TURBIDITY CURRENTS AND SEDIMENTATION IN
CLOSEDJEND CHANNELS
By
CHUNG PO LIN
A. DISSERTATI!:r! PRESENTED TOJ ThEr GRADUATE SCHOOLII
OF TH~-E UNIVERSITY OF FLORIDA IN
PART~IAL FULLFT~,:ILLMEN Oj;CF THE HEQUIRE.ENt~TS
FOE "HE DEGREE OF DOCTOR OF PHILOSOFEY
UNjIVERSITY OF FLORIDA
1987
ACKNOWLEDGEMENTS
The author would like to express his sincerest appreciation to his
advisor and supervisory committee chairman, Dr.. Ashish J. Mehta,
Associate Pr~ofessor of Civil Engineering and Coastal and Oceanographic
Engineering, for his continuing guidance and support throughout this
research. Appreciation is also extended for the valuable advice and
suggestions of Dr. B. A. Christensen, Professor of Civil Engineering,
as well as the guidance received from Dr. R. G. Dean, Graduate Research
Professor of Coastal and Oceanographic Engineering, Dr. D. M. Sheppard,
Professor of Coastal and Oceanographic Engineering, and Dr. A. K.
Varma, Professor of Mathematics.
Sincere thanks also go to Dr. B. A. Benedict, Dr. Y. Peter Sheng,
and Mr. J. W. Lott for their suggestions and help in this study.
Special thanks go to the staff of the Coastal Engineering Laboran
tor'y at thre Uniiversity of Florida, Marc Perlin, Vernon Sparkman, and
Jimr Join~er, and to Messrs. E.C. McNair, Jr. A. Teeter, and S. Heltzel
at the Ui. S. Alrmy Corps W~aterways Experiment Station, Vicksburg,
Mississippi, for their cooperation and assistance with the experiments.
The iut~hor wishes to thank Ms. L. Peter for the drafting of figures,
and Ms. L. Lehmann and Ms. H-. Twedell of the Coastal Engineering
Archlives for their assistance.
The support of the National Science Foundation, under grant number
CEE;84001490, is sincerely acknowledged.
ii
Finally, the author would like to thank his wife Ferng Mei-Chiang
for her love, moral encouragement and patience, and his par-ents for
their love and support.
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1 INTRODUCTION..................................
1.1 Turbidity Currents and Sedimentation in
a Closed"End Channel...........................
1.2 Approach to the Problems.......................
1.3 Scope of Investigation.........................
2 LITERATURE REVIEW...................................
2.1 Front Behavior of Turbidity Current............
2.2 Stationary Sediment Wedge......................
2.3 Mathematical Model.............................
2.0 Depositional Properties of FineMGrained
Sediments.................................
3 METHODOLGY....................................
3.1 Dimensional Analysis...........................
3.2 Development of Mathematical Model..............
3.3 Analytical Developments........................
EXPERIMENTAL EQUIPMENT AND PROCEDURES...............
4.1 Experiments at WES................,.............
11.2 Experiments at COEL............................
5 RESULTS AND DISCUSSION ................... ...........
5.1 Typical Results................................
5.2 Characteristic Test Parameters.................
5.3 Characteristic Results.........................
Results of Tidel Effect Tests................,..
TABLE OF CONTENTS
Page
11
vi
vii
xi
xix
1
1
5
8
11
11
18
21
24
31
40
57
69
69
82
96
96
127
127
165
ACKNOWLEDG EMENTS............................
LIST OF TABLES..................................
LIST OF FIGURES.................................
LIST OF SYMBOLS.................................
ABSTRACT.................................
CHAPTERS
6 CONCLUSIONS AND RECOMMENDATIONS..................... 173
6.1 Summary of Investigation........................ 173
6.2 Conclusions................................. 174
6.3 Recommendations for Future Study................ 184
APPENDICES
A CRID SYSTEM, FLOW CHART, AND STABILITY CONDITION
FOR MATHEMATICAL MODELING. ................... .............187
B FLOW REGIME IN A CLOSED END CHANNEL...................... 193
C COMPUTATION OF MAXIMUM SURFACE RISE...................... 201
D CHARACTERISTICS OF CLOSED-END CHANNEL DEPOSITS........... 203
REFERENCES......................................... 205
BIOGRAPHICAL SKETCH....................................... 212
lIST OF TABLES
Table Page
4.1 Chemical Composition of a Water Sample
from the 100 m Flume...............................,... 77
4.2 Summary of Test Conditions at WES..................... 83
4.3 Chemical Composition of the Tap Water................. 90
4.4 Summary of Test Conditions at COEL.................... 95
5.1 Basic Test Parameters for WES Tests................... 128
5.2 Basic Test Parameters for COEL Tests.................. 129
5.3 Critical Deposition Shear Stress of Kaolinite......... 156
C.1 Computation of Maximum Surface Rise................... 202
D.1 Characteristics of ClosedBEnd Channel Deposits........ 203
LIST OF FIGURES
Figure Page
1.1 Schematic of ClosedBEnd Channel and Turbidity
Current. (a) Plan View. (b) Elevation View.......... 4
2.1 Front Shape of Gravity Current From
Theoretical Model.................................... 13
2.2 Schematic of Mean Flow Relative to a Gravity
Current Head Subject to No*Slip Conditions at
Its Lower Boundary.................................. 13
2.3 The Front Position as a Function of Time for
Different Releases.................................. 19
2.4 Median Settling Velocity vs. Concentration
for Severn Estuary Mud................................ 27
2.5 Suspended Sediment concentration vs. Time
for Three Cohesive Sediments.......................... 27
2.6 (a) Median Dispersed Grain Size vs. Distance.......... 30
(b) Suspended Sediment Settling Velocity
vs. Distance................................... 30
3.1 Front Shape of a Propagating Turbidity current........ 32
3.2 Schematic of Theoretical TwoFLayered Model............ 58
3.3 Schematic of Stationary Sediment Wedge and
Water Surrface Rise.................................... 66
4.1 Schematic of Hydraulic System at WES.................,. 70
4.2 A Digital Electric Thermometer at WES................. 73
4,3 Suspension Sampling Apparatus Deployed
in the WES Flume..................................... 73
4.4 Particle Size Distributions of
Fine-Grained Sediments................................ 74
4.5 Two Different Size Settling Columns with
a Mixing Pump....................................... 76
vii
4.6 Median Settling Velocity vs. Concentration............ 78
4.7 Locations of Measurements in WES Tests (Plan View).... 81
4.8 Schematic of Hydraulic System at COEL (Plan View)..... 84
4.9 Locations of Measurements and Sampling Apparatus.
Measurement Locations (a) Plan........................ 87
Suspension Sampling Apparatus (b) Plan,
(c) Elevation.................................... 87
4.10 Electromagnetic Current Meter......................... 89
4.11 Water Surface Variation for Tide Effect Tests......... 94
5.1 Main Channel Horizontal Velocity vs. Elevation........ 98
5.2 Main Channel Concentration vs. Elevation.............. 98
5.3 Front Position vs. Elapsed Time....................... 99
5.4 Instantaneous Front Speed vs. Distance
From Entrance..................................... 101
5.5 Horizontal Velocity at z = 2.5 cm vs. Elapsed
Time at Five Locations................................ 103
5.6 Horizontal Velocity at z = 2.5 cm vs. Elevation,
for Four Times After Gate Opening..................... 105
5.7 Horizontal Velocity at Steady State vs.
Elevation for Three Locations......................... 106
5.8 Horizontal Velocity Contours at Steady
State in CloseddEnd Channel........................... 107
5.9 Concentration at z = 0.7 cm vs. Elapsed Time
for Five Locations. (calibration Results)............. 109
5.10 Concentr~ation at z = 0.7 cm vs. Elapsed Time
for Five Locations. (Verification Results)............ 110
5.11 Concentration at Steady State vs. Elevation
for- Five Locations. (Calibration Results)............. 112
5.12 Concentration at Steady State vs. Elevation
for Five Locations. (Verification Results)............, 113
5.13 Concentration at Front Head vs. Distance
From Entrance..................................... 115
viii
5.4Mean Conicentration Below Interface vs. Distance
From Entrance......~.................... ................ 117
5.15 Mean concentration Below Interface vs. Distance
From Entrance. (Verification Results)................. 118
5.16 Concentration Contours at Steady State
in1 ClosedrEnd Channel................................., 120
5.17 Surface Elevation Difference vs. Distance
From Entrance............................,......., 121
5.18 Sediment Deposition Rate vs. Distance From
Entrance. (Calibration Results)......................., 123
5.19 Sediment Deposition Rate vs. Distance From
Entrance. (Verification Results)......................, 124
S.20 Particle Size Distributions of Deposits
at Three Locations..............................,..., 126
5.21 Front Nose Height to Head Height Ratio vs.
Local Head Reynolds Number............................, 131
S.22 Front Head Height to Neck Height Ratio vs.
Local Neck Reynolds Number............................, 132
5.23 Initial Front Speed to Densimetric Velocity
Ratio vs. Densimetric Reynolds Number................., 134
5.24 Dimensionless Front Speed vs. Dimensionless
Distance From Entrance................................, 138
5.25 Local Densimetric Froude Number vs. Local
Neck Reynolds Number.................................., 140
S.26 Dimensionless Front Position vs. Dimensionless
Elapsed Time......................................., 142
5.27 Dimensionless Excess Density at Front Head
vs. Dimensionless Distance From Entrance.............. 144
5.28 (a) Dimensionless Mean Concentration Below Interface
vs. Distance From Entrance for Kaolinite Tests.... 145
(b) Dimensionless Mean Concentration Below Interface
vs. Distance From Entrance for Flyash Tests..... 146
5.29 (a) Dimensionless settling Velocity vs. Dimensionless
Distance From Entrance for Cohesive Sediments..... 149
(b) Dimensionless Settling Velocity vs. Dimensionless
Distance From Entrance for Cohesionless Sediments. 150
ix
5.30 Dimensionless Median Dispersed Particle Size of
Deposit vs. Dimensionless Distance From Entrance...... 152
5.31 Settling Velocity and Median Dispersed Particle
Size vs. Main Channel Concentration................... 150
5.32 Flocculation Factor vs. Distance From Entrance........ 157
5.33 Flocculation Factor vs. Median Dispersed
Particle Size....................................... 159
5.34 (a) Dimensionless Deposition Rate vs. Dimensionless
Distance From Entrance for Cohesive Sediments..... 161
(b) Dimensionless Deposition Rate vs. Dimensionless
Distance From Entrance for Cohesionless Sediments. 162
5.35 Mean Sediment Flux into the Side Channel Through
Entrance vs. Mean Sediment Concentration at Entrance.. 164
5.36 Dimensionless Front Speed vs. Dimensionless Distance
From Entrance for Tide Effect Tests................... 166
5.37 Dimensionless Front Position vs. Dimensionless
Elapsed Time for Tide Effect Tests.................... 169
5.38 Dimensionless Mean Concentration Below Interface
vs. Dimensionless Distance From Entrance for Tide
Effect Tests.........................,.........,.... 171
A.1 Spatial Grid System for Numerical Modeling............. 187
B.1 (a) Dimensionless Vertical Velocity Profile
at x = 1.8 m (Unsteady State)...................... 199
(b) Dimensionless Vertical Velocity Profile
at x = 4.0 m (Unsteady State)...................... 199
B.2 (a) Dimensionless Vertical Velocity Profile
at x = 1.8 m (Steady State)........................ 200
(b) Dimensionless Vertical Velocity Profile
at x = 4.0 m (Steady State)........................ 200
LIST OF SYMBOLS
A A positive coefficient in linear parabolic equation
ao Amplitude of tide
B Width of closedaend channel
C Suspended sediment concentration
C' Mean concentration at front head
Co Concentration in main channel
Co Depth-mean concentration in main channel
C1 Concentration just inside closeddend channel entrance
C1 Depth mean concentration just inside closed-end channel entrance
Cb Depthhmean concentration below zerodvelocity interface
Cb1 Depth mean concentration below zero -velocity interface just
inside closedkend channel entrance
CD Drag coefficient
Cp Constant in the expression of initial front speed (eq.2.1)
OC Degree Celsius
d Sediment deposition rate
d, Median dispersed particle size
d,: Median dispersed particle size of deposit just inside closed-end
channel entrance
d85 Eightynfive percent finer than particle size
e Base of natural logarithm
e, Horizontal momentum diffusion coefficient
e, Vertical momentum diffusion coefficient in stratified flows
eo Vertical momentum diffusion coefficient in homogeneous flows
X1
f A continuous function
fo Friction factor
F Flocculation factor
F1 Hydrostatic force acting on the cross section of closed end
channel entrance
F2 Hydrostatic force acting on the cross section of stationary
sediment wedge toe
Fi Component force
Fk Complex fourier coefficient of function f
Frg Densimetric Froude number
g Acceleration due to gravity
Gs Specific gravity of sediment particles
h Total water depth in closedwend channel
H Total water depth in main channel
hi Height of front nose
h2 Height of front head
h3 Height of front neck
AH Water surface elevation difference with reference to H
1&,,, Maximum AH in closediend channel
i Column number in spatial grid system for numerical simulation
j Layer number in spatial grid system for numerical simulation
k Component in complex Fourier series
k1 Coeffloient of settling velocity expression (eq.2.4)
k2 Coefficient of settling velocity expression (eq.2.5)
k3 Coefficient of deposition rate expression (eq.2.7)
L Length of closed~iend channel
L' Wave length of tide in shallow water, (gH)1/2T
m Deposited (dry) mass per unit bed area
xti
mf Mass of turbidity front
M Total dry mass of sediment deposited in closednend channel
during test
M, Total mass of waterzsediment mixture
MD Mass deposited in closed end channel due to turbidity current
over a tidal period
Ms Total mass of sediment
MT Mass deposited in closed end channel due to tidal motion over a
a tidal period
M, Total mass of fresh water
n Manning's roughness coefficient; also, time step number of
numerical simulation
N Top layer in spatial grid system for numerical simulation
n1 Exponent of settling velocity expression (eq.2.4)
n2 Exponent of settling velocity expression (eq.2.5)
p Pressure force
gy Constant in vertical momentum diffusion coefficient expression
(eq.3.36)
Q2 Constant in vertical mass diffusion coefficient expression
(eq.3.37)
Q1 Discharge of waterasediment mixture in upper layer at closedhend
channel entrance
Q2 Discharge of water-sediment mixture in lower layer at closed~end
channel entrance
r Sediment erosion rate
R Hydraulic radius
Re Reynolds number
Reg Densimetric Reynolds number
Reh Head Reynolds number
Re, Neck Reynolds number
Ri Gradient Richardson number
xiii
Amplification factor
Mean net sediment flux into closed;-end channel through entrance
Stagnation point at front nose
Water surface slope
Time
T Tidal period
A~t Time step in numerical simulation
u Horizontal velocity
uo Horizontal velocity in main chann~
U1 Horizontal velocity at closedcend
uo Depth~mean velocity in main chann~
ul Depth~mean velocity above zero~ve:
channel entrance
u2 Depth mean velocity below zero;;ve.
