UFL/COEL TR/079
VERTICAL STRUCTURE OF ESTUARINE FINE
SEDIMENT SUSPENSIONS
BY
MARK ALLEN ROSS
1988
I
A
REPORT DOCUMENTATION PAGE
I. Report No. 2. 3. Recipient's Accessioo No.
4. Title nod Subtitle 5. Report Date
VERTICAL STRUCTURE OF ESTUARINE FINE SEDIMENT June. 1988
SUSPENSIONS 6.
7. Author(s) 8. Performing Organization Report No.
Mark Allen Ross FL/COELTR/079
9. Performing Organization Name and Address 10. Project/Task/Work Unit Ho.
Coastal and Oceanographic Engineering Department
University of Florida
University of Florida 11. Contract or Crant No.
336 Weil Hall DACW 3987P1064
Gainesville, Florida 32611
13. Type of Report
12. Sponsoring Organization Name and Address Technical
U.S. Army Engineers
Waterways Experiment Station
Vicksburg, MS 391800631
14.
15. Supplementary Notes
16. Abstract
Fine sediment suspension concentrations in estuaries vary with depth depending
on sediment settling and mixing processes, which are in turn dependent on the turbu
lent flow field and the type of sediment. Two important phenomena, fluid mud and
lutoclines, are characteristic of high concentration suspensions. Understanding the
significance of these phenomena is of paramount importance to quantifying the mixing
process and the rate of material transport advected with the prevailing currents.
This research investigated the physical characteristics (vertical structure) of
estuarine fine sediment suspension profiles within a comprehensive descriptive frame
work. Suspension related mechanisms of erosion, entrainment, diffusion (in the pre
sence of buoyancy stabilization), advection, settling and deposition were examined in
this context. A vertical mass transport model developed from functional relationships
between the above processes was used to explain some of the important physical
characteristics.
Lutoclines, sharp steps (gradients) in the concentration profile, are regions
where the local mixing rate is minimal. The mechanisms for their formation have been
shown to be the nonlinear relationships between 1) vertical diffusion and concentra
tion gradient and 2) vertical settling and concentration. The effect of sediment is
to further stabilize the lutocline layer thereby making it much more persistent in
high energy environments than other pychnocline (e.g., haloclines). Application of
the vertical transport model to data from settling column tests, wave flume resuspen
sion tests, and estuarine field investigations provided reasonable agreement for
lutocline dynamics. Continued
17. Originator's Key Words 18. Availability Statement
Cohesive sediment Fine sediments
Concentrated suspensions Fluid mud
Concentration profiles Sediment transport
Estuaries Sedimentation
19. U. S. Security Classif. of the Report 20. U. S. Security Classif. of This Page 21. No. of Pages 22. Price
206
Fluid mud, a nearbed high concentration layer with negligible structural
integrity, results from high bed erosion or fluidization rates relative to upward
entrainment fluxes and from rapid deposition. Sensitive pore pressure and total
pressure measurements made in a laboratory flume have been used to demonstrate
that waves, for example, provide one mechanism for fluid mud formation by rapid
destruction of effective stress in the sediment bed. The upper interface of the
fluid mud layer, by definition a lutocline, represents a local maximum in net
downward settling flux (i.e., maximum settling minus diffusive flux). The fluid
mud layer thus forms (and grows) from rapid deposition whenever the depositional
flux at the bed exceeds the rate at which the sediment can develop effective stress
(usually very low).
Fluid mud has been shown to be either horizontally mobile or stationary de
pending on the depth of horizontal momentum diffusion vertically downward into the
high concentration layer. Fluid mud tends to occur over a density range between
1.01 1.1 g/cm but due to the dependence on hydrodynamic action near the bed a
precise definition cannot be made on the basis of density alone.
VERTICAL STRUCTURE OF ESTUARINE
FINE SEDIMENT SUSPENSIONS
BY
MARK ALLEN ROSS
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1988
I
ACKNOWLEDGEMENTS
My deepest and most heartfelt appreciation is extended to my
chairman, Dr. Ashish J. Mehta, Professor of Coastal Engineering. In his
capacity as advisor, educator and friend, he has shown me many lofty
values by example. My cochairman, Dr. Robert G. Dean, Graduate Research
Professor, an individual of unmatched character and inspiration, receives
a lion's share of my gratitude.
Special thanks are extended to my committee members and teachers,
Drs. Dave Bloomquist, Wayne Huber, Jim Kirby, and Dan Spangler, who
served so patiently and were responsible for many fruitful ideas.
My Ecuadorian research partner and friend, Eduardo Cervantes, proved
to be a source of much assistance, insight and camaraderie.
Honorable mention must be made of the tireless crew at the Coastal
Engineering Laboratory especially Vernon Sparkman and Chuck Broward for
their technical assistance. Helen Twedell of the Coastal Engineering
Archives also was very helpful.
Perhaps, most importantly for me is the great sense of honor to
which I have been imbued by my family. The result of love, patience,
encouragement and support shown by my beautiful wife and parents. Their
belief in me never faltered.
Finally, financial support for this work was derived from a research
grant extended by the U.S. Army Engineers, Waterways Experiment Station,
I _
Contract No. DACW 3987P1064. Particularly, technical and
administrative assistance and input provided by Allen Teeter is
gratefully acknowledged.
TABLE OF CONTENTS
page
ACKNOWLEDGEMENTS. . . . . ... .ii
LIST OF TABLES. . . . . ... . vi
LIST OF FIGURES . . . . ... . .viii
LIST OF SYMBOLS . . . . ... . xii
ABSTRACT. . . . . .. ... xviii
CHAPTER
1 INTRODUCTION. . . . . ... .. 1
1.1 Problem Significance . . . 1
1.2 Objective and Scope. . . . 4
1.3 Outline of Presentation. . . . 5
2 VERTICAL STRUCTURE OF SUSPENSIONS . . 8
2.1 Introduction . . . . 8
2.2 Typical Concentration Profile. . . 9
2.3 Problems Related to Defining The Bed . ... .12
2.3.1 Bed Formation Concepts. . . .. .13
2.3.2 Effective Stress. . . .. .16
2.4 Fluid Mud. . . . ... .. .20
2.4.1 Stationary Fluid Mud. . . .. 22
2.4.2 Mobile Fluid Mud. . . .. .27
2.5 Lutoclines . . . ... .30
3 TRANSPORT CONSIDERATIONS. . . . .. .34
3.1 Introduction . . . . ... .34
3.2 Mass Conservation Equation . . .. .35
3.3 Diffusive Transport. . . . .. .39
3.3.1 Turbulent Diffusion . . .. .39
3.3.2 Gravitational Stabilization . .. .43
3.4 Settling . . . . ... .. .49
3.4.1 Free Settling . . . .. .50
3.4.2 Flocculation Settling . . .. 52
3.4.3 Hindered Settling . . .. .54
3.5 Vertical Bed Fluxes. . .
3.5.1 Bed Erosion . .
3.5.2 Deposition. . .
3.6 Fluid Mud Entrainment. .
3.7 Horizontal Fluid Mud Transport
4 LABORATORY EXPERIMENTS. . . . .
4.1 Introduction . . . .
4.2 Flume Study. . . . .
4.2.1 Objectives. . . . .
4.2.2 Mud Characterization. . .
4.2.3 Equipment, Facilities and Techniques.
4.2.4 Summary of Test Conditions. . .
4.2.5 Results . . . .
4.2.6 Discussion. . . . .
4.3 Settling Column Tests. . . .
4.3.1 Historical Approaches . .
4.3.2 Concentration Profile Approach .
5 MODELING RESULTS AND DISCUSSION . .
5.1 Introduction . . . .
5.2 Settling . . . .
5.2.1 Quiescent Settling . .
5.2.2 TurbulenceEnhanced Settling.
5.3 Wave Resuspension . . .
5.4 Lutocline Evolution in Severn Estuary.
5.5 Fluid Mud Transport. . . .
5.5.1 Wave Tank Fluid Mud Transport .
5.5.2 Avon River Fluid Mud Transport.
6 CONCLUSIONS AND RECOMMENDATIONS . .
6.1 Conclusions. . . . .
6.2 Recommendations. . . .
APPENDIX
A DIMENSIONAL ANALYSIS OF TRANSPORT EQUATION.
B DATA ON WAVE RESUSPENSION TESTS . .
C MODEL SOURCE CODE . . . .
REFERENCES . . . . .
. . .113
.113
.113
.114
.126
.127
.130
.137
.138
.142
.147
. . .147
. . .153
. . .156
. . .159
. . .169
.177
. 76
. 76
. 76
. 76
S 77
S 78
. 85
. 86
. 99
. .100
. .101
. .103
LIST OF TABLES
Table Page
21 Fluid Mud Definition by Density/Concentration .... .21
31 Summary of Coefficient Values for Turbulent Vertical
Diffusion of Momentum in Continuously Stratified Flow 46
Ai Wave Data (Period, Length, Height and MWS Elevation),
Run 1 . . . . . 159
A2 Visual Bed Elevations (cm), Run 1 . ... .159
A3 WaveAveraged Bed Pressures (kPa), Run 1. . .160
A4 Dynamic Pressure Amplitudes (0.1 kPa), Run 1. . .160
A5 Sediment Bed Concentrations (g/1), Run 1. . .160
A6 Sediment Concentrations Station A (g/l), Run 1. .161
A7 Sediment Concentrations Station B (g/l), Run 1. .161
A8 Sediment Concentrations Station C (g/l), Run 1. .161
A9 Sediment Concentrations Station D (g/l), Run 1. .161
A10 Sediment Concentrations Station E (g/l), Run 1. .162
A11 Wave Data (Period, Length, Height and MWS Elevation),
Run 2 . . . .. . . . .162
A12 Visual Bed Elevations (cm), Run 2 . ... .163
A13 WaveAveraged Bed Pressures (kPa), Run 2. . .164
A14 Dynamic Pressure Amplitudes (0.1 kPa), Run 2. . .165
A15 Sediment Bed Concentrations (g/l), Run 2. . .165
A16 Sediment Concentrations Station A (g/l), Run 2. .166
A17 Sediment Concentrations Station B (g/l), Run 2. .166
vi
A18 Sediment Concentrations Station C (g/1), Run 2. .167
A19 Sediment Concentrations Station D (g/l), Run 2. .167
A20 Sediment Concentrations Station E (g/1), Run 2. .168
LIST OF FIGURES
Figure Page
21 Typical Instantaneous Concentration and Velocity
Profiles in High Concentration Estuarine Environments 10
22 Schematic Representation of Bed Formation Process 14
23 Bed Formation Process According to Imai (1981). .. .. 15
24 Definition Sketch of Bed Stress Terminology . .. 17
25 Effective Stress Profiles in a Settling/Consolidation
Test (reprinted with permission from Been and Sills,
1986) . . . . . . 19
26 Mud Dynamic Viscosity Variation with Concentration. 24
27 Bingham Yield Strength Variation with Concentration 27
28 Settling Velocity Variation with Concentration
Severn Estuary Mud (adapted from Mehta, 1986) . 28
29 Vertical Settling Flux Variation with Concentration
(reprinted with permission from Ross et al., 1987). 29
210 Typical Suspended Concentration Profile Showing
Multiple Lutocline Stability Over 10 min. Period
(Kirby, 1986) . . . . .. ... 32
31 Diffusion Flux vs. Concentration Gradient ...... 49
32 Ratio C/Co of Instantaneous to Initial Suspended Sediment
Concentration Versus Time for Kaolinite in Distilled
Water (after Mehta, 1973) . . . 61
33 Simplified Description of Density Stratified
Entrainment (after Narimousa and Fernando, 1987). 64
41 Grain Size Distribution of Hillsborough Bay Mud . 78
42 Flume Configuration . . ... . 80
viii
43 Example of Pressure Gage Calibration. . .. 83
44 Example of Wave Gage Calibration. . . ... 84
45 Suspended Sediment Siphon Sampler . . .. 85
46 WaveAverage Bed Pressures at Various Times for Run 1 .87
47 WaveAverage Bed Pressures at Various Times for Run 2 .88
48 Temporal Response of Effective Stress for Run 1 . 90
49 Temporal Response of Effective Stress for Run 2 . 90
410 Structural and Visual Bed Elevations for Run 1. .. 91
411 Structural and Visual Bed Elevations for Run 2. .. 91
412 Concentration Versus 1 Pa Effective Stress Elevation. 92
413 Bed Concentration Variation With Time . .. 94
414 Visual Bed Elevation Variation With Time for Run 1. .. 95
415 Visual Bed Elevation Variation With Time for Run 2. .. 95
416 Bed Dynamic Pressure Amplitudes With Time for Run 1 96
417 Bed Dynamic Pressure Amplitudes With Time for Run 2 96
418 Concentration Profiles at Station C for Run 1 .. .98
419 Concentration Profiles at Station C for Run 2 . 98
420 Local Mean Settling Velocity as a Function of Time for
Bentonite Clay and Alum in Water (adapted from
Mclaughlin, 1958) . . . . .. 105
421 Scale Drawing of Settling Column. . . ..107
422 Grid Index used in the Settling Velocity Calculation
Program . . . . . .. .109
423 Settling Velocity Variation with Concentration of
Tampa Bay Mud . .... ...... . ... 110
51 Settling Velocity and Flux Versus Concentration
for Tampa Bay Mud . . . .. .115
52 Model Simulated vs. Measured Settling Column
Concentrations Initial Concentration, Co= 1 g/l .119
53 Model Simulated vs. Measured Settling Column
Concentrations Initial Concentration, Co= 2 g/l .
54 Model Simulated vs. Measured Settling Column
Concentrations Initial Concentration, Co= 4 g/l .
55 Model Simulated vs. Measured Settling Column
Concentrations Initial Concentration, Co= 5.5 g/l
56 Model Simulated vs. Measured Settling Column
Concentrations Initial Concentration, Co= 7 g/l
57 Model Simulated vs. Measured Settling Column
Concentrations Initial Concentration, Co= 8 g/l
58 Model Simulated vs. Measured Settling Column
Concentrations Initial Concentration, Co= 12 g/l.
59 Model Simulated vs. Measured Settling Column
Concentrations Initial Concentration, Co= 17 g/l.
510 Conceptual Model of Concentration "Thinning" in Low
Concentration Flocculation Settling . .
