it does not provide the physical insight into the kinematic flow field and
crosssectional distribution of substance that Taylor's method does.
Further support of the general applicability of Taylor's dispersion
analysis to different flow geometries was given by Elder (4) who applied
Taylor's analytical technique to obtain a predictive expression for the
dispersion coefficient in two dimensional unidirectional shear flows in
infinitely wide open channels. In his analysis Elder obtains the following
expression for the longitudinal dispersion coefficient:
h z z
EL I uh v e1 u" dz'dz'dz' (113)
0 0 0
where z' is a variable of integration; z is the vertical coordinate; h is the
depth of flow; and e is either the vertical molecular diffusivity or the
vertical eddy diffusivity depending upon whether the flow is laminar or
turbulent. This expression indicates, in a manner similar to (18),
that in either laminar or turbulent unidirectional shear flow the longi
tudinal dispersion varies directly with the time scale of crosssectional
mixing. To obtain an expression for u" Elder assumed the velocity distri
bution to be logarithmic which upon substitution into (113) and correcting
for longitudinal eddy diffusion as done by Taylor, yields
EL = 5.93 uh (114)
where u, is the shear velocity.
The importance of the flow cross section geometry on the velocity
shear profile, transverse diffusivity, and ultimately the longitudinal
dispersion is demonstrated by (112) and (114) which if expressed
in terms of the hydraulic radius, RH, for the respective flows yield values
of EL/RHu, of 20.2 for the pipe and 5.9 for the infinitely wide channel.
Bowden (5) developed expressions for EL using several different velocity
and eddy diffusivity distributions with depth and showed that EL/u,h
ranged from 5.9 to 25 for the cases analyzed. Thus, the particular velocity
shear profile used along with the crosssectional eddy diffusivity have
a significant effect on the magnitude of the predicted longitudinal
dispersion coefficient.
While the works of Taylor, Aris, and Elder proved the existence of
an effective dispersive mass transport mechanism and there was reasonable
agreement between laboratory experimental data for pipe and infinitely
wide open channel flow geometries, other investigators were finding that
observed values of the longitudinal dispersion coefficient for natural
rivers and streams were considerably higher than those predicted by theory.
As reported by Fischer (6) observed values for EL in natural watercourses
ranged from 50 to 700 hu, as compared with the 5.9 to 25 hu, range predicted
by Elder and Bowden. To account for this increased effect Fischer concluded
that the dominant mechanism in longitudinal dispersion was the interaction
of the velocity shear profile and turbulent mixing time scales across the
finite width of the channel as opposed to the vertical variations treated
by the previous investigators for infinitely wide channels. The effect
of asymmetrical flow cross sections on longitudinal dispersion was first
examined by Aris (3) who showed that the dispersion in a circular tube is
less than that in an elliptical one of the same area. This in addition to
the fact that all real watercourses have a finite width and that width
todepth ratios for natural channels are usually significantly greater
than unity makes Fischer's hypothesis very reasonable. For wide channels
the transverse mixing time, k /Ky, over some characteristic zs would tend
to be greater than the corresponding vertical mixing time scale, h2/Kz.
Fischer thus argued that the velocity shear profile across the channel
would have a greater longitudinal dispersive effect since an increased
mixing time would produce larger values of the spatial correlation u'c".
Using this as the basis for his analysis Fischer applied Taylor's technique
to a rectangular coordinate system and assumed that vertical variations
in c" were negligible compared to the transverse variations to obtain
u" a (K (115)
where y is the coordinate axis across the channel.
Fischer then integrated (115) over the crosssectional area of the
channel to obtain an expression for c" which he then correlated with u"
for the dispersive mass transport and using Taylor's definition of a
Fickian flux obtained a longitudinal dispersion coefficient of
w y
EL = q"(y)dy K hy T dy q"(y)dy (I16)
0 0 0
where the depth, h, is a function of y; K is the transverse eddy diffusi
vity; w is the width of the channel; and q"(y) is the depth integrated
flow defined by
h(y)
q"(y) = S u"(y,z) dz (117)
0
To obtain a more useful expression for EL Fischer defines a Lagrangian
time scale for crosssectional mixing which he obtains through an extension
of Taylor's work on diffusion by continuous movements (7), and relates it
to the Eulerian time scale for crosssectional mixing previously discussed.
Then, using Elder's (4) experimental determination that the transverse
eddy diffusivity in an infinitely wide channel with a steady unidirectional
flow can be expressed as
K = 0.23 hu, (118)
Fischer arrives at an alternate expression for EL given by
2
EL = 0.3 u" (119)
L RHU.
where s in a natural channel is the distance from the point of maximum
surface velocity to the most distant bank. To verify his theory Fischer
conducted a detailed set of laboratory experiments. Using widthtodepth
ratios ranging from 9 to 15.7 Fischer obtained good agreement between
predicted and observed values of EL using (116). However, (119) over
predicted in each case, with a maximum error of 75 per cent.
Fischer (8) also applied his analysis to data obtained by other investigators
in 6 natural watercourses including Copper Creek, Va. and Clinch River,
Tenn. (2 locations each), Powell River, Tenn., and Coachella Canal, Cali
fornia. Widthtodepth ratios for these sites range from 15 to 62 with
most being less than 40. To carry out his analysis Fischer numerically
integrated (116) using measured data and found that in uniform channels
the predicted values for EL were within 30 per cent of the observed
values whereas in nonuniform channels the predicted value varied from
the observed by as much as a factor of 4.
Thus, while Fischer's hypothesis regarding the dominance of lateral
effects on longitudinal dispersion in unidirectional flows yields reason
able results his analysis provides very little insight into the physics
of the three dimensional problem. Both (116) and (119) make no attempt
to describe the velocity distribution over the flow cross section thereby
requiring detailed measurements for their application. Moreover, by neg
lecting the vertical shear effects it remains impossible to obtain from
his analysis the conditions for which either the vertical or lateral
effects would be the dominant mechanism for longitudinal dispersion.
IB.2 Oscillatory Flow
The analysis of dispersive mass transport in oscillatory flow is
more complex than the corresponding problem in steady unidirectional flow
because of the unsteady nature of the governing equations of motion and
transport diffusion. Since, however, many important water quality problems
involve the predicted distribution and transport of substances within
tidal waters the analysis of dispersive processes in oscillatory flows
has attracted considerable interest. There is in fact a reasonable doubt
that one is justified in representing the convective mass transport u"c"
as a Fickian flux in an oscillatory flow. However, it will be shown in
Chapter II that under the assumptions of periodicity and uniform flow the
analyses of Taylor and Aris apply.
Bowden (5) was the first investigator to look at longitudinal dis
persive mass transport in oscillatory flow. Bowden generally followed
Taylor's original analytical technique and began by assuming a monochro
matic flow in an infinitely wide channel in which u" can be expressed as
u"'z,t) = Ai(z) cos at + Bi(z) sin at (120)
He likewise assumed that c" is simple harmonic in time and of the form
c"(z,t) = P1(z) cos at + Q1(z) sin at (121)
conditional upon satisfying the transport diffusion equation expressed as
( ) + u" (Kz c) (122)
(d t=a z (z z S9z )
This is a very proper formulation of the problem, allowing for the
periodic variation of u" and c" with time and the existence of temporal
phase shifts for u" and c" as a function of position within the water
column. Unfortunately, Bowden proceeded by means of his assumptions to
reduce the problem to one which applies only to the case for which the
period of oscillation is infinitely long, ( = 0, and for which the
temporal phase shift as a function of position within the water column
has been eliminated. For these conditions Bowden found that the longitu
dinal dispersion coefficient in an oscillatory flow is onehalf the value
of the same coefficient for a corresponding unidirectional flow having the
same surface velocity, the same shear profile, and the same vertical eddy
diffusivity.
Holley and Harleman (9) treated the case of oscillatory turbulent pipe
flow and obtained an expression for the longitudinal oscillatory flow dis
persion coefficient, Ex, using Aris' method of moments. In their analysis
they assumed that at each instant in time the lateral distribution of
velocity and eddy diffusivity are the same as they would be if the flow
were steady with the same crosssectional mean velocity. This assumption
effectively removes the oscillatory nature of the flow and reduces the
problem to essentially the same one solved by Aris (3) and Taylor (2).
Thus, it is not surprising that Holley and Harleman predicted a longitu
dinal dispersion coefficient of
Ex (t) = 10.la u, (t) (123)
where, as before, a is the radius of the pipe, and u, (t) is the friction
velocity now periodic in time. Predicted values of Ex were compared
with observations made from a series of laboratory experiments in which
an oscillatory flow was produced in a 11/2 inch pipe by a piston genera
tor. Superimposed upon the oscillating flow was a small unidirectional
flow to simulate the net fresh water discharge of an estuarine type flow.
The experimental data, however, showed that due to the small magnitude
of the unidirectional flow compared to the velocity amplitude of the oscilla
tory flow, its effect on longitudinal dispersion could be neglected. The
data also showed that (123) generally underpredicted the dispersion
coefficient with better agreement between predicted and observed values
coming with higher Reynolds number flows. The authors attributed this
behavior to laminar sublayer effects.
Okubo (10) obtained a less restrictive solution which for the first
time demonstrated that in a two dimensional oscillatory shear flow the
period of oscillation and the time scale of vertical mixing played an
important role in the dispersive process. Okubo confined his analysis
to vertically bounded and unbounded flows with no transverse shear. Only
the formerwill be discussed. He assumed a linear, monochromatic flow
superimposed upon a linear unidirectional flow, or
u(z,t) = Uo (1) + Vo (1 ) sin at (124)
where Uo and Vo are the surface amplitudes of the unidirectional and
oscillatory flow, and a is the angular frequency of oscillation. Okubo
then assumed a constant eddy diffusivity both in the direction of flow
and over the depth, and zero diffusive flux conditions at the surface and
bottom. Using the method of moments as applied by Aris and the assumptions
discussed he obtained an expression for the longitudinal dispersion coeffi
cient, Ex, functionally expressed as
Ex = X (Uo, Vo, h, T, ) (125)
x z
h2
where T is the period of oscillation, and the time scale of vertical
z
mixing. To interpret his results Okubo considered two extreme cases.