channel entrance
el
channel entrance
el
locity interface at closed end
locity interface at closed"end
Depth mean velocity at front neck
Horizontal velocity of bottom layer
Instantaneous front speed
Initial front speed
Initial front speed id mean of local speeds between
x = 0.4'2.0 m
Horizontal velocity at the j th layer
Horizontal velocity at the top layer
Densimetric velocity
Flow velocity of river opposing the advancing saline wedge
Horizontal velocity at water surface
Horizontal velocity at the top of a layer
Total volume of water sediment mixture
xiv
aFs Volume of sediment occupied
w Vertical velocity
Wb Vertical velocity at the bottom of a layer
WN Vertical velocity at the bottom of the top layer
ws Particle settling velocity; also, computed (floc) settling
velocity
wso Reference settling velocity
Ws1 Particle settling velocity inside entrance of closediend channel
Wsd Stokes settling velocity of solid spherical particle with
diameter equal to d,
wsm Median quiescent column settling velocity
WT Vertical velocity at the top of a layer
x Horizontal position coordinate along longitudinal axis of
closed end channel
xf Instantaneous position of front head along closed';end channel
xo Lock length in look exchange flow
Ax Longitudinal grid size in numerical simulation
z Vertical elevation coordinate
Az Vertical grid size in numerical simulation
Azb Thickness of the bottom layer
Azj Thickness of the j th layer
AzN Thickness of the top layer
a Constant in sediment flux formula (eq.3.61)
aq Coefficient in vertical momentum diffusion coefficient express;
sion (eq.3.36)
a2 Coefficient in vertical mass diffusion coefficient expression
(eq.3.37)
8 Constant in the expression for mean concentration below
interface (eq.3.52)
Sq Coefficient used to express longitudinal distribution of
settling velocity
B2 Coefficient used to express longitudinal distribution of
horizontal velocity
7 parameter used in eq.5.1
R Constant in tidedinduced deposition formula (eq.3.66)
6 Deposition rate (dry sediment massedeposited per unit bed area
per unit time)
61 Deposition rate inside closedhend channel entrance
nHeight of interface (zero~velocity elevation)
SPi term notation used in dimensional analysis
p Density
pl Density of sedimenthladen water just inside closedwend channel
entrance
pl Depthamean density just inside closed-end channel entrance
p' Density in front head
pb Density at the bottom layer
ph Density at water surface
p, Mean density between turbid lower layer and clear upper layer in
closedaend channel
pp Density at the top layer
p, Density of sediment particles
p, Density of water
pp Density at interface
PlL Depth mean density in the lower layer at channel entrance
plU Depth mean density in the upper layer at channel entrance
Ap Density difference
Apo Density difference between sediment'laden water and fr~eshh water
in main channel
Apl Density difference between sedimentrladen water and fresh water
just inside closedi~end channel entrance
A~p' Local density difference, based on concentration in front head
Apd Local density difference, based on concentration in front head
at x= 2.1 m
v Kinematic viscosity of water
v1 Kinematic viscosity of suspension
r Ratio of vertical elevation to total water depth (z/h)
5 Ratio of interface elevation to total water depth (n7/h)
Tb Bed shear stress; also, shear stress at the bottom of a layer
Tod Critical deposition shear stress
TT Shear stress at the top of a layer
cyHorizontal mass diffusion coefficient
CoVertical mass diffusion coefficient in homogeneous flows
EzVertical mass diffusion coefficient in stratified flows
81 Parameters used in the derivation-of horizontal velocity,
1 = 1fi5
X Spatial interval of periodicity in complex Fourier series
xvii
Abstract of Dissertation presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
TURBIDITY CURRENTS AND SEDIMENTATION IN
CLOSED END CHANNELS
By
CHUNG PO LIN
May, 1987
Chairman: Dr. Ashish J. Mehta
Major Deparment: Civil Engineering
As a result of concern over sedimentation and water quality in
closedkend channels such as docks, pier slips and residential canals,
an investigation on the front behavior of turbidity currents and
associated sedimentation in a closed~end channel was carried out. The
focus of the present study was on the case of a closedeend channel of
rectangular cross section and horizontal bed connected orthogonally to
a main channel, where turbulent flow and high suspension concentration
existed. Experiments were conducted in two specially designed flume
systems to investigate front characteristics of turbidity currents,
flow regime, and sediment deposition.
A group of dimensionless parameters for each physical aspect of
concern derived from dimensional analysis provided the basis for
selecting the measurement items in laboratory experiments and for
presenting th:e r~esults. A twondimensional, explicit, coupled finite
difference numerical model for simulating vertical flow circulation and
sedimentation in a closedbend channel was made. In addition, analytiN
xix
cal developments for the longitudinal distribution of mean concentrab
tion below interface,-flow field, the maximum rise of water surface,
sediment flux through the entrance, and tide-induced deposition were
also attempted.
No significant differences were found between the front shapes of
turbidity currents and non~settling gravity currents. The rate of the
decrease of front speed of turbidity currents with distance was faster
than that of nonctsettling gravity currents. Characteristics which
showed an exponential type decrease with distance in the closed~end
channel include front speed, mean concentration below interface, front
concentration, settling velocity of suspension, dispersed particle size
of deposit, and deposition rate. The relative settling velocity,
i.e. the ratio of particle settling velocity to the densimetric
velocity, was found to be the best parameter to interpret the different
ces between tests. Durations of front propagation in the initial
adjustment phase, inertial selfbsimilar phase, and viscous self~similar
phase were found to be mainly dependent on sediment type and the
relative settling velocity. A relationship for predicting the sediment
flux into a closed end channel through entrance was found. According
to the relationship, the sediment flux is proportional to H1/2 and
3/2
C1 where H is the total water depth and C1 is the depthcmean
concentration at the entrance of the closed~end channel. It was shcwn
that the numerical model could simulate the suspension concentration
distribution in the channel satisfactorily.
CHAPTER 1
INTRODUCTION
1.1 Turbidity Currents and Sedimentation in a ClosedrEnd Channel
Sedimentation problems in pier slips, tidal docks, and closed-end
channels have been mentioned by many investigators (e.g., Simmons,
1966; O'Dell, 1969; Wanless, 1975; Hoffman, 1982). In order to
maintain the required depth of navigation in these shoaled channels,
frequent costly dredging becomes inevitable. Recently, increases in
dredging costs and the problem of dredged spoil disposal have motivated
the search for other methods of sediment control (Hoffman, 1982;
Bailand et al., 1985). Because of the complexity and variety of the
factors involved in the shoaling processes at each shoaled area, thus
far no efficient and standard method of sedimentation control is avail-
able. In fact, the prevention of sedimentation at a particular site
must be based on a knowledge of the local dominant forces and associat-
ed mechanisms.
The main causative factors of the sedimentation problem, for
example, in coastal closed--end residential canals of southwest Florida,
suggested by van de Kreeke et al. (1977), are density and wind-induced
flows. In tidal docks on the Mersey Estuary, England, investigated by
Halliwell and O'Dell (1970), tidal flows and density currents are the
primary causes of sedimentation. Accordingly, tide, wind, and density
currents are the most likely physical factors involved in shoaling
processes in areas such as pier slips, tidal docks, and closed end
channels. Density currents include currents induced by salinity,
temperature, and turbidity gradients. There is a large body of papers
describing mechanisms related to tide- and wind driven circulation as
well as salinity- and thermally-induced density currents in bays and
estuaries. However, less consideration seems to be given the mechanism
of turbiditycinduced density currents (turbidity currents) in shallow
water environments. In some areas with high suspended sediment
concentrations, turbidity currents may play an important role in the
sedimentation problems of channels.
Inside a closed-end channel, suspended sediments tend to settle
onto the channel bottom because of the quiescent conditions. Subsequi
ently, the suspension concentration inside the channel will become
lower than that of the outside water body. Consequently, a turbidity
gradient between outside waters and inside waters appears, resulting in
the intrusion of a turbidity current into the channel along the bottom.
During the penetration of the turbidity current, the clear water
overlying the layer of turbid water is forced to flow out of the
channel due to the water surface gradient generated by the intrusion of
denser fluid. In essence, therefore, a vertical circulation of fluid
is set up. Such a circulation pattern can provide exchange between
outside and inside waters, and may affect the water quality in the
channel.
One significant feature of the turbidity current is that the
density current caused by sediment in suspension itself tends to keep
sediment in suspension (Bagnold, 1962). The type of turbidity current
r-depositing, eroding, or autosuspension (neither deposting nor
eroding) -- depends on! the properties of suspended sediment, the frontr
speed of the tur~bidity current, aInd the period of consolidation of the
bed material (Pallesen, 1983). In the present study, the turbidity
currents considered are depositing currents, which carry suspended
sediments from the outside area into the closed-end channel and deposit
them on the channel bed.
In this study, turbidity currents and associated sedimentation in
closed'end laboratory flumes were investigated. Prior to the start of
a test, steady flow and sediment concentration were established in the
main channel with the entrance gate closed, separating the quiescent
clear water in the closed-end channel from turbid water in the main
channel. The gate was lifted; then measurements were made as the
intrusion turbidity front propagated. Characteristics of resulting
deposits were determined subsequently. A schematic diagram showing the
geometry of the laboratory flume and an intruding turbidity front in
the closediend channel is shown in Fig. 1.1(a) and (b). The gyre zone
shown in the figure indicates a circulation cell developed just inside
the entrance. This circulation was induced by the lateral shear force
exerted by the main flow at the flow boundary near the entrance. This
gyre zone was relatively short in longitudinal extent. The suspended
sediment was fairly well mixed vertically in the gyre. Beyond the
gyre, the turbidity front wras stratified. Definitions of the other
terms indicated in the figures are given in section 2.1. It suffices
to note here that the major influencing parameters are the sediment
properties, channel geometry, and suspension concentration at the
U S ,
~~Entrance Gyre Zn
Trbidit Fr nt
I: (a)
_j
Fl
T
(b)
Figure 1.1 Schematic of ClosedHEnd Channel and Turbidity Current. (a) Plan View. (b) Elevation View.
f +0
entrance of the closedhend channel, which is connected to a main flow
channel.
1.2 Approach to the Problem
In order to study the front behavior of the turbidity current and
the associated sedimentation in a closed~end channel, dimensional
analysis was first carried out to relate the phenomena of interest to
their influencing dimensionless parameters. Next, physical and
mathematical models were developeded to investigate the general
characteristics of the problems of concern. An attempt was made to
solve the equations of convectivepdiffusion and momentum for concene
tration distribution and flow regime in the channel under a steady
state condition.
Dimensional analysis of several important aspects of this invest;
gation was carried out using the Buckingham a n theorem which provided
a group of dimensionless parameters. These dimensionless parameters
can aid understanding of the phenomena of concern in a physical sense,
to reduce the number of measurement items in laboratory experiments,
and to express the investigation results in a compact format.
The study of the problem by physical model includes two sets of
laboratory experiments, which were conducted in two differenthsized
flume systems. Each measurement taken in the flume tests and the
presentation of the results were based on the dimensionless parameters
obtained from the dimensional analyses mentioned above. Two types of
fine-grained sediments, cohesive and cohesionless, were used. The
settling velocities and median particle sizes of these sediments were
determined by settling column' analysis (McLaughlin, 1959) and hydroc
meter or pipette analysis (ASTM, 1983; Guy, 1975). Owing to the
complexity involved in scaling the properties of finergrained sediment
and internal density conditions, no presentlyc~known model technique can
reproduce the field phenomena in the smallnscale models (Tesaker,
1975). Therefore, the results of this flume study are regarded as
indicative of a qualitative approach to the prototype scale problem.
A two-dimensional (x and z directions), explicit, finite differs
ence mathematical numerical model was developed. In the model, the
coupled momentum and convective~diffusion equations govern the movement
of water and suspended sediments. A common procedure used for simulate
ing tidek and wind~induced sediment transport in estuaries by a
numerical model is to solve the flow field directly from the momentum
equation, then the transport of suspended sediment can be obtained by
solving the convectivekdiffusion equation with the known flow field.