511 Conceptual Model for Constant Settling in Moderate
Concentration Range of Flocculation Settling .
512 Simulated Field Settling of Parrett Estuary
Suspensions . . . . .
513 Model Simulated Versus Measured Concentrations 
Run 1 . . . . . .
514 Model Simulated Versus Measured Concentrations 
Run 2 . . . . . .
515 Model Simulated and Measured Lutoclines  Severn
Estuary . . . . . .
516 Model Simulated and Measured (Kirby, 1986)
Concentration Profiles 0900 hrs . . .
517 Model Simulated and Measured (Kirby, 1986)
Concentration Profiles 1100 hrs . . .
518 Model Simulated and Measured (Kirby, 1986)
Concentration Profiles 1300 hrs . . .
519 Model Simulated and Measured (Kirby, 1986)
Concentration Profiles 1530 hrs . . .
S.119
S.120
.120
.121
.121
.122
S.122
S.124
S.124
S.127
S.129
S.129
S.133
S.134
S.134
S.135
S.135
520 Model Simulated and Measured (Kirby, 1986)
Concentration Profiles 1700 hrs . . ... .136
521 Normalized Velocity Profiles  Severn Estuary
(data from Kirby, 1986) . . . ... .137
522 Total Fluid Mud Transport in Five Minutes  Run 1. .139
523 NonDimensional Bed Shear Stress (t;) versus Wave
Steepness (H/Lo) (reprinted with permission from Dean,
1987) . . . . . .. 140
524 Calculated Fluid Mud Velocity Profile  Run 1. .142
525 Measured Fluid Mud Concentration, Velocity and
Horizontal Flux  Avon River (data from Kendrick and
Derbyshire, 1985) . . . . .144
526 Calculated and Measured Horizontal Fluid Mud
Velocities. . . . . . .146
LIST OF SYMBOLS
Symbol
b Buoyancy jump across density interface
b Body force per unit mass tensor
C Sediment suspension concentration (mass/unit volume)
Ca Concentration at upper fluid mud interface
Cb Concentration at mobile/stationary fluid mud interface
Cc Concentration at bed surface
Cd Drag coefficient for sphere fall velocity
Ceq Equilibrium concentration during deposition
Ch Interference settling velocity concentration
ChT Hindered settling (flux) concentration
Ci Concentration of class i for deposition
Cm Characteristic maximum concentration
Css Steady state concentration after deposition
CT Total concentration (sum of components)
Co Initial concentration for settling; deposition
C1 Cohesive (class) sediment concentration
C2 Noncohesive (class) sediment concentration
C Time mean concentration
C' Instantaneous concentration component about mean
C' Nondimensional concentration C/Cm
d Equivalent spherical diameter of sediment grain
D Molecular diffusivity
d50 Sediment grain size diameter of 50% greater than fraction
Eij Turbulent diffusivity components in i,j direction
Em Turbulent momentum diffusion rate (eddy viscosity)
E* Entrainment coefficient, ue/u*
f DarcyWeisbach friction factor
Fb Vertical sediment bed flux (Fe+Fp)
Fd Vertical sediment flux from diffusion
Fe Vertical sediment flux from erosion
Fp Vertical sediment flux at the bed from deposition
Fpi Class i vertical sediment flux from deposition
F, Vertical sediment flux from settling
g Acceleration of gravity
h Water depth
H Wave height
Hb Breaking wave height
Ho Deep water wave height
k Wave number (2n/L)
K Turbulent mixing tensor
Kn Local neutral mixing rate
Ks Local mixing rate in presence of stratification
Kx,y,z Turbulent mixing components (cartesian)
Kx,y,z' Nondimensional turbulent mixing components (cartesian)
kI Flocculation settling velocity constant
k2 Hindered settling velocity constant
xiii
1 Mixing length scale of turbulence
L Lutocline layer; wave length
Lo Deep water wave length
m Mass flux of sediment across bed boundary
nI Flocculation settling velocity constant
n2 Hindered settling velocity constant
P Pressure variable used in the horizontal momentum equation
P Relative Probability for deposition rate expression
Pi Relative Probability for deposition (class i)
Ph Hydrostatic pressure
Ppw Pore water pressure
P' Nondimensional pressure P/yH
q Mass flux vector
R Reynolds' number of sediment grain (wsd/v)
Ri Gradient Richardson number
Ri* Bulk Richardson number (bh/u)
Riu Richardson number based on average velocity
Rw Wave Reynolds number
Rt Shear stress ratio, ty/lo
Sc Turbulent Schmidt number
t Time variable
to Characteristic time scale
t' Nondimensional time t/to
T Wave period
T Stress tensor
u Velocity component in xdirection
xiv
uo Characteristic velocity scale
U Imposed velocity on the sheared turbid layer
Ue Entrainment rate
ub Maximum nearbed orbital velocity
Ue Entrainment rate (dh/dt)
Au velocity jump across stratified layer
u* Friction velocity (/to/p)
u Time mean velocity
u' Nondimensional velocity u/uo
u' Instantaneous velocity component about mean
v Velocity component in ydirection
w Velocity component in zdirection
ws Sediment settling velocity
ws' Nondimensional settling velocity
wsm Characteristic maximum settling velocity
wso RichardsonZaki reference settling velocity
Wsol Stokes' settling velocity
wso2 Reference settling velocity for average floc size
x Longitudinal (horizontal) cartesian coordinate direction
x' Nondimensional horizontal direction x/L
y Lateral cartesian coordinate direction
z Elevation variable (positive upwards)
z' Nondimensional vertical direction
Za Upper fluid mud interface elevation
Zb Mobile/stationary fluid mud interface
zc Bingham plastic yield elevation
xv
Z Bed elevation
a Wave diffusivity constant
as r27/1
ay Yield strength calculation constant
a Viscosity/concentration constant
a' Munk and Anderson constant
al,a2 Erosion rate constants
ag Viscosity ratio, p24/1
B Settling velocity constant
Bi Settling velocity constant for sediment class i
Be Exponential diffusivity constant (mass diffusivity)
BH Holtzman constant
BMA Munk and Anderson constant
BOR Odd and Rodger constant
BRM Rossby and Montgomery constant
By Yield strength calculation constant
Pp Viscosity/concentration constant
B' Munk and Anderson constant mass diffusivity
Bg Coefficient used in fluid mud calculations
S Intermediate entrainment layer
6c Similarity variable (zc(t)/2/t)
8' Scl/as
Sfm Mobile fluid mud thickness
6i Upper entrainment layer thickness
Ss Shear layer thickness; Similarity variable (z/2vr/t)
s S,/,as
yFluid shear rate (au/az)
K von Karman constant (0.4)
xvi
p Density of water
Pb Bulk density of suspension
Po Fluid reference density for stratification
Ps Granular density of dry sediment
Pw Density of suspension fluid (water)
p' Nondimensional density p/Pm
PW', Dynamic viscosity of suspension fluid (water)
Im Dynamic viscosity of mud suspension
vm Kinematic viscosity of mud suspension (pm/p)
v' Nondimensional kinematic viscosity
T Odd and Rodger peak gradient Richardson number
c Munk and Anderson constant mass diffusivity
Co Erosion rate constant
a' Bed effective stress
o Total stress; wave frequency (2n/T)
tb Applied (timemean) bed shear stress
tbm Critical bed shear stress for partial deposition
tcd Critical bed shear stress for total deposition
to Bed shear stress
Cs Bed shear strength for erosion
tcz Shear stress component acting in xdirection on zface
Ty Yield strength of bed deposits
X Log average of sediment concentration
V Vector operator
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
VERTICAL STRUCTURE OF ESTUARINE FINE SEDIMENT SUSPENSIONS
By
Mark Allen Ross
August 1988
Chairman: Ashish J. Mehta
Major Department: Civil Engineering
Fine sediment suspension concentrations in estuaries vary with depth
depending on sediment settling and mixing processes, which are in turn
dependent on the turbulent flow field and the type of sediment. Two
important phenomena, fluid mud and lutoclines, are characteristic of high
concentration suspensions. Understanding the physical significance of
these phenomena is of paramount importance to quantifying the mixing
process and the rate of material transport advected with the prevailing
currents. This research investigated the physical characteristics
(vertical structure) of estuarine fine sediment suspension profiles
within a comprehensive descriptive framework. Suspension related
mechanisms of erosion, entrainment, diffusion (in the presence of
buoyancy stabilization), advection, settling and deposition were examined
in this context. A vertical mass transport model developed from
functional relationships between the above processes was used to explain
some of the important physical characteristics.
Lutoclines, sharp steps (gradients) in the concentration profile,
are regions where the local mixing rate is minimal. The mechanisms for
their formation have been shown to be the nonlinear relationships
between 1) vertical diffusion and concentration gradient and 2) vertical
xviii
I 
settling and concentration. The effect of sediment settling is to
further stabilize the lutocline layer thereby making it much more
persistent in high energy environments than other pychnoclines (e.g.,
haloclines). Application of the vertical transport model to data from
settling column tests, wave flume resuspension tests, and estuarine field
investigations provided reasonable predictive agreement for lutocline
dynamics.
Fluid mud, a nearbed, high concentration layer with negligible
structural integrity, results from high bed erosion or fluidization rates
relative to upward entrainment fluxes and from rapid deposition.
Sensitive pore pressure and total pressure measurements made in a
laboratory flume have been used to demonstrate that waves, for example,
provide one mechanism for fluid mud formation by rapid destruction of
effective stress in the sediment bed. The upper interface of the fluid
mud layer, by definition a lutocline, represents a local maximum in net
downward settling flux (i.e., maximum settling minus diffusive flux).
The fluid mud layer thus forms (and grows) from rapid deposition whenever
the depositional flux at the bed exceeds the rate at which the sediment
can develop effective stress (usually very low).
Fluid mud has been shown to be either horizontally mobile or
stationary depending on the depth of horizontal momentum diffusion
vertically downward into the high concentration layer. Fluid mud tends
to occur over a density range between 1.01 1.1 g/cm3 but due to the
dependence on hydrodynamic action near the bed a precise definition
cannot be made on the basis of density alone.
xix
CHAPTER 1
INTRODUCTION
1.1 Problem Significance
Finegrained, cohesive sediment is transported in suspension from
fluvial and marine sources to depositional environments including
navigation channels and harbors. This sediment affects water quality by
transport of sorbed nutrients (or pollutants) and light penetration
(Hayter, 1983). Shoaling is often one other critical issue. In the
continental United States alone, the cost of maintenance dredging of
coastal waterways, including estuarine ports and harbors, is
approximately onehalf billion dollars per year (Krone, 1987). Estimates
of contaminant removal or dredging requirements are dependent upon a
knowledge of the rates of horizontal transport of the suspended material
over periods ranging from days to years. The accuracy of predictions,
typically via numerical solutions of the sediment mass transport
equation, is therefore strongly contingent upon an understanding of the
structure of the vertical profile of sediment concentration and
interaction with the turbulent flow field.
Present day modeling of cohesive sediment transport is limited by
knowledge of the fundamental transport processes of erosion, entrainment,
settling, deposition and consolidation of these sediments. In
particular, the dynamics of estuaries with relatively high concentration
suspensions typical of macrotidal (tidal range > 4 m) environments are
1
poorly understood (Parker, 1987). In this context engineers and other
scientists are beginning to deal with the important question of fluid
mud, loosely defined as a high concentration slurry transported in the
form of a relatively thin suspension layer near the bed by the prevailing
currents. At present, there are difficulties associated with measuring
the slurry concentration and transport velocity. The result is that
large errors often occur in calculating the associated horizontal flux of
sediment transport over the water column.
Fluid muds also occur in meso (24 m) and microtidal (< 2 m)
estuaries and along the open coasts where waves play a more important
role than in macrotidal environments. Wells and Kemp (1986) observed
that waves traveling over nearshore mud shoals principally acted as an
agent for softening and fluidizing the muddy bed. Maa and Mehta (1987)
made similar observations in laboratory flume tests. In nearshore areas
waves can thus significantly assist currents in transporting fluidized
material to sites prone to sedimentation. Consequently, in microtidal
waters the generation and transport of fluid mud is far more episodic
than under macrotidal conditions.
Understanding the dynamics of fluid mud is central to the issue of
understanding the response of the vertical concentration profile to
hydrodynamic forcing by currents and waves. Unlike the boundary of beds
composed of cohesionless material (e.g., sandy beds), the cohesive bed
boundary is often poorly defined as it is not evident, e.g., from echo
sounder data, at what depth the nearbed suspension ends and the bed
begins. Parker (1986) noted ambiguities when lead lines, echo sounders
or nuclear transmission or backscatter gauges are used to identify the
3
bed. In fact, Ross et al. (1987) noted that due to the dynamic nature of
the cohesive bed boundary which responds significantly to hydrodynamic
forcing, the density of the suspension by itself cannot be used either to
identify the cohesive bed boundary or the fluid mud layer which occurs
immediately above this boundary. An understanding of the interaction
between the concentration (or density) profile with the flow field is
critically important.
Kirby (1986) recently published a summary of extensive field
observations made in the Severn estuary, a macrotidal estuary (maximum
tidal range 14.8 m) on the west coast of England. Large mass transport
rates via fluid mud generation regularly occur in this estuary. The
dynamic interaction between the concentration field and flow field are
further complicated by the extremely high concentrations. Surface
concentrations reach 1000 mg/l easily, which may be compared with s 20
mg/l in Florida's coastal waters (Mehta et al., 1984). A significant
observation in the Severn was the generation of rather sharp gradients in
concentration termed lutoclines, which rise and fall through the water
column depending upon the flow condition. According to Kirby (1986),
lutoclines, which are analogous to other types of pycnoclines, e.g.,
haloclines, seem to occur where the suspension concentration exceeds 
500 mg/l. However, they differ from other pycnoclines by the added
process of sediment settling. Sediment settling further supports
stabilization and resulting high density gradients. For this reason
lutoclines are much more persistent than for example haloclines in high
energy environments. An example of a lutocline is the upper level of
fluid mud within which concentrations can typically exceed 10,000 mg/l.
4
Above this level, lutoclines often show up as multiple "steps," which
represent local complex imbalances between diffusive and settling fluxes.
Kirby (1986) observed that lutoclines often are not simulated properly by
numerical models with resulting errors in the estimates for the rates of
mass transport.
