• TABLE OF CONTENTS
HIDE
 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














Title: Dispersive mass transport in oscillatory and unidirectional flows
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Permanent Link: http://ufdc.ufl.edu/UF00098181/00001
 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 )
One-dimensional flow   ( lcsh )
Civil and Coastal Engineering thesis Ph. D   ( lcsh )
Dissertations, Academic -- Civil and Coastal Engineering -- UF   ( lcsh )
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis -- University of FLorida.
Bibliography: Bibliography: leaves 140-142.
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
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        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
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        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
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        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
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        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
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        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/370-165.









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 . . . . . . . . . . . .


I-A General Background . .
I-B Discussion of Previous Work
Mass Transport . . .


on D spearss .ve
on Dispersive
. . . . . . .


I-B.1 Unidirectional Flow . . . . . .
I-B.2 Oscillatory Flow . . . . . . . .

I-C Description and Scope of the Present Work . . .

II CONSIDERATIONS FOR APPLICATION OF TAYLOR'S
DISPERSION ANALYSIS TO FREE SURFACE FLOWS. . . . .


II-A General . . . .
II-B Unidirectional Flow . .
II-C Oscillatory Flow . . .
II-D Applications of Dispersion
by Taylor's Method . .


22


. . . . . 22
. . . . . . 27
. . . . . . 30
Coefficients Predicted
. . . . . . 33


III DISPERSIVE MASS TRANSPORT IN AN INFINITELY
WIDE RECTANGULAR CHANNEL . . . . . . . . .

III-A Steady Unidirectional Flow . . . . . .


III-A.1 Velocity Distribution. .
III-A.2 Concentration Distribution
III-A.3 Dispersive Mass Transport.


III-B Oscillatory Flow. . . . . . . . . .


III-B.1 Velocity Distribution. .
III-B.2 Concentration Distribution
III-B.3 Dispersive Mass Transport.


III-C Discussion of Two Dimensional Shear Flow Results.

III-C.I Dispersion in Unidirectional Flow as
a Limit of the Oscillatory Flow Case . .
III-C.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

IV-A Oscillatory Flow. . . . . . . . ... 66

IV-A.1 Velocity Distribution . . . . .. 67
IV-A.2 Concentration Distribution. . . . .. 71
IV-A.3 Dispersive Mass Transport . . . ... 74

IV-B Steady Unidirectional Flow . . . . ... 90

IV-B.1 Velocity Distribution . . . . .. 90
IV-B.2 Concentration Distribution. . . . ... 93
IV-B.3 Dispersive Mass Transport . . . ... 95

V COMPARISON OF RESULTS WITH FIELD AND 100
LABORATORY DATA . . . . . . . . . . .

V-A Unidirectional Flow. . . . . . . . ... 100
V-B Oscillatory Flow . . . . . . . .... 106

VI SUMMARY AND CONCLUSIONS . . . . . . .... 116

APPENDIX

A DERIVATION OF NORMALIZING EXPRESSIONS FOR TWO DIMENSIONAL
OSCILLATORY SHEAR FLOW SOLUTION . . . . .... 120

A-1 Surface Amplitude of Velocity, umax . . . .. 120
A-2 Surface Excursion Length, L . . . . . . 124

B MATHEMATICAL DETAILS FOR THREE DIMENSIONAL
OSCILLATORY SHEAR FLOW DISPERSION . . . . .... 125

B-1 Velocity Solution . . . . . . . .. 125
B-2 Concentration Solution . . . . . . .. 128
B-3 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 WIDTH-TO-DEPTH RATIOS FOR APPLICATION
OF THE TWO DIMENSIONAL OSCILLATORY SHEAR FLOW
DISPERSION COEFFICIENT. . . . . . . . 87

3 OBSERVED DISPERSION DATA AND PREDICTED T
VALUES USING (111-59) . . . . . . .... 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 NON-DIMENSIONAL 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 SEMI-DIURNAL
PERIODS OF OSCILLATION WITH SAME umax . . . ... 58