In the present study, since the motivating force in the momentum
equation is the density gradient resulting from suspended sediment
concentration, the flow regime and the distribution of suspended
sediment closely affect each other. Hence the two governing equations
must be coupled and solved simultaneously. The four coefficients RP
longitudinal and vertical momentum diffusion coefficients, and longitu-
dinal and vertical mass diffusion coefficients isF were calibrated until
the model's predictions showed a reasonable agreement with the measured
data from a selected laboratory test. With the chosen coefficients,
the model's predictions of sediment distribution and deposition rate
were compared with the corresponding experimental results for other
tests.
Five physical aspects of concern in the closedkend channel at the
steady state condition are discussed in the analytical development
section. They include 1) the longitudinal variation of mean suspension
concentration below the interface; 2) the flow regime in the channel;
3) sediment flux into the channel; 4) the maximum rise of water surface
in the channel; and 5) the tidekinduced to turbidity current~induced
deposition ratio. The longitudinal variation of mean concentration
below the interface is obtained by solving the verticallywintegrated
form of the convective~difffusion equation at steady state. Based on
this result, the steady state equation of motion is solved for the flow
field inside the closedbend channel. The formula for predicting the
sediment flux into the channel through the entrance is obtained as a
product of the mean velocity and suspension concentration below the
interface at the channel entrance; furthermore, the mean velocity is
related to the densimetric velocity, which is the characteristic
velocity induced by density differences (Keulegan, 1957).
A stationary sediment wedge can be established by a force balance
between the force induced by the dense fluid at the entrance of the
closed-end channel and the hydrostatic pressure of clear water with an
elevated water surface in the region beyond the toe of the stationary
sediment wedge. A formula for predicting the magnitude of the rise of
water surface needed for the establishment of a stationary sediment
wedge in the closed end channel is derived based on the conservation of
momentum.
The ratio of tide-induced deposition to turbidity currenttinduced
deposition in the closed end channel can be computed based on the
concept of tidal prism and the sediment flux of turbidity current
mentioned above.
1.3 Scope of Investigation
Three major foci of this study were 1) front behavior of the
turbidity current, 2) mechanics of a stationary sediment wedge, 3)
turbidityeinduced sedimentation.
Front behavior of turbidity currents (which play a significant
role in sediment transport) was examined primarily by laboratory flume
tests. The investigation included initial front speed, characteristics
of front propagation, and front shape. Results are compared with the
corresponding front behavior of salinitycinduced density currents
reported in previous studies.
For the stationary sediment wedge, the longitudinal variation
of mean concentration (mean sediment concentration in the layer below
the interface) along the closed~end channel was investigated experii
mentally and analytically. The flow pattern in the channel was
analytically derived from the equation of motion. In addition,
formulas for predicting sediment flux into the closedciend channel and
for computing the magnitude of the water surface rise in the region
beyond the toe of sediment wedge were derived, and the predictions were
compared with experimental data.
Characteristics of turbidity~induced sedimentation in a closedrend
channel were investigated experimentally and by means of the numerical
model. Herein, the longitudinal variations of the deposition rate,
median dispersed particle size and settling velocity of deposits along
the flume are presented and discussed in Chapter 5.
A number of publications are reviewed inl Chapter 2Z. These papers
describe the front behavior of density currents (most are salinity-
induced density currents), the steady-state sediment wedge in a
closed-end channel, mathematical models for estuarial sediment transe
port, and the depositional behavior of cohesive sediments.
Chapter 3 presents and discusses a dimensional analysis for
several subjects of concern in the present study. Finite difference
formulations of the coupled governing equation in the mathematical
model are developed and the stability conditions of the numerical
scheme are examined. Analytical solutions for mean sediment concentra-
tion below interface and water velocity at steady state are obtained
from the equations of convective~diffusion and momentum. In addition,
the solutions for sediment flux into the closed-end channel, the rise
of water surface in the closed end channel at steady state, and
tide-induced to turbidity current-induced deposition ratio are present-
ed.
In Chapter 4, the setup, equipment, test materials, and procedure
of two sets of laboratory experiments are described. The first set was
performed in a 9.1 m long flume at the U. S. Army Corps of Engineers
Waterways Experiment Station (WES), Vicksburg, Mississippi; the second
set was conducted in a 14.7 m long flume at the Coastal Engineering
Laboratory (COEL) of the University of Florida.
Chapter 5 shows typical experimental observations of a selected
laboratory test. It also includes the comparison between experimental
results and the corresponding predictions obtained by the calibrated
numerical model and/or analytical approach. Subsequently, characteris-
10
tic results, which include the data of all experimental tests, are
presented mostly in a dimensionless format. Some of the characteristic
results are compared with the corresponding results reported by
previous studies and/or from the present analytical approach. In
addition, the results of two tidal effect tests, (one for "flood tide,"
one for "ebb tide"), and a reference test are discussed.
In Chapter 6, a summary of this study, conclusions, and recommen-
dations for future work are presented.
In Appendix A, the spatial grid system, flow chart, and stability
condition for the mathematical modeling are presented. Appendix B
includes the derivations for the relationship between suspension
concentration and density of water-sediment mixture, and the flow
regime in the closedeend channel at steady state. In Appendix C, a
computation of the maximum water surface rise in the closed-end channel
at steady state for a select laboratory test is given. The characteri-
stics of closed-end channel deposits are presented in Appendix D.
CHAPTER 2
LITERATURE REVIEW
2.1 Front Behavior of Turbidity Current
2.1.1 Classification
Gravity currents, sometimes called density currents, are formed by
fluid flowing under the influence of gravity into another fluid of
different density. Due to the greater weight of the denser fluid, a
larger hydrostatic pressure exists inside the current than in the fluid
ahead, which provides the motive force to drive the gravity current.
Gravity currents occur in many natural and man made situations, for
example, a turbidity current resulting from a landslide at the sea
bottom; oil spillage on the sea surface; spreading of warm water
discharged from power plants into rivers; a cold air front confronted
by a warm air front in the atmosphere; and accidental release of dense
industrial gases. Because of the wide and frequent occurrence of
gravity currents, investigations of the front behavior of such current
r
have continued to the present time (0'Brien & Cherno, 1934; Von Karman,
1940; Ippen & Harleman, 1952; Barr, 1963a, 1963b, 1967; Middleton,
1966a, 1966b, 1967; Benjamin, 1968; Simpson, 1982; Akiyama & St~efan,
1985). In these studies, the front shape, which indicates the coverage
range of the current, and the front speed, which shows the spreading
rate of the current, were of primary concern.
12
The gravity current is characterized by a "head wave" where the
head height, 62, is higher than neck height, h3 (see Fig.1.1(b)). On
the rearward side of the head wave there is a highly turbulent zone
suggestive of wave breaking and mixing. These features appear to be
common to many physical phenomena that may be classified as gravity
currents.
2.1.2 Front Shape
Von Karman (1940) investigated the front shape of the gravity
current based on Stokes' deduction of the extreme sharp~scrested shape
of water waves, and concluded that the stagnation point, denoted as S'
in Fig.2.1, lies on the no-slip bottom boundary, and a tangent line
drawn from the point S' on the interface of the front has a slope of
w/.The same result was derived in Benjamin's (1968) extensive
treatment which was based on inviscid fluid theory. However, from
laboratory experiments, the stagnation point was observed to be
elevated (Fig.2.2), unlike the analytical predictions by Von Kirmin
(1940) and Benjamin (1968).
From their laboratory experiments, Ippen and Harleman (1952),
Keulegan (1958), and Middleton (1966) have suggested a universal
dimensionless front shape profile which will approximately fit all
gravity current heads of different sizes and velocities. In these
profiles, estimates of the ratio of nose height to head height hl/h2,
where h1 is the nose height or the distance from the bottom to the
point S', and h2 is the head height, shown in Fig.2.2, vary from 0.1 to
0.27. This result reveals that in real flows, because of friction
resistance at the stationary boundary, the lowest streamline in the
uf
Figure 2.1.
Front Shape of Gravity Current From Theoretical Model
(after Von Karman, 1940).
Figure 2.2 Schematic of Mean Flow Relative to a Gravity Current
Head Subject to NoHS11p Conditions at Its Lower
Boundary (after Simpson, 1972).
flow relative to the head must be towards the rear, and the stagnation
point S' then is raised some small distance above the floor (Fig.2.2).
Simpson (1972) further found that the ratio, hl/h2, where h1 is
the nose height, is not a universal constant, but is dependent on a
local head Reynolds number, Reh = ufh2/v, where uf is the front speed
of the gravity current and v is the kinematic viscosity of water. Bas-
ed on his experiments, he proposed a relationship between hl/h2 and
Reh, according to which hl/h2 decreases as Reh increases. In Chapter
5, the trend between h /h2 and Reh, observed in the laboratory measure-
ments of the present study, is compared with the results of previous
investigations.
Another important aspect of front shape is the ratio of h2/h3,
where, as noted, h2 and h3 are the heights of the head and the neck,
respectively. An approximately constant value of 2.08 was obtained by
Keulegan (1958) in the range of Ren between 103 and 105, where Ren
ufh3/v designates the local neck Reynolds number, using the characte-
ristic length, h3. Simpson and Britter (1979) found empirically that
the ratio h2/h3 is not a universal constant, but depends upon the
fraction depth h3/H, where H is the total water depth. A comparison
between the variation of h2/h3 with Ren, obtained from the present and
previous investigations, is given in Chapter 5.
The existence of a "head wave" in the gravity current is due to
the excess mass flux of the lower fluid brought forward to the head
from the neck region and lifted upward within the head, then returned
back toward the neck region by means of interface waves breaking just
behind the head. In order to explore the intensity of mixing between
15
the two fluids at the head Interface, which is related to the front
head height, the ratio uf/u3, where u3 is the depth-averaged velocity
in the neck, was examined by Komar (1977). Based on his experimental
results, Komar pointed out that uf/u3 is dependent on the densimetric
Froude Number, Frg = uf/[ApgH/pw]1/2, where pw is the density of fresh
water and g is the acceleration of gravity. Note that u3 is greater
than uf. As the ratio uf/u3 becomes smaller with increasing Frg, the
ratio h2/h3 will become larger (i.e. more intensive mixing between
two fluids will take place). Since the rate of mixing at the head
between the two fluids results from the excess mass flux into the head;
therefore this rate can be computed as (u3 uf)h3.
2.1.3 Front Speed
The initial front speed, uf1, in look exchange flow of saline
water has been investigated by, among others, O'Brien and Cherno
(1934), Yih (1965) and Keulegan (1957). These investigations found
that in a mutual intrusion between two liquids of a look exchange flow,
a stagnation point appears at the elevation of one half of the total
water depth at the cross section of the barrier (used to separate two
fluids of different densities before the test). The initial front
speed, uf1, can be expressed as
Ap 1/2
uffl Cp [ -- gH ] ( 2.1 )
Pm
where Cp is a "universal" constant, and Ap and p, are the density
difference and mean density between the two liquids, respectively. In
eq.2.1 the square-root term was denoted as the densimetric velocity,
up, by Keulegan (1957):
Ap 1/2
us = [ -- gH ] ( 2.2 )
Pm
The coefficient, Cp, in eq.2.1 was found to be 0.50 by Yih (1965) using
an approach based on the conservation of energy, and 0.46 from the
laboratory experiments conducted by Yih (1965) and Keulegan (1957).
Initial front speeds were examined in laboratory experiments of the
present study, and the results are compared in Chapter 5 with those
obtained by Keulegan (1957).
Benjamin (1968) pointed out that the initial front speed which
occurs after removing the barrier in look exchange flow will remain
unchanged over a distance, during which the gravity current propagates
without energy losses. After the front propagates beyond a certain
distance, energy conservation no longer holds, because of interfacial
as well as bottom friction. During energy dissipation, a "head wave"
at the front is formed, and both front speed and head height decrease.
A dimensionless factor, Fra = uf/[Ap'gh /p,]1/2, termed the local
densimetric Froude number, is used to express the relationship between
the front speed, uf, and local densimetric velocity, where Ap' is the
local density difference between the liquid in the front head and the
liquid ahead. Keulegan (1958) found that Frg is independent of the
local neck Reynolds number ufh3/v when the number is greater than 400,
but is a function of ufh3/v when ufh3/v < 400. A similar conclusion
that the dimensionless front speed is strongly dependent on the
densimetric Reynolds number, Reg = ugH/v, as Reg < order(103), and
Reynolds number independent as Reg > order(103) was made by Barr (1967)
17
based on a largei~scale flume study. In the present investigation,
values of Frg were computed from measured data in the range of 324 <
Reg < 3,830 and are compared in Chapter 5 with those obtained by
Keulegan (1958).