The aforementioned issues illustrate the strong need to examine the
entire question of the vertical structure of concentration and its
interaction with the flow field within a comprehensive framework. An
attempt is made in this thesis to approach the problem via analysis of
laboratory and field measurements within a descriptive framework for the
vertical concentration structure. New definitions are proposed and the
dynamics of the concentration profile are adduced through relatively
simple mathematical models which are verified by laboratory and field
data. The objectives and scope are accordingly as follows.
1.2 Objective and Scope
The objectives of this study were to
1. Define the physical characteristics of fine sediment
suspension profiles in estuaries including lutoclines,
fluid mud, and the cohesive bed within a comprehensive
descriptive framework.
2. Determine the important physical mechanisms and processes
which influence these characteristics.
3. Develop simple but useful qualitative and quantitative
descriptions for these processes which could be used in a
predictive capacity to model suspensions in the prototype
environment.
To meet these objectives the scope of this research was as follows:
1. Laboratory tests were conducted, using natural estuarine
5
sediment, to measure the parameters important to cohesive
bed and suspension profile definitions.
2. For the simple vertical structure model development, only
vertical transport fluxes were considered. Analysis of
turbulent diffusion was based on classical mixing length
approximations and gradient Richardson number bouyancy
stabilization relationships. Sediment settling velocity
expressions were concentration dependent.
3 Horizontal transport in the fluid mud layer was calculated
from consideration of momentum diffusion resulting from
applied interfacial shear stress.
4. Verification of model applicability was limited to
comparisons with selected field and laboratory data (e.g.,
time series concentration profiles).
1.3 Outline of Presentation
The study is presented in the following order. Chapter 2 can be
regarded as a description and definition chapter. Specific justification
is presented for delineating processes influencing vertical suspended
sediment structure. Physically based, qualitative definitions are given
for lutoclines, stationary and mobile fluid mud layers, and bed
elevation. This chapter also describes many of the complexities
associated with defining the cohesive bed from theoretical and applied
perspectives.
Chapter 3 presents the theoretical development of the vertical
transport and momentum diffusion models. For the transport model, the
advectiondiffusion equation is given and the individual terms are
discussed. Entrainment, diffusion, settling and bed fluxes are
addressed. For the fluid mud momentum diffusion model, theoretical
formulations are presented with assumptions concerning theological and
temporal responses.
6
Chapter 4 presents the objectives, procedures and results of
laboratory experiments with three specific themes: cohesive bed dynamics
associated with waveinduced bed fluidization and delineation of the
cohesive bed boundary; wave resuspension with emphasis on the evolution
of the suspension profile with time; and settling velocity determination.
The natural estuarine sediment used in the tests is characterized in
Section 4.2.2. Historical approaches toward settling velocity
determinations are discussed in the context of strengths and weaknesses.
Section 4.3.2 presents an improved procedure for determining cohesive
sediment settling velocity concentration relationships using settling
columns.
The application of the vertical transport model is presented in
Chapter 5. An early attempt was made to progressively verify the
individual routines in the model, before concurrent simulation. Thus,
for example, the settling routine was first tested by reproducing
quiescent (column) settling results (a nearly pure settling condition,
see, for example, Yong and Elmonayeri, 1984, for diffusion in quiescent
settling). Next, the diffusion, erosion and deposition routines were
added and tested. Predictions of lutocline formations in field
conditions together with fluid mud layer development in a wavetank
illustrating the model's ability to handle sediment fronts (i.e., sharp
concentration gradients) are also shown in Chapter 5.
The fluid mud horizontal transport model results also are included
in Chapter 5. Steady and unsteady simulations of wavetank data
(presented in Chapter 4) and field data (published by Kendrick and
Derbyshire, 1985) are shown.
7
Conclusions, recommendations for future research and miscellaneous
closing comments are given in Chapter 6.
Appendix A presents a dimensional analysis to determine the
important terms in the transport equation. Appendix B contains the
tabulated laboratory data taken during wave resuspension tests. Appendix
C is a printout of the lD vertical transport model developed for this study.
CHAPTER 2
VERTICAL STRUCTURE OF SUSPENSIONS
2.1 Introduction
Suspended sediment concentration in estuaries varies greatly with
depth, the highest concentrations being usually found nearest the bed.
Simply stated, this variance is because gravitational flux (associated
with settling) counteracts mixing and prevents the sediment from becoming
uniformly mixed as is the case with neutrally buoyant or dissolved
constituents. In an equilibrium profile (profile not changing with time)
the vertical flux associated with settling is everywhere equal to the
vertical flux associated with upward diffusion (typically turbulent
mixing). For sediment with a constant settling velocity and mixing based
on a Prandtl/von Karman mixing length approximation, analytical solutions
for the concentration profile follow the classical works of O'Brien
(1933) and Rouse (1937). In a fully developed turbulent flow in the
absence of significant density gradients the mixing rate, which is
directly proportional to the eddy scale of turbulence, is smallest near
the bed and increases upward reaching a maximum approximately at mid
depth (Schlichting, 1979). However, sediment in suspension can greatly
increase the bulk density of the water. If high concentration (density)
gradients develop, turbulent mixing will be greatly damped locally
(Fischer et al., 1979). This has been well documented for stratified
flows associated with dissolved salt and temperatureinduced density
9
gradients (Turner, 1973), but has been generally overlooked by classical
solutions of vertical sediment transport (Rouse, 1937; Raudkivi, 1967).
This chapter presents the physically based definitions for the
vertical structure (or vertical characteristics) of finegrained
suspended sediment profiles. Much of the terminology and descriptions
used have been liberally applied in broad contexts in previous related
and unrelated studies. Most were disparate in their objective. The
following pages will help clarify the usage.
2.2 Typical Concentration Profile
Figure 2.1 shows a typical instantaneous concentration profile as
might be observed in a high sediment load environment such as the Thames
River (UK), San Francisco Bay or the Severn Estuary. While the values
are assumed, they are representative of those commonly reported in the
literature (Parker and Kirby, 1979).
It is noted that there is a 45 order of magnitude range in
concentration from water surface to bed surface. While most sediment
transport models focus primarily on calibrations of the upper water
column concentrations, the significance of neglecting the near bed layers
should be obvious, but will be shown in detail.
The largest layer is the mobile suspension layer which extends down
to reference level, Za. This is the layer that is most often turbulent.
It is also generally, dominated by pressure gradient driven flow
associated with water surface elevation gradients resulting from tides
and freshwater discharge. Concentrations in the upper mobile suspension
10
CONCENTRATION, C (mg/I)
1 2 3 4 5 6
10 10 10 10 10 10
0 ^  I  I 
L3 U
C
E 2
N
Mobile L2
LL Suspension
S4 Lutocllne
4D Layer
0
SZA L1
m +
6 Mobile Fluid Mud
LL Z, Br.L
Stationary "Fluid" Mud
Zc
8 Bed
0 0.25 0.50 0.75 1.00 1.25
VELOCITY, u(m/s)
Figure 21. Typical Instantaneous Concentration and Velocity Profiles
in High Concentration Estuarine Environments
11
layer are usually 11000 mg/l but in rare cases exceed 10,000 mg/l (in
the lowest portions) during extreme tides or storm conditions (Parker and
Kirby, 1979).
At various levels in the mobile suspension layer there can exist
sharp increases in concentration which result from and further support
local minima in mixing and upward vertical diffusion. These are termed
lutocline layers which are one form of pycnoclines (regions of sharp
density gradients). There can be multiple lutocline layers but more than
23 is rare. Such multiple layering in salinity or thermal structure is
called finestructure (Posmentier, 1977).
Below Za there is a sharp increase in concentration above 10,000
mg/l to 100300 g/l. This is the socalled "fluid mud" layer defined in
Section 2.4. Thus, Za represents a lutocline between upper column mobile
suspensions and nearbed fluid mud. This is often mistaken as the bed on
echo sounder records (Kirby, 1986). Depending on the theological
properties of the mud, the magnitude and duration of the applied
interfacial (lutocline) shear stress, and/or the internal pressure
gradients, a portion of this fluid mud layer is mobilized to flow in a
direction with the applied forcess. The interface between the mobile
and stationary mud suspensions is labeled reference level Zb. The
symbol, Zb, will not necessarily be identifiable from concentration
profiles but instead must be identified from accurate measurements of the
velocity profile. Below the fluid mud layer at reference level Zc there
exists a definable sediment interface below which the sediment exhibits
bed properties based on classical soil mechanical definitions. This is
the cohesive bed elevation, above which only suspension occurs (discussed
12
in Section 2.3). Strictly speaking, the stationary fluid mud layer (Zb
Zc) may not necessarily behave as a fluid (i.e., not supporting shear
stresses), but since it fits the general definition for fluid mud (i.e.,
nearbed, highconcentration layer) the terminology is nevertheless
retained.
A typical velocity profile is also shown in Figure 21 for
reference. It is shown to be of almost logarithmic form in the mobile
suspension layerindicative of turbulent flow. Near the fluid mud layer
turbulence is dampened out and there is a transition layer which,
proceeding down with depth, gives way to a shear flow viscous layer.
This is analogous to stratified flows of salts (Yih, 1980; Narimousa and
Fernando, 1987) and is described further in Chapter 3.
2.3 Problems Related to Defining the Bed
When trying to determine the bed elevation, Zc, to do so on the
basis of concentration only is imprecise. As pointed out by Sills and
Elder (1986) and in this report, bed properties (i.e., development of
effective stess as defined in Section 2.3.2) can exist in concentrations
as low as 7080 g/l, depending primarily on the fluid suspension and bed
dynamics (stress, strain and strain rate) at any particular time. Thus,
under field conditions, a precise identification of the cohesive bed
interface would not be possible without dynamical data (e.g.,
measurements to determine effective stress). Bed definition is also
dependent on previous formation conditions, wave and current actions and
consolidation properties of the particular sediment. These phenomena
provide justification for a brief discussion of bed formation and
13
consolidation concepts followed by a subsection (2.3.2) on the concept of
bed definition related to effective stresses.
2.3.1 Bed Formation Concepts
To characterize the process of bed formation in a laboratory or
field setting it is important to distinguish the mode of deposition.
According to Parchure and Mehta (1985), in the laboratory, bed formation
can be in the form of a "placed" or "deposited" bed. Placed beds are
those developing from high concentration slurries. Deposited beds result
from lower concentration, particle by particle deposition. Placed beds,
therefore, are more uniform vertically, whereas deposited beds are non
uniform and dewater relatively rapidly. The specific character of each
bed type is most pronounced earliest after formation, decaying with time
until the properties are nearly indistinguishable. Because of the time
scales involved, placed beds are probably more typical of laboratory
conditions; however, rapid fluid mud deposition in an estuary would have
similar characteristics.
In the field, an alternative to considering the bed based on
depositional mode is to examine in detail the physics of mud deposition
and bed formation. A schematic representation of the bed formation
process is shown below (Figure 22).
There are basically two mechanisms responsible for bed formation:
sedimentation (deposition) and consolidation. Sedimentation can be
defined as the process by which particles or masses of particles leave
suspension and settle onto the bed under gravity. Consolidation in a
fully saturated environment results from particle framework (mineral
Dilute
Suspension
Concentrated
Suspension
Decreasing
Increasing Rate of
SoftRapidly Concentration Vertical
Consolidating Movement
Mud
Consolidated
Bed
Figure 22. Schematic Representation of Bed Formation Process
skeleton) deformation under applied stress. The applied mechanical
forces can be either due to net negative buoyancy (selfweight) or
imposed overburden (surcharge) loading.
Imai (1981) gave a description and graphical model of the bed
formation processes. Figure 23 shows this description.
The flocculation stage in Figure 23 actually includes the complex
process of particle destabilization by doublelayer suppression in the
presence of available cations and subsequent aggregation by interparticle
collision and cohesion. The floc formation process takes place under
settling conditions as pointed out by Krone (1962). The settling zone
shown in Figure 23 would be more appropriately labeled hindered (or
zone) settling. No further discussion of settling will be given here as
the settling process is discussed further in Chapter 3. Between times tl
C i*..n.:.n. inTerrace
o
SSettling I ... Soil Formation Line
> Flocculation 'Zone." .' ..
S.'.' "" I
Consolidation Zone
0 t t2
Time
Figure 23. Bed Formation Process According to Imai (1981)
and t2, sediment flocs settle to form a soft bed. The bed is continually
built up by continuous deposition of these flocs but simultaneously
undergoes dewatering and consolidation. During this time, bed properties
begin to change with depth due to particle rearrangement and larger floc
breakdown (Krone, 1962). After the settling stage, consolidation
continues and the bed slowly begins to "harden" as depthvariation in bed
properties (i.e., density, effective stresses, etc.) become more
pronounced. This selfweight consolidation approaches a steady state
condition exponentially.
Since it is not within the scope of this study to discuss the
details of consolidation, it will suffice to conclude this section by
stating that a vast amount of geotechnical literature on
16
sedimentation/consolidation theories is available. The pioneering work
by Terzaghi (1923) using onedimensional finite strain theory now has
evolved into complex multidimensional, nonlinear finite strain
theories. The reader is directed to the paper by Schiffman et al.
(1986), which presents a noteworthy historical, theoretical, and applied
account of onedimensional sedimentation and consolidation.
2.3.2 Effective Stress
Given that the porous medium (the cohesive bed) is a twophase
system consisting of a deformable mineral skeleton filled with an
incompressible liquid (water), the effective stress, o', is defined as
the difference between the total stress, a, and the pore water pressure,
Ppw, at any point:
a' = a Ppw (2.1)
Empirically, it is found to be the controlling parameter in determining
soil strain, deformation and strength (Schiffman et al., 1986).
Classically, one type of soil failure is defined as a "quick" condition
in which the effective stress tends toward zero (Sowers, 1976).
Another important parameter is excess pore pressure, Au. This is
the difference between actual pore water pressure, Ppw (e.g., as measured
by a manometer), and hydrostatic pressure, Ph. Under dynamic conditions,
if the sum of excess pore pressure, Au, and hydrostatic pressure, Ph,
approaches the total stress, a, liquefaction occurs (Perloff and Baron,
1976),
(Liquefaction)
Figure 24 is a definition sketch for these terms.
Water Surf,
Mobile Suspension
Ph=
I \Fluid Mud Surfs
o'=0
< Fluid Mud
"> \/Bed Surft
LIJ
LLI
PRESSURE
Figure 24.