(1) >> T Neglecting longitudinal eddy diffusion his solution becomes
U h2 V2 T2K
x z) + ( ) (I26)
unidirectional oscillatory
From this Okubo correctly concluded that for situations in which the verti
cal mixing time is large compared with the period of oscillation the con
tribution to the longitudinal dispersive mass transport due to the oscilla
tory flow is small compared with the contribution due to the unidirectional
flow. However, (126) also suggests that the functional behavior of dispersion
in an oscillatory flow is significantly different from the same process in
unidirectional flow. One of the principal findings of the present work
concerns this point and will be treated in depth in Chapters III and IV.
h2
(2) T >> Again neglecting longitudinal eddy diffusion Okubo's result
may be written as
U0 h2 V h2
Ex ) +2 ( (127)
z z
This case is comparable to Bowden's analysis for a very long period of
oscillation and supports his finding that Ex 1/2 EL for large T.
Holley et al. (11) successfully applied Taylor's original method of
analysis to obtain an analytical expression for T in a two dimensional
periodic, uniform shear flow. For their analysis they assumed the spatial
variation component of velocity to be
u" (z,t) = a z sin at (128)
where a is a constant. This is identical to the linear oscillatory flow
profile assumed by Okubo. This expression for u" is then used to force
the following form of the transport diffusion equation to obtain a solution
for c" (z,t)
(=) Kz, =" (129)
The solutions obtained for u" and c" are then used to determine the depth
integrated time mean convective mass transport with respect to the depth
mean velocity. The resulting expression for the mean longitudinal dispersion
coefficient over the period of oscillation is
a2 T2 h4 1
S z n=l (2nli)2{[ (2n1)2 T' 2+
TK
where T' = h. Holley et al. then proceed to relate T in an oscillating
flow to the dispersive process in an oscillating flow of infinite period
with the same velocity profile as given by (128). Solving for c" using
the steady state form of (129) and proceeding in the usual manner they
obtain an expression for T which they call E It is shown that E.
is onehalf the corresponding value for EL in agreement with Bowden (5),
and is subsequently used to compare T with the corresponding steady
unidirectional flow coefficient. Apparently the investigators' primary goal
was not to describe longitudinal dispersion in an oscillatory flow as a
separate distinct physical process but rather to describe it in terms of a
"corresponding" process in a unidirectional flow. To do this the solution
for T given by (130) was normalized by E, and the result plotted against
TK
the nondimensional time, T' = , suggested by Okubo's (10) results.
Numerical simulations were also carried out for other velocity profiles
in which the velocity was again assumed to be temporally in phase over
depth during the period of oscillation. Based upon their analytical and
numerical results Holley et al. concluded that:
(1) T'>1 The longitudinal dispersion coefficient in an oscillatory
flow is independent of T' and equal to onehalf the value of the dispersion
coefficient in a "corresponding" unidirectional flow.
(2) T'<0.1 The longitudinal dispersion coefficient in an oscilla
tory flow rapidly becomes insignificant compared to the dispersion coefficient
in a "corresponding" unidirectional flow and is functionally described by
T ~ T'2 (131)
These results have been subsequently applied by Fischer (12, 13), Awaya
(14), and Fischer and Holley (15) in analyses of dispersion in oscillatory
flows. Fischer (12) attributes the functional behavior of oscillatory flow
dispersion to a temporal phase shift between u" and c" as a function of T'.
However, it will be shown in the present work that while these conclusions
T
are applicable for the nondimensional parameter they are incorrect
and misleading in their description of the longitudinal dispersive mass
transport process in oscillatory flow.
Fukuoka (16) obtained analytical expressions for E from several cases
of two dimensional and axisymmetric oscillatory shear flows using the method
of moments. For his analysis he assumed a velocity of the general form
u = us (z) + ut (z) sin at (132)
similar to Okubo (10), implying once again no temporal phase shift of the
velocity over the flow cross section. In specifying the forms of us (z)
and ut (z),Fukuoka assumed that the spatial variation at each instant of
time is the same as that for an equivalent unidirectional flow. Specific
cases treated in depth are:
(1) linear velocity profile over depth with constant K
(2) parabolic velocity profile for axisymmetric flow with constant
Kz
(3) linear velocity profile over depth with Kz = Ko sin ot .
In interpreting his results Fukuoka follows Holley et al. by plotting
Ex
vs. T' and arrives at the same conclusion regarding the functional be
havior of dispersion in oscillatory flows. For Case 3 he shows analytically
that the dispersion coefficient corresponding to a time varying eddy diffu
sivity differs by a factor of 8/72 from one that is considered to be
2
constant, provided the constant Kz is Ko. This small effect of a periodic
eddy diffusivity supports results obtained numerically by Holley et al.
It is interesting to note that none of the analyses dealing with
oscillatory flow discussed thus far have attempted to solve analytically
the velocity distribution over the flow cross section. Moreover, it has
been further assumed that the velocity profile at each instant in time is
the same as an equivalent unidirectional flow profile and that no temporal
phase shift as a function of spatial position exists. These assumptions
are particularly unrealistic when one considers the definition of dispersive
mass transport as defined by Taylor, i.e.A u"c" dA. By restricting the
form of u" in this manner one is also restricting the form of c" since by
(129) it is seen that the distribution of substance is in part forced by
the convective flux. Thus, any artificiality regarding the assumed form
of u" appears in the dispersion coefficient through both u" and c". The
significantly different nature of a truly oscillatory flow as compared with
a unidirectional one is illustrated bySchlichting (17) in his treatment
of the fluid motion above an oscillating flat plate, and Lamb (18) in his
treatment on the effect of bottom friction on long waves in two dimensional
shear flow. Although the forcing functions in these two cases are different
(oscillating boundary shear inSchlichting as opposed to an oscillating
pressure gradient in Lamb), the result of a shear wave propagating through
the fluid in a direction normal to the axis of flow is similar, causing a
continuously changing phase shift of the flow over the crosssectional area.
Awaya (14), Segall (19), and Segall and Gidlund (20) have obtained
analytical expressions for the longitudinal dispersion coefficient in
oscillatory flow situations using realistic velocities obtained directly
from solutions to the appropriate forms of the equations of motion. Awaya
addresses the case of an oscillatory laminar flow in a circular cross section
for which he obtains an expression for T using Taylor's technique, and
20
normalizes it by E, after Holley et al. His results are similar to those
of Holley et al. and his interpretation the same. Segall and Segall and
Gidlund use a simplified version of Lamb's previously mentioned solution
for u corresponding to small values of the vertical eddy viscosity, ez
which they apply to the transport diffusion equation and solve using the
method of moments. In both of these analyses the interpretation of the
functional behavior of the dispersive process in oscillatory flows is the
same as that of previous investigators although the matter is not pursued
in any detail.
IC Description and Scope of
the Present Work
In the chapters that follow four sets of boundary value problems will
be solved yielding analytical solutions for u", c", EL, and T for the
following cases:
(1) Steady unidirectional flow in an infinitely wide rectangular
channel, Ez and Kz constant
(2) Monochromatic periodic flow in an infinitely wide rectangular
channel, ez and Kz constant
(3) Steady unidirectional flow in a rectangular channel of width
w; EZ, E KZ, and K constant
(4) Monochromatic periodic flow in a rectangular channel of width
w; EZ, Ey, K, and K constant.
These solutions deal exclusively with dispersive mass transport in homo
geneous fluids and uniform or nearly uniform flows.
In Chapter II a discussion is presented of the conditions under which
Taylor's definition of dispersive mass transport is valid for unidirectional
and oscillatory flows. Chapter III treats Cases 1 and 2 above yielding
solutions for EL and T in two dimensional shear flows while Chapter IV
extends the analysis to three dimensional shear flows through Cases 3 and 4.
In Chapter V predicted values of EL and T using expressions developed
in Chapters III and IV are compared with available data. Chapter VI presents
a summary of the present work and some conclusions drawn from it. The mathe
matical details associated with portions of the boundary value problem
solutions are reserved for Appendices A, B, and C.
CHAPTER II
CONSIDERATIONS FOR APPLICATION OF TAYLOR'S
DISPERSION ANALYSIS TO FREE SURFACE FLOWS
IIA General
In order to provide a solid base upon which the analyses presented in
Chapters III and IV are built, it is necessary to set forth at this juncture
a detailed development of Taylor's dispersion analysis as it applies to
both unidirectional and oscillatory free surface flows. By doing this it
is hoped that all assumptions made in obtaining solutions for u", c", Ex
and EL, and the conditions under which a dispersion coefficient may be defined
and applied consistent with Aris (3) and Taylor (1,2), will be clearly
understood.
Consider a three dimensional shear flow, either unidirectional or
oscillatory, in a rectangular channel of width w and depth h. Let the
coordinate system be selected as shown in Figure 1 with the x coordinate
taken along the longitudinal axis of the channel denoting the direction
of the
u (y,z,t)
z z=0
yrw/2 \Y = f/2
S= h"
FIGURE 1 DEFINITION SKETCH
velocity component, u(y,z,t); the y coordinate denoting lateral position
across the channel; and the z coordinate acting positive upward from the
free surface. To begin the analysis it shall be assumed that:
(1) Longitudinal mass transport due to eddy diffusion along the flow
axis is small compared to convective transport mechanisms.
(2) Eddy diffusivities K and Kz can be represented by their mean
values over the flow cross section.
(3) Uniform flow conditions along the principal axis of the channel
exist as indicated by the functional notation for u.
Applying these assumptions to the channel shown in Figure 1 the
general form of the transport diffusion equation for a conservative sub
stance may be written as
ac aC 2 + 2(Il)
+ u K K(111)
at 3x y y2 Z az2
After Taylor (1,2), transform (II1) to a coordinate system traveling
with the crosssectional mean velocity using
u = u + u" (112)
S= x ut (113)
so that the functional transformation of the concentration c is given by
c(x,y,z,t) + c(S(x,t),y,z,t).
The transformation relationships are then
3c c +3c C
=t (1) +
ac ac 95
ax 3E ax
c ac
and 2 unchanged
3y az
Using (113),
(114)
(115)
Equations (112), (114),
yield
and (115) are then substituted into (II1) to
+ C 1 ac 32C 92C
at)c + ull K y K Z@Z
K+Ky2 a2
(116)
Next, decompose c into its crosssectional mean and variation components
by the expression
c = c + c"
and substitute into (116). For uniform flow this produces
(117)
ac ac c1 a2 2c" (u"c")_ u" 3W 8)
at t y ay2 Z 2 a
which is the form of the transport diffusion equation as seen from a
coordinate system traveling with the mean crosssectional velocity.