A series of recent studies on the front propagation of salinity'
induced gravity currents in look exchange flow have been carried out
by Huppert and Simpson (1980), Didden and Maxworthy (1982), Huppert
(1982), and Rottman and Simpson (1983). According to their studies,
there are three distinct phases which may exist during front propagat-
tion along the flume. In each phase the front movement is controlled
by different combinations of dominant forces and can be characterized
by a specific function of the elapsed time t. In the first phase
(initial adjustment phase), during which the initial conditions are
important, the front position, xf, changes in linear proportion to the
elaped time, t. This was empirically and-numerically verified by
Rottman and Simpson (1983). The second phase is the inertial selfrsim-
ilar phase, during which the gravity force (or buoyancy force) is
balanced by the inertia force. That xf moves as t2/3 was theoretically
derived by Huppert and Simpson (1980), and was experimentally and
numerically confirmed by Rottman and Simpson (1983). At the stage when
the viscous force becomes more important than the inertia force, a
viscous self-similar phase is reached. In this third phase the force
balance is governed by gravity and viscous forces. That xf moves as
tl/5 was obtained by Didden and Maxworthy (1982) by estimating the
order of magnitude of forces involved. Also, Huppert (1982) found the
same characteristic relationship between xf and t for the viscous
18
self similar phase after studying the viscous gravity current usingg a
lubrication theory approximation. Within. the first two phases (when
the Reynolds number of the gravity current is high), the front movement
is locally controlled by the conditions at the front. In the viscous
phase (low Reynolds number gravity current), the front shape and front
speed are independent of the conditions at the front. A figure
(Fig.2.3) given by Rottman and Simpson (1983) clearly illustrates the
existence of these three distinct phases as the salinity front propagar
ted along a flume. In the figure, x0 is the lock length, and tO is a
characteristic time parameter. Evidence of these three phases was
found from the laboratory measurements in the present study, and a
discussion in detail is given in Chapter 5.
The effect of the internal stratification in the gravity current
front on the front movement was of particular interest for this study
because a strong stratification in the turbidity front can exist due to
the settling of suspended sediment. Stratification considerably
reduces front speed, according to the findings of Kao (1977).
2.2 Stationary Sediment Wedge
If the leading wedge (toe) of the turbidity current has propagated
to the point where all suspended sediment at the front head has
deposited, a steady or quasi steady state sediment wedge can be
established in a closed-end channel where the force balance is achieved
by a raising of the water level in the region beyond the toe. McDowell
(1971) mentioned that such a stationary sediment wedge may exist in an
enclosed, quiescent basin connected to open water. He proposed a
simple means of determining the sediment flux from outside waters into
100
z x/xo
o o 6 Slope= 3
6 12
Fz0 0 25
B 10 -
mY Slope= I~
v, Y Slope = 5
H 10 100 1000
o DIMENSION LESS ELAPSED TIME,t /to
Figure 2.3. The Front Position as a Function of Time for Different
Releases (after Rottman & Simpson, 1983)
20
the basin, as the product of the mean settling velocity of particles
and the arrested wedge length. In addition, based on analytical
considerations, he found that for the case of particles with small
settling velocity and high suspension concentration, inflow velocity in
the lower layer would be approximately equal to 0.71ug, where up is the
densimetric velocity. Yet, no evidence was provided to verify his
findings.
Gole et al. (1973) conducted an experimental investigation of a
stationary sediment wedge in a two-dimensional (i.e relatively narrow)
closedcend channel by permitting the denser siltpladen saline water to
intrude into siltkfree saline water. Under the equilibrium condition,
the surface outflow velocity of silthfree water at the entrance of the
channel was measured and found to be approximately 0.35ug, which is
only oneehalf of McDowell's (1971) finding. Based on the volumetric
conservation of water, the magnitudes of volume flux in lower and upper
layers should be identical. Accordingly, if the zerocivelocity interface
(stagnation point) occurs at the midrdepth, it can be assumed that the
depthhaveraged velocities in both layers at the channel entrance are
equal. Once the inflow velocity is known, then the sediment flux into
the channel can easily be computed based on the inflow velocity, inflow
depth, and sediment concentration at the entrance.
One of the objectives in the present study was to find out the
longitudinal suspended sediment distribution, sediment flux into the
closedbend channel, and the flow regime in a stationary sediment wedge
by means of experimental, analytical, and/or mathematical model
approaches. The results are discussed in Chapter 5.
2.3 Mathematical Model
Fischer (1976) attempted to simulate density currents in estuaries
by a numerical model and found that numerical instability or large
numerical errors which occur in solving the convective diffusion
equation will be encountered if a common numerical method (e.g.
leapfrog method) is used to solve the eqution for the density current
with a fairly sharp density discontinity (i.e. at the density front).
A higheriorder difference approximation, Hermite interpolation func-
tion, was suggested by Fischer to be used to formulate the convec-
tiveadiffusion equation. A conclusion resulting from comparing the
solutions obtained by a semi-analytical method, the higherborder
Hermite approximation, and other methods, is that the numerical errors
were considerably reduced by using the higher-order Hermite function
approximation.
Miles (1977) developed a multi-layered, one dimensional model to
study the detailed vertical structure of the flow and salinity distri-
butions in estuaries. Non-uniform layer thicknesses can be deployed
over the water depth to give thinner layers at the locations where
vertical velocity or salinity varies rapidly. In the equations of
motion and convectivesdiffusion equation, a mixing length turbulence
model was adopted to express the vertical momentum and mass transfer in
the stratified flow with stratified buoyancy effects. A semi implicit
numerical scheme was used to formulate the governing equations in which
explicit finite differences for the convective terms and implicit
finite differences for the remaining terms were used in the equations.
Subsequently, the double sweep method (Abbott, 1979) was applied to
22
solve the finite difference formulations with known boundary condli"
tions.
Perrel~s and K~arelse (1981) developed a two -dimensional, laterally'
averaged model to study salinity intrusion in estuaries. A finite
difference method was selected, in combination with a coordinate
transformation, for the numerical integration of the system of differ
ential equations. An explicit technique was used in the longitudinal
direction and an implicit technique in the vertical direction for both
equations of momentum and convectivecdiffusion. For the continuity
equation a central difference scheme was used. The calibration and
verification of the model were carried out by comparing model results
with experimental data obtained from laboratory flume tests.
One of the first mud transport models was developed by Odd and
Owen (1972). It is a twohlayered, oneridimensional coupled model
which simulates both the tidal flow and mud transport in a well-mixed
estuary. The two layers can be of unequal thickness, with uniform
properties (e.g. flow velocity, suspension concentration) assumed for
each layer. The equations of motion and continuity for each layer are
solved using a finite difference formulation, while the convective'
diffusion equation governing the transport of suspended sediments in
two layers is solved using the method of characteristics. Erosion
and deposition are included in this model.
Ariath~ural (1974) and Ariathurai and Krone (1976) developed an
uncoupled twc'dimensional, depth averaged cohesive sediment transport
model which used the finite element method to solve the convective-dife
fusion equation. The model simulates erosion, transport, and deposit
23
tion of suspended cohesive sediments. Aggregation of cohesive sediment
was accounted for by determining the sediment settling velocity as a
function of the suspension concentration. Ariathurai et al. (1977)
modified the model to solve the two-dimensional, laterally-averaged
suspended sediment transport problem. The model was verified using
field observations from the Savannah River Estuary. Required data for
the model include the two-dimensional, laterallynaveraged velocity
field, diffusion coefficients, and sediment deposition and erosion
properties.
Kuo et al. (1978) developed a two-dimensional, laterally averaged,
coupled model which simulates the motion of water and suspended
sediment near the turbidity maximum of an estuary. The vertical dimen-
sion was divided into a number of layers, and a finite difference
method was used to solve the equations of motion, continuity and
convective~diffusion for each layer. Stratification effects on the
vertical momentum and mass diffusion coefficients were considered in
this model. Furthermore, an empirical formula was used for the relah
tionship between the vertical momentum or mass diffusion coefficient
and the local Richardson number, suggested by Pritchard (1960). The
longitudinal momentum and mass diffusion coefficients were obtained by
multiplying a constant (=105) times the vertical momentum and mass
diffusion coefficients, respectively. Erosion and deposition were
accounted for in the convective difffusion equation for the bottom
layer.
241
2.4 Depos~itional Properties of Fine*Grained Sediments
Two different types of fine-grained sediments, cohesionless and
cohesive sediments, were used in the present investigation of sedimen-
tation in closedfiend channels. Understanding the sedimentation
characteristics in the channel requires knowledge pertaining to the
settling characteristics of suspended sediment and the depositional
properties of sediment. Several important depositional properties of
fine grained sediments are reviewed as follows.
2.4.1 Settling Velocity
For cohesionless fine sediments, the settling velocity can
practically be assumed to be concentrationcindependent and can easily
be determined by Stokes' formula (Daily & Harleman, 1966):
d, (G/(1) g
ws = --- ( 2.3 )
18 v
where ws is particle settling velocity, d, is median grain size, and Gs
is the specific gravity of sediment.
However, for cohesive sediments (particle sizes less than about 20
microns), the settling velocity is dependent on floc or particle size,
suspension concentration, local physico-chemical conditions, and
microbiological activity in the water or at the particle surface. Of
these factors, the effect of suspension concentration on the settling
velocity was found to be very significant due to the change of the
frequency of inter particle collision in suspension with changes in
concentration. Collision, which influences the rate and degree of floc
aggregation, is caused by Brownian motion, the presence of a velocity
25
gradient, and differential particle settling velocities (Hunt, 1980).
K~rone (1962) has discussed the effect on inter particle collision in
suspension by each mechanism. It was noted that when settling occurs
under static or quiescent flow conditions, i.e. the conditions of the
present study, Brownian motion and differential settling velocities
mainly determine the frequency of collision and, consequently, the
degree of aggregation.
Mehta (1986) summarized the relati-onship between settling velocity
of cohesive sediment and initial suspension concentration in three
different concentration ranges. At very low concentrations, the rate
of aggregation is negligible, and the settling velocity, ws, does not
depend on the suspension concentration. At moderate concentrations,
aggregation causes ws to increase with concentration, and a relations
ship was found, from laboratory settling column analyses, of the form
ws = k1 C ( 2.4 )
where k1 depends on the sediment composition, while n1 was theoretical"
ly and experimentally found to be equal to 1.33 by Krone (1962) for San
Francisco Bay sediment, and to be less than unity by Teeter (1983) for
Atchafalaya Bay sediment.
At high concentrations, ws decreases with increasing concentration
because ws is hindered by the mutual interference of particles and by
the upward flux of fluid escaping through the small spaces among the
network of aggregates, Brownian motion becomes important under these
conditions. The following relationship has been suggested by the work
of Richardson and Zaki (1954) to be applicable in this range:
ws Ws0 (18k2C) ( 2.5 )
Where ws0 is a reference settling velocity, k2 is a coefficient which
depends on the sediment composition, and n2 is a coefficient which has
been analytically derived to be equal to 11.65 by Richardson and Zaki
(1954) and was empirically found to be equal to 5.0 by Teeter (1983).
An example of the variation of the settling velocity, ws, with
concentration is shown in Fig.2.4, which is based on measurements in a
settling column using mud in salt water from the Severn Estuary,
England (Thorn, 1981).
Settling column tests for the settling velocities of the fine
grained sediments used in the present flume study were carried out at
the Coastal Engineering Laboratory of the University of Florida. In
these tests, the settling velocities of kaolinite were examined over
the concentration range from 0.45 g/1 to 10.0 g/1, and the results are
presented in Chapter 4.
2.4.2 Deposition Rate
Krone (1962) and Mehta (1973) conducted laboratory deposition
experiments in which they monitored the variation of suspension
concentration with time under a given applied bed shear stress,
'b. In several tests Tb was selected to be smaller than the critical
deposition shear stress, TEd, i.e. the shear stress below which all
initially suspended sediment deposits eventually.
With reference to the timenconcentration curves in Fig.2.5 (Mehta,
1973), the linear portions corresponding to the range of low concentrate
Sediment Co Depth rb
Mor~ocobo 14456 20 0.02
Boy Mud 720 30 0.05
* Koolinite 1078 15 0.15
1E 000
so
0 1
2"
I
(3
z
v,
Figure 2.4.
Median Settling Velocity vs. Concentration for
Severn Estuary Mud (after Thorn, 1981).
TIME ( mins)
40 60 80 IOO 120 140 160
O
O 20 40 60 80
100 120 140 160
TIME (hrs)
Figure 2.5. Suspended Sediment Concentration vs. Time for
Three Cohesive Sediments (after Mebta, 1973).
CONCENTRATIONC g/Q)
tions, where aggregation is negligibly slow, leads to the relation
dm 'b
=wsC (1 --- ) ,Tb < ed ( 2.6 )
dt 'od
where m is the mass of suspended sediment per unit bed area over the
depth of flow. Eq.2.6 is a mass balance equation which essentially
represents dilution of the suspension with time. Eq.2.6 can also be
used to compute the deposition rate of suspended particles.
When aggregation is proceeding at a significant rate, as in the
initial phase of the curves of Fig.2.5, the dilution rate of suspension
has been described by Krone (1962) as
dm Tb 1
----= r [k3C (1 8 )] rb Tod ( 2.7 )
dt Tod t
where k3 is a factor that must be determined empirically, and t is the
time since the beginning of aggregation.
Mehta (1973) also carried out extensive deposition tests under
steady flows using kaolinite as well as muds from San Francisco Bay and
Maracaibo Estuary, Venezuela, and in addition reanalyzed some previous
flume data. The critical deposition shear stress, 'od, was found to
depend on .the sediment composition, and varied from 0.04 to 0.15 N/m2.
Note that Tod of kaolinite was equal to 0.15 N/m2. which will be
referred to in comparison with Tod of kaolinite obtained from the
laboratory measurements of the present study.
2.4.3 Sorting of Deposited Sediments
Dixit et al. (1982) analyzed a series of data from deposition
experiments conducted by McAnally (Mehta et al. 1982) in a 100 m long
29
flume at the U.S. Army Corps of Engineers Waterway Experiment Station,
Vicksburg, Mississippi. In one test, a kaolinite-water mixture was
introduced from the headbay of the flume with an initial concentration
on the order of 10 g/1. The depth of flow was 0.17 m, the bed shear
stress was 0.11 N/m2 and Manning's roughness coefficient was approxi-
mately 0.01. The test was run for one hour, and a smooth surface
without any ripple-scaled bed features was found at the end of the
test. The variation of median grain size, dmt of the dispersed
sediment from the deposit with distance from the flume headbay is shown
as Fig.2.6(a). Comparing these values with d, = 0.001 mm of the
injected sediment, a sorting effect is evident. In Fig.2.6(b) the
settling velocity of the same deposit computed from the mass balance
equation is plotted against the distance from the headbay. Similar
sorting trends of d, and ws with distance, x, were found from the
deposits of the present laboratory experiments, and are presented in
Chapter 5.