6 COMPARISON OF DISPERSION FOR DIURNAL AND SEMI-DIURNAL
PERIODS OF OSCILLATION WITH SAME SURFACE EXCURSION
LENGTH . . . . . . . . . .. . 60

7 NON-DIMENSIONAL LONGITUDINAL DISPERSION COEFFICIENT
E', AS A FUNCTION OF T' FOR TWO DIMENSIONAL
x z
OSCILLATORY SHEAR FLOW . . . . . . . .. 61

8 NON-DIMENSIONAL VELOCITY AND CONCENTRATION VARIATION
COMPONENT PROFILES FOR T' = 0.5. . . . . . 63

9 NON-DIMENSIONAL VELOCITY AND CONCENTRATION VARIATION
COMPONENT PROFILES FOR T' = 1.58 . . . . . 64

10 NON-DIMENSIONAL VELOCITY AND CONCENTRATION VARIATION
COMPONENT PROFILES FOR T' = 5.0. . . . . . 65

11 DEFINITION SKETCH FOR THREE DIMENSIONAL SHEAR FLOW 67

12 NON-DIMENSIONAL LONGITUDINAL DISPERSION COEFFICIENT
E',AS A FUNCTION OF T' FOR FULL RANGE OF T' VALUES ... 79
x z C
13 NON-DIMENSIONAL LONGITUDINAL DISPERSION COEFFICIENT
E', AS A FUNCTION OF T' FOR T' = 1.0, 0.67, 0.33
xz c
AND 0.1. . . . . . . . . . . . 81

14 NON-DIMENSIONAL 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 10-6. . . . .............. 82










LIST OF FIGURES (Continued)


Figure Page

15 NON-DIMENSIONAL 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 NON-DIMENSIONAL 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 cross-sectional 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 cross-sectional
mean value

c" Separation variable representing spatially dependent
s portion of c"

cND Non-dimensional 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' Non-dimensional time averaged longitudinal dispersion
x coefficient for oscillatory flow

T Longitudinal dispersion coefficient averaged over
period of oscillation

E' Non-dimensional longitudinal dispersion coefficient
for unidirectional flow

f Summation index

fDW Darcy-Weisbach 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 cross-sectional value of K
z z
K Mean cross-sectional 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 Semi-diurnal tidal period, 44,712 sec.

T' Non-dimensional 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' Non-dimensional lateral mixing time
y
T' Non-dimensional 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 cross-sectional mean value









LIST OF SYMBOLS AND ABBREVIATIONS (Continued)


UND Non-dimensional 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 Non-dimensional 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 non-dimensional 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

I-A 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 cross-sectional mean concentration

of substance by the mean cross-sectional 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.









I-B Discussion of Previous Work on
Dispersive Mass Transport

I-B.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 (I-1)

u = u + u" (1-2)

where u is the cross-sectional 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 ) (1-3)
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 (1-3) is integrated over the cross-sectional area of the tube

and the concentration, c, decomposed into its cross-sectional mean and

variation component, by the expression

c = E + c" (1-4)

then (1-3) reduces to

- T (I-5)


For standardization throughout the text overbars shall be considered

to represent cross-sectional averaging whereas the symbol < >T shall be

considered to represent temporal averaging over the periodic interval T.

The form of (1-5) led Taylor to define the convective transport shown as

a Fickian flux which he called dispersion, or


E 1 u"c"dA (1-6)
EL


where EL is the longitudinal dispersion coefficient for steady unidirec-

tional flow and A is the flow cross-sectional area. As indicated by (1-6)

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 (1-3), decomposed c by (1-4)

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 (1-3) to

u r -3c ) (1-7)


which Taylor then solved for c", performed the integration of u"c" in

(1-6) and produced the following expression for the longitudinal disper-

sion coefficient

a2u2
E max (1-8)
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 (1-6) for EL, (1-5) may then be written as