Proceeding with the analysis, (118) is integrated over the cross
sectional area,
Sw/2 h w/2 h w/2
( ) dydz + ( a) dydz K dydz
w/2 0 w/2 0 w/2
h w/2 h w/2 h w/2
z dydz (u"c") dydz u" dydz (119)
0 w/2 0 w/2 0 w/2
It is next assumed that:
(1) The time rate of change of the depth is small.
ac (9)
af at aca
ac Dc
5 T TE
(2) There is no turbulent diffusion of substance across the enclosing
boundaries of the flow cross section.
To apply the condition of zero diffusive flux across the boundaries
of the channel it is helpful to use Green's Theorem in the plane which for
the third and fourth terms on the left hand side of (119) states
a__ ac" a c" ac" c
'(K ) + z(K dA = dzdy} (II10)
ay y 9y at y ay z
A u
where u is the perimeter of the area A. Since by assumption (2) it is
required that there be no diffusive flux across u, the left hand side of
(II10) is equal to zero. Using this result, assumption (1), and noting
that
u" dA = c" dA = 0
A A
by definition, (119) is reduced to
( = (u ) (II11)
As noted in I.B.1 the form of (II11) induced Taylor to define the spa
tially averaged convective flux u"c" as a Fickian flux thus transforming
(II11) to the one dimensional heat equation form
(C) = EL (1112)
where,
Su"c" (1113)
EL _c
ac
Referring again to I.B.I, Aris showed that (1112) and (1113) are
valid formulations if the following necessary and sufficient conditions
are satisfied: (1) steady uniform flow; and (2) steady state concentration
with respect to a point traveling at the mean velocity, i.e. (c) = 0.
at t
In the sections that follow it will be shown that these conditions are
necessary and sufficient not only for unidirectional flows but for oscilla
tory flows as well provided that steady state is defined in the periodic
sense. It will also be shown that Taylor's assumptions regarding the
functional dependence of c and c" on 5, y, and z as stated in I.B.1
follow directly from the more general requirements of steady state u and
c, and uniform flow.
IIB Unidirectional Flow
To obtain an expression for EL as defined by (1113) it is necessary
first to have solutions for u" and c". Ideally a solution for u"(y,z) is
obtained from the governing equation of motion in the x direction. This
solution is then used to force a simplified form of (118) to obtain a
solution for c" which is then correlated with u" according to (1113) for
the determination of the longitudinal dispersion coefficient. As stated
in I.B.1 Taylor reduced an expression similar to (II8) to a solvable
form by assuming:
(1) c" is not a function of 5.
(2) is constant.
(3) Steady state conditions with respect to the moving coordinate
system exist, i.e. (3c) 0.
3t E
Aris, however, showed that assumption (3) along with the requirements
of steady uniform flow were necessary and sufficient for the definition
of a dispersion coefficient as stated by (1113), and it will now be
demonstrated that assumptions (1) and (2) follow from these conditions.
For steady state conditions to exist (II11) reduces to
3C A
= I u"c" dA
A
Assuming the channel to be prismatic, the order of differentiation and
integration may be reversed yielding
0 = A T (u"c") dA
A
which for uniform flow becomes
0 = 5 u" dA (1114)
A
To examine the functional dependence of c", let
U" = g2 (y,z)
3c"
= f2 (y,z,S)
Equation (1114) may then be written as
0 = : g2 (y,z)f2 (y,z,S) dA (1115)
A
which upon integration yields
G() = 0 (1116)
From (II16) it may be inferred that Tc= 0 or c" is not a function of
thereby justifying Taylor's first assumption restated as
c(x,y,z) = c(x) + c"(y,z) (1117)
The above result is then used to substantiate Taylor's second assump
tion that is constant. To do this (118) is differentiated with
respect to E. For steady state conditions and uniform flow the result
is
32= 0 (1118)
which upon integration yields
j= constant (1119)
ot.
The application of (1117) and (1119) reduces (118) to
K + z = u" (1120)
where c is constant.
Thus it is shown that the definition of the longitudinal dispersion
coefficient as stated by (1113), which implies the use of solutions for
u" and c" obtained from the governing equation of motion and (1120)
respectively, is valid for free surface flows provided the necessary and
sufficient conditions of uniform steady flow and steady state concentration
distributions with respect to a coordinate system traveling with the mean
velocity are satisfied. Taylor's assumptions as stated by (1117) and
(1119) follow from these conditions.
IIC Oscillatory Flow
The same analysis carried out in IIB for steady unidirectional flow
can be extended to the oscillatory flow case if it is assumed that steady
state conditions in the oscillatory sense can be satisfied by requiring
both the velocity, u, and the concentration, c, to be temporally periodic.
This, of course, implies that u, c, u", and c" are also temporally periodic.
To show that Taylor's same assumptions apply to dispersion in oscilla
tory flows it is again necessary only to require that the flow be uniform
and that the velocity and concentration be steady state functions in the
oscillatory sense, i.e.,periodic. If this can be shown to be true then
it is reasonable to assume that Aris' argument could be extended to periodic
functions thereby justifying the definition of a longitudinal dispersion
coefficient in oscillatory flow using Taylor's method.
The analysis is begun by averaging (II11) over one full period of
oscillation which for a periodic c yields
< (u"c > = 0 (1121)
where C is now defined by
C = x \ (t') dt' (1122)
0
Next, it is assumed once again that the time rate of change in depth is
small so that the implied order of integration and differentiation in
(1121) may be reversed
h w/2 T
h w/2 5 u"c" dt dy dz = 0 (1123)
S0 w/2 0
Since it is required that the flow be uniform and that u and c be
temporally periodic, assume
u" = g1(y,z) cos at
c" = fl(y,z,') cos(at a,)
where ai is a constant. It then follows that the temporal integration
of u"c" is of the form
T u"c"dt = B(y,z,S) cos ct (1124)
0
Substituting (1124) into (1123)
h w/2
h Sw B(y,z,S) cos ai dy dz = 0
0 w/2
which upon integration reduces to the form
cos ai x) = 0 (1125)
From this it is seen that X() is a constant and thus c" is not a
function of . This is the same result obtained in IIB and is consistent
with Taylor's first assumption. However, for the oscillatory case there
is an additional distinction to be made. Recognizing that uniform flow
is required and c" is not a function of E, (II11) reduces to
( = 0 (1126)
and (118) becomes
(at KKz 2 = u" 3 (1127)
Differentiating (1127) with respect to E yields
u" 0
32
or, D= constant, which is again consistent with Taylor's assumption for
the unidirectional flow case. Equation (1127) with constant is the
comparable form of (1120) for oscillatory flow. It is from the solution
of this equation that c"(y,z,t) is obtained for determination of the
longitudinal dispersion coefficient, Ex.
Based upon the above discussion it is concluded that Taylor's analysis,
under the conditions of uniform flow and temporal periodicity of u and c,
is capable of being extended for the determination of longitudinal dispersion
coefficients in oscillatory flows and remains consistent with the condi
tions for which a dispersion coefficient may be defined as shown by Aris.
IID Application of Dispersion Coefficients
Predicted by Taylor's Method
It is not the purpose of the present work to provide a detailed anal
ysis of the conditions under which values of EL and Ex predicted by
Taylor's method may be applied for the determination of concentration
distributions. However, a comment regarding this matter is considered
useful here in order to retain a proper perspective on the work presented.
Equation (1112) indicates that in applying values of the dispersion
coefficient, steady state conditions for the crosssectional mean concen
tration are not satisfied whereas in Taylor's analysis for the determina
tion of the dispersion coefficient it is assumed that these steady state
conditions exist. Aris (3) has shown that the steady state conditions
required for the convective mass transport to behave as a Fickian flux are
approached asymptotically at a rate determined by the ratio of the time
of convective flow to the time of turbulent crosssectional mixing. This
interpretation was also given by Taylor (1) on a qualitative basis. Fischer
(21) then reasoned that a necessary condition for the application of coeffi
cients obtained in this manner for the prediction of concentration is that
c" be everywhere much less than c. This in effect strikes a compromise
between the strict steady state requirement for the definition of a dis
persion coefficient and the conditions for which it may be used. Fischer
(21) among others has developed detailed criteria for the application of
dispersion coefficients in (1112).
CHAPTER III
DISPERSIVE MASS TRANSPORT IN AN INFINITELY
WIDE RECTANGULAR CHANNEL
IIIA Steady Unidirectional Flow
The coordinate system has been selected (Figure 2) with origin at
the bed of the channel, z coordinate positive upwards, and x coordinate
positive in the direction of flow, u.
z=h
_^_____ z
u(z) c(x, z)
FIGURE 2. DEFINITION SKETCH FOR TWO
DIMENSIONAL SHEAR FLOW
The general procedures used throughout Chapters III and IV will be to
first obtain analytical solutions for u" and c" and then use these to
calculate the dispersive mass transport u"c" from which the dispersion
coefficient is determined.
IIIA.1 Velocity Distribution
It is assumed that the turbulent shear stress may be expressed in
terms of the Boussinesq approximation and that the eddy viscosity, ez,
is constant and equal to its mean value over depth. The equation of
motion for two dimensional steady uniform shear flow is then
0 P U (III1)
p ax z z2
where p is the density of the fluid and P is pressure. It will be
assumed in all analyses that turbulent fluctuations have been averaged
and incorporated into the remaining terms. The pressure gradient term
for this case may then be expressed as
1 2P
K (III2)
p Tx
where K is constant. Substituting (III2) into (III1) and requiring the
boundary conditions of no slip at the bed and zero shear stress at the
surface the flow boundary value problem is then
DE: d2u K
Sdz2 E (1113)
z
BC's: u(0) = 0
du
Tu7 z=h = 0
The velocity shear profile is obtained by integration of (1113) and
the application of the boundary conditions. The resulting expression is
u(z) =  (z2 hz) (III4)
The spatial variation of the velocity as defined by (112) is then
obtained by averaging (III4) over depth and subtracting out the depth
mean component to yield
u"(z) = K( hz +2) (III5)
Cz
The portion of the unidirectional flow analysis remaining in this
section and portions of the analyses in later sections treat the expres
sion of the variations of velocity and concentration and the dispersion
coefficient as functions of normalizing parameters other than the pressure
gradient modulus, K. In some cases these alternate formulations are
considered to be more useful for application purposes, whereas in other
cases they are presented for the comparison of longitudinal dispersive
mass transport in oscillatory and unidirectional flow or in oscillatory
flows of differing periods of oscillation. Therefore, before proceeding
further a brief discussion follows describing these alternate formulations
and the rationale used in their selection.