Furthermore, the flocculation factor, F, of cohesive sediment is
defined as the ratio of the median settling velocity of the flocs to
the settling velocity of the primary constitutive particles. According
to the relationship between F and the median primary particle diameter,
d,, found by Migniot (1968) and confirmed by Dixit et al. (1982), the
dependence of cohesive sediment aggregation on primary particle size
can be demonstrated in a quantitative manner. A similar relationship
between F and dm was found in the present study and is discussed in
Chapter 5.
o IOOm Flume
- -
'
0.009
0.007
E
E 0.00 5
m 0.003
DISTANCE FROM HEADBAY, x(m)
0.04
0.02
0
0.00(
do
do
8o
DISTANCE FROM HEADBAY,x:(m)
Figure 2.6. (a) Median Dispersed Grain Size vs. Distance;
(b) Suspended Sediment Settling Velocity vs.
Distance (after Dixit et al. 1982)
CHAPTER 3
METHODOLOGY
3.1 Dimensional Analysis
3.1.1 Front Shape
The height of front nose. The geometry of the experimental
channel system is given in Fig.1.1 (a) and (b). The channel has the
same cross section from the entrance to the closed end. The bottom
of the channel system is in the same horizontal plane, and the water
depths at main and closed~end channels are the same before removing the
gate located at the connection between both channels. The suspended
sediment source at the main channel remains constant. Here L, B, and H
denote channel length, channel width, and water depth, respectively.
Let p, and v be respectively the density and kinematic viscosity of
fresh water, and p, + Ap and v1 be the corresponding quantities for the
waterbsediment mixture. Because the density and temperature variations
occurring throughout a test were relatively small, the average value of
the densities can be expressed as p,, and of the kinematic viscosities
as v.
As mentioned in Chapter 2, the front shape of gravity currents
has interested many investigators. During the present laboratory
experiments, a typical front characteristic of turbidity current was
observed and is shown in Fig.3.1. Let h1 and h2 be the heights of
front nose and front head, respectively (Fig.2.2). Then an expression
Figure 3.1. Front Shape of a Propagating Turbidity Current.
for h1 ,.s given as
h1 = hi (h2, H, uf, v) ( 3.1 )
where uf is the front speed. An operation using Buckingham L sr theorem
(Streeter; & Wylie, 1975) leads to the desired form:
h1 ufh2 h2
=f ( ) ( 3.2 )
h2 v H
which shows that the ratio of nose height to head height may be depends
ent on the local head Reynolds number, ufh2/v, and fraction depth,
h2/H.
The height of front head. One significant feature of the front
shape is that the height of the front head is larger than the height of
front neck, and an intensive mixing takes place at the interface right
behind the front head (Fig.2.2).
A series of photographs of the front shape taken in the present
experiments indicated that the size of the advancing front and the
degree of front mixing decrease as the front speed decreases. The
front head completely disappears after the front travels for a relatic
vely long distance. Let h3 denote the height of front neck, then the
expression of the height of the front head is
h2 = h2 (H, h3, uf, V, ws, B) ( 3.3 )
where wa is the particle settling velocity at the front. It leads to
the following expression:
h2 ufh3 h3 ws B
h3 v H uf H
The quantity h2/h3? was found to be independent of B/H by Keulegan
(1958), but dependent upon the fraction depth, h3/H, by Simpson and
Britter (1979). A final form of h2/h3 can be written
h2 ufh3 h3 a,
-=Ef(-,, ) ( 3.5 )
h3 v H uf
which indicates that besides the fraction depth h3/H* h2/h3 may be
dependent on the local neck Reynolds number, ufh3/v and the relative
settling velocity of suspended sediment, ws/uf.
3.1.2 Front Speed
Initial front speed. Let ufl denote the initial front speed of
turbidity front, that is the velocity immediately after the gate is
opened. Ideally the opening is very smooth and sudden. Within such a
short time, the effect of the sediment settling velocity on the front
movement is negligible. The expression of ufl with the related
geometrical and physical quantities is given as
ufl = ufl ( g, Ap pw, v, H, B ) ( 3.6 )
where Apl is the density difference between the main channel and
closedrend channel waters. One important velocity characteristic,
densimetric velocity ua was introduced in eq.2.2. Another signifii
cant dimensionless parameter was noted as the densimetric Reynolds
number, Rea by Keulegan (1957):
up H
Reg '- ( 3.7 )
A final form for eq. (3.6) is
ufl ugH B
= f ( --- ) ( 3.8 )
up v H
which suggests that the dimensionless initial front velocity, uf /ua,
may depend upon the densimetric Reynolds number, ugH/v, and the
width to'depth ratio, B/H.
Instantaneous front speed. A major concern with front behavior
of advancing turbidity current in the present study was how these
physical parameters (e.g. channel geometry, boundary suspension
concentration, and sediment properties) influence the front speed. It
will be discussed in this section from two viewpoints; one is by
relating the front speed to the channel geometry and suspension
concentration at the entrance, and the other is by relating the front
speed to the local front shape and suspension concentration at the
front.
The front speed, uf, and these possibly influencing parameters are
grouped as follows:
uf = uf (x, H, B, g, Aple Pw, v, Ws ) 3*9 )
This equation can be reduced to be in a final form
uf x B uCgH ws
-- = f ( --, )( 3.10 )
uA H H v uCp
which shows that the dimensionless front speed may be a function of
front position, widthhto~depth ratio, densimetric Reynolds number, and
36
relative settling velocity. Eq.3.10 is similar to the dimensional
analysis result of front speed of a saline wedge, which is advancing
against the river flow, obtained by Keulegan (1971):
uf x 8 upH u,
-- = f ( -, ---, -- )( 3.11 )
except for the term u,/ug, where u, is the velocity of the river
opposing the advancing wedge.
Barr (1967) investigated look exchange flow of saline water in a
large scale flume (i.e. flume width, B, much greater than water depth,
H), and found that the dimensionless front speed, uf/ug, is a function
of the dimensionless distance, x/H, and the densimetric Reynolds
number, Reg:
uf x ugH
= f( -, -- ) ( 3.12 )
up H v
For a small--scale flume, it would be expected that uf/ug would also be
a function of B/H. Therefore, the dimensionless grouping of instantar'
neous front speed for the look exchange flow of saline water would be
the same as eq.3.10 except for the relative settling velocity term,
ws/ug.
It was noted in the experiments during the front propagating along
the channel that whenever a decrease in the front speed was observed, a
decrease in the front height was also observed. This suggested that
possibly a relation between the front speed and the.front height
existed. The local density of the front, denoted as py+6p', decreases
37
by dropping a portion of suspended sediments to the bottom as it
propagates downstream. Thus the effect of density decrease of the
front on the instantaneous front speed must be taken into account. The
expression of the front speed can be written as
uf = uf ( h3, 8* AP', pw* v, ws, H, B ) ( 3.13 )
here the neck height, h3, is used as a height characteristic of the
front. Equation 3.13 can be reduced to
uf ufh3 ws h3 B
Frg 1/2 = -, - ) ( 3.14 )
[ap'gh3/p ] v uf H H
where Frg designates local densimetric Froude number. Here, the
parameter of ws/uf will introduce a stratification effect of suspension
concentration at the front on the dimensionless front speed, as found
by Kao (1977). Besides the internal stratification in the front, the
local densimetric Froude number, Fra, may also be influenced by the
local neck Reynolds number, ufh3/v, the fraction depth, h3/H, and the
widthrto-depth ratio, B/H.
3.1.3 Front Position
The identification of three phases, each controlled by specific
dominant forces, in the front propagation along a flume were mentioned
in section 2.1.3. In each phase the front position, xf, was found to
be a specific function of the elapsed time, t. In the present study an
attempt using dimensional analysis was made to provide the proper
dimensionless forms of xf and t for analyzing measured data and
presenting the results. An expression of the front position may be
stated as
xf = xf ( g, H, B, t, Aple Pw, *, Ws ) ( 3.15 )
An operation using Buckingham h n theorem leads eq.3.15 to a final form
with relevant dimensionless parameters:
xf B uat ws uaH
= f ( - - )( 3.16 )
which indicates that the dimensionless front position is dependent on
the width tohdepth ratio, B/H the relative settling velocity, ws/ub,
the densimetric Reynolds number, ubH/v, and a dimensionless elapsed
time, unt/H.
Barr's (1967) study on the lock exchange flow of saline water in a
large scale flume provided a group of dimensionless influencing
parameters for the dimensionless front position, xf/H, i.e.
xf uat ubH
= f( -, -- ) ( 3.17 )
Comparing eq.3.16 with eq.3.17, it can be seen that two additional
dimensionless factors are included in eq.3.16. They are: 1) the
width tocdepth ratio, B/H, which will introduce an effect on the front
position xf/H when the gravity current propagates in a small scale
flume, and 2) the relative settling velocity, ws/ug, which is a
particular characteristic of turbidity currents.
3.1.4 Deposition Rate
One of the major concerns in the present study was the deposition
of fine-grained sediments in a closed end channel. The sediment
39
deposition rate in each subsection of the channel bed can be computed
using the deposited sediment mass divided by the effective duration
(i.e., the time elapsed from che passing of the turbidity front to the
end of the test) and deposited area. The local deposition rate would
be expected to vary with distance due to the distance variations of
mean suspension concentration and of settling velocity of suspended
sediment. The variation of deposition rate with distance and other
relevant parameters are expressed as
6 = 6 ( x, H, ws, g, B, p,, Apl) ( L1 )
where 6 is the local deposition rate. Eq.(3.18) can be further reduced
to a desired form as
6 x B ws bp1
= f --, - )( 3.19 )
61 H H uh pw
where 61 is the deposition rate at the entrance of the closedrend
channel. The normalized deposition rate, 6/61, is thus observed to be
a function of x/H and B/H. Also, that 6/61 depends upon the relative
settling velocity, ws/ug, is obvious. The influence of Apl/Pw on 6/61
can be made apparent by introducing different sizes of the flocculated
particles through the entrance.
The dimensionless parameters obtained in eq.3.19 have also been
used to examine the experimental results on the local sediment settling
velocity, ws, and the median dispersed particle size, d,, since all
data on 6, ws, and dm were obtained by analyzing deposited samples from
the flume tests.
ri0
3.2 Developement of Mathematical Model
3.2.1 Model Description
Basic governing equations. The mathematical model developed is a
time varying, two-dimensional, coupled finite difference model that is
capable of predicting the vertical and temporal variations in the
suspension concentration of finergrained sediments and in flow velocii
ties in a coastal closedlend waterway with a horizontal bottom. In
addition, it can be used to predict the steady'state or unsteady
transport of any conservative substance or nonkconservative consti'
tuent, if the reaction rates are known.
The governing equations, which control the flow regime and
sediment transport, are expressed in conservation forms (Roache, 1972)
and shown as
--+ -- = ( 3.20 )
au au2 8(uw) 1 ap a au
+-- + -- = -- + -(e, -)
at ax az p ax ax ax
+ -- (ez --). ( 3.21 )
1 ap
0 = -g - ( 3.22 )
aC 8(uC) a 8 aC 8 aC
-- + ----- + -- [(w ws)C] = -- (EX --> + -- (Ez --)
at 3x 8z ax ax 8z 8z
Ud + ( 3.23 )
in which:
u, w =- velocity components in the x and z directions,
respectively
p = pressure force
e,, ez = momentum diffusion coefficients in the x and z directions,
respectively
EX, EZ = mass diffusion coefficients in the x and z directions,
respectively
d = sediment deposition rate
r =- sediment erosion rate
In fact, the physical quantities involved in these equations, such
as u, w, p, and C, are time~averaged quantities over a period which is
greater than the time scale of turbulence fluctuation when the flow
is turbulent.
Equation 3.20 is the continuity equation for an incompressible
fluid. Equation 3.21 is the equation of motion for an incompressible
fluid, and represents the longitudinal momentum conservation of the
flow. Equation 3.22 is the hydrostatic equation which results when the
vertical components of the flow velocity and acceleration are smaller
relative to the horizontal flow velocity and acceleration. Equation
3.23 is the convective diffusion equation for suspended sediment, with
resuspension and deposition as source and sink, respectively.
To obtain the time-varying solutions of the longitudinal and
vertical velocity field, eq.3.21 must be solved with the continuity
equation (eq.3.20). With the velocity field solved, it may be substir
tuted into eq.3.23 to solve for the timeavarying concentration field of
suspended sediment.
42
Vertical integration of basic equations. Since the physical
quantities in the channel can change rapidly over a short vertical
distance, they require a grid size that is much smaller in the vertical
direction than in the longitudinal direction. The fluid motion will be
considered in horizontal slices with an exchange of mass and momentum
between these slices. A so-'called sparse grid system (Miles, 1977) in
space used in the model and the location of physical quantities within
the grid are shown in Fig.A.1 of Appendix A. At the top layer, the
free surface is allowed to occupy any position within the layer, but
the thicknesses of the others are fixed.
Integration over the height of the j th layer can be performed by
assuming that all variables are practically constant through the depth
of any layer, and that the fluxes of momentum and mass normal to the
bottom of the channel and to the surface are zero.