D ELC 2 c (1-9)
t ) = L S2


Equation (1-9), with EL given by (I-8), shows that the cross-sectional 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 (1-9) the time scale of convective

transport associated with u" must be significantly greater than the time

scale of cross-sectional 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) (1-10)



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 (1-7) he obtained a solution for c" and numerically performed the

integration of u"c" shown in (1-6). The result was the following expression

for the longitudinal dispersion coefficient in turbulent pipe flow

EL = 10.06 a u, (1-11)


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

(1-11), the corrected dispersion coefficient becomes

EL = 10.1 a u, (1-12)

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 cross-sectional 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

cross-sectional 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 cross-sectional 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 cross-sectional 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 (1-13)

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 cross-sectional 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

cross-sectional 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' (1-13)
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 (1-8),









that in either laminar or turbulent unidirectional shear flow the longi-

tudinal dispersion varies directly with the time scale of cross-sectional

mixing. To obtain an expression for u" Elder assumed the velocity distri-

bution to be logarithmic which upon substitution into (1-13) and correcting

for longitudinal eddy diffusion as done by Taylor, yields


EL = 5.93 uh (1-14)

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 (1-12) and (1-14) 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 cross-sectional 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-

to-depth 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 (1-15)

where y is the coordinate axis across the channel.

Fischer then integrated (1-15) over the cross-sectional 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 (I-16)
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 (1-17)
0

To obtain a more useful expression for EL Fischer defines a Lagrangian

time scale for cross-sectional 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 cross-sectional 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, (1-18)

Fischer arrives at an alternate expression for EL given by

2
EL = 0.3 u" (1-19)
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 width-to-depth
ratios ranging from 9 to 15.7 Fischer obtained good agreement between

predicted and observed values of EL using (1-16). However, (1-19) 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. Width-to-depth ratios for these sites range from 15 to 62 with

most being less than 40. To carry out his analysis Fischer numerically

integrated (1-16) 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 non-uniform 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 (1-16) and (1-19) 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.

I-B.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 (1-20)

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 (1-21)

conditional upon satisfying the transport diffusion equation expressed as

( ) + u" (Kz c) (1-22)
(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 one-half 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 cross-sectional 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) (1-23)

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 1-1/2 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 (1-23) 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 (1-24)

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, -) (1-25)
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) + (- ) (I-26)


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, (1-26) 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 ( (1-27)
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 (1-28)

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, =-" (1-29)

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 (2n-li)2{[ (2n-1)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 (1-28). Solving for c" using

the steady state form of (1-29) and proceeding in the usual manner they

obtain an expression for T which they call E It is shown that E.

is one-half 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 (1-30) was normalized by E, and the result plotted against
TK
the non-dimensional 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 one-half 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 (1-31)

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 non-dimensional 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 (1-32)

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

(1-29) 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 cross-sectional 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.








I-C 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

II-A 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(I-l)
+ u K K(11-1)
at 3x y y2 Z az2

After Taylor (1,2), transform (II-1) to a coordinate system traveling

with the cross-sectional mean velocity using

u = u + u" (11-2)

S= x ut (11-3)

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 (11-3),


(11-4)



(11-5)


Equations (11-2), (11-4),
yield


and (11-5) are then substituted into (II-1) to


+ C 1 ac 32C 92C
at)c + ull K y K Z@Z
K-+Ky2 a2


(11-6)


Next, decompose c into its cross-sectional mean and variation components
by the expression


c = c + c"

and substitute into (11-6). For uniform flow this produces


(11-7)


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 cross-sectional velocity.
Proceeding with the analysis, (11-8) 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 (11-9)
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 (11-9) states
a__ ac" a c" ac" c
'(K -) + z(K dA = dzdy} (II-10)
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

(II-10) is equal to zero. Using this result, assumption (1), and noting
that

u" dA = c" dA = 0
A A

by definition, (11-9) is reduced to

( = (u ) (II-11)


As noted in I.B.1 the form of (II-11) induced Taylor to define the spa-

tially averaged convective flux u"c" as a Fickian flux thus transforming

(II-11) to the one dimensional heat equation form

(-C) = EL (11-12)

where,

Su"c" (11-13)
EL _c
ac









Referring again to I.B.I, Aris showed that (11-12) and (11-13) 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.