Definition Sketch of Bed Stress Terminology
In a nonfluidized sandy (or any large porosity) sediment bed the
effective stress everywhere is nonzero. The total pressure is the
integral of the density profile over depth (acted on by gravity) and the
pore pressure is everywhere hydrostatic. For finer sediments which are
much less permeable, pore pressures easily increase to above hydrostatic.
In the upper bed where the pore water pressure is equal to the total
pressure, the sediment is in suspension and the water bears the weight of
the sediment (increased bouyancy through higher bulk densities). When
Au + Ph a
(2.2).
18
the pore pressure drops below the total vertical stress, there is
particle interaction. Thus, a weak structure forms that is able to
support some of the weight of the sediment. Therefore, the development
of effective stress provides a fundamental distinction between suspension
and structural bed, i.e.,
o' = 0 ; Ppw = : Suspension
o' > 0 ; Ppw < a : Bed (2.3).
The elevation of the (structured) bed, Zc in Figure 21, therefore,
should be based on the development of effective stress below this
elevation. Within the bed, the interaction between sediment flocs
provides a resistance to erosion due to frictional and electrochemical
bonding. A reduction in effective stress, therefore, leads also to the
reduction in the yield strength and the critical shear stress for
erosion. No effective stress means no interaggregate contact or
friction. This important distinction was pointed out by Sills and Elder
(1986).
Been (1980) and Been and Sills (1981) made extensive laboratory
measurements of the development of effective stress in quiescent
settling/consolidation of fine sediments. A representative plot of
measured effective stress profile is shown in Figure 25.
In their experiments on Combwich mud using different initial
concentrations, no unique concentration was found at which effective
stress developed. The concentration range over which structural
development occurred was between 80 and 220 g/l depending on the initial
Pressure plotted obove
hydrostatic
S800 800 o Totol stress
E E X Pore pressure
 0I
400 400
0 0
1.0 11 12 0 2 0 06 08 10 12
Density Mg/m3 Pressure kN/m2
Initial density 109 Mg/m rn hour profile
Figure 25. Effective Stress Profiles in a Settling/Consolidation Test
(reprinted with permission from Been and Sills, 1986)
concentration of the slurry. One significant observation was that
effective stress existed always in concentrations greater than 220 g/1.
This observation seems to imply that structural phase development is
dependent on sedimentation rate especially in low concentration quiescent
conditions. It is noted in the following sections that structural phase
development is also dependent on hydrodynamic agitation. This dependence
is demonstrated in laboratory tests of wave erosion as described in
Chapters 4 and 5.
2.4 Fluid Mud
As stated in Chapter 1, fluid mud is defined as a nearbed, high
density, cohesive sediment suspension layer (Ross et al., 1987). In
20
areas with large bed slopes, fluid mud loosened by currents or waves can
flow down the slope by gravitational forces similar to mudslides and
debris flow on hillsides (Odd and Rodger, 1986). For this reason,
navigation channels and basins are especially vulnerable to this type of
sedimentation.
Many investigators have identified fluid mud in terms of a range of
bulk density (or concentration) of the sedimentfluid mixture, as noted
in Table 21. It should be pointed out that these investigations were,
in general, disparate in terms of their aims, dealing with field
observations or laboratory tests. Nevertheless, there seems to be some
agreement amongst the proposed densities initially suggesting perhaps an
approximate range of 1.03 to 1.20 g/cm3 (concentration range of 10 to 320
g/1).
To provide a quantitative definition for fluid mud based on a
discrete concentration range is not possible because, as pointed out in
the previous section, the effect is not simply dependent on concentration
but on the flow conditions and sediment settling properties. Therefore,
the values given in Table 21, without qualifying the particular flow
conditions and sediment settling behavior under which the ranges apply,
are not amenable to developing a general definition applicable in all
cases. All that can be deduced from the tabulated data is that fluid mud
seems to occur within a rather wide concentration range of between 3 and
500 g/l (two orders of magnitude).
Fluid mud can form during rapid erosion or deposition. During
erosion, if initially the erosion rate greatly exceeds the turbulent
entrainment rate, i.e., the rate at which sediment is mixed by turbulence
21
into the upper column mobile suspension layer, the nearbed high
concentration further dampens turbulent mixing and the nearbed
Table 21. Fluid Mud Definition by Density/Concentration
Conversion between density and concentration
density of 2.65 g/cm3.
based on assumed sediment
suspension can be stabilized as a stratified flow. This effect is often
the case in wave erosion (Maa and Mehta, 1987). This is discussed in
Chapter 3 and later shown in laboratory wave resuspension tests
documented in Chapter 4. During deposition, if the sediment deposition
flux exceeds the rate of pore fluid transport upward (dewatering rate of
the suspension), dense nearbed suspensions are formed that grow upward
and only slowly consolidate (see Section 2.3.1).
As shown in Figure 2.1, fluid mud can occur as a mobile or
stationary suspension. This distinction was first made by Parker and
Kirby (1977). Both conditions are discussed in the following sections.
Density/Conc. Range
Investigators)
Bulk Degsity Concentration
(g/cm ) x 10 (mg/1)
Inglis and Allen (1957) 1.03 1.30 10 480
Krone (1962) 1.01 1.1a 10 170
Wells (1983) 1.03 1.30 50 480a
Nichols (1985) 1.003 1.20 3 320
Kendrick and Derbyshire (1985) 1.12 1.25a 200 400
2.4.1 Stationary Fluid Mud
Within the fluid mud layer there are typically two distinct regions
separated by a level below which no horizontal motion takes place. In
the definition sketch (Figure 21) this was elevation Zb. For instance,
this elevation might be considered to be the applicable elevation of the
bottom boundary condition for a horizontal transport model. However,
this level is quite sensitive to the theological response of the mud from
imposed stress (e.g., the lower extent of the vertical momentum diffusion
resulting from an applied horizontal shear stress at the upper fluid mud
interface). For the purpose of describing why this layer exists and how
it is differentiated from the mobile layer above, several simple
arguments are presented here and are more formally posed in Chapter 3
(Section 3.7).
It is possible to estimate the stationary layer elevation, Zb, by
making several simplifying assumptions. As a first approach, analogy can
be made between flow in the fluid mud layer and unsteady couette flow
development beneath an infinite plate moving with a constant velocity
after being started from rest. A shear stress results on the upper fluid
mud interface, elevation Za in the definition sketch (Figure 21),
because of an imposed velocity in the upper column (mobile suspension).
Momentum diffusion then occurs over a finite thickness, 6fm, in the fluid
mud layer. For a constant kinematic viscosity, vm of the mud, the
temporal response (for relative time t after imposing the shear stress)
of the mobile fluid mud layer thickness, 8fm, is
6fm = P6 Vmt
(2.4)
where pg is a constant (Eskinazi, 1968). However, this leads to
calculations of layer thicknesses of many meters after periods of
minutesunreasonably large values. Also, the viscosity of these high
concentration layers is not constant but concentration (and thus depth)
dependent (Krone,1962).
Another approach is to consider the slurry to have a concentration
dependent dynamic viscosity, pm(C). The density of the bulk suspension,
Pb, is given by a simple linear relation to the (mass/unit volume)
concentration, C, as
(2.5)
Pb Pw + C (1 )
Ps
where p, is the density of the suspending fluid (water) and ps is the
granular density of sediment (typically 2.65 g/cm3).
A summary of empirical relationships for dynamic viscosity variation
with concentration is shown in Figure 26. The trend, in the fluid mud
range (10 C 200 g/1), seems to be of an exponential or polynomial
form: i.e.,
apC
Pm " w e
Pm w(1 + Pp CaC )
(2.6a)
(2.6b)
where pm is the dynamic viscosity of mud suspension, pj is the dynamic
100
80 Krone(1963)
S60 
L 40 
S20 EngelundZhaohul(1984)
) 20 (Bentonlte)
( 10 / Delft Hyd. Lab (1985)
> 8
W 6
S4 EngelundZhaohul(1984)
(Kaolinite)
cc 2
0 100 200 300 400 500
SUSPENDED SEDIMENT CONCENTRATION (g/I)
Figure 26. Mud Dynamic Viscosity Variation with Concentration
viscosity of clear (free from suspension) water, C is the concentration
of suspended sediment, a a, a and 0p are empirical coefficients.
Engelund and Zhaohui (1984) proposed a relationship of the form of Eqn.
(2.6b) for kaolinite suspensions. They found 1/a 1 = 0.206 and a. = 1.68
for kaolinite concentration (in percent) and fresh water (p~ = .001 N
s/m2). Equation 2.6b represents a truncated approximation of a power
series expansion of pm(C). For a more general form for pm(C), the reader
is directed to the discussion by Krone (1963).
However, it must be pointed out that data published by Krone (1963)
showed that multiple values for Bp, ap and aP are possible for a
particular sediment, depending on the shear rate and degree of
25
aggregation. Therefore, caution must be advised concerning the validity
of Eqn. 2.6 for more detailed application.
By considering the mobile fluid mud as depth varying viscous
Rayleigh flow (Stokes' first problem with variable viscosity; see
Schlichting, 1979), a numerical solution of the flow and boundary layer
thickness can be obtained. This is a nonsteady state approach to
determining the horizontal transport layer detailed in Section 3.7.
An alternative approach to determining the mobile/stationary
interface is considering the nonNewtonian theological properties of high
concentration suspensions. Past research has indicated that
concentrations in the fluid mud range behave as Bingham plastics or
pseudoplastics (Krone, 1963; Kirby and Parker, 1977; Faas, 1981; 1987;
Nichols, 1985). Over short (tidal) time periods the designation of
effective yield strengths may be appropriate. In this case, the data
seem to suggest a concentration power law relationship. Figure 27 shows
a very approximate linear (on loglog paper) relationship between yield
strength, ty, and concentration through the data sets shown. The
expression to relate this functional dependence is of the form,
ay
Ty = By C (2.7)
where y and ay are empirical constants. The data in Figure 27 suggest
that By = 8.7 x 107 Pa (1 Pa = 1 N/m2) and ay = 2.55. With this
assumption it is possible to estimate the lower penetration distance
(lower extent of horizontal motion), the mobile/stationary fluid mud
layer interface, Zb, based solely on equating the applied bed shear to
1037
Q San Francisco Bay Mud
O Wilmington Mud
SBrunswick Mud
2 Grundlte
1 02 Kaollnite Suspension
110
/
E 10
1000
5 /
z
102 
S/ /
SEDIMENT CONCENTRATION, C(g/l)
Figure 27. Bingham Yield Strength Variation with Concentration
27
the level of equal yield strength. However, field observations of fluid
mud flows (Kendrick and Derbyshire, 1985) do not seem to support this
approach (see Section 5.5). This is because flow occurs when the applied
shear stress is less than the reported shear strength (Figure 27). This
suggests that the behavior is more pseudoplastic than Bingham. Further
discussion of possible means of determining the elevation, Zb, is
described in Section 3.7.
2.4.2 Mobile Fluid Mud
The mobile fluid mud layer as described in Section 2.2 is that part
of the fluid mud layer which is advected along with the mobile suspension
layer current. It may also be gravitational slump flow along a sloping
bed (Kendrick and Derbyshire, 1985). The elevation, Za, (Figure 21)
which defines the upper bound of the layer represents a local maximum in
net downward vertical flux.
The settling velocity of cohesive sediments varies with
concentration in suspension, ws(C). Initially constant, the velocity
rises with increasing concentration (due to flocculation) to a level
where it becomes constant again then rapidly drops. An example of the
settling characteristic of a natural estuarine sediment is shown in
Figure 28. The point beyond which no further increase in settling
velocity occurs has been termed "hindered settling" (Owen, 1970; Imai,
1980; Teeter, 1986a). For purposes that will become clear with the
following arguments, it is important to distinguish hindered settling
velocity from hindered settling flux. The details of free, flocculation
and hindered settling velocity are discussed in Chapter 3 (Section 3.4).
28
100.0
kI = 0.513
E n, = 1.29
S10.0 W n Wso= 2.6 mm/s
Ws = kC k2 = 0.008
n2 = 4.65
1.0 .* *
> 0.1
o V
Ws = Wso (1k2 C)n
0.01 
0.001 1 I
0.01 0.10 1.0 10.0 100.0
CONCENTRATION, C (g/.)
Figure 28. Settling Velocity Variation with Concentration
Severn Estuary Mud (adapted from Mehta, 1986)
The vertical flux of sediment (mass per unit area per unit time)
from settling, Fs, is the product of the local settling velocity and
concentration as
Fs(C) = ws(C) C (2.8)
For the data of Figure 28 (source: Thorn, 1981) the vertical flux, Fs,
is plotted against concentration, C, in Figure 29 below.
From Figures 28 and 29 it is observed that the peak flux occurs at
a much higher concentration (i.e., 20 g/l) than that at which the peak
settling velocity occurs (i.e., 3 g/1). This is due to settling
velocity being either constant or only slightly decreasing from the
60 2
Fm= 40 g/ms
CA= 2x104 mg/I
E 40 m
20
0 II
103 10 4 CA 105
CONCENTRATION, C(mg/l)
Figure 29. Vertical Settling Flux Variation with Concentration
(reprinted with permission from Ross et al., 1987)
maximum (* 23 g/1) over a wide concentration band (210 g/1). The peak
settling flux (i.e., 20 g/l) represents a more reasonable definition for
hindered settling than that based on the peak settling velocity. Beyond
this point, the actual vertical mass flux from settling diminishes
rapidly with increasing concentration.
The upper elevation, Zg, of the fluid mud layer under settling
conditions therefore occurs at the "hindered" (defined on the basis of
flux) concentration. A discrete interface forms because the sediment
accumulates at this level because the flux is increasing above and
decreasing below this interface.
After all the sediment in the upper suspension layer has settled
onto the fluid mud layer, the interface settles according to the
30
interfacial settling region (shown in Figure 23). When the flow in the
upper suspension layer is turbulent, diffusion and entrainment at the
interface reduce the overall downward vertical flux and the interfacial
concentration, Ca, drops from that given by pure settling conditions.
Thus, Ca has a maximum value given by the hindered (flux) concentration.