Employing Leibnitz's rule for the vertical integration of equa;-
tions 3.20, 3.21, and 3.23 over the j th layer and/or the surface
layer, results in the following equations:
ah 8
wN '- -(uN~zN) ( 3.211 )
at 8x
wT = wb -- (uj~zj) ( 3.25 )
ax
auj 8 2 1 1 ap
--+ -- (uj) (wyuT wbub) = - (-)j
at ax Azj pj ax
8 au 1
+ (e, -) j + -- (+T Tb) ( 3.26 )
ax ax pj ~zj
aCj a 1
--- + -- (Cju ) + --- [(wT ws)CT (wb ws)Cg]
at ax AzJ
a aC 1 aC aC
= -- (Ex -)j + -- [(EZ -)T (EZ -)bi
d r
--+ ---- ( 3.27 )
Azj Azj
where:
uj, Az pj = longitudinal velocity, height, and density for the
j th layer, respectively
uT, wT, 'T = longitudinal velocity, vertical velocity, and shear
stress at the top of a layer, respectively
ubs Wb, 'b = longitudinal velocity, vertical velocity, and shear
stress at the bottom of a layer, respectively
uN, WN, AzN = longitudinal velocity, vertical velocity, and height
for the top layer, respectively
h = water surface elevation.
Equations.3.2ll and 3.25 are the continuity equations for the top
layer and all other layers, respectively. Equation 3.26 is the
longitudinal equation of momentum, and eq.3.27 is the convective diffu-
sion equation for suspended sediment.
Further derivation of the pressure term, (1/pj)(aP/ax)j in eq.3.26
is necessary. The hydrostatic pressure at an elevation z in the fluid
can be obtained from eq.3.22
p(z) g Ihp dz ( 3.28 )
and its longitudinal gradient is
=pz -- (g p dz) gr -- dz + gph -- ( 3.29 )
ax ax z z, 3x ax
where ph is the density at water surface.
The mean pressure gradient in the top layer can be derived thus
3p 1 h" ap
( -- )N =(--) dz
ax AZN h"AzN 3x
g 3pN 3h
= AzN --+ pN ( 3.30 )
2 ax 3x
where pN is the average density in the top layer. For two adjacent
layers, j and j+1, the relationship between the layer'averaged pressure
gradients can be derived from eq.3.29, and the final expression is
ap ap g 3pj+1 apj
( ) j = ( )j+1 + (zj+1 z ( 3.31 )
ax 3x 2 3x 3x
where all of pj, (3p/ax)j, pj+l, and (8p/ax)j+1 are layer-averaged
quantities. From eq.3.30 the mean pressure gradient at the top layer
can be computed with a known mean density gradient at the same layer
and the slope of water surface. Once the mean pressure gradient at the
top layer is solved, the mean pressure gradient at the other layers can
be calculated, based on eq.3.31.
Boundary conditions. At free surface boundary, no flow will occur
across the free surface, and the vertical velocity on the water surface
must be equal to zero. The shear stress induced by wind acting on the
water surrface is not considered in this study; thus the shear stress at
surface boundary can be regarded as zero. No mass flux .of suspended
sediment can flow through the free surface. This condition is express>
ed as
aC
C, --= at z = h ( 3.32 )
At the bottom boundary, no flow can pass the bottom; therefore,
the vertical velocity at the channel bed must be zero. The relations
ship between the friction factor, fo, and Manning's roughness coeffi-
cient, n, can be obtained from the formulas of Manning and of Darcy'
Weisbach (Daily & Harleman, 1966) as
8gn2
o 8/ ( 3.33 )
where R is hydraulic radius. Accordingly, the bottom shear stress, -rb,
can be expressed as
gn2
Tb = ---" Pb ub ubl ( 3.34 )
Azb
where pb, ub, and A~zb are the density, horizontal velocity, and
thickness of the bottom layer, respectively. Like the free surface
boundary, no mass flux of suspended sediment is allowed to pass through
the bottom boundary, that is
3C
C, -- = 0 at z = 0 ( 3.35 )
az
i16
In addition, two boundary conditions at the entrance of the
closedrend channel are necessary: 1) concentration profile of suspended
sediment, and 2) the water surface elevation. The data from either
condition can be time-varying quantities. To investigate the problem
under tide effects, the surface elevation at the entrance of the
closed end channel can be specified with a periodic function including
prototype tidal amplitude and period. The concentration profile of
suspended sediment and water surface elevation involved in the present
study did not vary with time. Also, the boundary conditions at the end
of the closed end channel include zero horizontal velocity and zero
horizontal concentration gradient at the vertical wall.
Momentum and mass diffusion coefficients. In turbulent flow of a
homogeneous fluid there are no buoyancy effects and lumps of fluid
moved by turbulence fluctuations have no restoring force to return to
their original position. When the flow is stratified, the buoyancy
effect tends to restore the moved lumps back to their original posi-
tion, and results in the reduction of the turbulent transfer of
momentum and mass. Bowden and Hamilton (1975), among others, have
pointed out the necessity of considering the buoyancy effects on the
momentum and mass transfer in stratified flows. In order to consider
the buoyancy effect of stratification, the momentum and mass diffusion
coefficients were related to a dimensionless parameter, which is a
function of vertical density and velocity gradients. Furthermore,
according to many investigations based on measurements, a typical
relationship between either vertical momentum diffusion coefficient,
ez, or mass diffusion coefficient, eg, and the dimensionless parameter,
117
Ri, is given as
41
ez= co ( 1 + al~i ) ( 3-36 )
or
42
cZ = EO a2 i ) 3.37 )
where eo and to are the vertical momentum and mass diffusion coeffici-
ents in a homogeneous flow, respectively; al, q1, u2, and q2 are
constants needed to be calibrated using data; and Ri is the gradient
Richardson number defined as follows:
ap
g az
Rii = ----- ( 3.38 )
p au 2
az
which is generally used as a stability index in stratified flows
(Turner, 1979). For the vertical diffusion coefficients, eo and al are
assumed identical tO Co and a2, respectively, while ql is taken to be
-1/2.
One well-known formula for the vertical mass diffusion coefficient
has been suggested by Pritchard (1960). The formula was developed from
a study of velocity and salinity distributions in the James River
estuary by fitting observational results. The formulas of vertical
mass diffusion coefficient in the stratified and homogeneous flow
fields are given respectively as follows:
Ez "E = CO *.27 Hi 3*39 )
-3 lujz2(H-z)2
"0 = 8.59 x 10 ------- ( 3.40 )
H3
where H is the total water depth and z is the elevation at which cz is
being calculated. In eq.3.39, ol2 and q2 are equal to 0.276 and "2,
respectively.
For the longitudinal momentum diffusion coefficient, e,, Festa and
Hansen (1976) changed the value of e, from ex = e, to ex = 106 ez with
negligible effects on the results of their tidal model. This indicates
that the results are not sensitive to the actual value of ex. Yet they
found that varying the longitudinal mass diffusion coefficient, eg,
from EX = E, tO EX = 107 EZ did produce significant changes in their
model results. Dyer (1973) and Kuo et al. (1978) suggested that
longitudinal momentum and mass diffusion coefficients are on the order
105 of the corresponding vertical diffusion coefficients.
Sediment settling velocity and deposition rate. In a mathematical
model of sediment transport, model prediction is very sensitive to the
settling velocity of suspended sediment given as input data. Thus the
determination of sediment settling velocities for a mathematical model
requires much care.
There are essentially four types of field or laboratory measure
ment of the suspension settling velocity mentioned by Mehta (1986): 1)
the use of an in situ tube for prototype measurement, 2) fitting
analytical solution to measured suspended sediment-depth profile from
49
the prototype measurement,, 3) use of laboratory settling column, and LI)
measurement of rates of deposition in the flume.
The measurement by means of a laboratory settling column for the
settling velocities of fine-grained sediments used in the tests was
made in the present study. The principle of this test is essentially
based on the known relationship between the downward settling flux of
suspended sediment and the dilution rate of suspension concentration in
the settling column. During the test, the suspension samples are
withdrawn at designated times from several vertically aligned taps,
which are attached to the side wall of the settling column. The
experimental facility and procedure of the settling column test
performed in the present study were designed with reference to the work
by Interagency Committee (1943), Mclaughlin (1959), Owen (1976), and
Christodoulou et al. (1974). The results of the median settling
velocity versus initial suspension concentration for each raw sediment
are presented in Chapter 4. However, these settling velocities cannot
appropriately represent the local suspension settling velocities in the
closed end channel due to the losses of a part of raw sediments before
they reach the closed end channel entrance and the nonuniform flow
regime in the closed-end channel.
The suspension settling velocity in the closed-end channel, which
is subject to local flow and concentration conditions, was computed
based on the measurement of deposition rate from each test. These
computed settling velocities are expected to be more realistic than
those obtained using settling column analysis for representing the
local settling velocities of suspension in the closedcend channel.
50
Therefore, the computed settling velocity was adopted as input data in
the mathematical model.
In the present study, the bed shear stress generated by turbidity
current was much smaller than the critical erosion shear stress which
is on the order of 0.2 0.6 N/m2 (Parchure and Mehta, 1985). Thus,
the term for resuspension rate of bed material in the convective-diffu-
sion equation was omitted.
The mass balance equation (eq.2.6) was used for computing the
deposition rate of suspended sediment in the mathematical model. In
this equation, the critical deposition shear stress, Tod, must be
known before any computation. Therefore, an attempt at finding 'od for
kaolinite was carried out based on the measured data in the main
channel and is discussed in detail in Chapter 5. Note that the term
for sediment deposition rate in the convectivebdiffusion equation is
only considered at the bottom layer of the numerical grid system.
3.2.2 Finite Difference Formulation
There are many ways to present the derivative terms of the
governing equations in the finite difference formulation for solving
the flow velocity and sediment concentration numerically. An explicit
numerical scheme was used to develop the mathematical model. In order
to gain better stability and accuracy of the solutions of the finite
difference formulations, the temporal, convective, and diffusive terms
in the equations of motion and convective~diffusion were time~ and
space-centered.
With reference to the grid system of Fig.A.1 of the Appendix,
three independent variables, x, z., and t denote longitudinal and
51
vertical space coordinates, and time, respectively. Using i and j to
represent the number of space intervals in the x and z directions,
respectively, and using n to denote the number of time intervals that
have elapsed, variables will be represented using i, j, n subscripts,
where i, j, or n = 0, 1, 2, 3, etc. The surface elevation, h, is a
function of x and t only, while the layer thickness, Az, is a function
of t for the surface layer and a function of z only for the other
layers.
The finite difference approximations of eqs.3.24, 3.25, 3.26, and
3.27 are:
the free surface elevation
n+1 n-l n n n n
hi hi n 1 Azi+1,N + Azi,N n AziN + Azi'1,N n
=w --- [ u u ]-------
2At i,N Ax 2 i,N 2i-,
( 3.41 )
the vertical velocity
( 3.42 )
n Clx
Azj
equation of motion
(~ uij-l ug~j) [ui + ui+1,j) (ui-ly + ui j) ]n
[(ui,j+ui ,j+1 '(i,j+1 +Wi*1 ,j+1 ) (ui,j-1 +ui,j l"i,j "i+1l,j j
26x2
1ij1 i+1,j+1 i,j+1 1j
+-[--
i,j i+, ,j i,jR1
nl n 1
(Azj + Azj-1)
n n n n n n
[(Ci,j + Ci,j+1)("i,j+1 Ws) (Ci,j-1 + Ci, j)(wig r ws)]
26zj
n n
n n
n n no ~
nc1
[(g +ey
i,j
n91 n41 n91 n 1
)(u Fu )-(
i+,j i+,ji,j iR1,j
n-1 nrr1 n 1
+ex )(u Ru
i,j i,j ibl,j
n-L nR1
n 1 91
n 1 n-1
S -u )
Cu) (ez
n 1
Azj
(Azj + Azj+1)
ap 2
ax
( 3.L13 )
and the convective-diffusion equation for sediemnt
n+1 n 1
n n n
n
n n
+ Ci )ui-1,j
26x
n-1 nr1 n-1
i,j
nc1 n-1 n-1
nH
nc1 nrl
n-C1
eg
i,j
nl n-1
2
+ ---- [
n-1
rlz
n"1 n-1
(6zj-1 + Az )
nP1 nnl
(Azj +6zj+1)
( 3.1111 )
in addition, the finite difference formulation of the gradient
Richardson number, eq,3.38, can be expressed as
2g (bzj+1l + bz j )pnj1" i
Ri ='
n nn 2 n n 2
( 3.45 )
The procedure of numerical calculation starts with all the
variables assigned at their initial values, and moves from the channel
entrance to the closed end in x direction, from the top layer to bottom
layer in z direction. With all variables known at the n th time step,
the continuity equation for the top layer (eq.3.41) is used to calcula-
n+1 n+1 n+1 n+1
te hi .With hi known, ui,j and Ci,j can be calculated for all i's
and j's using equations 3.43 and 3.44. Knowing all the u's for the
(n+1)st time step allows all the w's for the (n+1)st time step to
be calculated using eq,3,41.
Next, the density, pressure gradient, and longitudinal and
vertical mass diffusion coefficients are calculated for the (n+1)st
time step. Subsequently, the entire procedure described is repeated to
calculate the values of the variables at the (n+2)nd time step. A
brief flow chart which summaries the sequence of numerical calculations
is shown in Appendix A.2.
3.2.3 Stability and Accuracy
As the derivatives in the governing equations are centered in time
and space, the finite difference scheme has second order accuracy.