II-B Unidirectional Flow

To obtain an expression for EL as defined by (11-13) 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 (11-8) to obtain a

solution for c" which is then correlated with u" according to (11-13) for

the determination of the longitudinal dispersion coefficient. As stated

in I.B.1 Taylor reduced an expression similar to (II-8) 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 (11-13), and it will now be

demonstrated that assumptions (1) and (2) follow from these conditions.

For steady state conditions to exist (II-11) 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 (11-14)
A

To examine the functional dependence of c", let

U" = g2 (y,z)

3c"
= f2 (y,z,S)

Equation (11-14) may then be written as

0 = : g2 (y,z)-f2 (y,z,S) dA (11-15)
A


which upon integration yields

G() = 0 (11-16)
From (II-16) 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) (11-17)
The above result is then used to substantiate Taylor's second assump-

tion that is constant. To do this (11-8) is differentiated with

respect to E. For steady state conditions and uniform flow the result

is

32-= 0 (11-18)

which upon integration yields


j-= constant (11-19)
ot.


The application of (11-17) and (11-19) reduces (11-8) to










K + z = u"- (11-20)


where c is constant.
Thus it is shown that the definition of the longitudinal dispersion

coefficient as stated by (11-13), which implies the use of solutions for

u" and c" obtained from the governing equation of motion and (11-20)

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 (11-17) and

(11-19) follow from these conditions.








II-C Oscillatory Flow

The same analysis carried out in II-B 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 (II-11) over one full period of

oscillation which for a periodic c yields

<- (u"c > = 0 (11-21)

where C is now defined by


C = x- \ (t') dt' (11-22)
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

(11-21) may be reversed
h w/2 T
h w/2 5 u"c" dt dy dz = 0 (11-23)

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 (11-24)
0

Substituting (11-24) into (11-23)

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 (11-25)

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 II-B 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, (II-11) reduces to

( = 0 (11-26)

and (11-8) becomes

(at KKz 2 =- u" 3 (11-27)

Differentiating (11-27) 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 (11-27) with -constant is the

comparable form of (11-20) 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.









II-D 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 (11-12) indicates that in applying values of the dispersion

coefficient, steady state conditions for the cross-sectional 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 cross-sectional 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 (11-12).









CHAPTER III
DISPERSIVE MASS TRANSPORT IN AN INFINITELY
WIDE RECTANGULAR CHANNEL

III-A 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.

III-A.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 (III-1)
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 (III-2)
p Tx
where K is constant. Substituting (III-2) into (III-1) 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 (111-3)
z

BC's: u(0) = 0
du
Tu7 z=h = 0

The velocity shear profile is obtained by integration of (111-3) and

the application of the boundary conditions. The resulting expression is

u(z) = -- (z2 hz) (III-4)


The spatial variation of the velocity as defined by (11-2) is then

obtained by averaging (III-4) over depth and subtracting out the depth

mean component to yield

u"(z) = K( hz +2) (III-5)
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 cross-sectional 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 one-half 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 (111-5) in terms of umax is easily accomplished by

solving (111-4) for K at z = h, or
2e u
K = 2 max (111-6)
h

which upon substitution into (111-5) yields
2u 2 h2
u,,(z) max z2 h2
u z2 hz + ( ) (111-7)


III-A.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 (III-5),
to force the one dimensional form of (11-20). Thus, the concentration

variation boundary value problem may be stated as
DE: d2c" E
dzK u (111-8)

dc"
BC's: z = O,h dc"= 0
dz








where,

ac
= constant

u"(z) is given by (III-5).