In the presence of mixing, the mobile fluid mud layer, 6fm = Zb Za,
does not necessarily become thicker (by becoming more diffuse). Due to
the sharp density gradients resulting from the high suspension
concentrations, turbulent mass and momentum diffusion across the fluid
mud layer is greatly damped. This results in a stable stratification,
often termed buoyancy or gravitational stabilization (Fischer et al.,
1979). In this case, upward entrainment, which is dependent on the
degree of stratification and relative turbulent intensity (Yih, 1980)
becomes the dominant mixing mechanism. Stratification development is
discussed in greater detail in the next section. Mixing in the presence
of gravitational stabilization is discussed in Section 3.3.2.
2.5 Lutoclines
Lutoclines are defined as pronounced "steps" in the vertical
concentration profile resulting from complex mixingsettling processes.
The upper fluid mud interface, by this definition, is also a lutocline
(shown as L1 in Figure 21). However, lutoclines can as well occur in
the mobile suspension layer (shown as L2 and L3). Lutoclines have a
vertical scale (distance between steps) dependent on the local vertical
scale of turbulence (Posmentier, 1977). Therefore, only a limited number
can exist and over limited periods. The origin of this term stems from
31
the Latin word lutum which means mud (Kirby, 1986). Lutoclines are
analogous to other types of density stratification (pycnoclines) from
sharp salinity gradients (haloclines) and temperature gradients
(thermoclines) with the exception that suspended sediment exhibits
settling independent of the fluid. They are easily recorded by high
frequency echo sounders and are characteristically observed in high
sediment (> 500 mg/l) environments.
Figure 210 shows a typical suspended sediment profile showing the
relative temporal stability of two lutoclines.
The velocity data also shown in the figure together with the
concentration profiles suggest turbulent, well mixed flow between
lutoclines.
The physics of lutocline genesis, growth, and decay is governed by
the dynamic interaction between the counteracting processes of turbulent
mixing and gravitational settling. Simply stated, lutoclines occur
because sediment is heavier than water and it tries to settle out under
quiescent conditions. Due to flocculation and hindered settling, fine
sediment suspended at large concentration settles as a sharp interface,
as opposed to concentration "thinning" (Bosworth, 1956). Turbulent
eddies impinging on the interface exchange "parcels" of sedimentladen
fluid. However, due to the potential energy difference of each "parcel"
with its surroundings, they are returned to near origin levels with only
modest mixing. This is in sharp contrast to the rapid mixing which takes
place in the low density gradient regions (qualitatively defined below).
Thus, the moderate mixing at the interface is counterbalanced by the
sediment settling, and the interface remains stable.
32
Water surface 
*
*
*
*
I 1 I I I
0 2 4
o .5 i i i i
0 .5
15:4 5
15:50 Time of profile
15:55
5 minute mean velocity
, (g/l)
6 8
' t (m/s)
1.0
Figure 210. Typical Suspended Concentration Profile Showing
Multiple Lutocline Stability Over 10 min. Period
(Kirby, 1986)
One means of relating the relative magnitudes of gradients in
kinetic energy, a(pu2/2)/az, to potential energy, 8(pgz)/8z, is through
the local gradient Richardson number defined as
(2.9)
Ri = R (au)2
p 8z 8z
where g is the acceleration of gravity, p is the fluid density, u is the
horizontal velocity and z is the vertical coordinate direction (positive
upwards).
33
Thus, the implications of equation 2.9 are
Ri > 0 High rate of kinetic energy dissipation relative
to low potential energy gradient 4 Rapid Mixing
Ri >> 1 Low rates of kinetic energy dissipation relative
to high potential energy gradient 4 Minimal
Mixing
Ri < 0 Density gradient, ap/az, > 0 the system is
unstably stratified 4 Overturning
As an example, letting the local mixing rate in a neutrally
stratified condition (no density stratification) be defined as Kn(z)
implying that it is variable with depth, the simplest relationship for
mixing in the presence of stratification, based on Richardson number, is
Ks(z) = Kn(z) (1 + Ri)1 (2.10)
where Ks is the local mixing rate dampened by stratification. The limits
on mixing meet the above requirements as it can be seen that Ks(z) 
Kn(z) for Ri 0 and Ks(z) 0 if Ri *. This is then one means of
quantifying the stratification dampened mixing.
A more general form for the above equation, a review of literature,
and a discussion of applicability of buoyancy stabilization are given in
Section 3.3.2.
CHAPTER 3
TRANSPORT CONSIDERATIONS
3.1 Introduction
The physics related to the vertical structure of fine sediment
suspension can be addressed by considering the important components of
the advectiondiffusion equation. This equation, simply an Eulerian
conservation of sediment mass expression, relates the temporal changes in
sediment concentration to the spatial gradients in fluxes. Simple
arguments show that for the present purpose the important coordinate in
the equation is the vertical, z (positive upwards from the bed),
direction. Furthermore, gravitational forces which influence the
diffusion and settling flux terms are responsible for the complex
structure of lutoclines and fluid mud as defined in Chapter 2.
Theoretical and rationally based relationships for settling
velocity, neutral turbulent diffusivity, and buoyancy stabilization are
presented in this chapter for explanation and predictive purposes. A
simple onedimensional numerical model, developed from these
relationships and the advectiondiffusion equation, is used to explain
laboratory and field data presented later in this report. Finally, to
distinguish the lower layer of mobile fluid mud, a simple numerical model
based on momentum diffusion is developed to evaluate the dynamic and
steady state characteristics of this layer and to estimate horizontal
sediment transport rates.
35
3.2 Mass Conservation Equation
In Cartesian coordinates (x, longitudinal; y, lateral; and z,
directed vertical upwards positive from the water surface), the
instantaneous Eulerian conservation of mass equation for (scalar)
sediment suspension concentrations C(x,y,z,t) (mass of sediment/volume of
suspension) can be written as
dC
d V.q (3.1)
dt
where q is the resultant mass flux vector (from diffusion and settling)
and V is the vector operator.
For Fickian molecular diffusion, the mean mass flux vector is
qm = DVC (3.2)
where it is assumed that the molecular diffusivity, D, is isotropic
(Fischer et al., 1979).
Since it is implausible to track particles in suspension on an
instantaneous, infinitesimal scale, and because natural flows are
typically turbulent, it is usual to express equation (3.1) in terms of
time averaged values (e.g., time average velocity, u, and concentration,
C) where the averaging time is sufficiently long to negate turbulent
fluctuations but short enough to track longer period temporal behavior
(Vanoni, 1975; McDowell and O'Connor, 1977). However, time averaging
greatly increases the diffusive mass flux vector. Fortunately, as an
36
approximation, turbulent diffusion can be expressed analogous to Fickian
diffusion in the form
qt = K.VC (3.2a)
where K is the turbulent mixing vector with Cartesian (x,y,z) coordinate
components (Kx,Ky,Kz). Since turbulent mixing is much greater (28
orders of magnitude) than molecular diffusion, the latter is often
neglected (McCutcheon, 1983). Simple perturbation analyses, i.e.,
letting the velocity (vector) and concentration (scalar) components be
represented by mean (e.g., u,C ) and fluctuating values (e.g., u',C'),
have been used to support this result mathematically. The reader is
directed to Hayter (1983) or French (1985) for this derivation.
The mass flux vector from settling is, simply,
s = Fs = wsCj (3.3)
where ws is the mean sediment settling velocity and j is the unit vector
directed along the z axis.
The resultant mass flux vector for suspended sediment is then
approximated by
q = qt + qs 0 < z < Zb (3.4)
away from the boundaries (water surface z=0 and bed surface, z=Zb).
37
For the purposes of considering vertical structure, only the
vertical transport terms need to be evaluated. Horizontal gradients in
concentration are (typically 34) ordersofmagnitude smaller than
vertical gradients.
Cl << C C C<<
ac ac ac ac
8x az y 8z
Nondimensional scaling arguments have been used to determine the
relative importance of the individual terms in Eqn. 3.1 This analysis is
included for reference in Appendix A. For typical estuarine conditions
(see Appendix A) horizontal and vertical advective fluxes and horizontal
diffusive fluxes can be neglected for first order analysis. The
governing equation for considering the vertical structure of fine
suspended sediments is now reduced to
ac = a a ac
at az a= sC + Kzaz 0 < z < Zb (3.5)
where qz is the resultant vertical flux from settling and vertical
diffusion away from the boundaries (z=0 and z=Zb) shown by the bracketed.
terms in Eqn. 3.5.
The boundary conditions which must be imposed on Eqn. 3.5 are
Bed Flux Boundary Condition. Application of Eqn. 3.5 at z = Zb
requires that a bed flux term, Fb (mass of sediment per unit bed area per
unit time), containing both erosion, Fe, and deposition, Fp, fluxes as
Fb = Fe Fp (3.6)
be defined. In addition, the diffusion and settling flux terms at the
bed are zero. Thus qz(Zb,t) = Fb and ws = Kz = 0 at z = Zb is the
appropriate bed boundary condition. Fb is dependent on sediment and
hydrodynamic conditions. Section 3.5 presents a detailed discussion of
bed fluxes (erosion and deposition) used in the vertical transport model.
Surface Boundary Condition. The boundary condition at the water
surface, z=0, is a no net flux boundary. This means that there is no net
transport across the free surface and diffusion flux is always
counterbalancing settling flux i.e.:
aC
qz(0,t) = (wsC) + {KzCz) = 0 (3.7)
az
The diffusion flux term Fd = [Kzz} must include entrainment and
gravitationally stabilized mixing. In the absence of well defined
hydrodynamics (i.e., perhaps the results from a full turbulence model
simulation), functional forms for the vertical turbulent diffusivity, Kz,
based on firstorder closure modeling using mixing length approximations
can be used (McCutcheon, 1983). This assumes that the mass diffusivity
can be related to the momentum diffusivity. Furthermore, due to
differences in time scales, spatial variability, and kinetic energy
dissipation, the functional forms for highly oscillatory currents (e.g.,
waves) are quite different from those for unidirectional flows.
Stratification, in general, dampens turbulent mixing by the
mechanisms described in Chapter 2. Through local gradient Richardson
39
number relationships of the Munk and Anderson (1948) form, buoyancy
(gravitational) stabilization can be modeled. Stabilized diffusivity is
treated separately in Section 3.3.2.
On the subject of mixing and stratification it must be pointed out
that surface waves can create interfacial waves which can build to
breaking, thereby greatly enhancing interfacial mixing (Yih, 1970; Dean
and Dalrymple, 1984). Due to the limited scope of this research and
because this phenomenon was not observed in laboratory or field data for
this study, no further discussion is provided. The reader can find
additional information on this topic in Lamb (1945), Yih (1976), and Yih
(1980).
The settling flux (wsC) as written in Eqn. (3.5) allows for spatial
variability in both unknowns, settling velocity and concentration. In
general, for both cohesive and noncohesive sediments, settling velocity
is a function of concentration, ws(C). The settling behavior of cohesive
and noncohesive sediment is covered in Section 3.4.
3.3 Diffusive Transport
3.3.1 Turbulent Diffusion
In turbulent flows mixing occurs mainly because the timeaveraged
products of the velocity and concentration fluctuations i.e., u1C', are
nonzero. Through adequately measuring the simultaneous fluctuations in
velocities and concentrations, turbulent mixing can be precisely
quantified. Then, for predictive purposes, correlations to flow
parameters such as bottom friction, mean velocity and pressure gradient
are required. Reasonable success is beginning to be achieved in the area
40
of turbulence modeling (Zeman and Lumley, 1977; Sheng, 1983). However,
in light of the difficulties in precise measurement of these
fluctuations, verification poses difficulties.
For fine sediment suspensions the turbulent diffusion of sediment
mass, Ks, is approximately equal to that of the diffusion of momentum,
Em. The turbulent Schmidt number, Sc (Daily and Harleman, 1966), which
is the ratio of mass to momentum diffusivity is equal to one (Teeter,
1986b) as
K
Sc E 1 (3.8)
m
In turbulent flows momentum diffusion is by Reynolds stress,
Tij = p uuuj, gradients where the time mean product of the velocity
fluctuations is nonzero. For mass diffusion, the time mean product of
the concentration and velocity fluctuation is nonzero analogously.
This observation (Reynolds' analogy) allows the use of a wide body
of literature on firstorder closure modeling based on the coefficient of
eddy viscosity, relating the Reynolds stress to mean velocity gradient as
au.
ij = pEij x (3.9)
J iJ 3^ax
where Eij is the i,j component of the momentum diffusivity (eddy
viscosity) tensor. It can be seen from Eqn. (2.9) that the eddy
viscosity, in general, must be a function of mean shear rate and shear
stress. It is also common to assume that turbulent diffusion is
41
isotropic (i.e., Eij = Eji = E, Fischer et al., 1979) in the absence of
stratification.
The most commonly applied expression of vertical variation in eddy
diffusivity is the formulation given by Rouse (Vanoni, 1975). By
following von Karman's assumptions of a linear shear stress distribution
with depth leading to a logarithmic velocity profile, the following
expression is found:
z
E(z) = KU*Z (1 ) (3.10)
h
where K is von Karman's constant, u* is the friction velocity (v1c7p) and
h is the flow depth. While this expression may be sufficient for
describing turbulentlogarithmic unidirectional flows, it does not
describe highly oscillatory flows such as under waves. Maa (1986),
Kennedy and Locher (1972), and Hwang and Wang (1982) have reviewed
currently popular expressions for diffusion coefficients under waves.
There seems to be little consistency in the forms. One of the most
promising expressions based on energy dissipation is that developed by
Hwang and Wang (1982). Their model, applicable outside the wave boundary
layer, is of the form
sinh2kz
E(z) aH*2 2sinh2kh (3.11)
where a is a constant, H is the wave height (twice the amplitude), a is
42
the wave frequency (2n/T, T = wave period), and k is the wave number
(2T/L, L = wave length).
Thimakorn (1984) found success using a coefficient similar to that
given by Hwang and Wang (1982) to predict vertical profiles of natural
clay concentration during resuspension in a wave flume. It should be
pointed out that the concentrations reported were small (<1000 mg/l) and
any buoyancy stability effects therefore were likely to be negligible.
Next to the bed boundary layer effects greatly increase vertical
mixing under waves due to the relatively large velocity gradients and
shear (Neilson, 1979). Orbital particle trajectories are significantly
altered from those predicted for example by linear wave theory (inviscid
potential flow) because viscous (or turbulent) effects dominate.