Unconditional stability does not exist because explicit finite differ;
ence formulations are used. A stability analysis of the full set of
governing equations is impossible due to the involvements of nonli-
near terms and of the coupling characteristic. Instead, parts of the
finite difference formulation representing different physical process-
es have been analyzed separately. The stability condition of each part
is essential for the overall stabitity of the model, but is not
sufficient to ensure that it is the stability criterion for the entire
equation system. The stability condition (Courant'Friedrichs-Lewy
condition) of the numerical solution when considering only temporal and
linear convective terms in the finite difference equations of motion or
convectiverdiffusion is
(Ax)min
At (- ( 3.46 )
umax
which is valid for the explicit discretization of a hyperbolic equati-
on. In eq.3.46, At is the time interval of each time step, (Ax)min
is the minimum longitudinal space interval used in the grid system, and
u,,, is the maximum velocity occurring in the study area.
A linear parabolic equation including the temporal and diffusive
terms, which are time- and space-centered with three successive time
levels, is expressed as
n+1 n-1 nrl n 1 n-1
fj aL fj fj+l 2fj + fj-l
=A ( 3.117 )
26t Ax2
where f denote velocity or sediment concentration, and A can be
momentum or mass diffusion coefficient. The stability condition for
eq.3.47, is derived using complex Fourier series analysis
A*At 1
< ---- ( 3.48 )
(6x)2 4
The derivation of the inequality condition (eq.3.48) is given in
Appendix A.3
The stability conditions found above provide guidance for select-
ing Ax, At, and momentum or mass diffusion coefficients.
3.2.4 Calibration and Verification
In the present study, calibration is defined as the adjustment of
coefficients based on an assumption that the relations used in the
model, in which the coefficients appear, adequately describe the
physical phenomena under consideration. The unknown coefficients in
the model are: 1) longitudinal and vertical momentum diffusion coeffir
clients, e,, ez; 2) longitudinal and vertical mass diffusion coeffi-
cients, EX, EZ; and 3) Manning roughness coefficient, n. Besides, some
physical parameters need to be determined empirically; these include
the settling velocity, ws, and the critical deposition shear stress,
'od. The test COEL-4 was used as a calibration test, such that its
conditions, i.e. water depth, water density, sediment specific gravity,
56
and concentration profile at the entrance of the closedcend channel,
were related as the input data in the mathematical model simulation.
In addition, for grid size (which is based on how detailed predictions
in space one likes to obtain) and time step (which is referred to the
stability conditions aforementioned), values of Ax=0.5 m, Az=1.0
cm, and At=0.15 see were adopted for simulating all tests of WES and
COEL. Also, EXI Z, are usually assumed to be identical to the longituk
dinal and vertical momentum diffusion coefficients, respectively, by
virtue of Reynolds analogy. Since the turbidity currents were laminar
in all the laboratory tests in the present study, the kinematic
viscosity of water was used for ez (i.e. e, = 10*6 m2/sec). The
longitudinal momentum diffusion coefficient, ex, was determined with
reference to the stability condition given as eq.3.48 and was founo to
be 5.0 x 10 3 m2/sec. Manning's roughness coefficient, n, was 0.03
sec/m/3 This n value selection can be justified by comparing with an
illustrative value n = 0.022 sec/ml/, which is obtained by utilizing
the Moody diagram (Daily & Harleman, 1966) and eq.3.31 for a turbidity
current with u = 0.5 cm and an interface thickness n = 5 em. The
calibration results are given in Chapter 5.
Model verification was carried out by checking how well the
calibrated model reproduced the phenomena in other tests. Here, COELR5
was selected as a verification test. Comparison between the results
using the calibrated numerical model and experimental measurements for
many aspects of test COELc5 is given in Chapter 5.
57
3.3 Analytical Developments
3.3.1 Mean Sediment Concentration Below Interface
In order to examine the longitudinal variation of mean sediment
concentration below interface in a stationary sediment wedge, an
attempt at solving the steady state convectiveidiffusion equation
analytically was made as follows.
The physical properties at each layer are assumed to be uniform.
That is, the upper layer is assumed to be sedimentufree, while the
suspension concentration in the lower layer is assumed to be uniform.
Also, the velocity profiles in both layers are assumed to be uniform.
A schematic figure is shown in Fig.3.2. In the lower layer, upward
vertical velocity of the fluid is small compared with the settling
velocity of suspended sediment. Mass diffusion in both x and z
directions is assumed to be negligible as compared with mass convection
by flow. Under these assumptions the steady-state convectiverdiffusion
equation can be expressed as
aC aC
u- ws -- = ( 3.49 )
3x 3z
Integrating eq.3.49 from bottom to interface, n, then using the
boundary conditions at bottom (C = Cb, at z = 0) and interface (C = 0,
at z = n), an equation for mean concentration gradient is obtained
a b "s
= - Cb ( 3*50 )
ax unl
where Cb is the mean suspension concentration in the lower layer.
~ r ~
r
Figure 3.2. Schematic of Theoretical Two-Layered Model.
59
Furthermore, eq.3.50 is integrated from x=0 (3.e. the entrance of
the close-end channel) to x. Since both ws and u are functions of x.
the integration of the ter~m on the right hand side of eq.3.50 withl dx
cannot be carried out unless explicit forms of the functions, ws(x)
and u(x), are known. According to the experimental observations from
the present study, both sediment settling velocity and flow velocity
decreased exponentially with distance from the entrance of the side
channel. Therefore, ws and u can be expressed as ws = wsl~exp( 6 x)
and u = ul~exp( 82x), respectively, where ws1 and ul are the sediment
settling velocity and the flow velocity at x = 0, and B1, B2 are
constants. The thickness of the lower layer, n, is assumed unchanged
because it decreases very slowly with x. Accordingly, the solution for
the integration of eq.3.50 can be expressed as
Cb Ws1 (B2 r B1)x
In ---- = e [--- e cl 1 ] ( 3.51 )
Cb1 ul n(B2 C B1)
where Cb1 is the mean sediment concentration at x=0. The term
exp[(B2 L81)x] can be expanded using Taylor series, i.e. 1+(B2 81)x+
[(82"B1)x]2/2+....., and its second and higher order terms can be
omitted if the exponent (B2~B1)x is much less than unity (i.e. the
local relative settling velocity ws/u is very close to the ratio of
ws1/ul at the entrance). Also, it is assumed that n = costant*H and ul
= constant~ug, to relate these parameters (n and ul) to the total water
depth H and the densimetric velocity us, respectively. Finally, the
dimensionless form of the relationship for the mean concentration below
interface can be obtained as follows:
Cb W"s1 x
-- = exp [ i' 6(---)(---) ] ( 3.52 )
Cb1 up H
where 6 is a constant and must be determined empirically. The investi-
gation of look exchange flow (O'Brien & Cherno, 1934; Yih, 1965) showed
that the interface at the gate and the initial front speed are equal to
one half of the total water depth and densimetric velocity, respective
ly. Therefore, the 8 value in eq.3.52 could be expected to be larger
than 4, since ul is smaller than the initial front speed and the
interface at any location in the closedkend channel is lower than that
at the entrance.
3.3.2 Flow Velocity
The unsteady state fluid motion in the closed-end channel is
impossible to obtain by solving analytically the full set of equation
of motion due to the nonlinear convection term involved (eq.3.21). As
previously mentioned, there are three distinct phases found in the
front motion of a gravity current, and the third phase is the so-Acalled
viscous selfcasimilar phase, in which the front motion is governed by
gravity and viscous forces. Huppert (1982) solved the flow velocity
within the gravity current under this phase in a deep water channel.
In the present study, a gravityrviscous force balance is also consir
dered for solving the flow velocity in a sediment wedge arrested
in a shallow water channel.
The channel may be assumed two-dimensional, and the length of the
wedge will be considered to be much greater than the still water depth
H. The layer above the interface will be assumed to be sediment-free;
61
all the sediment being confined to the wedge. Thus, momentum diffusion
in the flow is more important in the vertical zrdirection than in the
horizontal x~direction. In addition, the closed end of the closediiend
channel will result in a water surface slope towards the entrance,
consequently, the hydrostatic head induced by the elevated water
surface will balance the excess pressure due to the density gradient of
sedimenthladen water in the channel as well as the shear stress
generated due to the vertical velocity gradient.
Under these conditions, the steady state equations of motion in x
and z directions can be expressed as follows:
1 ap 92u
0 = k - + ez -( 3.53 )
p ax az2
1 ap
S=; gP --- ( 3.5'1 )
The horizontal pressure gradient term in eq.3.53 can be substituted
by the result obtained by taking the horizontal derivative of the
hydrostatic pressure from eq.3.54. In the upper layer, the horizontal
pressure gradient is related only to free surface slope, but in the
lower layer, this gradient is related to both the free surface slope
and the horizontal density gradient. Therefore, eq.3.53 can be split
into two equations corresponding to both layers, which are thus
a2u ah
ez --- = 8 -- n < z < h ( 3.55 )
az2 ax
a2u ah p, pp n py all g ri ap
ez = - + g -- + dz 0 < z < n7 (3.56 )
8z2 ax p "w ax p z 8x
where h(x) is the elevation of free surface, and pp is the density of
sedimentcwater mixture at the interface. The second term on the right
hand side in eq.3.56 can be neglected, since the fluid density at the
interface, pp, is very close to the water density, p,. Here, the
sediment concentration, C, is defined as the sediment mass per unit
volume of waternsediment mixture. Therefore the density of the
water sediment mixture can be calculated from p = p,+C(141/G,), where
G, is the specific gravity of sediment. The derivation of the relaW
tionship between p and C is given in Appendix B.1. Then the integral
tion of 8p/8x in eq.3.56 can be carried out by using the expression of
eq.3.52. In order to solve eqs.3.55 and 3.56, the following conditions
are utilized; 1) no shear stress at the water surface (i.e. au/8z = 0,
at z = h), 2) notslip boundary condition at the bed (i.e. u = 0, at
z = 0), 3) the horizontal velocity is continuous at the interface, 4)
the slope of vertical velocity profile is continuous at the interface,
and 5) the volumetric rate of inflow in the lower layer is equal to the
rate of outflow in the upper layer at every location in the closedeend
channel (i.e. h~ udz = 0 ). Details of solving equations 3.55 and 3.56
are given in Appendix B3.2.
The final solution of flow velocity in upper and lower layers are
obtained as follows:
gsoH2
u = ---- [ 0.5 52 5 +" ]- 3 < i < 1 ( 3.57 )
ez 3(1 r 0.255)
gsoH2 ,3 i
u = ------- [ (0.5 ) 4
ez 353(1 0.255) (2(1 ; 0.255)
+ (14T + -------) 5 ] 0 < s < 5 ( 3.58 )
5(1 0.255)
where the water surface slope so = ah/ax, the dimensionless vertical
coordinate 5 = z/h, and the dimensionless interface elevation 5 r r7h.
Eqs.3.57 and 3.58, in which so and 5 are the function of x, are valid
everywhere in the closedcdend channel except at the entrance where the
water body is connected to open water, and the flow condition there
is critical as found by Schijf and Schonfeld (1953). Furthermore,
normalizing eqs.3.57 and 3.58 by the local surface outflow velocity,
us, these equations become
u (1.5 h 0.3755) 52 (3 A 0.755) 5 + 1
--- = ( < < 1 ( 3.59 )
us P0.5 + 0.3755
u (r/S)3+(r3+1.52o755280.3752(3)(/)+335+0.7 )gg
--- = o < ~c <
us '0.5 +0.3755
( 3.60 )
Note that us is always negative because the flow is towards the
entrance.
3.3.3 Sediment Flux
Since analytical velocity solutions of eqs.3.57 and 3.58 are not
applicable to the entrance of the closedbend channel, an attempt to
develop a practical method for determining the sediment flux at the
entrance of the channel is made in this section.
64
According to present and previous in~vestigatio~ns, the initial
front spe~ed, uf i.e., the particle velocity at the entrance immediate
ely after the gate is removed, is approximately equal to one half of
the densimetric velocity up. When the front of the turbidity current
proceeds downstream along the closed~end channel, the flow velocity at
the entrance, ul, decreases and rapidly reaches a steady state. The
particle velocity variation with the elapsed time at several locations
in the closeddend channel is shown in Fig.5.5 in Chapter 5, which
presents the simulation results obtained by the numerical model. From
the results, it is observed that the period between the maximum
velocity, i.e. during the front passing, and the steady state velocity
is much shorter compared with the overall duration of the experiment,
and the velocity at steady state is proportional to the maximum
velocity. Therefore, it is reasonable to assume that a constant
particle velocity at the entrance of the closedrend channel is estah
blished since the removal of the gate and ut = constant~ufl. There-
fore, ut can be expressed as ul = at us, since uft = 0.5 us, where a is
a constant and the densimetric velocity us = [AptgH/pg]1/2. The
sediment influx through the lower layer at the entrance per unit area
is
gH 1 1/2 3/2
S = uq *C1 = a [ -(1 re -) ] Ct ( 3.61 )
Pw Gs
where C1 is the depth averaged concentration at the entrance of the
closed(*end channel. Note that the sediment flux into the closed~iend
3/2
channel is proportional to Ct
Prediction of sediment flux computed by eq.3.61 is compared with
laboratory experimental results in Chapter 5.
3.3.4 Water Surface Rise
While a stationary sediment wedge is established in the closed end
channel, the water surface in the region beyond the toe of the sediment
wedge must be raised in order to' balance the excess force induced by
density difference at the entrance. In this section, an attempt is
made at solving for the maximum water surface rise, AHmax, based on
momentum conservation. A schematic diagram appears in Fig.3.3.