As noted in Chapter I the approach taken by most investigators in
describing dispersive mass transport in oscillatory flows has been to
first assume a velocity profile that is at each instant in time similar
in form to a corresponding unidirectional flow, and then use this to
obtain an expression for the dispersion coefficient, Ex, either directly
through the method of moments or indirectly through Taylor's method of
describing the convective mass transport uc" as a Fickian flux. The
dispersion coefficients thus obtained have then been analyzed only as
they relate to a "corresponding" unidirectional flow coefficient. The
approach taken here is that it is not realistic to assume a priori that
the dispersive mechanism in oscillatory flow is functionally similar to
the same mechanism in unidirectional flvo because of the distinct dif
ferences in the flow characteristics of the two cases. This argument
may also be applied to the more general case of comparing dispersive
mass transport characteristics of oscillatory flows of different periods.
It will later be shown that such comparisons are easily misleading and if
done at all they must be done with a full understanding of the mechanism
involved. To illustrate this point, there are many ways in which
unidirectional and oscillatory, or two oscillatory mass transport cases
of different periods, can be considered to be "corresponding." Normal
izing parameters considered in the present analysis for comparison of
"corresponding" dispersion coefficients are:
(1) pressure gradient modulus, K,
(2) maximum crosssectional velocity, umax' which for the most
general case of three dimensional oscillatory shear flow would
represent the amplitude of the periodic velocity function located
on the centerline of the channel at the surface,
(3) surface excursion length, L, experienced by a particle on the
free surface over a period of time equal to onehalf the period of
oscillation.
For the case of a unidirectional flow, such as that being considered
in this section, the concept of an excursion length as described above
has little physical meaning. Thus, excursion length normalization of
u", c", and Ex solutions will be made for oscillatory flows only. The
expression of (1115) in terms of umax is easily accomplished by
solving (1114) for K at z = h, or
2e u
K = 2 max (1116)
h
which upon substitution into (1115) yields
2u 2 h2
u,,(z) max z2 h2
u z2 hz + ( ) (1117)
IIIA.2 Concentration Distribution
A solution for c"(z) as a function of K is obtained by using the
solution for the spatial variation of velocity, u"(z), given by (III5),
to force the one dimensional form of (1120). Thus, the concentration
variation boundary value problem may be stated as
DE: d2c" E
dzK u (1118)
dc"
BC's: z = O,h dc"= 0
dz
where,
ac
= constant
u"(z) is given by (III5).
Substituting for u", integrating once with respect to z, and applying the
boundary condition at z = 0 produces
dc" K BE z3 hz2 h2z (119)
d * ( 6 [ + ] (III9)
dz  (JL6 2 3
SzKz
It should be noted that (III9) implicitly satisfies the boundary condi
h
tion at z = h since by definition S u"(z) dz = 0. Equation (III9) is
0
then integrated once more over z and the constant of integration is
evaluated using the requirement that c"(z) must have a zero mean over
depth. The resulting solution for the concentration variation is then
c"(z) = K c z4 hz3 h2z2 h4
c"(z) ( 2 + l (III10)
c K 24 6 6 t5
zKz
Using (III6) to substitute for K in (III10) the expression for
c"(z) as a function of umax is
c"(z = a 4 hz3 h2z2 h'
c"(z) = max (T ) r4 6 + T (IIIii)
K h2
z
IIIA.3 Dispersive Mass Transport
The dispersive mass transport over the depth of flow is given by
S= S u"c"dz (III12)
Substituting for u" and c" using (III5) and (III10) respectively,
(I1I12) becomes
K2 aBE z h2 z4 hz3 h2z2 h
1 =  () (2 hz + 3 24 6 6 45 dz
ezKZ 0
Carrying out the integration the mass transport is then
=( ) 2K2h (III13)
945EzKz
It shall be assumed for this analysis and for the analyses that
follow that Reynolds analogy relating the transfer of mass and momentum
by turbulent processes is applicable so that
E = K (III14)
z z
This assumption is not necessary to obtain solutions for EL and Ex using
the method presented here; however as will be seen later its use simpli
fies the forms of solutions considerably and facilitates interpretation
of the physical processes involved. Thus, with the aid of (III14) and
the introduction of the vertical mixing time defined by
T h2 (III15)
K
z
Equation (11113) may be written as
2K2T3 h
S( z4 (III16)
Assuming next that the required conditions for defining a longitudinal
dispersion coefficient as discussed in IIB are satisfied, then
m
L a)h
( 
2K2T'
cz
EL =~9 (III17)
Using (III6) and (11115) in (11117) the longitudinal dispersion
coefficient as a function of uax is
8u2 T
max cz
EL 4 (11118)
The functional form of EL given by (11118) is the same as that
obtained by Taylor for laminar unidirectional flow in a tube,(I8). In
both cases the longitudinal dispersion varies directly as the product
of the square of the velocity and the crosssectional mixing time.
Thus, for a given unidirectional velocity profile the longitudinal mass
transport of a substance increases linearly with the time required to
mix that substance over the flow cross section by diffusive processes.
IIIB Oscillatory Flow
IIIB.1 Velocity Distribution
The same assumptions used in IIIA regarding selection of coordinate
system, and the use of the Boussinesq approximation for expressing the
viscous term in the equation of motion will be applied here. The govern
ing equation of fluid motion for uniform unsteady two dimensional shear
flow is then
au 1 a2u (III19)
at px+ E z az2
For the oscillatory flow case it will be assumed that long wave phenomena
are of primary interest and therefore the pressure distribution over depth
is hydrostatic. It is then reasonable to assume that the pressure gradient
in (III19) is periodic and of the form
1 P KeiOt (III20)
p ax
where K is constant, as before, and a is the angular frequency of oscilla
tion. Note that only the real part of (III20) has physical meaning.
To obtain a solution assume that u(z,t) is periodic in time and of
the form
u(z,t) = us(z)eiot (III21)
where u (z) is a complex function. The boundary value problem is then
formulated by incorporating (III20) and (III21) into (III19) which
along with the boundary conditions of no slip at the channel bed and
zero shear stress at the surface becomes
dau
DE: z dS ius = K (III22)
z dz2 s
BC's: u (0) = 0
du
Iz z=h =
This problem has been previously solved by Lamb (18) and Segall (19)
who obtained
u(z,t) iK El cosh B(l+i)(hz) iat (11123)
u(,t ac 1 cosh B(l+i)h
where,
B= =/ (11124)
with only the real part of (11123) having any physical meaning. Its
form is clearly that of a damped progressive shear wave propagating
upward through the water column and causing a temporal phase shift in
the velocity as a function of z. It was also pointed out by Segall and
Gidlund (20) that the solution as expressed by (11123) correctly predicts
flow reversals in the lower momentum layers of the fluid near the bed
prior to a shift in flow direction of the higher momentum layers near
the free surface. These effects are extremely important to the disper
sive transport mechanism.
The spatial variation of the velocity, u"(z,t), is obtained by
averaging (11123) over depth and subtracting out the mean from the
total velocity according to
u"(z,t) = u(z,t) u(t)
The result is then
u"(z,t) = iKeiot { sinh (l+i)h cosh B(1+i)(hz)} (11125)
a cosh B(1+i)h (l+i)h
For reasons that will become obvious in the next section the velocity
variation solution as given by (11125) will now be expanded in a Fourier
cosine series of the form
u"(z,t) = an cos nT eiat (III26)
n=l
where coefficients an are complex. The a0 term in the series has been
omitted to satisfy the zero mean requirement for u". It is noted by
Hildebrand (22) that any piecewise differentiable function may be com
pletely represented by a cosine series of the form a cos z over the
closed interval 0 < z < h. Proceeding then,
h
S2 iK sinh B(1+i)h nrz d
a h (1+i)h cosh B(1+i)h cos dz
0
h
2 ( iK niz
h a cosh (1+i)h cosh B(1+i)(hz) cos n dz
o
The first term on the right hand side integrates to zero while the second
term must be integrated by parts twice which yields for an,
2K Bh(1i) sinh B(l+i)h
n o cosh B(1+i)h [(ni)2 + i2(Bh)2] (III27)
Since only the real part of u"(z,t) is of any interest, (III27) will now
be put in polar form for incorporation into (III26). Considering each
complex term separately,
(1i) = v2 e'T' (1128)
sinh B(1+i)h = (cosh 26h cos 2Bh) eial (11129)
cosh B(l+i)h = (cosh 2Bh + cos 2gh) e1i2 (11130)
[(nT)2 + i2(Bh)2] = [(nr)4 + 4(Bh)4]2 eia3 (11131)
where,
et = tan (cosh 0h sin h) (11132)
sinh Ah cos Bh
2 = tan1 (sinh h sin Bh) (11133)
cosh h cos 6h
a3 = tan1 C2 )] (11134)
(n) 2
44
To express u"(z,t) in its final form it is again assumed that
Reynolds analogy is applicable allowing the use of (III15). A non
dimensional time relating the period of oscillation to vertical mixing
time is then introduced by
T cz (III35)
z T
With the aid of Reynolds analogy and (III15), (III24), and (III35)
the Bh arguments appearing throughout an become
h =V/irT (III36)
The final form for u"(z,t) as a function of the pressure gradient is
arrived at by combining (III26) through (III36) to yield,
2T' (cosh 2 V71 cos 2/VrT) )
u"(z,t) = KT { Z z z
7 (cosh 2V/7rT + cos 2viT'/ )
Scos 2 e(t + L t a3(n))
n=1 [(nT)4 + (2Tz)2] (III37)
where,
cosh v rT sin VT
ai = tanI { (III38)
sinh I/TT cos /i
z z
Ssinh vV/T sin vv/T~
ap = tan {i z (III39)
cosh v/TT cos ViT
Z Z
i 2Trr
a3(n) = tan' { z2} (11140)
Expressions for u"(z,t) as a function of the surface amplitude,
umax, of the periodic velocity function and the surface excursion length,
L, experienced by a particle during onehalf of the period of oscilla
tion are obtained by solving for K as a function of umax and L using
max
(III23). Details of this development are presented in Appendix A, the
results of which are
VT Tr(cosh 2 v i + cos 2 vN/ T )2
K zZ u (III41)
T (cosh VTJ cos A/_F?) max
and 2 I L (cosh 2 2;' + cos 2 VJiTf)4
K = z z (III42)
T2 (cosh vTYT cos /'i)
z z
Substitution for K in (11137) using these two expressions then yields
2u (~T' (cosh 2 v/T' cos 2 V; )}
u"(z,t) max z
(cosh vTT cos /TTz')
nc z ei(ot T + al a2 a3)
)h] (III43)
[(nt)" + (2rTz)2
n=1
and,
2'iL { TzT' (cosh 2 v/' cos 2 /7T)}
u"(z,t) = 
T (cosh
cos n ei(ot + 0iC 2 03)
(III44)
n=1 [(nT)4 + (2T )2]'2
IIIB.2 Concentration Distribution
For temporally periodic u and c, uniform flow, and constant eddy
diffusivity it was shown in IIC that the transport diffusion equation
as described from a coordinate system moving with the crosssectional
mean velocity could be written as
ac" K c" = u"c (11145)
3t z azz 'l
where the rate of change with respect to C will hereafter be implied.