However, this layer is small (o/(2v) < 1 cm) and is often neglected (Maa,
1986). Further upward, the velocity amplitude gradients increase with
distance above the bottom to a maximum at the surface. This is the basis
for the Hwang and Wang (1982) form shown above. Maa (1986) conducted dye
diffusion tests under waves which showed larger lateral spreading rates
near the surface and immediately near the bottom. This is indicative of
higher energy dissipation in those regions which would support the
proposition of higher vertical mixing rates.
In the presence of density stratification the form of the neutral
diffusivity is not as important as the form of the stability coefficient
(French, 1985), which provides the basis for a discussion of mixing in
the presence of density stratification.
43
3.3.2 Gravitational Stabilization
In the previous section, theoretical and empirical based expressions
for the vertical turbulent diffusivity under current and waves were
mentioned. In a continuously, stably stratified flow the vertical
diffusion of both momentum and mass is inhibited by stratification, and
significant modification of the turbulent diffusivity occurs.
Furthermore, the diffusivity of momentum and mass are not affected in the
same manner. In the presence of density stratification, the eddy
viscosity (i.e., the turbulent momentum diffusity) is larger than the
eddy diffusivity of heat and mass (French, 1985). Progress has been made
towards estimating values and obtaining expressions for mass and momentum
diffusion in a continuously stratified flow. However, it must be
emphasized that at the present time an expression does not exist for
either eddy viscosity or diffusivity which is considered universally
valid. French (1985) provides a summary of several popular forms
developed for unidirectional flow only. A brief review of those plus
others is given here for the purpose of explaining vertical structure.
Rossby and Montgomery (1935) first proposed an equation relating
vertical eddy viscosity for stratified flow, Es, to the corresponding
value for homogeneous or neutral conditions, En, of the form
E
s I
S (1 + BRMRi) (3.12)
En
where PRM is an empirical coefficient and Ri is the local gradient
Richardson number, Eqn. (2.9). They assumed that the change in kinetic
energy per unit mass in going from a neutral or unstratified condition to
44
a stably stratified condition is equal to the potential energy change due
to displacement over the mixing length from the equilibrium position with
a different density.
Holzman (1943) suggested a somewhat different relationship
E
En (1 PHRi) (3.13)
where BH is a coefficient. Note the change in sign of the coefficients.
Munk and Anderson (1948) proposed a generalized form of the Rossby
and Montgomery (1935) and Holzman (1943) equations as
E aMA
E (1 + OMARi) (3.14)
En
where PMA and aMA are free coefficients.
Kent and Pritchard (1957) also used a conservation of energy
argument to develop an equation of the Munk and Anderson (1948) form;
however, they argued that a=2 on a theoretical basis.
Delft Hydraulics Laboratory, (DHL) (1974), reported that the ratio
of Es to En should decrease exponentially with increasing values of Ri or
E B Ri
s e
E e (3.15)
m
where Pe is an empirical coefficient.
Finally, Odd and Rodger (1978) used the original hypothesis of
Rossby and Montgomery (1935) to define equations applicable for two
specific cases:
45
Case 1. Stratified flow with a significant peak in the vertical profile
of Ri at a distance z = z0 from the bottom boundary where T is the peak
gradient Richardson number, then
E
s I
E (1 + ORT)
n
E
E (1 + OR)
En
for T < 1
for T > I
(3.16)
(3.17)
where BOR = a coefficient.
Case 2. No significant peak exists in the vertical profile of Ri: then
E
s 1
 = (1 + BORRi)
n
s 
En
for Ri 1
for Ri > 1
Equations (3.16) through (3.19) are applied throughout the vertical
dimension, but near the boundaries, if Es > En then En is used. Note
that Es/En = constant (not a function of depth) for all cases except
conditions when Eqn. (3.18) applies. This is significantly different
from the previously proposed forms (Eqns. 3.12 3.15) which are
everywhere depth variable.
(3.18)
(3.19)
46
The problem with all the above methodologies is that, in general,
they cannot be shown to be universally valid. Suggested values for some
of the coefficients used in the above equations are summarized in Table
31 below.
Table 31. Summary of Coefficient Values for Turbulent
Vertical Diffusion of Momentum in Continuously
Stratified Flow
Equation a Source
3.12 2.5 Nelson (1972)
5.0 DHL (1974)
30.3  French and McCutcheon (1983)
3.14 10 0.5 Munk and Anderson (1948)
30 0.5 DHL (1974)
3.163.19 0.31 0.747 French (1979)
0.062 0.379 French and McCutcheon (1983)
140180 Odd and Rodger (1978)
With regard to the data summarized in Table 31, the following should be
noted:
1. Nelson (1972) used published oceanographic, atmospheric, and
pipe flow data for his analysis, and the same was true of the
analysis by the Delft Hydraulics Laboratory (1974). Thus,
these investigators had no control over the quality of their
data.
2. The data used by French (1979) were taken under laboratory
conditions, but the flume used for these experiments had a
small widthtodepth ratio, and the results may have been
unduly affected by this fact.
3. Odd and Rodger (1978) used field data from a reach of tidal
channel. Their data set is perhaps the best data presently
47
available regarding the turbulent vertical diffusion of
momentum under stratified conditions.
4. French and McCutcheon (1983) used the Odd and Rodger (1978)
data set for their analysis. The coefficients for Eqns. (3.16
3.19) used in their work differ from that of Odd and Rodger
(1978) due to differences in the definition of reasonable fit.
5. In the past, Eqn. (3.12) has been the most commonly used method
of estimating Es (Nelson, 1972). It is more theoretically
justifiable than the methods of Odd and Rodger (1978), French
and McCutcheon (1983) or French (1979).
6. Delft Hydraulics Laboratory (1974) concluded that when Ri <
0.7, the scatter of the data available is so great that no
bestfit equation can be selected.
A number of models for the eddy (mass) diffusivity in stratified
flow have also been proposed. Most have been based on the results from
momentum diffusion; however, under stratified conditions, questions arise
as to the applicability of this assumption (e.g., see Oduyemi, 1986).
One of the most frequently used expression is of the form
K
s = c(l + B'Ri)a' (3.20)
Kn
where Kn and Ks are the vertical mass diffusivities for homogeneous and
stratified flows, respectively, and c, 0', and a' are coefficients. Munk
and Anderson (1948) estimated that c = 1, a' = 1.5, and B' = 3.33.
It is interesting to note that stratification apparently also acts
to reduce the value of the turbulent transverse diffusion coefficient by
48
turbulence damping; however, the results presently available in this area
(see, for example, Sumer, 1976) are inconsistent and are not relevant for
vertical structure considerations.
When gravitational stability is considered (e.g., by Eqn. 3.20),
nonlinearity between diffusive flux, Fd, and vertical concentration
gradient, C,z (note ,z denotes differentiation with respect to z),
develops. Without regard to stabilization (Kz = Kn), by Fickian
diffusion, the diffusive flux is linearly proportional to concentration
gradient, Fd = Kz.C,z. However, from theoretical results presented for
gravitational stabilization, the turbulent mass diffusion coefficient (Kz
= Ks) was shown to be inversely proportional to the gradient Richardson
number, Ri (given by Eqn. 2.9), to a power, a' > 1. The gradient
Richardson number, of course, is directly proportional to the density
gradient, P,z, which is a function of the concentration gradient because
bulk density is a function of sediment concentration (Eqn. 2.5). The
resulting dependence of diffusion flux to concentration gradient is
therefore highly nonlinear.
Figure 31 shows an example of the nonlinearity resulting between Fd
and C,z using Eqn. 3.20 with Munk and Anderson values for stability
coefficients (i.e., c = 1, a' = 1.5, and B' = 3.33). For this case the
flux initially increases with C,z reaches a maximum and then slowly
decreases. For a given flux below the maximum two values of C,z
(corresponding to the two roots) satisfy the equation. In the absence of
settling, discontinuities in concentration profile (two distinct C,z's)
are theoretically possible (because Fd,z = 0) and relatively stable as
long as Fd is constant with time (e.g., near the bed during steady
4
L.0 2 kn
x0 (1 + 4.17Ri?
U. / (1+3.33Ri')1.5
0
0.0 0.2 0.4 0.6 0.8
Concentration Gradient, .C (kg/m2)
az
Figure 31. Diffusion Flux vs. Concentration Gradient
erosion). For salinity concentrations in estuarine environments, this
has been pointed out to be a likely cause of salinity finestructure
(Postmentier, 1977). For suspended fine sediment, the nonlinearity in
diffusion flux, Fd, (with C,z) has the effect of promoting lutocline
growth and stabilityin addition to the nonlinearity between settling
flux, Fs, and concentration, C (pointed out in Section 2.4.2). The
settling properties of estuarine fine sediment is presented in the
following section (3.4).
3.4 Settling
The predominant distinction between fine sediment suspensions and
other density altering constituents (e.g., salt, temperature, etc.) is
50
that suspended sediment is negatively buoyant and settles independent of
the suspending fluid which surrounds it. This counteracts mixing to the
extent that under quiescent conditions partial or total clarification is
possible only to be later well mixed again under high flow conditions.
While the settling characteristics of noncohesive sediments (e.g.,
sand) are reasonably well behaved, i.e., not so strongly dependent on
concentration, salinity, etc., cohesive sediments are very sensitive to
these variables.
It is convenient to start by discussing the settling characteristics
of individual particles and work into high concentration (>20,000 mg/l)
settling suspensions.
3.4.1 Free Settling
Free settling was defined in Chapter 2 as the concentration range
over which individual settling sediment particles (both dispersed primary
particles and aggregates) do not physically interfere with one another.
For cohesive sediments, the upper concentration limit is in the range of
300500 mg/l (Krone, 1962) but for noncohesive sediments it is one to
two ordersofmagnitude higher (McNown and Lin, 1952).
Individual sediment particles settle at a terminal fall velocity
which results in a force balance between form and skin friction (viscous)
drag and net negative buoyancy. For a spherical particle of diameter, d,
settling in a viscous fluid with kinematic viscosity, v, the settling
velocity, ws, is
s (ps Pw) (3.21)
3 CD Pw
51
where g is the acceleration of gravity, CD is the drag coefficient and pg
and p, are the sediment and fluid densities, respectively. The
coefficient of drag, CD, is a function of the Reynolds' number of the
sphere (R = wsd/v), but cannot be determined analytically for R > 1 (see,
for example, Vanoni, 1975).
In the viscous or Stokes' settling range (R < 0.1) the drag
coefficient is given by CD = 24/R and the settling velocity is
gdl (Ps Pw)
Ws = d2 P (3.22)
s 18v p,
Fine estuarial sediment in dispersed or quiescent conditions typically
falls well within this range. Therefore, no further discussion of the
deviations from Stokes settling will be presented here with one minor
exception: fine estuarine sediment is not generally spherical. In
dispersed form, cohesive size sediment is platelike with a large surface
area to volume ratio (Van Olphen, 1963). This results in a higher drag
coefficient and slower settling velocity than spherical sediment of the
same volume. Very fine (d < 1 pm) dispersed sediment may not settle at
all due to the increased relative importance of Brownian motion.
Aggregates, although irregularly shaped, are generally more spherical
(and substantially larger than dispersed primary particles). For both
particles, it is typical to define an "effective" particle diameter based
on measured settling velocity and specific density (ps). A more thorough
discussion of the effects of particle shape on settling velocity as well
as other deviations from Stokes settling can be found in Vanoni (1975).
3.4.2 Flocculation Settling
In the presence of small amounts of dissolved salts (< 1 ppt NaCI)
cohesive sediment in suspension can flocculate greatly, thus changing the
settling properties. Flocculation of cohesive sediment particles is the
consequence of interparticle collision and cohesion. Cohesion and
collision are discussed in detail by Einstein and Krone (1962), Krone
(1962), Partheniades (1964), O'Malia (1972), and Hunt (1980) and reviewed
by Hayter (1983).
Cohesion depends primarily on the mineral composition and the
availability and charge of cations in the suspended fluid. Colloidal
particles have both attractive and repulsive forces (Van Olphen, 1963).
The attractive forces predominate when the coulombic repulsive forces are
suppressed by sorbed cations near the particle surfaces. A measure of
the relative cohesiveness of a particular colloidal sediment is the
cation exchange capacity, CEC. A high CEC indicates a highly cohesive
sediment. Montmorillinitic sediments have a higher CEC and thus are more
cohesive than illitic or kaolinitic sediments with lower CECs.
Collision intensity and frequency are dependent on three mechanisms:
Brownian motion, fluid shearing, and differential settling. Brownian
motion is the natural thermal agitation of the sediment particles in the
suspending medium. Particle movement from Brownian motion is erratic,
the collisions are weak and the resulting flocs are "fluffy" (of
relatively low density and weakly bound). This motion becomes much less
apparent as the floc size grows. Brownian motion in estuaries is the
least significant collision mechanism of the three (Krone, 1962).
Particle collision from fluid shearing, however, becomes much more
53
significant as the size of the flocs grows. The result is a greater
intensity of collision and stronger flocs. Differential settling becomes
increasingly more important as the distribution of the size widens.
Under quiescent conditions e.g., at the time of slack water, with a
natural nonuniform sediment this becomes the primary collision
mechanism. The frequency of all three means of collision increases with
increasing concentration.
Two characteristics of flocculated sediment which differ from the
dispersed form and which affect the settling velocity are particle
density and shape. First, because of interstitial trapped water, the
relative particle densities are reduced. This effect alone would lead to
reduced settling velocity in the flocculated state. However, because of
the larger size and more spherical shape, a decrease in viscous drag
results. Since the reduction in drag is much more significant than the
reduction in density, the settling velocity of the flocs are up to 4
orders of magnitude larger than dispersed particles (Bellessort, 1973).
This can result in rapid sedimentation and shoaling in upper estuaries
where flocculation (by introduction of dissolved salts) is first
stimulated.
Krone (1962) reasoned that the average (median, by weight) settling
velocity of flocculating Mare Island Strait (San Francisco Bay) sediment
for equal flocculation time was proportional to the sediment
concentration raised to the 4/3 power,
(3.23)
ws.C4/3
54
His reasoning was based on consideration of collision probability and
average floc size. He further supported this argument with data taken in
settling column and flume studies.
Burt (1986) used a general relationship for flocculation enhanced
variation with concentration as
n1
ws = k1 C (3.24)
where k1 depends on sediment composition and n1 can vary from about 1 to
2.