According to the equation of momentum conservation, the fol3owing
expression can be deduced
F1 + P1L 2u2 = F2 + PlU 1U1 ( 3.62 )
in which:
Fl, F2 = hydrostatic forces acting on the cross sections at
the entrance and wedge toe, respectively
plU, P1L = depthieaveraged densities in upper and lower layers
at the entrance, respectively
Q1, 42 = discharges in upper and lower layers at the entrance,
respectively and
ul, u2 = deptheaveraged velocities in upper and lower layers
at the entrance, respectively.
However, Q1 = Q2 must be satisfied due to the volumetric conservar'
tion of water at steady state. Accordingly, ul is approximately equal
to u2, since the interface elevation at the entrance is about one half
of the water depth. Therefore, eq.3.62 reduces to
F2 F1 (P1L P1U) 2u2 ( 3*63 )
in which:
F1 = B ( pl~1z~gdz) dz
Figure 3.3. Schematic of Stationary Sediment Wedge
and Water Surface Rise.
B 2
F2 = (H + aHmax) pwg, and
8 = channel width.
In eq.3.63, the term on the right hand side is assumed to be
negligible because (plL 1U) is small. Also, for F2, the term with
the order of (dH,,,)2, is neglected, since A3Imax is very small.
Consequently, the rise of water surface at the region beyond the toe of
the stationary sediment wedge, AHmax, is given as
A~eg = { -- [ pi(z)dz ] dz 4 } ( 3.611 )
H p, O z 2
where pl(z) is the density profile at the entrance of the closednend
channel.
3.3.5 TiderInduced to Turbidity CurrentcZInduced Deposition Ratio
In many coastal regions, sedimentation occurring in closed~end
channels have resulted from the combined effect of turbidity currents
and tidal motions. It is of concern from an engineering viewpoint to
distinguish the contribution of tide~induced deposition from total
deposition. In this section, an attempt to estabilish a relationship
between tide-induced deposition and turbidity current induced deposit
tion is made. A relationship for sediment flux into the closed~end
channel due to turbidity currents has been derived as eq.3.61. The
mass of sediment deposition over a tidal period can be computed as
MD = S 8 (H/2) T
( 3.65 )
6i8
where MD is the mass of depositioni induced by turbidity currents, and I
is the tidal period.
Regarding tide induced deposition, a simple way to compute the
deposited mass over a tidal cycle is using the concept of tidal prism.
This assumes that suspended sediment moves into the channel during
flood tide, then parts of It settles at slack water, during ebb tide
the remaining suspended sediment flows out of the channel. According>
ly, the deposition mass induced by tidal motion over a tidal cycle, MT,
is obtained as
MT = 20 ao L B C1 ( 3.66 )
where ao is the tidal amplitude and R is the portion of sediment
depositing inside the channel at slack water. R = 1 if no sediment-
outflow during ebb tide, and R < 1 if some sediment flows out of the
channel. Consequently, the ratio of tide;(induced to turbidity current
induced deposited mass (for R = 1 case) can be obtained, by dividing
eq.3.66 by eq.3.65, as
MT hp1 "1/2 ao L
= 1.4 ( ) (-) (-)( 3.67 )
MD Pw H L'
where Apl is the excess density at the channel entrance, and L' is the
tidal wave length in shallow water, L'=(gH)1/2T. In eq.3.65, three
basic dimensionless parameters are included, i.e. Ap1/Pw, the relative
excess density at the channel entrance, ao/H, the relative tidal
amplitude, and L/L', the channel length to tidal length ratio.
CHAPTER 4
EXPERIMENTAL EQUIPMENT AND PROCEDURES
Two sets of experiments were conducted in the present study. The
first was performed in a flume system at the U. S. Army Corps of
Engineers Waterways Experiment Station (WES), Vicksburg, Mississippi.
These experiments served as the preliminary study for an understanding
of turbidity current and sedimentation in a closed-end channel. The
experimental results have been presented by Lin and Mehta (1986).
During the tests, considerable temperature gradients in the flume were
observed. The second set of experiments was conducted in a smaller
size of flume system in a room with temperature control at the Coastal
Engineering Laboratory (COEL) of the University of Florida. Observe
tions in detail on the behavior of turbidity current and associated
sedimentation were made.
4.1 Experiments at WES
4.1.1 Experimental Set up
The laboratory tests at WES were performed in a T-shaped flume
(Fi.1.).A 9.1 m long, 0.46 m deep, and 0.23 m wide closed'end
channel was orthogonally connected to a 100 m long, 0.46 m deep,
and 0.23 m wide main channel at the approximate midpoint of the main
channel. Fig.4.1 shows the experimental set-up schematically. Both
main and closed end channels were horizontal and made of 6 mm thick
lucite. The walls of the main channel were artificially roughened
69
e
O,
111(
2 oT
o .-
como
DD O
oo o
CI 1-(L (
- N rr Y
using the lucite elements. Previous measurements in the main channel
showed that the main channel had a Manning's roughness coefficient n =
0.022 with the artificially roughened walls (Dixit et al. 1982).
Other components of the experimental set up comprised:
1) A 6.1 m square and 1.5 m deep basin at each end of the main
channel. The upstream basin including an overflow weir structure
served as the headbay and the basin at the downstream end as the
tailbay.
2) Fluid injection apparatus consisting of a water tank of 114 liter
capacity and a slurry tank of 4500 liter capacity. As shown in
Fig.4.1, a quick action valve controlled the water supply between
the water tank and the slurry tank. Four sets of 15 cm propellers
driven by a 945 rpm electric motor and a mixing pump were provided
in the slurry tank. One end of a 2.5 em diameter hose was connectF
ed with the outlet of a rotameter, and the other was placed in
the main channel 6.0 m upstream from the entrance center of the
Closed-end channel.
3) Venturi meter attached to the water supply pipe towards the headbay
in order to monitor the flow rate from the supply sump to the
headbay.
4) V notch weir to measure discharge from the main channel.
4.1.2 Auxiliary Equipment
1) Point gauges for the measurement of water surface and bed eleva-
tions at the main and closedrend channels.
2) Digital electric thermometers, shown in Fig.4.2, to monitor the
temperature variation of fluid in the main and closed~end chan-
72
nels. The thermometers could read up to the second decimal place
in degrees Fahrenheit.
3) A rotating cup current meter for measuring the distribution of the
horizontal velocity over the vertical in the main channel. The
frequency of wheel revolution could be converted to the flow
velocity from a calibration curve.
4) The suspension sampling apparatus consisted of several sampling
devices and a vacuum pump. Each sampling device included four taps
aligned vertidally, as shown in Fig.4.3; the inner diameter of each
tap was 3 mm. One end of each tap was connected by a long,
flexible plastic tube to a set of 50 ml plastic jars located in a
closed rectangular plastic box. In all, 30 jars were placed in the
closed box for collecting the suspension samples from different
locations and elevations during each test. The outlet of the
closed box was connected to a vacuum pump.
4.1.3 Test Materials
Five kinds of sediment were used in the laboratory experiments at
WES: 1) kaolinite, 2) silica flour, 3) flyash I, 4) flyash II, and 5)
Vicksburg loess. The kaolinite was available commercially (Feldspar
Coorporation, Edgar, Florida). The size distribution of the kaolinite
is given in Fig.Ll.4. The median particle diameter was 1.2 pm. The
cation exchange capacity was found to be approximately 6.0 milliequiva-
lents per 100 grams (Parchure, 1983). Its specific gravity was 2.57
(Lott, 1986).
The particle size distribution of silica flour obtained by a
Sedigraph Particle Size Analyzer, ~which uses an x r~ay beam to detect
Figure 4.2. A Digital Electric Thermometer at WES.
Figure 11.3. Suspension Sampling Apparatus Deployed in the WES Flume.
80
o\\1
i;S O ~\~ ~Flyash II
~JSilica Flour
40 FlyOSh I .
V Cedar Key Mud
** 2 Flyash Ill
200- 100 50 10 5 0.5 ~0.2
GRAIN SIZE IN MICRONS, pzm
Figure 4.4. Particle Size Distributions of Fine Grained Sediments
Koolinite
75
the concentration decrease due to settling in a sample cell, is shown
in Fig.4.4. The median particle diameter was 7 um. The specific
gravity of the sediment was 2.65. No organic matter was found.
The particle size distribution of flyash I was obtained using the
Sedigraph Analyzer and is shown in Fig.4.4. The median particle
diameter was 14 pm. The specific gravity of the sediment was 2.45.
Loss on ignition was found to be 3.7 percent.
The particle size distribution of flyash II was also obtained by
the Sedigraph Analyzer and is shown in Fig.4.4. The median diameter
was 10 pm. The specific gravity of the sediment was 2.37. Loss on
ignition was found to be 1.0 percent.
The particle size distribution of the Vicksburg loess was obtained
by Hydrometer analysis and is shown in Fig.4.4. The median particle
diameter of the sediment was 18 pm. Xcray diffraction analysis was
conducted at WES in order to identify the clay and nonclay minerals of
the loess. It was found to consist of five clay minerals, which are
chlorite, illite, kaolinite, montmorillonite, and vermiculite, as well
as three major nonclay minerals, which are quartz, feldspar, and
dolomite. The cation exchange capacity was found to be approximately
19 milliequivalents per 100 grams. The specific gravity was 2.68, and
the organic content was found to be 0.6 percent.
As mentioned in Chapter 3, the settling velocities of raw fine
sediments used in both WES and COEL flume tests were determined by
settling column analysis. Two different sizes of settling column were
used, as shown in Fig.4.5; one was 1.15 m high, 10 em square, (desig~
nated as column no.1), and the other was 1.9 m high, 10 em in diameter,
Figure 11.5. Two Different Size Settling Columns with a Mixing Pump.
77
(designated as column no.2). A mixing pump was used to help fully mix
the water and fine sediment by passing air bubbles into the column
prior to the beginning of a test. Inasmuch as the settling velocity
is dependent upon sediment concentration for cohesive sediment, several
tests were conducted with different initial concentrations for each
sediment. The relationship of median settling velocity to the initial
sediment concentration for each sediment is shown in Fig.4.6. A
detailed description of the column test procedure is given by Lott
(1986).
The chemical composition of water used in the tests at the WES
flume is given in Table 4.1 (Dixit et al. 1982).
Table 4.1 Chemical Composition of-a water sample from
the 100 m flume (after Dixit et al. 1982)
Ions
++
Ca
++
Mg
Na
K
++
Mn
+++
Fe
Cl
Electrical
Conductivity
PH
ppm
1.9
0.7
6.6
0.0
0.0
18.3
0.19 mmho/cm
7.8
78
0.20
0.1050
oo
ws=000 0.79
0.010001
z
<1 Sediment Column No.
o 0.005
u3 o Koolinite
t a Silica FlourI
Flyosh I
I Flyosh TI
a Vicksburq Loess
*Koolinite 2
Flyosh Tm 2
0 Cedor Ky Mud 2
0.001 e
200 500 1000 5000 10000
INITIAL SUSPENSION CONCENTRA'T10N,C (mg/l)
Figure 4.6. Median Settling Velocity vs. Concentration.
4.1.ii Experimental Procedure
1) The flow rate of fresh water to the headbay was adjusted until a
calibrated reading of the venturi meter was attained. Thereafter,
the flow was kept running for about 1.5 hours in order to establish
a steady flow with a desired water depth and velocity in the main
channel. Within the period of establishing the steady flow, the
gate located at the entrance was kept open.
2) The appropriate amounts of test sediment and water were mixed in
the slurry tank to obtain a desired slurry concentration. The
mixing propellers and the pump started operating before the
sediment was added to the tank. For easy observation of the
movement of turbidity current in the closed end channel, Rhodamine
dye (red) was added to the slurry tank.
3) Fresh water from the water tank was pumped into the main channel
through the injection hose at a flow rate equal to the slurry
injection rate required. The injection flow rate was selected to
obtain the desired suspension concentration in the flow of the main
channel.
4) Until the flow in the main channel became stable, the gate of the
closed~end channel was closed, a vertical velocity profile at the
main channel, 1.5 m dowmstream from the entrance center of the
closed~end channel, was obtained.
5) A quick action valve was switched to change the injection from
fresh water to slurry, then the sediment slurry injection to the
main channel was kept at the constant flow rate throughout the
test.
80
6) With a constant flow rate of slurry in the main channel, the gate
of the closed'end channel was removed allowing the sediment-laden
water to intrude into the closed~end channel along the lower
layer. The interface between the fresh water and the sediment
front was traced several times during a test on transparencies,
which were attached on the side wall with a scale reference.
7) Thirteen point gauges deployed in the main and closedpiend channels
are shown in Fig.4.T. The distance scale of the point gauge
location is the same as the length scale of the closedbend cha"
nnel. Water surface and bed elevations were measured using point
gauges in the main and closed~end channels at three different times
during a test, (i.e. during a constant flow rate of fresh water in
the main channel with the entrance gate closed, a constant flow
rate of fresh water in the main channel with the gate open, and a
constant flow rate of sediment~laden water in the main channel with
the gate open).
8) Three probes of digitalized electric thermometers were mounted
on point gauge supports in the main and closedkend channels as
shown in Fig.4.7. Vertical temperature profiles were taken
periodically at all three locations.
9) A dye injection technique and floating tracer particles were used
to measure vertical and lateral velocity profiles in the closed ~end
channel.
10) Seven suspension sampling devices were deployed in the main and
closedc'end channels as shown in Fig.ll.7. The distance scale of the
sampling device location was the same as the length scale of the