The concentration variation, c"(z,t), shall be assumed to be of the form
iat cos mnz iat
c"(z,t) = c1(z)eit a cos eit (III46)
m=l
where a' is complex. This expression for c" is seen to satisfy the require
ments of periodicity and zero diffusive flux of substance across the flow
boundaries. Moreover, if (III46) is substituted into (III45) and
operated on, the left hand side is composed of cos nz terms only. Thus,
to satisfy the conditions of equality, the expansion of (III25) for u"
in a cos z series is justified. The formal statement of the boundary
value problem for c"(z) is then
Sd2c"
DE: K 2 ic = u" (III47)
z dz2 s s 
dc"
BC's: dc ,h = 0
dz z=0,h
where is constant and u (z) is defined by (III26). Substitution for
c" and u" transforms (III47) to
s s
{ + iuala cos z(z ()C) cos niz
Sm h 3 mn h
m=1 n=1
which due to the orthogonal properties of cos mT and cos n requires
h d cos requires
that m=n for a nontrivial solution. The Fourier coefficient a' can
m
then be written as
S h2a
a' m ( m (III48)
SKz (mr)2 + i(h2)]
Kz
The required solution for c"(z,t) is arrived at by: (1) using
(III26) and (III37) to determine the polar form of am; (2) transforming
the denominator of (III48) to polar form; (3) applying the definitions
for T and T' where necessary; and (4) substituting the resulting expression
into (III46). The variation component of concentration as a function of
the pressure gradient is then,
3 2(cosh 2 r/TP cos 2 v/iT)
c"(z,t)= ( C) KT T'T {  z rz .
T z ir(cosh 2 Vn 27 + cos 2 vi/T )
Z cos i(t 1 + ai 2a (m))
Cos  e 4 (III49)
m=1 [(mr)" + (2TTz)2
where ai, a2, and a3(m) are given by (III38) through (III40) with m
replacing n in (III40).
Solutions for c"(z,t) as a function of umax, and L are obtained by
applying (11141) and (11142) to (III49), to yield
c"(z,t) = 2uax T T() (cosh 2 cos 2
max z 
(cosh v T cos VIT)
z z
Scos z i(at + c 1 z2 2ca3(m))
[(mr)4 + (23rT')2 (III50)
m=1
and
3 (cosh 2 v/TT cos 2 VTi)
c"(z,t) = 2L(TT) ) z z
(cosh VnTV cos v'7 )
mo z ei(at : + c~ 2 2a3(m))
C(m+)* + (2nTz.)2 (III51)
m=1
IIIB.3 Dispersive Mass Transport
For the oscillatory flow case the mass transport considered will be
the time mean transport averaged over one period of oscillation. Thus,
the dispersive mass transport is given by
<> r = Re(u") Re(c") dz T (III52)
Using (III37) and (III49) for u"(z,t) and c"(z,t), the integra
tion over depth of the product R (u") R (c") reduces the implied
double summation over m and n to a single sum through the orthogonal
properties of the functions cos nz and cos TZ since,
h h
h n h m
cos hr cos  dz =
00 n m
Performing this integration and averaging the results over T yields,
h K2T3T12 (cosh 2 V7i cos 2 2vj)
T 27r (cosh 2 V/ + cos 2 VTV)
cos a3(n)
> Cos a(n) (11153)
n=l [(nT)4 + (2+ T)2 2
The longitudinal dispersion coefficient is next introduced using
Taylor's analogy defining the convective mass transport T as a
Fickian flux so that
S < E h()> (11154)
Noting from (11140) that
cos a3 (n) = (nr)2
[(ni)4 + (2rT )23]
and combining (11153) and (11154) the time mean longitudinal dispersion
coefficient as a function of the pressure gradient is expressed as
SKT3T'2 (cosh 2 V/vz cos 2 /'T)
T = z z
S (cosh 2 VTr/ + cos 2 VJ/~i)
>J (nir)2
[(nir)4+(2rT)22] (III55)
n=l
Solutions for T as a function of umax and L may be obtained by
substituting directly for K2 in (III55) using (III41) and (III42).
The resulting expressions are as follows:
(cosh 2 VT7 cos 2 V"_7)
= 'u2 T T'2 z z
x T max z (cosh vi cos I I)2
z z
> (nn)'2 (III56)
n [(n) 4+(2rTz)212
n=1
and,
iT L2T'2 (cosh 2 /rT cos 2 /vT')
= Z Z Z
x T T (cosh /T7 cos /'TT)2
(nm)2 (III57)
[(nr)4 + (2Tz)22
n=1
Equation (11156) is considered to be the most useful and descrip
tive of the solutions presented for T in two dimensional oscillatory
shear flow. Therefore, to facilitate a discussion of these results in
the next section, (11156) is nondimensionalized by defining
T
E x (III58)
x 2T
max
50
and restating it as
ST'2(cosh 2 V T'z. cos 2 Vi )T'
E; =    .
X (cosh v/rTTz cos v rT')2
z z
(ns)2
n=1 [(nw)' + (2wTz)2]2 (III59)
IIIC Discussion of Two Dimensional Shear Flow Results
IIIC.1 Dispersion in Unidirectional Flow as a Limit of the Oscillatory
Flow Case
This discussion shall begin by relating the oscillatory flow disper
sion coefficient, T to its "corresponding" unidirectional flow
coefficient, EL, as has been previously done by Holley et al., Awaya,
and Fukuoka; and shall then proceed to show how the unidirectional process
is in fact a limiting case of an oscillatory process which has its own
distinct characteristics. Thus, solutions obtained in IIIB.3 for
T as a function of the pressure gradient and the surface amplitude
of the velocity are normalized by the "corresponding" solutions for EL
obtained in IIIA.3. Dividing (11155) by (11117) and (11156) by
(III18) yields
EL 4z (cosh 2 + cos 2 v f)
Z (n)2 (11160)
n=1 [(ni)" + (2T)2]2
for the same K, and
T 945irT (cosh 2 V T' cos 2 / rT')
A I z z
EL 8 (cosh viT T cos / iT)2
z z
Z (nT)2
n=1 [(nT)4 + (27TT)2]2 (11161)
for the same umax. All calculations involving infinite summations for
two dimensional shear flow dispersion use 200 terms in each sum, well
within fourth place accuracy.
The expressions given by (11160) and (11161) are plotted in Figure
3 vs. the nondimensional time 1/Tl corresponding to T' defined by
z
Holley et al. Also included in Figure 3 is the solution obtained by
Holley et al. for T/EL for the oscillatory linear velocity profile
discussed in IB.2. It is noted that all three solutions approach the
limiting value of onehalf for situations when the period of oscillation
is greater than the vertical mixing time in accordance with the findings
of Okubo and Bowden. However, the difference in behavior of the two
solutions given by (III60) and (III61) is dramatic and immediately
raises the question of whether or not a plot such as Figure 3 is the
most meaningful method of illustrating the characteristics of dispersive
mass transport in oscillatory flow. For the case of the pressure
gradient normalized solutions the ratio T/EL decreases rapidly
with increasing Tz because:
(1) For unidirectional flow the pressure gradient and viscous
forces are in equilibrium so as to produce a constant unidirec
tional shear flow for the dispersive transport of substance.
(2) In oscillatory flow the pressure gradient is assumed to be
simple harmonic in time and therefore is in constant balance
with the varying inertia and friction forces. As the period
of oscillation decreases the inertial effects become very large
such that in the limiting case little or no flow would be
induced. This results in little or no dispersive transport. This
effect of the period of oscillation on velocity can be seen
from (III23).
The solutions normalized by umax exhibit a much slower decrease of
T/EL with increasing T' simply because in this case the surface
A 1 L t
I0" I0" 10
T' TK
h2
FIGURE 3 NONDIMENSIONAL DISPERSION COEFFICIENT
T L'
velocities of the unidirectional and oscillatory flow are required to be
the same. In terms of pressure gradients this would correspond to the
situation where
Kosc >> Kunidir
Based upon these results it becomes apparent that due to the very
different hydrodynamic characteristics of unidirectional and oscillatory
flow systems, the interpretation of the nature of dispersive mass trans
port in an oscillatory system as compared to the same process in a uni
directional system is misleading and can vary widely depending upon how
one chooses to relate the two systems. This is demonstrated by Figure 4.