3.4.3 Hindered Settling
As the concentration of sediment in suspension increases beyond the
flocculation settling range, the mean sediment settling velocity begins
to drop. Aggregates are so closely spaced as to form a continuous
network, and the interstitial fluid is forced to escape through smaller
and smaller pore spaces. This is commonly termed "hindered settling" in
the literature (Mehta, 1986, Lavelle and Thacker, 1978). However, the
inadequacies of this definition were pointed out in Section 2.4.2. The
pioneering work of Richardson and Zaki (1954) on the settling of uniform
glass spheres resulted in a widely accepted relationship for the settling
velocity as a function of concentration of the form,
n2
ws = Wso (1 k2C) (3.25)
55
where wso is the initial or reference settling velocity, k2 is a
coefficient which depends on the sediment composition and n2 a 5. The
coefficient k2 can be considered to be the reciprocal of the hypothetical
concentration where hindered settling gives way to primary, firststage
consolidation. This is typically in the neighborhood of 120160 g/l
(Mehta 1986, Einstein and Krone, 1962). For fine sandcoarse silt the
reference velocity, wso, is given by Stokes' Law. For cohesive
flocculated sediment the reference velocity, wso, is the maximum velocity
of the flocculation range. Teeter (1986b) found that most natural fine
bay sediments fit this relationship well.
Lavelle and Thacker (1978) used an expression of this type in
steadystate analysis of the high concentration data of Einstein and
Chien (1955) for coarsegrained sediment. Including a term of (1C)a in
the Rouse (1938) equation allowing for finite and reasonable
concentrations at the bed (z=0), they found success in predicting the
nearbed high concentration data of Einstein and Chein (1955).
3.5 Vertical Bed Fluxes
The bed flux boundary condition for solution of Eqn. (3.5) plays a
critical role in the evolution of the vertical suspension profile as the
overall source and sink component of sediment mass in suspension. Bed
fluxes can be either erosional or depositional. Both are discussed in
the following paragraphs. It is important to point out that defining the
elevation at which the erosion or deposition process takes place is, in
itself, a formidable task. From a practical viewpoint, simultaneous
continuous profiling of concentration, velocity and bed stresses
C I
56
(pressures and shear) are required in the upperbed to near bottom layers
to define the interface elevation with time and hydrodynamic action. As
was pointed out in Chapter 2, it is very important to distinguish the
stationary bed material from the fluid mud layer. Additionally, erosion
relationships developed for bed/mobile suspension interfaces may not be
adequate for erosion and fluidization of the bed beneath a fluid mud
layer. This is a possible limitation of the proposed erosion/deposition
functions used in the vertical structure model and presented in the
following subsections.
3.5.1. Bed Erosion
Bed erosion occurs when the resultant hydrodynamic lift and drag
forces on the sediment at or below the bed interface (Zc in Figure 21)
exceed the resultant frictional, gravitational and physicochemical
bonding forces of the sediment grain or particle. Continuous inter
particle contact ceases and individual or groups of aggregates become
resuspended.
There are two modes of erosion (Mehta, 1986), surface or particle by
particle erosion and mass or bulk erosion. In surface erosion,
individual particles break free of the bed surface as the hydrodynamic
erosive force (i.e., instantaneous turbulent shear stress acting on the
particle surface) applied to them exceeds the resultant gravitational,
frictional and cohesive bed bonding force. Under mass erosion, failure
occurs well below the bed surface resulting in large chunks of sediment
being broken from the bed structure and, subsequently, resuspended. Bed
fluidization is mass erosion where large structural breakdown occurs with
57
an initially minimum change in density. Surface erosion is more typical
of low concentration, low energy environments while mass erosion occurs
under higher flow and higher concentration conditions (Mehta, 1986).
Surface waves and other highly oscillatory currents have a
particularly pronounced influence on erosion in comparison with
unidirectional currents. Because of the increased inertial forces (e.g.,
"added mass" drag) associated with a local change in linear momentum, the
net entrainment force is much greater than with turbulent unidirectional
flows. Much more significant is the effect bed "shaking" and "pumping"
can have under highly oscillatory flows. "Shaking" or bed vibrations
occur because of the oscillatory bed shear stress which is transmitted
elastically (while at the same time damped) down through the bed.
"Pumping" occurs from oscillatory fluid hydrostatic pressure at the bed
which, given the low permeability of cohesive sediments, can lead to
internal pore pressure build up and liquefaction, similar to earthquake
failure of saturated terrigenous soils (Seed, 1976). This effect can
cause destruction of effective stress in larger layers depending on the
bed characteristics leading to mass erosion and fluid mud formation
(Alishahi and Krone, 1964; Wells et al., 1978; and Maa and Mehta, 1987).
The destruction of effective stress under waves is documented, perhaps
for the first time, in laboratory measurements presented in Chapter 4.
Erosion (particles leaving the bed surface) precedes scour
(resulting decrease in bed elevation) which will continue under constant
loading until the bed shear stress and the bed shear strength are equal.
The bed shear strength is a function of the deposition and consolidation
58
history plus the physicochemical characteristics of the sediment. The
shear strength, in general, increases with depth into the bed.
The rate of erosion (= flux of sediment from the bed), Fe, from
surface erosion is linearly related to the "excess shear" stress, tbts,
for spatially and temporally uniform bed properties (Kandiah, 1974) as
( b s)
F = a1 (3.26)
e I T s
where al, is an empirical rate constant, tb is the applied (timemean)
bed shear stress and ts is the bed shear strength for erosion. For a
given al, which is related to the type of flow and sediment
characteristics, the erosion rate, Fe, is constant. For nonuniform beds
(e.g., soft, partially consolidated) the rate of erosion can be found by
(Parchure 1984, Parchure and Mehta, 1983).
Fe = o exp{a2 [tb s(z)]1/2} (3.27)
where co and a2 are constants (determined empirically). Since ts
increases with depth below bed, the erosion rate, Fe, decreases as scour
proceeds.
No currently unique expression exists for mass erosion since it must
involve dynamic bed data (i.e., bed stresses and pressures) as well as
imposed shear.
For mass erosion under waves the practice is to increase the
coefficients to account for the larger magnitude erosion. Maa (1986)
showed success with this procedure and demonstrated that the coefficients
59
were as much as an orderofmagnitude larger for wave erosion than for
what has been found for the unidirectional case. Under pure wave flow
condition it is difficult to distinguish bed erosion from fluid mud
entrainment. Even though wave erosion has a greater ability to break the
bonding forces, without high momentum diffusion or turbulent entrainment
rates the fluid mud may not become mobile.
3.5.2 Deposition
Sediment particles or aggregates in suspension will redeposit on the
bed if the bed shear stress drops below some critical threshold value,
tcd. tcd is the shear stress below which all initially suspended
sediment deposits eventually. In general, it takes lower turbulent bed
shear stress to keep cohesive sediment in suspension than it does to
erode it (i.e., tbm < ts). tbm is the shear stress above which no
deposition occurs and it is generally larger than the limit for total
deposition, tbm > rcd, (Mehta, 1986). This is because after deposition
interparticle bonding and orientation are timedependent, as well as
dependent on consolidation mechanics (e.g., overburden, etc. as discussed
briefly in Chapter 2) and the critical shear stress for erosion increases
with time. For a uniform sediment tcd = tbm.
For uniform sediment, in a depositional environment (i.e., tb <
tcd), the rate of sediment deposited (= flux of deposited sediment), Fp,
on the bed is related to the average aggregate settling velocity, ws, the
nearbed concentration in suspension, C, and the relative probability, P,
that the sediment will stay on the bed as
Fp = ws C P tb < tcd (3.28)
60
The probability, P, that the sediment will stick to the bed is related to
the relative shear stress (Krone, 1962) as
P = (1 (3.29)
tcd
As observed, this relationship indicates no deposition when tb i tcd and
rapid settling when the bed shear goes to zero (tb = 0). Krone (1962)
and Mehta (1973) conducted deposition experiments under steady flows
using natural estuarine sediments and commercial kaolinite. tcd was
found to depend on sediment composition, varying from 0.04 to 0.15 N/m2
Mehta (1986) made the distinction for critical shear stress for
deposition of nonuniform sediment. He pointed out that while deposition
proceeds when tb < tbm, not all of the sediment in suspension deposits
when tb > Tcd. This is illustrated by the data in Figure 32. Mehta
(1986), in reanalyzing earlier data (Mehta, 1973), pointed out that even
after long periods the ratio Ceq/Co of ultimate equilibrium
concentration, Ceg, to initial concentration, Co, was only a function of
tb (i.e., Ceq/Co = f(tb)), not of Co.
This, then represents a fundamental distinction between cohesive and
cohesionless sediment since for cohesionless sediment the equilibrium
concentration, Ceq, is dependent on tb and independent of initial
concentration, Co, (i.e., Ceq = f(tb)). For cohesionless sediment, the
equilibrium concentration represents a balance between the rates of
erosion and deposition, whereas for cohesive sediment simultaneous
erosion and deposition did not occur under test conditions relative to
U 0.6O o
" 
0.4
0.2
0 I I I 
0 2 4 6 8 10 12 14 16 18 2
TIME (hrs)
Figure 32. Ratio C/Co of Instantaneous to Initial Suspended Sediment
Concentration Versus Time for Kaolinite in Distilled Water (after Mehta,
1973).
Figure 32. Thus Ceq, for cohesive soils was the steady state
concentration, Css. For cohesive soils, winnowing (coarser material
settling out first) is a likely cause of the variable steady state
ratios, Css/Co (Mehta and Lott, 1987). Thus, the steady state
concentration, Css, results in a suspension with a mean particle grain
size finer than the original suspension. For modeling purposes,
discretizing nonuniform suspended sediment into a finite number of
classes, Ci, and treating erosion and deposition for each class
separately would be one means of handling the winnowing (and resulting
bed layering) phenomena. The vertical structure model considers
independent settling and deposition of multiple classes of suspended
62
particles after it was found to be significant in settling column tests
of natural bay sediment (see Section 4.3). For discretizing the non
uniformity of the deposition Eqn. 3.28 (originally developed for uniform
sediment) is assumed to be valid as
FPi = Wsi Ci P tb < tcdi (3.30)
where the i subscripted variables must be defined for each class.
3.6 Fluid Mud Entrainment
Once a fluid mudlayer is formed, either from high erosion or
deposition rates, entrainment of this high concentration sediment
suspension can occur at the upper, mobile fluid mud interface (see Figure
21). Entrainment is markedly distinguished from bed erosion in that the
sediment is already in suspension. Fluid mud entrainment results from
interfacial instabilities and dissipation of kinetic energy with, as yet,
limited theoretical analysis. However, because it is believed to behave
analogously to twolayer density stratified flows associated with salt or
temperature gradients, a relatively larger literary and theoretical base
exists for these cases (Yih, 1980).
Velocity shear at the interface accounts for the primary mixing
mechanism. Unlike mixing in homogeneous or weakly stratified shear
layers, strong stratification characterized by a high Richardson Number
is composed of events such as interfacial wave generation and breaking,
interchange of energy between waves and the mean flow, and local shear
instabilities (Narimousa and Fernando, 1987).
I
63
Kato and Phillips (1969) in laboratory experiments of entrainment of
linearly stratified fluids found that the entrainment coefficient,
E* = Ue/u*, where ue is the entrainment rate (dh/dt) and u* is the
friction velocity, decreased with increasing stratification. They found
an inverse relation between entrainment rates and bulk Richardson Number,
Ri* = Abh/u*2 in which Ab = g(ppo)/Po is the buoyancy jump, p is the
fluid density, po is a fluid reference density, g is the gravitational
acceleration, and h is the average depth of mixed layer. They suggested
E* = Ri*1/2. Other research indicates that it should be related to mean
velocity, u, in the mixed (i.e., upper) layer (Price, 1979; Thompson,
1979) as
u
Ev = e = f(Riu)4 (3.31)
u
where Riu = Abh/u2.
Still other researchers (Phillips, 1977; Price, 1978; Narimousa and
Fernando, 1987) showed supporting evidence for using the velocity jump,
Au, across the interface defined as the difference between the mean flow
velocities in each layer. Later researchers reasoned that the major
portion of the energy for turbulent mixing at the density interface
results from shear production at the entrainment zone itself and,
therefore, Au is the significant velocity scale to obtain a measure of
the energy dissipation rate.
Narimousa and Fernando (1987) presented a graphical depiction of the
entrainment process which is qualitatively descriptive enough to warrant
reproduction here.
Mixed layer
I
I
Nonturbulent layer
Figure 33. Simplified Description of Density Stratified Entrainment
(after Narimousa and Fernando, 1987)
Figure 3.3 shows the entrainment process based on experimental
observations (Narimousa and Fernando, 1987). The upper turbulent layer
of thickness, h, is well mixed and the lower layer is initially
stationary. An intermediate entrainment layer, 6, separates the two
layers and is the region characterized by high energy dissipation and
buoyancy gradients. In the upper entrainment layer, the mean shearing
rate, du(z)/dz, increases downward reaching a maximum at 6i, and then
decreases as viscous dominant momentum diffusion penetrates deeper and
deeper into the stationary layer. The shear layer thickness is shown as
6s. The highest density gradients occur in the entrainment layer of
thickness 6, which is inside the shear layer, 6s, where turbulence
dampening is sufficient to eliminate turbulent penetration into the
65
lowest layer. The momentum diffusion (viscous) layer, of thickness 6,,
can be dynamic (growing with time) or relatively constant with respect to
the interface. Also shown are the flattening of large eddies (with
turbulent velocity components ul and wl) at the density interface and
local scouring and internal waves of height 8w in the intermediate
entrainment layer by the mixed layer eddies of mixing length scale, 1
(proportional to the mixed layer depth, h).
As can be deduced from the number of characteristics in the above
description, entrainment of density stratified flows of single phase
fluids is, in itself, an interesting and challenging field. Add to this,
particle settling associated with the twophase sediment/fluid mixture
and one can see that fluid mud entrainment deserves fundamental research.
No effort has been made to distinguish fluid mud entrainment from general
lutocline mixing in this research. Nevertheless, despite this
limitation, reasonable success has been achieved in explaining the
observed physical behavior of prototype and field vertical profiles, as
shown in Chapter 5. Further research in fluid mud interfacial
entrainment is required before a more refined understanding and usable
results are obtained.