Substituting for u", integrating once with respect to z, and applying the
boundary condition at z = 0 produces

dc" K BE z3 hz2 h2z (11-9)
d *- (- 6- -[ + ] (III-9)
dz -- (JL6 2 3
SzKz

It should be noted that (III-9) implicitly satisfies the boundary condi-
h
tion at z = h since by definition S u"(z) dz = 0. Equation (III-9) 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- (III-10)
c K 24 6 6 t5
zKz

Using (III-6) to substitute for K in (III-10) the expression for

c"(z) as a function of umax is

c"(z = a 4 hz3 h2z2 h'
c"(z) = max (T ) r-4- -6 + T- (III-ii)
K h2
z

III-A.3 Dispersive Mass Transport
The dispersive mass transport over the depth of flow is given by


S= S u"c"dz (III-12)









Substituting for u" and c" using (III-5) and (III-10) respectively,

(I1I-12) 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 (III-13)
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 (III-14)
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 (III-14) and

the introduction of the vertical mixing time defined by

T h2 (III-15)
K
z

Equation (111-13) may be written as
2K2T3 h
S( z4 (III-16)

Assuming next that the required conditions for defining a longitudinal

dispersion coefficient as discussed in II-B are satisfied, then

m
L a)h
(- -









2K2T'
cz
EL =~9 (III-17)

Using (III-6) and (111-15) in (111-17) the longitudinal dispersion

coefficient as a function of uax is

8u2 T
max cz
EL ----4 (111-18)

The functional form of EL given by (111-18) is the same as that

obtained by Taylor for laminar unidirectional flow in a tube,(I-8). In

both cases the longitudinal dispersion varies directly as the product

of the square of the velocity and the cross-sectional 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.









III-B Oscillatory Flow

III-B.1 Velocity Distribution

The same assumptions used in III-A 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 (III-19)
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 (III-19) is periodic and of the form
1 P KeiOt (III-20)
p ax

where K is constant, as before, and a is the angular frequency of oscilla-

tion. Note that only the real part of (III-20) 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 (III-21)

where u (z) is a complex function. The boundary value problem is then

formulated by incorporating (III-20) and (III-21) into (III-19) 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 (III-22)
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)(h-z) iat (111-23)
u(,t ac 1 cosh B(l+i)h

where,

B= =/ (111-24)

with only the real part of (111-23) 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 (111-23) 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 (111-23) 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)(h-z)} (111-25)
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 (111-25) will now be expanded in a Fourier

cosine series of the form

u"(z,t) = an cos nT eiat (III-26)
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)(h-z) 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(1-i) sinh B(l+i)h
n o cosh B(1+i)h [(ni)2 + i2(Bh)2] (III-27)

Since only the real part of u"(z,t) is of any interest, (III-27) will now
be put in polar form for incorporation into (III-26). Considering each
complex term separately,

(1-i) = v2 e'T' (11-28)

sinh B(1+i)h = (cosh 26h cos 2Bh) eial (111-29)

cosh B(l+i)h = (cosh 2Bh + cos 2gh) e1i2 (111-30)

[(nT)2 + i2(Bh)2] = [(nr)4 + 4(Bh)4]2 eia3 (111-31)
where,
et = tan (cosh 0h sin h) (111-32)
sinh Ah cos Bh

2 = tan-1 (sinh h sin Bh) (111-33)
cosh h cos 6h

a3 = tan-1 C2 )] (111-34)
(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 (III-15). A non-
dimensional time relating the period of oscillation to vertical mixing
time is then introduced by

T cz (III-35)
z T

With the aid of Reynolds analogy and (III-15), (III-24), and (III-35)

the Bh arguments appearing throughout an become

h =V/irT (III-36)

The final form for u"(z,t) as a function of the pressure gradient is

arrived at by combining (III-26) through (III-36) 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] (III-37)


where,
cosh v rT sin VT
ai = tan-I { (III-38)
sinh I/TT cos /i
z z
Ssinh vV/T sin vv/T~
ap = tan {i z (III-39)
cosh v/TT cos ViT
Z Z
i 2Trr
a3(n) = tan-' { z2} (111-40)


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 one-half of the period of oscilla-
tion are obtained by solving for K as a function of umax and L using
max