In this figure the longitudinal dispersion coefficient, T' as given
by (11156) is plotted against the vertical mixing time, Tcz, for four
periods of oscillation ranging from 22,536 to 223,560 seconds and
a umax of 1 ft/sec. Also plotted is the solution for the unidirectional
flow coefficient, EL, as given by (III18). As shown by Figure 4 the
behavior of EL and T are clearly different. The unidirectional flow
coefficient varies directly with the vertical mixing time which for little
or no turbulent mixing over the water column allows the shear flow to
transport higher concentrations of substance far downstream. The behavior
of the oscillatory flow coefficient, however, is governed by a type of
resonant interaction between the period of oscillation and the vertical
mixing time. As the period of oscillation is increased the resonant
peak shifts to the right and likewise increases until the limiting case
is reached where the peak is infinitely large and the values predicted
for T are exactly onehalf those predicted for the unidirectional
flow coefficient EL.
 600
..D A
 5I
S500 ,
400 I
400
C 81
0 I l \
10 102 104 10106 10
c 300 447 (sec)sec
^ (Semi Diurnal Tide) I
FIGURE 4 DISPERSION IN UNIDIRECTIONAL FLOW AS A LIMIT
OF THE OSCILLATORY FLOW CASE
200
100
10 102 10o 104 10e 10e 107
FIGURE 4 DISPERSION IN UNIDIRECTIONAL FLOW AS A LIMIT
OF THE OSCILLATORY FLOW CASE
The resonant characteristic of T may also be used to explain
the abrupt decrease in the ratio T/EL as shown in Figure 3. The
value of T' at which the knee of the curve occurs in Figure 3 corresponds
to the value of Tcz in Figure 4 at which T for a given T begins to
diverge from the unidirectional solution. At this point EL continues
to increase whereas T peaks and begins to decrease.
IIIC.2 Characteristics of Dispersive Mass Transport in Oscillatory Flows
The physical reasoning behind the resonant behavior of
straightforward and is begun by considering the extreme cases of T << Tc
and T >> Tcz:
(1) T <
small during one period of oscillation that an elemental volume initially
residing in the water column at elevation z, and containing an initial
concentration c, would remain at this elevation thus being transported
over the closed pathline of flow, and returned to its initial position
with no net longitudinal dispersion having occurred. Conversely,in a
unidirectional flow with little or no vertical mixing the longitudinal
dispersion would be very large.
(2) T >> Tcz Here the rate of vertical mixing is so rapid that
there is no time for the velocity shear profile to transport the sub
stance longitudinally before it loses its identity through vertical
mixing. In this case the oscillatory and unidirectional flow dispersive
processes behave in a similar manner and are both small.
Thus, it is seen that for an oscillatory flow the longitudinal
dispersive mass transport becomes small for both T << T and T >> T,
cz cz
whereas for a unidirectional flow the longitudinal dispersive mass trans
port varies directly with Tcz. This resonant nature of T can be
further explained on physical grounds by beginning with Case 1 above and
gradually allowing the vertical mixing time to decrease. In this example
the velocity shear profile in conjunction with the increased vertical
mixing causes a net longitudinal mass transport to occur over a period
of oscillation. This effect continues to increase as Tcz is decreased
until the optimum ratio between the vertical mixing time and the period of
oscillation is reached. At this point the longitudinal dispersive mass
transport has reached its maximum value. A further decrease in Tcz
begins to introduce the effect noted in Case 2 where the vertical
mixing is now too rapid thus causing T to decrease.
Additional physical insight into the behavior of T can be ob
tained by considering the relative longitudinal dispersion associated
with two oscillatory flows, one having twice the period of oscillation
of the other. The two periods considered are 44,712 and 89,424 seconds
corresponding to the periods of a semidiurnal and diurnal tide
respectively.
The first of the two comparisons is illustrated by Figures 5a and
5b. Once again the expression used for T is given by (III56) with
umax = 1 ft/sec for both the semidiurnal and diurnal cases.
As shown in Figure 5a after an elapsed time of t = 89,424 sec the
semidiurnal tide will have completed two full cycles of excursion
length L,, while the diurnal tide will have completed one full cycle of
excursion length L2 = 2L,. In Figure 5b the favorable effect of the
longer excursion length on longitudinal dispersion is seen for flows of
different periods of oscillation with the same umax. The second compari
son between the semidiurnal and diurnal period flows uses excursion
length as a normalizing parameter and is illustrated by Figures 6a and
58
x+
______
T=To; Umx I ft/sec
ST=2To Uax = I ft/sec
L2= 2L.
(a.)
Tcz = h (sec)
(b.)
FIGURE 5 COMPARISON OF DISPERSION FOR DIURNAL AND SEMIDIURNAL
PERIODS OF OSCILLATION WITH SAME umax
6b. In this comparison the velocity for the diurnal tide is scaled
downward so as to produce the same excursion length as the semidiurnal
tide. Figure 6a shows that after an elapsed time of t = 89,424 sec the
semidiurnal tide will have completed two full cycles of excursion
length Lias before. However, the diurnal tide, because of the reduced
velocity, has completed one full cycle of the same excursion length.
Values of T are plotted vs. Tcz using (III57) in Figure 6b. As
seen from this plot the semidiurnal dispersion has remained unchanged
from Figure 5b whereas the diurnal curve is now everywhere less than
the semidiurnal curve. This is not surprising since from (III57) the
fixing of excursion length and vertical mixing time leaves only the
decrease in the period of oscillation as a means of increasing longitudinal
dispersion.
The results of this analysis on the predicted behavior of T
in two dimensional oscillatory shear flow are summarized in non
dimensional form by Figure 7. In this figure the nondimensional
longitudinal dispersion coefficient as defined by (III59) is plotted
vs. Tz. Thus, by knowing the period of oscillation, the surface
amplitude of velocity, the water depth, and the vertical eddy diffusi
vity, one could use Figure 7 to obtain a predicted value for the
longitudinal dispersion coefficient. The resonant value of Tz for
which T is a maximum is seen from Figure 7 to be
T' = 1.58 (1162)
This relationship is applicable to all oscillatory flows in in
finitely wide rectangular channels provided the assumptions made in this
analysis are reasonably well satisfied.
A look at the kinematic structure of the flow field and the
T = To; L To/ Umax
+
T= 2To; L=TO/r Umax
itllf/if /_/l//ff/F//////lrr ^
10 I
106 I0.
FIGURE 6 COMPARISON OF DISPERSION FOR DIURNAL AND SEMIDIURNAL
PERIODS OF OSCILLATION WITH SAME SURFACE EXCURSION
LENGTH
61
350 I ii I I
300
250
x 150 
100
50
II
I T = 1.58
10 I0 10 I 10 102 l
Tc z
T
TZ T
FIGURE 7 NONDIMENSIONAL LONGITUDINAL DISPERSION COEFFICIENT
E;, AS A FUNCTION OF T' FOR TWO DIMENSIONAL
OSCILLATORY SHEAR FLOW
concentration distribution over depth is presented for three values of
Tz in Figures 8, 9, and 10. In these figures, nondimensional forms of
the solutions for u"(z,t) and c"(z,t) given by (III43) and (III50)
respectively are plotted over the nondimensional depth, z/h. The non
dimensional forms used are
UND Umax
C1 = (III64)
( )u T
DS) max
Each figure shows the u" and c" profiles for maximum flow and slack
water. Several interesting points are illustrated here. First, the
superior correlation of u" and c" over depth in Figure 9 when compared
to the same correlation in Figures 8 and 10 demonstrates qualitatively
why T is greater for T' = 1.58 than for the other values of T'
considered. Second, the phase dependency of the velocity with depth is
clearly shown in all figures near slack water with flow reversals
occurring in the lower momentum layers near the channel bed prior to
a shift in the upper higher momentum layers. Finally, it is interesting
to note that the shear profile is confined to a decreasingly thinner
layer near the bed with the upper portion of the profile becoming
flatter as T' is increased (increasing eddy viscosity by Reynolds
analogy).
analogy).
1.0 lI I 1.0
0.8 '0.8
NOD NO
0.6 CN / C  .6
z z.
0.4 / 0.4
0.2 .2
WlO 0.75 0.5 0.25 0 0.25 0.5 0.25 1.0 .1.0 0.75 0.5 0.25 0 0.25 0.5 0.75 1.0
I I II I I I D I I I I I I L I I
8 6 4 2 0 2 4 6 8 8 6 4 2 0 2 4 6 8
CN1 X 102 CND X 102
a. Maximum Flow (ot = (ot)max) b. Slack Water (crt = (t)max + 7'/2)
FIGURE 8 NONDIMENSIONAL VELOCITY AND CONCENTRATION VARIATION
COMPONENT PROFILES FOR T' = 0.5
z
U NO
"UNO
 CN' 
II
I
/
/
/I
iI
UND
CNo
/ I
/2/ //
/
K
10 0.75 05 0.25 0 0.25 0.5 0.75 1.0 1.0 0.75 05 0.25 0 0.25 0.5 0.75 1.0
8 6 4 2 CO 10 2 4 6 8 8 6 4 2 0 22 4 6
Maximum Flow b. Slk Wter
a. Maximum Flow ( 0t = (0t)max) b. Slack Water (ot = (0t)max + )
FIGURE 9 NONDIMENSIONAL VELOCITY AND CONCENTRATION VARIATION
COMPONENT PROFILES FOR T' = 1.58
Z
1.0
0.8
0.6
z
h
0.4
0.2
0,
I I I '
I
I I
1.0 \ 1.0
\ II
0.8 0.8
uND \ UNO /
0.6 CND. N C' / 0.6
z / z
h 0.4 / 0.4h
0.2 l 0.2
/ /
O0 I I 0
1.0 075 0.5 0.25 0 0.25 05 075 1.0 1.0 075 0.5 0.25 0 025 0.5 0.75 1.0
U" U "
I I I I IND I I I I I I IND I I I
8 6 4 2 0 2 4 6 8 8 6 4 2 0 2 4 6 8
CN, X 10 CN X 102
a. Maximum Flow (ot = (ot)max) b. Slack Water (oat = (ot)max + "/2)
FIGURE 10 NONDIMENSIONAL VELOCITY AND CONCENTRATION VARIATION
COMPONENT PROFILES FOR T' = 5.0
z
CHAPTER IV
DISPERSIVE MASS TRANSPORT IN RECTANGULAR
CHANNELS OF FINITE WIDTH
IVA Oscillatory Flow
The physical problem and the general analytical approach used to
solve it are the same in this chapter as those discussed in Chapter III
with the added feature of a finite channel width. The effect of this
feature is to introduce a three dimensional shear flow with vertical and
transverse velocity variations which are used to force the transport
diffusion equation to obtain a solution for c"(y,z,t). Solutions for
u"(y,z,t) and c"(y,z,t) are then correlated over the flow cross section
and, in the case of oscillatory flow, averaged over the period of
oscillation to obtain the longitudinal dispersive mass transport.