3.7 Horizontal Fluid Mud Transport
Several approaches to solving for the horizontal transport of mobile
fluid mud and the relative thickness of the mobile layer are available.
These approaches are based on different simplifying assumptions
concerning the theological and temporal behavior of the fluid/sediment
system. The solution approximations (for velocity profile in the fluid
66
mud layer) together with limitations are presented in order of increasing
complexity, beginning with the analytical solution of viscous boundary
layer development under an imposed shear stress. The following titles
are given for solution approaches:
A. Constant Viscosity Rayleigh Flow
B. Constant Viscosity Unsteady Bingham Flow
C. Variable Viscosity Steady Bingham Flow
D. Variable Viscosity Rayleigh Flow
E. Variable Viscosity Unsteady Bingham Flow
Applicable solution techniques were applied to field and laboratory data,
the results of which are presented in Chapter 5.
The constitutive equations which govern fluid mud transport are the
conservation of momentum (Cauchy's Equation) and mass (continuity)
equations. The Cauchy Equation of motion written in tensor notation is
(Malvern, 1969)
du
p pb + V.T (3.31)
where p is the local fluid density u is the velocity vector, b is the
body force per unit mass vector, T is the stress tensor, and V is the
vector operator. The first term in Eqn. (3.31) is the time rate of
change of momentum per unit volume. The other terms are the body force
per unit volume and stress tensor gradient, respectively.
For an incompressible viscous fluid, the conservation of horizontal
momentum equation in Cartesian coordinates is
I i
67
du aP a8 au a u
P = + ~I t + a}1 (3.32a)
dt Tx z a z ay ay
where P is the pressure. The dynamic viscosity, p, is assumed here to be
isotropic but, in general, a function of concentration, p=p(C). Together
with the continuity equation,
au 8v aw
 + + 0 (3.32b)
ax Ty 0z
sufficient boundary and initial conditions (outlined below), the problem
is said to be closed and formally defined.
For a tractable solution to the horizontal flow problem, somewhat
far reaching assumptions must be made. First, onedimensional horizontal
flow in the x direction is assumed (no v and w components in the velocity
vector). Next, the assumption of lateral uniformity is made. Then, by
continuity, the horizontal velocity component must only vary in the z
direction, u = u(z,t). The third assumption is by far the most
stringent. It is assumed, analogous to the laminar sublayer next to a
boundary (Schlichting, 1979) and the shear layer in a stratified fluid
(Narimousa and Fernando, 1987), that the horizontal pressure gradient is
much smaller than the vertical shear stress gradient,
S< axz (3.33a)
ax 8z
To more formally show the conditions under which this assumption is
valid, scaling arguments are used to evaluate the relative magnitude of
68
the terms of Eqn. 3.32a for dynamic momentum diffusion into the fluid mud
layer. First, defining nondimensional (primed) variables as
t u x z
to u L 6
P' P p= P = (3.33b)
yH Po o
where to and uo are the characteristic maximum time and velocity, L is
the length scale of the estuary, 6 is the length scale of the fluid mud
layer depth, H is the differential height of the water surface over L, y
is the specific weight of mud, po and po are the characteristic mud
density and dynamic viscosity. Substituting the above variables in Eqn.
3.32a and considering only vertical shear gives
uo du' Y H 1 aP' [oUo 1 ,u'
[ dt . + ][ i 81 + 1' (3.32c)
to dt' poL p'ax' po 2 p' az' az
where all terms not in brackets, [ ], are order 1. Multiplying
Eqn. 3.33c through by [o] gives,
du' gHto 1 8P' tolo 8 ,2u'9
[11 d [ ] + [ o az (3.32d)
dt' L uo p'ax' po p az, az'
Substituting typical numerical values for fluid mud layers in estuaries
of g = 101 m/s2, H = 100 m, to = 103 s, L = 105 m, uo = 100 m/s,
vo = Po/Po = 104 m2/s, and 6 = 101 m, the order of magnitude of the
pressure gradient term is
(101 m/s2)(100 m)(103 s) = [0 1 (3.33d)
(10 m)(10 m/s)
The viscous shear term is
(104 m2/s)(10 s) = [101 (3.33e)
(10 m)
which, for the particular set of conditions, is two orders of magnitude
greater than the pressure gradient magnitude. Hence, neglecting
horizontal pressure gradients in Eqn 3.32a for qualitative understanding
of the dynamic momentum diffusion depth is justified, albeit weakly. It
must be emphasized that under fully developed steady flow, the order of
magnitude of the viscous shear stress and horizontal pressure gradient
terms are the same (since they are the only two nonzero terms in the
equation).
Under the above constraints the momentum equation becomes
au 1 (a (u
t p 8z 8z) (3.34)
The equation is now in a form in which analytical and simple numerical
solutions are possible with careful specification of initial and boundary
conditions and theological behavior.
70
A. Constant Viscosity Rayleigh Flow. For the case of constant mud
viscosity, pm, and unsteady shear flow, an analytical solution proposed
by Stokes (see Schlichting, 1979) is appropriate. With the boundary
conditions, 1) imposed velocity, U, at the upper interface u(z)=U @ z=Za
(in Figure 21), and 2) U(z)=0 @ z4. The solution for the horizontal
flow velocity is (Eskinazi, 1968)
u = U (1 erfcSs) (3.35)
where 6s is the similarity variable z/2/vt and erfc is the complementary
error function defined as
erfcS, = 1 e dn (3.36)
The penetration depth of the mobile fluid mud layer (Zb defined in Figure
21) can be found by considering the boundary layer thickness,
defined by u/U = 0.01 which is
6 = 3.64 /v (3.37)
The inadequacy of this solution is that even for viscosities ten
times higher than water (i.e., 105 m2/s), the predicted boundary layer
thickness over several hours is too large; e.g.,
S = 013.64(105.104)1/2} = 0(3.64 m)
(3.38)
71
Additionally, the approach does not adequately represent the rapid depth
variation in concentration (i.e., increasing concentration with depth) of
the fluid mud.
B. Constant Viscosity Unsteady Bingham Flow. For the steady state
flow of a Bingham plastic with constant viscosity (and constant yield
strength), an analytic similarity solution has been presented by Phan
Thien (1983). He assumed a two layer system with properties as
P2Y I, I^ Ty
(3.39)
where j is the time rate of shearing, u, and y is the Bingham yield
strength. Denoting the velocities in layer i (i=1,2) as
ui(z,t) = o 2/vji Ui(Ss)
Pi'
(3.40)
where to is the imposed shear stress, vi, pi are the kinematic and
dynamic viscosities and Ss is the similarity variable.
z
(3.41)
2s /
2,,vi
Ui(6s) is the similarity solution of Eqn. 3.34 given as
U1(s) = s (1R )(
sQ 2
6 z2
Se dz
0
(3.42)
and
8' z2 2
[sis 1 s
[6 e dz +
z
e dz
c
where Sc = zc(t)/2vtit a8 = 92/91 (ag for an ideal Bingham fluid),
86 = 8s,/Ja, S6 = Sc/Ia, and R = y/o. Additionally, since the
velocity field is continuous at z = zc(t) and is represented as
U2(6c) = a8 UI(Sc) (3.44)
a value for 8c is determinable and consequently for zc(t). But,
unfortunately, the inability to treat concentration variation with depth
as was the case in approach A is still undesirable.
C. Variable Viscosity Steady Bingham Flow. Neglecting the obvious
error involved with omission of the pressure gradient term, for the
steady flow from applied shear stress of an ideal Bingham plastic with
concentration dependent yield strength and viscosity, the constitutive
equation is
to = p(z) 1 to y
(3.45)
y = 0, u(z)=0 to < ty
In the region where the yield strength is exceeded, the flow velocity is
analogous to Couette flow with depth varying viscosity. For the region
I
73
where the shear strength exceeds the applied shear stress, no motion
occurs (the mud behaves as a solid).
For a solution to Eqn. 3.34 for this case, a concentration
relationship for the viscosity and yield strength must be specified. For
example, one approximation for the viscosity/concentration relationship,
based on data presented in Chapter 2 (see Figure 26), is of the form
a
P(C) = Pw(l + OPC) (3.46)
where p, is the viscosity of the suspending fluid (water), and P and aP
are empirical constants.
A power law expression for yield strength, ty, (also presented in
Chapter 2, Figure 27) is
a
ty = PyC (3.47)
where ay and a are empirical constants.
The boundary conditions are 1) an imposed shear stress at the upper
interface, t = to @ z = Za and 2) noslip at the lower yield elevation,
u = 0 @ z = Zb.
Additionally, for steady flow the depth varying shear stress, ty(z),
is everywhere equal to the imposed interfacial shear stress, to, down to
the stationary interface, zb, where ty = to.
This approach has been presented for comparison only since the
aforementioned error would be appreciable under quasisteady flows in
estuaries. The absence of timedependence is also a drawback to this
I
74
approach. Depending on the depth of the fluid mud layer and the imposed
shear stress, the velocity profile can take minutes to hours to reach
steady state form. For imposed shear which is continuously changing, as
is the case in tidal flows, steady flow is never reached.
The last two approaches offer the most promise in providing for
realistic spatial and temporal variability of horizontal momentum
diffusion into a fluid mud layer.
D. Variable Viscosity Rayleigh Flow. This approach describes
unsteady flow of a fluid mud layer with depth varying viscosity initially
subjected to horizontal motion at the upper interface, Za. The governing
equation is still Eqn. (3.34) but no specification is made regarding the
overall lower extent of the boundary layer, Sfm, In general, a
numerical solution is warranted. An explicit, finite difference
approximation (with j time and i direction index), for example,
j+l = At I j
u1 u+ a p ( ) i (u+ u )( (3.48)
1 i Az2Pi 1 i i
where pi = (i+l+ pi)/2, gives an easily obtainable solution path. The
boundary conditions are those given in A. Additionally, proper concerns
for numerical stability and convergence must be addressed (i.e., At
Az2p/2Pmax).
The last approach offers the most realistic simulation of
theological and temporal variability of the approaches presented thus
far.
E. Variable Viscosity Unsteady Bingham Flow. A numerical solution
of a form similar to Eqn. 3.48 above is employed for the region where the
75
mud is sheared. Additionally, the lower interface is tracked by
considering the temporal response of the shear stress and yield strength
at each layer. This is written
Eqn. (3.48) for t(z,t) 1 ty(C,t)
u(z) = (3.49)
0 for t(z,t) < ty(C,t)
The boundary conditions are the same as those of C. Comments concerning
the numerical technique, stability and convergence mentioned in D, also
apply for the shear layer here.
With regard to Bingham plastic vs. Newtonian fluid (with viscosity
which varies with concentration) behavior, the data in Figure 27 suggest
yield strengths which are sufficiently large to preclude flow under mild
bed shear stress (e.g., C = 100 g/l corresponds to ty = 0.1 N/m2).
However, field and laboratory data used to verify the above approaches in
Chapter 5 (Section 5.5) show evidence of relatively high flows under very
mild imposed shear stress. For this reason, care must be taken in
application of the above approaches using a functional relationship for
yield strength such as Eqn. 3.47, that the empirical coefficients ay and
By fit a particular sediment behavior. It is suggested that non
Newtonian pseudoplastic behavior (where viscosity is a function of shear
rate) may be a more reasonable model than Bingham plastic for fluid mud
flows. However, no further supporting arguments or discussion are made
in this report since application of the Newtonian models showed
reasonable results.
Mm
CHAPTER 4
LABORATORY EXPERIMENTS
4.1 Introduction
Laboratory experiments were conducted at the University of Florida's
Coastal Engineering Laboratory. These experiments consisted of two flume
tests and settling column tests. The flume tests were designed to
evaluate the dynamical effects of wave action on a partially consolidated
natural estuarine sediment bed. Bed erosion (by fluidization) and upper
column suspension concentrations were measured. Settling tests were
performed to obtain the concentration dependent settling properties of
natural flocculating fine sediment. New settling column tests were
devised to provide development and verification data needed for the
vertical profile model.
4.2 Flume Study
4.2.1 Objectives
The objectives of the wave flume study were as follows:
1. To use advanced pressure sensor instrumentation to measure and
document the effective stress breakdown (fluidization) in a
partially consolidated cohesive bed subjected to wave loading.
2. To observe, record and determine factors characterizing fluid
mud formation and stability (during wave erosion) presented in
Chapters 2 and 3.
77
3. To measure wave resuspension concentrations related to
hydrodynamical data (i.e., wave height, water depth, fluid mud
and bed thickness) for the purpose of verification of the
descriptive vertical transport model.
4. Investigate the role of wave resuspension in the overall
sediment transport process in the prototype setting.
4.2.2 Mud Characterization
The estuarine sediment selected for use in the flume and settling
column studies was mud from Tampa Bay, Florida. Collection was from a
site adjacent to a Hillsborough Bay navigation channel. It was
predetermined by a bay mapping study (City of Tampa, 1986) to be an area
of predominately fine sediment (clay and fine silt) and relatively high
sedimentation rates (0.31 m/year). Grain size distribution of dispersed
free particles, obtained by standard ASTM hydrometer method, is shown in
Figure 41. It can be seen that d50 = 2.6 pm, which indicates that 50%
of the sediment sample was finer than the upper limit of clay size
particles (2 pm). Furthermore, less than 10% by weight of the sediment
sample was coarse silt to fine sand.
The flocculated sediment was pumped into 55 gallon drums in the
field then to washing and storage tanks in the laboratory. The sediment
was then mixed and decanted several times to equilibrate with tap water
until a slight background salinity (1 ppt) remained. Details of this
procedure can be found in Cervantes (1987). The slight salinity was
sufficient to maintain the flocculated state of the cohesive (< 20Pm)
particles. Characterization tests were conducted at the University of
U.nI 0.010 0.10
Equivalent Grain Size (mm)
Figure 41. Grain Size Distribution of Hillsborough Bay Mud
Florida Soils Science Laboratory. XRay diffraction revealed that the
clay size fraction was primarily made up of montmorillinite (91%) and
very small amounts of kaolinite (4%) and quartz (5%). A cation exchange
capacity (CEC) test reported 197.2 meq/100g (an unrealistic and suspected
erroneously high value). Percent organic carbon content, determined by
standard combustion technique (e.g., ASTM 500C incineration) indicated
that 5% by weight of the sediment sample was of detrital (organic)
origin. Chemical composition of the fluid (tap water) can be found in
Dixit (1982).
I