(III-23). 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 (III-41)
T (cosh VTJ cos A/_F?) max

and 2 I L (cosh 2 2;' + cos 2 VJiTf)4
K =- z z (III-42)
T2 (cosh vTYT cos /'i)
z z

Substitution for K in (111-37) 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] (III-43)
[(nt)" + (2rTz)2
n=1
and,
2'iL { TzT' (cosh 2 v/' cos 2 /7T)}
u"(z,t) = -
T (cosh
cos n ei(ot -+ 0i-C 2 03)
(III-44)
n=1 [(nT)4 + (2T )2]'2

III-B.2 Concentration Distribution
For temporally periodic u and c, uniform flow, and constant eddy
diffusivity it was shown in II-C that the transport diffusion equation
as described from a coordinate system moving with the cross-sectional
mean velocity could be written as

ac" K c" = -u"c (111-45)
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 (III-46)
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 (III-46) is substituted into (III-45) and

operated on, the left hand side is composed of cos nz- terms only. Thus,

to satisfy the conditions of equality, the expansion of (III-25) 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" (III-47)
z dz2 s s -

dc"
BC's: dc ,h = 0
dz z=0,h

where is constant and u (z) is defined by (III-26). Substitution for

c" and u" transforms (III-47) 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 non-trivial solution. The Fourier coefficient a' can
m
then be written as
S h2a
a' m ( m (III-48)
SKz (mr)2 + i(h2)]
Kz

The required solution for c"(z,t) is arrived at by: (1) using

(III-26) and (III-37) to determine the polar form of am; (2) transforming









the denominator of (III-48) to polar form; (3) applying the definitions
for T and T' where necessary; and (4) substituting the resulting expression
into (III-46). 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 (III-49)

m=1 [(mr)" + (2TTz)2
where ai, a2, and a3(m) are given by (III-38) through (III-40) with m
replacing n in (III-40).
Solutions for c"(z,t) as a function of umax, and L are obtained by

applying (111-41) and (111-42) to (III-49), 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 (III-50)
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 (III-51)
m=1


III-B.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 (III-52)


Using (III-37) and (III-49) 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) (111-53)
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()> (111-54)

Noting from (111-40) that

cos a3 (n) = (nr)2
[(ni)4 + (2rT )23]









and combining (111-53) and (111-54) 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] (III-55)
n=l

Solutions for T as a function of umax and L may be obtained by
substituting directly for K2 in (III-55) using (III-41) and (III-42).
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 (III-56)
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 (III-57)
[(nr)4 + (2Tz)22
n=1
Equation (111-56) 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, (111-56) is non-dimensionalized by defining
T
E x (III-58)
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 (III-59)









III-C Discussion of Two Dimensional Shear Flow Results
III-C.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 III-B.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 III-A.3. Dividing (111-55) by (111-17) and (111-56) by
(III-18) yields

EL 4z (cosh 2 + cos 2 v f)


Z (n-)2 (111-60)
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 (111-61)

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 (111-60) and (111-61) are plotted in Figure

3 vs. the non-dimensional 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 I-B.2. It is noted that all three solutions approach the

limiting value of one-half 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 (III-60) and (III-61) 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 (III-23).

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 NON-DIMENSIONAL 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 (111-56) 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 (III-18). 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 one-half 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.

III-C.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 semi-diurnal 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 (III-56) with

umax = 1 ft/sec for both the semi-diurnal and diurnal cases.

As shown in Figure 5a after an elapsed time of t = 89,424 sec the

semi-diurnal 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 semi-diurnal 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 SEMI-DIURNAL
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 semi-diurnal

tide. Figure 6a shows that after an elapsed time of t = 89,424 sec the

semi-diurnal 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 (III-57) in Figure 6b. As

seen from this plot the semi-diurnal dispersion has remained unchanged

from Figure 5b whereas the diurnal curve is now everywhere less than

the semi-diurnal curve. This is not surprising since from (III-57) 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 non-dimensional

longitudinal dispersion coefficient as defined by (III-59) 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 (11-62)