The coordinate system selected has its origin at the center of a
rectangular section whose sides correspond to z = h, and y = w/2. As
shown in Figure 11, the open rectangular channel is mathematically
represented by the lower half of the section. Selection of the coordinate
system in this manner preserves the symmetry of the problem about the
origin and, as will be shown later, correctly predicts the same dispersive
mass transport for equal degrees of skewness in channel geometry in either
the vertical or lateral directions.
z=h
u(y, z,t)
zz
y=w/2 y=w/2
^ /^ y
Iy=w 20
z=h ,,ry=w/2
z=h
FIGURE 11. DEFINITION SKETCH FOR THREE
DIMENSIONAL SHEAR FLOW
IVA.1 Velocity Distribution
Using the same assumptions that were made in Chapter III regarding
eddy viscosity, time averaging of turbulent fluctuations, and the form
of the pressure gradient term, the equation of motion for uniform flow
in the x direction may be written as
au iot 2u + 2u
= Ke +E +e (IV1)
at z z2 3y2
If a solution of the form
u(y,z,t) = us(y,z) eiat (IV2)
is assumed to be valid for (IV1) then the three dimensional shear flow
boundary value problem for oscillatory flow may be formally stated as:
a2u a2u
PDE: E + E iau = K (IV3)
Sz2 y aSy s
BC's: u (y,h) = 0
us( ,z) = 0
It is noted that physical considerations require that u (y,z) be
an even function in y and z. Thus, a solution to (IV3) of the form
us(y,z) a' cos (1 (n 2m+1 1) y (IV4)
m=0 n=O
is assumed. An examination of (IV4) with the statement of the boundary
value problem above shows that the assumed solution satisfies the no
slip conditions on the perimeter of the section. Moreover, (IV4) also
satisfies the condition of zero vertical shear at the free surface, and
zero lateral shear at the centerline of the channel. The assumed solution
is then substituted into (IV3) and the Fourier coefficients, amn, are
determined in the usual manner through the use of the orthogonal proper
ties of the cosine function. Details of this analysis are presented in
Appendix B. Solving for amn and thus determining the form of us(y,z),
the complete solution for the velocity function u(y,z,t) using (IV4)
and (IV2) is then
Z V m+nc (16K2m+1)6rz s( (2n+l)gTy igt
u(y,z,t) = 216K( hmc 2 cos w w (IV5)
j (2m+1) ( rCo [2m+1h2 + C (2n+1 )T,
m=O n=O (2m+1)(2n+l)Tr{ E 2i 1 2 + + io}
To obtain the spatial variation, u"(y,z,t), the crosssectional mean
velocity, u(t), must be subtracted out of (IV5). The mean component of
u is defined by 2
iot
u(t) = e2h us(y,z) dy dz (IV6)
h w/2
Performing the integration in (IV6) and subtracting the result from
(IV5) yields
u"(y,z,t) = n {cos (2m+)z cos(2n+1)y g(m,n)}eit (IV7)
m=O n=O
where,
m+n
g(m,n) = 4(1)m
'(2m+1)(2n+1)
and,
16K (1)m+n
amn n 2(2m+1)(2n+l) { z [(2E 12 + Ey 2n+ 1 i
(IV8)
(IV9)
Since only the real portions of these expressions have any physical
meaning, (IV7) is transformed to polar form as was done in Chapter III
for the two dimensional case. In addition, it will be assumed once again
for simplification of results that Reynolds analogy applies thereby
allowing the introduction of the vertical mixing time Tcz and the defini
tion of a comparable lateral mixing time as
T = (/2)2
cy K
y
The solution for u"(y,z,t) as a function of the pressure
then be written as
u(yzt) 8K ()m+n ei(ctai(m,n))
u"(y,z,t) = 7.)
m = 0 n = 0 (2 m+1)(2 n+l) R0 (m,n)
(2 m+)z cos (2 n+l)y g(mn)
(IV10)
gradient may
K
(IV11)
where,
T m+l)2 T +I +)712)
R1(m,n) = T [ (2 m+1)12 2 T E (2 n+l1)72 + 1
1 c cy
al(m,n) = tan { [(2 mlh12 T [ ( nlL 1
T(mn) = tan(T 2 m+l)721 T (2 n+1)Tn 2
2T ^ 2 2] T T L 2 J
(IV12)
(IV13)
In order to put (IV11) in a more usable form, (IV5) is used to
obtain a relationship between the pressure gradient, K, and umax, where
umax is now defined as the amplitude of the periodic velocity function at
the free surface centerline. Thus, from (IV5)
u(0,0,t) = .
m = 0 n= 0
16 K()m+ ei (14)
T2(2 m+l)(2 n+1) {Ez[2 m+) 2+ [(2 n+l)l 2 i w
which upon maximization with repsect to time yields the relationship
umaxir3
K max (IV15)
8T
where,
S = ,)m+n(R(m,n)l)
m 0 n 0 (2 m+l)(2 n+l)Rl(m,n
+ ()m+n (IV16)
m = 0 n = (2 m+l)(2 n+l) R(m,n)
The details of developing (IV15) from (IV14) are presented in Appendix
C. Substitution of (IV15) into (IV11) for K yields the following
expression for u" as a function of umax
metUm)max)
u"(y,z,t) umax (Im+ni(t(mn))
m = 0 n = 0 (2 m+1)(2 n+l)R (m,n)
cos (2 m+) cos (2 n+l)y g(m,n) (IV17)
2h w
IVA.2 Concentration Distribution
The form of the transport diffusion equation used to solve for
c"(y,z,t) is identical to (1127), or
c" a2cl' a2c a3 2
K z K = u" (IV18)
at z z2 Y y2 aT
where u"(y,z,t) is given by (IV7). It is assumed that c"(y,z,t) is of
the form
c"(y,z,t) = c" (y,z) eiot (IV19)
s
where c"(y,z) is complex following the same approach used with u"(y,z,t).
Substituting (IV19) into (IV18) and requiring zero diffusive flux across
the boundaries of the flow, the boundary value problem is then stated as
a2c" 92c"
PDE: K s + K ic" = u" (IV20)
z @2 y 9y2 s s 9E
ac"
BC's: I = 0
z =h
y = w/2
Now, c" must also be an even function in y and z based on physical
arguments and is therefore assumed to be of the form
c (y,z) = a'k COST z cos 2kY (IV21)
Z=0 k=0
Equation (IV21) is seen to satisfy the conditions of zero diffusive flux
across the flow boundaries. Substituting for c" using (IV21) and for
u" using (IV4) and (IV7), (IV20) becomes
Kz ()2 + ( )2+ ic} a cos z cos 2 ry
Z (Lh. y w Zk h w
a =0 k=O
7 (2 m+l),z (2 n+l) g(m,n))
( ) amn {cos (2 m+1),z cos ( n 1)ny
m = 0 n = 0 (IV22)
where amn is given by (IV9). The unknown Fourier coefficients ask are
solved for in the same manner as was used to determine amn. The detailed
mathematics of the solution are presented in Appendix B, the result of
which is given by
at 256K 2i (IV23)
aik = 4{_ 2 ) 2 O 2
Kz() y+ K( + i p = 0 q = 0
(1)(P+q)
[(2 p+1) (2Z)2][(2 q+l)2(2k)2]{[E (2 ]2 r(2 q+1z12 + io}
Thus, the solution for c"(y,z,t) includes four infinite summations
arising from the nonorthogonal nature of the cosine functions
representing u"(y,z) and c'(y,z). Again introducing Reynolds analogy
and the expression for the vertical and transverse mixing times, the
solution for c"(y,z,t) in polar form as a function of the pressure
gradient is
c"(y,z,t) = (c) 64KT
= 0 k = 0p= 0 q =0
pk()(+k)cos _z cos 2kRy e ie2
h w (IV24)
[(2 p+1)2(2P)2][(2 q+1)2(2k)2]R (p,q)R (l,k)
where,
68 = at a((p,q) az(k,k) (IV25)
R,(p,q) is given by (IV12) with p replacing m and q
replacing n.
R,(,k) = {T (T)2 + 2T (k7)2}2 + 1 (IV26)
cz cy
ac(p,q) is given in (IV13) with p replacing m and q
replacing n.
c2(Z,k) = tan' { T 1 T (IV27)
2 F 2rT (kiT)2
cz cy
(0, k = =
=k = , k = 0 or R = 0 (IV28)
1, k f 0,, 0
The factorPZik is necessarily included in (IV24) to properly account
for the a6k and a'o terms, and to insure that c"(y,z,t) has a zero mean
over the flow crosssectional area. Detailed mathematics for arriving
at (IV24) are given in Appendix B.
The expression of c"(y,z,t) as a function of umax is determined by
substitution for K in (IV24) using (IV15) to yield,
c"(y,z,t) = ( aI 8
k=0 k=0 p=O q=0
S(l)(+k)cos 7z cos 2ktry ei,2
h w (IV29)
[(2 p+1)2(2)2][(2 q+1)2(2k)2]R (p,q)R,(R,k)
IVA.3 Dispersive Mass Transport
The time mean dispersive mass transport averaged over one period of
oscillation for the full rectangular section as shown in Figure 11 is
given by
w/2
T = R R[U"(y,z,t)].Re["(y,z,t)] dz dy > T (IV30)
w/2 h
The integration in (IV30) are carried out in Appendix B and it is noted
that the nonorthogonality of the cosine functions representing u" and c",
arising from the form of their respective arguments, once again produces
two additional infinite summations. Thus, using (IV11) and (IV24) for
u" and c" the following expression for the longitudinal dispersive mass
transport over the region h < z < h and w/2 < y < w/2 is determined
= 2048 wh K2T>
S0 k= 0 m= 0 n= 0 p 0 q = 0
Upk{(R2(o,k)l) cos X + sin X}
61( ,k,m,n,p,q) (IV
where,
X = al(m,n) al(p,q) (IV32)
61(a,k,m,n,p,q) = [(2p+l)2(22)2][(2q+1)2(2k)2][(2n+l)2 (2k)2]
(IV33)
[(2m+l ) _(2P)2] R I 1(m,n)R'lj'(p,q)R2(P,k)
Now the dispersive mass transport associated with the open rectangu
lar channel comprising the lower half of the full box is T. Therefore,
the Fickian flux representation of the dispersive mass transport in the
open rectangular channel is simply
 (wh) c= (IV34)
Replacing T in (IV34) with the right hand side of (IV31) and solving
for T yields
T 1024K2T3 Z z 7
T ZTr Z Z
=0 k= 0 m= 0 n= 0 p = 0 q = 0
uZk{(R2(,k)l)2 cos X + sin X}
(IV35)
61(Z, k,m,n,p,q)
The longitudinal dispersion coefficient is expressed as a function
of umax by substituting for K in (IV35) using (IV15) to give
T max
C=O k=O m=O n=O p=O q=O
T l = 0 k = 0 m = 0 n = 0 p = 0 q = 0
PZk(R2(z,k)l)2 cos A + sin X}
(IV36)
61(Z,k,m,n,p,q)
To facilitate the presentation of results and the application of
(IV36) for predictive purposes a nondimensional relative mixing time,
Tc, is introduced as
T' T
Tc =T' (IV37)
y cy
where,
T
T' = cy (IV38)
y T
Thus, as shown by (IV37), T' is a measure of the relative effects of
vertical and lateral shear for a given widthtodepth ratio, and vertical
and lateral eddy diffusivities. With the aid of (IV37) the following
definitions can then be made:
ri(m,n) = (2nTI )i(m,n) = {[ 2m + T [ ]2}+(2T)2 (IV39)
r2(z,k) = (2TTT)2R2(Z,k) = {( +)2 +Tc(ki)2}2+(21T')2 (IV40)
c,(m,n) = tan [2ml 2]2+Ti[ ]2 (IV41)
2
2 ()m+n [rl(m,n)(2,Tz)2 1
(2 2= 2iT (2m+l)(2n+l)r1(m,n) I
m = O n = 0
+ 21z T 2m+l (2n r(mn (IV42)
z2aT (2m+l)(n Zn)r)1 ,)
m =0 n= 0
These relationships are then applied to (IV36) to obtain an expression
for the nondimensional longitudinal dispersion coefficient, E defined
by (11158). This expression represents the final form of the solution
for the case of three dimensional oscillatory shear flow and is there
fore presented along with a summary of the associated terms as follows:
T 32T
max = 0 k = m = 0 n = p = q =
sk{[r2(Z,k)(2Tz )2] cos x + (2rT') sin x}
z(IV43)
62(Z,k,m,n,p,q)
where,
62(k,k,m,n,p,q) = (2rTz)461(t,k,m,n,p,q)
= [(2p+l)2(2Z)2][(2q+l)2(2k)2][(2m+l)2(2)2]
[(2n+l)2(2k)2]r(m,n)rpq)r( 2(,k) (IV44)
r1(m,n) {[(2m+1)1]2 T 2+ 2}2 + (2TT')
r1(p,q) = r1(m,n) with p replacing m and q replacing n
r2(.,k) ={(pr)2+Tc(k7)2}2 + (2iT )2
27rT'
1(m,n) = tan 'gi+l){
al(p,q) = al(m,n) with p replacing m and q replacing n
(l)m+n[rc(mn)(21T)2] 1 2
2 = (2m+l)(2n+l)r1(m,n)
+ 2 0 nz = (2m+I)(2n+l)r1(m,n)
m = 0 n =
S= a1(m,n) ca(p,q)
1 k f 0, 0
k! = k = 0 or Z = 0
0 k = 0
The symmetry of this solution can now be seen. If, for example,
T, had been inversely defined as Tcy /Tcz, then (IV36) would have
reduced to (IV43) with T' replacing T' everywhere, and the newly defined
y z
T' multiplying all complementary terms in rl, r2, and a,. This form of
(IV43) would predict the same value for E' as the original formulation
x
provided that u and T were the same and that the new T' equaled the
max y
old T' and the new T' equaled the old T'. Stated another way, for the
z c c
same umax and T, a channel whose halfwidth was twice its depth would
produce the same dispersive mass transport as a channel whose depth was
twice its halfwidth provided that the ratio K /K for the first case
equaled K /K in the second case.
The functional behavior of E' as given by (IV43) for three dimen
x
sional oscillatory shear flow is illustrated by Figure 12. In this
figure E' is plotted against the nondimensional vertical mixing time,
x
Tz, for fixed values of the relative mixing time, Tc. Because the
solution for E' contains six infinite summations nested in series some
x
limitations were necessary in carrying out the required computations.
By varying the upper bound on each of the sums it was demonstrated that
the solution converges upward to its limiting value. The convergence
occurs reasonably rapidly; however, above an upper limit of 10 terms for
each sum the rate of convergence is slowed considerably. The results
400
350 
300
=0.01
S= 1.0
250 5 / \\ c
Tc = 0.67
S= 0.33
200 c// \\
=0 0.1
x c
 X
w 0 0.0c
50 
T' X 0 10 and X1
Tc': 0.001
T0 0 o 
50
15)3 102 1(' 10 102
T :Tcz
TT =
FIGURE 12 NONDIMENSIONAL LONGITUDINAL DISPERSION COEFFICIENT
E', AS A FUNCTION OF Tz FOR FULL RANGE OF T' VALUES
x C
presented in Figure 12 and in subsequent figures were computed using an
upper bound of 10 on each sum; therefore, each value of E' includes
approximately 1.2 x 106 terms. An estimate of the error between the com
puted values of E' and the values predicted by the complete series
x
represented by (IV43) is obtained by comparing the limiting curve for a
very wide channel in Figure 12 with the corresponding curve for an
infinitely wide channel shown in Figure 7. The limiting curve in Figure
12 is identified by T' < 1 x 10.6 on the left hand side of the peak and
c 
by T' < 0.001 on the right hand side of the peak. A comparison of the two
C 6
curves suggests that for T' < 1 x 10 the solution for E' given by
ix
(IV43) very nearly approximates for all T' the solution given by (III59)
for E' in an infintely wide channel. The general shapes of the two curves
are nearly identical with both peaks occurring at T' = 1.58. For the
infinitely wide channel the peak value of Ex is 3.27 x 103 whereas for
the limiting curve in Figure 12 it is 3.07 x 103 or 6 per cent below
the infinitely wide case. Thus, it appears that the three dimensional
solution for E' approaches the two dimensional solution for E' in the
x x
limit as T' 0, and it is estimated that values predicted for E' in
c x
Figure 12 are low but generally within 10 per cent of their actual
values as determined by (IV43). This does not apply to the range
T' > 100 where the Figure 12 values are as much as 70 per cent low.
However, this is not a region of practical significance since values
of Tz in nature are generally <100.
To aid in the interpretation of results, the eight curves shown in
Figure 12 have been separated into two groups and are replotted in
Figures 13 and 14. Figure 13 shows the E' resonance curves for T' = 1.0,
S
0.67, 0.33, and 0.1 while Figure 14 shows the corresponding curves for
81
400 I I
350 
300
250 
200 
x
A D x
150
Tc' 0. I
100 Tc= 0.33
Tc'.= 0.67
50 Tc= 0
10" 102 I I I0 I02 103
Tcz
Tz T
FIGURE 13 NONDIMENSIONAL LONGITUDINAL DISPERSION COEFFICIENT
E' AS A FUNCTION OF T; FOR T' = 1.0, 0.67, 0.33
AND 0.1
To' 0 .
x
100 Tc' =0.01j
(C = 0.001\
50 TC' =0.0001
6
Tc' IXIO
103 102 101 I 01 102 03
S Tcz
Tz T
FIGURE 14 NONDIMENSIONAL LONGITUDINAL DISPERSION COEFFICIENT
E' AS A FUNCTION OF T' FOR T' = 0.1, 0.01, 0.001,
x z c
0.0001, and <1 x 106
6
T' = 0.1, 0.01, 0.001, 0.0001, and < 1 x 106. Note that the curve for
c
T' = 0.1 is repeated to provide a common reference. If for the purposes
c
of interpretation it is assumed that K = K then T' may be written as
z y c
T' = [ h]2 (IV45)
c (w/2r
which is simply the inverse square of the channel halfwidth to depth ratio.
Referring to Figures 12, 13, and 14 with the aid of (IV45) it is seen
that as the halfwidth of the channel is increased from a value equal to
the depth the resonant peak shifts from T' = 4.5 leftward toward Tz = 1.58
z
at which it occurs when T' < 0.001. Concurrent with the shift is an ini
C 
tial increase in the peak value of E' as T' is decreased from 1.0 to 0.25
x c
followed by a decrease in the peak value as Tc is further decreased to
0.001. Decreasing Tc below this value has no effect on EL for T' > 1.2.
c x z
A summary of the maximum value of E' achieved and the value of T' at which
x z
it occurs for the corresponding T' is given in Table 1. These data are
also presented graphically in Figures 15 and 16.
TABLE 1 PREDICTED RESONANT PEAK DATA FOR
LONGITUDINAL DISPERSION COEFFICIENT
IN THREE DIMENSIONAL OSCILLATORY SHEAR FLOW
NonDimensional Peak Value
S. Vertical Mixing of NonDimensional
Relative Mixing Time, T' at which Longitudinal Dispersion
Time, T' z Coefficient, E' x 105
c Maximum Occurs x
1.00 4.50 339
0.67 3.75 340
0.40 2.40 350
0.33 2.20 353
0.25 1.95 354
0.20 1.80 352.5
0.10 1.65 346
0.01 1.60 325
0.001 1.58 310
0.0001 1.58 307
0.00001 1.58 307
0.000001 1.58 307
84
4.0
3.0
Tz 2.0
.2 0T z = 1.58
1.0
0 10I 102 103 i0o 10 5
Td
FIGURE 15 NONDIMENSIONAL VERTICAL MIXING TIME, T;, AT
WHICH MAXIMUM E' OCCURS AS A FUNCTION OF T'
x C
360
o 340
0
x
Li
FIGURE 16 PEAK E' AS A FUNCTION OF T'
x C