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 SEMI-DIURNAL
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 NON-DIMENSIONAL 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, non-dimensional forms of

the solutions for u"(z,t) and c"(z,t) given by (III-43) and (III-50)

respectively are plotted over the non-dimensional depth, z/h. The non-

dimensional forms used are


UND Umax


C1 =- (III-64)
(- )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 I-I 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 = (o-t)max) b. Slack Water (crt = (-t)max + 7'/2)


FIGURE 8 NON-DIMENSIONAL 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 ( 0-t = (0t)max) b. Slack Water (ot = (0t)max + )


FIGURE 9 NON-DIMENSIONAL 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 NON-DIMENSIONAL VELOCITY AND CONCENTRATION VARIATION
COMPONENT PROFILES FOR T' = 5.0
z













CHAPTER IV
DISPERSIVE MASS TRANSPORT IN RECTANGULAR
CHANNELS OF FINITE WIDTH

IV-A 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

IV-A.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 (IV-1)
at z z2 3y2

If a solution of the form
u(y,z,t) = us(y,z) eiat (IV-2)

is assumed to be valid for (IV-1) then the three dimensional shear flow

boundary value problem for oscillatory flow may be formally stated as:
a2u a2u
PDE: E + E iau = K (IV-3)
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 (IV-3) of the form










us(y,z) a' cos (1 (n 2m+1 1) y (IV-4)
m=0 n=O
is assumed. An examination of (IV-4) 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, (IV-4) 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 (IV-3) 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 (IV-4)
and (IV-2) is then


Z V m+nc (16K2m+1)6rz s( (2n+l)gTy igt
u(y,z,t) = 2-16K(- hmc 2 cos w w (IV-5)
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 cross-sectional mean
velocity, u(t), must be subtracted out of (IV-5). The mean component of
u is defined by 2
iot
u(t) = e2h us(y,z) dy dz (IV-6)
-h -w/2

Performing the integration in (IV-6) and subtracting the result from
(IV-5) yields

u"(y,z,t) = n {cos (2m+)z cos(2n+1)y g(m,n)}eit (IV-7)
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


(IV-8)



(IV-9)


Since only the real portions of these expressions have any physical
meaning, (IV-7) 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(ct-ai(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)


(IV-10)


gradient may




K


(IV-11)


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


(IV-12)


(IV-13)








In order to put (IV-11) in a more usable form, (IV-5) 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 (IV-5)

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 (IV-15)
8T

where,

S = ,-)m+n(R(m,n)-l)
m 0 n 0 (2 m+l)(2 n+l)Rl(m,n


+ (-)m+n (IV-16)
m = 0 n = (2 m+l)(2 n+l) R(m,n)

The details of developing (IV-15) from (IV-14) are presented in Appendix
C. Substitution of (IV-15) into (IV-11) 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) (IV-17)
2h w









IV-A.2 Concentration Distribution

The form of the transport diffusion equation used to solve for

c"(y,z,t) is identical to (11-27), or

c" a2cl' a2c a3 2
K z K = u" (IV-18)
at z z2 Y y2 aT

where u"(y,z,t) is given by (IV-7). It is assumed that c"(y,z,t) is of

the form

c"(y,z,t) = c" (y,z) eiot (IV-19)
s

where c"(y,z) is complex following the same approach used with u"(y,z,t).

Substituting (IV-19) into (IV-18) 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" (IV-20)
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 (IV-21)
Z=0 k=0

Equation (IV-21) is seen to satisfy the conditions of zero diffusive flux

across the flow boundaries. Substituting for c" using (IV-21) and for

u" using (IV-4) and (IV-7), (IV-20) 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 (IV-22)


where amn is given by (IV-9). 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 (IV-23)
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 non-orthogonal 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 (IV-24)
[(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) (IV-25)

R,(p,q) is given by (IV-12) with p replacing m and q
replacing n.

R,(,k) = {T (T)2 + 2T- (k7)2}2 + 1 (IV-26)
cz cy

ac(p,q) is given in (IV-13) with p replacing m and q
replacing n.

c2(Z,k) = tan' { T 1 T (IV-27)
2 F 2rT (kiT)2
cz cy

(0, k = =
=k = , k = 0 or R = 0 (IV-28)
1, k f 0,, 0

The factorPZik is necessarily included in (IV-24) 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 cross-sectional area. Detailed mathematics for arriving
at (IV-24) are given in Appendix B.
The expression of c"(y,z,t) as a function of umax is determined by
substitution for K in (IV-24) using (IV-15) 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 -(IV-29)
[(2 p+1)2-(2)2][(2 q+1)2-(2k)2]R (p,q)R,(R,k)









IV-A.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 (IV-30)
-w/2 -h

The integration in (IV-30) are carried out in Appendix B and it is noted
that the non-orthogonality 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 (IV-11) and (IV-24) 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) (IV-32)



61(a,k,m,n,p,q) = [(2p+l)2-(22)2][(2q+1)2-(2k)2][(2n+l)2- (2k)2]


(IV-33)


[(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= (IV-34)


Replacing T in (IV-34) with the right hand side of (IV-31) 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}
(IV-35)
61(Z, k,m,n,p,q)

The longitudinal dispersion coefficient is expressed as a function
of umax by substituting for K in (IV-35) using (IV-15) 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}
(IV-36)
61(Z,k,m,n,p,q)

To facilitate the presentation of results and the application of
(IV-36) for predictive purposes a non-dimensional relative mixing time,
Tc, is introduced as

T' T
Tc =T' (IV-37)
y cy








where,
T
T' = cy (IV-38)
y T


Thus, as shown by (IV-37), T' is a measure of the relative effects of
vertical and lateral shear for a given width-to-depth ratio, and vertical
and lateral eddy diffusivities. With the aid of (IV-37) the following
definitions can then be made:


ri(m,n) = (2nTI )i(m,n) = {[ 2m + T [ ]2}+(2T)2 (IV-39)


r2(z,k) = (2TTT)2R2(Z,k) = {( +)2 +Tc(ki)2}2+(21T')2 (IV-40)





c,(m,n) = tan [2ml 2]2+Ti[ ]2 (IV-41)

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 (IV-42)
z2aT (2m+l)(n Zn)r)1 ,)
m =0 n= 0


These relationships are then applied to (IV-36) to obtain an expression
for the non-dimensional longitudinal dispersion coefficient, E defined








by (111-58). 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(IV-43)
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) (IV-44)

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 (IV-36) would have

reduced to (IV-43) with T' replacing T' everywhere, and the newly defined
y z
T' multiplying all complementary terms in rl, r2, and a,. This form of

(IV-43) 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 half-width was twice its depth would

produce the same dispersive mass transport as a channel whose depth was

twice its half-width 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 (IV-43) for three dimen-
x

sional oscillatory shear flow is illustrated by Figure 12. In this

figure E' is plotted against the non-dimensional 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 NON-DIMENSIONAL 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 (IV-43) 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
(IV-43) very nearly approximates for all T' the solution given by (III-59)

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 10-3 whereas for

the limiting curve in Figure 12 it is 3.07 x 10-3 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 (IV-43). 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 NON-DIMENSIONAL 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' =0.01---j

(C = 0.001\

50- TC' =0.0001
-6
Tc' IXIO



103 102 10-1 I 01 102 03
S Tcz
Tz T

FIGURE 14 NON-DIMENSIONAL 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 10-6








-6
T' = 0.1, 0.01, 0.001, 0.0001, and < 1 x 10-6. 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 (IV-45)
c (w/2r


which is simply the inverse square of the channel half-width to depth ratio.

Referring to Figures 12, 13, and 14 with the aid of (IV-45) it is seen

that as the half-width 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

Non-Dimensional Peak Value
S. Vertical Mixing of Non-Dimensional
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 10-I 10-2 10-3 i0o 10 5
Td

FIGURE 15 NON-DIMENSIONAL 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




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